’ ■ nbsp;nbsp;nbsp;■-^ ‘■ijis- ^- ' ^' ■ '' '*

••.S nbsp;nbsp;nbsp;- -.nbsp;nbsp;nbsp;nbsp;T-' -V

-•fgt;.'' nbsp;nbsp;nbsp;'i

' ;.:. ■

*1%, .,-r^ r '-^.r

R'CKT^'S�nrrE GlSTELLiMG, VOLGENS GEN'iL-i^EK'S ACLLN-.Ei'-rnbsp;D.D. 29 lAEi 1961.

STiCNTING

UNIVERSiTniTSMUSEUM

ELEMENTS

PHILOSOPHY.

PREFACE.

The principal charaSeriftic which dif-^ tinguilhes the human being from thenbsp;reft of the animal creation, is the inheritancenbsp;of knowledge, which the individuals of eachnbsp;generation are able to derive from their pre^nbsp;deceflbrs.

Xhe bee of modern times forms the cells of its hive exactly of the fame fliape as thenbsp;bee of the remoteft antiquity ; each Ipecies ofnbsp;birds builds its neft after the fame unalterablenbsp;pattern, and lings the fame invariable melody.nbsp;The fheep of the prefent day has no betternbsp;defence againft the wolf; nor has the fly againftnbsp;the fpider, nor the fmaller birds againft thenbsp;eagle, than the like animals of former times.nbsp;quot;The fame wants, fimilar dangers, the like de-and unalterable cuftoms, are the conftantnbsp;attendants of each ditferent tribe ; nor is anynbsp;A 2nbsp;nbsp;nbsp;nbsp;individual

individual benefited by the experience or by the improvements of all its predecefibrs.

Man .alone has received from his Divine Creator the ineflimable advantage of beingnbsp;benefited by the knowledge of his foi'efathers,nbsp;and of his being able to bequeath that knowledge, together with his own improvements,nbsp;to his poflerity.

The -accumulated experience of a long fe-ries of years, accurately recorded in a vaft many books, or traditionally imparted from one generation to the other, gradually exalts thenbsp;flate of human beings, fupplies their wants,nbsp;increafes their fecurity, and promotes theirnbsp;happinefs. The plough, the loom, the forge,nbsp;the prefs, the glafs-houfe, and innumerablenbsp;other ufeful inventions of ot;r predeceflbrs,nbsp;luccefiively improved by conftant ufe and experience, form the invaluable advantages ofnbsp;modern times; and their combined effe�l,nbsp;affually-elevates the individuals of a modernnbsp;civilized nation, lb far above the uninftrublednbsp;favas-es, as might almofl feera to render themnbsp;of a different fpecies.

That experience, properly difpofed under ^iftin�l heads, forms the various fubje61s ofnbsp;knowledge. The arrangement, and the elu-^nbsp;cidation of each particular fubje�l, is called anbsp;Science. The ultimate or the pra�tical application of it is called an Art.

Arts and fciences are too numerous and too extended, to be comprehended in their greateftnbsp;extent by each hngle individual: hence isnbsp;derived the divhion of labour, or the adoption of a particular branch by each hnglenbsp;individual. But all thofe branches derive their 'nbsp;origin from the fame natural powers, theynbsp;i^re all in their principles regulated by the famenbsp;general laws of Nature, and almoft all theirnbsp;applications may be fubje61ed to calculationnbsp;and demonflration. The invefligation of theirnbsp;oingiu, and of their mutual dependence on eachnbsp;other, the illuftration of their principles, thenbsp;methods of enlarging their limits by means ofnbsp;experiments and calculation, and their apj)!!-cation to our various wants, fall under the titlenbsp;of Natural or Experimental Philosophy, the Elements of which form the fiib-jea of this Work.

In the courfe of the laft twenty or thirty centuries, during which time (as v/ritten documents inform us) more or lets attentive ob-fervations ha\e been made on the propertiesnbsp;of natural bodies, various theories have beennbsp;formed, or different ideas have been entertained concerning the nature of thofe bodies,nbsp;or concerning the general fubjecl of Naturalnbsp;Philofophy ; but the fmall proportion of realnbsp;fafts, and the vaftly greater proportion ofnbsp;vague and unwarrantable ideas which formednbsp;thofe theories, rendered them always iniuffi-cient, and frequently abfurd ; whence confu-fion of ideas, and retardation of fcience, naturally enfired.

The nature and the fate of thofe theories gradually cautioned the judicious part of thenbsp;inquifitive world, and fhewed them the ne-cellity of tubftituting experiments and flrictnbsp;mathem.atical reafcning to the I'uggeftions ofnbsp;the imasrination. This rational reform, ornbsp;cautious mode of proceeding, fince the i6thnbsp;century, has been produdtive of a vaft number of ufeful difco\ eries ; and, by its havingnbsp;placed the progrefs of fcience in the right

cJ^atinel, has enabled philofophers to trace out ^he principles of feveral of its branches, tonbsp;iuveftigate divers new fubjeds, and to opennbsp;wew paths to the inexhauftible treafures ofnbsp;nature.

The progrefs of experimental inveftigation, and the mathematical mode of reafoning, arenbsp;both flow and laborious; but they are fafe,nbsp;and produdive of true and ufeful knowledge;nbsp;nor has the human being any other means ofnbsp;feeling his way through the dark labyrinth ofnbsp;Nature. It is wonderful to obferve what manual labour, and what exalted exertions of thenbsp;^nman mind, have been beftowed upon the various branches of Natural Philofophy, Thofenbsp;profound inquiries, fometimes fruitlefs, and atnbsp;other times either diredly or indiredly fuccefs-ful, alternately difplay the ftrength and thenbsp;weaknefs of the human underftanding; butnbsp;npon .the whole, it mufl; be acknowledged thatnbsp;Wonderful improvements have undoubtedlynbsp;been derived from thofe extraordinary exertions ; and the progrefs of fcience within thenbsp;laft two centuries has certainly advanced withnbsp;increafing velocity.

A 4 nbsp;nbsp;nbsp;It

It is not my intention to deceive the reader by aflerting, that I have rendered all the principles of Natural Philolbphy intelligible to thenbsp;meaneft capacity ; for in that cafe, I Ihouldnbsp;either have been obliged to omit the morenbsp;abftrufe branches of philofophy, or the fallacynbsp;of the aflertion would be rendered daringlynbsp;manifeft in feveral of the following pages.nbsp;Original difcoveries of fa�ls, or principles ornbsp;laws of nature, are generally made throughnbsp;intricate and perplexed paths. By fubfequentnbsp;revifion and confideration, the fuperfluous isnbsp;removed, the defedlive is lupplied, and thenbsp;confufed materials are properly arranged ;nbsp;whence the, train of reafoning frequently becomes fhorter and more natural, or thenbsp;nature of the fubjeft is rendered more evident and more intelligible. But this fimplifi-cation has a limit which differs in differentnbsp;fubjedts; nor can the comprehenfion of whatnbsp;depends upon a vad number of previous ideas,nbsp;mathematically connedled, be rendered attainable to fuch perfons as are deditute ofnbsp;fuch ideas, or whofe mind is incapable of detaining the neceffary chain of reafoning.

By following the example of the cleareft �'Vriters, and by confidering each, particularnbsp;fubjedl in different points of view, I have endeavoured to explain it with all the fimplicitynbsp;and the clearnefs which my llendcr abilitynbsp;could faggeft. In feveral places I have avoidednbsp;fome abflrufe technical formalities of ordernbsp;or phrafeology, and have preferred familiarnbsp;expreffions wherever it appeared prafficable ;nbsp;but when the fubjeft feemed lefs likely tonbsp;be comprehended by the greateft number ofnbsp;readers, I have always placed it in the Notes,nbsp;where thofe only who are competently qualified may read it. And here it mufi: benbsp;obferved, that, for the fake of diftindion,nbsp;tbe references from the text to thofe notes,nbsp;confift of the common numerical figures;

taming quotations, additional remarks, amp;c. confift of afterifms, or fuch like marks.

A few repetitions, which the reader will meet with in the courfe of the work, will, Inbsp;truft, be ealily excufed, confidering that theynbsp;bave been thought neceffary for promotingnbsp;tbe elucidation of particular fubje�ts. With

relpe�l to the termination of certain words of an entire Latin origin, it muft be obferved thatnbsp;I have indifcriminately written them eithernbsp;with a Latin or with an Englifh termination,nbsp;fuch as radii and radiufes, media and meidums,nbsp;amp;c. for having found them uled both waysnbsp;by different writers, I was unwilling to adoptnbsp;a decided partiality for either mode.

With refpedl to the difpofition of the materials throughout the work, it may perhaps be neceffary to mention, that my rule hasnbsp;been to begin with the general properties ofnbsp;matter, or fuch as conftant experience fhewsnbsp;to belong to bodies of every kind. I havenbsp;afterwards proceeded to examine thofe whichnbsp;belong to a particular fet of bodies, and thennbsp;thofe of fewer or of hnarle bodies,

The aftronomical part has been naturally placed after the flatement of the above-mentioned properties, fnce the knowledgenbsp;of the appearances of the celeftial bodies isnbsp;not fo immediately concerned with our welfare, as that of the fubflances which nearlvnbsp;*nbsp;nbsp;nbsp;nbsp;furround

In the illulfration of the various branches, a multiplicity of experiments and extendednbsp;hiftorical accounts have been carefully avoided, left the flatement of hiperfiuities Ihouldnbsp;have occupied the place of ufeful materials.nbsp;The different fubjefts of Natural Philofophynbsp;cannot be rendered fufficiently intelligiblenbsp;without a certain extent of explanation ; butnbsp;at the fame time their number would rendernbsp;the work too extenfive, if the limits of ab-folute neceffity were not carefully preferved.nbsp;In this, however, the Author is expofed to anbsp;dangerous dilemma, as the fame illullrationnbsp;which proves prolix to certain readers, is in-fufficient for others. Different views of thenbsp;fame abllrufe fubjedl, though tedious to thenbsp;proficient, are undoubtedly of great affiftancenbsp;to the novice. In this cafe the limits ofnbsp;fufficiency or of infufficiency are vague andnbsp;indeterminate; and whilft they tend to perplex the author, they afford, according to thenbsp;inclination of the reader, ample fcope fornbsp;cnticifm or fatisfa�lion. Natural order, accuracy

curacy of ftatemcnts, perfpicuity, and coii-cifenefs, have been the conftant objefts of my views in the compilation of this work. Inbsp;have endeavoured to feled from multiplicity,nbsp;and to remove obfcurity. In certain placesnbsp;I have added new fa6ts, in others I havenbsp;fearched for new and true explanations ofnbsp;natural effe�is. I have pointed out the defers of feveral particulars, and have recommended the elucidation of the fame to thenbsp;diligence of zealous ftudents. But whether ornbsp;not the performance is fufficiently conformable to thofe views, I humbly fubmit it to thenbsp;deciiion of the impartial and diftinguifhingnbsp;part of my readers.

As this work is likely to fall into various hands, it may perhaps be ufeful to add a fewnbsp;remarks and a few dire6tions for the ufe, notnbsp;of the proficient, but of thofe to whom thenbsp;fubjeft is either partially or entirely new, innbsp;order that unprofitable labour, or extravagantnbsp;expeftations, may in great meafure be avoided.

Of the various readers of books in general, I fhall briefly attempt to difcriminate the following:

iowing clafles. There are feme who imagine that the fame Velocity of reading is fufficientnbsp;for a novel, or a poetical, or an hiftorical, ornbsp;a fcientific book ; and when they find thatnbsp;they are not able to comprehend the latter,nbsp;they conclude either that the author is ob-fcure, or that they themfelves have not capacity fufficient for it. Others imagine that anbsp;fingle careful perufal of a fcientific book isnbsp;fufficient to inftraft them in a new fubje^f.nbsp;Laftly, there are others who never proceed tonbsp;the next page, unlefs they have thoroughlynbsp;underftood the preceding part of the work.nbsp;This method, in the reading of natural phi-lofophy, though very proper, is by no meansnbsp;very pleafing, and generally tires the fludentnbsp;before he has read a quarter of the work.

Where a great many new ideas mufi: be acquired, -much attention mufi: be neceflarilynbsp;befiowed. Therefore, in ,thc reading ofnbsp;novels or poetry, the only exertion of thenbsp;mind which is required for a fafisfadorynbsp;perufal, is the connexion of the differentnbsp;parts or accounts, and a tolefable degree ofnbsp;attention to the beauties pf the performance

M^hich arif� from the ftyle and the imagination of the writer ; for with reipedt to fads andnbsp;the meaning of words, they are fo much. likenbsp;the occurrences of common life, as never tonbsp;demand any exertion of the underflanding.nbsp;Nearly the fame thing may be faid with re-fpeiSl to the reading of hiftory. But withnbsp;fcientific fubjecls the cale is quite different;nbsp;for in them the great variety of new thingsnbsp;and new ideas, to Vv^hich w'ords of uncommonnbsp;ufe have been appropriated, and their dependance upon each other, or upon fafts of unufualnbsp;occurrence, demand a continual exertion bothnbsp;of the memory and of the underftanding ?nbsp;which, unlefs it be relieved by means ofnbsp;order, patience, and a competent allowancenbsp;of time, will certainly prove irkfome to moftnbsp;ftudents.

Therefore on thofe accounts I beg leave to recommend the following method. Let thenbsp;novice in the ftudy of natural philofo-phy readnbsp;this work a firft time, rather (lowly, butnbsp;without perufing thofe notes which, as hasnbsp;been remarked above, have a numericfi reference,' npr caring, as he proceeds, whether

much more copious than is cuftomary for books of this kind, Laflly, the ftudent maynbsp;read over a third timCj or oftener, fuch partsnbsp;only of the work as his particular Inclination,nbsp;or his uuderftanding or his memory may render neceflary.

T. C.

CONTENTS.

Chapter!. Of NATURAL PHILOSOPHY.�Its name; its objefl; its axioms, and the rules of philo^nbsp;fophizingnbsp;nbsp;nbsp;nbsp;page i

Chap. II. Containing a general idea of matter, and its properties. Of the elements, and the definitions of words that are principally ufed in Natural Philofophy - lO

Chap. VIII, Of compound motion; of the compofition and refolution of forces ; and of oblique impulfes p. 114

VOLUME

CONTENTS.

Chap. I. Containing an enumeration of the various known bodies of the univerfe, under general and comprehenfivenbsp;appellationsnbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;quot; P- 5

Chap. VIII. Of pneumatics, or of permanently elaftic fluids; of the atmofpherical air, and of the barometer 198nbsp;Chap. IX. Of the denfity and altitude of the atmofphere,nbsp;together with the method of meafuring altitudes by meansnbsp;of barometrical obfervations -nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;_ 226

Chap. XIV. The defcription of the principal machines which depend upon the foregoing fubjedls of fluids - 426 �nbsp;Chap. XV. Containing the principles of chemiftry, andnbsp;particularly th� defcription of the principal operations andnbsp;apparatus -nbsp;nbsp;nbsp;nbsp;^nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;- 495

Chap. XVI. Containing a fketch of the modern theory of chemiftry .nbsp;nbsp;nbsp;nbsp;_nbsp;nbsp;nbsp;nbsp;_nbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;,nbsp;nbsp;nbsp;nbsp;, 514

b2 nbsp;nbsp;nbsp;VOLUME

m nbsp;nbsp;nbsp;CONTENTS.

VOLUME III.

Chap. I. The theory of heat; or the general effefls of a fuppofed calorific fluid -nbsp;nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;- p. ^

Chap. IV. Of the inflexion of light, the colours of thin tranfparent bodies, and of coburs in general -

Chap. VIII. Natural phenomena relative to light - 301

CONTENTS.

VOLUME

PART

CONTENTS.

PART V.

Containing a few unconnected Subje�ts, p. 315.

Sedlion II. Of Meteors, and of the Jlany fubftances which at various times are faid tonbsp;have fallen from the fky, p. 357.

DIRECTIONS /or the BooUinder,

And the laft FOUR Plates ; viz. XXVI. to XXIX. indufively -nbsp;nbsp;nbsp;nbsp;- miff be placed at the end

The Astronomical Table (a Quarter of a Sheet of Letter Prefs) to face - p. 190 tir Fourth Volume.

Natural things means all bodies j and the affem-blage or fyftero of them all is called the tmiverfe.

The word phenomenon fignifies an appearance, or, in a more enlarged acceptation, whatever is perceived by our (�nfes*. Thus the fall of a ftone,nbsp;the evaporation of water, the folution of fait innbsp;water, a flafh of lightning, and fo on; are allnbsp;phenomena.

As all phenomena depend on properties peculiar to different bodies; for it is a property of a ftonenbsp;to fall towards the earth, of the water to benbsp;cvaporable, oft the fait to be foluble in water, amp;c.nbsp;therefore we fay that the bufinefs of natural phi-lofophy is to examine the properties of the various.nbsp;bodies of the umverfe, to inveftigate their caufes,nbsp;and thence to infer ufeful deduftions.

dom, from the words piMj, a lover ov friend, and crotploi, of knoivledge or wifdom. Moral philo'i�ophy is derived fromnbsp;the latin mos, or its plural mores, fignifying manners ornbsp;behaviour, it has been likewife called eihies,iiom thenbsp;Greek r/ocj, mos, manner, hehavtoiir. Natural philofophynbsp;has alfo been called phyfscs, phyfiology, and .experimental phi-ifophy. The frfl: of thofe narries is derived fromnbsp;nature, or pvaaKss, natural; the fecond is derived from givaic,nbsp;nature, and Aeyof, a dijcourfe -, the lall denomination, which-v,?as introduced not many years ago,, is obvioutly derivednbsp;from the jull method of experimental inveftigation, whichnbsp;has'been univerfally adopted njice the revival of learningnbsp;m Europe.

* Phenomenon, whofe plural is phenomena, owes its origin to the Greek v/or.d ^aheyte appear.

�Agreeably to this, the reader will find in the courf� of this work, an account of the principalnbsp;properties of natural bodies, arranged under dif-tindt heads, with an explanation of their elfeds,nbsp;and of the caufes on which they depend, as farnbsp;3-s has been afcertained by means of reafoning andnbsp;experience; he will be informed of the principalnbsp;hypothefes that have been offered for the explanation of, fafls, whofe caufes have not yet. beennbsp;dernonflratively proved; he will find a flatementnbsp;tif the laws of nature, or of fuch rules as havenbsp;deduced from the concurrence of fimilarnbsp;fadts ; and, laftly, he will be inftrufted in the ma-iiagement of philofophical inftruments, and in thenbsp;^ode of performing the experiments that may benbsp;thought necefla ry either for the illuftration of whatnbsp;has been already afcertained, or for the farthernbsp;inveftigation Qf properties of natural bodies.

We need not -fay much with refpeft to the end defign of natural phiiofophy.-�Its applicationnbsp;and its viies, or' the advantages which mankindnbsp;niay derive therefrom, will be eafily fuggefted by anbsp;'^'-'�y fuperficial examination of whatever takesnbsp;place about us. The properties of the air wenbsp;breathe; the aftion and power of our limbs; thenbsp;it� the found, and other perceptions of ournbsp;es; the aftions of the engines that are ufed innbsp;, ^fbandry, navigation, amp;c.; the viciffitudes ofnbsp;e feafons, the movements of the celeftial bodies,nbsp;*nd fo forth j do all fall under the confideration of

A very flight acquaintance with the political flate of the world, will be fuflicient to fliew, thatnbsp;the cultivation of the various branches of naturalnbsp;philofophy has aflually placed the Europeans andnbsp;their colonies above the reft of mankind. Theirnbsp;difeoveries and improvements in aftronomy, optics,nbsp;navigation, chemiftry, magnetifm, mineralogy, andnbsp;in the numerous arts which depend on thofe andnbsp;other branches of philofophy, have fupplied themnbsp;with innumerable articles of ufe and luxury, havenbsp;multiplied their riches, and have extended theirnbsp;powers to a degree even beyond the expectationsnbsp;of our predeceflbrs.

The various properties of matter may be divided into two clafies, viz. the general properties, whichnbsp;belong to all bodies, and the peculiar properties, ornbsp;thofe which belong to certain bodies only, exclu-fively of others.

In the firft part of this work we lhall examine the general properties of matter. Thofe whichnbsp;belong to certain bodies only, will be treated ofnbsp;in the fecond. In the third part we fliall examinenbsp;the properties of fuch fubftances as may be callednbsp;hypothetical\ their exiftence having not yet beennbsp;hitisfadlorily proved. In the fourth we fliall extend our views beyond the limits of our Earth,nbsp;and lhall examine the number, the movements,,nbsp;and other properties of the celeftial bodies.

The fifth, or laft part, will contain feveral detached articles, fuch as the defcription of feveral additional experiments, machines, amp;c. which cannot conveniently be inferted in the precedingnbsp;divifions.

The axioms of philofophy, or the axioms which have been deduced from common and conftantnbsp;experience, are fo evident and fo generally known�nbsp;that it will be fufficient to mention a few of themnbsp;only.

Some perfons may perhaps not readily admit the propriety of this axiom; feeing that a greatnbsp;many things appear to be utterly deftroyed by thsnbsp;adion of fire; alfo that water may be caufed tonbsp;difappear by means of evaporation, and fo forth.nbsp;But it muft be obferved, that in thofe cafes thenbsp;fubftances are not annihilated; but they are onlynbsp;difperfed, or removed from one place to another,nbsp;or they are divided into particles fo minute as tonbsp;elude our fenfes. Thus when a piece of wood isnbsp;placed upon the fire, the greateft part of it difap-pears, and a few adres only remain, the weightnbsp;and bulk of which does not amount to the hundredth part of that of the original piece of wood.nbsp;Now in this cafe the piece of wood is divided into

its component fubftances, which the aOrion of the fire drives different ways: the fluid part, for in-ftance, becomes fteam, the light coaly part eithernbsp;adheres to the chimney or is difperfed througli thenbsp;air, amp;c. And if, after the combuflion, the fcatterednbsp;materials were colledled together, (which may innbsp;great meafure be done), the fum of their weightsnbsp;would equal the weight of the original piece ofnbsp;wood.nbsp;nbsp;nbsp;nbsp;'

It may in general be obferved with refpeft to thofe axioms, that we only mean to affert what hasnbsp;been conftantly (hewn, and confirmed by experience, and is not contradidted either by reafon, ornbsp;by any experiment. But we do not mean to affertnbsp;that they are as evident as the axioms of geometry jnbsp;nor do we in the leaft prefume to prefcribe limitsnbsp;to the agency of the Almighty Creator of everynbsp;thing, whofe power and whofe ends are too far removed from the reach of our underftandings.

Having ftated the , principal axioms of philolb-phy, it is in the next place neceflary to mention the rules of philofophizing, which have been formed after mature confideration, for the purpofe ofnbsp;preventing errors as much as poffible, and in ordernbsp;to lead the fludent of nature along the Ihortefl andnbsp;fafeft wa}'^, to the attainment of true and ufefulnbsp;knowledge.-viz.

I. nbsp;nbsp;nbsp;We are to admit no more caufes of naturalnbsp;things, than fuch as are both true and fufficient tonbsp;explain th� appearances.

II. nbsp;nbsp;nbsp;Therefore to the fame natural effedls wenbsp;muff, as far as poffible, alTign the fame caufes.

III. nbsp;nbsp;nbsp;Such qualities of bodies as are not capablenbsp;�of increafe or decreafe, and which are found to belong to, all. bodies within the reach of our experiments, are to be efteemed the univerfal qualities ofnbsp;ab bodies whatfoever.

JV. In experimental philofophy w^e are to look upon proportions collefted by general indudtionnbsp;from phenomena, as accurately or very nearly true,nbsp;notwithfttinding any contrary hypothefes that maynbsp;be Imagined, till fuch time as other phenomenanbsp;occur, by which they either may be corredled, ornbsp;may be fhewm to be liable to exceptions.

With refpedl to the degree of evidence which ought to be expected in natural philofophy, it isnbsp;neceffary to remark, that phyficai matters cannotnbsp;general be capable of fuch abfolute certainty asnbsp;the branches of mathematics.�The proportions ofnbsp;the latter fcience are clearly deduced from a fet ofnbsp;axioms fo very fimple and evident, as to conveynbsp;perfed convidion to the mind; nor can any of-them be denied without a manifeft abfurdity. Butnbsp;m natural philofophy v/e can only fay,, that becaufenbsp;forne particular efFeds have been conftantly produced under certain circumftances; therefore theynbsp;Will mofl; likely continue to be produced as long

as the fame circumftances exiflj and likewife that they do, in all probability, depend upon thofenbsp;bircumfiances. And this is what we mean by lawsnbsp;of nature-, as will be more particularly defined innbsp;the next chapter.

We may, indeed, afllime various phyfical principles, and by reafoning upon them, we may ftriftly demonftrate the deduction of certain confequences.nbsp;But as the demonftration goes no farther than tonbsp;prove that fuch confequences mull neceffarily follow the principles which have been alTumed, thenbsp;confequences them.felves can have no greater degreenbsp;of certainty than the principles are pofTelfed of; fonbsp;that they are true, or falfe, or probable, accordingnbsp;as the principles upon which they depend are true,nbsp;or falfe, or probable. It has been found, for in-ftance, that a magnet, when left at liberty, does always direbl itfelf to certain parts of the world; uponnbsp;W'hich property the mariner�s compafs has beennbsp;conftmfted and it has been likewife obferved,nbsp;that this diredive property of a natural or artificialnbsp;magnet, is not obftruded by the interpofition ornbsp;proximity of gold, or filver, or glaft, or, in fhort,nbsp;of any other fubftance, as far as has b�en tried,nbsp;excepting iron and ferrugineous bodies. Nownbsp;afluming this obfervation as a principle, it naturallynbsp;follows, that, iron excepted, the box of the mariner�s compafs may be made of any fubftance thatnbsp;may be moft agreeable to the workman, or thatnbsp;may bell anhyer other purpofes. Yet it mull benbsp;'nbsp;nbsp;nbsp;nbsp;confefTed,

confeffed, that this propofitton is by no means fo certain as a geometrical one ; and ftridlly fpeakmg itnbsp;may only be faid to be highly probable; for thoughnbsp;all the bodies that have been tried with this view,nbsp;iron excepted, have been found not to afTcift thenbsp;directive property of the magnet or magnetic needle;nbsp;yet we are not certain that a body, or fome combination of bodies, may not hereafter be difcovered,nbsp;which may obftru6l that property.

Notwithftanding this obfervation, I am far from meaning to encourage fcepticifm; my only obje^nbsp;being to drew that juft and proper degree of con-vi�tion which ought to be annexed tophyficalnbsp;knowledee ; fo that the ftudent of this fcience maynbsp;become neither a blind believer, nor a ufelefsnbsp;feeptic¹.

Eefides a ftrid adherence to the abovementioned rules, whoever willies to make any proficiency innbsp;the ftudy of nature, lliould make himfelf acquainted with the various branches of mathematics; atnbsp;leaft with the elements of geometry, arithmetic,nbsp;trigonometry, and the principal properties of the

Scepticifm or (kcpticirtn is the doftnne of ihe fci-ptics, ancient let of philolbphcrs, whofe pecuf.ar tenet was, thatnbsp;all things are uncertain and incomprehenfible; and thatnbsp;mind IS never to alient to any thing, but to remain in an ab-folute ftate of hefitaticn and imlifFerence. � The wornbsp;feeptic is derived from the Greek lt;rxedliK�-y which fig-nifies conjiderate, and inquifttive.

conic feftions; for fince almoft every phyfical eifeft depends upon motion, magnitude, and figure, it isnbsp;jmpoflible to calculate velocities, powers, weights,nbsp;times, amp;c. without a competent degree of mathematical knowledge; which fcience may in truth benbsp;called the language of nature.

Of tlie Elements ; �and the Definitions of JVords that are principally ufed in Natural Philofophy.

rr'^HE matter or fubftance of the bodies which A we fee, feel, tafte, or, in fliort, that affecT:nbsp;our fenfes, becomes known to us merely by itsnbsp;properties. We know that the fun exifts becaufenbsp;w,e fee its luminous and circular fiiape; becaufenbsp;we feel its heat. We know that the ground exiftsnbsp;becaufe we fee it, and feel it with our limbs. Wenbsp;acknowledge .the exiftence of air, becaufe we feelnbsp;the refiftance it offers to the motion of other bodies,nbsp;amp;c. Now the fun, the ground, the air, and allnbsp;Other bodies, muft, agreeably to the firft axiom,nbsp;confift of fomething. That fomething is callednbsp;matter �, yet we are perfedlly ignorant of the intimatenbsp;nature of that matter j fince we are unable to fay

whether it confifts of particles of any definite fize, fliape, and hardnefs; and whether all the bodies ofnbsp;the univerfe confift of the fame fort of matter differently modified, or of different forts of matter;nbsp;and in the latter cale, we can form no conje�turenbsp;refpe�ting the poflible number of thofe differentnbsp;forts.

Some philofophers have fteadily fupported that there is one fort only of original matter, and thatnbsp;the variety of bodies arifes from the various arrangements of that primitive matter; like pafte, bread,nbsp;and bifeuits, which may be faid to confift of thenbsp;fame matter, viz. flour.

Other philofophers have believ^ed that the forts of primitive matter, or elements, are two. Others,nbsp;that they are three. Others again,' that they arenbsp;four or five, or fix or feven, and fo on. But thenbsp;hiftory of fuppofitions muft riot be miftaken for thenbsp;knowledge of fafts ¹.

The truth is, that the prefent ftate of knowledge does not furnifli us with reafons fufficient to determine

Whoever wifhes to examine the various ideas that have been entertained concerning original matter, and thenbsp;number of the elements, muft confult the works of the following authors; but his labour will not be compenfated bynbsp;any material information: Ariftotle; Plato; Boyle, on thenbsp;principles of natural bodies ; Newton�s Optics; Wood-Ward�s Nat. Hift. of the Earth, p. v.; Muffehenbroek�snbsp;Elements of Phyf. � 6i, 83, 383; Keill Introd, to Nat.nbsp;Phil. L��f. viii.; Higgins on l-ught; Chambers�s Cyclop;nbsp;2nd Hutton�s Mathem. Di�f. Art. Element.

inine the number of the elements. Modern che-mifts, indeed, find from the refult of their numerous analyfes, that all the bodies which have been fub*nbsp;jeflied to experirnents, are either the fubftances thatnbsp;are mentioned in the following lift, or they are anbsp;combination of fome of them. Yet no great confidence fhould be placed upon their number; fornbsp;befides there being great fufpicion that feveral ofnbsp;them are ftill refolvable into fimpler components;nbsp;new fubftances are almoft daily difcovered by thenbsp;prefent rapid progrefs of philofophical inveftigation ;nbsp;and fome of them are merely hypothetical.

The following lift contains the bare names of thofe elehtentary fubftances which are at prefentnbsp;acknowledged by the philofophical chemifts, or fuchnbsp;as chemifts have not yet been able to decom-pofe; but a full explanation of the fame will benbsp;found in other chapters of this work; and tillnbsp;then the reader is requefted not to endeavournbsp;to inveftigate the meaning of their names, or tonbsp;take any farther notice of them.

Befides thofe which are mentioned in the prefenf chapter, there are feveral other words which defervcnbsp;likewife to be particularly defined ; but thofe wenbsp;fhall explain occafionally in the courfe of thenbsp;work, and when the mind of the reader may benbsp;better difpofed to comprehend their meanings.

Space (though it'be incapable of a proper de* finition) may be faid to be that univerfal and unlimited expanfe in which all bodies are contained;nbsp;and that part of fpace, which is occupied by anynbsp;particular body, is called the place of that bbdy'.

Space is diftingui'fiied into abjolute, and relative. . Abfolute fpace is that which is referred to nothingsnbsp;and remains always fimilar and immoveable. Relative fpace is the fame with abfolute fpace in magnitude and figure, but not in fituation. Suppofe,nbsp;for example, that a fliip flood perfectly immoveablenbsp;in the univerfe, the fpace which is contained withinnbsp;its cavity, would be ca:lled abfolute fpace. But ifnbsp;the fliip be in motion, then the fame Ipace withinnbsp;it will be called relative fpace.

Place h likewife diftinguiamp;ed into abfolute txnd relative-, the former being immoveable and permanent ; whereas the latter refers to other bodies.nbsp;Thus if a man be fcated in a corner of a fliip whilflnbsp;the fiiip is failing along, he is faid to remain in thenbsp;fame place relatively to the parts of the fiiip; yetnbsp;he is continually changi.ng, his abfolute place.

Rejl is the permanence of any body in the fame place, and it is called abfoliiie reft, or relative reft,nbsp;as the place, which the body occupies, is eithernbsp;abfolute or relative. -

Motion, on the contrary, is a continual and fuc-ceffive change of place. And it is called abfolute motion or relative motion, according as the change ofnbsp;-fituation is made in abfolute or in relative place.

Thus, if a fliip were to remain immoveable in the univerfe, a man fitting in a corner of it, wouldnbsp;be faid to be abfolutely at reft; but if the fliip benbsp;in motion whilft the man remains fitting, then thisnbsp;man will be faid to be at reft relatively to the partsnbsp;of the (liip, though he is adually or abfolutely innbsp;motion.�Farther, fuppofe that the fhip were tonbsp;move equably forward over ad iftance equal to itsnbsp;length, and that at the fame time the man in hisnbsp;chair were drawn from the fore to the back part ofnbsp;the Ihip, with the fame equable motion, then thenbsp;man would be in motion relatively to the parts ofnbsp;the Ihip; yet he would remain in the fame abfolutenbsp;place.

With refped to the words matter and body, we fliall for the prelent only remark the following difference between them; viz. that the v/ord matternbsp;has no relation to any determinate' figure; whereasnbsp;the word body more generally means fome feparatenbsp;3-nd determinate quantity of matter. Thus we faynbsp;'''ith propriety, that the movements of the celeftialnbsp;bodies are difficidtly determined, and the matter which

forms the atmofphere is heterogeneous; whereaS it Would be improper to change the places of thenbsp;words hodj and matter, by faying that the movementsnbsp;of the xeleftial matters are difficultly determined, andnbsp;the body zvhich forms the atmofphere is heterogeneous.

'Time, ftridtly fpeaking, is incapable of definition, and the only thing we can remark with refpeft tonbsp;it, is the difference between abfolute and relativenbsp;lime. Abfolute time flows equably, but does notnbsp;refer to the motion of bodies. Relative time is thatnbsp;portion of abfolute time, during which a Certainnbsp;movement is performed, and we affume fome ofnbsp;thofe movements, when they are equably andnbsp;fteadily performed, as the meafures of time; Thusnbsp;that portion of abfolute time which the fun employs in performing its apparent revolution roundnbsp;the earth, is called a day; the 24th part of thatnbsp;day is called an hour-, 365 times that day is callednbsp;jyrar, and fo on.

The properties of a thing are thofe qualities and operations which beldng to that thing, and, bynbsp;which it is diftinguiflied from other things that donbsp;not pofleis the fame properties. It is, for inftance,nbsp;a property of the fun to be luminous, of the magnetnbsp;to attract iron, amp;c.

. The hardnefs of a body is that degree of refiftance which the body offers to any power that may benbsp;applied for the purpofe of feparating its parts.nbsp;Whereas fuidity is the want of that refiftance;nbsp;fo that a perfeft fluid is that body whofe parts may

It will appear from the fequel that we are not acquainted with any perfeffly hard, or any perfe�tly fluid, body fo that we can only examine the intermediate gradations, which exift between thofenbsp;extremes; but thofe gradations which are expreffednbsp;by the words hardnefs, rigidity, brittlenefs, toughnejs^nbsp;fofinefs, clamminefs, fluidity, amp;c. are incapable ofnbsp;precife definitions or limits.

Cau/e_and effeti are relative terms; the effeSi being that which is produced by the cauie, and the Cdufe that which produces the effect.

Caules as well as effects are diftinguiftied into pTimary, Jecondary, amp;c. or into immediate and remote.nbsp;Thus when the heat of the fun rarefies the air, thatnbsp;� I'arefadtion produces wind, and that wind impels anbsp;Ibip forward. In this cafe the heat of the fun isnbsp;the ca�fe of the wind ; the wind is the effe�l of thenbsp;tarefaftion, and is at the fame time the caufe ofnbsp;the (hip�s motion; the motion of the (hip is effed:-

by the adion of the wind, fo that the wind is the immediate, ^nd the heat of the fun is the remote,nbsp;caufe of the (hip�s motion.

nri

Hus we know from conftant and univerfal experi-that whenever a body is left to itfelf, it always

Mechanical means fomething that i elates to, or is regu-ated by, the nature and laws of motion.

VOL. I. nbsp;nbsp;nbsp;C

ways falls towards the centre of the earth, unlefs fotne other body prevents it; we therefore aflurnenbsp;this obfervation as a law of nature, and exprefs itnbsp;by faying, that the various bodies of the earth tend, ornbsp;gravitate, towards the centre of it.

The exiftence or non-exiftence of a vacuum, meaning an extenlion entirely void of matter, has often been difputed amongft philofophers; their arguments always depending upon fome affumed hy-pothefis concerning the intimate nature of matternbsp;or of its ultimate particles gt; but as we are utterlynbsp;ignorant of the nature and properties of thofe particles, their arguments cannot determine the queftionnbsp;one way or the other.�The only conclufions wcnbsp;can make with refpeft to a vacuum, are ift. thatnbsp;the poffibility of its exiftence can be eafily imagined ; adly. that we are not certain whether itnbsp;really exift or not; and laftly, that if it be admitted that the figure of the leaft particles of matter is unchangeable, the motions of bodies, fuch asnbsp;continually take place in the univerfe, cannot benbsp;underftood without admitting the exiftence of a.nbsp;vacuum.

The word infinity has likewife been produdtlve of numerous difputes. Many odd pofitions have beennbsp;allumed' for the fupport of fpecious arguments,nbsp;and feveral abfurd confequences have been deducednbsp;from them. Thofe errors have principally arifennbsp;from the idea of fomething determinate, which hasnbsp;been annexed to the words infinite, or infinity, in-

mid its Properties. nbsp;nbsp;nbsp;19

ftead of fomething indefinite or indeterminate. In coni�quence of this idea, infinites have often beennbsp;compared together, and one infinite has been faidnbsp;to be the double, or treble, or the half, amp;c. ofnbsp;another infinite; whereas infinites, (in the truenbsp;fenfe of the word, which means fomething greaternbsp;cr lefs tlian any affignable quantity, but not determinate) are incapable of coniparifon j fince annbsp;^determinate quantity cannot bear any affignablenbsp;Pfoportion to another indeterminate quantity; andnbsp;cif courfe one mfinite cannot be faid to be greaternbsp;dian, equal to, or lefs than, another infinite. ,:

It has been ufually alledged, that if a line be infinitely,extended one way only, and another linenbsp;I�s infinitely extended both ways; the latter in-finite line muft be double the former Infinite line,nbsp;quot;'hich evidently implies a limited or determinatenbsp;length; namely, that the latter line has been extended on either fide as much as the former linenbsp;lias been extended one way only.

Again; take the length of one inch, and fup-Pofe it to be divided into an infinite number of parts. Take alfo the length of a foot, and fuppofenbsp;this to be divided into an infinite number of parts.nbsp;Here, they fay, it is evident that the latter infinitenbsp;is exactly equal to twelve times the former. Butnbsp;this, in nay humble opinion, feems to be a miftakennbsp;conclnfion; for the expreffions of infinity do not restquot; to the extenfions oi one foot and one inch; butnbsp;^ the numbers of the parts into which thofe exten-

c 2 nbsp;nbsp;nbsp;fions

fions have been divided j and thofe numbers can bear no affignable proportion to each other; juft becaufcnbsp;they are infinite.

The faft then is, that one foot is equal to twelve times one inch �, and if each of thofe extenfions benbsp;divided into any number of parts equal to each othernbsp;in length, the number of parts in the extenfion ofnbsp;one foot will be equal to twelve times the number'nbsp;of the parts that are contained in the extenfion ofnbsp;one inch; but this is not the meaning of dividingnbsp;a foot or an inch into an infinite number of parts;nbsp;therefore when the foot and the inch, are eachnbsp;divided into an infinite number of parts, thofenbsp;numbers have no affignable proportion to eachnbsp;other; though the fum of the former is undoubtedly equal to twelve times the fum.of the latter¹.

Numerous inftances of an infinite number of quantities having a finite or determinate fum, occur both in arithmetic and in geometry. In geometry it is fliewn, thatnbsp;a finite line may be divided into an infinite number ofnbsp;parts; and it is evident that the fum of all thofe parts muftnbsp;be equal to the line itfelf gt; viz. a finite quantity. In arithmetic it is fhewn, befides many other inftances, that if younbsp;take one half, and one half of that half, and one half of thenbsp;laft half, and fo on without end, the fum of them all isnbsp;equal to one; that is i j. rt Ts w� amp;c.=:i.

IT has been already- remarked, that a body is diftinguiQied from other bodies by means ofnbsp;peculiar properties. Thus we know water bynbsp;Its fluidity, and by its want of tafte, fmell, andnbsp;oolour; gold is known by its great weight andnbsp;peculiar colour; fait is known by its particularnbsp;t3.ftc; and fo forth. But there are certain properties, which belong equally to water, to gol,d, tonbsp;and to all other bodies. Extenfion for inftancenbsp;a property which belongs to them all; for theynbsp;all are extended. So likewife is weight; for theynbsp;a 1 are more or lefs heavy. Such then are callednbsp;General, or Common, Properties of. Matter ; and, asnbsp;far as we know, they are fix in number ; viz. ex-tenfion, diviJibiUty, impenetrability, mobility, vis iner-^^^5 or pajji^je^igj^^ and gravitation.

We have faid above, as far as we know, becaufe Matter in general may pofTefs ether properties,nbsp;^kat are not yet come to our knowledge. And thenbsp;fame obfervation may be made with refpe�l to thenbsp;^niverfality of thofe properties; viz. that they arenbsp;to be general, becaufe no body was ever foundnbsp;anting any one of them. But mankind is notnbsp;acquainted with all the bodies of the univ^erfe, and

even feveral of thofe which are known to exift, cannot be fubjefted to experiments,

�xtenfwn of a body is the quantity of fpace which a body occupies; the extremities of which, limitnbsp;or circumfcribe the matter of that body. It isnbsp;otherwife called the magnitude, or fize, or bulk ofnbsp;a body.

A certain quantity of matter may indeed be very fmall, or fo fine as to penetrate the pores of moftnbsp;other bodies j but yet fome extenfion it muft have;nbsp;and it is by the comparifon of this property thatnbsp;one body is faid to be larger than, equal to, ornbsp;fmaller than, another body. The mealurement ofnbsp;a body confifts in the comparifon of the extenfionnbsp;of that body with a certain determinate extenfion,nbsp;which is alTumed for the ftandard, fuch as annbsp;inch, a foot, a yard, a mile ; and hence we fay thatnbsp;a certain body is three feet long, another body isnbsp;the hundredth part of an inch in length, andnbsp;fo on*.

A body is not only extended, but it is extended three different ways, viz. it has length, breadth, andnbsp;tliicknefs. Thus an ordinary flieet of writing papernbsp;is about fixteen inches long, fourteen inches broad,

�* The attempts, which have been made for the purpofe of eftablifliing an invariable ftandard of meafure, togethernbsp;with the difficulties which obftru� the perfeil accomplifh-nient of that obje�f; as alfo the principal meafures whichnbsp;are now in ufe, will be mentioned in another part of thisnbsp;work.

by the mathematicians, that which has no parts nor magnitude. Thus if you divide a line intonbsp;two parts, the divifion or boundary between thenbsp;two parts is a point.

Having ihewn above that there cannot exift, or that our fenfes cannot perceive, a furface without anbsp;body, it evidently follqws that neither a line nor anbsp;point can be perceived without a body. We fpeaknbsp;of the line or path of a planet; we alfo fay that anbsp;ftone thrown horizontally defcribes a curve line;nbsp;but in thofe cafes the meaning is, that the planet,nbsp;or the ftone, has paffed through certain places;nbsp;not that thofe lines do adually exift as any thingnbsp;fubftantial. When we look on a fh^et of paper,nbsp;we fee its furface, the edge of which is a line, andnbsp;the extremity of the line, or corner, of the paper,nbsp;is a point. But if you remove the paper j the furface, the line, and the point, vanilh from our light,nbsp;and they can only remain in our imagination,

Dmifibility of matter is the property of its being divifible into parts. Some philpfophical writersnbsp;have confidered it as a diftind; prof)erty of matternbsp;itfeif; but it may with more propriety be con-fidered as a property of extenfion; for we cannbsp;eafdy conceive that a given extenfion may benbsp;divided into any number of parts, let it be evernbsp;fo great; but it is by no means known whethernbsp;matter is, or is not, capable of being divided adnbsp;infinitum, that is, without any limit.

That a certain extenfion, as an inch, or any pther length, be it ever fo fmall, is capable of in-6nbsp;nbsp;nbsp;nbsp;finite

finite divifion, may be rendered evident by means of arithrpetic or of geometry. We may take, fornbsp;inftance, the halves of the propofed extenfion, thennbsp;the halves of thofe parts, then the halves of thofenbsp;halves, and fo on without end j for if you proceednbsp;tn this manner ever fo far, there will after all Hillnbsp;remain the halves of the laft parts, wdrich may benbsp;alfo divided into other halves, amp;c. Again, fup-pofe the line AB in fig. ift. plate i. to be the pro-Pofed extenfion. Through the extreme points ofnbsp;this line draw two indefinite lines EF, and CD,nbsp;parallel to each other. In one of thofe lines, asnbsp;fi-Fj take a point L, and from this point drawnbsp;ftraight lines to any parts of the line BD, everynbsp;one of W'hich lines will evidently cut the propofednbsp;extenfion AB into a different point. Now as thenbsp;fine Bjynbsp;nbsp;nbsp;nbsp;produced towards D without

fimitation, and ftraight lines may be drav/n from to an infinite number of points in the extendednbsp;fine BD; therefore the extenfion AB may be divided W'ithout end, or beyond any afiignable number of parts.

. Thus far we have fhewn that extenfion may be ^ivilible into an unlimited number of parts; butnbsp;V'ith refped to the limits of the divifibility of mat-itfelf we are perfeftly in the dark. We cannbsp;indeed divide certain bodies into furpnfingly finenbsp;nnd numerous particles, and the works of naturenbsp;� nt many tinids and folids of wonderful tenuity;nbsp;both our efforts, and thofe naturally fmall

20 nbsp;nbsp;nbsp;Of the General

cbjeds, advance a very fliort way towards Infinity. Ignorant of the intimate nature of matter, we cannot alTert whether it may be capable of infinitenbsp;divifion, or whether it ultimately confifls of particles of a certain fize, and of perfedt hardnefs.

I fhall now add fome inftances of the wonderful tenuity of certain bodies, that has been producednbsp;either by art, or that has been difcovered by meansnbsp;of microfcopical obfervations amongft , the ftupen-dous works of nature.

The fpinning of wool, filk, cotton, and fuch like fubftances, affords no bad fpecimens of this fort jnbsp;- fince the thread which has been produced by thisnbsp;means, has often been fo very fine as almoft tonbsp;exceed the bounds of credibility, had it not beennbsp;fufficiently well authenticated. Mr. Boyle mentions, that two grains and a half of filk was fpun intonbsp;a thread 300 yards long.

A few years ago a lady of Lincolnfliire fpun a fingle pound of woollen-yarn into a thread 168000nbsp;yards long, which is equal to 95 Englifli miles¹.nbsp;Alfo a fingle pound weight of fine cotton-yarn wasnbsp;lately fpun, in the neighbourhood of Manchefter^nbsp;into a thread 134400 yards long.

The dudility of gold likewife furniflies a flriking example of the great tenuity of matter amongflnbsp;the produdions of human ingenuity. A finglenbsp;grain weight of gold has been often extended into

^ furface equal to 50 fquare inches. If every fquare inch of it be divided into fquare particles ofnbsp;the hundredth part of an inch, which will be plainlynbsp;vilible to the naked eye, the number of thofe particles in one inch fquare will be 10000; and, multiplying this number by the 50 inches, the productnbsp;is 500000; that is, the grain of gold may benbsp;a(5lually divided into at lead half a million of particles, each of which is perfedtly apparent to thenbsp;naked eye. Yet if one of thofe particles be viewednbsp;in a good microfeope, it will appear like a largenbsp;furface, the ten-thoufandth part of which might

An ingenious artift in London has been able to ^ravv parallel lines upon a glafs plate, as alfo quot;uponnbsp;hlver, fo near one another, that 10000 of themnbsp;Occupy the fpace of one inch.�Thofe lines can benbsp;feen only by the affiftance of a very good microscope.

Another workman has drawn a fiiver wire, the diameter of which does not exceed the 750th partnbsp;of an inch.

But thofe prodigies of human ingenuity will appear extremely grofs and rude', if they be compared with the immenfe fubtility of matter whichnbsp;naay every where be obierved amongft the works ofnbsp;nature. The animal, the vegetable, and even thenbsp;niineral, kingdom, furnifli numerous examples ofnbsp;this fort.

Wnat mull be the tenuity of the odoriferous parts of mulk, when we find that apiece of it will feent

a whole room in a flrort time, and yet it will hardly lofe any fcnfible part of its weight. Butnbsp;fuppofing it to have loft one hundredth part of anbsp;grain weight, when this fmall quantity is dividednbsp;and difperfed through the. whole room, it muft fonbsp;expand itfelf as not to leave an inch fquare ofnbsp;fpace where the fenfe of ftnell may not be affected by fome of its particles. How fmall muftnbsp;then be the weight and fize of one of thofe particles ?

The human eye, unaffifted by glaffes,- can frequently perceive infeCts fo fmall as to be barely dif-cernible. The leaft reflection muft fhew him, that the limbs, the veflels, and other neceffary parts,ofnbsp;fuch animals, muft infinitely exceed in finenefs everynbsp;endeavour of human art. But the microfeope hasnbsp;difeovered wonders, that are vaftly fuperior, andnbsp;fuch indeed as were utterly unknown to our forefathers, before the invention of that noble in-ftrument.

InfeCts have been difeovered, fo fmall as not to exceed the looooth part of an inch: fo thatnbsp;loooooooooooo of them might be contained within the fpace of one cubic inch; yet each animalculenbsp;muft confift of parts connected with each other;nbsp;with veffels, with fluids, and with organs neceffarynbsp;for its motions, for its increafe, for its propagation,nbsp;amp;c. How inconceivably fmall muft thofe organsnbsp;be ? and yet they are unqueftionably compofed ofnbsp;other parts ftill fmaller, and ftill farther removednbsp;from the perception of our feijfes,

Properties of Matter, nbsp;nbsp;nbsp;29

We might eafily fill a great many pages v/ith examples and calculations relative to this fubjefl:;nbsp;but as the pleafing narration of fuch wonderfulnbsp;fafts is not likely to give any real information concerning the general properties of matter, whichnbsp;form the fubjedt of this part of the book, I mullnbsp;refer the inquifitive reader to other works¹. Thenbsp;confideration of this divifibility does alfo lead thenbsp;mind to certain curious fpeculations. (i)

Boyle�s book of Effluvia; Keill�s Iiitroduaioii to Nat. Phil.; Rohault�s Phyficks; Phil. Tranf. N. 194;nbsp;s�Gravefand�s Phil, j Mufichenbroek�s Phil. amp;c.

(l.) Several writers, when treating of the divifibility of matter, have mentioned two curious theorems, which Inbsp;ftall fubjoin in this note, as they may be of ufe to the fpe-culative philofopher. Thofe theorems are eftablifned onnbsp;^hc fuppofitign that matter is divifible without end.

Theorem I. A quantity of matter however fmall, and any finite fpace however large^ being given', it ispojfible thatnbsp;that matter may be diffufed through ail that fpace, and fo fillnbsp;St, as not to leave in it a pore, whofe diameter will exceed anbsp;given right line.

P. I. fo that the cube be equal to MfP and let the quantity of matter be reprefented by i�; alfo let the linenbsp;D be the limit of the diameter of the pores.

The fide AB being a finite quantity, may be conceived to be divifible into parts equal to the line D. Let the num-fsr of thofe parts be reprefented by �, fo that nDr:AB, andnbsp;�nbsp;nbsp;nbsp;nbsp;�AB)^. Conceive the given fpace to be divided

50 nbsp;nbsp;nbsp;Of the General

Tlie contemplation of thofe wonders of nature, cannot fail of impreffing on our minds a ftrongnbsp;iflea of humility as well as of aftoniQiment.'�A vaftnbsp;gradation of animals perfect in their kind, butnbsp;fmaller than the human being in fize and duration,nbsp;defcends as far down as our eyes can poffiblynbsp;difcern, even when they are affitled by the moilnbsp;powerful microfeopes. This vaft gradation, infteadnbsp;of exhaufting the powers of nature, fhews the probable

D, and the number of thofe cubes wil! be ft*, which cubes are reprefented in the fig. by E, F, G, H. Again, let thenbsp;particle P be fuppofed to be divided into parts whofe number be ft*} and in each cubic fpace let there be placed onenbsp;of thofe particles; by which means the matter P will benbsp;diffufed through all the given fpace. Befides each particlenbsp;being placed in its cell, may be formed into a concavenbsp;fphere, whofe diameter may be equal to the given line D ;nbsp;whence it will follow, that each fphere will touch thatnbsp;which is next to it; and thus the quantity of matter P, benbsp;it ever fo fmall, will fill the given finite fpace, howevernbsp;large, in fuch a manner as not to leave in it a pore largernbsp;in diameter than the given line D.

Corollary. 1 here may be a given body, whofe matter if it be reduced into a fpace abfolutely full; that fpace may benbsp;any given part of the former magnitude.

I heorem II, There may be two bodies equal in bulky whofe quantities of matter may be very unequal, and thoughnbsp;they have any given ratio to each other, yet the fums of thenbsp;fores or empty fpaces in thofe bodies may ahnoft approach thenbsp;ratio of equality.

Properties of Matter. nbsp;nbsp;nbsp;31

bable exiftence of animated beings vaftly fmaller than thofe; nor have we the leaft reafon to fix anbsp;limit to the ferles.

If we contemplate the ftate of exiftence of thole animals j of one, for inftance, out of a large number of the fame fpecies, that has been born in anbsp;glafs of dirty water; whofe life lafts but a few hours,nbsp;and whofe fize is lefs than the 5000th part of annbsp;mch; for fuch animals have been actually feen.nbsp;If we indulge our fancy by confidering what knowledge, or what ideas, can he poffibly entertain ofnbsp;man, � of the earth, � of the univerfe; we maynbsp;'''ithout difficulty conclude, that, far from havingnbsp;precife notions of our exiftence, he may in all

The demonftration of this theorem is eaflly derived from the foregoing, for fmce the matter of a body may be connived to be condenfable into any part of the original bulk ;nbsp;therefore fuppofing two bodies, A and B of equal bulk, tonbsp;he fuch that the matter of A be 100 times the matter of B;nbsp;the matter of B may be conceived to be condenfed into onenbsp;tooooooth part of its original bulk, and of courfe thenbsp;matter of A will be condenfed in one hundred loooooothnbsp;parts of the fame bulk; in which cafe the fpaces left in thenbsp;original bulk of B will be to the fpaces left in the originalnbsp;hulk of A as 999999 to 999900, which numbers arenbsp;nearly equal to each other.

Inftead of the above-mentioned numbers, the proportion, of the quantities of matter may be increafed at pleafure,nbsp;�nd fo niay the proportion of the original bulks of thenbsp;bodies to the fpaces into which they may be conceived to

probability look upon the glafs of dirty water as the boundary of the habitable world. Out of thatnbsp;water, tradition or his own experience, flrews himnbsp;nothing but tlie inevitable deftfudion of his fpecies,nbsp;and a confufed aflemblage of immenfe objeds,nbsp;whofe nature and whofe motions are utterly inexplicable to him. Yet he may poffibly fufped thatnbsp;thofe very objeds have powers infinitely fuperiornbsp;to thofe of his own fpecies.nbsp;nbsp;nbsp;nbsp;y

Let us now follow the analogy, and let us briefly apply the fame contemplation to ourfelves. Thenbsp;planets, the ftars, the comets, and perhaps an infinity of other bodies that are far beyond the reachnbsp;of our knowledge, ruanifeft the exiftence of powersnbsp;infinitely above us, and perhaps even lefs compre-henfible to us than we are to the above-mentionednbsp;animalcule. Confined to the globe of this earth,nbsp;yvhich is only a fpeck in the univerfe; and, withnbsp;refped to us, not much better nor worfe than thenbsp;glafs of dirty water is with re{y:)ed to thole infeds;nbsp;how infignificant are our powers, ^and how imper-fed is our knowledge of nature 1 How little likely'nbsp;are we to comprehend tiie real order of things, andnbsp;the Great Wifdom that regulates the whole ! Innbsp;this fublime inquiry the afliftance of our reafoningnbsp;faculty is trifling indeed; the clew of analogy isnbsp;flrort and imperfed j and our imagination loonnbsp;lofes itfelf in the boundlefs extent of immenfity.

Impenetrability is that property, by which a body excludes every other body from the place which

Properties of Matter.

itlelf occupies. Thus one cannot drive a cubic inch of gold into a cubic inch of filver. You maynbsp;indeed melt and incorporate the two metals intonbsp;one lump; but then the lump will meafure tw'Onbsp;Cubic inches; wdiich proves not that the gold occupies the fame cubic inch of fpace which is occupied by the filver ; but that the particles of thenbsp;two metals are placed contiguous to each other.nbsp;Thus alfo, if a quantity of water be put into anbsp;ftrong veffel, for Inftance, of iron, and the veflelnbsp;he accurately Ihut up, it will not be poffible tonbsp;prefs the fides of the veflel towards each otjier j thenbsp;Gutter w'hich fills the cavity of it being fufficientnbsp;to refill; any degree of preflure.

Though impenetrability be admitted as a ge-neral property of matter, it muft, however, be observed, that in certain mixtures of two or more bodies of different natures, a lofs of bulk does ac-tually take place ; thus if a cubic inch of fpirit ofnbsp;^'^ine be mixed with a cubic inch of water, the bulk

bic inches ; yet the weight of the mixture (prh-Mded no evaporation be allowed to take place) ^'ill be equal to the fum of the weights of the twonbsp;fluids; which indicates that one of ine fluids mufhnbsp;have filled up fome of the pores or vacuities of thenbsp;^ther fluid.. It is befides not unlikely that fomenbsp;other finer fluid may have efcaped in the alt;S ofnbsp;mixing the tw'o bodies.

lu other parts of this work we fhall take notice ''OL. r.nbsp;nbsp;nbsp;nbsp;onbsp;nbsp;nbsp;nbsp;of

of the lofs of weight and other phenomena, that take place in many cafes of mixture j but with re-fpedt to impenetrability itfelf, we may rather confi-der it abftraftly as a property of the real quantity ofnbsp;matter which exifts in bodies, independently ofnbsp;pores and vacuities, than as a general property,nbsp;without exception, of bodies in their ufual ftate ofnbsp;exiflence.

Mobility of matter is that effential and general property, whereby any body is capable of beingnbsp;moved from one part of abfolute fpace to anothernbsp;part of if. Experience conftantly fliews, that thenbsp;force, which is required to move a botty, is proportionate to its weight j therefore we conclude withnbsp;faying, that all bodies are capable of being moved �,nbsp;provided an adequate force be employed to. putnbsp;them in motion.

It is a faft proved by conftant and univerfal experience, that the progrefs of a body in motion is retarded precifely in proportion to the obftrudlionnbsp;which the body meets with in its way. Thus ifnbsp;two bodies, A and B, exadlly alike in fliape, weight,nbsp;and fubftance, be put in motion by equal impulfes,nbsp;and meet with equal obftruftions } by moving, fornbsp;* inftance, throu^ the fame medium, or by rollingnbsp;over the fame fort of plain furface, thofe two bodiesnbsp;will run over equal fpaccs in equal times j but if thenbsp;t)ody A meets with half the obRrudtion that thenbsp;body B meets with, then A will go as far again asnbsp;the body B i when A meets with a quarter of thenbsp;5nbsp;nbsp;nbsp;nbsp;obftruilion,

'^bftruftioDj it will go four times as far as B ; and fhort, A will percur a fpace longer than B, bynbsp;much as its obftruftion is diminilhed ; and consequently when the obflrudlion to A�s motion is en-brely removed, A will go infinitely farther than B ;nbsp;^bat is, it will continue to move for ever. It therefore appears, that a body once put in motion has nonbsp;power to flop itfelf j nor can its motion ceafe,nbsp;Onlefs fome force is exerted by fome external powernbsp;^gainft it.

By the fame fort of reafoning, we prove that a body at reft has no power to put itfelf in motion,nbsp;^od of courfe that it will continue for ever at reft,nbsp;'^olefs it be impelled by fome external power; fornbsp;f'Oce we find that a certain impulfe is required tonbsp;^ove a body with a certain quicknefs, viz. fo as tonbsp;it run over the fpace of a mile in one minute ;nbsp;diat with half that impulfe it will percur half a mile;nbsp;quot;''fh the hundredth part of the original impulfe itnbsp;quot;'�b percur the hundredth part of a mile; it willnbsp;^3-turally follow, that without any impulfe at all, itnbsp;not move in the leaft : a body therefore has nonbsp;power either to put itfelf in motion if it be at reft,nbsp;ftop itfelf if it be in motion : and this paffive-^^fs of matter is called the vis inert ice, or wafit ofnbsp;^^ivify^ of bodies.

^ novice in philofophy m,ay perhaps be Induced fnfpect the truth or generality of this property ofnbsp;by obferving that a man, or other animal,nbsp;eafily move' himfelf from reft, or ftop his

motion : but m this cafe it muft be remarked, that the animal receives a general impulfe at the commencement of his life, and that all his aftions, asnbsp;long as he exills, are the confequence of that original impulfe. I fliall endeavour to illuftrate thisnbsp;matter by an inftance of a much lefs complicatednbsp;nature.

It is very well known that a common eight-day dock, when it is once wound up, will continue tonbsp;move its pendulum for a whole week, and at thenbsp;end of every hour it will ftrike a number of ftrokesnbsp;on the bell. It is evident likewife that thofe motions of the pendulum, the hammer, amp;c. are owingnbsp;to the original power or impulfe which was communicated to the machine by the perfon who woundnbsp;it up} yet an ignorant man might fay, if bodies cannot put themfelves in motion, nor can they flopnbsp;thcmfelves when they are aftually in motion ; hovvnbsp;d�es it happen that the ftriking part of the docknbsp;puts itfelf in motion, and then flops itfelf at the endnbsp;of every hour ? The anfwer is, that the power whichnbsp;was communicated to the fpring or weight of thenbsp;clock, is fo regulated by the mechanifm, as to aftnbsp;by little and little, fufficiently to keep the pendulum and the w'heels in motion ; and that when anbsp;particular part of one of thofe wheels comes againftnbsp;a certain machinery, it then difengages a portionnbsp;of the other pow'qr, viz. of the fpring or weigh�dnbsp;of the ftriking part, which puts the hammer innbsp;aftion. .

Whft

What has been faid of the clock will perhaps be fufficient to remove the difficulty refpecling the apparent felf-moving power of more complicated me-^^hanlfms, fuch as that of an animal or vegetablenbsp;body. But though we are led by the analogy ofnbsp;^luch fimpler movements, to admit the dependencenbsp;^f animal and vegetable motion on an original im-Pulfe j we do not, however, prefume to explain thenbsp;*^dgln, dependence, and poffible modifications ofnbsp;that impulfe; our underftandings, and our know-^^dge, being as yet infufficient to explain the naturenbsp;^r^d the laws of that original energy.

AttraElmi is that property whereby one body or Part of matter attracts, or endeavours to get near,nbsp;^f^other body. There are feveral forts of attraction ;nbsp;b^ch as the magnetic attraction, which takes placenbsp;between magnets and iron ; the electric attraction,nbsp;^'hich is obferved amongft bodies in certain circum-^ances, See. Thefe attractions, however, belongnbsp;certain bodies only, and of courfe they muft benbsp;examined in other parts of this work. But there isnbsp;^ fort of attraction which belongs to bodies of everynbsp;^ind; it is mutual among them, and it feems tonbsp;P^J'vade the unlverfe. It is that property wherebynbsp;bodies tend, or fall, towards the centre of the earth,nbsp;^^d it fias been called gravitation, becaufe thenbsp;^i^antity of that tendency in different bodies, is thenbsp;^eafure of their weight or gravity.

.^^P^^^soce, reafoning, and analogy, Ihew that �s gravitation exifts not only between the globe of

the earth and the furrounding bodies, but between all parts of matter. One terreftrial body gravitatesnbsp;or tends towards another terreftrial body ; thenbsp;moon gravitates towards the earth ; the moon,nbsp;the earth, and all the planets, gravitate towardsnbsp;each other, and towards the fun; and probablynbsp;the fun, with all its planetary fyftein, may gravitatenbsp;towards fome other objeft.

The motion of certain bodies which leem to fly away from the earth, muft not be confidered as annbsp;exception of this general law; for in thofe cafes thenbsp;bodies only give way to other furrounding bodies ofnbsp;a heavier nature, viz. that have a greater tendencynbsp;towards the earth. Thus fmoke, when extricatednbsp;from burning bodies, goes upwards, or from thenbsp;centre of the earth, becaufe the furrounding air,nbsp;which is heavier than fmoke, takes its place : butnbsp;if the air be removed, or at leaft it be fo far rarefiednbsp;as to become lighter than fmoke, then the fmokenbsp;will defcend like a ftone or other heavier body.nbsp;Thus alfo if you drop a piece of cork into an emptynbsp;vetTel, the cork will go downwards or to the bottomnbsp;of the veflTel; but if afterwards you pour water intonbsp;the vefTel, the cork will afcerfd in order to make waynbsp;for the water, which has a greater tendency towardsnbsp;the centre of the earth than an equal bulk of cork.

Dally and conftant experience fliews to every perfon, that near the furface of the earth, all bodiesnbsp;tend towards the centre of it, unlefs they are hindered by other bodies. But the reader may naturally

turally afk, how is it known that the planets and the earth gravitate towards the fun ? The anfwer is,nbsp;that from the accurate meafurements of the motions of thofe planets, they are found to follow thenbsp;fame laws that bodies do, which are projefted in anbsp;certain manner near the furface of the earth, andnbsp;^hofe motion is undoubtedly determined by thenbsp;power of gravitation; we therefore, according tonbsp;the rules of philofophizing, attribute fimiiar caufesnbsp;to fimilar effedts, and conclude that the planets gra-'^itate towards the fun, in the fame manner asnbsp;ftones, water, and other terreftrial bodies, gravitatenbsp;towards the earth.

What is the cauie of gravitation, or how can a body adt upon another body through a certainnbsp;^Pace ? is a quellion which naturally prefents itfelfnbsp;to the inquifitive mind ; but which we are utterlynbsp;^^capable to anfwer.

A variety of conjectures have been formed, and ^any hypothetical fuppofitions have been offered,nbsp;for the elucidation of this queftion ; but as they arenbsp;^11 involved in abfurdity and obfcurlty, I fliall notnbsp;detain my reader with any account of them. Allnbsp;1^0 Can fay is^ that the effedl is certain, the know-edge of its laws is highly ufeful to mankind ; butnbsp;ds caufe is hidden amongft the myfteries of nature.

OF the general properties of matter, the firll three may be prefumed to have been fuffici-ently illuftratcd in the preceding chapter; but thenbsp;other three, viz. mobility, vis iverfi^, and gravitation, are the foundation of the extenfive doftrine ofnbsp;itiotion, or of mechanics; and are therefore deferv-ing of a full and particular examination.

Almofl all the phenomena of nature are owing to motion. The appearance and difappearance of thenbsp;coeleflial bodies; the increa'e of animals and vegetables ; the compofition and decompofition of complex fubftances, fire, amp;c. are all effefled by motion.nbsp;Therefore the laws of motion muft. be looked uponnbsp;as the foundation of natural philofophy ; fo thatnbsp;without a clear comprehenfion of thofe laws, it willnbsp;be impoflible to make any proficiency in the fludynbsp;of nature.

The importance and extent of the fubjeft, render it neceffary to divide the materials into feveralnbsp;chapters, in each of which fuch particulars will benbsp;arranged, as are more immediately connefted withnbsp;each other, and more conducive to concifenefs andnbsp;perfpicuity.

It is a natural confequence of the vis inerlia of matter, that whatever body is in motion, muft benbsp;fnppofed to have been put in motion by fomc �nbsp;aftive force; viz. fome external impulfe.

This impelling force may be of two forts. It may either communicate the impulfe at firft, andnbsp;then ceale to aft, like the impulfe which is givennbsp;to a bullet by the difcharge of a gun; or it maynbsp;aft irremittedly on the body in motion, like thenbsp;force of gravity on a ftone that is dropped fromnbsp;any height. For diftinftion fake we (hail call thenbsp;hrft (imply an impulfe, and the latter an accekrafivsnbsp;force.

A body may be put in motion by one, two, or more forces at the fame time, and thofe forcesnbsp;may be either all fimple, or all accelerative, or fomenbsp;may be of one fort, and others of the other fort.

Moft of the movements that commonly take place in the world, are the efFeff of more than onenbsp;�mpulfe; and they are never performed wnth per-fefl freedom, fmee they are always performed innbsp;refilling mediums. However, in order to prefervenbsp;perlpicuity as much as it lies in our power, wenbsp;fliall In the firft place examine the motions arifmgnbsp;from a fimple impulfe in a non-refilling medium,nbsp;�^rid fliail then proceed in the examination of thenbsp;more intricate caufes of motion.

1. Every body will continue in its flate of reft,)nbsp;or of moving uniformly in - a ftraight line ; unlefsnbsp;it be compelled to change that fta^e by forces im-

II. nbsp;nbsp;nbsp;The change of motion is always proportionalnbsp;to the moving force imprefled, and is always madenbsp;according to the right line, in which that force isnbsp;imprefled,

III. nbsp;nbsp;nbsp;Adion and re-adion are always equal andnbsp;contrary to each other; or the adions of tw'onbsp;bodies mutually upon each other, are always equal,nbsp;and direded towards contrary parts.

The firft of thofe laws is evidently nothing more than the vis inertia of matter, announced in a different manner; excepting only the aflertion of thenbsp;body moving in a ftraight, and not in a cuiwe,nbsp;line, which particular may perhaps be defervingnbsp;t)f fome explanation.

The proof of this particular property has like-wife been deduced from conftant experience; for we find that whenever a body moves in a curvenbsp;fine, there always is fome fecondary power whichnbsp;forces it to deviate from the redilinear courfe;nbsp;and that deviation is exadly proportional to thatnbsp;fecondary power. Thus a ftone which is throv/nnbsp;horizontally would proceed horizontally in anbsp;ftraight line, were it not drawn downwards by the

force of gravity; and we find by computation, that the deviation from, the horizontal direftion isnbsp;exaftly proportional to the force df gravity.

Hence the fecond law has been deduced, in whicli it is afierted that the change of motion isnbsp;always proportional to the moving force imprefled,nbsp;and is made according to the right line in which,nbsp;that force is imprefled; for if it were made in anbsp;crooked line, it would imply the a�fion of a thirdnbsp;force ; and if it were not proportional to the movingnbsp;force, the efFedt would not be adequate to thenbsp;caufe, '

The third law may be eafily illuftrated by means of examples; apd the lead refledtion on the phenomena, which commonly occur, will be fufficientnbsp;to manifeft the truth and unlverfality of it.

When a man ftrikes one of his hands againfl; the other, the blow is felt equally by both hands. Ifnbsp;you ftrike a glafs bottle with a fleel hammer, thenbsp;blow will be received equally by the hammer andnbsp;by the glafs bottle j and it is immaterial whethernbsp;the hammer be moved againft the bottle at reft,nbsp;or the bottle be moved againft the hammer at reft;nbsp;yet the bottle will be broken, whereas the hammernbsp;will not, becaufe tlie fame blow, which is fuffi-oient to break glafs, is not fufficient to break anbsp;lomp of fteel.�It is for the fame reafon, that if anbsp;ftrike his fift againft another man�s face, thenbsp;�w, which is equally received by the fift and by

the

If a Hone be tied to a horfe by means of a rope, the horfe in dragging the ftone will exert a degree of force equal to the refiftance of the ftone;nbsp;for the rope which is ftretched both ways willnbsp;equally pull the horfe towards the ftone, and thenbsp;ftone towards the horfe. And, in fa�f, the ftonenbsp;will not follow the horfe, unlefs the power ofnbsp;the horfe be greater than the refiftance of thenbsp;ftone.

Experience likewife thews, that if a loadftone and a piece of iron be placed on feparate pieces ofnbsp;cork, and be fuffered to float on the lurface ofnbsp;water, the attraftion between them will be mutual,nbsp;and they will move towards each other fo as tonbsp;meet in a place between their two original fitua-tions. If the loadftone only be held faft in itsnbsp;place, the iron will come all the way to meetnbsp;it; and if the iron only be held faft in its place,nbsp;the magnet will advance towards the iron until itnbsp;comes in contafl with it.

The motion given to a boat by oars is likewife a convincing illuftration of the third law; for bynbsp;the adion of one extremity of each oar againft thenbsp;water one way, its other end re-ads upon the boat,nbsp;and impels it the contrary way.

We flrall now examine the motion which is produced by a Angle impulfe, which ads at firft only,

It has been already (hewn, that in this cafe the body will continue to move uniformly �, that is, itnbsp;would run over equal fpaces in equal portions, ofnbsp;time; and fuch would be the cafe of a bullet fliotnbsp;out of a gun, or of a ftone thrown out by a man�snbsp;hand, were they not impeded by the refiftance ofnbsp;the air, and were they not adled upon by thenbsp;force of gravity. But it is now neceflary to takenbsp;. notice of feveral particulars relative to this fort ofnbsp;motion.

In the firft place it may be alked, how does the impelling force put the body in motion, or wdiatnbsp;does it communicate to the body ?. The anfvvernbsp;. IS, that the moving force does not communicatenbsp;^riy thing to the bodyj but it only moves thenbsp;body through a certain fpace in a certain time,nbsp;after which the body, being left to itfelf, will con-bnue ro move at the fame rate, viz. will continuenbsp;lo run over like fpaces in the like portions ofnbsp;time; and that merely in confequence of its visnbsp;^ne)tits \ of which vis inertiie, however, we do notnbsp;pretend know any thing more, than that it hasnbsp;been found to be a general property of matter.

All the particulars which can be remarked with rcfpedt to the above-mentioned limple motion, arenbsp;the relations between the time, in which a certainnbsp;pace is defcribed; the fpace which is percurrednbsp;a certain time, the quantity of wliich Ihews

the velocity i.the quantity of matter in motion; and laftly, the momentum, by which word we meannbsp;the force of the body in motion, and reckon itnbsp;equivalent to the impretEon that the body innbsp;motion would make on another body at reft,nbsp;that fliould bie prefented to it precifely in the di-reftion of its motion.

The momentum has been often call�d the quantity of motion, or limply the motion; but we fliall notnbsp;make ufe of the laft word in this fenfe, left itnbsp;fhould be miftaken for the velocity, in whichnbsp;fenfe it has been likewife ufed. We, fhall alfonbsp;exprefs the above-mentioned four particulars bynbsp;their initial letters, viz. T for the time, S for thenbsp;{pace, V for the velocity, Q for the quantity ofnbsp;matter, and M for the momentum.

By the word velocity we mean nothing more than the ratio of the quantity of fpace which is runnbsp;over in a certain portion of time. Thus it isnbsp;faid that a body moves with the velocity of threenbsp;feet per fecend ; alfo that the velocity of a bodynbsp;A is to the velocity of another body B, as two tonbsp;three; meaning that if A goes over a certain fpace,nbsp;as for inftance, four miles, in a certain time, thenbsp;body B will percur fix miles in the fame time;nbsp;lince two is to three as four is to fix.

It is therefore evident, that in equal times the velocities are as the fpaces; but if the times be 'nbsp;unequal, then the velocities are as the quotients ofnbsp;the fpaces divided by the times refpe�lively. Thus

fupp. ft

tuppofe that a body A paffes over ten feet' in two minutes, and another body B paffes over eightnbsp;feet ill four minutes, the velocity of A will be to

two } for by dividing the ten feet by the two mi-iiutes, we find how many feed the body A runs over in one minute, and likewife by dividing thenbsp;eight feet by four minutes, we find how manynbsp;feet the body B runs over in the tame time; viz,nbsp;^ne minuie; fo that by the operation of dividing thenbsp;fpaces by the times refpedtively, we do nothingnbsp;more than find Out the fpaces that are percurrednbsp;^ the two bodies in efial times, and then comparenbsp;^hem together.

Before we proceerl any farther, it is neceffary to obferve, that whenever it is faid that certain thingsnbsp;as certain other things, we only affert the rationbsp;the former to the latter; viz. that the formernbsp;mcreafe or decreafe according as the latter do in-oreafe or decreafe; but from fuch affert ions nothing real and determinate can be deduced, unlefsnbsp;We have reeourfe to experiments, in order to af-*^ertain fome of thofe particular things with whichnbsp;others are compared. Thus in the preceding pa-^^graph, it has been afferted that the velocities arenbsp;the quotients of the fpaces divided by the times;nbsp;this affertion will not enable us to determinenbsp;Velocity, or the fpace run over, or th� time.

unlefs fome of thofe particulars be prevloully known. Hence if we learn from aftual experiment (viz. by meafuring the fpace with a rulernbsp;and the time by a watch), or are otherwife informed, that a tody has been moving throughnbsp;�ten feet in two feconds; then dividing the ten bynbsp;two, the quotient five gives the velocity ; whichnbsp;means that the body moves at the rate of, ornbsp;percurs, five feet per fecond. If by the above-mentioned proportional exprefiion we wifli to findnbsp;the fpace, we mull previoufly know the velocitynbsp;and the time ; and if we with to afcertain the time,nbsp;we muft previoufly know the velocity and thenbsp;fpace. Therefore, in general, the ufe of fuch proportional expreflions is to render certain particularsnbsp;deducible, by computation, from other particularsnbsp;which belong to the fame exprefiion, and whichnbsp;have been previoufly afcertained by means ofnbsp;aftual experiments. We fhall now proceed tonbsp;explain the other particulars which relate to thenbsp;aboverinentioned Ample or equable motion.

ThQ fpace is as the velocity multiplied by the time ; (that is, S is as V T) for if a body move with thenbsp;velocity of three feet per minute, it is evidentnbsp;that it muft pafs over twice three, or fix, feet, in twonbsp;minutes; three times three, or nine, feet, in threenbsp;minutes; four times three, or twelve feet, in fournbsp;minutes ; and, in fhort, the fpace is as the produdtnbsp;of the velocity, or rate of going, multiplied bynbsp;the time.

The time is as the fpace divided by the velocity j {viz. T is as ~) for if a body, for inftance, runsnbsp;Over 12 feet when its velocity is three feet per minute, it is evident that in order to find the numbernbsp;of minutes, that the body has employed in pafiingnbsp;over 12 feet of fpace, we muft fay, by the commonnbsp;rule of three, if the body pafles over three feet innbsp;one minute, how many minutes will it employ innbsp;paffing over 12 feet; which proportion is ftatednbsp;thus; 3 : 1: M2 ;, and as the fecond term is unity,

've need only divide the 12 by 3; (viz. the fpace by the velocity) and the quotient 4 is the

The momentum, and the quantity of matter, are the two laft particulars which remain to be examinednbsp;with refpecfi; to this fort of motion. It has alreadynbsp;been mentioned, that the momentum is the force ofnbsp;the body in motion, and is equivalent to the im-preffion it would make on another body that fhould

According to the fourth axiom, every effect muft be produced by an adequate caufe; therefore if anbsp;body be caufed to move with a certain velociiy bynbsp;rueans �f a certain impulfe, the double of thatnbsp;rrnpulfe will be required to make it move with thenbsp;fiouble of that velocity 5 three times that impulfenbsp;to let It � move with three times the original velo-VOL, I.nbsp;nbsp;nbsp;nbsp;�nbsp;nbsp;nbsp;nbsp;city;

city; and, in fliort, the moving force or impulle mufl. be proportionate to the velocity. And fornbsp;the fame reafon, the refiftance, which muft be op-pofed to the faid body in order to flop it, mullnbsp;like wife be proportionate to the velocity of thenbsp;body.

Now let two diftin�t bodies, A and B, move with equal velocities j but let the quantity ofnbsp;matter in B be the double of the quantity ofnbsp;matter in A; and it is evident that the momentumnbsp;of B mufl be double the momentum of A ; fornbsp;if we imagine B to be divided into two equal parts,nbsp;each of thofe parts mufl have a momentum equalnbsp;to the momentum of A; (A being equal to thenbsp;half of B) and of courfe both halves together muftnbsp;have a niomentum double of the momentumnbsp;of A.

If the body B be fuppofed to move as fiift again as A, or with the double of its former velocity, it follows, from what has been mentioned,nbsp;above, that its momentum muft be double of itsnbsp;�former momentum; but before its momentumnbsp;was double the momentum of A, therefore nownbsp;its momentum muft be quadruple the momentumnbsp;of A j that is, it muft be multiplied by two onnbsp;account of its double quantity of matter, and againnbsp;by two on account of its double velocity; whichnbsp;is as much as to fay that the momentum is as the'nbsp;produd of the c|uantity of matter multiplied by

tt.e velocity; (viz. M is as Q V.)�Or we itiay confider it as a definition, and fay that by the mo-gt;�entum we mean the produft of the quantity ofnbsp;Matter by the velocity

If the quantity of matter in E; inftead of being double, be fuppofed. to be treble, or quadruple, ornbsp;the half, or other multiple, of the quantity ofnbsp;matter in A j the fame mode of reafoning willnbsp;fliew that its momentum muft be treble, or qua*-druple, or the half, or any other multiple refpec-tively of the momentum of A, when the velocities of A and B are equal; but that thofe mo-mentums muft be multiplied by the velocitiesnbsp;^hen the velocities of the bodies A and B arenbsp;�Unequal; which proves that the propofition isnbsp;�univerfally true.

cnt velocities, produced fotne years ago a long a difpute amongft the learned in Europe, ^he intricacynbsp;the arguments would render a fiatei^ent of the qnbsp;too long for this work, and it woul^ befid� be J

relatively to this queftion, to two excellent tra ; firft of which is entitled Jn EJpgt;y ^onUty, y tnbsp;verend Mr. Reid, in the 45th vol. of the Phil- Tranfi thenbsp;fecond is An Inquiry into the Meafure of the Force of Bodusnbsp;in Motion, by Dr. Irwin, Ph'il. Tranf. for

g 2 nbsp;nbsp;nbsp;LafWy.1

Laftly, the quantity of matter is as the momentum divided by the velocity j (viz. Q is as ^ }

for let V in the preceding proportional expreffion (M as V Q) be reprefented by the numbernbsp;2; then that proportional expreffion will become M as 2 Q; meaning that the momentumnbsp;is as twice the quantity of matter j but if the momentum is as twice the quantity of matter, therefore, taking the halves of thofe quantities, (for thenbsp;halves, or the quarters, or any other like parts,nbsp;or multiples of two quantities, have the famenbsp;proportion to each other as the quantities them-.nbsp;felves. Euclid. Elem. B. v. prop. 15.) half the

velocity be reprefented by any other number, as by 12, the proportion M as V Q, will becomenbsp;M as 12 Q, and, taking the 12th part of thofenbsp;two quantities, we fay, that fince the momentumnbsp;is as 12 times the quantity of matter, thereforenbsp;the �2th part of the momemtum is as the quantity of matter, which is expreffed thus; Q as

12 in the laft fuppofition; by the number ^ i*^ the preceding fuppofition, and may be reprefentednbsp;by any other number; therefore, univerfally, thenbsp;quantity of matter is as the quotient of the momentum divided by the velocity.

I ffi^U

I fiial�l now coilefl: all the propofitions, or laws, ^vhich belong to fimple motion, under one pointnbsp;of view, and, for the fake of perlpicuity, 1 Aral}nbsp;cxprefs them both in the concife way, by ufing'thenbsp;initial letters, and in words.

The fame exprefTed in w'ords.�In fimple motion, t'iz. zvhen a body is put in motion by a finale impulfe,nbsp;quot;^hicli a5ls (it firjl, and then leaves the body to proceednbsp;h itfelf in a non-refjUng medium-, or when feveralnbsp;bodies are thus feparately put in motion; the velocitiesnbsp;^^e as the fpaces divided by the times; the fpaces arenbsp;the velocities multiplied by the times j the times arenbsp;ike fpaces divided by the velocities j the momentumsnbsp;as the velocities multiplied by the quantities ofnbsp;Matter; and, lafly, the quantities of matter are as thenbsp;�^�^aentums divided by the velocities,nbsp;nbsp;nbsp;nbsp;' '

Thus, confidering the importance of the fubjefV, J have endeavoured to demonftrate the particularsnbsp;i'elative to fimple motion, in as familiar a manner,nbsp;^nd as little encumbered with mathematical ex-hieffions, as the fubjeft feemed tp admit, pur-hnfely tQ adapt them to the capacity of beginners.

t muft earneftly entreat the reader to make nnfelf matter of the contents of this chapter |3^-gt;nbsp;he proceeds to the next.

OF THE MOTION ARISING FROM CENTRIPETAL, AND CENTRIFUGAL, FORCES i AND OF THEnbsp;CENTRE OF GRAVITF.

^ Centripetal force is that power which com--pels bodies to move, or to tend towards a point, which is called the centre of attraBion. Anbsp;centrifugal force, on the contrar}', is that powernbsp;which compels bodies to recede from a point,nbsp;which is called the centre of repulfion. Gravitation^nbsp;or that power, by which bodies are forced to fallnbsp;towards the centre of the earth, is a centripetalnbsp;force, and will ferve us as an example for th� il-luftration of the general theory.

But though bodies direct their courfe towards the centre of the earth, yet the attraftive powernbsp;muft not be confidered as a peculiar property ofnbsp;that centre, or of any particular body near it*nbsp;Attraction is a property which belongs to matter innbsp;general, and is proportionate to the quantity of it*nbsp;The parts of the earth mutually gravitate towards, ofnbsp;attradt, each other ;�a ftonc at trails another fton^gt;nbsp;or any other body j the earth attracts a ftone,nbsp;well as the latter attrafts the former, and all bodies�nbsp;in fliort, mutually attradl each other ; nor arenbsp;acquainted with any particle of matter which nr^y

be faid to be deftitute of attraftion towards the 'whole aflemblage of terreftnal bodies. That, caterisnbsp;paribus, the attraft/ve force is proportionate to thenbsp;^juantity of matter, may be eafily proved ; for letnbsp;A, B, and C be three bodies equal in every relpeft;nbsp;^od if A attraft C with a certain force, (for in-ilance, a force equal to one ounce) it is evidentnbsp;that B, its equal, muft likewife attract C with thenbsp;force of one ounce; and, of courfe, A and B together, or a body equal to thofe tv/o, muft attract C with the force of two ounces. Again, ifnbsp;v/e take ten equal bodies, it is evident, that twonbsp;of them will attraft another diftindl body withnbsp;twice the force of one of them only, as alio thatnbsp;four, or five, or fix of thofe equal bodies will at-traft the other body with four, or five, or fix timesnbsp;refpeQ-Jvely the force of'One of them only, and fonbsp;forth; which evidently Ihews the generality of the

Propofition.

It is in confequence of this truth, that when a body A prevents another body B from falling to-'^'ards the centre of the earth, the former is preflednbsp;by the latter, and that preflure is proportionatenbsp;fo the quantity of matter in B. Now, thatnbsp;preflure is called the weight of the body B, andnbsp;fbe quantity of it is exprefled by comparing it withnbsp;^ Certain arbitrary ftandard weight, which maynbsp;be called an ounce, a pound, a grain, he. Sonbsp;that when a certain Body A is faid to v/eigh threenbsp;pounds, w'hilft another body B weighs one pound,nbsp;E 4nbsp;nbsp;nbsp;nbsp;the

the meaning is, that the quantity of matter in A, and of courfe its attraftion towards the earth, isnbsp;treble the quantity of matter in B, or the attraction of B towards the earth.

Since attraftion is a general property of matter, it may be alked, why do we not perceive any at-traftion between the bodies which ufually fur-Tound us, as for Inftance between two flints, ornbsp;two pieces of lead ? The anfwer is, that the attractive force of matter in general is too fmall tonbsp;become perceptible, excepting when the bodies,nbsp;or one of them. Is very large, as is the cafe betweennbsp;the earth and a flint, or other body ; for if younbsp;fuppofe that a flint ftone A be equal to thenbsp;lOGOOooooooooooth part of the whol� earth,nbsp;and likewife fuppofe that another body B is attracted by the earth with a force equivalent to onenbsp;pound; then it follows that the body B muft benbsp;attracted by the flint ftone A with a force equivalent to the losooooooooooooth part of a pound;nbsp;which is too fmall to produce any fenfible effeCt.nbsp;Yet, notwithftanding this, the accuracy and improvements oftheprefent age, have found means dfnbsp;rendering the attraction between bodies of no greatnbsp;fize, fufficiently fenfible; but the account of fuchnbsp;experiments will be found in another part of thisnbsp;work.

Confidering that the attraction is mutual between bodies, as between a ftone and the earth, it may be alked, why does not the earth move to-*nbsp;nbsp;nbsp;nbsp;wards

wards the ftone at the fanie time that theftonemoves towards the earth ? The airfwer is, that the earth,nbsp;agreeably to the theory, muft aftually move towards the ftone, but its motion is too ftnall to benbsp;perceived by our fenfes; for if vve fuppofe that the

times larger than the ftone, the attraction of the earth for the ftone, muft be to the attraction of thenbsp;latter for the former, as that immcnfe number isnbsp;to unity. Now fince the efteCts are always proportionate to their caufes, it follows, that if in a,nbsp;certain time the ftone moves through 1000 feet innbsp;its defeent towards the earth, the earth muft in thenbsp;fame time move towards the ftone throughnbsp;---------- parts of a foot �,

(which is the fame thing) through the iQoooooooooooooooooooth part of a foot; anbsp;RUantity vaftly too fmall for our perception.

Were the two bodies not fo difproportionate^ they would both be feen to move towards eachnbsp;other. Thus if two equal bodies, as A and B figs' Plate I. be placed at a certain diftance of eachnbsp;other, and be then left at liberty, viz. free fromnbsp;any obftruCtion, they will move towards eachnbsp;Pther, and will meet at a point C midway betwe^nbsp;^helr original fituations. But if the bodies benbsp;Unequal; for inftance A in fig. 4, Plate I. benbsp;three times as big as B, then they will meet at anbsp;point C, which is as much nearer the originalnbsp;fluation of A, than that of B, as the body A is

bigger than the body B ; viz. AC will be equal to one third part of BC j for fince the quantity ofnbsp;matter in A is equal to three times the quantity of matter in B, the attradtion of the formernbsp;muft be three times as great as the attraction ofnbsp;the latter, confequently the fpace run over by thenbsp;body B muft be three times as great as the fpacenbsp;run over by the body A, in the fame time.

It is evident that the like reafoning may be applied to bodies that bear any proportion tonbsp;each other ; hence we conclude that the diftancesnbsp;of the original fituations of the bodies fromnbsp;the point C, where if left at liberty they will meet innbsp;confequence of their mutual atiraSlion, are inverfely asnbsp;their quantities of matter; viz. as the quantity ofnbsp;matter in A, is to the quantity of matter in B, fonbsp;is the diftance BC, to the diftance AC.

The point C is called the centre of gravity of thofe two bodies; being in fa�l the point, ornbsp;centre, towards which they gravitate, and wherenbsp;they will adcually meet, if not difturbed by anynbsp;external force or impediment.

What h^s been obferyed with refpedt to the two bodies, may be eaftly applied to the mutual at-traftion of three, or four, or, in flrort, of any number of bodies; there being always a centre ofnbsp;gravity which is'common to them all. Such alfonbsp;is the cafe with a fingle body ; viz. there is a pointnbsp;in any fmgle body, which is its centre of gravity,^nbsp;towards which, if the body were divided into ditr

ferent parts, thofe parts would gravitate. The nature and properties of the centre of gravity will be farther noticed in the next chapter.

Since the attradlive power is proportionate to the quantity of matter, it follows, that all forts of bodies,nbsp;however different they may be in their weights, ifnbsp;they begin to move towards the earth from the famenbsp;height, at the fame time; they mnfl be equally accelerated -, that is, they nmfl all defend through the likenbsp;/pace in the fame portion of time', for though a body Anbsp;be twice as heavy as another body B, if you imaginenbsp;that the former is divided into two equal parts, eachnbsp;of thofe parts muft be equal to B, and of courfe itnbsp;luuft move through an equal fpace, as B, in the famenbsp;time. Now it is evident, that when the two partsnbsp;of A are joined together, the elFedt mufl; be thenbsp;f^me. The like reafoning may be extended tonbsp;bodies, whofe quantities of matter bear any othernbsp;proportion to each other. Hence all forts ofnbsp;bodies, when left at liberty, would fall from thenbsp;fame height to the ground precifely in the famenbsp;time, were they not unequally refifted by the airnbsp;through which they move. I fay unequally refifted, becaufe that refiftance is in proportion notnbsp;to the quantity of matter, but to the furface, yrhennbsp;the quantities of matter are equal. This may benbsp;firtisfodlorily proved by a variety of experiments.nbsp;Take, for inftance, a fmall quantity of cotton,nbsp;fipread it as much as you can, then let it fall fromnbsp;your Irand to the ground, and you will find that

the cotton will employ three, four, or more, leconds of time in that defcent. But if you take up thatnbsp;cotton and coraprefs it into a very fmall compafs,nbsp;you will find that on repeating the experiment, thenbsp;fame quantity of cotton will defcend to the groundnbsp;in lefs than a fecond. Thus alfo if you drop fromnbsp;the fame height at the fame time a guinea andnbsp;a common gold leaf, the guinea will come to thenbsp;ground incomparably quicker than the gold leaf.nbsp;But if you comprefs the leaf fo as to form it intonbsp;a fmall lump, and repeat the experiment with thisnbsp;lump and the guinea, they will be found to touchnbsp;the ground nearly at the fame moment

The converfe of the laft propofition is likewife evident; namely, that if bodies, in falling from thenbsp;fame height towards the centre of the earth, defcribenbsp;equal fpaces in the fame portion of time, the attraSlion

* This propofition is confirmed in a manner lefs eafy indeed, but more evident and conclufive, by means of anbsp;tall glafs receiver, having a mechanifin at its upper end,nbsp;from which a guinea and a feather, or other light body,nbsp;may be dropped at the fame time. When this glafs receiver is fet llraight up, and is exhaufted of air, in thenbsp;manner which will be defcribed hereafter, the above-men-�nbsp;tioned guinea and feather, will, on being difengaged, arrive at the bottom of the receiver at the fame moment pre-cifely. But if the receiver be not well exhaufted of air,nbsp;then the feather will arrive at the bottom later than thenbsp;guinea} and much more fo when the receiver is quite fullnbsp;of air.

invfl be proportionate to their quantities of matUf otherwife the fpaces, See. would not be equal.

Hitherto we have taken notice of the properties which naturally arife from the attraftion beingnbsp;proportionate to the quantity of matter. It is nownbsp;neceffary to examine the aftual motion of bodiesnbsp;which move towards a centre of attradion.

The great diiference between the fimple impulfe, mentioned in the preceding chapter, and a centripetal, or centrifugal, force, is that the formernbsp;produces equable motion; that is, fuch as compels bodies to deferibe equal fpaces in equal portions of time j whilft the latter produces unequablenbsp;lotion; viz. it compels bodies to deferibe une-lt;lual fpaces in equal portions of time.

This inequality arifes from the continual adion the latter power j for a centripetal, or centri-force, does not ad at firft only; but itnbsp;continually ad upon, and impel, the bodiesnbsp;motion; that is, the centripetal, towards thenbsp;'Centre of attradion, and the cehtrifugal, from thenbsp;centre of repulfion.

The attradion of the earth, or gravitating power, has been found, from a variety of fads,nbsp;Wnich will be mentioned hereafter, to decreafe innbsp;proportion as the fquares of the diftances from thenbsp;centre of the earth, increafe ; or, in other words,nbsp;force of gravity at diirerent heights is inverfelynbsp;the fquares of the diftances from the centre ofnbsp;earth. At a height, for inftance, as far from

the furface of the earth as the furface Is from ths centre, the force of gravity is a quarter of whatnbsp;it is at the furface for the diftances being as onenbsp;to two, their fquares are one and four; therefore,nbsp;as one is to four, fo is the force of gravity at thenbsp;above-mentioned height, to the force of gravity atnbsp;the furface.

This diminution of intenfity in the proportion of the fquares of the diftances from the centre ofnbsp;emanation, feems to take place not only with thenbsp;force of gravity, but likewife with all forts ofnbsp;cxnanations from a centre, fuch as light, found,nbsp;amp;c. as far however as we are able to judge fromnbsp;the prefent ftate of knowledge; for with the de-creafe either of found or of light, this law has notnbsp;been afcertained to any great degree of accuracy.

But, independently of aftual experiments, it may be ftriftly demonftrated, that emanations^nbsp;which proceed in Jiraight lines from a centre, and donbsp;not meet ivith any obJlruElion, muft decreafe in intenftynbsp;inverfely as the fquares of the diftances from thenbsp;centre, (i)

(1) Let A, fig. 5. Plate I. be the centre of emanation (for inftance the flame of a candle.) Let OPEt; be anbsp;fquare hole, and drawing Ifraight lines from A to thenbsp;corners of this fquare, produce them indefinitely towardsnbsp;I, H, E, r.

In the firft place It is evident that the light which pafl�s through the fquare hole OPBt;, will fill all the fpace between

Ci^ntripeial, and Centrifugal, Forces, i^c. 6g

Bodies that are left to fall from any height, will niove fafter and fafter the nearer they come to thenbsp;furface of the earth; for if t^he force of gravitynbsp;a�ted upon a body only at the commencement of

tween the four ftraight lines AH, AI, Ar, and AE. Se-r Condly, it is alfo evident that if a plane furface be placed atnbsp;B, parallel to the fquare OPB�;, all that part of it whichnbsp;lies between the aforefaid ftraight lines; viz. IHEr, willnbsp;illuminated by the light which paffes through OPBw;nbsp;l^ut as the plane IHEr is larger than OPBu, the light uponnbsp;It cannot be fo denfe as at OPBt/; and for the fame reafon,nbsp;a plane be fituated at D, parallel to OPBt/, the lightnbsp;tJpon it will be lefs denfe than at OPBv, but more denfe,nbsp;than at IHEr, amp;c. Thirdly, it is alfo evident that thenbsp;planes IHEr, KGD^, LFCr, are fquare figures, fincenbsp;the hole OPBv has been fuppofed to be a fquare. There-^�te, the only thing which remains to be proved; is, that ifnbsp;thediftanceAC be equal to twice the diftance AB, thenbsp;area of the fquare LFCat is four times as large as the areanbsp;OPB^y; that if AD be equal to three times AB, the areanbsp;KGDj is nine times as large as OPBu ; or, in fhort, thatnbsp;the areas OB, LC, KD, amp;c. are as the fquares of thenbsp;diftances from A, which is eafily done ; for ABP, ACF,nbsp;Being equiangular triangles (Eucl. p. 2g. B. I.) we havenbsp;(Eucl. p. 4.B. VI.) AB: AC; : PB: FC; but PB and

OPBt;, LFCa-; and (Eucl. p. 20. B. VI.) thofe figures ^re as the fquares, or in the duplicate proportion, of theirnbsp;homologous fides; therefore OPB'w; LFCa :; PBp: t^Cl . �

its defcent, the body would, (according to the ' laws of limple motion, Chap. IV.) continue to de-fcribe equal fpaces in equal portions of time. Butnbsp;the very next moment the force of gravity impelsnbsp;the body again, in confequence of which thenbsp;body�s velocity muft be doubled; fince the fecond,nbsp;impulfeis equal to the firft, and the firft remainsnbsp;unaltered. For the fame reafon on the third moment the body�s velocity will be trebled, ar d fo on.nbsp;Or, fpeaking more properly, the velocity will in-creafe as the .time increafes, viz. the velocity willnbsp;be as the time; the meaning of which is, that thenbsp;velocity at the end of two feconds is to the velocity¹nbsp;at the end of three feconds, as two to three; ornbsp;the velocity at the end of one minute is to thenbsp;velocity at the end of one hour, as one is tonbsp;lixty, amp;c. ¹.

The^paces defcribed by fuch defending bodies cannot be proportionate limply to the times ofnbsp;defcent j for that wovrld be the cafe if the velocitynbsp;remained unaltered; but, the velocity increaling

The velocities are as the times when the gravitating power/ remains unaltered, or with the fame gravitatingnbsp;power ; but if two diftindt gravitating powers be compared together, then the velocities will be as the produdtsnbsp;of the times multiplied by the gravitating forces refpec-tively ; it being evident that a double force will produce anbsp;double effedt, a treble force will produce a treble elFedfjnbsp;amp;c, flence when the times are equal, or in the fame time,nbsp;the velocities are as the gravitating, or the impelling,nbsp;forces.

'Continually, it is evident that the /paces tnuji as ike times tnultiplied by the velocities; for anbsp;double velocity will force the body to movenbsp;through a double fpace in an equal portion ofnbsp;and through a quadruple fpace in twice thatnbsp;time; alfo a quadruple velocity will force thenbsp;body to move through a quadruple fpace in annbsp;^qual portion of time, and through eight timesnbsp;that fpace in twice that time ; and fo on in anynbsp;proportion. But it has been flrewn above that thenbsp;Velocities are as the times; therefore to fay thatnbsp;the fpaces are as the times multiplied by the velo-^�ties, is the fame thing as to fay that the fpacesnbsp;^re as the times multiplied by the times, or as thenbsp;hduares of the times j and for the fame reafon it isnbsp;the fame thing as to fay that the fpaces are as thenbsp;vsiocitles multiplied by the velocities, or as thenbsp;^riares of the velocities

This property of defcending bodies, (viz. that they run through fpaces which are as the fquaresnbsp;the times) has been ufually demonftrated in anbsp;nerent way, by the pbilofophical writers. Theirnbsp;demonftration may, perhaps, appear more fatisfac-trgt;ry than that of the preceding paragraphs to fomenbsp;^f my readers; I lhall therefore fubjoin it, efpe-cially as it proves at the fime time another law re-^dve to the velocity of defcending bodies.

equal times the fpaces are as the itn-t)r gravitating, forces. See the laft note,

Let AB, fig. 6. Plate I. reprefent the time, during which a body is defcending, and let BC reprefent the velocity acquired at the end of that time. Complete the triangle ABC, and thenbsp;parallelogram ABCD, Alfo fuppofe the time to benbsp;divided into innumerable particles, et, im, mp, po^nbsp;amp;c. and draw ef, ik-, mn, amp;c. all parallel to thenbsp;bafe BC. Then, fince the velocity of the defcending body has been gradually increafing from thenbsp;commencement of the motion, and BC reprefentsnbsp;the-ultimate velocity; therefore the parallel linesnbsp;ef, ik, mn. See. will reprefent the velocities at thenbsp;ends of the refpeftive times Ae, At, Am, amp;c.nbsp;Moreover, fince the velocity during an indefinitelynbsp;fmall particle of time, may be confidered as uniform ; therefore the right line ef will be as the velocity of the body in the indefinitely fmall particlenbsp;of time ei; ik will be as the velocity in the particlenbsp;of time im, and fo forth- Now the fpace palTednbsp;over in any time with any velocity is as the velocity multiplied by the time; viz. as the redtangknbsp;under that time and velocity; hence the fpacenbsp;pafled over in the time ei with the velocity ef,nbsp;will be as the reftangle if-, the fpace pafled over innbsp;the time im with the velocity ik, will be as thenbsp;re�tangle mk; the fpace pafled over in the timenbsp;mp with the velocity mn, will be as the reftangknbsp;pn, and fo on. Therefore the fpace palTed overnbsp;in the fum of all thofc times, will be as the fumnbsp;of all thofe redangles. But fince the particles of

time

time are infinitely fmall, the fum of all the rectangles will be equal to the triangle ABC. Now fince the fpace patTed over by a moving body innbsp;the time AB with a uniform velocity BC, is as thenbsp;feftangle ABCD, (viz. as the time multiplied bynbsp;the velocity) and this redtangle is equal to twicenbsp;the triangle ABC (Eucl. p. 31. B. I.) therefore thenbsp;^Pace paffed over in a given time by a bodynbsp;falling from reft, is equal to half the fpace paffed.nbsp;over in the fame time with an uniform velocity,nbsp;^qual to that which is acquired by the defcendingnbsp;tgt;ody at the end of its fall.

Since the fpace run over by a falling body in time reprefented by AB, fig. 7. Plate I. withnbsp;tfie Velocity BC is as the triangular ABC, and thenbsp;fpace run over in any other time AD, and velocitynbsp;is reprefented by the triangle ADE; thofenbsp;fpaces muft be as the fquares of the times AB,

� for the limllar triangles ABC, and ADE, are fftuares of their homologous fides, viz. ABCnbsp;to ADE as the fquare of AB is to the fquare ofnbsp;(Eucl. p. B. VI.)

In fig. the 8th. Plate I. the fpaces, which are cnbed by defcending bodies in fucceffive equalnbsp;portions of time, are reprefented, for the purpofe ofnbsp;'?�^preffing w�ith greater efficacy on the mind of thenbsp;^ader, the principal law of gravitation. The Hirenbsp;B reprefents the path of a body, which is let fallnbsp;on A, and defcends towards the ground at B.nbsp;^ divifions on the line AB denote the places ofnbsp;r znbsp;nbsp;nbsp;nbsp;the

the body at the end of one fecond, two feconds, amp;c. which equal portions of time are marked on thenbsp;left hand fide 5 whilft the numbers on the right ex-prefs the feet percurredi or real diftances from A tonbsp;the firft divifion, from A to the fecond divifion,nbsp;and fo oni It appears, therefore, that in one fecondnbsp;the body has defcended through 16,087 feet, thatnbsp;in two feconds it has defcended through four timesnbsp;16,087, or 64,348 feet, amp;c.

It may alfo be obferved, that the fpaces run thtough during each fingle fecond, are as the oddnbsp;numbers i, 3, 5, 7, amp;c.; that is, if the fpace per-curred in the firft fecond be called one, the fpacenbsp;percurred during the fecond fecond only will benbsp;three times as great, the fpace percurred in thenbsp;third fecond w'ill be five times as great, and fo on.nbsp;In fa6t, if we fubtradt 16,087 from 64,348, thenbsp;remainder, 48,261, is equal to three times 16,087 �nbsp;ifwelubtraft 64,348 from 144,783, the remainder,nbsp;80,435, equal to five times 16,087,

It has been Ihewn above that the force of gravity at equal diftances from the centre of the earth is proportionate to the quantity of matter; but it muftnbsp;be obferved, that when the diftances are unequal,nbsp;then the gravitating forces, or weights, of bodies,nbsp;are as the quotients of the quantities of matter dirnbsp;vided by the fquares of the diftances refpedtively,nbsp;or, which is the fame thing, the weights of bodiesnbsp;are faid to be as the refpedlive quantities of matternbsp;diredly, and the fquares of the refpedtive diftances

Ceniripeta!, and Cenirifugal, Forces, nbsp;nbsp;nbsp;69

inverfely ; fince the gravitating force has been {hewn ^ecreafe inverfely as the fquares of the diftancesnbsp;from the centre of attraftion. Thus if a body A,nbsp;'^hich is hve times as big as another body B, isnbsp;^diated at the diftance of 4000 miles from thenbsp;'^ontre of the earth, whilft B-is fituated at 6000nbsp;^iies diftance, then the weight of A will be tonbsp;the Weight of B as the quotient of five divided bynbsp;^^0 fquare of 4000, is to the quotient of one di-

* Suppofe it be required to find how much a leaden ball, on the furface of the earth weighs twenty pounds.nbsp;Weigh at the top of a mountain which is three milesnbsp;high,

The femidiameter of the earth is known to be about 3985 *^des, to which we add the height of the mountain, viz.

and we have the two diftances j that is from centre of the earth to the furface, 3985 miles, and fromnbsp;^ centre to the top of the mountain 3988 miles. Thenbsp;^^ares of thofe numbers are 15880225 and 15904144.

cn fay as 15^04144 is to 15880225, fo is twenty pounds to a fourth proportional, which by the commonnbsp;c of three (viz. by multiplying 158S0225 by 20, andnbsp;^ iding the produift by 15904144) will be found to benbsp;�9^9, or jg pounds and 15I ounces, which is thenbsp;leaden ball at the top of the mountain, viz.nbsp;y I'alf an ounce lefs than on the furface of the earth.

�s balanced by a counterpeife of twenty pounds in y 3nbsp;nbsp;nbsp;nbsp;a pair

In the preceding explanations and examples, the fpaces and velocities of defcending bodies havenbsp;been calculated on the fuppofition that the force actsnbsp;uniformly; viz. that during the defcent of the bodynbsp;from A, fig. 8. Plate I. towards the ground, the attraction of the earth does not increafe; which fuppofition, ftridtly fpeaking, is not true; for it has alreadynbsp;been (hewn, that the force of gravity decreafes in-verfely as the fquares of the diftances from the centrenbsp;of the earth; fo that the nearer the body comes tonbsp;the ground, the ftronger its gravitation will be.nbsp;However, in ihort diftances from the furface of thenbsp;earth, that increafe of gravity is fo very trifling, thatnbsp;for common purpofes it may be fafely neglecfted.nbsp;But as the fame theory is applicable to all forts ofnbsp;gravitating powers, and as very great diftances maynbsp;fometimes enter the calculation, it W'ill be proper tonbsp;fubjoin the method of calculating the velocitiesnbsp;whiich are acquired by bodies defcending towards a

a pair of fcales on the furface of the earth, will appear lighter at the top of the mountain ; for this will not be thenbsp;cafe, becaufe the counterpoife itfelf will lofe an equalnbsp;portion of its weight by being fituated on the top of thenbsp;mountain ; and of courfe the equilibrium of the fcales willnbsp;not be difturbed. But if the leaden ball in queflion benbsp;weighed in one of thofe weighing inftruments which arenbsp;made with a fpiral fteel fpring, then indeed the decreafe ofnbsp;its weight at the top of the mountain will be clearly perceived, provided the weighing infirument be fufficientlynbsp;accurate.

Ceniripelal, and Centrifugal, Forces, tele. 71

centre of attradion, when the incrcafe of the attractive power is taken into the account (2).

It is evident that, fince the continual adion of the force of gravity accelerates the motion of anbsp;lt;iefcending body, it mufl: continually retard the

Let S. fig. 9. Plate I. be the centre of the earth, B any point in its furface, and let the force of gravity in allnbsp;places be reciprocally as the fquares of their diftances fromnbsp;*1^0 centre of the earth ; it is required to determine the vek~nbsp;^ity of a htauy body at the furface of the earth, which it ac-i'ctres in falling from any given altitude AB.

Let ^ be any indeterminate diftance from the centre of ^l^e earth, and let v be the velocity of the falling body atnbsp;that difiance. Let reprefent the force of gravity at

sbi

Ljand confequentlyits force at the diftance x, FIrft th � �nbsp;nbsp;nbsp;nbsp;^

^n It is plain, that after the falling body is arrived at the ^iftance x, and then defeends further through any infinitelynbsp;'^dl fpace as x, the time of that infinitely fmall defeentnbsp;'''^ill be as �; that is, it will be as the fpace diredly, andnbsp;the velocity inverfely; and the infinitely fmall acqui-*^�on rrtade by the velocity in that defeent will be as the

'yz nbsp;nbsp;nbsp;Of the Motton arijing from

motion of an afcending body. A ball, for inftance, which is projedled upwards, will be gradually retarded by the gravitating force, which adls in anbsp;contrary diredlion,

The foregoing explanations relatively to the laws of gravity, or of a centripetal force, may be ealilynbsp;applied to the explanation of the properties of z.nbsp;repulfive, or centrifugal, forte; for in faft the fame

This being difcovered, let DB be the height from which a body will fall to the lurface of the earth in onenbsp;fecond of time; and fince during fo fmall a defcent, thenbsp;force of gravity may be looked upon as uniform, it is evident that a body falling from D to B will acquire a velocitynbsp;which will carry it uniformly through the fpace zBD innbsp;a fecond of time. Let 2BD reprefent this velocity ; thennbsp;muft every other velocity be reprefented by the fpacenbsp;through which it will carry a body in a fecond of time.nbsp;Now to find the velocity acquired in falling from A to B,

. Centnpeta�i ond Centrifugal, Forces, 73

feafoning will do for the one as for the other, changing only the word attra^ion for repuljion, andnbsp;the word acceleration for retardation. Thus thenbsp;�'Velocity of a body which is receding from a centrenbsp;^f repulfion, is retarded in proportion as the timenbsp;tncreafes ; or the velocities are laid to be inverfelynbsp;the times, Alfo the fpaces decreafe, or are, in-t�erfely, as the fquares of the times.

If a body be thrown perpendicularly ufnvards, that is, in a diredion from the centre of th� earth,nbsp;��''ith the velocity which it acquired by falling in anbsp;E'Ven timcj it will arrive at the fame height from

DB gnd DS having been afcertained by of experiments, 2 m is thereby found to be aboutnbsp;Engiifh miles.

�mes equal to one, hence it goes out of the queftion; nd therefore a body falling from an infinite height, willnbsp;Acquire at the furface of the earth but a finite velocity;nbsp;'2. fuch a Velocity as will carry it uniformly throughnbsp;''2n miles in one fecond of time.

C-oroll. 2. Therefore, # converfo, if a body be thrown �^PWards with fuch a velocity, it will never return, but itnbsp;'*''ill afeend for ever.

3� After the fame manner we may determine velocity of a falling body, whatever be the law ofnbsp;erayity.

which it fell, in the fame time^ and will lofe all its momentum. And when bodies are thrown perpendicularly upwards, the heights of their afcentsnbsp;are as the fquares of their velocities, or as the fquaresnbsp;of the times of their afcehding.

N. B. Throughout this chapter no notice has been taken of the 'refiftance, which the air offers tonbsp;the motion of bodies.

tEe method of ascertaining the situation OF THE CENTRE OF GRAVITY, AND AN ENUMERATION OF ITS PRINCIPAL PROPERTIES.

The definition and the nature of the centre of gravity having been (hewn in the precedingnbsp;pages, we fliall in the prefent chapter fhew the'nbsp;method of finding its fituation in a fyftem of bodies,nbsp;as well as in a fingle body or figure; after whichnbsp;we fliall ftate its various properties, the knowledgenbsp;of which is of the utmofl; importance in the ftudynbsp;of natural philofophy, and efpecially in mechanics.

When the common centre of gravity of two bodies is to be determined, their quantities of matter and diftance from each other being known;

draw a flraight line from the centre of gravity of one of the bodies, as A, fig. 10. Plate I. to thenbsp;centre of gravity of the other, B. (which centres wenbsp;dull, for the prefent, fuppofe to be known; for itnbsp;��'^nll prefently be fhewn how to find the centre ofnbsp;gravity of a fmgle body.) Then divide this linenbsp;in E, fo that its parts BE, AE may be to each othernbsp;in the proportion of A to B, and E is the centrenbsp;iought. For example, let A weigh 3 pounds,

^ 2 pounds, and let the diftance AB be 20 feet. Say as 3 is to 2, fo is BE to AE 5 then bynbsp;compofition (Eucl. p. 18. B. V,) fay as 3 plus 2,nbsp;^'iz. 5, is to 2 fo is BE plus AE, viz. 20 feet, to anbsp;fourth proportional, which, byihe common rule ofnbsp;^iiree, is found to be 8, and is equal to AE s fo thatnbsp;centre of gravity E is 8 feet diftaiit from A.

''Then the centre of gravity of three bodies, as and D, fig. 11. Plate I. is to be determined,nbsp;^i^cir quantities of matter and diftances beingnbsp;known; you muft in the firft place find the centrenbsp;nf gravity E between any two of thofe bodies, asnbsp;of A and B, after the manner mentioned above,nbsp;'k'hen imagine that the two bodies A and B arcnbsp;f�oth colledted in the point E, and laftly find thenbsp;Centre of gravity between E and D, which will benbsp;the common centre of gravity of the three bodies;

draw the ftraight line DE, then as the fum of ^he matter in A and B, is to the matter in D, fo isnbsp;to CE; and, by compofition, fay as the fum ofnbsp;�he matter in A, B and D, is to D, fo is DE to CE;

wliicb gives C, the common centre of gravity of the three bodies.nbsp;nbsp;nbsp;nbsp;'

In the fame manner the centre of gravity of four, or more, bodies may be determined; viz. by conceiving the matter of three of thofe bodies to benbsp;collefted in the common centre of gravity of thofenbsp;three bodies, and then finding the common centrenbsp;of gravity of the iafl mentioned centre and thenbsp;fourth body^ kc.

The centre of gravity of a fingle body may be eafdy determined by the following general method,nbsp;viz. by fuppofmg the body to be divided into twonbsp;or more parts, and then finding the common centrenbsp;of gravity of thofe parts, which will be the centrenbsp;of gravity of the body itfelf. But in certain regularnbsp;figures, luch as a circular furface, a fphere, a cube,nbsp;kc. it is evident that the centre of the figure muftnbsp;coincide with the centre of gravity; for if in thenbsp;circle, for intlance, tig. 12. Plate I. you divide thenbsp;area into two equal parts A�B and AEB, it isnbsp;evident that the common centre of gravity of thofenbsp;two equal parts muft be fomewhere in the line AB;nbsp;and if, by cutting the circle in any other diredlionnbsp;ED, you divide the area into two other equal partsnbsp;EAD, and DBE, it is evident that the commonnbsp;centre of gravity of thofe two parts muft be fome-wbere in the diameter ED; therefore the centre ofnbsp;gravity of the circle muft be in the interfcdtion ofnbsp;the two diameters AB, ED i viz. at C, which is thQnbsp;centre of the circle.

� nbsp;nbsp;nbsp;In

In a right lined plane trianglcj as ABD, fig. 13. Plate I. the centre of gravity may be eafily foundnbsp;by dividing any two of its fidcsj as AD and BD,nbsp;each into two equal parts at F and E, and bynbsp;drawing ftraight lines from thofe points of divifionnbsp;to the oppofite angles ; the interlefliori C of thofenbsp;two lines being the centre of gravity of the triangle;nbsp;for fince the line AE divides the triangle into twonbsp;equal parts, (Eucl. p. i. B. VI.) the commonnbsp;centre of gravity of thofe two parts muft be fome-'vhere in the line AE, and for the fame reafon thenbsp;Common centre of gravity of the two parts ABF,nbsp;snd FED muft be fomewhere in the line BF; therefore the centre of gravity of the whble triangle muftnbsp;at C, the interfeftion of the two lines AE, BF.

If the figure be terminated by more than three ff�^aight linef, as ABODE, fig. 14. Platei, its centrenbsp;gravity m.ay be found by dividing it into anynbsp;convenient number of triangles, as ABC, BCF,nbsp;f^FE, then by finding the centre of gravity ofnbsp;each triangle, and laftly by finding the commonnbsp;Centre of gravity of all the triangles, which is to'nbsp;f�e done in the fame manner as the centre of gra-'^ity between three or more bodies was determinednbsp;above. In a fimilar manner the centre of gravitynbsp;^f irregular folids may frequently be found.

There ate how-ever feveral figures in which the � centre of gravity cannot be eafily found by thenbsp;^bove deferibed methods, at leaft not with greatnbsp;accuracy. But a more general, and accurate

yS nbsp;nbsp;nbsp;The Method of nfcertaintng the

method of finding the centre of gravity, is derived from the dodrine of fluxions, which will be foundnbsp;explained in the note (i).

{I) Imagine that at D,E, F, G, H, fig. 15. of Plate I. there are fo many weights affixed to the inflexible linenbsp;AB j and let C be the centre of gravity of the faid linenbsp;and weights together; fo that when the point C refts uponnbsp;a fulcrum, neither end will preponderate, and of courfe thenbsp;whole loaded line will remain perfectly balanced.

It has been Ihewn in the preceding pages that the mo--mentum or force of any weight, as H, (vix, the body fuf-pended at H) to raife the oppofite end A of the line, or lever, is expreffed'by the produft of its quantity of matternbsp;multiplied by its diftance from the fulcrum, or centre ofnbsp;gravity C; viz. by H X HC (for by the letters D, E, F,nbsp;amp;c. we exprefs the weights of the refpedive bodies).nbsp;Therefore, fince the line with all the weights is perfectlynbsp;balanced, it follows that the fum of the momenta of all thenbsp;weights which lie on one fide of C, muft be equal to thenbsp;fum of all the momenta, which lie on the other fide of C ;

viz. H x HC G X GC F xFC =: D xDC E X EC ; that is H X BC�BH G x bC � Bcj F x BC�BFnbsp;E X �B�BC D X DB � BC; or HxBC � H x BH nbsp;GxBC �GxBG FxBC �FxBF=ExEB�EXnbsp;BC fD X DB�D X BC. Then by tranfpofitionnbsp;we have H X BC G X BC F X BC � x BC D Xnbsp;BC = H xBH G xBG FxBF ExEB DXnbsp;DB; which equation by divifion becomes EC =-HxBH GxBG FxBF ExEB DxDB

h g f e d nbsp;nbsp;nbsp;�

Centre of Gravity, i^c, nbsp;nbsp;nbsp;79

!� If two bodies be conneBed together by means of

extremity B, is equal to the quotient of the fum of the Pfodufts of all the weights multiplied each by its diftancenbsp;hoiti B, divided by the fum of all the weights.

Hence we derive the following general rule for finding Centre of gravity in a fyftem of bodies; viz. AJfume anbsp;at one extremity of the fyftem ; multiply the weight ofnbsp;body by its diftance from that point; divide the fum ofnbsp;*he produets by the fum of the weights, and the quotient will

Take notice that if the above-mentioned point be aflumed at the extremity of the fyftem, but any where betweennbsp;bodies, as between D and H, fig. 15. plate I; then thenbsp;by their refpeflive diftanccs, of the bodies on onenbsp;' ^ of the alTumed point, muft be confidered as negative,nbsp;^'�bilft the other are confidered as pofitive ; and they nvuftnbsp;� added together agreeably to the common algebraicalnbsp;for adding pofitive and negative quantities together,nbsp;'0 refult then, according as it turns out pofitive or nega-gt; Will fhew the diftance of the centre of gravity fromnbsp;0 a(run:jgj pointy either on the pofitive or on the negativenbsp;� of that point.

*'^^3nce, for finding the centre of gravity of a triangle, benbsp;nbsp;nbsp;nbsp;fphere, amp;c. by imagining the faid figure to

infinite number of parts; for by the of fluxions, the fum of the momenta, as alfo of thenbsp;b tts of all thofe pari*, may be eafily afeertained.

So 7he Method cf. ajcertaining the

afi inflexible line or rod, as A and B, fig^ 2 2. Plate �/ and the line or rod be fupported by a prop, or (as It is

commonly'

Thus let the figure be a plane, as ADC, fig. i6. Plate I. whofe axis is A B, and whofe parts are fuppofednbsp;to be endued with gravity. Imagine this figure to be re-folved into an infinite number of weights F G, fg, Sianbsp;all perpendicular, or all alike inclined, to the axis A Bjnbsp;and let x reprefent the diftance A E of the little weightnbsp;F G from Ai Then the breadth of one of thofe weightsnbsp;is denoted by i (the fluxion of the axis A E) ; thereforenbsp;one of thofe infinitely frhall weights is expreffed by FG X � ;nbsp;the fluent of which, when x becomes equal to the wholenbsp;axis A B, is the fum of all the weights. Farther, if onenbsp;of thofe weights be multiplied by its diftance from A, thenbsp;product, FG XxXX, will exprefs its momentum ; and thenbsp;fluent of this expreflion, when x becomes equal to thenbsp;whole axis A B, is the fum of all the momenta. Therefore, agreeably to the general rule, if the fluent of FG Xnbsp;be divided by the fluent of FGxx-, the quotient will exprefs the diftance of the centre of gravity from A on thenbsp;axis A B.

If the figure be a folid, imagine it to be divided into art infinite number of fe�iions, or fmall weights, all perpendicular, or all alike inclined, to the axis ; put the expreffioAnbsp;which denotes one of thofe fe�lions, inftead of F G in th*nbsp;above fluxional expreflions ; then proceed as above d�-re�bed. Or, which is the fame thing, call one of thgt;gt;ftnbsp;feffions S ; then divide the fluent of 3 .v Aquot;, by the fluentnbsp;of S X, and the .quotient will exprefs the diftance of th�nbsp;centre of gravity.

The only praftical difficulty confifts in finding the vsD� of F G i or of the ?,bcve mentioiie^fedion S in a foli'd figr't��

corhtnonly called) a fulcrum, placed under the centre ^f gravity C �, the bodice zvill remain motionlefs.

In �which Value muft, by means of the equation of the figure,nbsp;he expreffed in terms wherein x is the only variable letter.

Example I. To find the centre of gravity of a firalglit bne, or very flender cylinder, AB, fig, 17. plate I, whofepartsnbsp;^nay h fuppofed to be endued with gravity.

Let its length AB be called �, and fuppofing the line to divided into an infinite number of little parts or weights.nbsp;Line of thofe parts is denoted fimply by a- ; (for the breadthnbsp;is nothing) and the fluent of x is x, Alfo the mo-^'^ntum of on,e of thofe particles is exprefled by xx, whofenbsp;is fj;*. Therefore, dividing the latter fluent by the

fhevvs that the centre of gravity is at C in the middle of ^heline; viz. its diflance from cither extremity is equalnbsp;*0 half tbs length of the line,.

Example II. find the centre of gravity in a triangle fig,nbsp;nbsp;nbsp;nbsp;p zvhere the axis AD~a, hafe BC~b-,

� rom the fimilarity of the triangles we have AD : BC :: LI.EF; \lz,a:l/: ; a:�~y. Therefore the infinitely fmall

Sz nbsp;nbsp;nbsp;The Method of afcertaining the

In this fituation no one of the bodies will have more tendency towards the earth than the other ;

for

which is the diftance of the centre of gravity from the vertex A; viz, f of the axis AD.

It needs hardly be mentioned that the centre of gravity mufl: neceflarily be in the axis.

Example III. To find the centre of gravity in a parabolic figure ABC, fig. 19, Plate I.

Put the axis AD~a-, abfeifs AF=x; and ordinate EF �y. By conics we know that the fquare of the ordinate EF is equal to the produft of the parameter multiplied

1 j, nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;,nbsp;nbsp;nbsp;nbsp;,nbsp;nbsp;nbsp;nbsp;1 i

Farther 2^^ A-^ xi; ot ip��-xic is the momentum of the little weight; whofe fluent, which reprefents the fum of

or (when x is equal to the whole axis AD) zz ^ a ; fo that the centre of gravity is diftant from the vertex A, r of thenbsp;wliole axis AD.

Centre of Gravity., ^c. nbsp;nbsp;nbsp;S3

for fince C is their common centre of gravit}'�, the foftances AC and CB, are inverfely as the weights

Put the axis or altitude AD �ff, diameter of the bafe ; abfcifs AO �r and ordinate EO=:y.

7, nbsp;nbsp;nbsp;lot

putting c for the circumference of a circle whofe dia-is unity, the circumference of a circle whofe diame-*sr is nbsp;nbsp;nbsp;-^yhofe area is icy X fy; viz.

rjji is the fum of all thofe weights. Alfo^^xx is

momentum of the little weight, and its fluent, , is fum of all the momenta. Then dividing the latternbsp;^uent by the former we havenbsp;nbsp;nbsp;nbsp;7Z^ =��*'; or (when

^ becomes equal to the whole axis AD) |lt;i; which fhews ^hat the diftance of the centre of gravity from A on thenbsp;uitis AD is equal to | of the whole axis AD.

Example V. To find the centre of gravity of an hemif-Phere ABO,fig. 2*. Plate I.

Put the axis or radius AD�3; DP�x; and MP, � parallel to the bafe, Then PMD being a

34- nbsp;nbsp;nbsp;�^he Method of ajcertahihig the

of tliofe bodies; viz. AC is to C B as the body' B is to the body A. Now fliould the rod benbsp;moved from its fituation AB into the fituationnbsp;F E,. the body B would defcribe the arch B E, andnbsp;the body A would defcribe the arch AF, whichnbsp;arches reprefent the velocities of thofe bodies ; fornbsp;they are the fpaces through which they m�ve ia

circle whofe diameter is unity, the circumference of a circle whofe diameter is ME, or will be icy, and itsnbsp;area will be icy X ~y ; viz. ry% or (by fubftituting for yy itsnbsp;value as found above, viz. �xquot;quot;) ca^�cjr*; and this is anbsp;fedtion of the hemifphere parallel to the bafe. Then

ca^�CA-* X A is one of the infinitely fmall weights into which the hemifphere is fuppofed to be divided; and its

__ .3

fhe faiTie time. But it has been denionflrated by the geometricians, that thole arches bear the famenbsp;l^oportion to each other as the radii or diftancesnbsp;^ C A ; viz. B E is to A F as B C is to A C jnbsp;it has been fliewn above, that BC is to ACnbsp;^ the body A is to the body B j therefore it fobnbsp;^hws, that the arch BE is to the arch AF as thenbsp;body A is to the body B. But of four proportionalnbsp;'1'^a-ntities, the produdl of the extremes is equal tonbsp;'�he produft of the means ; therefore the produdtnbsp;the body B, multiplied by the arch B E (whichnbsp;^�^�iftitutes the momentum of B) is equal to thenbsp;P^odudt of the body A multiplied by the arch A Fnbsp;(quot;'hich conftitutes the momentum of A) ; fo thatnbsp;'�heir momentums being equal, thofe bodies will

^�^nce each other, and of courfe they will remain reft. jj. is evident that the fame reafoning isnbsp;^PPhcable to the common centre of gravity of anynbsp;^rimber of bodies, as alfo to the centre of gravity ofnbsp;^ hngle body; viz. that if a fyftem of bodies,nbsp;'�hat are connefted together, or a fingle body, benbsp;Placed -with the centre of gravity on a fulcrum,nbsp;fyftem, or fmgle body, will remain perfedlynbsp;a.anced thereon, and as fteady as if it were placednbsp;^Pon a fiat horizontal furface.

� t he Jlate^ zvhether of reft or motion, of the ^mon centre of gravity of two bodies, zvill not benbsp;otJ^^�*nbsp;nbsp;nbsp;nbsp;rnutual aSlion of thofe bodies upon each

gravity is at reft; and that the two bodies, in con-fequence of their mutual attraftion, approach each other in a certain time; it follows from the foregoing theory, that the fpaces through which theynbsp;move muft be inverfely as their weights, and ofnbsp;courfe, the remaining fpaces between their laftnbsp;fituations and the centre of gravity, will remain innbsp;the fame proportion to each other as the originalnbsp;cliftances ; therefore the centre of gravity will notnbsp;be moved from its original place.�An examplenbsp;will render this explanation more evident.

Let the body A, fig. 23, Plate I. weigh 2 pounds, and B, 6 pounds. The diftances of thofenbsp;bodies from the centre of gravity C, are inverfelynbsp;as the weights of thofe bodies; viz. BC, is 10 feet,nbsp;and AC, 30 feet (that is, as one to three) becaufenbsp;the w'eights are as three to one. Now fuppofe thatnbsp;in confequence of their mutual attraftion, thofenbsp;bodies begin to move towards each other; and if innbsp;one minute B comes to the place D, having paflednbsp;over the diftance B D,, equal to one foot; the bodynbsp;A muft in the fame time have pafled over 3 feet,nbsp;and muft have arrived at E ; then taking away BPnbsp;from BC; viz. one foot from 10 feet, there remains D C, equal to 9 feet; alfo taking away A Enbsp;from AC; viz. 3 feet from 30, thye remains Cnbsp;equal to 27 feet. Now thofe two remaining dif'nbsp;tances; viz. 9, and 27, are the one to the othernbsp;as one to three ; therefore, amp;c.

In the lecond place, if the two bodies, togethegt;^

SS nbsp;nbsp;nbsp;^he Method of ajceriaining the

V. If tzvo bodies be carried tozvards contrary partSy

gravity, and vvhilft A moves as far as let B move as far as b, and C as far as c. Since the fpaces ka, andnbsp;Cf, exprefs the refpedive velocities of the two bodies, andnbsp;of their centre of gravity, all we have to prove is, that thenbsp;fum of the produdfs of A multiplied by A lt;?, and of B multiplied by B by is equal to the produdf of A plus B, multiplied by Cf; viz. that A X A a B x Bi=: A B X Cf.

Since C is the centre of gravity, A is to B as B C is to A C, and zs b c is to a c. Then alternately B C : be �nbsp;AC : a Cy and converfely BC:BC � ^lt;r::AC:ACnbsp;� a c. But B C � ^ r is equal to C r � B 5 ; and A C �*nbsp;rt r is equal to A �-*- C r ; therefore B G : A C : : C e ��nbsp;Bb : A a �� C c. But it has been fhewn above, that A inbsp;B ; : B C : A C ; therefore A:B;:Cr � B�:A�--'nbsp;C c. Now, fince of four proportional quantities, the pro-dudf of the extremes is equal to the produdf of the means�nbsp;we have AxAa-� A X Cc = BxCf � BxB�, andnbsp;by tranfpofition AxAa BxB�arA -J-B x Cc.

When the bodies do not move in the fame ftraight line� the demonftration is the fame j excepting only that in tbi*nbsp;cafe the velocity is to be reckoned not upon the path whichnbsp;is aftually deferibed by the bodies, but upon the path ofnbsp;their common centre of gravity. Suppofe, for inftancc�nbsp;that the bodies A and B, fig. 2. Plate II. move towardsnbsp;D, whilil their common centre of gravity C moves in th^nbsp;line CD ; alfo that in a certain time thofe bodies hav�nbsp;moved as far as the places a and b refpedlively, at the faif�nbsp;time that their coRituon centre of gravity has moved fron^nbsp;C to e,'

Front�

paris {which is eq^uivalenl lo the Jum of their momen-tiims townrdsjhe fame part') will be equal to the momcn^ ii�-m that would arije if both the bodies were carried to-quot;^vards the fame part with the velocity of their commonnbsp;centre of gravity {^f).

From A, B, a, and-^, drop B H, A F, ag and bK, perpendicular to H D, the diredlion of the centre ofnbsp;Suavity. Then Fg will reprefent the velocity of A, andnbsp;Hk, the velocity of B ; for thofe are the real advancesnbsp;bodies have made towards D. Notv from the propertynbsp;the centre of gravity we have A:B ::BC;AC;:nbsp;(ftt'.ce the right angled triangles ACF, BCH, are equi-^ttgular,'and coiifequcndv fi-iiilar, by Eucl. p. 15? B. I. andnbsp;P* 4* B. VI.) H C : C F ; : Kf :nbsp;nbsp;nbsp;nbsp;Then the demon-

(3) Suppofe that the body A, fig. 3. Plate II. moves Pfomnbsp;nbsp;nbsp;nbsp;fame time that the body B moves in a

A B X Cr.

^ From the nature of the centre of gravity, we have A :

� � B C ; A C : : b c : ac hence alternately BC : be \ C ; f, and converfely BC:BC � ^lt;:::AC:

'^�^ntrary diredlion from B to i, whilll their common centre gravity moves from C to f. Then their refpeaive ve-^rgt;cities are reprefented by A tt, B b, and C r. Now, innbsp;�rder to demonftrate the propolition, we mufl: prove thatnbsp;triultiplied by Aa, minus B, multiplied by Bi, is equalnbsp;A multiplied by C c, plus B multiplied by C c ; or,nbsp;^xAa

� nbsp;nbsp;nbsp;� It lt;.� OJUC 11. U�� UV-wiinbsp;nbsp;nbsp;nbsp;Vijt*�. 4nbsp;nbsp;nbsp;nbsp;^

VI. What has been faid in the preceding paragraphs with refped; to the centre of gravity of two bodies, may be applied to the centre of gravity ofnbsp;three, or four, or, in Ihort, of any number of bodies jnbsp;for it follows from the preceding propofitions, thatnbsp;two or more of the bodies may be conceived to benbsp;concentrated into their common centre of gravity;nbsp;hence the cafe m.ay always be reduced to that ofnbsp;two bodies only.

If it be aiked why, in the computation of the centre of gravity, we took no notice of the decreafenbsp;of the attraftivc force according to the fquares of'nbsp;the diftances; the anfwer is, that in that cafe thenbsp;diftance being one and the fame j viz. (the diftancC'nbsp;of the body A from the body B, Fig. 22. Plate I. isnbsp;the fame as the diftance ofB from A,) the computation is not altered by it¹ ².

Farther, BC � he is equal to B ^ C r, and A C � f is equal to A 0 � C lt;r j therefore A ; B :: B � C rnbsp;A� � C c i of which four proportional quantities the pro-ducl of the extremes mull be equal to the produdl of thenbsp;means ; viz. Ax ha � A x Cf = BxBigt; BXnbsp;Cf j and, by tranfpofition, AxAa �BxBi = A Bnbsp;X Cr.

Though the prefent part of this work treats exprefsly of fuch properties as belong to allnbsp;tgt;odles, without noticing the particular qualitiesnbsp;'�''hich diftinguifli one body fronj another y yet innbsp;^his chapter it will be neceflary to take notice ofnbsp;peculiarity only; namely, of the differentnbsp;effects which, in the collifion of bodies, are produced by their being e/q/iie or non-elajiic-, the mean-^Ug of which words will be explained in the foilow-definitions.

A body perfeBfy hard is that whofe figure is Hi the leaft altered by the flroke, or collifion, ofnbsp;^uother body.

A body perfeBly foft is that wdiofe figure is altered by the leaft impreffion, and which isnbsp;^^ftitute of the power of recovering its originalnbsp;^gure.

_ 3* An elaflic body is that which yields to the !^prefiion of another body, but afterwards recoversnbsp;Its figure. And,

4. nbsp;nbsp;nbsp;It is called perfeSlly elafic when it recovers itsnbsp;^original figure entirely, and with the fame force withnbsp;which it loft it; otherwife it is called imperfectlynbsp;elajlic.

5. nbsp;nbsp;nbsp;One body is faid to ftrike direEtly on anothernbsp;body, when the right line, in which it moves,nbsp;pafl'es through the centre of gravity of the othernbsp;body, and is perpendicular to the furface of thatnbsp;other body.

Though there are innumerable gradations from a body perfeftly hard, to one perfedtly foft; or between the latter and a body perfectly elaftvc; yetnbsp;we cannot fay with certainty that a body perfe�llynbsp;pofiefled of any of the above mentioned qualitiesnbsp;does adtually exift. It is however certain that ournbsp;endeavours have not been able to deprive certainnbsp;lx)dies of the leaft degree of their elafticity, by mechanical means.

The objeft of the theory of percutient bodies is to determine the momentums, the velocities, andnbsp;the diredlions of bodies after their meeting; whichnbsp;we fiiall lay down, and explain, in the followingnbsp;propofitions. But it muft be obferved, thatnbsp;throughout this chapter we only fpeak of bodiesnbsp;which move with equable motion, that is, of fuchnbsp;as deferibe equal fpaces in equal portions of time ;nbsp;and we do likewife fuppofe that the bodies move innbsp;a non-refifting medium, and that they are not influenced by any other adlion, excepting the finglenbsp;impulfe, which puts them in motion: for though

Perciitient Bodies, amp;c. nbsp;nbsp;nbsp;95

Aich fmipJe and regular movements never take place in nature; yet when their theory is once eftablillied,nbsp;tke complicated cafes, wherein the refiftances ofnbsp;mediums and other interfering caufes, are compre-hended, mav be more commodioufly examined jnbsp;^fid proper allowances may be made agreeably to thenbsp;nature of thofe caufes.

I- If bodies moving in the fame flraight line,Jirihe each other, the fate of their common centre ofnbsp;S'^avity �will not thereby be altered; viz. it �will eithernbsp;^^�'Hain at reft, or it 'will continue to move in the famenbsp;fit'aight line, exaSily as it did before the meeting of thenbsp;bodies.

This propofition is fo evidently deduced from properties of the centre of gravity, as men-tioned at N� II and III. in the preceding chap-that nothing more needs be faid about it innbsp;place.

Let there he tzvo non-elajlic bodies; and if one of ^^feni move in a flraight line, �whilfl the other is at refinbsp;that line, or is moving in the fame diretlion, but at anbsp;fovoer rate, or is moving in the contrary diretlion �, viz.nbsp;towards the body firjl mentioned; then thofe bodies mufinbsp;^tecejfarily meet orjlrike diretlly againfi each other, andnbsp;^fter the Jlroke they �will either remain at rejl, or theynbsp;quot;^^ll move on together, conjointly zvith their commonnbsp;of gravity.�Lheir momentum after the firokenbsp;he equal to the fiim of their momentums before thenbsp;if they both moved in the fame dir edtion, but itnbsp;'d^tll he equal to the difference of their momentums if they

moved in contrary direElions.�Their velocity after the Jlroke will be equal to the quotient that arifes from dividing the fum of their momentums, if they both movednbsp;the fame way, or the difference of their momentums, ifnbsp;they^moved in contrary direSlions, by the fum of theirnbsp;quantities of matter.

That in any of the above mentioned cafes the two bodies muft meet, and ftrike againft each other, isnbsp;fo very evident as not to require any farther illuf-tration.

That after the ftroke thofe two bodies muft either remain at reft, or they muft move together, conjointly with their common centre of gravity, isnbsp;likewife evident for as the bodies are not elaftic,nbsp;there exifts no power that can occafion their fepa-ration.

With refpeel to the momentum, it may be ob-ferved, that when the two bodies meet, whatever portion of momentum is loft by one of them muftnbsp;be acquired by the other; fince, according to thenbsp;third law of motion, adlion and re-a6lion are alwaysnbsp;equal and contrary to each other; therefore, if before the ftroke the bodies moved the fame way,nbsp;their joint momentum after the ftroke will be equalnbsp;to the fum of their momentums before the ftroke.nbsp;If one of the bodies was at reft, then, as its momentum is equal to nothing, the joint momientum willnbsp;be equal to the momentum of the other body before the meeting. If the bodies moved towardsnbsp;each other, then their momentum after the meet-qnbsp;nbsp;nbsp;nbsp;ing

irig v.ill be equal to the difference of their former TOomentums; and if in this ca.fe their momentumsnbsp;equal, then their difference vaniflies; hence thenbsp;bodies will remain motionlefs after their meeting.

The laft part of the propofition is likewife evi-*^ient; flnce it has alread)^ been fhewn, that in Equable motion, the velocity is equal to the quo-bent of the momentum divided by the quantity

When the weights and velocities of the tv/o bo-before their meeting are knov/n, their velocity ^ber the meeting may be determined by the following general method.

Let A and B, in fig. 4, 5, 6, and 7, of Plate IL quot;'bich reprefent the above mentioned cafes, be thenbsp;bodies; let C be their common centre of gra-'^by, and D the place of their meeting. Make DEnbsp;to DC; fo that the point D may be be-Ween C and E; then D E will reprefent the ve-'ocity of the two bodies after their meeting ; for,nbsp;the bodies after the concurfe move togethernbsp;'�^^jointly with their common centre of gravity;

fince it has been proved in the preceding pro-b^btion, and at N� II. of the preceding chapter, biat tl)e ftate of the common centre of gravity ofnbsp;two bodies is not altered by their mutual a�lionnbsp;bpon each other; therefore the velocity of theirnbsp;^�fnmon centre of gravity after their meeting, mufi:nbsp;Dnbsp;nbsp;nbsp;nbsp;velocity before the meeting; viz.

Velocity of the two bodies after their meeting, be�quot;gt; caufe then they move together with their commonnbsp;centre of gravity.

Of the above mentioned figures, it may be eafily perceived, that the 4th flrevrs when both the bodies move the fame way ; the 5th reprefents thenbsp;cafe in which B is at reft before the ftroke, and ofnbsp;courfe the two points B and D coincide ; the 6thnbsp;thews when the two bodies move towards eachnbsp;other; and the ytli flrews when the two bodiesnbsp;move towards each other with equal momentum?�nbsp;in which cafe, after their meeting, they will remainnbsp;at reft. The refpeftive velocities of thofe tvvnnbsp;bodies are reprefented in all the four figures, bynbsp;A D and B D ; for they run over thofe diftanct?nbsp;in the fame time; and A B is the difference ofnbsp;thofe velocities. Alfo their refpedtive momentuniSnbsp;are reprefented by the produdt of the weight of Anbsp;multiplied by AD, and the produ�t of the weightnbsp;of B multiplied by B D. The momentum ofnbsp;both the bodies together after their meeting, is re'nbsp;prefented by the produdt to their joint weight mul*nbsp;tiplied by DE (i).

Since

(i) The following is an example of the numeric�^ c�mputation of the firft cafe, fig. 4, which will be fafi�quot;nbsp;cient to indicate the manner of calculating the oth^^nbsp;cafes.

PercHtknt BodieSflSc. nbsp;nbsp;nbsp;97

Since when one of the bodies is at reft, the velocity after the meeting is equal to the quotient of the velocity of the moving body, divided by thenbsp;^urn of the quantities of matter of both the bodies;nbsp;�t follows that the larger the body at reft is, thenbsp;htialler will the. velocity be after the meeting. For

BDrrr, and AD will be equal to 32 ;f. Then the th ^ ^'*�P'�yed by A in moving from A to D, is equal tonbsp;Quotient of the fpace 32 y, divided by its velocity;

inftance, if the moving body A weigh one pound� and move at the rate of one foor per minute, whilftnbsp;the body B at reft weigh one pound alfo, the vC'nbsp;locity after the concurfe will be half a foot pe*'nbsp;minute; half a foot being the quotient of one foodnbsp;divided by the fum of their quantities of matter jnbsp;viz. 2 pounds. If cateris paribus B weighnbsp;pounds, then the velocity after the concurfe wh^nbsp;be the i ith part of a foot per minute. If B weighnbsp;looooo pounds, then the velocity after the con*nbsp;curfe will be the looooith part of a footnbsp;minute; and in Ihort, when B is infinitely biggc-than A, the velocity after the concurfe will be iO'nbsp;finitely fraall, which is the fame thing as to fa/�nbsp;that in that cafe, after the ftroke, the bodies wi^.nbsp;remain at reft. And fuch is the cafe when ^nbsp;non-elaftic body ftrikes againft an immoveable oh'nbsp;ftacle.

III. If a body in ^notion frikes direElly againft other body, the magnitude of the ftroke is proportioii^^nbsp;to the momentum loft, at the concurfe, by the more poivt^'nbsp;fill body.

According to the third law of motion, aftion an^ re-adtion are equal and contrary to each othegt;� �nbsp;therefore whatever momentum is loft by one ofnbsp;bodies, is acquired by the other. Or the mag'^'nbsp;tude of this acquired momentum (which isnbsp;effedl of the ftroke) is as the momentum loftnbsp;the more powerful body j it being by the quant**-^.

IV. fP/ien a given body Jirikes direBly againft mother given body, if the latter be at reft, the quantitynbsp;of the Jiroke is proportional to the velocity of the formernbsp;body.�If the fecond body be moving in the fame direction zvith the firft, but at a flower rate, the magnitudenbsp;of theJiroke will be the fame as if the fecond body floodnbsp;flill, and the jitftl impinged upon it zvith a velocity equalnbsp;10 the difference of their velocities.�And lafily, if thenbsp;bodies move direllly tozvards each other, the magnitudenbsp;of the Jiroke is the fame as if one of the bodies flood atnbsp;^oft, and the other fir tick it with the fum of their velo-oities.

The momentum of a given body fs proportionate to its velocity ; for with a double velocity the momentum is double, with a treble velocity the momentum is treble, and fo on �, therefore, as long asnbsp;the body remains the fame, the magnitude of thenbsp;fti'oke, being proportional to the momentum, mufl;nbsp;hkewife be proportional to the velocity. And whennbsp;One of the bodies is at reft, the magnitude of thenbsp;ftroke is evidently proportional to the velocity ofnbsp;the moving body.

V. It follozvs from the foregoing theory, that the mitual aElions of bodies, which are' inclofed in a certainnbsp;Space^ are exalllv the fame, zvhether that [pace be at

way, it takes away an equal portion of velocity from thofe bodies within it which move the contrarynbsp;way; thus, all the motions of the bodies in a fbipnbsp;are performed in the fame manner, and the famenbsp;cffedts are produced on each other, whether thenbsp;111 ip be at reft or move uniformly forwards.

The attentive reader muft have perceived, that in the explanation of the preceding cafes the dif-tances have been reckoned from the centres of thenbsp;bodies; whereas it is evident that the thlckneffesnbsp;of the bodies muft be deduefted from thofe dif-tances; for the bodies do not ftrike againft eachnbsp;other with their centres, but with their furfaces.nbsp;It muft be obferved, however, that when the dif-tances are very great, and the bodies proportionately very fmall, it is immaterial whether the diftancesnbsp;be reckoned from the centres or from the furfacesnbsp;of the bodies. But If great accuracy be wanted, thenbsp;thicknefs of the bodies may be eafily allowed for innbsp;the computation.

A limllar obfervation may be made with refpedc to the fliapes of the bodies; viz, that they havenbsp;been reprefented as being globular, for the pur-pofe of rendering the explanation Ihort and per-fpicuous.

Having, hitherto treated of unelaftic bodies, that is, of fuch bodies as are either perfeftly hardnbsp;or perfedly foft, it is now neceffary to ftate thenbsp;rules of congrefs which take place with elaftilt;^nbsp;bodies.nbsp;nbsp;nbsp;nbsp;gt;

At the commencement of the. prefent chapter, clafticity has been faid to be that property bynbsp;'''^hich bodies yield or fufFer their figure to be altered by the preffure of other bodies ; but which,nbsp;the removal of the preflure, recover their originalnbsp;%ure of their own accord.

This recovery of the figure is performed with greater or lefs quicknefs, and with more or lefs ex-aftnefs, in. different bodies ; which differences con-ftitute the various degrees of elafticity. And thofenbsp;bodies, which recover their figure completely, andnbsp;quickly as they loft it, are faid to be -perfeSlly

Though we are acquainted witlr - the effefts and laws of elafticity, in a manner fufficient to ren-that property fubfervient to our purpofes, yetnbsp;caufe of that property is by no means under-; nor has any hypothefis been offered in expla-^^don of it, which may be faid to be fufficientlynbsp;P^aufible.

^ Let a firing, AB, fig. 8. Plate II. be ftretched ^^Ween, and be firmly faftened on, two immovc-^ Supports at A and B ; and if, by applying anbsp;^^ger at C, this ftring be pulled towards D, thenbsp;will be found to refift that effort with a forcenbsp;�''^3-ter and greater, the farther it is pulled from itsnbsp;th ftraight fituation. When difengaged fromnbsp;^ finger, the ftring will not only return to its ori-Viz^nbsp;nbsp;nbsp;nbsp;fituation, but it will go beyond it;

� towards E, and nearly as far from the fl.raigbt H 3nbsp;nbsp;nbsp;nbsp;fituation

fituation ACB, as AD B is from it, after which it will again bend itf^lf towards D, and fo on, vibratenbsp;ing backwards and forwards, but deviating continually.-lefs and lefs from the ftraight fituation, untilnbsp;at laft it remains at reft in its original fituationnbsp;ACB.

Now when the ftring is firft difengaged from the finger, its elaftic force draws the part D towards C;nbsp;but this force decreafes in proportion as the part Dnbsp;comes nearer to C, and when the part D' is arrivednbsp;at C, that is, when the ftring is in its ftraight fituation, the above-mentioned force is infinitely littlenbsp;or is equal to nothing ; but by that time the partnbsp;D, having been impelled by a continual thoughnbsp;decreafing force, will have acquired a riaomentumnbsp;, which carries it towards Ej viz. beyond the ftraightnbsp;fituation; but as foon as the ftring goes beyond C,nbsp;the elaftic force begins to adl again in a diredlionnbsp;contrary to that of the momentum. This actionnbsp;becomes ftronger and ftronger the farther the middle part of the ftring goes from the ftraight direction,' and of courfe it gradually diminiflies thenbsp;above-mentioned momentum, until at laft the momentum being entirely fpent, the ftring beginsnbsp;again to move towards C, in virtue of the elafticnbsp;force, and fo on.

The extent of the vibtations becomes continually fliorterand flaorter, on account of the refiftance of the air, and of the want of perfedt pliability,nbsp;pf perfedt elafticity, in the parts of the firing�

If the firing be moved out of its ftraight direction, not by the application of a finger, but by a ^ody E, fig. 9. Plate 11. falling upon it at C fromnbsp;toe height H, it will be readily underftood thatnbsp;the momentum of the body will eafily impel thenbsp;ftring towards D ; but the re-a�lion of the firingnbsp;^0 the body will gradually diminifli the momentumnbsp;the latter, and the farther the firing is carried,nbsp;^mm the ftraight fituation, the ftronger will its re-^ftion be, until at laft the body, having loftnbsp;its momentum, will be carried back againnbsp;towards C by the elaftic force of the ftring; andnbsp;m its way back, the conftant though decreafing ac-bon of this elafticity from D to G, will give it anbsp;momentum which will carry it up towards H;

the body would afcend precifely up to H, were ^t not for the above-mentioned caufes of obftruc-; viz. the refinance of the air, amp;c.

It is almoft fuperfluous to obferve that the great-bie height is from which the body E falls upon firing, the farther will the ftring be removed fromnbsp;ftraight fituation, and of courfe the ftronger will,nbsp;^ts re-aftion be.

This explanation of the elafticity of the ftring be applied to all forts of elaftic bodies; for thenbsp;j ^nbsp;nbsp;nbsp;nbsp;every one of them will be bent more or

^ ^ by any ftroke or preffure, and will afterwardg ^co\er its original form by re-adling the contrarynbsp;H 4nbsp;nbsp;nbsp;nbsp;way

way with a force, which, in perfe�tly elaftlc bodies, is equal to the preffure orftroke received.

Jf inftead of the firing, whofe ends are immoveable, amp;c. we imagine that two equal bodies like A and B, fig. lo. Plate II. being impelled by equal andnbsp;contrary forces, direftly ftrike againft each other atnbsp;C, it is evident that the contiguous furfaces of bothnbsp;will be bent inwardly, and that the elaftic force ofnbsp;A will drive the body B back from C towards B,nbsp;(as the firing did in the preceding cafe) at thenbsp;fame time that the elaflic force of the body B willnbsp;impel A back with equal force from C towards A,nbsp;fo that in this cafe the bodies after the ftroke willnbsp;recede from each other; whereas had they been non-elaftic, they would have remained flationary at thenbsp;place of their congrefs C.

We muft now determine the effedls produced after the congrefs of bodies that are pe/fehr/y ehHic;nbsp;from which the laws of congrefs amongfl thofe thatnbsp;are hnperfeHly elaftic may afterwards be eafily deduced.

VI. When two bodies that are perfeEily elaftic ftrike direElly on each other, their relative velocity (bynbsp;which is meant the excefs whereby the velocitynbsp;the fwifter body exceeds that of the flower) will benbsp;the fame before and after theftroke-, viz. they will recedenbsp;from each other with the fame velocity with whebnbsp;they approached before the ftroke.

It has been fliewn above, that when two given bodies ftrike on one another, the magnitude of the

the definition, a perfectly elaftic body has been feid to be that which recovers its figure with thenbsp;fame force with which it loft it. Therefore in perfectly elaftic bodies the reftoring force is equal tonbsp;the compreffing force; fo that if the momentumsnbsp;the bodies produced a certain comprefiion, thenbsp;elaftic force muft re-ad on the bodies with the likenbsp;force; hence the bodies will be forced to recedenbsp;from each other with the fame velocity wherewithnbsp;they approached each other.

It is a natural confequence of this deraonftra-tron, that in equal times taken bejore and after tht J^^'oke, the difiances of the bodies from one another willnbsp;equal, and therefore in eqtal times their dijlancesnbsp;f'oni their common centre of gravity will liketvife benbsp;equal,

'I'hus much being premifed, the laws of congrefs bodies that are perfedly elaftic may be eafily determined. But in order to comprehend the folu-tion of the following cafes, the reader Ihould recol-and keep in view the following particulars,nbsp;'^hich have all been fufficiently proved in thenbsp;preceding pages; viz. tft, that the diftances of twonbsp;bodies from their common centre of gravity are in-'^erfely as their weights : adly, that the ftate of reftnbsp;�r of uniform motion of the centre of gravity ofnbsp;bodies is not altered by the mutual adion of thofenbsp;bodies on each other: 3dly, that in bodies that arenbsp;Perfe.dly elaftic, the reftoring is equal to the comprefling

preffing force : 4thly and laftly, that the diftanccs of the bodies from each other, and from their common centre of gravity, are equal in, equal timesnbsp;taken before and after the ftroke.

Now let A and B, fig. lo. Plate II. be two equal bodies perfectly elaftic; C is their common centrenbsp;of gravity, which, fince the bodies are equal, ftandsnbsp;midway between them. Let both the bodies benbsp;impelled with equal force, direfilly towards eachnbsp;other, in confequence of which they will move i.nnbsp;a certain time (for inftance, a minute) fromnbsp;their refpefilive places to C, where they meet-After the impulfe their elafticity will impel eachnbsp;body back towards its original place, fo that at thenbsp;end of one minute from the time of their meetingnbsp;they will be found precifely where they were a minute before their meeting, viz. at A and B.

Let A and B, fig. 11. Plate II. be two equal and perfefilly elaftic bodies, as in the laft cafe, andnbsp;one of them, for inftance B, be at reft, whilft thenbsp;other body A moves towards it, fo as to reach it i�'*nbsp;one minute�s time. Here AB reprefents thenbsp;velocity of it, and C B reprefents the velocitynbsp;of the centre of gravity G j for in the fame timenbsp;that A comes from the place A to the place Egt;nbsp;C muft come from C to B, therefore at the end ofnbsp;one minute after the ftroke the centre of gravitynbsp;muft be at F, viz. as far from B as its original fitO'nbsp;ation C was from the place B a minute before thenbsp;ftroke �, but a minute after the ftroke the body ^

�Tiuft be a� far from the centre of gravity, viz. from as it flood a minute before the ftroke; therefore itnbsp;'^uft ftand at B. Now as the fame reafoning maynbsp;be applied to any other equal times taken before andnbsp;after the ftroke, fuch as half a minute before andnbsp;half a minute after, amp;c. therefore in this- cafe thenbsp;body A after the ftroke will remain ftationary at B,nbsp;and the body B will move on with the velocity thatnbsp;^be body A had before the ftroke.

Having thus explained two of the fimpleft cafes ^f congrefs in a feparate manner for the fake of perspicuity, I fhall now give one general rule for thenbsp;Solution of all the other cafes, which are delineatednbsp;Plate II. from fig. the 12th to fig. the 20th in-^lufively, in which figures A and B r�prefent thenbsp;ftvo bodies ; C is their common centre of gravity ;

the place at which they meet. A D expreffes the '�^locity of A; BD the velocity of Bj and CDnbsp;Sbat of the centre of gravity.�Then the rule for de-^�rniinino- the velocities after the ftroke is as follows:

Take a point E in the line A B, produced if ne-^offary, fo that the diftance C E be equal to C D ; ftien after the ftroke the right line E A will exprefsnbsp;ftie velocity of the body A from E towards A, andnbsp;ftie right line EB will exprefs the velocity of Bnbsp;^��om E towards B.

In any one of thofe cafes the centre of gravity nruft move from C to D, and after the ftroke,nbsp;ftonr D forwards to q diftance D P' equal to D C in

a portion of time equal to that in which A and B employ for coming from the places A and B to D.

Make Fa equal to C A ; and lince in equal times� taken before and after the ftroke, the diftances ofnbsp;the bodies from the common centre of gravity arenbsp;equal, therefore when the centre of gravity is at Fnbsp;the body A will be at � j fo that after the ftroke it 'nbsp;will move from D towards a, and Da, which it hasnbsp;palled over in that time, will reprefent its velocity.nbsp;But becaufe C E is equal CD, or to F D, and C A,nbsp;is equal to Fa, the difference of the right lines C Egt;nbsp;C A will be equal to the difference of the right lines,nbsp;FD, Fa; viz. EA will be equal to Da. But repre-fents the velocity of the body A after the impulfe,nbsp;therefore its velocity will alfo be reprefented by E A.nbsp;And fince the relative velocity of the bodies beforenbsp;and after the ftroke is the fame, and E A reprefentsnbsp;the velocity of the moving body A ; therefore thenbsp;Velocity of the body B, moving from E towards B,nbsp;is of courfe reprefented by the right line EB,

For the better illuftration of this theory, I fliall briefly mention the meaning of the figures whichnbsp;exhibit the various cafes of the congrefs of bodiesnbsp;that are perfectly elaftic, to every one of which thenbsp;preceding explanation is equally applicable.

Fig. 12. is the cafe when B is larger than A� (which is indicatedby C, the fituation of the centrenbsp;of gravity) B is at reft, and A ftrikes againft it*nbsp;In this cafe, after the ftroke, both the bodies will

13. thews when A is larger and runsagainft ^ reft j in which cafe, after the ftroke, both bodiesnbsp;move towards F.

�^ig. 14. is the cafe when A is larger than B, '^^th bodies are in motion the fame way, and meetnbsp;^'0, amp;c.

In fig. 17. after the ftroke the equal bodies and B recede with interchanged velocities.

In fig. 18. the bodies are proportional to their ''^locities, in which cafe the points C, F, D, and E,nbsp;'^'quot;'ncide.

In fig. 20. though the bodies A and B meet at ^ between the places A and B, yet after the ftrokenbsp;bodies will move towards F. (2)

^^{2) The method of making the numerical computation ^^hofe cafes will be fhewn by the following example,nbsp;IS adapted to the cafe reprefented in fig. 14! tonbsp;'Vatit^ reader is requefted to dire�t his eye j though fornbsp;, room the parts of that figure be not drawn in th�

no nbsp;nbsp;nbsp;^he Theory of

After the explanatiorS of the preceding cafes, tb� method of determining the . velocities after tb�nbsp;ftroke, when the bodies are not perfe6tly elaftic, mafnbsp;be eafiJy underftood.

Let A and B be two perfeAly elaftic bodies. A weigi�* 2 pounds, and moves at the rate of 8 feet per fecond. ^nbsp;weighs one pound, and moves the fame way at the ratenbsp;5 feet per fecond ; and let the diftance A B be J2 feet.

1. nbsp;nbsp;nbsp;To find the centre of gravity C, we have A B'nbsp;B AB: CA, viz. 3; i12: 4; fo that AC =4, and C'Bnbsp;=AB�AC=:8.

2. nbsp;nbsp;nbsp;To find the diftance BD, putBOar^;; and fm^�nbsp;the diftances ADand BD are run over in the fame tim^'nbsp;the former at the rate of 8, and the latter at the rate of i

3. nbsp;nbsp;nbsp;If the diftance BD, viz. 2C, be divided by thenbsp;locity of B,^viz. 5, the quotient 4 is the number of 1�'nbsp;conds, during which the bodies moved from their refpe'-'nbsp;live places A and B, to the place of their congrefs D-

4 (the number of fcconds found above) gives 6 for the locity of A after the ftroke, in the diredtion from Enbsp;wards A.

Alfo EB = EC CB = 28 S = 36 ; which, beinS divided by 4 (the number of feconds, amp;c.) gives- 9nbsp;the velocity of F, after the ftroke in the diredlion ftoa� (nbsp;towards B, So that after the ftroke, the bodies A and Bnbsp;will both continue to move the fame way, but the

Thus let A and B, fig. 21 and 22, Plate II. be two bodies imperfedtly elaftic ; C their 'common centrenbsp;of gravity ; D the place of their meeting. Dividenbsp;AC in a, fo that A C may be to aC, as the forcenbsp;. compreffing the body A is to tbe force whereby itnbsp;reftores itfelf. Alfo divide BC mb-, fo that ECnbsp;tnay be to C as the force compreffing the body Bnbsp;is to the force whereby it reftores itfelf. Take C Enbsp;equal to C D ; and laftly the right line Ert will ex-Prefs the velocity of A after the ftroke, in the di-i^etftion from E towards a, and the right line E^ willnbsp;oxprefs the velocity of B after the ftroke in the di*nbsp;leflion from E towards D.

There being perhaps no body in nature which iiiay-be faid to be perfectly elaftic, the rules givennbsp;for determining the velocities of bodies that arenbsp;perfectly elaftic, cannot be verified experimentally;nbsp;but the deviation of the experimental refult fromnbsp;rules is proportionate to what the bodies want

In the like manner may the other cafes be calculated.� I have given this method of adapting the calculation tonbsp;the figures, or of expreffing the parts of the diagrams, bynbsp;nieans of numbers, in preference to the complex rulesnbsp;'''hich have been given for this purpofe by certain learnednbsp;'''nters, becaufe the latter are feldotn remembered,nbsp;^nd arc difficultly applied to' the folution of the variousnbsp;impadl amongft bodies that are pofi�fl�d of perfect

of perfeft elafticity; and this deviation is taken Int� the account after the manner mentioned above.

The precife degree of elafticity of which any particular body is poflefled, muft be afcertained by means of aftual experiments on the body itfelf,nbsp;which experiments differ according to the variousnbsp;nature of the bodies.

When more than two bodies move in the fame ftraight line, the determination of the velocity ofnbsp;each body,after the various impadls with each other,nbsp;cannot be comprehended under any general rules,nbsp;the variety of cafes being too great, and fometimesnbsp;very intricate; yet when any particular cafe pre-fents itfelf, the preceding rules will be found fuffi-cient for the folution of it, viz. for afcertaining thenbsp;velocities,' amp;c. But in the folution of fuch cafes,nbsp;the operator muft take care to calculate, in the firftnbsp;place, the velocities of thofe two bodies which appear from the circumftances of the cafe to meetnbsp;firft; then to fubftitute thofe new velocities whichnbsp;are the real velocities of thofe two bodies after theirnbsp;meeting, and with them to calculate (according asnbsp;any one of thofe bodies is concerned with the fe-cond ftroke) the velocities after the fecond congrefs,nbsp;and fo forth.

Moft of the foregoing cafes, both of pcrfeclly and of imperfeiftly elaftic bodies, might be ex-preffed in the form of canotjs, (that is, of particular rules) and by ftating bodies of various weights,nbsp;and moving v/ith various velocities, the numbersnbsp;Inbsp;nbsp;nbsp;nbsp;o�

thofe canons might be increafed without end. he following ihrOe paragraphs contain three fuchnbsp;canons by way of examples, which the reader willnbsp;^afil^ perceive to be nothing more than particularnbsp;'^afes exprefled in words.

dl'lien tivo bodies, perfeclly elajlic and equal, are tozvnrds the Jayne part, after the meeting (ifnbsp;Pieir velocities be fuch as to admit of their Jlrikingnbsp;^S^inji each other') they zvill continue to move in thenbsp;f^'me direBion but zvith interchanged velocities. But ifnbsp;^hey hg carried tozvards contrary parts, then after thenbsp;^^eting^ they zuill go back zvith interchanged velocities.nbsp;If any number of equal and perfcBly elajlic bodiesnbsp;at reft, contiguous to each other in the, fame firaightnbsp;and another body equal to one of them ftrike thenbsp;of them in the fame firaight line zvith any vcIq-'i then after the ftrpke the Jlriking body and all thenbsp;^ tvill remain at reft, and the laft body only zvill movenbsp;zvith the velocity of the Jlriking body.�In thisnbsp;the bodies aft as if they were feparate ; viz.nbsp;A, fig. 23, Plate II. ftrikes direftly againft B,

5 and remains itfelf at reft; and in the like manner C communicales the fame velocity to D,nbsp;^t^dl3 to E; which laft body E will in confequencenbsp;'t be forced to move towards F with tlie velo-that A had at firft.

If every thing remain, as in the preceding cafe, except'' ing that notv tzvo bodies A and B, fig. 24, Plate 11- 'nbsp;contiguous to each other, be impelled towards C, then �nbsp;after thefiroke. A, B and C tvill remain at reft, andOnbsp;and E will move off towards F tvith the velocity thatnbsp;A and B had atfirft.

OF COMPOUND MOTION; OF* TIT� COMPOSITION AND RESOLUTION OF FORCES; AND OF OS' 'nbsp;LIQVE IMPULSES.

By compound motion is underftood that move-, ment of bodies which arifes from more thaonbsp;one impulfe; for in fuch cafes the velocity and th^'nbsp;direftion of tlie body put in motion, arife froiT*nbsp;the concurrence of all the impulfes, and participat� inbsp;of them all, under certain determinate laws, whichnbsp;will be fpecified in the following propofitions.

J. J'Vhen a body is impelled at the fame time by forces in different direliions, the body wiPmove notnbsp;any one of them, but in a dirediion between thojtnbsp;tzvo.-

Thus if a body A, fig. 25, Plate IL be impefic^^ by two forces, viz, one which by itfelf would drivednbsp;in thediretfion AB, and another, which by ' �nbsp;would drive it in the diredion AD ; then the body

A. being impelled by both thofe forces or imptilfes, the fame time, will move in a direftionACjnbsp;between AD and AB; for hnce, ^.ccording to thenbsp;Second law of motion, the change of motioh isnbsp;always made in the direftion of the right line innbsp;'^'hich that force is imprefled, therefore the motionnbsp;�f the body along the line AD is altered from thenbsp;^ire�lion AD, to another direftion towards AB^nbsp;by the other impulfe, which a�fs in the diredfionnbsp;ABi And for the fame reafon, the motion of the 'nbsp;body along the line A B is changed for another di-gt;�20:1011 towards A D by the impulfe which ads innbsp;^hat diredion. Therefore the motion arifing fromnbsp;dhofe two impulfes muft have a diredion betweennbsp;A D and A B.�But it will be fliewn in the follow'nbsp;gt;rig propofition, how much this new' diredion w'illnbsp;'^^oviate from A D, and from A Blt;

f h I'VheH a body is impelled at the fame time hy two h^ces in different directions, if tzvo lines be drawnnbsp;the place in which the body receives the doublenbsp;^^pulfe, in the direSiioHs of thofe impulfes; and thenbsp;^^^igths of thofe lines be made proportionate to the ini-Pdling forces 5 alfo through the end of each of thofenbsp;^^^^es a line he draivn parallel to the other, a parallelo-i�ani zvill thereby he formed', and if a diagonal linenbsp;drazvn froili the place where the body receives thenbsp;�^^^��bie impulfe to the bppofite corner of thepdrallelo-I'am, the length and ftuatkn of that diagonal zvillnbsp;f' efent the velocity and the direEllon of the body'snbsp;arifing from the double impulfe.

Thus, fuppofe that the body A, fig. 26, Plate II. be impelled in the direftioh A D, by a force whichnbsp;would enable it to move at the rate of 4 feet pernbsp;lecond; alfo that at the fame time the. fame bodynbsp;be impelled by another force in the direction AB,nbsp;which would enable it to move at the rate of 3 feetnbsp;per fecond. Make AD equal to four, and ABnbsp;equal to tliree (for inftance, inches; or you maynbsp;life any other dimenfion toreprefent feet). Throughnbsp;D draw DC parallel to AB; and through B drawnbsp;B C parallel to A D ; by which means the parallelogram A B D C will be formed. Laftly, draw thenbsp;diagonal A C, and A C is the direftion in w'hich thenbsp;body which is impelled by the aboye-mentionednbsp;two impulfes, will move. Alfo the length A C willnbsp;exprefs the velocity of the body ; fo that if AC benbsp;foimd, either by calculation or by meafuring thenbsp;diagram, to be 5 inches long (1); w-e conclude

(i) The length and direflion of A C ; viz. the angles it makes with'AD and AB, may beeafily found by trigonometry; it being the folutlon of a plane triangle, in whichnbsp;two fides, and the angle between thofe two fides, arenbsp;known.

The direction of the impulfes being given, the angle DAB is alfo known ; for it is the angle which the directions of the two impulfes make with each other-'j'he angle ADC is likewife known, becaufe it is thenbsp;complement of the angle DAB to two right angles. Thenbsp;lines AD and DC (^AB) are to each other in the proportion

Of Compound Motion, ^c. nbsp;nbsp;nbsp;117

that the body will move at the rate of 5 feet per fecond, fince in the dimenfions of AB and AD,nbsp;inches were employed for reprefenting feet.

That the body thus impelled by the two forces iTiufl; move along the line A C, is eafily deduced fromnbsp;the fecond law of motion; for fince the change ofnbsp;iiiotion is proportionate to the moving force im-Preffed ; if from any point c in the diagonalnbsp;Ac you draw two right lines, viz. d c parallelnbsp;to A B, and b c parallel A D, thofe two linesnbsp;�'vill reprefent the deviations of the body�s mo-tion from the diredtions A D and AB; fince bynbsp;iaw the 2d, the chan2;o of motion is made in the di-^^ftion of the moving force imprefled. And thofenbsp;lines are proportional to the impelling forces,

Portion of the two impulfes, and may be reprefented by dimenfions, as inches, feet, amp;c. Therefore in thenbsp;triangle ADC the fides AD, DC, and the included anglenbsp;are known. Hence by trigonometry we have AD

tgt;C;

______ ____ 4

�tber angle DAC, which gives the direftion of AC : thus

the angles will be known, Laftly, fay, as the fine of

the angle DCA is to the fine of the angle ADC, fo is

to a fourth proportional, which is equal to AC.

both the direction and the length of AC will be

�iiown.

I q nbsp;nbsp;nbsp;01'

ii8 ' Of Compound Motion, tdc. or to the lines A B and A D, which reprefent thoicnbsp;forces; viz. dc is to be, as AB is to AD; becaui�nbsp;(by Eucl. p. 24. B. vi.) the parallelogram Kdcbnbsp;is fimilar to the parallelogram ABDC.

If it be livid that the body thus impelled will at an)' time be found at any other place 0 out of thenbsp;diagonal AC, draw om parallel to A D, and oinbsp;parallel to AB; then om and od, which reprefent thenbsp;deviations, amp;c. ought to be proportional to thenbsp;forces which occalipn thofe deviations, viz. om oughtnbsp;to bear the fame proportion to sv/ as AD docs tonbsp;AB. But this is not the cafe, bccaufe the parallelogram A v/wo is not fimilar to the paralielogtaninbsp;ADBC. Therefore the body, amp;c. muft movenbsp;along the diagonal AC, and in no other diredion.

III. When a body is impelled at the fame time bj three forces in three different directions^ the velocitynbsp;and the direction of the body's motion, which ariffnbsp;therefrom, muft be determined by firft afeertaining tUnbsp;courfe which would be produced by any two of thoffnbsp;forces, according to the preceding propofitioii; and thp^

' by finding the courfe I aft found, and the third foretj which zvill be the courfe foiight.

iiute in the dke�tio^ A D. Make the lengths of the right lines proportionate to the forces, viz. A Bnbsp;four, AC three, and A D five, inches, or feet, he.nbsp;long. Then imagine as if the body were impellednbsp;by the firft and fecond forces only, and, by the preceding propofition, find the compound motionnbsp;^filing therefrom, viz. through B draw B E parallel to A C, and draw C E through C parallel to AB;nbsp;^tid the diagonal A E will reprefent the directionnbsp;^rid the velocity of the motion refulting from thofenbsp;t^vo forces. Then after the lame manner find thenbsp;Compound motion refulting from the force repre-fcnted by A E, and the third force reprefented bynbsp;AD; viz. by drawing through E and D the linesnbsp;and DF, refpeftively parallel to AD and tonbsp;A E, as alfo the diagonal A F; and this diagonalnbsp;AF will reprefent the courfe of the body, viz, thenbsp;''clocity and diredion of its motion, arifing fromnbsp;^fie above-mentioned three impulfes.

The demonftration of this propofition is fo evi-*^ent a confequence of the preceding propofition, *^hat it will be needlefs to detain the reader with anbsp;Repetition of almoft entirely the fame words.

ft appears likewife, that the like reafoning may f'e extended to the cafe of four or five, or in Ihort,nbsp;cf any number of impulfes.

Notwithftanding the apparent multiplicity and '^tricacy of fuch cafes, an obvious remark will fur-a general rule, by means of which the placenbsp;the body at any time may be eafily deter^-I 4nbsp;nbsp;nbsp;nbsp;mined

mined in all cafes; viz. whether the impulfes be fingle, or accelerative like the force of gravity,nbsp;or whether fome of them be of the former, andnbsp;others of the latter fort.

The obfervation which furnilhes the rule is, that at the end of a certain time the body which is im-pelled by two forces, will be found precifely at thenbsp;place where it would be found if the two forces actednbsp;one after the other j the time however mufc not benbsp;doubled. For inftance, in the cafe of fig. 26, Platenbsp;II. the body A is impelled by two forces, viz. onenbsp;in the difeftion AD, which alone would drive itnbsp;as far as D in one fecond, and aqother force innbsp;the direftion A B, which alone would drive it to Bnbsp;in one fecond. Now if you imagine that thofe twonbsp;forces be applied one after the other, viz. thatnbsp;when the body is at D, the other force impels it innbsp;the dire�lion DC parallel to AB, and as far frontnbsp;D as B is from A ; then C is the place where thenbsp;body will be driven in one fecond by the compoundnbsp;action of both the forces.

This obfervation is evidently applicable to the cafe of four or more impulfes; and hence we derive the following general rule for finding the placenbsp;or fituation of a body after a certain time, whennbsp;the body is impelled by any given number of givennbsp;forces.

Rule. Imagine as if the body were impelled by given forces, not at once, but fuccejftvcly one after thinbsp;other, in directions parallel to their original direCiiontnbsp;1nbsp;nbsp;nbsp;nbsp;'nbsp;nbsp;nbsp;nbsp;an^

Of Cc�}po7ind Motlcn, nbsp;nbsp;nbsp;121

each in an equal porlioji of time; and the lajifitua-IS the place zvheve the body zvill be driven in the ^^ke portions of time, bp the joint action of all the forces

Thus in the cafe of fig. 27, Plate 11, the firfl; force by itfelf would impel the body to B in onenbsp;��ilnute; the fecond force would by itfelf impel itnbsp;from B to E in one minute (B E being equal andnbsp;parallel to A C); and the third force alone wouldnbsp;irttpel it from E to P' in one minute (EF beingnbsp;parallel and equal to A D); therefore the joint cac-fion of all the three forces will drive the body fromnbsp;to F in one minute.

If, inftead of one minute, any other portion of little be made ufe of, the figures arifing therefromnbsp;^ill always be fimilar, fo that whether the figurenbsp;Ite larger or fmailer the point F will always be innbsp;the right line AF; which likewife flrews that whennbsp;^ body is impelled by fingle impulfes (viz. fuch asnbsp;produce equable motion)' let their ^number benbsp;^hat it may, the courfe of the body between itsnbsp;^tiginal place A and the place F, where it will benbsp;found at the end of any time, is always reflilinear;nbsp;hence the right line A F repretents, as we have al-ready obferved, the direftion and velocity of thenbsp;body�s motion.

Sometimes the diredlions and the ftrength of the riupuifgg are fo circumftanced as to produce no mo-bon on the body; in which cafe the forces are faidnbsp;be balanced in oppofite directions �, and to thofe

cafes the above-mentioned rule is equally applicable. Thus if a body A, fig. 28, Plate JI. be impelled by a force which in one minute�s timenbsp;would drive it in the direction A B as far as B, andnbsp;likewife by another force equal to the former, whichnbsp;by itfelf would drive it in the direction A C op-pofite to the direction A B, to a place C, as farnbsp;from A as B is from A; it is evident that thofe twonbsp;equal, but oppofite, impulfes, adling at the famenbsp;time, will not produce the lead motion in thenbsp;body, for they deftroy each other.

Likewife if a body be impelled by three powers in three different directions, and the compoundnbsp;courfe which w^ould be produced by tw'o of thofenbsp;forces be equal and oppofite to the third force, thenbsp;body will not be moved by thofe three forces.��nbsp;Thus if the body A, fig. 2^, Plate II. be Impellednbsp;^ in the direction 'A B by a force which in one minute w'ould enable it to go as far as B; alfo by a fe'nbsp;cond force, which in one minute would enable itnbsp;to go in the diredion A C as far as C; and laftb%nbsp;by a'third force, which in one minute w^ould enablenbsp;it to move in the diredion AE as far as E. Find thenbsp;courfe of the body which would arife from the jointnbsp;adion of the two farces AB and AC, viz. by drawingnbsp;B D parallel to A C, and C D parallel to A B, andnbsp;joining A D. Now if this diagonal A D happensnbsp;to be equal to AE and oppofite to it, that is, in thenbsp;fame right line, then the body A will not be movednbsp;by the joint action of thofe three forces j f^r

the two forces A B and A C are equivalent to the force reprefented by A D ; but this force A D isnbsp;oqual and oppofite to the force A E ; therefore thenbsp;t�m of the two forces A B and A C, is llkewifcnbsp;oqual and oppoiite to A E ; hence the body willnbsp;^^ot be moved from its original place A.

Since A B D C is a parallelogram, the line B D Js equal and parallel to A C, as alfo the line D Cnbsp;to AB ; and, in cafe of an equilibrium or balancenbsp;the three forces, A D ha,s been fhevvn to benbsp;^qual to, and in the fame right line with, A E,nbsp;''^hich is the fame thing as to fay that A D is pa-^allel to A E. Therefore we eftablifh the followingnbsp;Propolition, which is of great ufe in mechanics.

IV. If a bod^ be impelled by three powers, or, is the fame thing) it bedrazvn by three pozvers,nbsp;three different direSiions, and thofe pozvers balancenbsp;^ach other fo as to leave the body at reft �, then thofe _nbsp;pegt;zvers muft have the fame proportion to each other asnbsp;have the right lines (A B, B D, and AD) drawnpa-'^allel to their direblions, and terminated by their mutualnbsp;^�ncoiirfe. And vice verfa, if the lines drawn pa~

^ allei to the direSiions of the three forces, and termi-^^ated by their mutudl concourje, bear to each other the fdme proportion that the forces bear to each other, then,nbsp;^he body will remain at reft (2).

(2) By trjgonornetry, the fides of plane triangles are as the fines of their oppofite angles. Therefore in the triangle-

It will be hardly neceflary to obferve, that a balance of forces may take place amongft any number of fuch forces; fo that a body may remain perfectly at reft, though aded upon by four or five, ornbsp;any other number of forces. In this cafe the forcesnbsp;� are fometimes called prejfures ;� for in fad they onlynbsp;prefs upon the body without being able to move itnbsp;out of its place.

V. As the joint atiion of fever al impulfes compel a body to perform a certain courfe, fo ivhenever we ob- 'nbsp;ferve any particular coiirj'e of a body, we may imaginenbsp;that courfe to be produced by the joint aSlion of two ornbsp;more impulfes that are equivalent to that fingle ini-pulfe.

Thus finding that a body A, fig. 26, Plate II. has moved from A to C, we may imagine either 'nbsp;that the body has been impelled by a fingle forcenbsp;in the diredion of A C, and proportionate to thenbsp;length of AC, or that it has been impelled by twonbsp;forces at once, viz. by one in the diredion of A D,

angle A B D, fig. 29, Plate II. A B is to B D as the fine of the angle ADB, or DAC, is to the fine of thenbsp;angle DAB; hence any two powers will be to each othernbsp;reciprocally as the fines of the angles, which the lines re-prefenting their directions, make with the direction of thenbsp;third power. Farther, A D is to A B as the fine of thenbsp;angle AB D is to the fine of BDA, or D AC ; and innbsp;like manner the power acting according to A E is to thenbsp;power adting according to A C, as the fine of the anglenbsp;A C D is to the fine of the angle ADC, or B A D.

3nd proportionate to the length of ADj and by Another force in the cliredtion of A B or DC, andnbsp;proportionate to A B or DC. Therefore, if twonbsp;�fides of any triangle (as A D and DC) reprefent bothnbsp;luantities and the direSiions of tzvo forces aSlingnbsp;fioni a given point, then the thirdfide (as AC) of thenbsp;will reprefent both the quantity and the direc-of a thirdforce, 'which, aEtingfrom the fame point,-'^^tll be equivalent to the other two, and vice verfa.

Thus alfo in fig. 27, Plate 11, finding that the ^ody v\ has moved along the line AF from A to Fnbsp;D a certain time j we may imagine, ift, that thenbsp;has been impelled by a finglc force in the di- 'nbsp;�^^'^ion and quantity reprefented by A F ; or adl}'-,nbsp;it has been impelled by two forces, viz. thenbsp;reprefented by A D, and the other reprefentednbsp;P A E j or thirdly, that it has been impelled bynbsp;^^^ree forces, viz. thofe reprefented by A D, A B,

A C ; or laftly, that it has been impelled by other number of forces in any direftions �, pro-^''^ed all thofe forces be equivalent to the finglenbsp;which is reprefented by A F.

^ This fuppofition of a body having been impelled ' y two or more forces to perform a'certain courfe.jnbsp;^ � on the contrary, the fuppofition liiat a bodynbsp;been impelled by a fingle'force, when the bodynbsp;knov/n to have been impelled by feveralnbsp;which are, however, equivalent to that finale

-?/

. In the preceding pages we have laid down the laws relative to the congrefs, or impaft, of bodies�nbsp;when the bodies ftrike in a direftion perpendicularnbsp;to each other. It is now neceflary to examine thofenbsp;cafes in which the bodies ftrike in an oblique direction, the elfects ol which could not have been conveniently explained prcvioully to the dodtrine ofnbsp;the conipofition and refolution. of forces, lince hnbsp;depends principally on that doctrine.

It has been fliewn, that if a body A, fig. 26� Plate I!, be ftiuck by two powers at the fame time�nbsp;viz. by one in the direction from A towards B, ani^nbsp;by the other in the direction from A towards P�nbsp;the body will thereby be forced to deicribc the diii'nbsp;gonal A C. Now let this motion be reverfed, vi^'nbsp;imagine that the two pow'ers or bodies K and P�nbsp;are at reft, and that the body A, advancing fron^nbsp;C, along the line C A, ftrikes againft thofe twOnbsp;bodies at the fame time ; the confequence will b^�nbsp;that both the bodies will be moved from therfnbsp;places, fince they are both ftruck j that thenbsp;pulfe will be divided amongft them in the propo''nbsp;tion of the line A B, which is perpendicular to th^nbsp;body K ; to the line A D, which is perpendicul^'^nbsp;to the body L ; and laftly, that the body willnbsp;impelled towards Q, whilft the body K is impd^'quot;�'^nbsp;towards P,

It is evident that the force of the impulfe muft be divided amongft the two bodies; fo that thenbsp;greater is the quantity of it which is communicatednbsp;to the one, the imaller muft be the quantity of itnbsp;t'diich is communicated to the other,; alfo each ofnbsp;thofe quantities muff be lefs than the whole original force of the body A ; otherwife there would benbsp;�n accumulation of force without any adequatenbsp;^aufe, which is not poffible. �

The force is not only divided amongft the two bodies K and L, but it is divided in the proportionnbsp;the lines BA to DA, which is eafily provednbsp;^hus r Since any force may be refolved into two ofnbsp;Jitore forces, therefore if we divide the force represented by the line A C into two forces, luch as thatnbsp;nne of them cannot poffibly act .upon the body K,nbsp;'''bilft the other afts diredlly againft it, we flrall bynbsp;^ba.t means determine the queftion. Draw A Dnbsp;Parallel to the furface of the body K at the pointnbsp;nf'congrefs; from C drop C D perpendicular tonbsp;^ D, and through the point of congrefs draw A Bnbsp;Parallel to D C, which A B being perpendicular tonbsp;D, mufl likewnfe be perpendicular to the bodynbsp;^ ^t the point of congrefs. Thus the force A Cnbsp;refolv'ed into the two forces A D and D C, ornbsp;^ ^ ; the former of which cannot have any actionnbsp;'^Pon the body K, whilft the latter acts entirelynbsp;'^Poti it. Por inftance, imagine that inftead of anbsp;body moving from C towards A, two bodies movenbsp;*^nwards that point, viz. one in the dirc�ion D Anbsp;the other in the diredfion BA, and it is evident

that of thofe two powers, the one in the direction BA vvill aft entirely and direftly upon the bodynbsp;K, whilft the other in the direftion DA paffes b}'nbsp;it, and of courfe cannot affeft it.

By the like reafoning it is proved, that when the force reprefented by C A is refolved into two othernbsp;forces, viz. A B, which is parallel to the furface ofnbsp;the body L at the point of congrefs, and BC,- c.rnbsp;its equal A D, which is perpendicular to it,the latternbsp;only will aft upon the body L; therefore the forcenbsp;which acts upon the body K is reprefented by A B,nbsp;or its equal DC; and the force which afts upon thenbsp;body L is rcprclcnted by the line AD, or its equalnbsp;BC; and thole two forces are equivalent to thenbsp;force A C. '

The inclination of the direftion of the Broke to the body K, or to thedine A D, which is parallel to the furface of it at the point of congrefs, isnbsp;reprefented by the angle D A C ; and the inclination of the Broke to tlie body L, or to the linenbsp;A B, is reprefented by the angle CAB. Now (bynbsp;trigonometry) when A C is made radius, D C, or itsnbsp;equal AB, becomes the fine of the angle of inclination DAC; and B C, or its equal AD, becomes the fine of the angle of inclination BAC�nbsp;therefore the effeCt of the oblique force C A, is to th^nbsp;effect that 'would be produced by the fame force coni'nbsp;ing in a perpendicular direSimi, as thefne of the cinp,dnbsp;oj inclination is to radius; which is a general aotlnbsp;ufeful law in the computation of. oblique in^'nbsp;puifes.

In the prefent inftance the proportion of th^ ob* iique force C A to move the body K, is to that ofnbsp;fame force coming in a perpendicular direftion,nbsp;^sthe fine D C is to radius AC ; and for the bodynbsp;^ it is as the fine B G to radius A C.

If in the above-mentioned cafe v/e imagine that of the bodies be removedj whilft the other isnbsp;^xedj we thall then form the cafe reprefented bynbsp;%� I. Plate III. in w'hich the body A, moving innbsp;diredtion A C, ftrikes obliquely at C on thenbsp;obftacle B F; where it is plain that the mag-^'tude of the obliq ue ftroke is to the magnitude of.

fame ftroke if it had come in a diredlion per-I^/^Rdicular to the obftacle^ as the fine of inclina-or of imidence, (viz. as the perpendicular A B) _ to the radius A C.

^^^Jlroke this body zvi� be reflekedfrom that obftacle;

direEtion C E, in fuch a manner as to form the angle ^ ^fieEtion E C F, equal to the angle of incidence

^CB.

Tlig oblique force AC being refolved into two �^tces, viz. D C perpendicular to the obftacle andnbsp;^ � parallel to it; the effedl on the plain is thenbsp;Rie as it the body had advanced towards it di-^^dtly from D, and according to the laws of (joninbsp;S^fs between perfedlly elaftic bodies, (chap, vii.)nbsp;th^ ^ ^fter the ftroke would be fent back innbsp;^ diredtion C D. But of the two forces intonbsp;Knbsp;nbsp;nbsp;nbsp;which

a body perfiS�ly elafiic as A, fig. I. Plate III. at C on the firm obfiacle B F, .then after

which the original force of the body 'A was refoIv�ecJ# this body retains the one reprefented by A D, finctfnbsp;this force was hot concerned in ftriking the ob-ftacle ; therefore after the ftroke the body A is actuated by two forces, viz. one reprefented by C D�nbsp;equal to A B, ecjual to E F ; and the other reprefented by C F, equal to D A, equal to D E, hencenbsp;it mull move in the diagonal CEi and fince thenbsp;lines CF, F E, are refpedively equal to the linesnbsp;CB, B A, and the angles at B and F are equal, be-caufe they are right-angles ; therefore (Eucl. p. iv'.nbsp;B. I.) the triangle EFC is in every refpeft equal t�nbsp;the triangle ACB �, confequently the angle of re-fledlion ECF is equal to the angle of incidencenbsp;ACB.

Some writers call the angle ACD the angle of incidence, and the angle DCE the angle of reflection ; viz. the angles made by the body with thenbsp;perpendicular DC �, this however does not alter thenbsp;propofition, for the angle ACD is likewife equal tonbsp;the angle DCE; thofe angles being the compk'nbsp;ments of the equal angles ACB, ECF, to twonbsp;right angles.

In any cafe, whenever two bodies ftrike obliquely againft each other, whether one or both be in motion, their diredlions, velocities, and momentumsnbsp;after the ftroke may be eafily determined from what'nbsp;has been explained in the laft paragraphs, togethetnbsp;with what has been delivered concernina the direftnbsp;iropad of claftic and non-elaftic bodies in Chap*

VII*

Imagine that two non-elaftic bodies, A and B' %� 2. Plate III. moving, the former in the direction AC, the latter in the direftion BD, do meetnbsp;CD. Let the line MG be drawn through theirnbsp;Centres and the point of contaft. From the ori-S�rgt;al fituations of thofe bodies, viz. from A andnbsp;^gt;drop AM and BN perpendicular on MG. Thennbsp;tile force of each body may be refolved into twonbsp;lorces, viz. that of A into A M, and M C; andnbsp;that of B into BN and ND.

Of its two forces, A retains the force A M, tvhilft the force MC is exerted againft the othernbsp;^eidy. Of the two forces belonging to the body B,nbsp;the force BN is retained by it, w'hilfl the forcenbsp;is exerted againft the other body. Thereforenbsp;the adtion of thofe bodies upon each other is ex-^'^ly the fame as if they moved diredtly one fromnbsp;h'l and the other from N; hence whether theynbsp;''^uld, after the ftroke, proceed both the famenbsp;or different ways, and at what rate, mu ft benbsp;^sterfnined by the rules of diredt impa�l (chapvii.)

�It when their Velocities have been thus deter-l^'ned j for inftance, it be found that if the bodies �ll! moved diredtly from M and N, after the ftrokenbsp;laody A would have moved as far as O, whilftnbsp;lie body B would have moved as far as G, Thennbsp;iscolledted, that, in the prefent cafe ofnbsp;collilion, the body A has retained the forcenbsp;K 2nbsp;nbsp;nbsp;nbsp;AIM1

A M j therefore after the ftroke the body A iS aduated by two forces, viz. one equal and parallel to A M, and another force, which is equal andnbsp;parallel to CO; in confequence of which thisnbsp;body muft run a compound courfe, which isnbsp;found thus: Through the centre C draw C I equalnbsp;and parallel to A M; through I draw IE equal andnbsp;parallel to C O; then the diagonal C E exhibits thenbsp;velocity and the diredion of the body A after thenbsp;oblique concurfe.

With refped to the body B, ic has been faid that at the concurfe this body retains the, force B Nfnbsp;and lhat, if the bodies had moved dircdly towardsnbsp;each other, B would, after the ftroke, have movednbsp;from D to G. Therefore through D draw Dl^nbsp;equal and parallel to BN, and through H draWnbsp;HF equal and parallel to DG; and laftly, the diagonal DF will reprefent the velocity and the direction of the body B after the oblique concurfe.

This is the cafe when the bodies are perfedlf hard or non,-elaftic. But if they be perfedly elaf*nbsp;tic, then fuppofe it be found by the rules for elafti^nbsp;bodies tlrat, after the/uppofed dired concurfe, thenbsp;body A would have been fent back to Q in thenbsp;fapre time that B would have been fent back to F-*nbsp;Then after the oblique ftroke the body A will benbsp;actuated by two forces, viz. one equal and parallelnbsp;to A M, and the other equal and parallel to CQ �nbsp;'And the body B will likewife be aduated by tw^nbsp;forces, viz. one equal and parallel to BN, and the

other equal and parallel to DR. Therefore in fig. 3gt; through Q draw QZ equal and parallel to AM,nbsp;through Z draw IZ equal and parallel to CQ;nbsp;the diagun^J CZ reprefents the direflion andnbsp;''Velocity 5f the elaftic body A after the oblique con-ourfe. '

Again through R draw the line RX equal and P3.rallel to BN, and through X draw the line XYnbsp;^fual and parallel to DR, then the diagonal DXnbsp;quot;'^il reprefcnt the velocity and diredlion of the elaf-body B after the oblique concurfe.

only in this, namely, that the bodies are fup-f�ofed to be perfectly hard in the former, but per-f^ftly elaftic in the latter.

might now proceed to examine the particu-relative to the congrefs of three or more bo-as alfo of bodies of different Ibapes, for hither

^^^der may, by a little exertion of his ingenuity, derive it from what has been already ex-PlS�ined; and fecondly, becaufe the particular ex-^^^ination of all the branches of compound motionnbsp;'Quid fwell the fize of the work far beyond thenbsp;'^^�ts of an elementary book *.

r or further information relative to this fubjecft, the ^�ler may confuk the 2d book of s�Gravefande�s Mat,nbsp;^ etn. of Nat, Phil, edited by Defaguliers,

T TITHERTO we have confidered the com-pound motion which arifes from fimple im' puifes, or fuch as produce equable motion. It wil^nbsp;now be neceffary to apply the above-mentionednbsp;rules to the cafes of that fort of compound motion, which arifes from the joint adlion of a fimplenbsp;and of an accelerative or continuate force ; in whichnbsp;cafe it will be found, that the body will defcribe notnbsp;a ftrai^ht courfe, as when it is impelled by fimplenbsp;impulfes, but it will defcribe curve lines, whichnbsp;differ according as the proportion of the forces dif*nbsp;fers ; excepting however when the forces ad in thonbsp;fame diredion, or diredly oppofite to each other,nbsp;in which two cafes, the motion of th� body w�^nbsp;always be redilinear.

Imagine that the body A, fig. 4, Plate III. impelled from A towards H, with fuch a forcenbsp;by itfelf would enable it to run over the eqoa^nbsp;fpaces AB, BF, FG, amp;c, in equal portions of time!nbsp;for inftance, each of thofe diftances in one minute*nbsp;Imagine likewife that an attradive (confequently-�nbsp;an accelerative) force, continually draws the fapr^nbsp;body A towards the centre. C, in, fuch a manud

by itfelf would enable it to run over the un^ �qual fpaces AT, IK, KL, LM, in equal portions ofnbsp;hnie, viz. a minute each.

Now, the joint aftion of both thofe forces, muft 'Compel the body A to run the compound and cur-'^ilinear courfe A N O P, amp;c.�Through B draw thenbsp;bne B C, that is, in the direftion of the centre ofnbsp;^ttra�iion ;�through I draw IN parallel to A B;nbsp;^od it is evident, from what has been faid above,nbsp;^hat at the end of the firlV minute the body willnbsp;found at N. Now if at this period the attrac-bve force ceafed to aft, the body would run on innbsp;direftion N R, by the firft law of motion,nbsp;fince the atti;aclive force continues to ad, thenbsp;^ody at the end of the fecond minute will be foundnbsp;O; for the like reafon, at the end of the thirdnbsp;tititiute it will be found at P, and fo on. The courfenbsp;*ben ANOP is not ftraight ; but it confifts of thenbsp;titles AN, NO, OP, amp;c. forming certain ^nglesnbsp;'^ith each other.

ff inftead of finding the place of the body at the Cfid of every minute, we had determined its placenbsp;the end of every half minute; then each of thofenbsp;^'ties an, no, amp;c. would have been refolved intonbsp;^''^0 lines containing an ansle. And in the fame

other 5 but fince the attraftive force afts not at intervals, but conftantly and unremittedly; therefore, the real path of the body is a polygonal courfe�nbsp;confifting of an infinite number of fides ; or morenbsp;juftly fpeaking, it is a continuate curve line, whichnbsp;paffes through the points A, N, O, P, amp;c. as isnbsp;fliewn by the dotted line,

The curvature of the path ANOP of a body� which is aded upon at the fame time by an equable�nbsp;and by an accelerative force, varies according to thenbsp;proportion of the two forces. Thus if the equablenbsp;impulfe be increafed or diminifhed, in ffich a manner as by itfelf .vould enable the body to pafs overnbsp;fpaces longer or Ihorter than AB, BF, FG, amp;c. innbsp;the like equal portions of time, as were fuppofetinbsp;above, the attradive force remaining the fameinbsp;then the curvature of the path will be increafed ornbsp;diminiflied accordingly, as is Ihewn in fig. 5 andnbsp;6. P. Ill,

When the two forces are in a certain ratio to each other, then the courfe or path of the body is a cir-cle; in other proportions within a certain liroi^nbsp;the path becomes elliptical, or an oval more or lei�nbsp;extended �, and in other proportions beyond thatnbsp;limit the path becomes an open curve, or fuch a�nbsp;never returns to itfelf. Such curves are callednbsp;r.aholas or hyperbolas, and their properties, as well aSnbsp;thofe of the eilipfis, are deferibed by all the writer�nbsp;on conic fedions,

In fig. 4, the cetitre of attraftion C, has been placed not very far from the diredlion AH of thenbsp;equable force. But when this centre is very farnbsp;it, the right lines CB, CF, CG, amp;c. will become nearly parallel, and in many cafes, they maynbsp;'vithout error, be conlidered as being actually parallel.

A cafe of this fort is reprefented in fig. 7, Plate III . where the centre of attraftion is fo remote fromnbsp;that the right lines BC, FC, GC, amp;c. whichnbsp;proceed from it, are not to be diftinguifhed fromnbsp;Parallel lines. In this cafe, if the fpaces AI, AK, AL,nbsp;be as the fquares of the times, viz. as thenbsp;Iquares of one minute, of two minutes, of three mi-brutes, Sic.; whilfl; the fpaces AB, AF, AG, amp;c. ornbsp;^Iroir .equals IN, KO, LP, amp;c. be fimply as thenbsp;brnes, then the curve or path of the body, ANOP,nbsp;a fort of curve called a �parabola, which is morenbsp;lefs open according as the projedlile or equablenbsp;force is more or lefs powerful. And fuch is thenbsp;path which is deferibed by all bodies that are pro-J^fted obliquely near the furface of the earth, viz.nbsp;Cannon balls, ftones thrown by the hand or othefnbsp;^��gine, and in fhort by all forts of projedtiles; ex-^^pting however that deviation from the parabolicnbsp;Curvature, which is occafioned by the refiftance of

For near the furface of the earth, the fpaces acribed by defeending bodies, are as the fquares

138 nbsp;nbsp;nbsp;Of Curvilinear Motions.

of the times, (according to what has been (hewn in Chap. V.) and the centre of attradion is aboutnbsp;4000 miles below the furface. (i.)

(i.) The admirable dodlrine of Curvilinear motion de-ferves the greateft attention of the philofopher, fince it unfolds the grandeft phenomena of nature. It comprehends almoft all the movements which take place in thenbsp;world. It meafures and afeertains every particular relativenbsp;to the motions of cceleftial bodies,�It leads the human mind,nbsp;through fafe paths, to the inveftigation and knowledge ofnbsp;the moft complicated appearances, and the moll abftrufenbsp;fubjedts. I fhall, therefore, in this place endeavour tonbsp;.explain this, dodirine in as concife and comprehenfive anbsp;manner, as the nature of the fubjedl, and- the limits of thenbsp;work, mayfeem to allow.

A centripetalforce^ in its full meaning, is that whereby a body in motion is continually drawn from its redtilinearnbsp;courfe, towards fome centre. This force may likewife benbsp;the adlion of a firing holding the body ; or it may be it?nbsp;coherence with another revolving body, or it may be thenbsp;gravitating power, he.

� A centrifugal force is the re-a�lion or refiftance, which a moving body exerts to prevent its being turned outnbsp;its way, and whereby it endeavours to continue its motionnbsp;in the fame direclion ; and as re-adlion is always equal anhnbsp;contrary to action, fo is the centrifugal to the centO'nbsp;petal force. 1'he centrifugal force arifes from the inerti*nbsp;�f matter; for the body that moves round a centre, wowldnbsp;fly oft in the direction of the laft moment, or laft particle

0/ Curvilinear Motions. nbsp;nbsp;nbsp;139

The paths of bodies that move round a centre of ^�ttradlion, are poffeffed of feveral remarkable properties ;

its curvilinear courfe, viz. in a tangent to the carve, ^ere the aflion of the centripetal force to be fufpended.nbsp;The equality of the two forces, viz. of the centripetal innbsp;^Ppolltion to the centrifugal force, may be more eafily con-Ceivcd, by imagining that a revolving body is detainednbsp;''''thin its circular orbit by a firing ; for this firing muftnbsp;^tlually endeavour to draw the body towards the centre ofnbsp;^�^traftion, and the centre of attraclion towards the body.

Since the centripetal force is proportionate to the fpace ''hich the body defcribes in a given time by the a�lion ofnbsp;''�at force, it is evident that the centripetal as well as thenbsp;'^^n.trifugal force, may be reprefented by the nafcent linesnbsp;fig. 8. Plate III. for whilft the body defcribes thenbsp;�quot;finitely fmall tangent AB, the fpace which the centripetal force compels it to pafs through in the fame time, isnbsp;^�lquot;al to BC.

B. The lines BC, b r, as well as AB, A are drawn large, merely for the fake of illuflration ; whereas bynbsp;quot;quot;fcent or evanafcent lines, we mean lines of the fame na-,nbsp;'quot;te, but indefinitely fmall, and near the point A.�Thenbsp;tquot;e thing mull be underftood of other lines or quantities,nbsp;quot;'hich are nafcent or evanefcent in the following propofi-I'ons.

^ The right-angled triangles ABE, and ACD, fig. 9, III. are fimilar; therefore, AE : AD :: BE : CD,nbsp;the arch BED, or angle BAD becomes very fmall,

140 nbsp;nbsp;nbsp;Pf Curvilinear Motiom,

perties; that is their periods, their velocities, their diftances from the centre of attradtion, amp;c. follownbsp;certain invariable laws, the knowledge of which isnbsp;exceedingly ufcful in the inveftigation of natural

or fmaller than any given quantity, the point E will approach the point D indefinitely near j fo that the difference between AE and AD will almoft vanifli, and of courfenbsp;the difference between BE (the fine) and CD (the tangent) will likewife nearly vanifli; viz. the fine and thenbsp;tangent will become nearly equal. And fince the chordnbsp;BD, and the arch BED, are each of them longer than thenbsp;fine, and fliorter than the tangent ; therefore in very fmallnbsp;arches, the fine, the tangent, the chord, and the arch iifelf,nbsp;are nearly equal.

Prop. II. In a circle the evanefeent, or infinitely fmall fub' tenfes of the angle of contail-, are as the fquares of the eon-'nbsp;terminal arches.

In fig. 8. Plate III. BC, and be drawn , perpendicular to the tangent A�, are the fubtenfes of the angle of contactnbsp;b Ac, made by the tangent A b, and the circumferencenbsp;ACD ; which fubtenfes muft be imagined to be very nearnbsp;the point A ; in which cafe we fliall prove them to be tonbsp;each other as the fquares of the conterminal arches AC,nbsp;A c.

In confequence of the parallelifm of the lines AD, BC, I c, and of Ab, mC,nc', the line BC is equal to A nhnbsp;and i f is equal to A ;z. (By Eucl. p. 8. B. VI.) AD : ACnbsp;; ; AC : Am ; and AD �. A c A c : An therefore ADnbsp;X A w = ACi^; and AD x An = t\ rlh Hence we havenbsp;Ac]�quot;: Act�' :: Am x AD : An X AD :: A m : Annbsp;BC: be.

�f Cuwilinedr MotmtSi nbsp;nbsp;nbsp;14!

plienomena. Thofe laws will be found mathema-dcally deduced from a few well eftablilbed principles,

Here AC, A f, may be taken for the arches as well as for the chords which fubtend thofe arches ; fliice, by thenbsp;Preceding propofition, they are nearly equal.

Prop. III. In thefirjl or nafcentfate of circular motion, projectile force infinitely exceeds the centripetal force.

In hg. 10. Plate III. the circle ACD reprefents the '^rbit of the body A, moving equably along the faid circumference ; vi/.. the body A is impelled by a projeftilenbsp;^urce, in the diredlion AH perpendicular to AN, and isnbsp;the fame time conftantly a�ted upon by an attradlivenbsp;in the direftion towards the centre N ; thofe twonbsp;forces bei.ng fo adjufted, or being in fuch proportion tonbsp;other, as to keep the body in the circular orbit

^CDA.

In the very fmajl arch AC, the line AB is to the line ^nbsp;nbsp;nbsp;nbsp;1 = BC) as the force of projeaion is to the attradive,

'Or Centripetal, force, at the diftance AN; for w'hilft AB *^^Prefents the equable movement which arifes from thenbsp;force in a certain time, BC reprefents the devia-^�un from that courfe, or the force whereby the body isnbsp;''�'awn towards the centre of attradion in the fame time.nbsp;I^ow, by the preceding propofition BC (~Am): AC::

: AD ; and w hen the arch AC becomes extremely or is in its nafeent Hate, then the diameter A D be-'�Urnes irifinitely greater than AC ; and of courfe AC, ornbsp;^ (which by p, 1. is nearly equal to ir) becomes infinitely

�^z nbsp;nbsp;nbsp;Of Curvilinear Motions^

pies, in the note immediately under this paragraph^ But the principal of them will be proved experi-*nbsp;mentally in the lequel. For the prefent, the

reader

greater than BC,or Km; viz. the projeftile force infinitely greater than the central force. '

In order to compare, and to demonftrate with more expedition, the proportions relative to the velocities, the forces, amp;c. of bodies revolving equably in different circular orbits, as ACD. and ILO, fig. lO, it will be ufeful tonbsp;fubftitute letters inftead of thofe particulars, and whilft thenbsp;capital letters are applied to the body A moving in the circular orbit ACD, the fmalHetters of the fame name will denote the fame things with refpedt to the body I moving irgt;nbsp;the circular orbit ILO. Therefore letnbsp;F,y^ ftand for the central forces.

V, w, for the circular velocities, or for any arches A Cgt; IL ; fince in equable motions the fpaces palled over in ^nbsp;giventime are as the velocities.

The meanings of thofe letters will be eafily remembered� ftnee they are the initials of what they are meant to repr^'nbsp;fent.

Prop. IV. The central forces are as the fquares of velocities directly, and as the diameters inverfely.

d nbsp;nbsp;nbsp;^ a d

Of Curvilinear Motiont, nbsp;nbsp;nbsp;143

deader would do well if he fixed in his mind two of thofe laws, which are as follows, and whofe ufe isnbsp;extenlive.

iff.

Prop. V. In different circular orbits the central forces are the diatneters direSily^ and as the fquares of the periodicalnbsp;times inverfely.

In equable motions the velocity is exprelTed by the quo-^'�nt of the fpace divided by the time j and in circular

quot; -�jr 5 therefore FDlf x T = P; and FDTquot; = ?\ Put periphery of any circle is equal to 3,1416 multipliednbsp;the diameter; therefore P* = 3,1416!^ X ^FDTquot;^;nbsp;^11^_^4i6Px D rc: FT hence we have the force F =

^ for any other circular orbit as I L O ; there-^ore p ; �. ; 3,14161 nbsp;nbsp;nbsp;: 3,1416! ^ d�� D .

this cafe the area is reprefented by VD, which being to the other area vd, we have V :nbsp;nbsp;nbsp;nbsp;: : 4: D, and V*

� � di--, D�. nbsp;nbsp;nbsp;prop. IV.) F ; �;: �nbsp;nbsp;nbsp;nbsp;�; hence

144 nbsp;nbsp;nbsp;Q/quot; Curvilinear Motions '.

ifl. When bodies revolve in equal circles^ the c�ft^ tral fortes are as the fqtiares of the velocities �, or atnbsp;double projeb�iie force balances a duadruple force of

Prop. VII. When the periodical times are equal, the central forces are as the radii, viz. as the diftances from the centre of attradiion, and vice verfa.

radii); the converfe of this propofition is allb evident) viz. that when F : �;: D : tbe periodical times muft benbsp;equal�

Prop. VIII. When the diameters are equal, the central forces are as the fquares of the velocities.

V��

�: �; therefore wheb D d

t) ~ d, this analogy becomes F : �:: V* : e;*; viz* whei' the circles are equal, or in the fame circle, the forces at�nbsp;as the fquares of the velocities*nbsp;nbsp;nbsp;nbsp;�'

Prop. IX. When the diameters, or diftances, and courfe the circles, are equal, the central forces are inverfllnbsp;as the fquares of the periodical times.

Since in that cafe the analogy of Prop. V. (viz. Ft/''

IL : i ]becomes F : �: : 4- : - : 11quot;: T*

'1 * f nbsp;nbsp;nbsp;1 ^

Prop. X. TFhen the diameters are equal, the perio times are inverfeh as the velocities.

^ nbsp;nbsp;nbsp;P ? � �

It appears from Prop. V. that V : w : 1 � : nbsp;nbsp;nbsp;'

the

Of Curvilinear Motions. nbsp;nbsp;nbsp;145

ffntripetal attraSlion^ For inftance, if a body, which b impelled with a certain velocity, and is attrafled

with

brne proportion as their peripheries). Now whenD zr i/jthen fte preceding analogy becomes V : v:: ^ : -L; or T : tnbsp;'��v-.V.

For in that cafe, the analogy of p. IV. (viz. F ; Jy ; ^ j becomes F ^nbsp;nbsp;nbsp;nbsp;; or F : �: : i; D.

Frop. XII. When the velocities are equal, the periodical ^�nies are as the diameters; or as the peripheries, which isnbsp;fame thing.

0 = 4.'

Frop. XIII. Wloen the central forces are equal, the perio-ttmes are as the fquare roots of the diflances, or of the �^��meters.

prop. V. ithasbeenfliewn thatF: �;: � = �. Now 'vhen F =�; then ^ z= or D r* =z 4 T� whichnbsp;^''es the analogy T* : F :: D : 4; and of courfe T: f::

Frop. XIV. When the central forces are equal, the fquares f^he Velocities areas the diflances-, and the periodical timesnbsp;as the velocities.

146 nbsp;nbsp;nbsp;Of Curvilinear Motions,.

with a certain central force, defcribe a circle rouncJ the centre of attraftion; then if the velocity benbsp;doubled, or tripled, the attraftive force mufl; be

. foul'

d '=� v'^ D �, which gives the

: A (the diftances being 2nbsp;nbsp;nbsp;nbsp;2

the halves of the diameters). Alfo by p. XUI. T; f. : D��

� d �, and by the lafl: analogy, V �. v : :T) r : nbsp;nbsp;nbsp;there

fore T: /: : V : �y.

Prop. XV. When the central forces are tnverfeJy as the fquares of the diameters, or of the diftances ; then the fquaresnbsp;of the periodical times are as the cubes of the diftances.

Imagine the central forces to be as fome power, m, of the diftances ; viz. F : �:: D�: d'�. Now by prop, V. F ; �: �

D '1'quot;	d t^	; therefore D� : d�^ ;: ^	d . f- '
and	Dquot; �	d^' or nbsp;nbsp;nbsp;: d	.	: nbsp;nbsp;nbsp;and
	-1 ,		\~m .
	2	2	2	%

Now when the forces are equal then the power, �?, va-nifhes, or w = 0, and then the laft analogy becomes D � � d^::T: t,�which is the fame thing as was fliewn in prop. Xllhnbsp;When the forces are as the diftances, then tn is the firft

7 =:� t, which is the cafe �f prop, VII.�Laftly, when th� forces are inverfely as the fquares of the diftances, then^''*nbsp;= �- 2 and the above-mentioned analogy, becorhes D* �

Of Curvilir.ear Motions. nbsp;nbsp;nbsp;147

four or iiiue times- as ftrong as it was before, in or-to let the body move in the fame circle.

2d. V/lien bodies nipv� hi nusqual circular orbits,

quot;The planets of our folar fyllem follow this grand law of ^3ture. The fquares of their periodical times are - as thenbsp;'^abes of th eic diftances from the common centie of attrac-hon, which is very near the centre of the fun, as will be fhewunbsp;^teafcer ; and thus Newton�s hypothecs of mutual and uni-''^ffal attradlion amongft thebodiesof the uniyerfe is fhewnnbsp;be foconfonant with the ftridteft mathematical reafoning,nbsp;with all the appearances, that none'but the ignorantnbsp;refufe their alTent to it.

Th�s dodtrine of circular movements, which 1 have ex-'oited in 15 proppfitions, might have been condenfed into ^ Narrower compafs, had not my principal objecl been tonbsp;*''^'ider the comprehenfion of it eafy to the reader ; I have

taught by experience,'that in many inftances it is far tt laborious to d�duce every particular cafe from onenbsp;^'^'hprchenfivc propofition, than to read a particular propo-bir every fingle cafe.

in the preceding propofuions, ftated the pro-ions between the forces, the velocities, and the peri-

148 nbsp;nbsp;nbsp;Of Curvilinmr Motions.

Jo that the fquares of the times of their revolutions as the cubes of their difances from the centres of thofnbsp;circles, then the central forces are inverfely as tidnbsp;fquares of the difances j and vice verfa.

attradlion ; for when this quantity becomes knoK'n eithe^ from experiments or by dedudlion from other knovf��nbsp;quantities, we may thereby eafily determine all the oth^^nbsp;particulars.

I. The attradlive force is meafured by the veloci^/ which may be uniformly generated in a given time, wh'^-^nbsp;time we fhall call one, (meaning one fecond, or onenbsp;nute, or, in fhort, the unity of any other divifion of tigt;^�

this force or vel�city by r�, (viz. an indeterminate n of the radius r of the orbit).

' nbsp;nbsp;nbsp;mnbsp;nbsp;nbsp;nbsp;�It

II. It has been fhewn in page 67, that a body, whi^

begins to move from reft and proceeds towards a cen of attraiftion, will at the end of any given time quot;nbsp;quire fuch velocity as would enable it to movenbsp;through twice that fpace in an equal portion of titrCj

Thef�'

Of Curvilinear Motions. nbsp;nbsp;nbsp;149

The foregoing theory of curvilinear motion is extenfive, fince it applies fo a great variety otnbsp;^^rreftrial as well as celeftial phenomena. But in

the quot;'Quid defcend through towards the centre of attra�icn N in

^ ctie produdl of the diameter, or of the radius, multiplied ^ ^be force; and according to the above-mentionednbsp;j. ^^'on,nbsp;nbsp;nbsp;nbsp;the force is exprefled by r�; therefore the

. 3

150 nbsp;nbsp;nbsp;Of Curvilinear Motions. ,

the calculation of the particulars which relate tO thofe phenomena, certain circumftances generally

interfere,

2, nbsp;nbsp;nbsp;,

fcribed with a uniform motion being as the times, it wd be r !L 'll� : arc (cz the whole circumference) :: tnbsp;2 c r-

VII. The fpace through which the body muft defcenlt;^ towards the centre of attradlion, in order to acquire anbsp;locify equal to that with which it revolves, is equalnbsp;half the radius, viz. | r. For in Chap. V. it has beertnbsp;fliev.n, that the fpaces .defcribed by defeending bodies are,nbsp;as the fquares of the times, or of the velocities. It b�^nbsp;alfo been Ihewn (�. II.) that the velocity rquot; is acquired bfnbsp;a defeent through f r�. At prefent we wifti to knov/ hoquot;^nbsp;low a body muft defeend, to acquire a velocity equal

Thus we have exprefTed the meafures of the velocit'^quot;� periodical times, amp;c. in a general yet fimple manner, tnbsp;may be applied to any attradxive power, and to any P^

Of Cdirvilinear Motions.

interfere, which render the relult of the cakula-hons fomevvhat different from the obfervations ; ^hat is, of the experiments. In terreftrial affairs,

the

^nown, beirrg the value of rquot;. But for the fake of illuf-^lation, vi'i fnall now apply it to the force of terreftrial gravity; in which cafe it is known, that a body near thenbsp;^iirface of the earth, will defcend from reft 16,087 feet innbsp;firft fecond of time, gt;( which is the time i); therefore,nbsp;16,087 feet, and 7-� = 32,174- (��nbsp;nbsp;nbsp;nbsp;) Hence by

thofe values for f r� and r� refpe�tively, the �^ove-mentioned meafares will be exprefled in known

�erms.

Example i. The velocity of a body that revolves round the 'arth but near the furface of it, is (by,�. V.) r 'iJli; which�,

�y fubftituting, 32,174 for r�, becomes 32,174^ | j and this '^^'^ot�ies (ftnce the femi-diameter or radius r of the earth isnbsp;^nown to be nearly 210OOOOO feet) 32,174xaioooooolfnbsp;^ ^5993,3 feet per fecond ; fo that a body moving withnbsp;iW Velocity, would revolve continually round the earth ;

velocity being juft fufficient to balance the force of �ravity; but this velocity is about 30 times as great as thenbsp;^�^I'rial velocity of a cannon ball.

B. No notice of the refiftance of the air has been �ri in this example, or will be taken in the followingnbsp;�ii^rnpleg of this note.

he periodical time of the fame body under the fame '^�^'lumftances, is (by �.VI.) 2lt;:rnbsp;nbsp;nbsp;nbsp;-

152 nbsp;nbsp;nbsp;Of Curvilinear Motions.

the refiftance of the air is one of the principal ob-truders. Tlie movements of the coeleftial bodies

arc

Example 2. By prop. IX. when the diftances are equal or in the fame circle, the central forces are inverfely as thenbsp;fquares of the periodical times ; and, by the preceding example, the velocity which near the furface of the earth isnbsp;equivalent to gravity, is = 25993,3 feet per fecond.nbsp;Therefore, we fay as the fquare of the earth�s diurnal rotation round its axis, is to the fquare of the periodical tifflOnbsp;of the body mentioned in the preceding example, (viz. of 1**nbsp;24'47quot;) 5j or nearly 85'); fo is the force of gravity (which W�nbsp;fhall call i)to the centrifugal force of bodies near the equator of the earth ; viz. 2073600' ( = the fquare of 24 hours)nbsp;: 7225' ;; i : 0,003485 the centrifugal force near thenbsp;equator; viz, the force by which bodies that are near thenbsp;equator, are attracted towards the centre, is to the forcenbsp;with which they endeavour to fly oft, in confequence of thenbsp;earth�s diurnal rotation round its axis, as i is to 0,003485�nbsp;or as 1000000 to 3485; viz. the former is almoft 30^nbsp;times more pov.�erful than the latter.

By this means we may determine the centrifugal force of bodies in different latitudes ; for as the earth turns roundnbsp;its axis, it is evident that thofe bodies on the furface of ifgt;nbsp;which lie nearer to the axis, dr, which is the fame thing, af�nbsp;nearer to the poles, perform circles fmaller than thofenbsp;which lie nearer to the equator ; though they are all p^�''nbsp;formed in the fame time, viz. 24 hours. Hence (by prop-VII.) the periodical times being equal, or the fame, thenbsp;central forces are as the radii of the circles, and as in dif'nbsp;ferent latitudes the radii are equal to the cofines of the 1�'nbsp;ti lades, therefore, as the radius is to the cofine of a

Of Curvilinear Motions, nbsp;nbsp;nbsp;i_53

generally influenced by more than one centre of S'ttra�tion, Thus the moon is attraded by thenbsp;^arth and likewife by the fun. The planets are

attraded

btitude, fo is the centrifugal force of bodies fituated at the ^^luator, to the centrifugal force of bodies at that given la-btude. Now as the cofines grow fhorter and Ihorter, thenbsp;�Nearer they come to the poles, fo the tendency of bodies tonbsp;off from the furface of the earth is greateft at the equator,nbsp;it diminifhes as you approach the poles ; and hence wenbsp;^te why the earth has been found by means of undoubtednbsp;�^eafurements and other obfervations, to he an oblate fphe-.nbsp;*^�'d, -whofe polar diameter js the fborteft. And this fur-.nbsp;|'�0ies a ftrong evidence of the earth�s daily rotation about

axis.

'1'ameters of the earth. Alfo the force of gravity at dif-^erent diftances, is inverfely as the fquares of the diftances, ^ud the radius of the earth is 21000000 feet; therefore, as-(quare of 126^200000, is to the fquare of 2100000c,nbsp;b the force of gravity at the furface of the earth, to thenbsp;^^fce of gravity at the diftance of the moon, viz. 160579nbsp;�5^5-0000000000 ; 441000000000000 :i '.0,000274; fonbsp;*^st the force of gravity at the furface of the earth, is to thenbsp;of gravity at the moon as i is to 0,000274 gt; or asnbsp;^000000 to 274. And fince near the earth falling bodiesnbsp;over 16,087 feet in the firft fecond of time ; therefore,nbsp;fay^ j000000 : 274 :: 16,087 : 0,0044 of a foot; whichnbsp;that the moon, ftiould its velocity ceafe at once,nbsp;'^oulo fall towardsnbsp;y^�ouid defceijd

�j4- nbsp;nbsp;nbsp;Of CtirviUmar Motions.

On this account we might now extend our examination to the cafes in which two or three, or

Farther. By prop. XV. when the central forces are in-verfely as the fquares of the diameters, then the cubes of the diftances are as the fquares of the periodical times.nbsp;Therefore the diftance of the body, which circulates nearnbsp;the furface of the earth (Example i.) being one femidiameternbsp;of the earth, and the diftancc of the moon being 6o femi-diameters ; alfo the period of the former being 84',8 wffnbsp;may find the period ofthelatter by faying i�: 60)^;nbsp;tQ the fquare of the moon�s period ; viz. i ; 216000 ;; 719Inbsp;: 1553256000 j the fquare root of which, viz. 39411quot;,^ ornbsp;27 days 8 hours 5i'j3,1s the period of the moon�s revolution round her orbit, whici� is nearly equal to what thenbsp;atrronomers reckon it, viz. 27^ 7quot; 34'; and it Wouldnbsp;have come out exadly like it, had the diftances been ftatednbsp;v.�ith exadtnefs ; and had we likewife taken into the accountnbsp;certain circumftances, which interfere with that period,nbsp;which however we have purpofely avoided in this example�nbsp;for the fake of brevity.nbsp;nbsp;nbsp;nbsp;^

Similar calculations may be inftituted with refpeiSl; to all the planets of our folar fyftem, and the refult of the calculations will be found to coincide wonderfully well- with thenbsp;appearances; which, as we have already remarked, is ^nbsp;ftrong confirmation of the Newtonian theory of univerfalnbsp;gravitation.nbsp;nbsp;nbsp;nbsp;.

Example 4. Let a ball of one pound weight be faftened to a firing 2 feet long, and be whirled about a centre fo asnbsp;to deferibe each revolution in half a fecond. In this ca(e

Of Curvilinear Motions. nbsp;nbsp;nbsp;quot; ^55

fnore centres of attraction aft upon the fame body; but this inveftigation we flrall omit on two ac-lt;^ounts, viz. firft, becaufe the fubjeft is too intricate

and

tiie orbit or circumference of the circle is 4x3)1416 12,5664. The velocity of the ball is 25,1328^ feet per fe-^on-.l. In order to 4ctern'!ine the centrifugal force of thenbsp;^sli thus revolving, viz. the force with which the firing isnbsp;Wretched by it, compared with the force of gravity (which

Jil ( nbsp;nbsp;nbsp;dnbsp;nbsp;nbsp;nbsp;,nbsp;nbsp;nbsp;nbsp;~ '�

16,087x0,25 nbsp;nbsp;nbsp;4,02175

meafure of the central force of the body in queflion; tnis force therefore is to to the force of gravity as 9,8 to i ;nbsp;f* that fince the body weighs one pound when quiefeent,nbsp;''�z. itflretches the firing with the weight of one pound ;nbsp;l^etefore when revolving according to the fuppofition, itnbsp;'''�h ftretch the firing with the force of 9,8 pounds.

FJow this central force may be called centripetal or cen-lUiugal^ according as it is applied to the tenacity of the of the firing, or to dse force of the body; fo that thenbsp;^�dy isnbsp;nbsp;nbsp;nbsp;retained by a centripetal force 9,8 times

great as the force of terreftrial gravity; or it may be faid the centrifugal force of the revolving body flretchesnbsp;^ firing as much as if a weight of g,8 pounds were fimpiynbsp;Upended to it.

� Of

156 nbsp;nbsp;nbsp;Of Curvilinear Motions,

and extenfive ; and fecondly, becaufe in moft natural phenomena, the difturbing caufe which arifes from the. action of a fecond or a third, or in general

Of the motion of bodies about a centre of attraciionf but in curves differing from circles.

It has been fufEciently fliewn that a- certain determinat� velocity is required to confine the movement of a body innbsp;a circular orbit round a centre .of attradlion; whence itnbsp;follows, that v/ith a greater or a leiler velocity bodiesnbsp;will move in curve lines dilFerent from circles. Thofenbsp;curves appear to be the conic fedtions; and fince, ftridtlynbsp;fpeaking, the circle is likewife a conic fedion, therefore itnbsp;may be concluded, that in general the movements of bodiesnbsp;round any centre of attraction are performed in curves ofnbsp;the conic kind, provided the bodies do not meet with anynbsp;pbftrucling medium, or other attradliqn, in their way; fornbsp;under fuch circumftances, their paths may degenerate intonbsp;fpirals, or other'curves of a more intricate nature.

The movements of the coeleftial bodies are not firidfly circular, though they do not deviate much from that figure inbsp;excepting however the comets which move either in verynbsp;eccentric'elipfes, or elfe in parabolas or hyperbolas; andnbsp;therefore in the lafl: two cafes they can never return to th�nbsp;fame parts of the heavens; but they mull continually recedenbsp;from the common centre of attradion, which, in our folafnbsp;fyftem, is not far from the centre of the fun.nbsp;y With refped to the theory of circular movements, I havenbsp;endeavoured to demonftrate the principles, and to iliuftratenbsp;t'ne practical operations in a manner fufficicntly extenfive�nbsp;being perfuaded that if that branch of compound rnotion benbsp;well underftood, the reader (provided he is acquainted with

Cy Curvilinear-Moiims. nbsp;nbsp;nbsp;ijij

�ieral of more than one centre of attia�tion, is not '�ery confiderable; yet in the courie of this work,nbsp;method of taking the above-mentioned cir-

principal properties of the conic fe�iions) will eafily 'Comprehend what follows; I fliall therefore endeavour tonbsp;'xplain the nature of the movements in curves of the conicnbsp;^*nd, in a manner more comprehenfive and concife.

In Sig. l u Plate III. ACD reprefentsa circular orbit, S reprefents an elliptical orbit, ArE a parabolic, andnbsp;'^KF an hyperbolic orbit, of bodies moving with certainnbsp;''elocities under the influence of the centre of attra{2;ionnbsp;which is the centre of the circle, and the focus of thenbsp;^onic fedtions.

Let AB, perpendicular to AD, reprefent the velocity '''Ifich is neceffary to retain the body in the circular orbit,andnbsp;Lt this velocity be called i; for \ve (hall compare the othernbsp;^^grees of velocity with this unity. Alfo let abodybe projedl-from A in the diredfion AI with any other degree of ve-Wity It Js jtotv neceflary to determine the nature of thenbsp;'Curve which will be deferibed with this other velocity k,nbsp;father it is required to afeertain what the value of nnbsp;''rufl: be in order to produce each particular conic feclion.'nbsp;Lraw otK parallel to Al, interfering the circle as wellnbsp;the other curves. Let AN be denoted hy d-, the femi-^fanfverfe axis of any of the conic fedlions, by a; the femi-'conjugate, by Zi; and Am (r:BCx:G%=:Hr=:IIx) by .v.nbsp;^hen the ordinate mC in the circle will be ~�dx�

-, j

^5^ nbsp;nbsp;nbsp;Q/� Curvilinear Motions.

It is however proper to obferve, that the various circumftaiices which obftru6l or influence the

locities in every point of their refpedtive curves in the di-redtion AI j and in the like proportion are the quantities

Now when the point in the curve approaches the point A fo near as to coincide with it, then Am vaniflies, of

parameter, and (fmee the parameter is a third proportional to the tranfverfe and conjugate diam'eters) 2a ; 2lgt;;: 2^ v

Rioveir.ents of bodies, are far from being all known, fully underftood. Eefides, even tliofe that are

aving determined thofe values of the tranverfe and con-\Vg nbsp;nbsp;nbsp;wherein n is the only indeterminate value,

by making certain fubfiitutions inftead of �, afcer-C�f nbsp;nbsp;nbsp;value of � muft be in order to produce one

^y making � zz: i, each of the above values be-^'dUal nbsp;nbsp;nbsp;therefore the two diameters become

t6a nbsp;nbsp;nbsp;Of Curvilinear Motionst.

known, are moftly fludtuating in the intenfity ^ their adlions. Much light has undoubtedly beennbsp;' �nbsp;nbsp;nbsp;nbsp;thrown

in fait the velocity which retains the revolving body in a circular orbit, has been called i, or unity.

pofitive; whereas, by the fame fubllitution, the vain� becomes impoflible; which fliews, that when

Laftly, if we make n equal to any thing greater than tb� fquare root of 2; then the values of a and b for the hypefquot;nbsp;bola become pofitive ; whereas thofe for the ellipfis becotn�nbsp;impoflible j hence in this cafe the curve muft be the hyp�*^'nbsp;bola.

We (hall conclude this fubjedl with the following neral propofition, which, together with its corollaries, is np'nbsp;plicable to a variety of natural phenomena.

In all determinate orbits-, defcribed by bodies revolving ^ certain velocities in non reftfling medinms, about a centrt '�Jnbsp;attraliion, the areas, which are defcribed by a Jlraight

Of Curvilhiedr Motions.

^Wown on this fubjefl by the ingenuity of fcientific perfons during the two laft centuriesgt;yet a great dealnbsp;ftill remains to be done, and a vaft field of fpecu-^nbsp;Nation offers itfelf to the induftry of future philosophers.

^'^nneSiing the centre of attraHion and the revolving body., lie one invariable plane, and are always proportional to thenbsp;^�ftes in uubich they are defcribed.

Imagine the time to be divided into equal particles, and a body moving round the centre of attradlion N, fig.nbsp;Plate III. runs over the fpace A B in the firih particle ofnbsp;^'me. It ig evident that, were the body left to itfelf, itnbsp;^ould proceed ftraight to H, defcribing BH, equal to A B,nbsp;1'^ tlgt;e fecond particle of time ; but at B imagine that thenbsp;receives a Angle inftantaneous impulfe from the centrenbsp;^ttraftion N in the diredlion BN, fufficient to changenbsp;'liredlion from BH to BC. Through H draw CH pa-^^'bl to Bn, which will meet BC in C ; and, agreeably tonbsp;� laws of compound motion, at the end of the fecondnbsp;I'�'tide of time, the body will be found at C in the famenbsp;Pane tvith the triangle ANB. Draw the lines NC,

th Hrice they ftand on the lame bafe NB, and between Same parallels NB, CH (Eucl, p. 37, B. I.) It willnbsp;^ �Wife be equal to the triangle ABN, fince they havenbsp;^4nal bafes and the fame altitude (Eucl. p. i. B. VI.) Bynbsp;th Same mode of reafoning it may be proved, that, ifnbsp;funbsp;nbsp;nbsp;nbsp;force adl upon the body at the end of each

� CD, DE, EF, amp;c. thofe Xpaces will all lie in the � plane ; the triangles ANB, BNC, CND, DNE, amp;c.

Of Curvilinear Motions^

In the prefent ftate of the world the improve'' inents of fcience feldom die with individuals. Thenbsp;accumulation of knowledge by leading the under-,Handing, and by furnifliing tools to the fenfcs, promotes the difcovery of farther truths, and the in-

Avill be all equal, and will be defcribed in equal times. Con-fequently two or three, or any number of them, will be de-fcribed in two or three, or the like number of particles of time, viz. they qre as the times.

Now imagine that thofe triangles are infinitely increafe^l in number, and diminiflied in fize; then the polygonal pathnbsp;ABCDEF, will become a continuate curve; for the con-ftant a�tion of the centre of attraction will be continuallynbsp;. drawing the body away from the direction of the tangent atnbsp;every point of the curve. And it is evident that the fcCto'nbsp;ral areas of the faid curve, or number of infinitely ftnallnbsp;triangles, muft be proportional to the times in which the/nbsp;are defcribed, and that the curve muft lie in one immoveable plain.

Corollary l. The velocities in different parts of the orid are inverfely as the perpendiculars dropped from the centre 4nbsp;attradlion on the tangents to the orbit at thofe parts or poinU'nbsp;Forfince the velocities are as the bafes AB, BC, CD,nbsp;of equal triangles, they muft be inverfely as the heights ofnbsp;thofe triangles, (Fuel. p. 15, E. VI. and p. 38, B.nbsp;which are the fame as the perpendiculars dropped from th�nbsp;centre N, on the tangents to the orbit at thofe points.

Corollary 2. The times in which equal parts., or arches 4 the orbit are defcribed, are diredily as thofe perpendicularsnbsp;the tangents. For when the arches, or bafes of the tfi'nbsp;angles, are equal, the triangles are as their altitudes;nbsp;nbsp;nbsp;nbsp;,

Of Curvilinear Motions. nbsp;nbsp;nbsp;163

quot;exhauftible fund of nature offers on all fides innu-nierahle objecls of inveftigation to the inquifitive

niind.

as the above-mentioned perpendiculars. But they are lilcewifeas the times; therefore, hz.

Corollary 3. If, by dravs^ing lines parallel to the chords AB, BC, of any two contiguous and evanefcent archesnbsp;defcribed in equal times, the parallelogram be completed,nbsp;the diagonal BG, when produced, will pafs through thenbsp;t^entre of attradliori N, which proves the converfe of thenbsp;proportion; viz. that when the areas., which are defcribed bynbsp;�Jiraight line, connedling a moving body and a certain point,nbsp;*�re proportional to the times in which they are defcribed, thennbsp;the body is under the infiuence of a centripetal force tending tonbsp;that point.

Corollary 4. In every point of the orbit the centripetal ��'�� is as the fagitta, or verfed fine, of the indefinitely fmallnbsp;^rch at that point.��The centripetal force at B is as BG, be-t^aufe BG is equal to CH, and CH is the deviation fromnbsp;*he ftraight direcSion AH, which has been occafioned bynbsp;centripetalforce. And the half of BG, viz. B O, isnbsp;the fagitta, or verfed fine, of the indefinitely fmall arch

ABC.

Prop. I. �T7HEN a body is placed upon an in~ � f dined plane, the force of gravitynbsp;�which urges that body downzvards, ads with a pozvernbsp;fo much lefs, than if the body defceiided freely and per-'nbsp;pendicularly downzvards, as the elevation of the plantnbsp;is lefs than its length.

If BD, fig. I, Plate IV. be an horizontal plane, and a body A be laid upon it, this body will re-main motionlefs; for though the powder of gravity,nbsp;or (which is the fame thing) its own weight, drawsnbsp;it towards the centre of the earth, yet the planenbsp;DB fupports it exadly in that direftion j hence nonbsp;motion can arife.

But if the plane be inclined a litfle to the horizon, as in fig. 2, Plate IV. then the body will de-fcend gently towards the lower end D. And if tf� inclination of the plane be increafed, as in fig- 3�nbsp;Plate IV. the body will run down towards D withnbsp;greater quicknefs.

In the two lafl cafes; or, in general, whenever the plane is inclined to the horizon, the adlion ofnbsp;gravity is not entirely but partially counteraffed b/nbsp;the plane. For if, from the centre A of the body�nbsp;in the figures 2 and 3, you draw two lines, viz. AV

perpendicular to the horizon, and AF perpendicular to the plane.; the whole force of gravity, which is reprefented by the line AE, is refolved into twonbsp;forces; viz. AF and EF, whereof AF being perpen-.nbsp;dicular to the plane, is that part of the gravitatingnbsp;power which is counteradfed by the inclined plane;nbsp;Or that part of the w'eight of the body w'hich isnbsp;fupported by the plane BD ; and EF reprefents thenbsp;other part of the gravitating power, which.urges thenbsp;tiody downwards along the furface of the plane.nbsp;Therefore the force of gravity which moves thenbsp;fgt;ody, is diminllhed in the proportion of AE to EF,nbsp;^ut the triangles AFE, EDG, and BDC, are equiangular, and of courfe fimilar (becaufe the angles atnbsp;Tj C, and G are right, and the angle AEF is equalnbsp;the angle DEG, by Eucl. p. 15, B. I.; as alfonbsp;^qual to the .angle DBC, by Eucl. p, 29. B. I.)nbsp;^ricncc we have AE to EF, as DB to BC ; viz. asnbsp;rile length of the plane is to its elevation, or as thenbsp;quot;'hole force of gravity is to that part of it whichnbsp;quot;''ges the body down along the inclined plane¹.

Prop. II. The /pace which is defcribed by a body ^foending freely from reft towards the earth, is to thenbsp;Space which it will dejcribe upon the furface of an in-

If (by trigonometry) DB be made radius, BC be-the line of the angle of inclination BDC ; therefore ^ quot;o^boh force of gravity is faid to be to that part of it uihich

dined plane in the fame time as the length of the plane is to its elevation, or as radius is to the fine of the plane�snbsp;inclination to the horizon.

The force of gravity, which urges a body down along the furface of an inclined plane, is diminilhednbsp;by the partial counteradlion of the inclined plane ;nbsp;but Its nature is not otherwife changed ; viz. it adfsnbsp;conftantly and unremittedly. Hence the velocitynbsp;of the body is continually accelerated, and thenbsp;fpaces it runs over are alfo proportional to thenbsp;fquares of the times; though thofe fpaces will notnbsp;be fo long as if the body defcended freely and.perpendicularly towards the ground.

Now in order to afcertain how much the fpace, which is defcribed by a body running down an in'nbsp;dined plane in a certain time, is (Irorter than the fpacenbsp;through which it would defcend freely and perpendicularly in the fame time, we muft recoiled what hasnbsp;been proved in page 64, relatively to the fpaces,whichnbsp;are defcribed in the fame time, by bodies that arenbsp;aded upon by different central forces; namely�nbsp;that in equal times, the fpaces are as the forces jnbsp;then, fince the whole force of gravity is to thatnbsp;force which draws a body.down the inclined plane�nbsp;as radius is to the fine of the plane�s inclination.nbsp;Therefore the fpace defcribed by a body which dc'nbsp;fcends freely, is to the fpace which a body will de-fcribe on an inclined plane, in the fame time ^nbsp;radius is to the fine of the plane�s inclination, ornbsp;the length of the plane is to its altitude.

Example. Let the length BD of the inclined piane, fig. 2, Plate IV. be 10 feet, and its elevation BC,

4 feet. It is known from experiment, that in the fecond of time, aBody will delcend freely fromnbsp;through 16,087nbsp;nbsp;nbsp;nbsp;Therefore, by the rule

�f three, we fay, as 10 feet are to 4 feet, fo are ^6,087nbsp;nbsp;nbsp;nbsp;a fourth proportional, viz. 10 : 4 : : ,

^hat a body running down the inclined plane BD, ''�ould pafs over little lefs than fix' feet and a half,nbsp;6,43^ feet, in the firft fecond of time.

Prop. III. If upon the elevation BC, fig. 4, Plate of the plane BD, as a diameter, the femicirclenbsp;^EGC be defcribed, the part BE of the inclined plane,nbsp;quot;^hicJi is cut off by the femicircle, is that part of thenbsp;plane over which a body will defcend, in the fame timenbsp;another body zvill defcend freely and perpendicu-^^ly along the diameter of the circle, viz. from B to C,nbsp;is the altitude of the plane, or fine of its inclina-to the horizon.

The triangle BEC is equiangular, and of courfe ^^ilar, to the triangle BDC (for the angle atnbsp;common to both, and the angle BEC isnbsp;foquot;nbsp;nbsp;nbsp;nbsp;P� 3�'- P- ^ angle, and there-

equal to the right angle BCD) hence we BD to BC as BC is to BE. But, by thenbsp;propolition, the fpace defcended freely

the plane is to its elevation, viz. as BD is to BC ; therefore, the fpace run freely and perpendicularly, is to the fpace run over the inclined plane,nbsp;likewife as BC is to BE. And fince BC is thenbsp;fpace freely defcended by a body in a certainnbsp;time, BE muft be the fpace which is run down bynbsp;a body on the inclined plane in the fame time.

Cor. A very ufeful and remarkable confequence is derived from, this propofition, namely, that a hodjnbsp;zvill defend from B over any chord whatfoever citnbsp;BE, or BF,�r BG, of the femicircle BEFC, exaSily igt;^nbsp;the fame time, viz. in the fame time that it would dt'nbsp;fcend freely from B to C. For if you imagine th^nbsp;inclined plane to be BH inftead of BD ; then bynbsp;this propofition, the body will defcend either froiT^nbsp;B to F, or from B to C in the fame time ; andnbsp;again, if you imagine the inclined plane to be B�gt;nbsp;then by this propofition, the body will defcendnbsp;cither from B toG, or from B to C, in the fatn^nbsp;time. And, in iliort, the fame thing may b�nbsp;proved of any other chord of the femicircle.

Prop. IV. Phe time of a body's defending the whole length of an inclined plane, is to the time ofnbsp;its defending freely and perpendicularly along the ciU^'nbsp;tnde of the plane, as the length of the plane is to its ttl'nbsp;titude or as the whole force of gravity is to that pt^^^nbsp;of it which aSs upon the plane.

The fpaces' run over the plane being as fquares of the times, we have the fquare of thenbsp;of pairing over BD, fig. 4, Plate IV. to the fquare

tiine of paffing over BE, as BD is to BE. But �D is to BC as BC is BE, viz. BD, BC, and BEnbsp;three lines in continuate geometrical propor-; therefore (Eucl. p. 20, B. VI.) BD is to BE,nbsp;the fquare of BD is to the fquare of BC. Itnbsp;been (hewn above, that the fquare of the timenbsp;paffing over BD, is to the fquare of the timenbsp;paffiing over BE, as BD to BE j therefore thofenbsp;ffuares of the times are to each other as the fquarenbsp;BD to the fquare of BC ; and of courfe thenbsp;^uare roots of thefe four proportional quantitiesnbsp;^�quot;elikewife proportional (Eucl. p. 22, B. VI.) viz.nbsp;time of a body�s defcending from B to D is tonbsp;time of its defcending freely and perpendicu-iarly from B to Ej or from B to C, as B D is tonbsp;He, or as the length of the plane is to its altitude;

(by the ift propofition of this chapter) as the ''^Hole force of gravity is to that part of it whichnbsp;upon the plane.

Prop. V. A body by defcending from a certain height ^he fame horizontal line, will acquire the fame velocitynbsp;'�whether the defeent be made ferpendicularly, or ob-^H'uely, over an, inclined plane, or over many fiiccejjivenbsp;^^^Ained planes, or laftly over a curve furface.

�ft. In page 64, it has been fhewn, that the �''�docity of a body defcending freely towards a cen-of attradion, is as the produft of the attrac-dve force multiplied by the time �, and by the pre-.^^ding propofition it has been proved, that on annbsp;^^clinetl plane the force of gravity is diminiffied innbsp;Pi'oportion as the time of the body�s running down

the whole length of the plane, is increafed, viz-when the force of gravity is half as ftrong as it would be in free fpace, the time is doubled; andnbsp;when the force is one-third as ftrong, the time isnbsp;trebled, amp;c. therefore theprodud; of the time bynbsp;the force is always the fame; for i- multiplied by '2nbsp;is equal to I- multiplied by 3, is equal to multiplied by 4, amp;c. hence the velocity being as thatnbsp;produft, muft, of courfe, be always the fame, ornbsp;a eonftant quantity. For example, fuppofe, thatnbsp;when the body defcends perpendicularly dowrtnbsp;from B to C, fig. 4, Plate IV. the whole force ofnbsp;gravity ads upon it. Let us call that whole fore�nbsp;1, and let the time employed by the body in com-ingdown from B to C be one minute, then the velocity acquired by that defeent is reprefented bynbsp;the produdof the time by the force, viz. i by I�nbsp;which makes one. Now when the body defcendsnbsp;from the fame altitude B, to the fame horizontalnbsp;line DC, over the inclined plane BD, the force ofnbsp;gravity which draws it dow'tiwards is diminiflied 7nbsp;for inftance, fuppofe it to ad with a quarter of it*nbsp;original power, then the time of the body�s defeend-ing from B to D will be four -minutes, and thenbsp;velocity acquired by that defeent, being as thenbsp;produd of the force by the time, is as the pro-dud of ^ by 4, which is one, or the fame as whennbsp;the body defcends perpendicularly down from ^nbsp;to C.

zdly. Suppofe that the body defcends from the fame altitude E to the fame horizontal line DC�

^0- 5, Plate IV. along the contiguous inclined planes EF, FG, GD; by the time it arrives at Dnbsp;will have acquired the fame velocity as if it hadnbsp;�^sfcended perpendicularly from B to C, or from Enbsp;Perpendicularly down to the horizontal line DC;

by the firft part of this propofition, it will ac-l^ire the fame velocity whether it defcends from ^ to F or from K to F, and by adding to both thenbsp;plane FG, it follows that the body will acquire thenbsp;Pame velocity whether it defcends along the Anglenbsp;plane KG, or along the two contiguous planes EF,nbsp;And by the like reafoning it will be proved,nbsp;the body will acquire the fame velocity whe-it defcends along th� Angle plane BD, or alongnbsp;contiguous two planes KG, GD, or along thenbsp;'�ontiguous three planes EF, FG, GD, amp;c.

3dly. If the number of contiguous planes be P^Ppofed infinite, and their lengths infinitely final},nbsp;will conftitute a curve line, like BH; whencenbsp;follow's, that a body by its defeent along thenbsp;^Urve line BH, or any other curve, will acquirenbsp;fame velocity as if it defeended perpendicularlynbsp;P''otn B to C.

Prop. VI. Let a circle be perpendicular to the ho-and if tzvo chords be drawn from any tzvo Points in the circimferejice, to the point in which thenbsp;�^^rcle touches the horizon; the velocities which are ac-'i^^ired by the defeents of two bodies along thofe

It has been fliewn by the preceding propohtioff� that.a body will acquire the fame velocity whethetnbsp;it defcends from B to D. fig. 6, Plate IV. or fromnbsp;E to D; D being the point of contact rvith the horizontal plane GI; and likewile the fame velocitynbsp;will be acquired by defcending from C to D, o''nbsp;from F to D i fo that the velocities, which are acquired by defcending along thofe chords, are re-fpedtively the fame as the velocities acquired bynbsp;defcending perpendicularly from E and F to P*nbsp;And (from what has been fhewn in p. 65) thof^nbsp;velocities are as the ft]uare roots of ED and FDnbsp;Now (Euch. p. 8. B. VI.) AD is to DB asnbsp;is to EDi therefore (Eucl. p. 20. B. IV.) AOnbsp;is to ED, as the fquare of AD is to the fquarenbsp;DB, and, for the fame rcafons, AD is to FD, as tlmnbsp;fquare of AD is to the fquare of CD. Hence, alternately, AD is to the fquare of AD, as ED is t*?nbsp;the fquare of BD and AD is to the fquare of AD�nbsp;as FD is to the fquare of CD j therefore ED isnbsp;the fquare of BD, as FD is to the fquare of CD inbsp;that is, alternately, ED is to FD as the fquare ofnbsp;BD is to the fquare of CD; and of courfe tb�nbsp;fquare root of ED is to the fquare root of Fl^nbsp;as BD' is to CD, and as the velocity acquired bynbsp;defcending along BD is to the velocity acquired by .nbsp;defcending along CD.

Prop. VIL If there be two -planes of unequal lengt^^^'gt; hit equally indhied to the horizon, the times of defc^�^^

the �whole lengths of ikofe planes will be as ike fpiare roots of their lengths refpeBively.

Let BD and EF, fig. 7, Plate IV. be two planes Unequal lengths, but equally inclined to the ho-''2on ; and it follows from prop. IV. of this chap-that the time of defcent -along the plane BDnbsp;the time of the perpendicular defcent alongnbsp;as BD is to BC; aifo that the time ofnbsp;^^fcent along E F is to the time of defcentnbsp;EC, as E F is to E C. The times of thenbsp;^^''Pendicular defcents along B C and E C arenbsp;Ihe refpeftive fquare roots of B C and ECnbsp;page 65.) Now the triangles BDC and EFCnbsp;equiangular, and therefore fimilar (EucLnbsp;4. B. VI.) we have BC to EC as BD to EF,nbsp;of courfe the fquare root of BC is the fquarenbsp;of EC, as the fquare root of BD is to thenbsp;^Uare root of EF: viz. as the time of defcentnbsp;BD is to the time of defcent along EF.

Lor. The fame thing muft be underftood (as eafily be derived from the above pro-l^^^tion) of two or more contiguous planes fimi-fituated, as BID, EHFj and likewifeofnbsp;^ Curve furfaces that are fimilar and limilarlynbsp;'^^ted; fince thofe curves may be conceived tonbsp;of an infinite number of planes ilmilarly

'lated.

we have f�ppofed the bodies to be fpherical, the planes as well as the bodies to be perfedll/nbsp;fmooth and not obftrudled, either by friction or h/nbsp;the refiftance of the air. We fhall now expla'*'nbsp;the properties of pendulums or pendulous bodied�nbsp;a pendulum being a body hanging at the end of ^nbsp;firing, like A, fig. 8, Plate IV. and moveable abo�*^nbsp;a fixed point of fufpenfion C. A pendulum howeV^'�nbsp;may conhfl of a Angle body fufpended withoutnbsp;firing, fuch as a rod of wood or other matter fufpei'^'nbsp;ed by one end, amp;c. but in the following propo^''nbsp;tions a pendulum muft be underflood to benbsp;cording to the former definition, viz. a bodynbsp;pended at the end of a firing, amp;c. and the flr*'^^nbsp;mufl be fuppofed to be void of weight, as alfonbsp;move with perfe�l freedom about the point ofnbsp;penfion, unlefs the contrary be mentioned.

tance from its natural and perpendicular direSlion,^^ there be let go, it will defcend towards the perptti^^'nbsp;cular, then it will afcend on the oppofite fide neatlynbsp;far from the perpendicular, as the place whence itnbsp;gan to defcend; after which it will again defit^'nbsp;towards the perpendicular, and thus it willnbsp;moving backwards and forwards for a confidetti^^^nbsp;time ; and it would continue to move in that mannednbsp;ever, were it not for the refifiance of the air, andnbsp;friSlion at the point of fufpenfion, zvhich always

Thus the pendulum, fig. 9, Plate IV. being ^^oved from the perpendicular dire�fion CB to thenbsp;fituation AC, and there left to itfelf, will defccndnbsp;^iongthe arch AB with an accelerated motion, in thenbsp;^3-rne manner as if it defeended over a curve furfacenbsp;; for it is evidently the fame thing whether anbsp;Body defeends along fuch a furface, or is confinednbsp;By the firing CB, fo as to deferibe the fame curvenbsp;^B. By the time the body arrives at the loweftnbsp;Point B, it will acquire the fame velocity as if itnbsp;Bad defeended perpendicularly from E to B, ^bynbsp;prop. V.) This velocity (if the retardation arifingnbsp;Bfotn the refinance of the air and the fridlion benbsp;*'etuoved) will carry it beyond the point B with anbsp;^'starded motion in an equal portion of time, as farnbsp;E (fee page 71) viz. as far from B as A is fromnbsp;It will then defeend again with an acceleratednbsp;potion towards B, and fo on. For fince the velo-of the pendulum in its afeent is retarded by the

�^re muft be the fatbe time employed in deftroy-as in generating any momentum.

It likewife followsfrom this confideration, that the ''^^^ght of the pendulum cannot alter its time of

^'^dies of different weights will move through equal in equal times, towards a centre of actraftion.nbsp;Provided the attractive force be the fame. And

The whole motion of the pendulum one way is called a vibration or ofcillation. Thus the motionnbsp;of the pendulum from A to D is one vibration jnbsp;from D to A is another vibration, and fo on. Thenbsp;body which hangs by the firing, is commonly callednbsp;the bob of the pendulum.

This property of the pendulum is fully confirmed by a variety of experiments. A pendulum, if oncenbsp;moved out of its perpendicular fituation, and thennbsp;left to itfelf, will move forwards and backwardsnbsp;a confiderable time (in fotne cafes, for many hours) gt;nbsp;but every vibration will be a little fliorter than thenbsp;preceding, until at laft the pendulum will entirelynbsp;ceafe to move. That this gradual retardationnbsp;entirely owing to the refiftance of ,the air, andnbsp;the friftion at the point of fufpenfion, is provednbsp;by obferving that the fame pendulum has beennbsp;found to continue its vibrations longer and longebnbsp;in proportion as thofe caufes of obftrudtion hav^nbsp;been diminilbed,; hence we conclude, that if thof^nbsp;caufes could be entirely removed, the pendulnn^nbsp;would continue to vibrate for ever.

Prop. IX. T/ie velocity of a pendidum in its lottttfi point is as the chord of the arch which it has defcti^^^^nbsp;in its defcentf

Thus if there be two pendulums of equal length�*� as CF and CA, fig. lo, Plate IV. and the forn^'-��'-of them deicends from F, whilft the latter defcend*

from

*'oni A; then at the loweft point B the velocity the former will be to the velocity of the latter,nbsp;the chord or flraight line FB is to the chord ornbsp;ftraight line AB, or as the velocities acquired bynbsp;perpendicular defcents GB EB; which is annbsp;^^�Ident application of the propofitions V. and VI.nbsp;this Chapter.

Frop. X. The very fmall vibrations of the fame Pendulum are performed in times nearly equal; but thenbsp;quot;^��brations through longer and unequal arches are per-,nbsp;farmed in times ftnftMy different.

It is evident (from cor. to prop. III.) that if the I^*^ndulous bodygt; inftead of vibrating along circularnbsp;^�'ches, could move along the chords of thofe arches,nbsp;femi-vibrationsj whether long or flrort, wouldnbsp;all performed in equal times; viz. each in the tim.enbsp;l^^at a body would employ in defcendingperpendicu-along the diameter of the circle, or twice thenbsp;�^gth of the pendulum. For inftance, in fig. 10,nbsp;l^late IV, the pendulous body would defcend from Fnbsp;^ Bor from A to B along the chords or ftraight linesnbsp;o or aB, exactly in the fame time, viz. the timenbsp;Wouij employ in the perpendicular defcent fromnbsp;^ to B j and fince the defcent from A to B, ornbsp;, ��Qm F .Qnbsp;nbsp;nbsp;nbsp;^ vibration, therefore each

But fince the body vibrates not along the chords hut ninbsp;nbsp;nbsp;nbsp;^nbsp;nbsp;nbsp;nbsp;.

178 nbsp;nbsp;nbsp;Of inclined Plants 1 and

IV.) �, yet in very fmall arches the chords are nearly equal to the arches that are fubtended by them�nbsp;(fee prop. T. of the note in p, 139.) therefore thenbsp;vibrations along very fixiall arches, though of unequal lengths, are performed in times nearly equal-Prop. X I. Js the diameter of a circle is to its ch�'nbsp;citmference,fo is the time of a heavy body's defcentfrontnbsp;reft through half the length of a pendulum to the tint^nbsp;of one of the fmalleft vibrations of that pendulum.

The demonftration of this propofitioh depend* upon certain difficult mathematical principles;nbsp;fhall therefore fubjoin it by way of a note, for thenbsp;information of thofe who are qualified to read'd)nbsp;and fhall now proceed to fliew the ufe of this curion^nbsp;propofition by means of examples (1).

rolled along the faid line, until the fame point A in the cumference, which firft touched the line AL, comesnbsp;in contadl with it in another point L; or till the cifnbsp;AB, by rolling along the line AL, has perfo/med a vfnbsp;revolution ; then the point A will, by its two-fold mod�*�nbsp;defcribe the curve ACDIL, which is called a cycloid,

The circle ABC is called tlie generating circle-, the b(fe, and DF, eredted perpendicularly in the miovi^

6 �

the DoBrint of Pendulums. nbsp;nbsp;nbsp;tyg

This propofition fliews the proportionality of four quantities; viz. the diameter of a circle, itsnbsp;circumference, the time which a heavy body employs in falling from reft through a certain fpace,

the bafe, and extended from the bafe to the curve, is the of the cycloid.�ABC, DGF, HIK, reprefent the ge-tteratinf circle in different fituations.

From this generation of the cycloid, the following particles are obvioufly derived.

I. nbsp;nbsp;nbsp;The bafe AL is equal to the circumference of the gent^nbsp;bating circle.

II. nbsp;nbsp;nbsp;The axis DF is equal to the diameter of the generatingnbsp;circle.

III. nbsp;nbsp;nbsp;The part KL of the bafe, viz. the part between onenbsp;*gt;etremity of it and the place which touches the generatingnbsp;^^rcle in any fttuation of it, is equal to the correfponding arch

IV. nbsp;nbsp;nbsp;FK, or its equal ME, is equal to the remaining archnbsp;in, or GD.

VI. nbsp;nbsp;nbsp;The chord IH, being perpendicular to IK, [for thenbsp;HIK in the femicircle is a right angle) is a tangent to

VII. nbsp;nbsp;nbsp;The tangent IH of the curve at /, or chord of thenbsp;^'rcular arch HI, is equal and parallel to the chord Lgt;G.

e diameter DF of the generating circle ; and any cycloidal *^t'ch ID, cut off by a line IE parallel to the bafe, is equal to

t8o nbsp;nbsp;nbsp;Of inclined Flanel, and

and the time of a fmall ofcillation of a pendulum� whofe length is equal to twice that fpace.

It is very well known that the diameter of a circle is to its circumference, as one is to 3,1415

twice the chord DG of the correfponding circular arch DG^ which is cut off by thefame line IE.

Draw PT indefinitely near and parallel to IE, whicH will cut the circle DGF in Join DQ_produce DG t�nbsp;meet TP in S; from Q_draw QO perpendicular to DS�nbsp;and draw GR, a tangent to the circle at G, and RD a tangent at D. Then, fince PT is indefinitely near to El�nbsp;GS is equal to the increment IT of the curve, whilft GOnbsp;is the increment of the chord DG; for DQ_being nearlynbsp;equal to DO, muft exceed DG by the increment, or additional part GO. And this increment or addition to thenbsp;chord has been made at the fame time that the curve Dlnbsp;has been increafed of the part IT, equal to GS.

Now the triangles DRG, GQS, being fimilar (fince DR is parallel to QS, and the angles at the vertex Gnbsp;equal), and DR being equal to RG, QS mull: be equalnbsp;to QG 5 hence GO is likewife equal to OS, and of courfunbsp;GS is equal to twice GO ; but GS is equal to the inet^'nbsp;ment of the curve, and GO is equal to the contemporaneousnbsp;increment of the chord DG; therefore the increment utnbsp;the curve is equal to twice the increment of the chord*nbsp;And as this reafoning is applicable to any point of the curvunbsp;from D to L, therefore we conclude, that fince fromnbsp;upper point D to the loweft L, the curve increafesnbsp;as fall as the correfponding chord of the circle DGF�nbsp;therefore any arch DTof the curve is equal to twice tl'�nbsp;correfponding chord DG ; and at L where the correfpoodiuS

the DoEtrine of Pendulums. nbsp;nbsp;nbsp;i8i

nearly; therefore if one of the other particulars be Jinown, we may find out the fourth by means ofnbsp;the common rule of three.

Example ift. The time in which a body will defeend from reft through 16,087 viz, (one

lt;:hord is DF, the curve or femicycloid DIL Is equal to tvvice DF, viz. twice the diameter of the generating-circle.

IX. Cyf, CBy fig, 12, Plate IV. rcprefier.t two equal fie-^icycloidal cheeks fet contiguous to each other with their ^ofes CE, CK, in the fame direkfion. BE)A is an invertednbsp;Myeloid equal to the cycloid of which' CA or CB is the half,nbsp;'^nd its bafe reaches from the vertex B of onefemicycloid to thenbsp;Vertex A of the other. At C fufpetid a pendulum CLf whofenbsp;is equal to one of the femicycloids. As this pendulumnbsp;quot;^Ihrates in the plane of the. cycloids,, its firing will apply itfelfnbsp;fifi to one and then to the ether of thofe cheeks, by which meansnbsp;end I of the pendulum will move preclfely in the curvenbsp;viz. in a cycloid.

It is evident from the conftruclion, that BA is the bafe '�f the cycloid BD.A; that BF = FA = CE = CK, ar,dnbsp;~ CLI = CLB = B.ID � twice the diameternbsp;'�tthe generating circle FGD, or EHB.

In any fituation of the pendulum, as CLI draw LH through the point where the eontadl beta'cen the firing ofnbsp;pendulum and the cycloidal cheek terminates, and dra\ynbsp;through the end of the pendulum, both parallel to thenbsp;^ c B A ; and Join the points B, H, G, F, with the linesnbsp;and GF.

Smee CLB is equal to CLI the difengaged part LI of n'g mull be equal to LB, and of courfe equal tonbsp;K 3nbsp;nbsp;nbsp;nbsp;twice

182 nbsp;nbsp;nbsp;Of inclined Planes, and

fecond) being given, to find the time in which a pendulum of twice that length (viz. of 32,174 feet)nbsp;will perform one of its leaft vibrations.

Here we have i : 3,1415 :: iquot;: to a fourth proportional, viz. to 3'^,i'4i5, .which is the tiriic in

twice the chord HB (by � VIII). But BH is equal and parallel to the tangent LM (by �. VII); therefore HB isnbsp;equal to ML, and confequently LM is equal to MI; hencenbsp;the parallels HL, IG are equidiftant from the bafe BAgt;nbsp;and cut off equal arches HZB, FSG, from the generatingnbsp;femicircles; therefore the chord FG is equal and parallelnbsp;to the chord HB, and to MI. Alfo MF is equal to IG�nbsp;thofe lines being the oppofite fides of a parallelogram-Now as BM is equal to HL, and (by � IV.) equal to thenbsp;arch HZB, or to the arch FSG ; the remainder MF, equalnbsp;to IG, will be equal to the remaining part GD of the ferni'nbsp;circle; which proves that the extremity I of the pendulunanbsp;is always in the cycloidal curve ADB.

For the fake of brevity we ftiail call the pendulum whicb vibrates in a cycloid) a cycloidal pendulum.

X. The velocity of a cycloidal pendulum in its Itvjef pd^^ is proportional to the fpace pafed through i vi%. to the m'dnbsp;of the cycloid which the pendulum has deferihed in its defeent.

, Thus in hg. 12, Plate IV. If the pendulum begin defeend from I; at D, its velocity will be as the arch 11^ �nbsp;and if it begin to defeend from B, then when it arrives *nbsp;the loweft point D, its velocity will be as the arch BlP�nbsp;which we are now going to prove.

the DoSlrine of Pendulums, nbsp;nbsp;nbsp;183

'''liich the pendulum of 32,174 feet will perform ^ach of its very fmall vibrations; viz. little morenbsp;^iian three feconds.

Example 2. The time in which a body will de-E:end from refl through 16,087 feet (viz. one fe-^ond) being given, to find the time in which a pendulum of four feet will perform one of its leafk '�brations.

Here ^J'om I to D, or perpendicularly from Y to D. Alfonbsp;fquare of the velocity of a falling body is as the fpacenbsp;Paffed through, or the velocity is as the Iquare rootnbsp;the fpace; therefore the velocity acquired by the pendu-I, in its defcent from I to D, is as the fquare root ofnbsp;viz. as Vm But (Eucl. p. 8. B. VI.) DY : DG

an invariable quantity, DY muft increafe or decreafc ^t^cording as the fquare of DG increafes or decreafes ; ornbsp;fquare root of DY (viz. the velocity in queftion) is asnbsp;^5 which is equal to half the cycloidal arch DI; hencenbsp;Velocity is as the cycloidal arch.

^ -dll the vibrations of a cycloidal pendtdum, whether t:rjhart^ are performed in equal times,

H all forts of motion, as we have abundantly fhewn, the is as the produft of the time multiplied by the velo-; viz. S is as TV, which gives the following analogygt;

^3fc of a cycloidal vibration, the fpace is as the velocity; therefore the time muft be as unity, or always the fame.

184 nbsp;nbsp;nbsp;Of inclined Planes, and

Here half the length of the pendulum is 2 feet 5 therefore in the firfl: place we mull find out whatnbsp;time a body will employ in defcending from reftnbsp;through 2 feet; and fince the fpaces pafled overnbsp;by defcending bodies, are as the fquares of thenbsp;times, (fee page 65) we fay as 16,087 feet are tonbsp;two feet, fo is the fquare of one fecond to thenbsp;fquare of the time fought; viz. 16,087 '� sl �

XII. If a cycloidal pendulum begin to defend from en) point L, fg, 13, Plate IF. to'Jjarch the vertex P';nbsp;velocity at any point M (viz. the velocity acquired 1'Jnbsp;defending from L to Ad) will be as the jquare root ynbsp;the difference of the fquares of the two arches f L, an-

FM{^ Vl%lt;. as TTf �. FM

fine of a circular arch whefe radius is equal to FL, and ugt;lgt;f coftne is equal to FM.

Through the points L and M draw LR, MS, parallel the bafe AB, which lines will cut the generatirig circlenbsp;O and Q. And draw the chords VO,

The velocity of the pendulum at the point M, after a ds^ feent from L, is equal to the velocity that would be acqW'^nbsp;by a body defcending perpendicularly from R to S (bynbsp;V. of this chap.) ; and this velocity is as the fquare root ^

the fpace RS ; or as RV�SV\�; or as nbsp;nbsp;nbsp;'

the

the Do5lrine of Tendulums. nbsp;nbsp;nbsp;185

2 feet. This time being found, we then fay, after manner of the preceding example; i : 3,1415

Example 3. The time in which a pendulum 39,1196 inches performs each of its fmall vibra-^'Qns (viz. one fecond) being given, to find thenbsp;^Pace through which a body will defcend from reftnbsp;the fame time.

Produce the axis DV towards Z; at V erciS VL perpendicular to DZ, and equal to the length of the cycloidal ^rch VML. Let the lengths VM, VL, in the ftraight linenbsp;^L, be made refpedlively equal to the lengths VM, VL, ofnbsp;cycloidal arch. With the centre V and radius VL,nbsp;'iraw the femicircle LZP. At M on the radius eredl MXnbsp;Perpendicular to it, which will meet the circumferencenbsp;X, and Lilly join VX.

quot;Phen MX is the fine of a circular arch, whofe radius VX or VL, which is equal to the cycloidal arch VL,nbsp;�nd whofe cofine is VM, which is equal to the cycloidalnbsp;V M. (By Eucl. p. 47. B. I.) MX is ecjual to

XlII. If zuhen the pendulum begins to defend from L ^long ijjg cycloid., another body be fuppofd to move in thenbsp;finicircle LZP from L tovjards Z %vith a uniform velocity,nbsp;^fot. to the pendnlunds greatef velocity ; [viz. that whichnbsp;icndulum acquirei by clef ending from L to the vertex V\)

186 nbsp;nbsp;nbsp;Of inclined Planes, and

time in which a body will defeend through a fpace equal to half the length of the pendulum, viz*nbsp;through 19,5598 inches.

Then,

then any circular arch XYlt;iuill he deferibed by the above-mentioned body with that uniform velocity, in the fame time that the cycloidal arch which is intercepted between the two corref-ponding points M and N, is run over by the pendulum wit^nbsp;its ufual accelerated velocity,

, Draw the line mx parallel, and indefinitely near, to th* fine MX. Through X draw X r parallel to the radiosnbsp;VL; and in the cycloidal arch take equal to M wnbsp;' the radius.

The arch X x, being indefinitely fmall, may be confidere^^ as a right line. Then the right angled triangles V Mnbsp;X A� r, being fimilar, (becaufe the angles r X a and M X ^nbsp;are equal, for each of them is the complement of V X fnbsp;a right angle), we have M X ; VX (or VZ, or VL) �'nbsp;X r (or M �2) : X x.

Now the velocity of the pendulum at,M (by � XH* this note) is as MX ; therefore the extremely finallnbsp;M OT in the arch, may without error be fuppofed to benbsp;feribed with that velocity. Aifo (by � X. of this nots)nbsp;the greateft velocity acquired by the pendulum in itsnbsp;feent from L to V, is as the arch LV, or as its equal, tb�nbsp;radius LV, and is the fame velocity with which the circul��-arch is equably deferibed ; therefore, the analogy of the pt^nbsp;ceding paragraph is, by fubftitution, converted into thenbsp;lowing : viz. as the velocity with which the circularnbsp;X X is deferibed, is to the velocity with which the f^�^nbsp;�cycloidal arch M m is deferibed by the pendulum, ^

the DoSlrine of Pendubims. nbsp;nbsp;nbsp;187

Then, fince the fpaces defcribed by defcending bodies, are as the fquares of the times, we fay, asnbsp;fquare of o^^,3183 is to the fquare of one fecond,nbsp;are 19,3598 inches to the fpace through which anbsp;^ody will defcend in one fecond ; viz. 0,1013'! 48 9nbsp;� I :: 19,5598 : 193,06 inches, or 16,083 feet; thenbsp;[pace through which a body will defcend from reftnbsp;One fecond.

Mw j fo that thofe fmall lines are as the velocities with '''hich they are defcribed. But when the fpaces are as thenbsp;'�elocines, the times muft be equal; therefore, the circularnbsp;X A- is defcribed in the fame time that the correfpond-''�S Cycloidal arch M ?n is defcribed by the pendulum. Nownbsp;fte fame thing may be faid of all other correfpondingnbsp;betw� een X and Y, and M and N ; therefore the wholenbsp;^*fcular arch XY is defcribed in the fame time in whichnbsp;Correfponding cycloidal arch MN is defcribed. Hencenbsp;whole cycloidal arch LV, and quadrant LZ, are de-^'^dbed in the fame time.

^IV'. The t'mie of a compleie ofcillatmi of a cycloidal is io the time in which a body would defcend per^nbsp;along the axis of the fame cycloid, as the circum~nbsp;'�quot;^Hce of a circle is to its diameter.

^ the firft place, it is evident that the time in which the ^Caicircig LZP is defcribed in the manner mentioned above,nbsp;the time in which the radius LV could be defcribednbsp;the fame equable velocity, as the circumference of anbsp;'tele is to its diameter. But the time in which the femi-t'^lcLZP is defcribed, is equal to the time in which thenbsp;^ 'tdulmji will make a complete cycloidal ofeiilation from L

188 nbsp;nbsp;nbsp;Of inclined Planes, and

In the preceding examples the calculations hav'C not been carried on to a great number of decitnalsnbsp;purpofely to avoid prolixity; the objed being onl/nbsp;to thew the' method of performing the calculations�nbsp;but in many cafes it will be necelTary to extend th^nbsp;operation to a greater degree of accuracy. Itnbsp;likewife neceffary that the reader be informednbsp;the real length of the, pendulum, which vibrates

to P. And the time in which LV (or its equal twice O^) could be defcribid with that fame velocity with whichnbsp;circle is dcfcribcd, is equal to the time of defcentnbsp;the chord OV, or along the axis DV (fee chap. V. 3�^nbsp;prop. VI. of this chap.); therefore, the above-mentionednbsp;analogy is, by fubftitution, converted into the followib�'nbsp;The time of a complete cycloidal ofcillation, is, to the tit�*�nbsp;in which a body would defeend perpendicularly alongnbsp;axis of the fame cycloid, as the circumference of a circle **nbsp;to its diameter.

This propofition evidently confirms prop, the nth oftl�** note; for fince every cycloidal vibration is in the fame rat'*�nbsp;to the time of defcent through the axis, as the invari^f^^nbsp;ratio of the circumference of a circle to its diameter,nbsp;muft be all performed in the fame time.

exceed two or three degrees, the curvature of the -cy coincides with the curvature of a circ !e whofe radiit^nbsp;equal to CV. viz. the length of the pendulum.

^^conds, or of the real fpace which is paffed over V defeending bodies in a given time ; (iince dienbsp;may be eafily deduced from the other) innbsp;^rder that he may ground his calculations on asnbsp;^^curatea foundation, as the prefent date ot know-can admit of.

�n different parts of the world, the pendulum vibrates feconds, is not of the fame length.

^'��ently (hewn by the figure itfelf; for when the pendulum ^'i�tates not far from the perpendicular CV, its firing does

^tdly touch the cycloidal cheeks CA, CB ; and of courfe Us �nbsp;nbsp;nbsp;nbsp;^

�Jluare roots of the lengths of the refpeSiive pendulums di-and as the fquare roots of the refpeiiive gravitating inverfely.

the pendulum), and inverfely as the Iquare root of the of Gravity; for when the gravitating force is inva-^'�ble, tire time of perpendicular defeent has been (hewn tonbsp;^ the fquare root of the fpace; and when the time is in-(viz. in the fame time) the fquare root of the fpacenbsp;been (hewn to be as the fquare root of the velocity, ornbsp;^the gravitating force; therefore when they are both va-the fquare root of the fpace or length is as the time

multiplied

^ % the preceding propofition the time of a cycloidal vi-'^^tion is to the time of perpendicular defeent along the gt; in an invariable ratio; that is, the former is as the lat-� Now the time of that perpendicular defeent is directlynbsp;the fquare root of the axis for of its double, viz. the lengthnbsp;quot;quot;fthenbsp;nbsp;nbsp;nbsp;^

3 go nbsp;nbsp;nbsp;O/ inclined Planes, and

It is a little longer on places that are fituate*^ nearer to the poles, and fhorter in fituations th^*-are nearer to the equator, (the reafon of whichnbsp;be (hewn hereafter). This difference, however,nbsp;known with fufficient accuracy. But moft ph�^'nbsp;fophical writers differ with i'efpe6t to the lengthnbsp;the pendulum which vibrates feconds innbsp;fame latitude; and of courfe with refpeft to

multiplied by the fquare root of the gravitating force j the time is as the fquare root of the fpace divided bynbsp;fquare root of the gravitating force; that is, as the fqu^^nbsp;root of the length of the pendulum direiSlly, and the fqii�^*nbsp;root of the force of gravity inverfely.

Independently of the cycloid, the time of any circt^l^^ ofcillation may be found out by means of the fo!loquot;��'�nbsp;problem, which is given by ProfelTor Saunderfon in his IVl� �nbsp;thod of Fluxions.

fnd, vjithout any fenfble error, alfa the time of any afillaiicn.nbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;^

Let a pendulum ND, fig. 14, Plate IV. vibrate in the ADC of a circle whofe diameter is ID; and fuppofe itnbsp;at the point E in its afeent from D to C. Let

prefs the velocity acquired by a heavy body in falling B to F, (AC,EE, being the parallel chords of the ar*''nbsp;ADC, EDE, which interfedl the diameter in B and F):nbsp;confequently the velocity of the pendulum at the pot*nbsp;Now I v^ID exprefi�es the velocity acquired bynbsp;ing through s and fmee a body with that

the DoBrine of Pendulums. nbsp;nbsp;nbsp;191

fpace which is paffed over by defcending bodies m a given time.�In the w^orks of the mod eminentnbsp;Philofophers of this country, I find the length ofnbsp;pendulum, which vibrates feconds in or nearnbsp;London, dated differently as follows: inches 39,2 ;nbsp;39,14; 39,128 ; 39,1^5;

The ^ould defcribe uniformly a fpace equal to f ID in thenbsp;time in which it would fall through | ID. Dividenbsp;fpace f ID by the velocity | V'lD, and the quotientnbsp;^nr, exprelTes the time wherein a heavy body would fallnbsp;through a I ID; viz. through half the length of the pen-

Draw ee indefinitely near to EE; then Er may be con-^'dered as the fluxion of the arch DE ; and nbsp;nbsp;nbsp;will ex-

-/bf

Prefs the time wherein the fmall arch Ee is defcribed by the Pendulum, or the fluxion of the time of a vibration. But,

'^ifxfd ) V if

192 nbsp;nbsp;nbsp;Of inclined Planes, and

The late Mr. John Whitehurft, an ingenious member of the Royal Society, feems to have con-,

trived

in K, and KD in L; and when the arch ADC is fmallj the quantity I F cannot diiFer fenfibly from IK, nor �^-5

y iF

X \/TD X

v/HF X FD

Upon the diameter BD defcribe the circle BGDG, cutting the chords EE, ee in G and g ; then will the fluxion oi

.; and therefore the fluxion of the time of vi' v'FB'xFD

bration through DE will be _^ x V^I D

t/BF I K

; which in fail is the time of the pendulum�s inovi*�� BD

from E to e.

^ ^ X VTD ^

J f ^ �

ihe quot;DoEtrine of Pendulums. nbsp;nbsp;nbsp;193

drived and performed the leaft exceptionable experiments relatively to this fubje�t. The refult of

When the arch of vibration is indefinitely final], the Quantity becomes i j and the time (T) of one of

. BGDGB.

j/iU X __ . T ; that is, as the

^�id therefore BD ; BGDGB :: v^lD diameter of a circle Is to its circumference, fo is the timenbsp;('^ID) of the defeent through half the length of the pendulum, to the time of one of the leaft ofcillations of thenbsp;Pendulum: which is the fame analogy as was derivednbsp;��oni the properties of the cycloid. Wherefore the time ofnbsp;^idillation in a cycloid, and in an indefinitely finall arch ofnbsp;^ circle, is the fame, viz. T r: i fecond, when the lengthnbsp;the pendulum is 39,r 196 inches; as has been proved ex-**^-imentally.

IK nbsp;nbsp;nbsp;IK

In order to adapt the preceding expreffions to the prac-^ � tralculation, it is neceftary to obferve that BD is the thnbsp;nbsp;nbsp;nbsp;CD, viz. of half the arch of vibration. DK is

^Ifof thatverfed fine j KL is a quarter of it j and ID is onbsp;nbsp;nbsp;nbsp;twice

194 nbsp;nbsp;nbsp;Of inclined Tlanes, arid

his experiments fliews, that the length of the pen-dulum which vibrates feconds in London, at

fwice the length of the pendulum ; hence if we call the verfed fine of half the arch of vibration a, and call th�nbsp;length of the pendulum b ; then the above ftated expreffioa

I K nbsp;nbsp;nbsp;�b-�{a

2)b � ia

Example i. Suppofe it be required to find the time ft* which a pendulum, that performs each of itsquot; fmalleft vibra'nbsp;tions in one fecond, will perform its vibrations in an archnbsp;of I ao�-.

In this cafe the length of the, pendulum is^ 39,119^ ^ the femiarch of vibration is 60�; and its verfed fine (whichnbsp;is taken from the trigonometrical tables, and is reduced 1�nbsp;the proportion of the tabular radius to the length of the pcf

the pendulum, fo is the tabular verfed fine to the verfed fif�� in queftion) is 19,5598 = a ; therefore the time fought

then if the number of feconds in 24 hours, viz, 86400'^ divided by the time of one vibration lafl: found; vi^'nbsp;1'''',07J4, the quotient 80735 is the number of vibratio'^*nbsp;which the pendulum will perform in 24 hours, when itnbsp;brates along the arch of 120�; whereas when the fame p��nbsp;dulum performs very fmal! vibrations, it will vibrate exa �'nbsp;feconds, viz. it will perform 86400 vibrations in 24 hout��nbsp;Example 2. Suppofe it be required to find the time of ft*nbsp;vibration, when the above-mentioned pendulum vibt^'quot;

X nbsp;nbsp;nbsp;throlt;

the DoBrine of Pendulums. nbsp;nbsp;nbsp;195

feet above the level of the fea, in the temperature of 60� of Fahrenheit�s thermometer, and when the

time of a fmall vibration of the pendulum, whofe length is 39,t Ig6 inches, is to the time of one of its vibrations alongnbsp;^ femicircle as I is to 1,16666, which is nearly in thenbsp;proportion of 6 to 7.

196 nbsp;nbsp;nbsp;OJ inclined Flanes, and

barometer is at 30 inches, is 39,1196 inches; whence it follows that the fpacc which is pafled over bf

bodies

r. If a certain line, as ACDB, be the line of fwifteft dc' fcent between two points A and B ; it follows that a body�nbsp;after its defcent from A as far as C, will continue to de-fcend quicker along the fame line from C to D, than alongnbsp;any other line, as CED; for if this be denied, then it muftnbsp;be admitted that the body will defcend falter along the linenbsp;ACEDB, than along the line ACFDB; confequently thenbsp;line ACFigt;B is not the line of fwifteft defcent, which i�nbsp;contrary to the hypothefis,

2. Let ADGB, fig. 16, Plate IV. be a curve between the two given points A and B ; let DE, EG, be two indefinite-1V fmall and contiguous portions of it. Through the pointsnbsp;D, E, and G,'draw DL, EO, GP, perpendicular to thenbsp;bafe AC; and through D draw DH parallel to the bafe-Now if this curve be fuch that the velocity with which thenbsp;indefinitely fmall portion D E is palfed over by a bod/nbsp;aft�r its defcent from A to D, be always proportional te

then the body after its defcent from A to D, will defcel^^^ along the curve from D to G in lefs time than along- a*�)nbsp;other way DFG; and of courfe this curve will be thenbsp;of fwifteft defcent.

Through F draw FQ_parallel to EG, and let FQJae ft'P' pofed to be pafled over with the fame velocity as EG r dra''^nbsp;FN perpendicular to DE, as alfo ME and GQ, perpend''nbsp;cular to FQ, then the triangle FNE being fimilar toPE^� �nbsp;as alfo FME fimilar to GEI, we have DE : DH : : EE '

jdeiic�

the DoElrine of Pendulums.

bodies defcending perpendicularly, in the firft fe-*^ond of time, is 16,087 feet.�This length of a

fecond

''docity with which NE is pafled over to the velocity with '''hich F'M is pafled over: whence NE, FM, are pallednbsp;�^''er in equal times. And fince MQ_ is equal to EG, thenbsp;^'aieof defcent through M Q_wil! be equal to the time ofnbsp;�^^(cent through EG; fo that the time of defcent throughnbsp;^Q.will be equal to the time of defcent through NEG.nbsp;fince the angle MQG is a right one, FG is greater thannbsp;fo that the time through FG will be greater than thenbsp;^'rtie through FQ_, or through NEGj and fince DF isnbsp;^�^^ater than DN, thetime through DF will be greater thannbsp;^ time through DN. Whence the time of defcent alonenbsp;FG, will be greater than the time of'defcent alongnbsp;NG. a heavy body therefore, after its fall from A�nbsp;1*^defeend from D to G along the curve DEG, innbsp;^ time than along any other line; confequently the curve

^ 3' Let ADEM, fig. 17, Plate IV. be a cycloid whofe is the horizontal line AG. Through any point D innbsp;^taw PQ_ parallel to the bafe AG, and cutting the ge-^tating circle at N and the axis at Draw the chordsnbsp;gt; NM ; through D draw DL perpendicular to the bafe;nbsp;ti^ E)E indefinitely near and parallel to LD. Nownbsp;^ ^ 'definitely fmall part DE of the curve may be confi-d as a right line coinciding with the tangent at D, andnbsp;03nbsp;nbsp;nbsp;nbsp;it

198 nbsp;nbsp;nbsp;Of inclined Tlanes, and

fecond pendulum is certainly not mathematically exaft, yet it may be confidered as fuch for all com'nbsp;monpurpofes; for it is not likely to difier from thenbsp;truth by more than x-s^s-s-th P^^t of an inch.*

XII.

it may likewife be fuppofed to be defcribed by a body dc' fcending from A, with the fame velocity wnich the body ha?nbsp;acquired by its defcent from A to D ; for the accelerationnbsp;of velocity through that indefinitely fmall fpace may benbsp;confidered as next to nothing. Now we fhall prove tbafnbsp;this cycloid has the property of the above-mentioned curvO'nbsp;viz. that the velocity with which the fmall portion DEnbsp;defcribed by a body falling from A, i? always proportional

From the above-mentioned properties of the cycloid, frriall line DE, coinciding with the tangent at D, is paraU^'nbsp;to the chord NM. Whence the triangles DHE, NQ^^�nbsp;and GMN, are equiangular and of courfe fimilar; ther*^'

is as the velocity which is acquired by the heavy body in defeent from G to Q_, or from L to D ; viz. as the veloc'^lnbsp;with which the indefinitely fmall line DE is paffed ovefgt;nbsp;therefore the cycloid, having the property of the above-incnnbsp;tioned curve, is the line of fwifteft defeent, drc.

: * See Mr. Whitehurft�s attempt towards obtaining riable meafures of length, capacity, and weight. Alfnnbsp;George Shuckburg Evelyrr�s excellent paper on thenbsp;ard of weigh t and meafure, in the Philofophical Tranfa*^*�*^ �nbsp;fpr the year 1798.

XII. the times in which fmilar vibrations {viz. quot;'Vibrations through arches of the fame number of degrees)nbsp;^f different pendulums are performed, are as the fquarenbsp;^oots of the lengths of the pendulums.

Tims if the pendulum AB, fig. 18, Plate IV. four times as long as the pendulum CD, thennbsp;time of a vibration of the former will be doublenbsp;time of a fimilar vibration of the latter. Fornbsp;(by cor. to prop. VII. of this chap.) the vibrations,nbsp;^Rd of courfe the femivibrations, being fimilar andnbsp;^'tnilarly fituated, the time of the pendulum�s de-^*^ent along the arch GB is to the time of the othernbsp;Pendulum�s defcent along the arch HD, as thenbsp;%uare root of GB is to the fquare root of HD.nbsp;^nt the circumferences of circles, or fimilar portionsnbsp;die circumferences, are as their radii; thereforenbsp;fquare roots of fimilar portions of the circum-^erences are as the fquare roots of the radii; continently the times of fimilar vibrations are as thenbsp;inare roots of th,e radii, or of the lengths of the

Pendulums.

quot;throughout the prefent chapter the force of �'avity has been fuppofed invariable; but whennbsp;^ IS not the cafe, as for inftance, when a pendu-vvhich vibrates near the furface of the earth, isnbsp;^^nrpared with a pendulum on the top of a verynbsp;� 1 naountain, or with a pendulum which vibratesnbsp;^ 3^n inclined plane; in which cafes the adion of thenbsp;b^9.vitating force on the pendulum�s is not the fame,nbsp;the time of vibration is as the quotient of thenbsp;o ^nbsp;nbsp;nbsp;nbsp;nar

fquare root of the length of the -pendulum divided hy the fquare root of the gravitating force.

The attentive reader muft undoubtedly ha''� remarked, that though in the precedingnbsp;chapter much has been faid with refpedt to th�nbsp;length of the pendulum, yet no mention hasnbsp;made of the point from which that length, or di^quot;'nbsp;tance from the point of fufpenfion, fhould benbsp;fured. The reafon of this omiffion is, that thenbsp;termination of that point, which is called thenbsp;lt;f ofcillation, requires a very particular confideration*nbsp;fuch indeed as could not without obfcurity benbsp;troduced in the preceding chapter. We fliallnbsp;endeavour to elucidate the nature of thatnbsp;and to lay down the methods of detennininS \nbsp;fituation or diftance from the point of fufpenfionnbsp;pendulums of different lengths and lhapes.

When the pendulum confifts of a fpherical body fattened to a ftring, a perfon unacquainted with thenbsp;hibjedl might at firfh fight imagine that the lengthnbsp;the pendulum muft be eftimated from the pointnbsp;�^ffufpenfion to the centre of the ball. But this isnbsp;tiot the cafe; for in fact the real length of the pendulum is greater than that diftance, the reafon ofnbsp;t'^hich is, that the fpherical body does not move iiinbsp;^ ftraight line, but it moves in a circular arch; in ^nbsp;^onfequence of which, that half of it which is fartheftnbsp;^�'otn the point of fufpenfion, runs through a longernbsp;^Pace than the other half which is nearer to thenbsp;point of fufpenfion; hence the two halves of thenbsp;^^11, though containing equal quantities of matter,nbsp;do adlually move with different velocities, thereforenbsp;^beir momentums are not equal; and it is in con-^^ffuence of this inequality that the centre of ofcil-^^don does not lie between the two hemilpheres jnbsp;^^at is, in the centre of the ball; but it lies withinnbsp;lower hemifphere, viz. that which has the greaternbsp;^Omentum. Now from this it naturally follows,nbsp;d^at if the ball of the pendulum could be concen-dated in one point, that point would be the centrenbsp;ofcillation; fo that the centre of ofcillation isnbsp;^^at point wherein all the matter (and of courfenbsp;forces of all the particles) of the body or bodiesnbsp;^^at may be joined together to form a pendulum,nbsp;*^ay be conceived to be condenfed.

The (entre of perctiffton is that part or point of a Pendulous body, which will make the greateft im-

preffion on an obftade that may be ojipofed to whilft vibrating; for if the obftacle be oppofed to' itnbsp;at different diftances from the point of �lufj)enfion�nbsp;the ftroke,or percuffion, will not be equally powerful�nbsp;and it will foon appear that thisoentre ofpercuHiunnbsp;does not coincide with the centre of gravity.

Let the body AB, fig. i, Plate V. N. I, confift' ing of two equal bails faftened to a fbff rod, mov'Snbsp;in a diret^ion parallel to itfelf, and it is evident thal^nbsp;the two balls muft have equal monientums, finccnbsp;their quantities of matter are equal, and they movcnbsp;with equal velocities. Now if in its way, as dnbsp;N. II, an obftacle C be oppofed exablly againft il�nbsp;middle E, the body wijl thereby be effeduallynbsp;floppedj nor can either end of it move forwards�nbsp;for they exadlly balance each other, the middlenbsp;this body being its centre of gravity. Now ftroul*^nbsp;an obftacle be oppofed to this body, not againftnbsp;middle, but nearer to one end, as atN. Ill, then th�nbsp;ftroke being not in the diredlion of the centre ^nbsp;gravity, is in falt;ft an oblique ftroke, in which cafe�nbsp;agreeably to the laws of congrefs which have bed^nbsp;delivered in chap. VIII. a part only of the momentutf'nbsp;will be fpent upon the obftacle, and the body ad'nbsp;vancing the end A, which is fartheft from the pb'nbsp;ftacle, as fliewn by the dotted reprefentation,nbsp;proceed wdth that part of the momentum whichnbsp;has not been fpent upon the obftacle; confequentlynbsp;in this cafe the percuffion is not fo powerful as ^nbsp;the foregoing. Therefore there is a certain point iu ^

�iioving body which makes a ftronger Impreffion ' On an obftacle than any other part of it.-r-In thcquot;nbsp;Preient cafe, indeed, this point coincides with thenbsp;Centre of gravity; becaufe the two ends of thenbsp;^ody before the ftroke moved with equal velocities. But in a pendulum the cafe is different; fornbsp;let the fame body of fig. i, Plate V. be fufpendednbsp;by the addition of aline AS, fig. 2, Plate V. which,nbsp;line we fltall fuppofe to be void of weight and flexi-biiitv, and let it vibrate round the point of fufpen-fion S. It is evident that now the two balls willnbsp;move with equal velocities j for the ball B, bynbsp;^lefcribing a longer arch than the ball A in the famenbsp;brne, will have a greater momentum; and of courfenbsp;point where the forces of the two balls balancenbsp;^ch other, which is the centre of percuffion, liesnbsp;Nearer to the lower ball B; confcquently this pointnbsp;^�Oes not coincide with the centre of gravity of thenbsp;bodyAB; but it is that poiiit wherein the forcesnbsp;cf all the parts of the body may be conceived to benbsp;Concentrated. Hence the centre of ofcillation andnbsp;Ibe centre of percuffion coincide; or rather they arenbsp;cxaftiy the lame point, whofe two names only allude,nbsp;^be former to the time of vibration, and the latternbsp;its ftriking force.

^f in fig. I, Plate V. the balls A and B be not ;Pual, their cornmon centre of gravity will not benbsp;the middle at E, but it will lie nearer to the hea-body, as at D, fuppofing B to be the heaviernbsp;podyj fg that_the diftances BD, AD, may be in-

vcrfely as the weights of thofe bodies. Now wheo the above-mentioned body is formed into a pendulum, as in fig. 1, though the weights A and B benbsp;equal, 3fet by their moving In different arches, vi2�nbsp;with different velocities, their forces or momentuiu�nbsp;become actually unequal j therefore in order tonbsp;find the point where the forces balance each other�nbsp;fo that when an obffcacle is oppofed to that point�nbsp;the moving pendulum may be effedtually ftopped�nbsp;and no part of it may preponderate, in which cafsnbsp;the obftacle will receive the greateft impreffion gt;nbsp;we muft find firft the momentums of the two bodies A and B, then the diftances of thofe bodiesnbsp;from the centre of percuffion, or of equal forces�nbsp;muft be inverfely as thofe momentums. Thus thenbsp;velocities of A and B are reprefented by the fimil^tnbsp;arches which they deferibe, and thofe arches are ^snbsp;the radii SA, SB. Therefore ihe momentum of ^nbsp;is the produft of its quantity of matter multiplie^^nbsp;by SA� and the momentum of B is the produdl ofnbsp;its quantity of matter multiplied by SBj confe'nbsp;quently AD muft be to BD, as the weight of ^nbsp;multiplied by SB is to the weight of A multiplio^nbsp;by AS. Then D is the centre of percuffion. Au*^nbsp;fince, when four quantities are proportional,nbsp;produdl of the two extremes is equal to the prO'nbsp;dufl: of the two means; therefore if the weight ofnbsp;A multiplied by AS, be again multiplied by A^�nbsp;the produdt muft be equal to the produft of tbo

% BD; that is, theproduSi of the body on otie fide of centre of ofcillation multiplied by ^both its diftancenbsp;the point of fufpenfion and its difance from thenbsp;^^ntre of ofcillation, is equal to the produEl of the bodynbsp;the other fide of the centre of ofcillation, multipliednbsp;^oth hy its difiance from the point of fufpenfion, andnbsp;difiance from the centre of ofcillation.

The fame reafoning may evidently be applied to ^ pendulum confifting of more than two bodiesnbsp;^'^nnected together, or to the different parts of thenbsp;^^tTie pendulous body �, hence we form the following,nbsp;^^neral law.

If the weight of each part of a fimple or compound ^^^^diilum be multiplied both by its difiance from thenbsp;^^�^tre of fufpenfion, and its diftance from the centre ofnbsp;^^^illation or percifiion, the fums of the produbis, onnbsp;^�^oh gj centre of ofcillation, %vill be equal tonbsp;other.

Trom this law the rule for determining the dif-of the centre of ofcillation from the point of f^^Penfion is eafily deduced j but the application ofnbsp;attended with confiderable difficulty, on whichnbsp;^'^Count we fliall fubjoin it in the note (i),andnbsp;^^11 now proceed to fhew an experimental or mechanical

(i) Let a pendulum confift of any number of parts or bodies A, B, C, D, E, joined together; let lt;7, b, c,.nbsp;hand ^or their refpedtive diftances from the point ofnbsp;*P�afion; and jf for the diftance of the centre of ofcilla-from the point of fufpenfion.

2oS Of th� Centre of Of dilation^ and

chanical method of finding the centre of ofciilatiort� which method is general and eafy, at the fame tirf'nbsp;that it admit? of fufficient accuracy.

The

The diftances of thofe parts, or bodies, from the centre cfcillation will be r � a, x � b, x�c, d � x, e � x -,Vnbsp;E being fuppofed to be farther from the point of fufper.fio''�nbsp;than the centre of ofcillation is. By multiplying every e*��nbsp;of thofe bodies, both by its diftance from the centre ofnbsp;pcnfion and its diftarice from the centre of ofcillation�nbsp;have, agreeably to the above-mentioned law, the equatio'*nbsp;Aax�Aaa Bbx�Bbb Ccx�Crr Ddd �

Eee � E� ; which, by tranfpofition and divifion, is folved into the following; viz.

^ Aaa Bbb Ccc -f Ddd -f- Er/r Aa Cr Dd Eenbsp;Should any of the bodies, as for inftance A and B, innbsp;preceding inftance, be fituated above the centre of fuip�*^

though their fq'uares aa, lb, are always pofitive. In this

, c � Aaa Bbb Ccc Ddd Err

the value of a' is =:-!-1_!_1��

� Aa � BA Cr Di Er

Since the centre of gravity of a body or fyftem of bodi^^�

is that point wherein all their matter may be conceived

be condenfed, therefore the produel of all the matter or

of the different weights A, B, C, D, E, multiplied by

diftance of the common centre of gravity ftom the poin*^

fufpenfion, is equal to the fum of the produdls of each bod/

multiplied by its diftance from the point of fufpen^l^

Hence the above ftated value of ;�� becomes Aaa -h ^

Ccc Ddd Eee divided by the produdl of the

body or fum of the weights, multiplied by the diftance �

th�

Cet�re of Percujfion.

The body whofe centre of ofcillation, or (which the fame) ofpercuffion, is to be afcertained, muff:nbsp;^ fufpended to a pin or other fupport, but as

freely

'�i'e centre of gravity from the point of fufpenfipn. And ^eing exprefled entirely in words, it forms the folfowingnbsp;general

Rule 1, If all the bodies or parts of a-body, that forms a ^�^dulum, be multiplied each by the fquare of its di/lance fromnbsp;point or axis of fufpenfion^ and the fum of the products benbsp;^'tiided by the produdl of the whole weight of the pendulum^nbsp;^^ItipUed by the diflance of the centre of gravity from thenbsp;^int tffttfpenfon ; the quotient will be the diflance of the cen-f ofcillation or percujfion from the point of fufpenfion.

The fituation of the centre of ofcillation may alfo be ^Ound by means of another rule, which we lhall likewife laynbsp;and (hall demonftrate ; fince in fome cafes this rulenbsp;^iil be found preferable to the firlf.

Rule 2. If the fum of the produlis of all the parts or ^��ghts, multiplied each by the fquare of its diflance from thenbsp;^�^�^tre of gravity, or from a line pajfng through the centre ofnbsp;^��avliy parallel to the axis of vibration, be divided by thenbsp;^^du�i of ilji ix^holo mafs or body, multiplied by the diflance ofnbsp;^ Centre of gravity from the point of fufpenfion, the quotientnbsp;dl be the diflance of the centre of ofcillation from the centrenbsp;Suavity ; which being added to the dijlance of the centre ofnbsp;^ �vity from the point of fufpenfton, will be the diflance of thenbsp;^^re of ofcillation from the point offufpenfion.

Cab fig. 4, Plate V. .reprefent any fort of body re-or irregular, fufpended at C j O its centre of ofcilla-centre of gravity; C O B its axis or right line ^ng through the point of fulpenfion, and centres of gra-

2o8 Of Ae Centre of Ofcillation, and

freely as may be pra�licable; and being once moveA out of the perpendicular fituation, muft be fulFerednbsp;to perform very fliort vibrations; viz. fo fmail a?

vity and ofcillation. This body may be conceived to con' lift of an indefinite number of extremely fmail part,snbsp;weights. Let W be one of thofe fmail weights ; joi�'nbsp;WC and WG, and from W drop WF perpendicularnbsp;CO. Then the produdl of W, by the fquare of its diftanc�nbsp;from C, is W xC WF. But (Eucl. p. 47. B. I.)nbsp;=:WF)' Cn*; and GWl^ =:Gr!�- WFlfi (Euc!*nbsp;p. 7. B. II.) UG)^ Gi']��=: aCG X GF 4- CFl* ; and h/nbsp;tranfpofition CFj* =: GF]^ CG *nbsp;nbsp;nbsp;nbsp;2 C G x G^'

Then by fubftitution (viz. by putting inftead of CFi% equrd GF]^ CG]^ �zCG x GF) the abovenbsp;equation becomes CWj* WFj^ GFp ' CGi*�^nbsp;aCG X GF = (putting GW)^ for its equal GF,* dquot;nbsp;WF,^) GW]* CG]* �z�G�TGF: And multiply'^�^nbsp;both fides by W, we have the fum of all the produdls W ^nbsp;CWj* =; the fum of all the W x GW|* all thenbsp;X CGl * � the fum of all the W x zCG xGF.

But by the nature of the centre of gravity the fum of ^ the W xGF is = 0; for thofe which are one fide of ^1�^

of courfe all the W X 2 CG X GF alfo become Therefore there remains the fum of all the W x CW)*^nbsp;fum of all the W x G * fum of all the W xnbsp;fum of all the W x GW]� the whole body X CGi'nbsp;Or (taking away the fum of all the W x GW]* from

the DoSirim of Pendulums. nbsp;nbsp;nbsp;200

a dock or watch with a feconds hand, the ob^-Server muft count the vibrations, and, if poffible, ^ven the part of a vibration, that are performed by

that

In the application of the above-mentioned rules, it is fre-^hently very difficult to find the fum of the produdfs of all ^¹16 Weights multiplied by the fquares of their refpedtive dif-^^hces. The method of fluxions is undoubtedly the moftnbsp;^^tenfive, as it may be applied to all fuch figures or bodiesnbsp;�^have fome regularity of fhape, or fuch as may be expref-I^d by axi algebraical equation. But in fome cafes the irregularity of form is fo very great, that the centre of ofcillationnbsp;'^^n Only be found out by means of the above-defcribed me-'^^atiical method.

In order to find the fum of the weights, amp;c. you muft ^'gt;nfider an indefinitely fmall part, or increment, or fluxion,nbsp;the figure, as being a fmall weight, and multiply it by thenbsp;^4uare of its diftance from the centre of fufpenfion or axisnbsp;vibration, according to rule the iff, or elfe multiply itnbsp;y the fquare of its diftance from the centre of gravity, ornbsp;�tn a line paffing through the centre of gravity, and pa-*^^Ucl to the axis of vibration, according to ri4,ls the 2d.; thennbsp;'^OL, r.nbsp;nbsp;nbsp;nbsp;pnbsp;nbsp;nbsp;nbsp;the

210 nbsp;nbsp;nbsp;Of inclined Flanes, and

that pendulum in one minute, and note the mini' ber. N, B. Should the pendulum appear likely t*!nbsp;flop before the expiration of the minute, a gentl�

the fluent of that expreflion will be the fum of the prodult;Sb of all the Weights, multiplied by the fquares of their refpe^^'nbsp;tive diftances, either from the axis of vibration, or from thenbsp;centre of gravity, amp;c. Laftly, this fluent muft be divide^nbsp;by the produdt of the whole body (to be had by comrnt�^nbsp;menfuration) multiplied by the diftance of the centre of gt^'nbsp;vity, from the point of fufpenfion ; and the quotient willnbsp;the diftance of the centre of ofcillation either from thenbsp;point of fufpenfion, dr from the centre of gravity, accordiaSnbsp;as the operation was performed either by rule the firft, ofnbsp;rule the fecond.

Example i. Let CB, fig. 5, Plate V. be a right line, very flendcr cylinder fufpended at C j and call it a, (meaniitSnbsp;either its length or weight, for the one is proportionate fonbsp;the other) G is its centre of gravity. Now if you callnbsp;part of this line x, reckoning from C, then the increment ofnbsp;fluxion of X is x, which k may be confidered as one of tb�nbsp;vaft many weights which form the whole line or flendofnbsp;cylufder. The produdl of this weight by the fquare ofnbsp;diftance from C is x'^x, and the fluent of this expreflioo

plied by the fquares of their refpedtive diftances from C-

the DoBrine of Pendulums. nbsp;nbsp;nbsp;211

and dexterous application of a finger once or twice, �'�ill increafe a little its vibrations, and prolong itsnbsp;aftion without altering the time of vibration.�

For

We muft now find the produdt of the whole line multiplied by the difiance of the centre of gravity G from C.

the centre of ofcillation O from the point of fufpenfion C ; that C� is equal to | of CB.

Example 2. Let AB, fig. 6, Plate V. be a right line or t'^ry flender cylinder faftened to a line GO void of weight,nbsp;^nd fufpended at O. The ends A and B are equidiftantnbsp;Eom O, and the axis of vibration is perpendicular to thenbsp;plane which paffes through ABOG ; fo that every part ofnbsp;given line from A to G, or from B to G, is at a dif-^�rent difiance from the axis of fufpenfion. Put OG = a,nbsp;�nd Qp _ -vvhofe fluxion is x, and is a particle or fmallnbsp;quot;^^ight of the given line, which multiplied by the fquaije ofnbsp;which is (Eucl. p. 47. B. i.) =:nbsp;nbsp;nbsp;nbsp;becomes,

quot;Ehe produdl; of the body GP by the diftance of the cen-of gravity G from O is (GP xOG) ax. Therefore the

;zi2 nbsp;nbsp;nbsp;Of inclined Planes^ and

For the fake of greater accuracy, the attentive obferver may count the number of vibrations for anbsp;longer time, as, for inftance, during two, or three,

3� nbsp;nbsp;nbsp;3OG

Example 3. Let the pofition be exaflly as in the pre* ceding example, excepting only that the axis of fufpenfiot*nbsp;or of vibration, which was then perpendicular, be now p3'nbsp;rallel, to the line AB,, as in fig. 7, Plate V. and in this cafe�nbsp;the centre of ofcillation will coincide with the centre 0^nbsp;^ gravity G ; for here, all the parts of the given line, as Arnbsp;G, P, B, amp;c. are equidiftant from the axis of fufpenfion tnbsp;fo that the weight x multiplied by the fquare of its diftanc�nbsp;from the axis of vibration DOC, becomes ; the fluen*^nbsp;of which is aV, and this fluent divided by av, quotes ^ fnbsp;that is OG for the diftance of the centre of ofcillation.

Example 4. Let the pendulum confift of an ifofcek� triangle ABC, fig, 8, Plate V. fufpended at A,nbsp;the axis of vibration parallel to BC. Put the altitiKJ�nbsp;AD =a a; bafe BCnbsp;nbsp;nbsp;nbsp;and AF = r. Through ?

the DoElrine of Pendulums. nbsp;nbsp;nbsp;213

or fou� minutes, and then raking the half, or third part, or fourth part of the number; for that part

will

The triangle ABC =: � ; and the diftance of its centre of gravity from A is r: �; hence the produdt of the tri-

Example 5. Let the pendulum confift of afpherical body ^''fpended at O, fig. 9, Plate V. by means of a line OD,nbsp;'''bich line weighs fo little with refpect to the body, that itsnbsp;^�*ght may be confidered as = o. Imagine DERD to benbsp;� Action of the fphere through its axis, and perpendicularnbsp;the axis of ofcilLtion KL. GE the radius perpendicularnbsp;^R. G the centre of gravity, and V the centre of of-

Let SFPS be any concentric circle; and put the ordinate j GPcca-; the circumference of a circle whofenbsp;^^dius is one, and draw NM parallel to GR. Suppofe anbsp;^ylindric furface to ftand on the circumference SFPS,nbsp;to be terminated by the furface of the fphere; then thenbsp;^quot;�cumference SFPS =cx, and the juft mentioned cylindricnbsp;^�'tface will be = 2eyx.

quot;Ehe diftance of the particles in each feiftion of this Cylindric furface, from the centre 0; gravity of the feiSion,

P 3 nbsp;nbsp;nbsp;or

, With

or of a line paffing through G parallel to KL, is therefore the fluxion of the weight or furface is 2nbsp;which multiplied by the fquare of the diftance GP� vi^*

In order to expunge from this expreffion one of the vfl' riablc letters, it mu ft be confidered that in a circularnbsp;the fine is to the cofine as the fluxion of the latter is to th�nbsp;fluxion of the former; for in fig. lo, Plate V, where

is the fine; AE = x is the cofine; if you draw Cf indefinitely near and parallel to BE, and BD parallel tonbsp;BD becomes i-, or the fluxion of the cofine, and CD be'nbsp;comes j), or the fluxion of the fine; and fince the right'nbsp;angled triangles ABE, BCD, are equiangular (the angl^*nbsp;CBD, EBA, being equal, becaufe each of them is the cpitt'nbsp;plement of ABD to a right angle) and fimilar, we have A�'

radius b^ called r?; fince the fquare of AB, or is equal tf Now by fubftitution the flue**^

. nbsp;nbsp;nbsp;...nbsp;nbsp;nbsp;nbsp;�snbsp;nbsp;nbsp;nbsp;VVnbsp;nbsp;nbsp;nbsp;9 yy

mer ; viz. zcyx^xzzT.cyx^ X nbsp;nbsp;nbsp;X '

X nbsp;nbsp;nbsp;ya'--

V a��y

the DoBrine of Penduhms. nbsp;nbsp;nbsp;215

With that number of vibrations, performed in one minute, the diftance of the centre of ofcillation

from

The folidity of a fphere, whofe radius is is exprefl'ed by which multiplied by the diftance GO=r/, in fig. 9.nbsp;^scomes | ca ^d.

Laftly, divide the above fluent by the laft product; viz. divide ca^ bynbsp;nbsp;nbsp;nbsp;and the quotient -y-'S the diftance

Should the point of fufpenfion be fituated clofe to the fur-^^ce, as at D; then the diftance between the centres of ^^^penfion and of gravity would become equal to radius,nbsp;^*2,. d-a-, and in that cafe the diftance between the centres

7�

plainly appears from the foregoing explanations and ex-�*^ples, thkt when the ftring of a pendulum is fliortcned, �''ery thing elfe remaining unaltered, the centre of ofcillationnbsp;^fiaiiggj its place; unlefs indeed the weight of the pendulumnbsp;bob be fuppofed to be condenfed in one point, whichnbsp;'�^fe can have place only in the imagination.

^onfequently what has been demonftrated refpe�ling the ^yololdal pendulum muft be confidered as a matter merelynbsp;^feful fpeculation, fince from it we derive the time innbsp;Ch a circular pendulum performs its vibrations. But innbsp;p 4nbsp;nbsp;nbsp;nbsp;practice

from the point of fufpenfion is determined by means of the following eafy calculation.

Divide fixty feconds by the number of vibrations� which the pendulum in queftion has performednbsp;one minute, and the quotient is the time of onenbsp;vibration. Square this time, (viz. multiply it bynbsp;itfelf) and multiply its fquare by the length of thenbsp;pendulum that vibrates feconds, viz. by 39,119^nbsp;inches, and the laft produdl fhews the diftancenbsp;inches of the centre of ofcillation or percuffiof*nbsp;from the point of fufpenfion in the pendulumnbsp;queftion.

Example i. Let a cylinderical ftick AB, fig- 3� Plate V. of about a yard in length, be fufpendednbsp;A, and be caufed to vibrate. Having obferved thti*^nbsp;it performs � 76_ vibrations in a minute, it is require'lnbsp;thereby to find the diftance of its centre of ofcill^'nbsp;tion from the point of fufpenfion A.

Divide 60 feconds by 76 vibrations, and th^ quotient, o'quot;',79 nearly (viz. 79 hundreths of ^nbsp;fecond) is the time in which the pendulum i*�nbsp;queftion performs one vibration. Then fincenbsp;lengths of pendulums are as the fquares of the tim^^nbsp;of vibration; therefore fay as the fquare of

praS;lce a cycloidal pendulum would not perform vibrations in equal times; becaufe by the application 0nbsp;firing to the cycloidal cheeks, the free part of the firingnbsp;be fhortened, and the centre of ofcillation would chaf^^nbsp;place continually,

Second, which is one, is to the fquare of 0,79 hundredth parts of a fecond, viz. 0,6241 ; fo is thenbsp;length of the pendulum which vibrates feconds,nbsp;''^iz. 39,1196 to the length fought; that is,nbsp;I : 0,6241 : : 39,1196; where fince the firft number is unity, you need, according to the precedingnbsp;^^le, only multiply 39,1196 by 0,6241; and thenbsp;Prodjudt 24,4 is the diftance fought; fo that thenbsp;Centre of ofcillation C in the flick AB is 24 inchesnbsp;3nd 4 tenths diftant from its extremity A; viz*nbsp;^bout two thirds of its length.

Example 2. An irregular body fufpended by end has been found to perform 20 vibrationsnbsp;a minute. Required the diftance of its centrenbsp;ofcillation from the point of fufpenfion ?

Here the time of one vibration is (I-�) 3 feconds; Ibe fquare of which is 9; and 39,1196, multipliednbsp;by 9, gives 332,0764 inche|, oV nearly 29 feet, fornbsp;*-be diftance fought.

T�HE preceding chapters contain the dodtrin� of motion in a manner rather extenfive fotnbsp;an elementary work. The abftradt mode in whid^nbsp;this fubjefl: has been delivered, may poffibly haV^nbsp;deterred the novice from the ftudy of natural ph''nbsp;lofophy. Perhaps he expedled that after ever/nbsp;theoretical chapter his attention fhouid be relievelt;inbsp;by fomc experimental application of the doctrine-But if fuch had been the plan, either the wor^nbsp;would have been protradled to an immoderatenbsp;length, or many ufeful branches of the theory woul*^nbsp;Jiave been fuppreffed.

The importance of the do�trine of motion, an^ its being the foundation of almoft all the phenO'nbsp;mena of nature, were the motives which placednbsp;before every other branch of natural philofophY�nbsp;and the reader may perhaps be pleafed to hear, tb^*-w'hoever underflands the leading principles of tb�nbsp;foregoing theory, will meet with very little dh^'nbsp;culty in the perufal of the following parts of p^^^'

lofophy*

^ofophy. He will alfo find that the do�lrine of iTiotion, which he may formerly hav^ looked uponnbsp;a difficult and almoft a ufelefs fubjecl of fpecu-^ation, is of general and extenlive application.nbsp;Every tool, every engine of art, every oecono-ftiical machine, all the inftruments of hufbandry,nbsp;^'�^d of navigation, the celeftial bodies, amp;c. arenbsp;�^C'-nftrucled, and act conformably to the laws ofnbsp;Motion.

The knowledge of this doftrine anfwers two ex-fenfive objedls. It ferves to explain natural appearances, and it furniffies the human being with ^feful machines, which enable him to accomplifiinbsp;Eich effefts, as without that affillance would benbsp;utterly out of his power.�The application to na-^'Jral phenomena will be inftanced in almoft everynbsp;chapter of this wmrk.�The fecond objeeft will Ivenbsp;^onfidered immediately.

Mechanics, in its full and extenfive meaning, is feience which treats of quantity, of extmfion, andnbsp;notion. Therefore it confiders the ftate �f bodiesnbsp;Either at reft or in motion. That branch of itnbsp;quot;'^dch confiders the ftate of bodies at reft, as theirnbsp;^�f'uilibrium when conneefted with one another,nbsp;*^heir preflure, weight, amp;c. is called Statics. Thatnbsp;'''hich treats of motion, is called Dynamics. Bothnbsp;expreffions are, however', uled in treating ofnbsp;^�lid bodies; for the mechanics of fluids has two

Hydroftatics, when it treats of the equilibrium 0^ quiefcent ftate, and Hydrodynamics or Hydraiaiciinbsp;when it treats of the motion, of fluids.

What belongs exclufively to fluids will be nO' ticed in the fecond part of thefe elements. Th^nbsp;equilibrium of folids has been fufficiently examine^nbsp;in the preceding pages, and will be taken farthernbsp;notice of in the following; fince in treating of mO'nbsp;tion, of adtions, of forces. Sec. it will naturally ap'nbsp;pear that when thofe forces are equal and oppolitenbsp;to each other, an equilibrium takes place.

The adlive application of the dodlrineof motio^^ confifts in the conftrudtion of machines for tb^nbsp;purpoies of overcoming refjftances, or of movingnbsp;bodies. Thus if a man wiih to remove a ftonenbsp;a ton weight from a certain place, for whichnbsp;pofe he finds his fhrength inadequate, he makes ul�nbsp;of a long pole, which being applied in a certati�nbsp;manner, adlually enables him to move the ftooe*nbsp;Thus alio another perfon may wifli to conveynbsp;heavy article to the top of his houfe, he makes ul�

, of a fet of pullies with a rope, See. and by tha*-means eafily apeomplilhes his objedl.

Infinite is the number, arid the variety of chines; but they all confift of certain parts ornbsp;pie mechanifms, varioufly combined and connedts^nbsp;with each other. Of thofe Ample machines wenbsp;reckon no more than fix or at moft feven;nbsp;the Lever, the IFheel and Hv/e, the moveable

The adlion or the efFedt of every one of thofe Mechanical powers, depends upon one and the lamenbsp;principle; which has been fully explained in chapternbsp;^^,V,andVI; butwelhall for the lake of perfpicuitynbsp;briefly repeat it in the following three or four paragraphs, wherein the attentive reader will find thenbsp;principles or analyfis of all torts of machines.

The force or momentum of a body in motion, to be derived not merely from its quantity ofnbsp;Matter, or only from its velocity, but from both.nbsp;Conjointly; for the heavier any body is, the greaternbsp;Power is required to ftop it or to move it; and onnbsp;other hand the fwifter it moves, the greater isnbsp;Hs force, or the ftronger oppolition mull be madenbsp;ftop it. Therefore, the force or momentum, isnbsp;^^0 produdl of the weight or quantity of matter bynbsp;Velocity. Thus if a body v/eighing lo pounds

The writers on mechanics do not agree with refpedt to number of the mechanical powers. Some exclude thenbsp;*^'^dned plane from the number; whilft others reckon itnbsp;of the principal, and confider the wedge and the fcrew-only fpecies of it. The balance has been likewife reck

a peculiar mechanical power. But it has been re-�l�0ed by others, either on account of its being nothing More than a lever, or becaufe by the ufc of a balar.ce no ad-�^hional power is obtained, which advantage ought in truthnbsp;M be the characleriftLc property of a mechanical power.

move at the rate of 12 feet per fecond, and anothe'*� body weighing 5 pounds move at the rate of ^�4nbsp;feet per fecond, their momentums will be eqoai?nbsp;that is, they will ftrike an obftacle with equal force?nbsp;or an equal power mull be exerted to flop them gt;nbsp;for the produdl of 10 by 12, viz. 120, is equalnbsp;the product of 5 by 24.

The forces of bodies acting on each-other by interpofition of machines is derived from thenbsp;principle. Thus the two bodies A and B, fig. i*�nbsp;Plate V. are connefted wfith each other by thenbsp;terpofition of an inflexible rod AB (the fimpleftnbsp;all machines) which refts upon the prop ornbsp;point F. If the rod move out of its horizont^'^nbsp;fituation into the oblique pofition CFE, the bocflnbsp;A will be forced to defcribe the arch AE, whil^nbsp;the body B defcribes the arch BC; and tfiof^nbsp;arches, being defcribed in the fame time, willnbsp;prefent the velocities of thofe bodies refpeftivelf �nbsp;therefore, the momentum of A is to the mornci^'

' turn of B, as the weight of A multiplied by the AE, is to the weight of B multiplied by the arc!'nbsp;BC.

The velocities of A and B are likewife reprefenl^ by their diftances from F; for the arches AE, 1^^�nbsp;are as their radii FA, FB. Thofe velocitiesnbsp;alfo reprefented by the perpendiculars E�,nbsp;for fince the triangles EFG, CDF, are equiang^^'^nbsp;and fimilar, (the angles at G and D being tig^'^�nbsp;and thofe at F being equal) we have EF to FC? ^

to CD. Therefore the refpeftivemomentums of A and B may be reprefented either by AxAE,nbsp;^nd BxBC; or by Ax AF, and B x BF; or laftlynbsp;% AxEG, and BxCD.

Note. The laft expreffion is ufed when the motion of bodies that are fo circumftanced, refults from the adion of gravity; viz. when one body recedes from, whilft the other approaches, the centrenbsp;of the earth �, becaufe gravity ads in that direction.

This is the principle of all forts of mechanifms; fo that in every machine the following particulars rauftnbsp;indifpenfably found, ifl. One or more bodiesnbsp;be moved one way, whilft one or more bodiesnbsp;*^ove the contrary way. One of thofe bodies ornbsp;^�^ts of bodies is called the weight., and the other isnbsp;the power, or they may be called oppofite.nbsp;hwers. adly. If the produd of the weight of onenbsp;thofe powers, multiplied by the fpace it movesnbsp;through in a certain time, be equal to the produdnbsp;�fthe weight of the oppofite power multiplied bynbsp;fpace it moves through in the fame time ; thennbsp;Oppofite momentums being equal, the machinenbsp;''^�11 remain motionlefs. But if one of thofe pro-or momentums exceeds the other, then thenbsp;^^cr is faid to preponderate, and the machinenbsp;^hl move in the diredion of the preponderatingnbsp;hovier; whilft the oppofite power will be forced tonbsp;^^ve the contrary way. And the preponderancenbsp;'�sprefented by the excefs of one momentum over

the other; for inflance, if one of the above-mentioned products or momentums be 24, and the other 12, then the former is faid to be double thenbsp;latter ; or that the former is to the latter as twnnbsp;to one.

By a ftrift adherence to thofe particulars, the attentive reader will be enabled to eftimate the power and effeft of every machine, excepting, ho'i'^^nbsp;ever, the obftruftion which arifes from the impel�'nbsp;feftion of materials and �f workmanlhip; as vviHnbsp;fully appear from the following paragraphs.

In the explanation of the properties of the mechanical powers, we fuppofe the rods, poles, planes� ropes, amp;c. to be deftitute of weight, roughnefs, ad-hefive property, and any imperfection; for whennbsp;the properties of thofe powers have been eftablidi'nbsp;ed, we fliall then point out the allowances propelnbsp;to be made on the fcore of fri�tion, irregularitynbsp;figure, amp;c.

A lever is a bar of wood, or metal, or oth^f fblid fubftance, one part of which is fupported hfnbsp;or refts againft a fleady prop, called thenbsp;about which, as the centre of motion, the levetnbsp;moveable.

The ufe of this machine is to overcome a obftacle, by means of a given power.�Thus itnbsp;Hone A, fig. 12, Plate V. weighing loco pound^�nbsp;be required tO be lifted up (fo as to pafs a rope nn

�er It, or for Tome other purpofe), by means of ordinary ftrength of a man, which may benbsp;^sckoned equal tO' 100 pounds vveight j a pole ornbsp;^6Ver CE is placed with one end under the ftonenbsp;E; it is refted upon a ftone or other fteady bodynbsp;B, and the man preffes the lever down at C. Innbsp;cafe the man�s ftrength is equal to thenbsp;*^5nth part of the ftone�s weight, therefore its ve-^ocity muft be ten times greater than that of thenbsp;ftone; that is, the part B'C of the lever muft be tennbsp;fttiaes as long as the part BE, in order that thenbsp;Power and the weight may balance each other;

if CB is a little longer than ten times BE, ftgt;en the ftone will be raifed. Indeed in this cafenbsp;part CB needs not be fo long; for as the ftonenbsp;Hot to be entirely lifted from the ground, anbsp;ftfler momentum is required on the part of thenbsp;Power at C.

In general, to find the proper length of the lever, ''o Heed only multiply the weight by that part ofnbsp;^^0 lever w'hich is between it and the fulcrum;

an equilibrium, and of courfe a little more ^n that length will be fufficicnt to overcome the

^^ftacle.

tt*

^ when the length of the lever is given,'you'

divide the produdl by the power; for the ft'^otient will be the length BC, w'hich is neceflarynbsp;form

oftie a known obftacle or weight; multiply the ight by that part of the lever which is between

it and the fulcrum, then divide the produft b/ the other part of the lever, and the quotient is thenbsp;anfwer.

The poffible different fituations of the weight, the fulcrum,quot;and the power, are not more than three?nbsp;hence arife three kinds of levers; to all of which?nbsp;however, the preceding calculations are equally ap'nbsp;plicable. Thofe fpecies are, i. when the fulcruninbsp;is placed between the weight and the power, asnbsp;the one already deferibed. i. When the hjbnbsp;crum is at one end, the power at the other encJ?nbsp;and the weight between them, as in fig, i3,-Plat^nbsp;V. And 3. When the fulcrum is at one end, th�nbsp;weight at the other end, and the power betweennbsp;them, as in fig. 14, Plate V.

Some writers add a fourth fpecies, viz. the bet*^ lever; but as this differs only in fhape from thenbsp;others, it does not conflitute a proper differencenbsp;kind.

Hitherto we have fuppofed that the weight an^ the power aft in direftions perpendicular to thenbsp;arms of the lever; but when this is not the cafenbsp;the diftances of the power and of the weight from thenbsp;centre of motion mufl; not be reckoned by thenbsp;diftances of the points of fufpenfion from thatnbsp;centre, but by the lengths of the perpendicularsnbsp;fall from the centre,of motion on the lines of th^nbsp;direftion of the forces. For inftance, in fig- �a�nbsp;Plate V. the power at P, afts by nreans of th*^nbsp;firing PBgt; on the end B of the lever, in adireft''^�^

oblique to the lever j and in eftimatlng the Momentum of the power, you muft multiply thenbsp;force or power applied to the ftring, not by thenbsp;fongth CB, but by the length CD, of the perpendicular, let fall from the centre of motion C, on thenbsp;iitie BP, which is the line of direction of the power.

Thus alfo in the bent lever ABC, fig. 16, Plate ^� whofe centre of motion is at B ; the momentums

D and E are the weight of D multiplied by Hg, and the weight of E multiplied by BF. Thenbsp;^^afon of the laft remark is eafily derived from thenbsp;^onipofition and refolution of forces (fee chap. VIII.)nbsp;quot;therefore we may in general fay, that in any fort ofnbsp;and ^in wlmtever direEiions the fozver and thenbsp;'^^ight aSl on it, if their (quantities be inverfely as thenbsp;'^^^pendicidars let fall from the centre of motion on theirnbsp;^^IpeElive direSlions, they will be in equilibrioi that is^nbsp;^laytce each other.

fo will be hardly neceflary to remark, that when foe lever is loaded with feveral weights at differentnbsp;^iftances from the centre of motion, the momen-fo'at on each fide of the centre of motion is equalnbsp;fo the fum of the produfts of all the weights onnbsp;fo^I fide multiplied each by its diftance from thenbsp;^*^titre of motion. Thus in fig. 17, Plate V. thenbsp;^tgt;mentum of the fide AD is ec^ual to the fum ofnbsp;fo^ produds of E multiplied by DA, F multipliednbsp;^ quot;^A, and H multiplied by OA; and the mo-of the fide AB is equal to the fum of Gnbsp;^fotiplied by .BA, and K multiplied by LA.

The ufe of the Jever is fo general and fo extenfivegt; that levers of all forts and varieties are to be foundnbsp;in almoft every mechanifmj�in the works of n^'nbsp;ture as well as thofe of human ingenuity.

Thp bones of a human arm, AC, fig. 18, Plat�d V. and irideed the greateft number of the move'nbsp;able bones of animals, are levers of the third kind-In fig. i8, D is the centre of motion; thenbsp;(viz. the infertion of the mufcle BC, the contractionnbsp;of which moves the arm) is at C, and the effeclnbsp;produced, or the �'eight is lifted, at A.

In this natural lever the power is not advantage' outly fituated; for as it lies very near the centt^nbsp;of motion, it muft be mnch greater than the weigh*'nbsp;which is to be lifted at A. But the lofs of povve'�nbsp;is abundantly compenfated by other advantag^^�nbsp;the principal of which is the compadnefs of th�nbsp;limb.

The iron crozv, fig. 19, Plate V. which is con^' monly ufed by carpenters, blackfmiths, ftone-m^^'nbsp;fons, amp;c. is a bent lever, flattened at A. It isnbsp;a little in order that the weight may be lefs aptnbsp;flip off; and it is flattened for the purpofe of d�nbsp;being more eafily admitted into narrow crevices.

The common balance, fig. 20, Plate V. or paif fcales, is a lever, whofc fulcrum or centre ofnbsp;is in the middle, and the rvetghts are fufpendednbsp;the two extremities; but as thofe extremities

Of the Mechanical Poioers.

^uft be equal; but when one of the weights exceeds the other, then that arm to which the former fufpended, will defcend, amp;c. And this is all thenbsp;that can be made of the balance, viz. to find ,nbsp;'''hen two weights arc equal or unequal*.

* Common balances are fubjedt to many ImperfeiSlions, 'he principal of which are as follows;

lit. A balance is frequently in equilibrio, when the oppo-hle Weights in its fcales are not equal. This arifes from 'he points of fufpenfion being not equidiftant from thenbsp;of motion; in v/hich cafe the empty fcales may benbsp;to balance each other; yet when equal weights arenbsp;in them, thofe weights will not balance each other ; fornbsp;, ^� lhey are fufpended at unequal diftances from the centre ofnbsp;*'^^fion, their momentums are adtually unequal. '

^dly. Xhe beam is frequently made too flight; in which it is apt to be bent more or lefs by the weights thatnbsp;put into the fcales; and of courfe the-apparent equili-cannot be depended upon.

3dly. Balances feldom are fufficiently fenfible. This arifes from various caufes, as from the great weightnbsp;Ihe beam, from rourrhnefs and friction at the point ofnbsp;�Penflon, from the centre of gravity of the beam beingnbsp;'''^tifiderably below the centre of motion, amp;c.

^^alances have been made in this country and elfewhere, ^ Wonderful degree of fenfibility ; viz. capable of having

Q 5 nbsp;nbsp;nbsp;The

The fteelyard, fig. 21, Plate V. (which many writers called by the latin name Jiatera romand)nbsp;a lever of the firffc kind, whof� fulcrum or centrenbsp;of motion is at A; the weight B is fufpended always at the fame diftance CA from the centre ofnbsp;motion ; but the power or counterpoife E may henbsp;Ihifted from one point to another all along the arnanbsp;AD; by v/hich means a great variety of weighl^*nbsp;may be balanced by the fame counterpoife E, whofonbsp;momentum increafes in the proportion of its dif'nbsp;tance from A. The whole length of the arm Al^nbsp;is marked with numbered divifions, each of whichnbsp;indicates the weight of B, which is balanced bynbsp;the counterpoife E, when E is placed at that p^^'nbsp;ticular divifion. Thofe divifions are afcertained bynbsp;trial; for the two arms of the fteelyard being un'nbsp;equal in weight, their momentums, when load^ianbsp;with the weights B and E, cannot be eftimatc^nbsp;merely by the products of thofe weights multipb^*^nbsp;each by its diftance from A.

The fteelyard was rendered more perfect by Af'quot;' B. Martin, a philofophical inftrument maker of v'cr/nbsp;diftinguiflred ability, who fixed a weight C tonbsp;fliort end of the beam (as is (hewn in fig 22,

V.) capable of juft balancing the oppofite arm A^ � in which cafe the momentums'of E and Bnbsp;equal to their weights multiplied by their refpe'^^^^nbsp;diftances from A; confequently the divifionsnbsp;the arm AD may be eafily determined bynbsp;fwrement.-

When a lever is fupported at its two extremities A and C, fig. 23, Plate V. and the weight W isnbsp;fufpended at a point B between A and C, tholenbsp;points A and C may be alternately confiderednbsp;as the power and the other as the fulcrum �,nbsp;froin which confideration it appears that the proportion of the w'eight which is fupported by one ofnbsp;diofe props, is to the other in the inverfc propor-Pon of the difhances AB, BC ; hence when a weightnbsp;tarried by means of a pole between two men, innbsp;manner commonly pradtifed by draymen whennbsp;'�'��ey carry a calk of beer, the weight may be madenbsp;bear harder, upon one of the men than upon thenbsp;'^Iher; by placing it nearer to the one than to thenbsp;�ther.

W/e wheel and axle (by fome called axis in pe-confifts of a cylinder, AB, fig. i, Plate and a wheel DF faftened to the cylinder, andnbsp;moveable round the common axis, which is fup-Pmted at its two ends B and G.

this mechanical power, the weight C is ralfed 1� ^ rope which coils about the axle, and the power

the velocity of the power, as the circumference the axis is to the circumference of the wheel, ornbsp;' ^^^ufe circles are as their diameters) as the dia-^�^ter of the axle is to the diameter of the wheel j

Inftead of the power E, the wheel may be fur-nilhed with little handles or fpokes, as reprefented in the figure, which may be moved by hand. Ofnbsp;long fpokes may be fixed ^through the axis, and thenbsp;hands of one or more men may be applied to thenbsp;ends of thofe fpokes, as in fig. 2, for the effedt wh^nbsp;be the fame as if there were a wheel ; which isnbsp;evident as not to need any farther illuftration.

Cranes for railing great weights, capftans, and windlafs, fuch as are ufed on board of (hips, arenbsp;engines of this fort.

Fig. 3, Plate VI. reprefents a very powerful en' gine, nearly of this fort. ABKI, and CIDH, arenbsp;two cylinders of unequal diameters, (but the dif'nbsp;ference of thofe diameters muft not be very great)nbsp;firmly connedted � together and moveable by mean*nbsp;of the handle F round the common axis EG, whoienbsp;extremities reft upon two fupports. The fa'^^nbsp;rope is faftened with one end at D, and is woun^Jnbsp;round the fmall cylinder CDj then it defeends andnbsp;paffes round the pulley to the frame of whichnbsp;the weight W is fufpended; and laftly, the othetnbsp;end of the rope is faftened at A to the larger ,nbsp;Under. Now by moving the cylinders round, th^nbsp;rope will unwind itfelf from the fmall cylinder, an

will coil itfelf round the large cylinder, as is cleadf

fliewn by fig. 4, which reprefents the cylinders as ieen by an eye placed in the dire�lion of the axis.nbsp;If the cylinders were of equal diameters, the lowernbsp;part Z of the rope, or the weight W which is fuf-pended to it, would not be moved; for in that cafe,nbsp;as much of the rope as is difengaged from one cylinder at each revolution, would be coiled roundnbsp;the other cylinder; but the cylinders beings of unequal diameters, it is evident that at each revolution of the handle F, .more of the rope will benbsp;Coiled round the cylinder ABIK than will be difengaged from the cylinder CDIH; and of courfenbsp;the weight W will be raifed.

The pulley is a thick circular piece of wood, or ^eUl, or Other folid matter, moveable round anbsp;Centre pin or axis, which is fixed in a block ornbsp;frame, in the manner reprefented by A. fig. 5,nbsp;�late VI. In this fig. the frame is faftened to anbsp;fteady beam; a rope is pafled over the pulley, tonbsp;end of which the weight W is fufpended, andnbsp;^he power P is applied to the other end of the rope,nbsp;fti this cafe it is evident, that in order to raife the

'''eight, the power P mull move downwards as much ��rs the weight W moves upwards; or in other words,nbsp;that their velocities are equal; hence no advantage

gained by this mechanifm, excepting the con-'cniency of changing the diredlion of the motion ;

lb that the aftion of this pulley'is exaffly analagous to that of the balance. Therefore the third mechanical power is not faid to be the pulley in general, but it is faid to confill of a moveable pulleftnbsp;or moveable pulleys, as fhewn in the figures 6, 7, andnbsp;8, Plate VI, for in thofe cafes, power is evidentlynbsp;gained.

In fig, 6, the rope is faftened to the hook at F; it paffes round the pulley BD, to the block of whichnbsp;the weight W is fufpended, and is then held by thenbsp;power .at E. When the power pulls the rope, thenbsp;block, with the weight, are raifed, and the rope isnbsp;fiiorcened on both fides; for inftance, when thenbsp;pulley, block, amp;cc. are at the dotted fituationnbsp;the rope has. been fliortened of the lengths AB,nbsp;CD; viz. double the height mD; and that quantitynbsp;of rope has been drawn by the power; therefore innbsp;order to pull the weight up from the fituation Wnbsp;to that of the dotted reprefentation, the power muftnbsp;have moved through twice that fpace; that is*nbsp;with double the velocity of the weight; hence thenbsp;equilibrium in this cafe takes place when the powe^'nbsp;is to the weight as one is to two.

Fig. 7, reprefents the fame cafe, excepting only that in this the direction of the power E is changednbsp;by the intetpofition of amp; fixed pulley F; fo that ifnbsp;W weigh two pounds, the power, or oppofite weig^^*-E, mult weigh one pound to balance it; and then*nbsp;if a little more weight be added to E, the weightnbsp;W will be raifed.

In fig. 8, there is a block or frame containing three pulleys, and having the weight W faftenedtonbsp;Its hook; there is alfo another block faftened to anbsp;fteady beam, and containing three other pulleys.nbsp;The fame rope paffes through them all, and is faftened with one end tolt;the upper block, whilft thenbsp;power E is applied to its other end. Here it isnbsp;evident, that in railing the w'^eight, the rope muffnbsp;be thortened at a, b, c, d, e,/and �; viz. fix timesnbsp;^s much as the weight is raifed; and of courfe tirenbsp;power E muff move with fix times the velocity ofnbsp;the weight; therefore the equilibrium takes placenbsp;tvhen E is the fixth part of W; viz. if W weighsnbsp;fix pounds, E needs not weigh more than onenbsp;Pound, in order to balance the weight W; but itnbsp;T Weigh a little more than one pound, then thenbsp;'Veight W will be raifed¹. The like reafoning maynbsp;be extended to any other number of pulleys.

The circumferences of pulleys are generally grooved, hollowed, in order to receive and retain the rope. Thenbsp;or centre pin, is fometimes fixed to the block, and thenbsp;Pulley rnoves round it; and at other times the axis is fixednbsp;the pulley, and its twg ends move in two holes made innbsp;hte block.nbsp;nbsp;nbsp;nbsp;i

^ great degree of fridlion is the principal defeff to which this mechanical power is liable, and which arifes from three,nbsp;'^^ufes; viz. from the diameter of the axis bearing a con-fiderab!e proportion to that of the pulley, from the pulley�snbsp;tubbing againft the fides of the block, and from the ropes

Fig. 9, of Plate VI. reprefents another variety of this mechanical power. It confifts of one fixt,' andnbsp;one moveable pulley; but in faft each of thofenbsp;pulleys performs the office of three pulleys, for itnbsp;confifts of three grooves of unequal diameters, as isnbsp;fliewn by the lateral reprefentation of one of themnbsp;at R. The fame rope which is faftened with onenbsp;extremity to one of the blocks, pafles fucceffivelynbsp;over the fix grooves, and the power is applied atnbsp;its other end E.

In order to underftand the aftion of this con-ftruftion, it muft be confidered, that in the combination of fig. 8, where the pulleys are all of the lame diameter, each pulley muft move fafter thannbsp;the preceding pulley, becaufe a greater length ofnbsp;rope muft pafs over each pulley than over the preceding pulley, as may be eafily comprehended by

The principal contrivances, which have been made for the purpofe of diminifhing thofe caufes of obftrudtion, wiHnbsp;be mentioned in the next chapter.

In the defeription of this mechanical power we have confidered the ropes as adting always perpendicular to tno horizon; but when that is not the cafe, as for inflance, gt;��nbsp;would be, if in fig. 7, Plate VI. the hook S and the pulk/nbsp;F were placed at a greater diftance from each other; thennbsp;the velocity of the weight is to be eftimated not by rhonbsp;length of the rope which is drawn, but by the perpendicular height to which the weight is raifed. And the faip�nbsp;thing muft be underftood VYith refpeft to the diredfiof*

Of the Mechanical Powers. ^37' Jnfpedling fig. the 7tb, where it is evident, that ifnbsp;the weight be raifed one foot, the ropes muft benbsp;fliortened of a foot each, viz. a foot from B to A,nbsp;^nd another foot from D to C i hence whilfl; thenbsp;length A B patfes over the pulley B D, twice thatnbsp;length muft pafs over the pulley F; fo that thenbsp;pulley F, if equal in diameter to BD, muft makenbsp;two revolutions, whilft the pulley BD makes onenbsp;^evolution. It is alfo evident, that if the pulleynbsp;P Were of double the circumference, or, which isnbsp;the fame thing, of double the diameter of BD, thennbsp;^ach of the pulleys would make one revolution innbsp;the fame time. Now returning to the conftrudfionnbsp;�f fig. 9, it will be eafily comprehended, that asnbsp;the three grooves of the upper pulley, as alfo thenbsp;three of the lower pulley, belong to one folid body,nbsp;they muft revolve in the fame time j therefore,nbsp;their ^diameters, or their circumferences, muft benbsp;^^�de in the proportion of the quantity of rope,nbsp;''�hich muft pafs over them in the fame time,nbsp;quot;^hus whilft one foot length of the rope paffes overnbsp;firft groove a, two feet of rope muft pafs overnbsp;the fecond groove b, three feet of rope muft pafsnbsp;the third groove c, and fo forth. Therefore,nbsp;the diameter of the fecond groove b, muft be twicenbsp;^l�'i diameter of the firft groove a ; the diameter ofnbsp;the third groove c, muft be three times that of thenbsp;hrft a; the diameter of the fourth groove J, muft,nbsp;four times that of a i amp;c. or, in other words,nbsp;-nbsp;nbsp;nbsp;nbsp;the

the diameters of the grooves a, h, r, d, e,f, mult be in arithmetic progreffion ; the ditFerence of tb�nbsp;terms bein^ equal to the diameter of the firft ornbsp;fmalleft groove.

It is evident, that in this conftfudlion, in order to raife the weight-, fix ropes muft be flrortened, andnbsp;of courfe the power muft move through fix timesnbsp;the fpace that the weight moves through, conic-quently the equilibrium takes place when the powernbsp;is equal to the fixth part of the weight W.

The principal advantage which is attributed to this conftrudtion, is the redudtion of fridtion fornbsp;in this, there are only two axes and four furfacesnbsp;which rub againft the blocks j whereas in the con-ftrudlion of fig. 8, where the pulleys are all fepa-rate, there are fix axes and 12 furfaces which rubnbsp;againft the blocks. But, in my opinion, this advantage is more than compenfated by the imper-fedtions which are peculiar to this conflrudlion gt;nbsp;for, in the firft place, if the grooves are not roadunbsp;exadlly in arithmetic progreffion, or if they become otherwile by the accumulation of dirt, itiC.nbsp;then the rope muft partly Aide over them, whichnbsp;will occafion a confiderable degree of fridtiou jnbsp;and fecondly, even when the grooves are of thenbsp;proper dimenfions, if the rope happens to ftretchnbsp;more in one place than in another, which is geue-rally the cafe, then the above-mentioned tlidir^nbsp;and fi'idwion will alfo take place.

A plane fuperficies inclined to the horizon, is S�tiother mechanical power; its ufe being to raifenbsp;^'Veights from one level to another, by the application of much lefs force than would be neceffary tonbsp;^aife them perpendicularly. Thus in fig. 10, Platenbsp;^I. AB reprefents a plane inclined to the horizontal plane AC ; where if the weight D be rolled up-^'ards from A to B, the force neceffary for the pur-Pofe will be found to be much lefs than thatnbsp;'^''hich would be required to raife it dlredly andnbsp;perpendicularly from C to B.

In this cafe the effedt which is produced, con-tifts in the railing of the weight from the level of to the level of B; but to effedt this, the powernbsp;have moved from A to B; (for the powernbsp;in that direction, whilft the weight or gravitynbsp;the body afts in the (Jiteftion of the perpendi-^'Jlar CB;) therefore the velocity of the weight innbsp;^'lis engine, being to the velocity of the power, asnbsp;perpendicular height BC of the plane is to itsnbsp;^^^gth AB, the equilibrium takes place when thenbsp;|Vl�ight is to th� power, as the length of the planenbsp;Its perpendicular height.

'This property may be clearly fhewn by the fol-jowing experiment :�Let AB, fig. ii, Plate VI. ^ a plane moveable upon the horizontal plane AC;nbsp;as to admit of its being placed at any required

quired angle of inclination, which is eafily accoIH'* pi idled by means of a hinge at A, and a prop between the two planes. The upper part of th^nbsp;plane mud; be furnidied with a pulley B, overnbsp;which a firing may eafily run. Let the cylindrical weight D be made to turn upon dender pinsnbsp;in the frame F, in which the hook e is faftened withnbsp;a fb'ing eBH, which paffing over the pulley B, holdsnbsp;the weight E fufpended at its other extremity.��nbsp;The pulley diould be fituated fo that the rope enbsp;may be parallel to the plane.

This plane may be fixed at any angle of inch' nation, and it will always be found, that if th�nbsp;weight of the body E be to the weight of thenbsp;body D, together with that of its frame F, as thenbsp;perpendicular height CB of the plane is to its lengthnbsp;AB, the power E will juft fupport the cylinder P�nbsp;with its frame F upon the plane, and the leaftnbsp;touch of a finger will caufe the cylinder D to afcendnbsp;or defcend ; the counterpoife or power E movingnbsp;at the fame time the contrary way.

It is evident, that the fmaller the angle of inch' nation is, the Jefs force is required to draw up th^nbsp;weight D ; and of courfe when the angle of inch'nbsp;nation vanrflies or becomes nothing, the lead:nbsp;will be fufficient to move the body ; that is, whei^nbsp;the plane AB becomes parallel to the horizon,nbsp;upon an horizontal plane, the heavieft body mJgh*quot;nbsp;he moved with the leaft power, were it not for

Of the Methanical Powers. nbsp;nbsp;nbsp;241

friction, v,'hich is occafioned by the irregularity of fire contiguous furfaces, amp;c. (i.)

(i.) The above-mentioned explanation of the property of fire inclined plane, applies only to one direction of thenbsp;power; namely when the power adls in a direction parallelnbsp;to the plane; but the general theory will be found in thenbsp;following propofition:

When a body or weight W h fuf anted upon a plane, '^hlch IS inclined to tke-horizon-, viz. when the poiver P isnbsp;Jnjf fuficient to balance the quot;Weight upon that plane-, then thenbsp;poiuer is to the quot;Weight, as the fine of the plane's inclination isnbsp;the fine which the db'eSlion of the power makes zvith a linenbsp;perpendicular to the plane.

Let AB fig. 12, Plate VI. be the plane inclined to the Irorizon AC, and let a weight at O be fupported partly bynbsp;*lre plane, and partly by a power which acts in the directionnbsp;Through O draw EOC perpendicular to AE, andnbsp;C, where EC meets the horizontal plane, eredt CV' perpendicular to the horizon, to meet the direction of thenbsp;Power as at V.

l^ow the body W, fituated at O, is balanced, or kept at by three powers, which (fee prop. IV. chap. VIII.)

^''0 the fame proportion to each other as have the right 'Oes parallel to their refpedtive diredtions, and terminatednbsp;their mutual concurfe; namely, by the power which isnbsp;OV ; by the gravitating power, which is as VC ; andnbsp;the readlion of the plane, which is as OC; hence thenbsp;Pt^wer is to the weight, viz� P : W :; OV : VC j or (fincenbsp;� fides of plane triangles are as the fines of their oppofitenbsp;�'^�les) P; W:: fin. OCV, or BAC : (for thofe angles arenbsp;^^lual fince the right-angled triangles BOC, and BACnbsp;I.nbsp;nbsp;nbsp;nbsp;Rnbsp;nbsp;nbsp;nbsp;have

Of the Mechanical Pozvers.

THE WEDGE.

The wedge has been juftly confidered as a fpe-cies of inclined plane j for it confifts of two

incline*!

have a common angle at B) : fin. VOE, or VOC; for thofe angles being the complement of each other to two rightnbsp;angles, have the fame fine.

From this propofition the following corollaries are evidently deduced:

I, Since P : W : : fin. BAG: fin. V O C; h

;; therefore if the

weight W, and the inclination of the plane, or fin. BAG remain the fame, the power muft increafe or decreafe if'nbsp;verfely as the fine of VOG; hence when the diredion of dwnbsp;power is perpendicular to EG, or parallel to the planenbsp;then the fine of VOG, being the fine of a right angle,nbsp;the greateft fine poffible, and, of courfe in that cafe th?nbsp;power P, which is required to fuftain the weight W, is thenbsp;leaf!: poffible; or, which amounts to the fame thing, the**nbsp;the greateft weight may be fuftained by a given powet-Alfo when the diredion of the power coincides with O^�nbsp;namely when the power ads in a diredion perpendicd^'^nbsp;to the plane, then the angle VOG vaniflies, and the poWe*�nbsp;muft be infinitely great.

2. If the diredion of the power be parallel to, or coni cide with the plane, then the equilibrium takes place wlW'*nbsp;the power is to the weight:: OB : BG ;; (Eucl. p*

B. VI.) BG : BA j viz. as the elevation of the plancis its length ; or as the fine of its inclination is to radius.

Of the Mechanical Pozveri. nbsp;nbsp;nbsp;243

inclined planes joined bafe to bafe, as (hewn in fig. I, Plate VII. where AB or GC is the thick-nefs of the wedge at its back, upon which the forcenbsp;or power is applied (be it the ftroke of a mallet, ornbsp;any other preti�ure) ; the middle line FD is the axisnbsp;Or heiglit of the wedge; DG and DC are thenbsp;lengtlis of itsHant fides; and OD is its edge,nbsp;'''hich is to be forced into the wood or other folid ;nbsp;fince the ufe of this inftrument is for cleaving ofnbsp;'''ood, ftone, and other folid fubftances; or, in ge-iieral, for feparating any tw^o contiguous furfaces.

power is to the weight :: OR : CR :; (fince the triangles f^RC and BAG are fimilar) BC ; CA; viz. as the eleva-hon of the plane is to its bafe.

4. The power muft fuftain the whole weight, when its ^itedlion is perpendicular to the horizon.

;: fin. OCV ; fin. OVC :: fin. BAG : fin. OVC. The preceding analogy,by alternation, becomes P: fin.

:; prelT. ; fin. OVC, from which it appears that when power and tne inclination of the plane, or angle BAG,nbsp;*^�tnaiii invariable, the prelTure on the plane mull increafe ornbsp;^-�Creafe according as the fine of OVC increafes or de-'^I'eafes; therefore when the direiSion of the power is pa-pillel to the bafe-, and OVC becomes a right angle, whofenbsp;'He is the greateft, then the prelihre on the plane will llke-''ile be greateft.

?� When the direflion of the power is parallel to the P: prelTure :; OB ; OC :: BC : AC.

Hence its application is very extenfive, and in faft? fcilTars, knives, nails, chifels, hatchets, amp;c. arcnbsp;nothing btit wedges under different fhapes.

Stri��y fpeaking, in the geometrical language, � wedge may be called a triangular prifm; for it maynbsp;be conceived to be generated by the motion of anbsp;plane triangle in a diredlion parallel to itfelf, as thatnbsp;of the triangle GCD, from GCD to ABO.nbsp;it is called an ifofceles or fcalene wedge, according asnbsp;the generating triangle, or face, GCD, is ifojcelt^nbsp;or fcalene.

The aftion of the wedge, is evidently derive^ from that of the inclined plane; yet a varietynbsp;circumftances has rendered the i'nvefligation ofnbsp;power of the wedge more perplexing than that o¹nbsp;any other mechanical power

The mo,ft rational theory fliews, i. 'That the prejfures on the Jides of the ifofceles wedge arenbsp;and aEl in directions perpendicular to thofe Jides,nbsp;equilibrium lakes place, when, the force on the backnbsp;' the wedge is to the fum of the preffiires on the Jides,nbsp;GF, viz. half the thicknefs of the back, is to either 4

The proportion between the power, which is appl'^^ to the back of the wedge, and the effefl: which is proda?^nbsp;on, the fides, has been Hated differently by different author�'nbsp;Thofe who wifli to examine the reafons of thofe diffcr^�|¹�nbsp;opinions, may confult Rownmg�.s Comp. Syft, of Phihnbsp;chap. lOj and Ludtards Matbm. EJjays,-

fures are equal, but aSl in direblions equally inclined to ^he Jtdes of the ifofceles wedge, the equilibrium takesnbsp;place when'the force on the back is to_ the fum of thenbsp;^'fiflances upon thefides, as th e produbi of the fine ofnbsp;half the vertical angle GDC of the wedge, multipliednbsp;h the fine of the angle which the direblions of the re-J'fiances make with the fides, to the fqudre of radius,nbsp;�^nd 3. that when in a fcalene wedge three forces a�l-perpendicularly upon its three fides, keep each othernbsp;equilibrio, thofe three forces are refpeSiively proportional to the fides.

The three parts of this propofition will be foundquot; ^ernonftrated in the note (2).

From

(2) In order to demonftrate the firft part of the above-*^entioncd propofition, let AKD, fig. 2, Plate VII. repre-^�^t the face of an ifofceles wedge. B and E are two obftacles, '''liich prefs upon its two fides in diredlions perpendicular tonbsp;^^ofe fides. Suppofe the wedge to be impelled downwardsnbsp;far as the dotted reprefentation GLF, in confequence ofnbsp;^hich the obftacles B and E muft be driven to the places

alfo O, M, with the lines IQ,, OM. Then it is evident �'ll the parallelifm of the lines, that OM is equal to IQ,;nbsp;�tice the part IQ_of the wedge muft have advanced as farnbsp;OM j therefore YN, or lO, or QM, reprefents the ve-�city Qf wedge (that is of the power) ; whilft BO andnbsp;1 reprefent the velocities of the obftacles.

R 3 nbsp;nbsp;nbsp;Now

2\6 nbsp;nbsp;nbsp;Of the Mechanical Powers.

From this it follows that by the addition of 3. little more force on the back of the wedge, thannbsp;that which is fufficient to form the equilibrium?nbsp;the retiftances will be overcome, amp;c.

Now the triangles lOB, and ACD are equiangular; (the angles at C and B being right, and the angle BlOnbsp;equal to CD A; Eucl. p. 29. and 32. B. I.' and ot courtsnbsp;firnilar (Eucl. p. 4. B. VI.) therefore contidering half thenbsp;wedge and one obftacle, OB:OI::AC:AD; thatnbsp;is, the velocity of the obftacle B is to the velocity of tbsnbsp;power, as half the thicknefs of the wedge is to its llant fids*nbsp;Likewife for the fame reafons we fay that the velocity of th�nbsp;prefling obftacle E is to that of the power, as half th�nbsp;thicknefs of the wedge is to its flant fide. Therefore,nbsp;adding thofe proportional quantities-, we fay that the velocitynbsp;of the obftacle B plus the velocity of the other obftaclenbsp;is to the velocity of twice the half wedge, (viz. ofnbsp;whole wedge) as the whole length AK of the back, isnbsp;the fum of the fides AD, DK; or as half the length of tb�nbsp;back is to one fide.

But when oppofite powers, which adl upon each othef� are inverfely as their velocities, they form an equilibrinin gt;nbsp;therefore when the power on the back of the wedge isnbsp;the fum of the refiftances on the fides, as half the lengrb �nbsp;the back is to one flant fide, the wedge remains motionl�^��nbsp;which is the firft part of the propofition.

In order to prove the fecond part of the propofition ABC, fig. 3, Plate VII. be the face of an ifofceles

Of the Mechanical Powers. nbsp;nbsp;nbsp;247

It alto appears that the fmaller the angle GDC ^^5 the lefs force will be required to drive the wedgenbsp;into any folid fubftance,

FD parallel, and DE perpendicular, to AC; then the former of thofe forces, being parallel to the fide of the wedge,nbsp;'^nnot have any power upon it; therefore the original forcenbsp;�iF will have juft the fame effecS upon the wedge as thenbsp;defier perpendicular force DE; the former being to thenbsp;Matter as radius to the fine of the inclination of the force EFnbsp;to the fide AC. But, by the firft part of this propofition,nbsp;this perpendicular force DE is to the power on the back ofnbsp;fhe Wedge which balances it, as AC is to AH, or as radiusnbsp;to the fine of the angle ACH, (viz. half the angle at thenbsp;''ottex of the wedge) therefore, by compounding thofe ratios,nbsp;xED ; power on the back x ED : : force EF : powernbsp;0fgt; the back :: fquare of radius : fine of half the verticalnbsp;^ogle X fine of the inclination of the refiftance.

quot;Fhe oblique force ef on the other fide of the wedge, being to EF, will require another power equal to the formernbsp;the back of the wedge, to balance it; therefore the fumnbsp;the refiftances on the fides of the wedge is to the wholenbsp;power on the back, as the fquare of radius is to the produdlnbsp;the fine of half the vertical angle multiplied by the fine ofnbsp;inclination of the refiftances to the fides of the wedge,nbsp;lo order to prove the laft part of the propofition, let G D,nbsp;andGE, fig. 4, Plate VII. reprefent the direffions ofnbsp;^he three forces perpendicular to the fides of the fcalenenbsp;^odge ABC. Produce � G ftraight towards O, andnbsp;rougij b draw FO parallel to DG. Then fince thofenbsp;'^oe forces balance each other, they muft be (by prop. IV.

K 4 nbsp;nbsp;nbsp;of

248 nbsp;nbsp;nbsp;Of the Mechanical Pozvers^

We flial! laftiy obferve, that when the wood fplits below the edge of the wedge, as is Ihewn bynbsp;fig. 5, Plate VII. vdiich is generally the cafe ; thennbsp;the fide of the wedge muft be confidered as equalnbsp;to either fide of the cleft; for in fact if we fuppofsnbsp;that the wedge is lengthened downwards to thenbsp;very apex of the cleft �, the effedl will be the fame.

The fcrew is the laft mechanical power that re' mains to be defcribed. This is likewife confiderednbsp;as a fpecies of inclined plane; it being in faft nO'

of chap. VIII.) refpeitively proportional to the thre^ fides of the triangle GOF; but this triangle GOFnbsp;equiangular, and therefore (Fuel. p. 4. B VI.) limilar,nbsp;the triangle ABC; therefore the three forces are likevvil�nbsp;refpedlively proportional to the three fides of the trianglenbsp;wedge ABC.

tbs

like manner is proved that the angle DBE is equal to th^ angle OGD, and likewife equal (Eucl. p. 29. B. I-)nbsp;�FOG. And lince the two angles at O and G of the'trinbsp;angle FGO are refpeflively equal to the two angles s�-and B of the triangle ABC ; the third angle of the fort��nbsp;mufi be equal to the third angle of the latter. .

The triangles GOF and ABC are equiangular; forth� four angles of the quatrilateral figure AEGF are equal t^nbsp;four right angles (Eucl. p. 32. B. I.) and fince the angk*nbsp;at E and F are right, the two angles FAE and FGE rntinbsp;be equal to two right angles; that is, equal to ,FGEnbsp;FGO. Therefore taking away the common angle FGE

filing more than an inclined plane coiled round a lt;^ylinder ; and the nut or perforated body whiclinbsp;ttioves up or down a ferew, moves up or down annbsp;inclined plane in a circular, inftead of a redlili-^lear, diredion.

Either the ferevv A may be moved forwards and backwards in a fixed nut as in fig. 6, Plate VII. ornbsp;the ferew A remains fixt, and the nut BC, or perfo- -J'ated piece, is made to move upon the ferew as innbsp;fig. y,�By way of diftindion A i's called the malenbsp;ferew, and the nut B with its perforation fhapednbsp;like an hollow ferew, is called the female ferew.�nbsp;quot;The Ipiral projedions e, f, g, kc. are called thenbsp;'threads oi the ferew.

The power which moves this moft ufeful, and hiofl powerful engine, is applied either to one endnbsp;the ferew, which is generally furnlfhed with anbsp;Ibrt of head or projedioii, or to the end of a lever �nbsp;^''hich is fixed either in the head of the ferew as innbsp;%� 6. or in the nut BC, as in fig. 7. And thennbsp;^ftdeed it may with more propriety be called annbsp;^^gine compounded of a ferew and a lever.

In all cafes the equilibrium takes place between die efFed which is produced at the end of the ferewnbsp;at the nut, and the power, when the former is tonbsp;latter as the circumference deferibed by thenbsp;Power in one revolution, is to the diftance betvyeennbsp;^'^0 contiguous threads of the ferew. Thus fup-Pofing that the diftance between the threads benbsp;half an inch and the length of the lever CD be la

inches ^ the circledeferibed by the end D of the lever , will be about 76 inches, or 152 times the diftancenbsp;between two contiguous threads; therefore if thenbsp;power at the end D of the lever be equivalent tonbsp;one pound, it will balance apreflure of 152 poundsnbsp;a�ting againfl; the end of the ferew in fig. 6 ; or itnbsp;will fupport a weight of 152 pounds on the boardnbsp;B, fig. 7, amp;c.

The reafon of this is fo evidently dependent on the properties of the inclined plane, that nothingnbsp;more needs here be faid about it.

Before we quit the prefent chapter it will be Proper to make the following remark, the objed ofnbsp;'vliich is to prevent the eftablifhment of wrong no-dons in the mind of the reader, with refped to thenbsp;Powers of the above-mentioned engines.

Beginners in this branch of natural philofophy frequently imagine that by means of the mechanical powers, a real increafe of power is obtained ;nbsp;whereas this is not true. For inftance, if a man benbsp;juft able to convey 100 weight from the bottom tonbsp;top of his houfe in one minute�s time, no mc-c'tanical engine will enabhe him to convey 300nbsp;quot;'eight to the fame height in the fame time; butnbsp;the engine will enable him to convey the 300nbsp;'quot;eight in three minutes; which amounts to the;nbsp;frrne thing as to fay that the man could, withoutnbsp;d'e engine, carry the 300 weight by going threenbsp;dtnes to the top of the houfe, and, carrying 100nbsp;quot;'eight at a time, provided the load admitted ofnbsp;being fo divided. Therefore the engine increafesnbsp;effed of a given force by lengthening the timenbsp;the operation; or (fince uniform velocity is pro-Pcgt;ttional to the time) by increafing the velocity ofnbsp;that force or power.

Thus again, if any adive force is able to raifo a quot;eight of 10 pounds with a given velocity, it willnbsp;'found impoffible, by the ufe of any inftrument,

to make the fame force ralfe a weight of 20 pounds� or in general a weight more than 10 pounds, withnbsp;the fame velocity; but it may, by the aid of thenbsp;inftruments, be made to raife the weight of 2^nbsp;pounds with half that velocity; or, which is tb^nbsp;fame thing, it may be made to raife it to half thenbsp;height in the fame time j for it is not the power ornbsp;force, but the momentum, (viz. the product of thenbsp;force by the velocity) that may be increafed or di'nbsp;minithed by the ufe of thofe engines.

The power, or acting force, is fo far from being increafed by. any machine, that a certain part of hnbsp;is always loft in overcoming the refiftance of mC'nbsp;diums, the fridion, or other unavoidable imperfections of machines. And this lofs in fome compound engines is veiy^ confiderable.

All the Inftruments or maclrines which conned an adive force with a certain effed, ^towever complicated they may be, will, upon examination, be found to confift of the already described mechanical powers. Thofe componentnbsp;Simple mechanifms are frequently varied in lhape jnbsp;tlielr connedtion is infinitely diverfified ; but theirnbsp;*^ature and their properties remain invariably the

Ss^me.

' Various are the powers which have been employed as firfl: movers of machines; but the prin-'^Ipal of them are, i. The natural ftrength of a man number of men. 2. The ftrength of other animals, and principally of horfes. 3. The force ofnbsp;^tinning water. 4. The force of the wind. 5. Thenbsp;^l^ftic force of the fteam of boiling water. 6. Thenbsp;^l^'ftic force of fprings. J. The fimple weight ofnbsp;S'cavy bodies.

-V great part of moft machines relates to the power itfelf � viz. it confifts of contrivances necef-S^�'y for the generation, application, prefervation,nbsp;^od renovation, of the adive pow er or force. The

' efied

effeft which is to be produced by that aftive force, is derived from the proper application of the above-mentioned fimple mechanical powers ; which divide, or concentrate, or regulate, the original force.nbsp;As for the effe�ts which are produced by machines,nbsp;it is impoffible to afcertain their number, or evennbsp;to arrange them under general and comprehenlivenbsp;titles.

The beft machine for the production of any particular effeft, is that, which (all circumftances of iituation, materials, amp;c. being confidered) will produce that effedt in the fimpleft, fteadieft, fafeft,nbsp;and cheapeft manner poffible.

It is not my intention to defcribe the principal machines, that are now in ufe amongft the enlightened nations of the world that being incompatible with the nature and the limits of the prefen^nbsp;work. Thofe perfons who m.ay be defirous of examining the peculiar conftrudlions of the varionsnbsp;engines of arts, manufadlures, navigation, aeco-nomy, amp;c. will find a great variety of books writteiinbsp;ekprefsly on the fubjecf, in almoft every languagenbsp;of Europe.; but in none more fo than innbsp;French.

In the prefent chapter, the methods of cooi-puting the powers and the effedts of machines general, will be briefly ftated j and the defcriptgt;�'^nbsp;of a few mechanifms will be inferted merely fornbsp;purpofe of exemplifying the application ofnbsp;methods j whence the reader may be enabled

iudg^

judge of the power and effedtof any other machine that may fall under his examination.

A compound engine either confifts of one limple mechanical power repeated two or more times; or,nbsp;confifts of feveral fimple mechanical powersnbsp;''^arioufly combined, and connedted with each other.

any cafe, the power and the effeft muft be efti-mated from the refult of the effedfs of all the component fimple mechanifms feparately confidered, tvhich is done in the following manner ;

Find w'hat proportion the power bears to the in each fimple mechanifm ; put all thofe ratiosnbsp;^ue under the other, and find their fum, which fumnbsp;'Vili exprefs the proportion between the power andnbsp;^he effe�t of the whole compound engine ¹. Thusnbsp;^�ppofe that a machine is compounded of threenbsp;^�mple mechanical powers, viz. a lever, an inclinednbsp;plane, and a moveable pulley ; and fuppofe that anbsp;Power applied to one end of the lever will producenbsp;^ double effedt at the other end ; for inftance, onenbsp;Pound will balance two pounds; then the propor-don of the power to the effedt, is as one to two.nbsp;^^Ppofe alfo that in the inclined plane, the power

The fum of two or more ratios is obtained by multi-Ply'og the antecedents together and the confequents toge-j and the two produdts will form a ratio, which is cal-*^he fum of the given ratios. Thus the ratio of 2 to 3, ratio of 2 to 5 ; plus the ratio of 4 to 7, is equal

256 nbsp;nbsp;nbsp;Q/quot; Compound. Engines, feV.

IS to the effect as three to feven �, and laftly, that i�t the pulley, the power is to the effe6t as one tonbsp;two. Now thofe three ratios being'written on^

' � nbsp;nbsp;nbsp;and

3:7

together, the two produdts thence arifing will eX' hibit the 'proportion which the power bears to tb^nbsp;effect in the whole compound engine �, viz. that ^nbsp;power of 3 pounds will balance a-weight ofnbsp;pounds.

Otherwife the effeft of a compound engine be computed by confidering the velocities of tb^nbsp;power and of the effedt; for they are to each otbc��nbsp;inverfely as their velocities, viz. the power is to tb�nbsp;effedt as the velocity of the latter is to that of tb�nbsp;former. Thus in a certain compound engine Inbsp;that the power muff move through 500nbsp;whilft the weight moves through 3 feet; I thetf'nbsp;fore conclude that a power of 3 pounds will '0^'nbsp;lance a w'eight of 500 pounds in that machine,nbsp;of courfe a little more than 3 pounds willnbsp;the 500 pounds weight.

Fig. 8, Plate VII. reprefents an engine of three levers CD, DG, GH, each of which rnoquot;'�^*nbsp;round a pin or axis fixed to a fteady poft, andnbsp;difpofed fo as to adt upon each other. Let

DE, GK be each one foot long, and AD, EG,

t)e each three feet long, then the weight I of one pound will produce a preflure of three pounds at Gnbsp;On the end of the next lever, and this force willnbsp;produce an effedl equal to 3 times 3^ or 9 pounds.nbsp;On the end D. Lallly, the end D of the thirdnbsp;^ever CD being preffed downwards with a force ofnbsp;9 pounds, will balance or keep fufpended at thenbsp;oppofite end C the weight W, of 3 times 9, viz-,nbsp;of zj pounds. And a little addition of power tonbsp;llie end H will enable the engine to lift up the

By increafing either the number of leversj or the 'difference of length between the two parts of eachnbsp;d^Ver, the effe�; at the end C may be i�rcreal�d tonbsp;^oy degree. But the great defeCl; of this engine is,nbsp;'1'at the end C with the weight W can be raifed anbsp;^^ry fhort Wayi

^ The efFeft of this compound engine maybe calculated ^'�'^erding to the preceding rule, by fetting down the pro-I�^ftion between the power and the efFe�l in each of thenbsp;'We levers; then multiplying the antecedents together andnbsp;^ � confequents together, their products will give the an-^ hus llnce the two parts of each lever are as one tonbsp;* therefore we have - - i ;nbsp;nbsp;nbsp;nbsp;3

Wich Ihews, that in order to form the equilibrium, the power I muft be to the weight W as I to 27, the fame asnbsp;�The fame th tno- moir n/� ixflfK ritllpr f'nCTineS.

258 nbsp;nbsp;nbsp;Of Compound Engines, iS'c.

Fig. 9, Plate VII. reprefents a combination four pulleys, three of which are moveable and onenbsp;is fixt. But this combination muft not be reckonednbsp;a Angle mechanical power, becaufe the lame ropenbsp;does not run over all the pulleys. It is, therefore, ^nbsp;repetition of one and the fame mechanical power,.

H, nbsp;nbsp;nbsp;and K. The firft rope goes round the pulleynbsp;A, to the block of which the weight W is faftened;nbsp;and is then tied to the hook of the block �f B-The fecond rope goes round the pulley B, andnbsp;faftened to the hook of the block of C. The third'nbsp;rope goes round the pulley C, as alfo round thenbsp;fixt pulley Dj and holds the counterpoife or power

The pulley D being fixt to th^ beam, does nothin^ more than change the direftion of the motion?nbsp;therefore if the power I weigh one pound, it wi^nbsp;balance a weight of two pounds affixed to th^nbsp;block of the pulley C. Then the pulley C acfl*^�nbsp;with a power �f two pounds, will balance a weigl��^nbsp;of tw�ice two, or of 4 pounds aflixed to the blool^nbsp;of the pulley B, and this will balance a weightnbsp;of twice 4, or 8 pounds, affixed to the block ofnbsp;pulley A ; fo that in order to pull up the weightnbsp;of 8 pounds, the power I needs be very little ho*^nbsp;vier than one pound.

This engine is fubjeft to the fame inconvenieti^� as the preceding; viz. the weight W can be rau^^nbsp;but a very little way.

Fig. 10, Plate VII. reprefents a combination of five Mieels A, B, C, D, E; each of which turns roundnbsp;^ centre-pin, which is fuppofed to be fixed to anbsp;fteady frame. Thofe wheOls are conneftcd withnbsp;other in the following manner: The wheel Anbsp;a fmall wheel or pinion o faftened to, and con-'^entric with, itfelf. This pinion is furnifhed withnbsp;^^eth, which move between the teeth on the circumference of the next wheel, which is likewifenbsp;^'irnilhed with a pinion which acts in a fimilarnbsp;'banner on the next wheel, and fo on, exceptingnbsp;laft, which has an axle inftead of a pinion,nbsp;round this axle a rope is applied, to which thenbsp;^^ight W is fufpended.�The power I is applied tonbsp;circumference of the firft wheel.

'This engine confifts of a repetition of the wheel axle; for the pinion of each wheel is in fadt itsnbsp;excepting that inftead of acting immediatelynbsp;!^P'On the weight by means of a rope, here it exerts

^et the circumference of each wheel be equal fo times the circumference of its pinion. Then ifnbsp;^''cight I of one pound be fufpended to the cir-'�^mference of the wheel A, the pinion o will adfcnbsp;the circumference of the fecond wheel with anbsp;^cce equal to five times the power I, viz. equal tonbsp;^ pounds, and this force of 5 pounds on the cir-l^'trference of the fecond wheel will enable its pi-to adl on the circumference of the third wheelnbsp;' s znbsp;nbsp;nbsp;nbsp;with

with a force equal to 5 times 5, viz. 25 pouni3^�' After the fame manner the force of 25 pounds 011nbsp;the circumference of the third wheel will enable itsnbsp;pinion to adl on the circumference of the fourthnbsp;wheel with a force equal to 5 times 25; viz.nbsp;pounds, in confequence of which force applied tothsnbsp;circumference of the fourth wheel, the pinion ofnbsp;wheel will aft on the circumference of the laft whee^nbsp;with a force equal to 5 times 125; viz. 625 pounds?nbsp;w'hich force will balance a weight W of 5 times 625�nbsp;viz. �f 3125 pounds. Therefore the power Inbsp;one pound will balance the weight W ofnbsp;pounds.

tbe axle E as 8 to one, then the power of 100 pounds on the circumference of the wheel C willnbsp;,act with a force equal to 8 times 100; viz. of 800nbsp;pounds on the circumference of the axle E, aboutnbsp;^'hich the rope of the weight W is wound. Therefore it appears that with this engine a weight ornbsp;Power of one pound will balance a weight W ofnbsp;^00 pounds.

A fcrew, like D, fituated fo as only to turn round axis, but without moving backwards and for-'^�ards, and always working on the circumference ofnbsp;^ 'vheel, as C, is ufually called an endlefs fcrew.

Pig. 12, Plate VII. reprefents an improved crane for railing of goods or heavy weights. This de-f-^iption has been taken from the appendix to Mr.nbsp;^quot;^rgufon�s Leftures, which I have preferred tonbsp;other defcriptions of fimilar engines; firft, on account of the improvements it contains, which willnbsp;'^^�turally (hew that a variety of collateral objedsnbsp;^0(1 be kept in view by the contrivers of fuchnbsp;^^ohines; and fecondly, for the purpofe of makingnbsp;reader acquainted with the meaning of thenbsp;h^'ticipal terms that are ufed in mechanics.

is the great wheel of this engine, and B its on which the rope C winds. This rope goesnbsp;^''or a pulley D in the end of the arm of thenbsp;* E, and draws up the weight F, as the

winch¹ G is turned round. H is the largeft trundlC'f gt; I the next, and K is the axis of the fmalleft trundle, which is fuppofed to be hid from view by thenbsp;upright fupporter L. A trundle M is turned by thenbsp;great wheel, and on the axis of this trundle is fixednbsp;the ratchet-wheel ^ N, into the teeth of which thenbsp;catch O falls. P is the lever, from which goesnbsp;rope QQ, over a pulley R, to the catch; one endnbsp;�f the rope being fixed to the lever, and the othetnbsp;end to the catch. S is an elaftic bar of wood, on^nbsp;end of which is fcrev.'ed to the floor ; and, fromth^nbsp;other end goes a rppe (out of fight in the figut^)nbsp;to the farther end of the lever, beyond the pinnbsp;axis on which it turns in the upright fupporternbsp;The ufe of this bar is to keep up the lever andnbsp;prevent its rubbing againft the edge of the whe^^

Winch or nvinder, an inftrument with a crooked the ufe of,which is to turn any thing round.

f A frnall wheel, which is turned round by the teeth n large wheel, which derives various denominations from ^nbsp;various fhapes. It is called pinibn when it is oblonge ^nbsp;the teeth are longer than the inftde folid part; the teeth � ^nbsp;then called the leaves of the pinion. When the fmallnbsp;is�fliapedlike that which is reprefented at H, it is then canbsp;a trundle, or fometimes a lantern, and even a drum- , ^

its teeth bent one way, wherein a folid piece, called a co click, falls, by which means the ratchet-wheel, when thenbsp;bears upon it, can turn one way only, but not the connbsp;way.nbsp;nbsp;nbsp;nbsp;�A

V, and to let the catch keep in the teeth of the ratchet-wheel. But a weight hung to the farthernbsp;end of the lever would do full as well as the elafticnbsp;bar and rope-

When the lever is pulled down it lifts the catch Out of the ratchet-wheel, by means of the rope QQ,nbsp;and gives the weight F liberty to defcend; but ifnbsp;the lever P be pulled a little farther down than.nbsp;Miat is fufficient to lift the catch O out of thenbsp;I'atchet-wheel N, it will rub againft the edge of thenbsp;'''heel y, and thereby hinder the too quick defcentnbsp;of the weight, and will quite flop the weight, ifnbsp;pulled hard. And if the man who pulls the levernbsp;^lould inadvertently let it go, the elaftic bar willnbsp;h-iddenly pull it up, and the catch will fall downnbsp;^ud ftop the machine-

WW are two upright rollers above the axis or ''Pper gudgeon * of the gib E, Their ufe is to letnbsp;^he rope C bend upon them, as the gib is turned

either fide, in order to bring the weight over the place where it is intended to be let down.

N. B. The rollers ought to be fo placed, that if ^he rope C be ftretched clofe by their utmoft fides,nbsp;*he half thicknefs of the rope may be perpendicu-larly over the centre of the upper gudgeon of thenbsp;�b, p'or then, and in no other pofition of the

re pins or extremities of an axle, which pins move in holes, amp;c. are called gudgeons in large works, and pevets ornbsp;t'lvots in fmall works.

rollers, the length of the rope between the pulley iw the gib, and the axle of the great wheel, will be abnbsp;ways the fame, in all pofitions of the gib, and thenbsp;gib will remain in any pofition to which ifnbsp;turned.

When either of-the trundles is not turned by the winch in working the crane, it may be drawn of\nbsp;from the wheel, after the pin near the axis of th^nbsp;trundle is drawn out, and the thick piece ofnbsp;wood is raifed a little behind the outward fupportefnbsp;of the axis of the trundle. But this b not mate^nbsp;rial; for, as the trundle has no friction ori its aXis,nbsp;but what is occafioned by its weight, it willnbsp;turned by the wheel without any fenfible refiftancenbsp;in working the crane.

This engine is to be fituated in a room with th? gib E projeding out of it, fo that the load maynbsp;raifed from the ftreet or other low'er fituation, bynbsp;turning the winches of the trundles as at G.

This crane has four different powers, The thr^^^ trundles H, I, K, are furnifhed with different iiutO'nbsp;bers of ftaves¹; the largeft has 24 ftaves, the ne^¹-12, and the fmalleft 6. The great w'heel A has 9^nbsp;CogSquot;fj therefore the largeft trundle makesnbsp;revolutions for one revolution of the wheel;

The flicks or cylindrical bars of trundles, which form the office of teeth, are called ftaves,

\vinGh G is occafionally put upon the axis of either of thefe trundles, for turning it; that trundle beingnbsp;then ufed which gives a power bed fuited to thenbsp;V^eight. The length of the winch is fuch, that innbsp;?very revolution its handle defcribes a circle equalnbsp;to twice the circumference of the axle B of thenbsp;Wheel. So that the length of the winch doublesnbsp;the power gained by each trundle.

If the winch be put upon the axle of the larged trundle, and turned four times round, the wheelnbsp;and axle will be turned once round; and the circlenbsp;defcribed by the power that turns the winch, beingnbsp;in each revolution double the circumference of thenbsp;^xle, when the thicknefs of the rope is addednbsp;thereto, the power goes round 8 times as muchnbsp;^Pace as the weight rifes through ; and thereforenbsp;(niaking fome allowance for fri(d:ion) a man willnbsp;laife 8 times as much weight by the crane as henbsp;''�'Quid by his natural flrength without it; thenbsp;Pnwer, in this cafe, being to the weight as 8 to i.

If the winch be put upon the axis of the next '^I�Undle, the power will be to the weight as 16 to i,nbsp;^^caufe it moves i6 times as fad as the weightnbsp;Proves.

If the winch be put upon the axis of the fmalled dundle, and turned round, the power will be to,nbsp;the Weight as 32 to i.

�^rit, if the weight Ihould be too great even for this power to raife, the power may be doubled, bynbsp;^wing up the weight by one of the parts of a

double

double rope, going under a pulley in the moveable block which is hooked to the weight below the armnbsp;of the gib. And fuch is the a�tual reprefentationnbsp;of the figure. Then the power will be to the weightnbsp;as 64 to I. Whilft the weight is drawing up, thenbsp;ratch-teeth of the wheel N flip round below thenbsp;catch or click that falls facceflively into them, andnbsp;thus hinders the crane' from turning backward, ornbsp;detains the weight in any part of its al'cent, if thenbsp;man, who works at the winch, fliould accidentlynbsp;quit his hold, or choofe to reft himfelf before thenbsp;weight be quite drawn up.

In order to let down the weight, a man pulb down the end Z of the lever, which lifts the catchnbsp;out of the ratchet-wheel, and gives the weight liber-ty to defeend. But, if the defeent be too quick�nbsp;'he pulls the lever a little farther down, fo asnbsp;make it rub againfh the outer edge of the roundnbsp;wheel quot;V, by which means he lets down the weightnbsp;as flowly as he pleafes; and by pulling a lid^^nbsp;harder, he may flop the weight, if neceflTary, in anynbsp;part of its defeent. If he accidentally quits holdnbsp;the lever, the catch immediately falls, and flop�nbsp;both the weight and the whole machine.

In the conftruftion of machines in general, th� queftions which occur in the firft place, relatenbsp;the choice of the power, and to the eftimationnbsp;its quantity; -viz. whether the force of wind ^rnbsp;water, or of a man, kc. fliould be preferred,

v\'hn*^

Of the principal aftive powers which have been enumerated at the beginning of this chapter, wenbsp;fhall briefly mention the common eftimation ofnbsp;their forces ; but we fliall take more particular notice of the force of wind, of water, and of fteam,nbsp;vvhen we come to treat of the properties of air, ofnbsp;the fheam of water, amp;c.

The power which can be applied as the firfl; mover' of a machine, in the eafieft manner, andnbsp;whofe a�llon is moft uniform, is the Ample weight,nbsp;fuch as is applied to clocks, jacks, and other machines ; but this fort' of power requires to be renewed after a certain period; that is, it muft benbsp;Wound up, or raifed,on which account it is moftlynbsp;tifed for flow movements; efpecially when a verynbsp;regular a�lion is required.

The force of running water, and that of the wind, where the fituafion of the place admits of their being ufed, are very powerful and advantageousnbsp;rnovers of machines, fuch as mills, pumps, flawingnbsp;engines, amp;cc.�They maybe applied to the workingnbsp;of the greatefl: engines. Running water is preferable to wind, on account of its afting with muchnbsp;tnore conftancy and uniformity.

The fleam of boiling water is likewife a mofl; powerful agent; and the recent Improvements whichnbsp;have been made by fevcral ingenious mechanics innbsp;^his country have extended the application of it

from

from the fmalleft to the largeft engines. The application of this power requires a very nice con-ftruftion of the mechanifm, and is attended with a confiderable cotifunipti-on of fuel, which particular?nbsp;are not to be obtained in every fituation.

A fpring is likewife a ufeful and commodious moving power ; but a fpring, like the weight, requires to be wound, or fet up, after a certain time ;nbsp;viz. when it is quite unbent; on which account itnbsp;is more commonly ufed for flow movements, fuchnbsp;as watches, table clocks, amp;c. But this fort ofnbsp;power differs from the weight in a very remarkablenbsp;circumflance; which is, that its aftion is never uni-forrn. Jt is ftrongeft when m�ft bent, and it de-creafes in proportion as it unbends. .

In order to reriiedy this defeft, and to render the adtion of a fpring uniform and effedlual, a curiousnbsp;contrivance has been long in ufe, and is as follows:nbsp;An hollow groove of a fpiral form is made round anbsp;folid piece of metal, fuch as is reprefented at fig-13, plate VIT. which is furnitbed with an axis ab,nbsp;round which it turns in the frame of the machine�nbsp;and is connected with a wheel g, whofe teeth achnbsp;upon the other wheels of the machine. This fpU^^nbsp;piece is called the fiifee, and ferves to render thenbsp;addon of the fpring equable or uniform. Bnbsp;connedled with the fpring by means of a firingnbsp;chain F, one end of which is faftened to the fpn^^�nbsp;which is not feen in the figure, and the other end

When the fufee is turned fo as to wind the firing or chain upon it, the fpring is thereby fet up, or bent,nbsp;and when afterwards the machine is left to itfelf,nbsp;the force of the fpring will, by pulling the chain ornbsp;firing, force the fufee to turn round its axis in anbsp;diredlion contrary to that in which it was woundnbsp;up. Now when the firing bears upon the fmalleftnbsp;part c of the fufee; viz. neareft to the axis wherenbsp;a greater force is required to produce- a certain ef-ftft, the fpring pulls the chain with its greateftnbsp;force, becaufe it is then bent moft; whereas when,nbsp;the firing bears upon the lower and larger part ofnbsp;the fufee, where lefs force is required to produce thenbsp;above-mentioned effc�l, there the fpring pulls thenbsp;ffring or chain with lefs force, becaufe then it isnbsp;bent lefs. Therefore the decreafing force of thenbsp;fpring is compenfated by the increafe of power withnbsp;tvhich the firing or chain adls on the axis AB;nbsp;hence the teeth of the wheel g aft always with thenbsp;fame degree of force upon the next wheel j andnbsp;^hus the motion of the mechanifm is rendered uniform. This mechanifm will ,be found almofinbsp;tiniverfally applied to pocket watches and fpringnbsp;clocks.

The natural firength of living animals is the lafl power that remains to be taken notice of; and herenbsp;flrall not extend our obfervations beyond thenbsp;force of men and horfes, concerning which the following particulars are deferving of notice. Thenbsp;different writers on mechanics do not quite agree

in the eftimation of the mean ftrength of a man J nor is it likely they Ihould, confidering how thenbsp;conftitution of men varies according to the difference of climate, of nourilhment, and of othernbsp;circumftances. Upon the whole, a man of ordinary ftrength is reckoned capable of raifing a weightnbsp;of 600 pounds avoirdupoife ten feet high in onenbsp;minute, and to be able to work at that rate for 10nbsp;hours out of 24; or to do any other work thatnbsp;may be equivalent to it. I am however inclinednbsp;to think that this eftimation is rather above thannbsp;below the real facft.

By means of a judicious application of the human ftrength, the effeft may in fome cafes be increafed,nbsp;and on the other hand an improper application ofnbsp;it will diminifli the effed. Thus if two men w'orknbsp;at a windlafs, or axle, by means of handles or levers, they will be able to draw a weight ofnbsp;pounds more eafily than one man can a weight ofnbsp;30 pounds, provided the handies or levers are atnbsp;right angles to each other.

A man is able to draw horizontally not above 70or 80 pounds; for in that cafe he can only employ half the weight of his body.

N. B. Here it is not to be underftood that a man cannot draw a cart or carriage that weighsnbsp;more than 80 pounds; for the weight of the cartnbsp;is fupported by the ground ; but we mean, that anbsp;man will not be able to draw fuch a cart as vviH

require more than 8p pounds to move it along.

If a man weigh about 140 pounds, he can exert Ho greater force in thruftiiig horizontally at a heightnbsp;even with his fhoulders, than what is equal to 27nbsp;Pounds.

A horfe draws with the greateft advantage when the line of direamp;ion is a little elevated above thenbsp;horizon ; and the power adts againft its breaft.

A horfe is reckoned capable of drawing againft a refiftance of 200 pounds at the rate of 2I miles annbsp;hour, and to continue that exertion for eight hoursnbsp;�^ut of 24.

The ftrength of the horfe, like that of a man, ^ay be rendered more or lefs efficacious, by meansnbsp;^f a proper or improper application of it. In gortig up a fteep afcent, five men can carry a muchnbsp;heater weight than a horfe. And in certain af-^^nts, n;en will be able to carry fome weight, wherenbsp;^ horfe will not be able to carry himfelf. In millsnbsp;^iid other machines, where the circular motion of anbsp;horfe is employed, the diameter of the circularnbsp;Ihould not be lefs than or 30 feet; other-the motion is neither very advantageous, nornbsp;h'eafant to the animal. With refpe�t to the quan-of power, it muft be obferved, that in thenbsp;hradfical application of a moving power to a ma-'^hine of any fort, it is not enough to employ anbsp;Power which is barely fufficient to overcome thenbsp;'^hftacle, or to produce the effedt; but fuch a

power muft be applied as will produce the defiref^ efFeft in the moft advantageous manner poflible}nbsp;for inftance if a power of a pound, applied to 3.nbsp;machine, will produce a certain etfecl in one hour inbsp;whereas if a power of two pounds v/ere applied Wnbsp;the fame m.achine, it would produce the like ef'nbsp;fedl in 20 minutes, it is evident, that the application of the latter power would be more advaH'nbsp;tageous than of the formerj for though the latte*'nbsp;power be double the former, yet the time of i**nbsp;performing the operation is lefs than half the tinisnbsp;of the former powers performing the fame operation. So that the nioji advantageous powernbsp;moving a machine^ is that, which being midtiplkd hjnbsp;the time of perfbming a determined effeSi, product-the leaf produH*.

With refpect to friction, two objects mud: oblerved ; viz. the lofs of power which is occa-floned by it, and the contrivances which have bee*^nbsp;made, and are in ufe, for the purpofe of diminifh'nbsp;ing its effects.

A body upon an horizontal plane drould be capable of being moved by the application of leaft force �, but this is not the cafe ; and the pri**'nbsp;cipal caufes which render a greater or lefs quantify

* For a farther inveftigation of the moft advantage�^^ application of powers to machines, fee Gravefand�snbsp;Elem. of Nat. Phil. B. I. chap. 21, and the foUovri*'�nbsp;fcholia j alfo alnioft all the writers on mechanics.'

cf force neceflary for it, are, ift, the roughnefs of ^he contiguous furfaces; adly, the irregularity ofnbsp;die figure, which arifes either from the imperfedtnbsp;'Vofkraanfliip, or from the� preflure of One bodynbsp;^pon the other; 3dly, an adhefion or attradlionnbsp;''�hich is more Or lefs powerful according to thenbsp;Mature of the bodies in queftion; and 4thly, thenbsp;^uterpofition of extraneous bodies ; fuch as moif-'^Ure, dull. See,

Innumerable experiments have been made for purpofe of determining the quantity of ob-ftruftion, or of friftion, which is produced in par-dcular circumftances*. But the refults of appa-^^�itly fimilar experiments, which have been madenbsp;different experimenters, do not agree ; nor is itnbsp;likely they Ihould, fince the leaft difference ofnbsp;ktioothnefs or polifh, or of hardnefs, or in fhoft ofnbsp;of the various concurring circumftances, produces a different refult. Hence no certain and de-^^JTtiinate rules can be laid down with refpedl: tonbsp;fubjecl of fridlion.

If a body be laid upon another body, and foon ^fter be moved along the furface of it, a leffernbsp;lurce will be found fufficient for the purpofe, thannbsp;fhe body be left fome time at reft before it benbsp;Uioved. This arifes principally from an a�lual

* See Mr. Coulomb�s Effay in the tenth vol, of the ^ �tnoires des Savants Etrangers. And M. de Prony�s Ar-^ kedtiire Hydraulique, � 1089, and following.

I. nbsp;nbsp;nbsp;Xnbsp;nbsp;nbsp;nbsp;change

change of figure, which is produced in a longer fhorter time according to the nature of the bodies*nbsp;Thus the maximum of adhefion between woodnbsp;and wood takes place in a few minutes� time ; be'nbsp;tween metal and metal it takes place almoft immS'nbsp;diately. A hard and heavy body laid upon a foftetnbsp;one will fometimes continue to increafe its adhe'nbsp;lion for days and weeks.

When a cubic foot of foft wood of eight pounds weight is to be moved upon a fmooth horizontalnbsp;plane of foft wood, at the rate of three feetnbsp;fecond, the power which is neceflary to move ibnbsp;and which is equivalent to the fridfion, amountsnbsp;between f and ^ of the weight of the cube.-^nbsp;When the wood is hard the fridlion amounts to be'nbsp;tween y and 4- of the weight of the cube.

In general the fofter or the rougher the bod'^* are, the greater is their friction. Yet when tquot;''^nbsp;pieces -of metal, extremely well poliflied, are 1^'^nbsp;one upon the other with an ample furface of co^'nbsp;tadt, they adhere to each other much more forciblynbsp;than when they are not fo well polifhed.

Iron or ileel moves eafieft in brats. Other tals, acting againft each other, produce more fi'**'nbsp;tion.

The fridtion, cateris paribus, inereafes w'ith th^ weight of the fuperincumbent body, and almoft *.nbsp;the tame proportion.

by a variety of circumftances, fuch as their peculiar qualitygt; the temperature of the atmofphere, and the diameter or curvature of the furface tonbsp;which they are to be adopted* But when othernbsp;circumftances remain the fame, the difficulty ofnbsp;bending a rope irtcreales with the fquare of its diameter, as alfo with its tenfion ; and it decreafes according as the radius of the curvature of the bodynbsp;to which it is adapted, increafes.

In a wheel, the fridion upon the axis is, as the Weight that lies upon it, as the diameter of the axis*nbsp;and as the velocity of the motion. But upon thenbsp;whole, this fort of fridion is not very great, provided the machine be well executed.�In commonnbsp;pulleys, efpecially thofe of a fmall fize, the fridionnbsp;is very great. It increafes in proportion as the diameter of the axis increafes, as the velocity increafes, and as the diameter of the pulley decreafes,nbsp;^ith a moveable tackle, or block, of five pulleys,nbsp;^ porver of 150 pounds will barely be able to drawnbsp;cip a weight of 500 pounds.

The fcrew is fubjed to a great deal of fridion j lb much fo that the power which muft be appliednbsp;lo it, in order to produce a given effed, is at leaftnbsp;double that which is given by the calculation independent of fridion. But the degree of fridionnbsp;m the fcrew is influenced confiderably by the nature of the conftrudion j for much of it is owing

to the tightnefs of the fcrew, to the diftance between its threads, and to the fhape of the threads j the fquare threads, like thofe of fig. 14, Plate Vll-producing upon the whole lefs fridfion than thofenbsp;which are fharp, as in the figures 6 and 7 of thenbsp;fame plate.

The fridlion which attends the ufe of the xvedgc, exceeds, in general, that of any other fimple mechanical power. Its quantity depends fo muchnbsp;upon the nature of the body upon which the wedgenbsp;adts, befides other circumftances, that it is impof-fible to give even an approximate eftimate of it.

The fridlion of mechanical engines does not only diminifli the efFedf, or, which is the fame thing�nbsp;occafion a lofs of power; but is attended with thenbsp;corrofion and wear of the principal parts of the machine, befides producing a confiderable degree ofnbsp;heat, and even adlual fire ; it is therefore of greatnbsp;importance in mechanics, to contrive means capable of diminifliing, if not of quite removing, th^nbsp;effedts of friction.

In compound engines, the obftrudlion which arifes from fridtion can be afeertained only by meansnbsp;of actual experiments. An allowance, indeed, maynbsp;be made for each fimple component mechanicalnbsp;power; but the error in eftimating the fridtion ofnbsp;any one fingle power is multiplied and increafe^nbsp;fo faft by the other parts, that the eftimate generally turns out very erroneous. Befides, much depends on the execution of the work; the quality

of which cannot be learned but by experience. Novices are generally apt to expeft too muchnbsp;or too little from any mechanifm,�In generalnbsp;It can only be faid, that in compound engines, atnbsp;lead one-third of the power is loft on account ofnbsp;the fri�tion.

The methods of obtaining the important ob-jed of diminhhing the fridion, are of two forts, viz. either by the interpofition of particular unftu-ous or oily fubftances bctv/een the contiguousnbsp;moving parts; or by particular mechanical contrivances.

Olive-oil is the beft, and perhaps the only fub~ ftance that can be ufed in fmall works, as innbsp;Watches and clocks, when metal works againft metal. But in large works the oil is liable to drain off,nbsp;tmlefs fome method be adopted to confine it.nbsp;Therefore for large works tallow is moftly ufed, ornbsp;greafe of any fort; which is ufeful for metal, asnbsp;Well as for wood- In the laft yafe tar is alfo fre-,nbsp;*l�ently ufed.

In delicate works of wood, viz, when a piece of ''^ood is to Hide into or over wood, and when anbsp;Wooden axis is to turn into wood, the fine powder

what is commonly called hlack~kad, when inter-, Pofed between the parts, eafes the motion ^onfider-and is at the fame time a clean and durablenbsp;^^bftance

Though olive-oil be the beft and the only fub-^ance that is ufed for delicate iwechanifms; yet it is

T j nbsp;nbsp;nbsp;far

far from being free from objeflions. Oil, when in contaft with brafs, is liable to grow rancid, in whichnbsp;ftate it llowly corrodes the brafs. In differentnbsp;temperatures it becomes more or lefs fluid; butnbsp;upon the whole it grows continually thicker, and ofnbsp;courfe lefs fit to eafe the motion of the parts, amp;c.nbsp;Trifling as thofe defects may at firft fight appear,nbsp;they are however of fuch moment in delicate works,nbsp;that in the greatly improved ftate to which watch-work has been brought in this country, the changeable quality of the oil feems at prefent to be thenbsp;principal, if not the only, impediment to the per-fedtion of chronometers.

The mechanical contrivances which have been made, and are in ufe, for the purpofe of diminlfliin�nbsp;the effe�s offridion, confift either in avoiding th^nbsp;contad of fuch bodies as produce much frictioo�nbsp;or in the interpofitlon of rollers, viz. cylindrical bodies, between the moving parts of machines, or between'moving bodies in general. Such cylindersnbsp;derive, from their various fize and application, th^nbsp;different names of rollers, friSiion-zvheels, and/^*^'nbsp;iion-r oilers.

Thus in mill-work and other large machines� wooden axes of large wheels terminate in iron g^^nbsp;geons, which turn in wood, or more frequentlynbsp;iron or brafs, which conftrudion produces lefs fti*-tlon than the turning, of wood in wood. Innbsp;fineft fort of watch-work the holes are jewelled,nbsp;many of the pivots of the wheels, amp;c. moye

holes made in rubies, or topazes, or other hard Hone, which when well finhbed are not liable tonbsp;Wear, nor do they require much oil.

In order to underftand the nature of rollers, and the advantage with which their ufe is attended, itnbsp;tnufl be confidered, that when a body is draggednbsp;over the furface of another body, the inequalities ofnbsp;the furfaces of both bodies meet and. oppofe eachnbsp;other, which is the principal caufe of the fridfionnbsp;or obftruftion ; but when one body, fuch as a calk,nbsp;a cylinder, or a ball, is rolled upon another body,nbsp;the furface of the roller is pot rubbed againft thenbsp;other body, but is only fucceffively applied to, ornbsp;laid on, the other j and is then fucceffively liftednbsp;Op from it. Therefore, in rolling, the principalnbsp;caufe of fri�tion is avoided, befides other advantages ; hence a body may be rolled upon anothernbsp;body, when the fhape admits of it, with incomparably lefs exertion than that which is required tonbsp;drag it over the furface of that other body. Innbsp;fafl; we commonly fee large pieces of timber, andnbsp;enormous blocks of done, moved upon rollers, thatnbsp;laid betw'een them and the ground, v/ith eafenbsp;^od fafety ; wheji it would be almofl impoffible tonbsp;roove them otherwife.

The form and difpofition of frldlion-wheels is re- � Prefented by fig. i. Plate VIII. which exhibits anbsp;front view of the axis d of a. large wheel, whichnbsp;Cloves between the friftion-wheels A, B, C* Herenbsp;^be end d of the axis (and the fame thing muft be

T 4 nbsp;nbsp;nbsp;underftood

tnuch weight as five men were barely able to ac-complifh with a limilar fet of common pulleys.

One of thofe pulleys is reprefented by fig. 2, Plate VlII. where the flraded part BBB is the pulley, Anbsp;is the axis, and c, c, c, c, c, c, are the cylindrical rollers, which are- fituated between the axis and thenbsp;infide cavity of the pulley. The ends of the. axisnbsp;h, are fixed in a block, after the ufual manner.nbsp;Tvery one of the rollers has an axis, the extremities of which turn in holes made in two brafs ornbsp;iron flat rings, one of which is vifible in the figure.

After having given a general explanation of the aftion of rollers, the advantage which Mr. Garnett�s pulleys muft have over thofe of the commonnbsp;fort, needs no farther illuftration. I fliall howevernbsp;Only obferve, that the friflion of the pivots of eachnbsp;roller in the holes of the brafs rings is very incon-fiderable j for thofe holes are made rather large,nbsp;the ufe of the axes to the rollers being only tonbsp;prevent their running one againft the other. Nornbsp;floes the addition of weight upon the pulley in-oreafe that friftion; for the addition of weightnbsp;^^pon the pulley will prefs the rollers harder uponnbsp;the axis A; but not upon their own axes, as may

DESCRIPTION OF THE PRINCIPAL MACHINES, WHICH ARE NECESSARY TO ILLUSTRATE THEnbsp;DOCTRINE OF MOTION, AND OF THEIR PARTICULAR USE.

The doftrine of motion in all its extenfive branches is derived, as we have alreadynbsp;{hewn, from a few general principles; and its application 'to particular circuraftances requires onlynbsp;the knowledge of a few experimental fafts, fuchnbsp;the natural defcent of bodies towards the earth?nbsp;the time of vibration of a pendulum of a detenu^'nbsp;nate length, amp;c. Then whatever relates to othefnbsp;complicated movements may be derived, by mean^nbsp;of ftrid and unequivocal reafoning, from thofenbsp;principles, and few afcertained fads or natural laquot;'�'nbsp;Yet notwithftanding the aflent which a ration^''^nbsp;being rauft give to the clear and evident demo^^'nbsp;ftrations that are derived from thofe principle��nbsp;it muft be allowed that an experimental confii'i^*'*'nbsp;tion of any theoretical propofition never failsnbsp;imprefs the mind with a pleafing, lading, and laonbsp;fadory convidion. Though the unavoidablenbsp;Derfedions of machines render the refult of exp^

^ nbsp;nbsp;nbsp;adiy

with the theory^ yet wlyen the error is not very great, and at the fame time it feems to be proportionate to the imperfedions of the mechanicalnbsp;conftrudion and operation, th^ mind of the ob-ferver will always feel itfelf fufficiently fatisfied.

In this chapter the reader will find the, defcription of the principal machines which have been contrived for the p'urpofe of confirming in an experi-Wiental manner the propofitions which relate tonbsp;motion.

The fpace defcribed, and the acceleration gained, by bodies which defcend freely towards the earthnbsp;lias been often attempted to be proved by means ofnbsp;dired experiments; but the refiftance of the airnbsp;Which oppofes a confiderable and fluduating impediment, and the difficulty of meafuring the time ofnbsp;defcent when falling bodies have acquired a greatnbsp;degree of velocity, which foonincreafes beyond thenbsp;Power of our fenfes to eftimate, have always rendered the refult of fuch experiments precarious andnbsp;tinfatisfadory. ¹ But we are indebted to Mr.

Dr. Defaguliers obferved the time that a leadeivball, of two inches in diameter, employed in defcending from reftnbsp;through 272 feet; that is, from the infide of the cupola ofnbsp;^t. Paul�s cathedral, wherein the experiment was tried, tonbsp;^he floor; and found it to be 4,5 feconds, whereas it fhouldnbsp;have been 4,1 feconds ; for in 4,5 feconds it ought to havenbsp;hefcended through fomewhat more than 325nbsp;nbsp;nbsp;nbsp;See his

contrivance, which obviates the abovementioiied impediments, and exhibits the phenomena of accelerated and retarded motion in a commodious andnbsp;fatisfaftory manner; or, in Mr. Atwood�s words,nbsp;� which will fubjedl to experimental examinationnbsp;� the properties of the five mechanical quantities jnbsp;� that is, the quantity of matter moved, the coO'

ftant force which moves it, the fpace defcribed � from reft, the- time of defcription, and the velo'nbsp;� city acquired.�

The reprefentation of this machine in fig. Plate VIII. is divided into two parts for the coH'nbsp;veniency of the plate, which however, can make nOnbsp;difference with refpe�l to the explanation; for thenbsp;reader needs only imagine that thefe two parts atenbsp;placed one upon the other, and are joined at thenbsp;places which are indicated by the fame letters,nbsp;GH, fo as to, form one entire figure.

tipdn the circumferences of two fridlion-'''heels, and the other end refts on the circumferences of the two others. The axes of the four ^ei^lion-wheels are fupported by, and move in, holesnbsp;rriade in the brafs frame which is fattened to thenbsp;rapper ttage, and whofe fliape is fufficiently indicatednbsp;the figure.

There is a groove all round the circumference of wheel a, b, c, for the reception of a fine flexiblenbsp;line, at the extremities of which the bodiesnbsp;B are fufpended. By this means the motion ofnbsp;wheel a, h, c, with 'the filk line is rendered fonbsp;free, that when the bodies A and B are equal,nbsp;�kone of them be gently impelled upwards ordown-'^'^rds, both bodies will readily move in contrarynbsp;^irettions; the friftion of the axis being almottnbsp;^fgt;tirely removed by the application of the friftion-'''keels.

kel is a fcale or rod, divided into inches and and is fo fituated that one of the bodies,nbsp;'^'2- A, may move very near the furface of it. C andnbsp;two little ftagcs, either of which may be fixed,nbsp;y means of the lateral fcrew M or N, on any partnbsp;''k the fcale LEK.- The former of thofe ttagesnbsp;^��Ves to flop the body A, when defcending, at anynbsp;^^^uired height. The latter ttage has a perforationnbsp;^'Jfiiciently large to permit the free paffage of thenbsp;'^'^y A; but its ufe is to fupport occafionally anbsp;I'^^'ght in the form of a bar, like the one feen uponnbsp;�^The perfpeftive reprefentation of this ttage I,

is confiderably ftrained for the purpofe of rendering its conftruftion more intelligible.

Upon the pillar of this machine there is adapter^ a fimple fort of time-piece, confifting of a pendu'nbsp;lum which vibrates feconds, and is kept in motionnbsp;for a few minutes by means of a wheel and weigh*-O. On the axis of the wheel there is a hand ofnbsp;index, which indicates the number of feconds onnbsp;the dial Z. The ufe of this time-piece is tonbsp;by the beats of the pendulum, the time whichnbsp;employed by the body A in afcending or defcend'nbsp;ing through a given fpace

The ufeful property of this machine is to diin^' niOi the force which a�ls upon, and occafions th�nbsp;defcent of bodies, in confequence of which a bodfnbsp;will defcend much flower; hence the obfervernbsp;be enabled to perceive the fpace it moves throug*��nbsp;as alfo its acceleration in a given time, amp;c. in ^nbsp;clear and commodious manner. I fliall endeavon'-

* It hardly needs beobferved tbatthofe who have aco^n mon dock, that beats feconds, may have the machine coHnbsp;flrucled without the laft-defcribed appendage. Befides th*^�nbsp;I fnall juft mention that the abovementioned machinenbsp;been improved, or rather altered, by fome philofopbica*

importance either with refuedt to its conftrudlion or ^ . . 1nbsp;performance, I have preferred Mr. Atwood�s original

to render the explanation of this property more intelligible, previoully to the narration of the experiments.

When the weights or bodies A and B are exaftly equal, they will balance each other, and of courfenbsp;will remain at reft. But if a body of little weightnbsp;be added to one of thofe bodies, as for inftance to

A, nbsp;nbsp;nbsp;then A will preponderate, and confequently willnbsp;defcend; the oppofite weight B afcending at thenbsp;fame time. Now in this cafe both the bodies A,

B, nbsp;nbsp;nbsp;and the wheels are put in motion by the gravitynbsp;of the fmall additional body; fo that the fum ofnbsp;all thofe bodies, being moved by a fmaller force,nbsp;muft move through a fhorter fpace in a given time,nbsp;than if the force were greater.

For inftance, imagine that the weiglits A and B, together with the weight which is required to putnbsp;the wheels in motion (which is equivalent to thenbsp;inertia of the wheels) amounts to 4 ounces, andnbsp;let the weight of the body which is added to A benbsp;half an ounce, then it is evident that a mafs of matter of 4f ounces is put in motion by the gravitynbsp;of a body of half an ounce; that is by the gravitynbsp;of a body equal to the 9th part of the matter whichnbsp;put in motion, which amounts to the famenbsp;thing as if that mafs of matter were attrafted bynbsp;the earth with the ninth part of its ordinary at-lra(5lion. But it has been thewn in page 64, thatnbsp;^�he fpace which is defcribed in a given time by anbsp;^ofcending body, is proportionate to the force of

gravity; therefore if in the natural way a body defcends from reft through 16,087nbsp;nbsp;nbsp;nbsp;the firft

fecond of time, in the above-ftated circumflanco the body A will defcend through the ninth part ofnbsp;16,087 feet, viz. through 21,4 inches, in the firftnbsp;fecond of time. Thus by adding a fmaller weightnbsp;to the body A, that body may be made to move asnbsp;llowly as the obferver pleafes.

The other properties of defcending bodies remain unaltered by this machine. Thus the fpaces whichnbsp;are defcribed by the defcending body A will benbsp;found to be as the fquares of the times; that is, ifnbsp;A defcribe 21,4 inches in the firft fecond of time,nbsp;it will defcribe 4 times 21,4 inches in the fecondnbsp;fecond of time, 9 times 21,4 inches in the third, Uc-Thus much may fuffice with refpeft to the princi'nbsp;pal effedl of this machine. I flaall now add Mf*nbsp;Atwood�s computation, and general mode of con-dufting the experiments.

In the firft place he afcertained the inertia of th^ wheels when the filk line with the bodies A,

� q�he

(i.) Having removed the weights A and B, with thek filk line, Mr. A. affixed a weight of 30 grains to a fdknbsp;(the weight of which was not fo much as | of a grain,nbsp;confequently too inconfiderable to have any fenfihlenbsp;the experiment); this line being wound round the whe^lnbsp;a�c, the weight of 30 grains, by defcending from re/f, coca-

� The refiftance to motion, therefore, arihng from the wheel�s inertia will be the fame as if they werenbsp;abfolutely removed, and a mafs of ai ounces werenbsp;Uniformly accumulated in the circumference ofnbsp;th� wheel ah c. This being premiled, let thenbsp;boxes A and B be replaced, being fufpended by thenbsp;tilk line over the wheel a b c, and balancing eachnbsp;other.

niunicated motion to the wheel, and by many trials it was obferved to defcribe a fpace of about 38,5 inches in threenbsp;leconds. From thefe data the inertia of the wheels maynbsp;be determined in the following manner:

If the weight of 30 grains had defcended through 9 times �93 inches in three feconds, as it would have done by itfeif,nbsp;^he inertia of the wheels would have amounted to nothing;nbsp;hut fince it moved through 38,5 inches in three feconds, itsnbsp;��etardation was occafioned by the inertia of the wheels,nbsp;bet the quantity of this inertia be called x-, then the attrac-force of the earth upon the mafs Ar 30 muft be lefsnbsp;�han upon the body of 30 grains alone; therefore *� 30nbsp;'^sfcends flower than the body of 30 grains would by itfeif;

properly fpeaking, the fpaces which are defcribed in ^he fame time, are inverfely as the mafles ; for the quantitynbsp;force being the fame, the efFedf upon 30 muft be asnbsp;'^Uch lefs than the effect upon 30, as 30 is lefs than 30 ;nbsp;^ence in the prefent experiment the fpace defcribed by thenbsp;l^ody of 30 grains in three feconds, is to the fpace dcfcrlbed

' *353)5. grains, and at = 1323,5 grains, or ounces. '^Oh. r,nbsp;nbsp;nbsp;nbsp;vnbsp;nbsp;nbsp;nbsp;� To

� To proceed in defcribing the conftruflion of the enfuing experiments. In order to avoid trou-blefome computations in adjufting the quantitiesnbsp;of matter moved, and the moving forces, fome determinate weight of convenient magnitude may benbsp;affumed as a ftandard, to which all others are referred. This ftandard weight in the fublequentnbsp;experiments is | of an ounce, and is reprefentcd bynbsp;the letter m. The inertia of the wheels beingnbsp;therefore equal to a I ounces, will be denoted bynbsp;wni. A and B are two boxes conftructed fo as tonbsp;contain different quantities of matter, according/'^nbsp;the experiment may require them to be varied-the weight of each box�, including the hook tonbsp;which it is fufpended, is equal to if oz. or 6/��nbsp;thefe boxes contain fuch weights as are reprefent^*^nbsp;by Q, ea�ch of which weighs an ounce, or 4�?: oth^tnbsp;weights of f an ounce � 2m, \ � rn, and aliqo^^*'nbsp;parts of m, may alfo be included in the boX^*�nbsp;according to the conditions of the different eXp^'nbsp;riments.

� If 4I: oz. or 15W, be included in either boX� this w'ith the w-eight of the box itfclf will benbsp;lb that when the weights A and B, each beif^nbsp;25 Kz, are balanced in the .manner above reprelentc gt;nbsp;tfeir whole mafs will be co;.^, which being addo

be added to A and one to B, the whole mafs will now become 63?�.', perfedtly in equilibrio, andnbsp;moveable by the lead: weight added to either (letting afide the effeds of fiidlion) in the fame manner precifely as if the fame weight or force werenbsp;applied to communicate motion to the mafs 63 ot,nbsp;etafting in tree fpace and without gravity.�

Since the natural weight or gravity of any given lubftance is conftant, and the exad quantitynbsp;of it eafily eftiraated, it will be convenient in thenbsp;fubfequent experiments to apply a weight to thenbsp;mafs A, as a moving force ; thus when the fyf-tem confitls of a mafs a::nbsp;nbsp;nbsp;nbsp;according to the

preceding defeription, the vvhole being perfedly balanced, let a weight of ^ oz. or �/, fuch as Y, be applied to the mats A, this will communicate motion to the whole fyflem.� But fince now thenbsp;whole mafs is 64, and the moving force is the gravity of one of thofe parts only; � therefore thenbsp;force which accelerates the defeent of A, is partnbsp;of the accelerating force by which bodies defeendnbsp;freely towards the earth�s furface.�

Thus by varying the weights, the moving force be altered without altering the mats; or thenbsp;nioving force may be mace to be in any requirednbsp;ratio to the mafs.

� The method of eftimating pra6tica!ly the fpace defcribed from qulefcence, is next to be confidered.nbsp;The body A defcends in a vertical line, and a fcalenbsp;of about 64 inches in length, graduated into inchesnbsp;and tenths of an inch, is adjufted vertically, and fonbsp;placed that the defcending weight A may fall in thenbsp;middle of the fquare ftage, fixed to receive it atnbsp;the end of the defcent; the beginning of the de-fcent is eftimated from o on the fcale, when thenbsp;bottom of the box A is on a level with o. Thenbsp;defcent of A is terminated when the bottom ofnbsp;the box ftrikes the ftage, which may be fixed atnbsp;different diftances from the point o, fo that bynbsp;altering the pofition of the ftage, the fpace de-feribed from qulefcence may be of any given mag'nbsp;nitude lefs than 64 inches.�

its defcent precifely at any beat of the pendului^quot;' r 9nbsp;nbsp;nbsp;nbsp;, theii

then the coincidence of the llroke of the box againft the ftage, and the beat of the pendulumnbsp;at the end of the time of motion, will fliew hownbsp;nearly the experiment and the theory agree ^ together. There might be various mechanical devices thought of for letting the weight A beginnbsp;�ts defcent at the inftant of a beat of the pendulum ; but the following method may perhaps benbsp;fufficient; let the bottom of the box A, when atnbsp;o on the fcale, reft on a flat rod held in the handnbsp;horizontally, its extremity being coincident withnbsp;o; by attending to the beats of the pendulum,nbsp;and with a little pra�lice, the rod which fupportsnbsp;the box A, may be removed at the inftant thenbsp;pendulum beats, fo that the defcent of A fliallnbsp;commence at the lame inftant.�

� It remains only to defcribe in what manner the velocity acquired by the defcending weight A,nbsp;any given point of the fpace through which itnbsp;has defcended, is made evident to the fenfes. Thenbsp;''clocity of A�s delcent being continually accelerated, will be the fame in no two points of the fpacenbsp;^eferibed ; this is occafioned by the conftant action of the moving force; and fince the velocity ofnbsp;at any inftant is meafured by the fpace whichnbsp;quot;^ould be deferibed by it, moving uniformly for anbsp;�iven time with the velocity it had acquired at

that inftant, this meafure cannot be experlmentallj' obtained, except by removing the force by whichnbsp;the defeending body�s acceleration was caufed.�^

� In order to fnew^ in what manner this is ef-fefted pradically, let us fuppofe that, according to a formec example, the boxes A and B zrnbsp;each, fo as together to be ~ 50^2; this with thenbsp;wheels � inertia ii;?2 will make 6i:n �. now let mnbsp;be added to A, and an equal weight m to B, thofcnbsp;bodies will balance each other, and the whole mafsnbsp;will benbsp;nbsp;nbsp;nbsp;If a weight m be added to A, mo

tion will be communicated, the moving force being tn, and the mafs moved 64W. In a formernbsp;example, the circular weight, equal ni, was madenbsp;ufe of as a moving force; but for the prefent pur-pofe of fhewing the velocity acquired, it will b^nbsp;convenient to ufe a flat rod (quot;like that wdiichnbsp;fliew'n at I on the perforated ftage) the weight ofnbsp;which is alfo equal to m. Let the bottom of thenbsp;box A be placed on a level with o on the fcale, thenbsp;whole mafs being asdeferibed above, G'^m, perfectlynbsp;balanced in equilibrio. Now let the rod, the weigh^_nbsp;of which � rn, be placed on the upper furface otnbsp;A; this body will defeend along the fcale preciie*ynbsp;in the fame manner as when the moving forcenbsp;was applied in the form of a circular w eight. S'Jp'nbsp;pofe the mafs A to have defeended by conftantnbsp;cpleration of the force m, for any given time, ornbsp;through a giveq fpacc : let the perforated

be To affixed to the fcale contiguous to which the weight defcends, that A may pafs centrally throughnbsp;it, and that this perforated ftage may intercept thenbsp;rod m, by which the body A has been acceleratednbsp;from quiefcencc. After the moving force m hasnbsp;been intercepted at the end of the given fpace ornbsp;time, there will be no force operating on any partnbsp;of the fyftem, which can either accelerate or retardnbsp;its motion ; this being the cafe, the weight A, thenbsp;inftant after m has been removed, mull proceednbsp;uniformly with the velocity which it had acquirednbsp;that inftant: in the fubfequent part of its defcent,nbsp;the velocity being uniform will be meafured bynbsp;the fpace defcribed in any convenient number ofnbsp;feconds.�

� The motion of bodies refifted by conftant forces are reduced to experiment by means of thenbsp;inftrument'above defcribed, with as great eafe andnbsp;precifion as the properties of bodies uniformly accelerated. A fingle inftance will be fuffident:nbsp;thus fuppofe the m.afs contained in the weights Anbsp;and B, and the wheels, to He him, when perfedlynbsp;equilibrio, as in a former example ; let a circularnbsp;^'�eight m be applred'to B, and Jet two Jong weightsnbsp;rods, each equal to m, be applied to A, thennbsp;'''ill A defcencl by the action of the moving fofcenbsp;the mafs moved being 64W : fuppofe that whennbsp;u 4nbsp;nbsp;nbsp;nbsp;it

it has defcribed any given fpace by conftant acceleration, the two rods m are intercepted by the perforated ftage, while A is defcending through it;nbsp;the velocity acquired by that defcent is known, andnbsp;when the two rods are intercepted, the weight Anbsp;w�ill begin to move on with the velocity acquired,nbsp;being now retarded by the conftant force m �, andnbsp;fince the mafs moved is 62�;, it follows, that thenbsp;force of retardation will be -g-V part of the forcenbsp;whereby gravity retards bodies thrown perpendicularly upwards. The weight A will thereforenbsp;proceed along the graduated fcale in its defcentnbsp;with an uniformly retarded motion, and the fpacesnbsp;defcribed, times of motion, and velocities deftroy-ed by the refifting force, will be fubjedl: to the famenbsp;meafures as in the examples of accelerated motionnbsp;above defcribed.�

Befides thofe properties, Mr. Atwood�s machine may be eafily adapted to other ufes, fuch as the experimental eftimation of the Velocities communicated by the Impact of bodies elaftic and non-elaftic; the quantity of refiftance occafioned bynbsp;fluids. See. Mr. Atwood alfo (hews its ufe in verifying pradically the properties of rotatory inO'nbsp;tion¹.

After the preceding fufficiently ample deferip' tion of the general mode of .ufing this inftrument.

We flial] by way of example fubjoin three or four experiments, and flrall leave Its further applicationnbsp;to particular cafes, for the exercife of the reader�snbsp;ingenuity. It is however neceflary, in the firftnbsp;place, to obviate fome doubts which may naturallynbsp;occur with refpedl to the performance of this machine ; the accuracy of which may be difturbed bynbsp;three caufes, viz. the fridlion of the axes of thenbsp;wheels, the weight of the filk line, and the refift-ance of the air.

� The effedts of fridtion are almoft wholly removed by the friction-wheels j for w'hen the fur-faces are well polilhed and free from dull, amp;c. if the weights A and B be balanced in perfedt equili-brio, and the whole mafs contifts of 63?�, according to the example already defcribed, a weight ofnbsp;I I grain, or at moft 2 grains, being added eithernbsp;to A or B, will communicate motion to the whole,nbsp;which Ihews that the effects of fridtion will not benbsp;fo great as a weight of 11 or 2 grains. In fomenbsp;Cafes, however, efpecially in experiments relating tonbsp;retarded motion, the effedts of fridtion becomenbsp;^enfible j�but may be very readily and exactly re-rtioved by adding a fmall weight of i f or 2 grainsnbsp;fo the defcending body, taking �care that thenbsp;Weight added is fuch as is in the leaft degreenbsp;frnaller than that which is juft fufficient to fet thenbsp;^�hole in motion, when A and B are equal, andnbsp;balance each other, before the rnoving force is applied.�

The filk line by which the weights are fufpent?quot; ed is 72 inches long, and weighs about 3 grains:nbsp;a quantity too fmall to affeft fenfibly the refult ofnbsp;the experiments.

Tlie effeft of the refinance of the air is likewile infenfible; for that refiftance increafes with the velocity, and in the experiments which are performednbsp;with this machine, the greateft velocity communicatee! to the bodies A and B, cannot much exceednbsp;that of about 26 inches in a fecond.

� Experiment ift. Let A and B, together with the 2' I oz. (which are equal to the inertia of thenbsp;wheels), amount to 16 oz. or 63?^; then add 3.nbsp;weight of J oz. that is w, to A, and A wall defeendnbsp;and will d'eferibe from reft three inches in the fitlfnbsp;ieeond; fo that if the fquare ftage be fixed evennbsp;with the 3 inches on the fcale, and A be permittednbsp;to defeend from o on the fcale juft when the pen'nbsp;dulum ftrikes, it will be found that exadly whennbsp;the pendulum ftrikes the next ftroke, the body ^nbsp;will ftrike againft the ftage. If the experimentnbsp;repeated with this variation only, viz. with thenbsp;ftage fixed even with the 12 inches on the fcalegt;nbsp;tlieti the weight A will ftrike the ftage exadtifnbsp;when the pendulum ftrikes the fecond ftrokenbsp;the commencement of A�s motion. And if tb�nbsp;ftagje be fixed even with the 27 inches, thenbsp;of A on the ftage will coincide with the thn'nbsp;ftroke of the pendulum; and fo on.

Hers

Here it is evident that the quantity of matter in motion is reprefented by 64 parts, and this quantity is put in motion by the gravity of one of thofenbsp;parts; therefore the moving force being the 64thnbsp;J'�art of what the earth would otherwife exert uponnbsp;the whole mafs, this mafs muft move through thenbsp;64th part of that fpace which, if defcending freely,nbsp;it would move through in the fame time. But innbsp;the natural way defcending bodies pafs throughnbsp;16,087 feet, or 193 inches, in the firfl fecond;nbsp;therefore in this experiment the body A muft de-feend through the 64th part of 193, viz. 3 inchesnbsp;nearly. In two feconds it muft defeend through 4nbsp;times 3, or 12 inches; in three feconds it muftnbsp;defeend through 9 times 3, or 27 inches, Sic. thenbsp;fpaces being as the fquares of the times. �

Experiment 2d. If the weight of A and B, together with the inertia of the wheels, be madenbsp;equal to 62�;, and a weight of ^m be added to A,nbsp;then the whole mafs in motion will be 64W, andnbsp;the moving force 2�z, viz. of the mafs; therefore in the firft fecond of time A will be found tonbsp;defeend through a fpace equal to the 32d part ofnbsp;^93, viz. 6 inches. In two feconds it will defeendnbsp;through four times 6, viz. 24 inches, and lo on.�nbsp;Thus the force may be varied at pleafure, and thenbsp;fpace deferibed by the defcending body in a givennbsp;time will be found proportionate to the force.

Experiment 3d. Let the quantity of matter be as in the firft experiment; add a bar of thenbsp;'''eight m to A, and place the perforated ftage even

with the 12 inches, and let the weight A commence its defeent when its upper furface is even with thenbsp;o on the fcale. It will be found that in two fecondsnbsp;the bar on the body A will ftrike againft, and remain on, the perforated ftage, after which the bodynbsp;A, not being any longer adled upon by any accelerative force, will continue to defeend with annbsp;equable motion, and will deferibe (according to thenbsp;law which has been mentioned and proved innbsp;page 6y.) a fpace equal to twice the above-mentioned defeent in the fame time, that is, 24 inchesnbsp;in 2 feconds. Thus the degree of velocity acquirednbsp;after any other defeent may be proved experimentally.

Experiment 4th. Let A be equal to 24! �?� and B equal 25!�;, and apply to the upper furfacenbsp;of A two rods, each of which is equal m, then wiUnbsp;the weight A preponderate and defeend by thenbsp;aftion of a moving force equal io ms the wholenbsp;mafs moved being equal to S-^m. Fix thenbsp;forated ftage at 26,44; then the weight A by dc-^nbsp;feending from reft through 26,44 inches, willnbsp;acquire a velocity equal to 18 inches per fecond :

that inftant the two rods, each of which is equal to nt, being intercepted by the ftage, the body A wiHnbsp;continue to defeend with an uniformly retardednbsp;motion; which will be precifely the fame as if ^nbsp;mafs o� (sim, without gravity, were projefted whh

a velocity of 18 inches in a fecond in free fpace, and a force or refiftance equal to tn were oppofed to itsnbsp;motion; wherefore A (with the other parts of thenbsp;fyftem) will lofe its motion gradually, and willnbsp;defcribe a fpace equal to 25,6 inches (that is,

ftroyed : A will therefore be obferved in the experiment to defcend as low as 52 inches, before it begins to afcend by the fuperior weight of B.

The next machine we fliall defcribe is called a 'ivhirling-table, and its ufe is for Chewing, in an experimental way, the nature and properties of centripetal and centrifugal forces.

The machine itfelf is exhibited by fig. i, Plate IX. and the apparatus is reprefented by the numbers I, 2, 3, amp;c. adjoining to it¹.

Upon the fteady table ��, the two ftrong pillars e, e are immoveably fixed, which are alfo fteadilynbsp;fcrewed to the crofs piece a b. Within this framenbsp;the two upright hollow axes are fituated fo thatnbsp;tach of them may turn with a pointed pin in a holenbsp;On the table, and with its upper extremity throughnbsp;^ hole in the crofs piece a b. The lower part of

Whirling tables have been varied more or lefs in fhape and hze by almoft all the different makers of thofe inltruments.nbsp;�fhat which I have preferred has confiderable advantage innbsp;point of fimplicity and durability. This machine was contrived and made by Mr. J. B. Haas.

each axis is immoveably conneded with a doubly grooved wheel or pulley KH, CY parallel to thenbsp;table. The grooved wheel B turns alfo parallel tt)nbsp;the table, round a ftrong pin or axis which is fixednbsp;to the table j and a catgut-firing is difpofed roundnbsp;the wheel B, and round the large or fmall circumference of the wheel at the bottom of each axis, irtnbsp;the manner which is clearly indicated by the figure-In this difpofition it is eafy to conceK-e that bynbsp;applying the hand to the handle at A, and turningnbsp;the wheel B, both the axes will be caufed to turJ^nbsp;round. A focket, or tube I, I is conneded with ^nbsp;circular brafs plate FG, ED, and Hides freely upnbsp;and down each axis. From the inlide of each ofnbsp;thofe tubes or Pockets a wire paffes through atinbsp;oblong flit, and projeds within the cavity of th^nbsp;axis, where it is fhaped like a hook; fo that a firingnbsp;may be tied to this hook, which palling upward^nbsp;through the aperture of the axis, may be pulled otnbsp;let down in fuch a manner as to let the platenbsp;focket move up or down the axis. Upon thol^nbsp;plates, femicircular leaden weights o o maynbsp;placed occafionally,�Thofe w'eights, being P^�quot;�nbsp;forated, are flipped over two wires which procee*^nbsp;from the plate ED, or F G, as they are indicatednbsp;by the figure;- by which means the weights at^nbsp;prevented from falling off.

To the upper p.art of each axis (viz. to the of it which projeds above the crofs piece ^nbsp;tariety of different mechanifms may be occariU'|

The oblong pieces which are reprelented in the figure as being aftualiy fixed to the axes, are callednbsp;hearers.�Their conftrudion being exactly the fame,nbsp;W'e need defcribe only one of them.

A perforated brafs plate with a ftrong fcrew which fits the fcrew at the top of the axis, is.fixednbsp;in the middle of the bearer ML; fo that wd'.en thenbsp;bearer is fcrewed to the axis, the hole in it communicates with the cavity of the axis. On onenbsp;fide of this hole, a perpendicular projedion T rilesnbsp;above the furface of the bearer, and afimilar pro-jedion rifes above the end L of the bearer, w'hidinbsp;is on the other fide of the central hole. Twonbsp;fmooth, ftrong, and parallel wires are ftretchednbsp;between thole two projedions by means of thenbsp;icrew-nuts at W. A cylindrical heavy body V isnbsp;�perforated with two longitudinal holes, throughnbsp;^vhich the abovementioned wires pafs, fo that thenbsp;body may be freely moved backwards and forwardsnbsp;'Upon, thole wires. On that fide of the cylinder V,nbsp;which lies towards T, there is a hook, to �which anbsp;�firing is faftened. This firing paffes through anbsp;bole in the projedion T; after which it goes roundnbsp;the grooved pulley S, which moves round au axisnbsp;an upright fraaie //, 'fixed to the bearer, andnbsp;whefe fituation is fuch that the firing in its dtlcentnbsp;T, may pals through the middle of the hole innbsp;bearer, and of the cavity of the axis, fo as to

be fattened with its extremity to the hook of tlic wire which proceeds from the focket of the plate-'nbsp;ED. After this defcription it is eafy to under-ftandj that the cylinder V and the plate ED arenbsp;connected together by means of the ftring, andnbsp;'that if V .be drawn towards W, in which fituationnbsp;it appears in the figure, the plate E D with itsnbsp;fuperincumbent weights will be pulled towards thenbsp;upper part of the axis; otherwife the weight ofnbsp;the plate ED will draw the' cylinder V tow'arclsnbsp;T, and will itfelf defcend towards the lower partnbsp;of the axis.

Either of thofe bearers may be removed from? and one of the following mechanifms may benbsp;fcrewed fatt upon, the axis.

No. I. reprefents a circular board, turned upfide down, having a ftrong fcrew in its middle, whichnbsp;fits the fcrew at the top of either axis of the ro^'nbsp;chine. There is a hole through the middle of thisnbsp;board and of its fcrew, which opens the communication with the cavity of the axis. But this, .holenbsp;in the middle of the board may be occafionaliynbsp;filled up v/ith a piece of wood in the form of ^nbsp;flopple, which is furnifhed with a ttiort pin, that�nbsp;when the piece of wood is fixed in the hole, prCquot;nbsp;jefts a little above the furface of the board.

No. 2. is an oblong bearer, which may be fcrewed� like any of the others, upon one of the axes of th^nbsp;machine. .It has an upright projelt;flion at eacu

tween thofe projeftions by means of the fcrew nutsj A, B. C and D are two perforated brafs balls, cfnbsp;Unequal weights, which are connefted together bynbsp;tneans of a brafs tube, and are freely moveablenbsp;Upon the wire AB from One end to the other*nbsp;On the outfide of the brafs tube which conneflrsnbsp;the two balls, there is fixed, Oxaftly at the common centre of gravity of thofe ballsy a fhort wirenbsp;E as an index, which ferves to fhew when thenbsp;comnion centre of gravity of thofe balls is placednbsp;exadtly againfl; the middle of the bearer.

No. 3. reprefents a board having at its lower end C, a fcrew which fits the fcrew at the top of onenbsp;of the axes of the machine, upon which it maynbsp;be firmly fcrewed j but this fcrew is fituated anbsp;Kttle aflant to the board, fo that when placed uponnbsp;the axis of the machine this board may ftand inclined to the horizon, making an angle of 30 or 40nbsp;degrees with it.

On the upper fide of this board are fixed two glafs tubes, A G and B F, clofe ftopped at bothnbsp;ends; and each tube is about three-quarters fullnbsp;�f Water. In the tube B F is a little quickfilver,nbsp;Which, in confequence of its weight, remains under the water at the end B; tn the other tube A Gnbsp;a piece of cork, which being lighter than water,nbsp;floats upon it towards the end G, and is fo fmallnbsp;not to fhick fall within the cavity of the tube.nbsp;1^0. 4. is an axis or ftrong wire fixed to a board,nbsp;�'^d having a fcrew at its lower part beneath thenbsp;''^OL. I,nbsp;nbsp;nbsp;nbsp;Tjnbsp;nbsp;nbsp;nbsp;board

board, which fits the fcrew-hole at the top of onS of the axes of the whirling table. Two circulafnbsp;brafs hoops, AC, BD, made very thin and pliable,nbsp;foldered to each other, and foldered or fcrewednbsp;to the axis, at I, have each a hole at the uppernbsp;part through which the axis paffes freely 3 fo thatnbsp;if a hand be applied to the upper part E of thofenbsp;hoops, they may be flattened down as far as thenbsp;pin O, which is feen acrofs the axis. In this cafenbsp;the hoops will change their circular form into annbsp;elliptical one 3 but, being elaftic, they will refumenbsp;their circular form as foon as the preffure is removed.

N�. 5. reprefents a hemifphere, which is to be fituated upon the board No.* i, when that is fixednbsp;upon one of the axes of the machine, in the following manner: A pin with a fcrew e (which isnbsp;not fixed to the hemifphere) is fcrewed in the middle of the board fo as to projeft a little above it�nbsp;and the hemifphere A is laid upon it, there being ^nbsp;cavity d i on the flat part of the hemifphere made onnbsp;purpofe to lodge the piri 3 but this cavity i*nbsp;an oblong groove, as is pretty well indicated bynbsp;the dotted line on the figure of the hemifphere�nbsp;and it is made fo that the hemifphere by Aidingnbsp;over the pin the' whole length of that groove, may'nbsp;be placed either concentric with the board, or outnbsp;of centre with it. The lateral wire C, with thenbsp;Irnall ball B, may be fcrewed occafionally on the

fide of the hemifphere A, in a divedlion oppofite to that of the abovementioned groove.

No. 6. reprefents a forked wire with a fcrew at its lower part, which fits the fcrew in the middlenbsp;of the board No. i. This fork ferves to fupport'nbsp;the wire C with the two unequal balls A and B;nbsp;but this wire being in no way connedled with thenbsp;forked wire, muft be balanced upon it, that is, itnbsp;rnutl be laid with the common centre of gravitynbsp;upon the fork; which is eafily done by trial.'

No. 7. reprefents a ball of about two inches in diameter, haying a hole from fide to fide, throughnbsp;which a wire BA patfes quite freely. This wirenbsp;out of the ball at A is {Taped like the letter T,nbsp;each of whofe projeflions is longer than the radius of the ball, and has a blunt termination. Oilnbsp;the other fide B the wire terminates in a ring, t�nbsp;which a firing is tied.�We lhall now proceed tonbsp;defcribe the experiments which are to be madenbsp;with the whirling table and its apparatus.

Experiment i. Fix the board No. i. upon one of the axes of the rriachine, knd put the piece ofnbsp;W�ood or Topper Vv'ith the pin, in the middle of it.nbsp;Take the ball apparatus. No. 7, make a loop on thenbsp;�nd C of the firing, taking care that the lengthnbsp;CA be not greater than the radius of the board,nbsp;i�ut the loop of the firing over th� pin in the middle of the board, and leave the ball upon thenbsp;t)oard. Then apply a hand at A, and turn th�nbsp;wheel B of the machine, which will give the boardnbsp;X 2nbsp;nbsp;nbsp;nbsp;a whirl-

a whirling motion. It will be found that the ball does not immediately begin to move with thenbsp;board ; but, on account of its inertia, it endeavoursnbsp;to continue in its ftate of reft, in which it floodnbsp;befote the machine was put in motion. But, bynbsp;means of the fridion on the board, that inertia isnbsp;gradually overcome; fo that by continuing tonbsp;whirl the board, the balbs motion .will becomenbsp;equal to that of the board; after which the ballnbsp;will remain upon the fame part of the board, itnbsp;being then relatively at reft upon the board. Butnbsp;if you flop the board fuddenly, by applying a handnbsp;to it, the ball will be found to go on in virtue ofnbsp;its inertia, and continu� to revolve, until the friction of the board, by gradually diminifhing its velocity, finally flops it. This fhews that matter isnbsp;as incapable of flopping itfelf when in motion, asnbsp;it is incapable bf moving itfelf from a ftate ofnbsp;reft.

got

B of No. 7. be about an ilich from the hole in the middle of the circular board. The weight of thenbsp;plate FG muft be very little more than fufficientnbsp;to draw the ball to the abovementioned fituation;nbsp;for which purpofe the leaden weights muft be removed from over the plate FG,. its own vyeightnbsp;alone being fufficient for the purpofe.

Having placed the ball fo that the ring B may be about one inch diftant from the centre of thenbsp;board, put the machine in motion by turning thenbsp;wheel B; and it will be found that the ball bynbsp;going round and round with the board, will gradually fly off to a greater and greater diftance fromnbsp;the centre of the board, raifing up the plate FGnbsp;at the fame time j which' thews that all bodiesnbsp;which revolve in circles, have a tendency to fly off,nbsp;fo that a certain power from the centre muft adtnbsp;.upon them in order to prevent their flying off. Ifnbsp;the machine be flopped fuddenly, the ball willnbsp;.continue to revolve for fome time longer; but thenbsp;fridlion of the board gradually diminiiliing its velocity, its tendency to fly off will alfo decrcafe, andnbsp;the weight of the plate FG will gradually draw itnbsp;nearer and nearer the centre, until its motion,nbsp;.ceafes entirely.

Experiment 3d. Let the apparatus remain as tn the preceding experiment, excepting only thatnbsp;the firing, being difengaged from the plate FGnbsp;niuft be let out of the flit in the axis, and the operator mUft hold its extremity in his hand. With

his other hand the operator muft throw the ball vipon the circular board as it were in a dire�lionnbsp;perpendicular to the firing, by which means the ballnbsp;will make feveral revolutions upon the board (thenbsp;machine being in this experiment at reft). But ifnbsp;whilft the ball is revolving you gradually pull thenbsp;lower end of the firing below the board, you willnbsp;find that the ball, in proportion as it comes nearernbsp;to the centre of motion, and of courfe it performsnbsp;its revolutions in fmaller circles, will revolve fafter;nbsp;whichfhew'S, asfaras fuch a machine can do it, thatnbsp;the fame moving force will enable a revolving bodynbsp;to defcribe a circular orbit fafter when the circle isnbsp;fmaller, and flower when the circle is larger. (Seenbsp;chap. VIII.)

Experiment 4th. Remove the circular board� and inftead of it, put the bearer on the axis; fonbsp;that both the bearers may be upon the machine, asnbsp;is reprefented in the figure.

Let the cylinders R V, be of equal weights j place equal weights upon the plates FG, ED; andnbsp;adjuft the lengths of the firings which conned!nbsp;thofe cylinders with thofe plates, fo that when th^nbsp;plates are quite down the cylinders may ftand a!nbsp;equal but fmall diftances from the centres of then'nbsp;refpedlive bearers. The catgut firing muft be pf!nbsp;either over both the large, or over both the fmallnbsp;circular grooves at the bottom of the axes (oAnbsp;which account it is neceflary to have two catgn!nbsp;firings, viz. one longer than the other) then put th�nbsp;4nbsp;nbsp;nbsp;nbsp;machine

machine in motion, and the cylinders R,V wjll be feen to recede from the centres of the bearers, andnbsp;to advance towards the ends X and W, raifing atnbsp;flre fame time, and to an equal height, the platesnbsp;FG, ED. This experiment proves that when equalnbsp;bodies revolve in equal circles, with equal velocities,nbsp;their centrifugal forces are equal.

Experiment 5th. Inllead of the C5'linder R, place another cylinder of half its weight, viz,, equalnbsp;to half the weight of V, on the wires of the bearer�nbsp;PN; adjuft the firings fo that when the platesnbsp;FG, ED are quite dowm, the diflanceof the cylinder V from the centre of the bearer ML may benbsp;half the diftance of the other cylinder from thenbsp;centre of its bearer, which is eafily fhewn by thenbsp;divifions which are marked upon the bearers; andnbsp;leave the reft of the'apparatus as in the precedingnbsp;experiment. Now when the machine is put in motion, there will be two bodies revolving, one ofnbsp;which is half the weight of the other, but the former revolves in a circle which is as large again asnbsp;the circle in which that other revolves. And itnbsp;will be found that the equal weights of the platesnbsp;ED will be equally raifed i which Ihews thatnbsp;the centrifugal forces of the revolving bodies (whichnbsp;mife the plates FG EDJ are equal as long as thenbsp;products of the bodies multiplied each by its velocity, viz. the momentums, are equall

The proportion of the weights of the two bodies iP'T-y be varied at pleafure; but the firings muft benbsp;X 4nbsp;nbsp;nbsp;nbsp;adj lifted

adjufted fo that their diftances from the centres of the bearers may be inverfely as thofe weights; andnbsp;the plates FG, ED, which are loaded with equalnbsp;weights, will always be lifted to equal heights; thenbsp;products of the bodies by their refpedtive velocitiesnbsp;being always equal,

Experiment 6tb. Repeat the preceding expe-_ riment, with this difference, that the cylinders be left at equal diHances from the centres of theifnbsp;refpective bearers; alfothat the weights on the platesnbsp;FG, ED be in the proportion of the weights ofnbsp;the cylinders, viz, when V weighs as much again aSnbsp;R, the weight of the plate ED muft be double thenbsp;weight of the plateFG, amp;c. Onputting themachinOnbsp;In motion it will be found that the plates FG, BVnbsp;are raifed to the fame height; which proves thatnbsp;%vhen revolving bodies move with the fame velo'nbsp;city, their centrifugal forces are proportionatenbsp;their refpe�tive quantities of matter.

catgut firing round the wheel B, the wheel Yj round the fmall wheel H, which is exaftly the dif*nbsp;pofition reprefented by the figure. Alfo, adjufl tb^nbsp;firings between the cylinders and the axes, f^nbsp;that when the plates FG, ED are quite down, tb^nbsp;cylinders may lie at equal 'diftances from the ceOnbsp;tres of the bearers. Farther, if the circumfot^^^^nbsp;pf the wheel Y be equal to twice that of the wbenbsp;yoti muft put four tini?s as much weight

the plate FG, as upon the plate ED. If the circumference of the wheel Y be equal to three times that of the wheel H, you mufl. put ninenbsp;times as much weight on FG as upon ED. Innbsp;fliort the weights on the plates mull be inverfelynbsp;as the fquares of the circumferences of the wheelsnbsp;Y and H, On putting the machine in motion itnbsp;will be found that the plates FG, jED are raifednbsp;to an equal height; which thews that when equalnbsp;hgt;odies revolve in equal circles with unequal velocities, their centrifugal forces are as the fquares ofnbsp;the velocities.

Experiment 8th. Let the catgut remain in the fame fituation as in the laft experiment, and let thenbsp;circumference of the wheel Y be to that of the wheelnbsp;H as two to one (in which proportion the circumferences of thofe wheels of whirling machines are generally conftru�ted), Alfo let the cylinders R and V benbsp;of equal weights, but adjuft the firings fo that whennbsp;the plates FG, ED are quite down, the diflance ofnbsp;the cylinder R from the centre, of the bearer FNnbsp;be two inches, whilft that of the cylinder V fromnbsp;the centre of the bearer ML be 34- inches¹. Thenbsp;circumference of the wheel Y being equal to twice

Inftead of inches, the difiances may be of any other �denomination; provided they be in that proportion. Thenbsp;hearers are generally divided into �qual parts which arenbsp;longer than inches; fo that the difhinces may be madenbsp;P^ual to 2 and 3^ of thofe divifionSj

the circumference of the wheel H, it follows that when the machine is put in motion, V muft makenbsp;one revolution wbiift R makes two; therefore theirnbsp;periodical times are as one to two, and the fquaresnbsp;of thofe times are i and 4; the former of which isnbsp;contained four times in the latter. But the dif-tance of R is 2, the cube of which is S; and thenbsp;diflance of V is 3'^gt; the cube of which is 32 nearly,nbsp;in which 8 is contained 4 times; therefore thenbsp;fquares of the periodical times are as the cubes ofnbsp;the diftances. Now let the weight of the plate EDnbsp;be 4 ounces, equal to the fquare of the diftance 2;nbsp;and the weight of the plate FG be 10 ounces,nbsp;nearly equal to the fquare of the diftance 3-^ ; thennbsp;on turning the wheel B, which will put the axes innbsp;motion, it will be found that the plates FG, EDnbsp;are raifed toan equal height.

This experiment proves that when equal bodies revolve in unequal circles, and the fquares of thenbsp;times of their going round are as the cubes of theitnbsp;diftances from the centres of the circles, then the'Jnbsp;centrifugal forces are inverfely as the fquares of theirnbsp;diftances.

Experiment 9th. Remove one of the bearers , from the machine, and place the mechanlfm No.nbsp;upon the axis. (See the defeription of this me-chanifm in p. 30/.) On turning the wheel B,nbsp;will be found that the contents of the glafs tubesnbsp;AG, BF vrill, in confequence of their centrifog�^nbsp;forces, run towards the outward and uppermoft

of tliofe tubes. And fince with equal velocities, the heavieft bodies have the greateft centrifugalnbsp;force, therefore the quickfilver in the tube BF willnbsp;go quite to the end F of the tube; its weightnbsp;being greater than that of an equal bulk of water;nbsp;but in the tube AG tlie piece of cork will be foundnbsp;at the bottom of the water; the water being muchnbsp;heavier than an equal bulk of cork.

Experiment loth. Remove the apparatus No. 3, and place No. 4. upon the axis. On putting thenbsp;machine in motion, the upper part E of the hoopsnbsp;ivill defcend towards the pin O, and the quicker thenbsp;machine is whirled, the nearer will the hoops comenbsp;to the pin, their middle parts receding at the famenbsp;time from the axis; fo as to aflume an ellipticalnbsp;form. This effedl arifes from the different centrifugal forces of the different parts of thofe hoops;nbsp;the centrifugal forces of thofe parts which are farther from the axis of motion being greater than ofnbsp;thofe which are nearer to it.

It appears therefore, that when globular bodies, \vhofe matter is fufhciently yielding, revolve roundnbsp;their axes, their figure cannot be perfedly fpherical,nbsp;but it is that of an oh/ate fpheroid.

Experiment nth. Remove the preceding apparatus from the axis of the whirling table; place the board No. i. upon it; fix the pin e of the machine No. 5. in the middle of the board, with thenbsp;hemifphere A upon it, but without the wire C.nbsp;Jf the hemifphere A be placed concentric with the

board, on whirling the machine, the hemirphere will be found to remain in its place upon the board;nbsp;the centre of gravity of the hemifphere coincidingnbsp;with the centre of motion. But if the centre ofnbsp;the hemifphere be placed a little on, one fide of thenbsp;centre of motion, then on whirling the machine,nbsp;the larger portion of the hemifphere, which lies onnbsp;one fide of the centre of motion, will acquire nnbsp;greater centifugal force, and confequently will draWnbsp;the hemifphere that way; fo that the pin Aidingnbsp;through the groove d i, the hemifphere will at laftnbsp;be found with the part d upon the pin.

Jf the wire C with the little body B be Icretye^ on the fide of the hemifphere, and the latter benbsp;placed upon the pin, concentric with the board ?nbsp;on whirling the machine, the fame thing as thenbsp;mentioned effed will take place^ for though thenbsp;hemifphere be placed concentric with the board?nbsp;yet when the body B is affixed to it, their commonnbsp;centre of gravity is different from the centre ofnbsp;gravity of the hemifphere alone.

Experiment jath. Nearly the fame thing fhewn by means of the apparatus No. 2. For whennbsp;this is ferewed upon the axis of the whirling table�nbsp;(the preceding mechanlfm being removed) ifnbsp;index E, viz. the centre of gravity of thenbsp;bodies C,D, be placed exadly over the centre 0nbsp;tbe bearer AB, the whirling of the machine wn.nbsp;not move the faid bodies upon the wire A��

little Qn one fide of the centre of motion, on whirling the machine, the two bodies will move towards that fide, as far as the upright projeftion A or B.

Experiment I3tk The fame thing may be Ihewn by means of the mechanifm No. 6. Placenbsp;the circular board No. i. upon one of the axes ofnbsp;the whirling machine j fix the forked wire D, �fnbsp;No. 6. in the middle of it; balance the bodiesnbsp;A,B, with their conneftiug wire, upon the fork inbsp;then put the machine in motion by turning thenbsp;wheel B, and the bodies A,B will remain balancednbsp;Upon the fork and will turn with it.

A vaft: variety of machines have been invented for the purpofe of illuftrating other branches ofnbsp;the dodrine of motion and equilibrium ; but asnbsp;the propofitions which relate to fuch other branchesnbsp;are very eafy and evident, the particular defcrip-tion of thofe machines would render the worknbsp;Voluminous, without proving of much advantagenbsp;to the reader. I fhall therefore only add a fhortnbsp;Account of the manner of (hewing, by means ofnbsp;Pendulumsy the principal phenomena which attendnbsp;the collifion of bodies; and the defcription of anbsp;Machine which ferves to (hew a few of the cafes,nbsp;''^hich relate to the compofition and refolution ofnbsp;forces, with which this chapter will be concluded,nbsp;The phenomena which attend the dired collifionnbsp;bodies, viz. when their centres of gravity lie innbsp;die diredion of their motion, may be very com-^�odioufly exhibited by means of pendulums, fuch

are reprefented in the figures 4, 5, 6 of Plate VIII; for if one of the pendulums as A in fig. 5, be removed to a certain diftance from the perpendicular, as to the fituation BC, and be then let go, thenbsp;impulfe which its ball gives to the next pendulumnbsp;D, vyill force the latter to move from its ftate ofnbsp;reft, and to defcribe a certain arch, which will benbsp;longer or fliorter according to the quantity of matter of the body which is ftruck, and according tonbsp;the momentum of the ftriking body, which mO'nbsp;mentum may be increafed or diminiflied by elevating the ftriking body to a greater or lefter angle^nbsp;and by varying its weight. The pendulums mafnbsp;alfo be made to ftrike againft, each other after having been both put in motion, either the fame way otnbsp;contrary ways.

The effefts of the collifion, viz. the diredtions of the bodies after the ftroke, and their velocities, maynbsp;be eftimated by obferving the arches which th^^^nbsp;defcribe after the impadt.

In this manner the experiments may be perform' ed on elaftic, as well as non-elaftic, bodies. Wh^^^nbsp;the bodies are�required to be elaftic, ivory bail^nbsp;are fufpended to the threads; but when the bodies are to be non-elaftic, the balls are made of f'^f^nbsp;of moift clay. And though the form^^

* White wax may be rendered fufficiently foft for purpote by melting it over a gentle fire and �ocorporaO^Snbsp;it with about one quarter of its weight of olive-oil;

be not perfeftly elaftic, nor the latter peifedly non-elafllc; yet the difference which arifes fromnbsp;their imperfedl properties, is fo trifling, that itnbsp;may be fafely neglefted in thefe experiments. Andnbsp;here it is proper to obferve that in performing fuchnbsp;mechanical experiments, wherein fome allowancenbsp;muft be made on account of fridtion, of refiftancsnbsp;of the air, of imperfect elafticity, amp;c. the refultnbsp;mull; be reckoned conclufive as long as the effedtnbsp;is fomewhat lefs than what it ought to be according to the theory; but if the efFedt is greater thannbsp;that which is determined by calculation; then fomenbsp;defedt in the machinery, or error in the theory, mufl;nbsp;be fufpedled.

In the abovementioned figures the threads of the pendulums are reprefented as being fixed to common nails; but a machine, or ftand, may be eafiiynbsp;f^ontrived (and many machines of this fort are described in almoft all the books of mechanical phi-lofophy *) upon which two or more pendulums maynbsp;be eafiiy fufpended; where the lengths of the pendulums might be accurately adjufted, and where anbsp;graduated arch, as in fig. 5, might be eafiiy applied.

afterwards, when cooled, be eafiiy formed into balls, and the %ure of the balls may be eafiiy reftored, after being alterednbsp;the courfe of the experiments.

* The beft defeription of the conftrudlion and ufe of

for the purpofe of meafuring the arches from which the pendulums are permitted to defcend, or thofenbsp;to which they aicend.

The facility with which fuch machines may h� contrived and conftrufted renders the particularnbsp;defcription of any of them in this place fuperfluous Jnbsp;one particular mechanifm cohcerning it is howevernbsp;defervingof notice; and fuch is reprefented by fig-7, Plate VIII. When a pendulous body; fufpendednbsp;by a Angle ftring, is railed to a certain height ifinbsp;order to give it motion, theleaft jerk or irregularitynbsp;of the hand is fufficient to make it deviate fromnbsp;the proper plane of vibration, in which cafe thenbsp;ftroke bn the other pendulous body will not benbsp;given in the diredion of its centre of gravity; hencenbsp;the efted will not turn out conformable to th*^nbsp;theory. Now the fufpenfion which is reprefentednbsp;in fig. 7, avoids the poffibility of that deviation Inbsp;and therefore fuch fufpenfion has been generallynbsp;adopted for experiments of the abovementionednbsp;nature. DE is a flip of brafs, the form of whichnbsp;fufficiently indicated b^�� the figure. It is faftenednbsp;to the ball by means �f a fcrew, and the threadnbsp;BDEC, whofe two .extremities are faftened at ^nbsp;and C to a bracket, or horizontal' arm of the rnanbsp;chine, pafles through two holes in the projedioi��nbsp;D,E of the brafs flip.�It is evident that thisnbsp;dulum mull vibrate in a plane perpendicularnbsp;the plane BD EC of the figure, without anynbsp;fible deviation?

Fig. 5. reprefents the cafe when the bodies are equal, and with this apparatus one of the bodiesnbsp;may be made to ftrike againft the other at reft ; ornbsp;they may be made to ftrike againft each othernbsp;when they are both in motion. The fame varietynbsp;of experiments may be performed with the pendulums fig. 4. but in thefe the quantities of matter are unequal.

Fig. 6. reprefents the cafe in which three equal elaftic bodies lie contiguous to each other, wherenbsp;if one of the outer bodies, as F,be lifted up to G, and.nbsp;then be permitted to defcend againft E, the ftrokenbsp;will be communicated from E to D; fo that Enbsp;will remain at reft, and D will be impelled up tonbsp;H.�For the various cafes of collifion which maynbsp;be exhibited by means of pendulums, fee chap.nbsp;VIE

Various mkchines have been contrived for the purpofe of illuftrating the compofition and refolu-tion of forces; and the weights fuftained by oblique powers¹. One of the cleareft methods ofnbsp;ftiewing the compofition of forces is the following:

Sufpend two pendulums ACI, BD, as reprefent-ed by fig. 8. Plate VIIL fo that their balls may be '^sry littie above the furface of a flat and fmooth

Such machines are particularly described in moft of the Works on ^mechanics and natural philofophy, efpecialjynbsp;pravefande�s Elem, of Natural Phil, and Muffchen-^roek�s Philofophy.

table LM RK. Place an ivory ball, E, upon tbtf table, in contaft with both the balls of the pendulums ; then if you draw the pendulum A a certain way out of the perpendicular diredion, andnbsp;then let it fall againft the ball E, in the diredionnbsp;the ball E will be forced to move from E tonbsp;O.�Replace the ball E in its former fituation;nbsp;raife the pendulum B fo as to make an angle withnbsp;the perpendicular, equal to. that made by the othernbsp;pendulum, then let it fall upon the ball E, whereby this ball will be forced to move from E to H*nbsp;Laftly,put the ball E once more in its former fituation j raife both the pendulums at the fame time�nbsp;and to the fame angle to which they were beforenbsp;raifed feparatelyj then let them go both at thenbsp;fame inftant, fo that they may both ftrike thenbsp;ball E at the fame time; and the ball E w'ill thereby be forced to move ftraight from E to G, whichnbsp;is the diagonal of the parallelogram GHEO, whofenbsp;fides are the diredions of the feparate impulfe*nbsp;EO, and EH.

The eiFed may be varied by increafing or di' minifhing one of the impelling forces, whichnbsp;be done by increafing or diminifhing the weigh*-of one of the balls A or B.

In the ufe of fuch a machine, care muft be had to let the two pendulums ftrike the ball E at thenbsp;very fame inftant i which requires a confiderabhnbsp;degree of dexterity. Mechanical means might

charging both the pendulums at the propesr time; but it is hardly worth while to conftrud a complicated machine for illuftrating fo evident a propo-fition.�In this machine the two pendulums are fuf-pended nearly after the manner of fig. 7, viz. eachnbsp;pendulum is fufpended by two threads, but thenbsp;flip of brafs is omitted.

CONTAINING THE APPLICATION OF SOME PARTS OF THE FOREGOING DOCTRINE OF MOTION Jnbsp;WITH REMARKS ON THE CONSTRUCTION OVnbsp;WHEEL CARRIAGES.

OF all the branches of mechanics the properties of the centre of gravity occur moft; frequently, and are of the greateft. confequence, to the human being.

Whatever body refts upon another body mull have its centre of gravity fupported by that othernbsp;body, viz. the line drawn from its centre of gra-''ity ftraight to the centre of the earth j or, which isnbsp;the fame thing, the line which falls from its centrenbsp;of gravity perpendicularly to the horizon, muft benbsp;�otercepted by, or fall upon, the other body; other-the former will not be fupported by the latter.

The abovementi�ned line, that is, a line drawn frorn the centre of gravity of a body, or fydem ofnbsp;bodies that are connefted together, perpendicularlynbsp;fo the plane of the horizon, is called the line of di'nbsp;rediion-, it being in faff the line along which thenbsp;body will direft its courfe in its defcent towardsnbsp;ihe centre of the earth ; and, of courfe, in order tonbsp;be fupported, it muft meet with an obftacle in thatnbsp;line. Thus in fig. 9. Plate VIII. the body CDOGnbsp;will reft very well with its bafe upon the groundnbsp;or other horizontal plane, becaufe its line of direction IF, drawn froth its centre of gravity I, perpendicular to the plane of the horizon, falls withinnbsp;the bafe Y G O, every point of which is fupport-ed by the ground; but if another body ABCPnbsp;be laid upon it, the whole will fall to the ground�nbsp;for in the latter cafe the centre of gravity of thenbsp;whole will be higher up, as at K, and the linenbsp;direction KH falls,out of the bafe. Thus alfo i*'nbsp;fig. to. Plate VIII. the body D will roll down thenbsp;inclined plane AB, becaufe its line of direhfio'^nbsp;falls without its bafe; whereas the body C, whof^nbsp;�line of dire�tion fails within its bafe, wall only did^nbsp;down that plane, unlefs the friction prevents it,nbsp;which cafe it will remain at reft; but friftion vvi^nbsp;not prevent the rolling down of the body D-� It is therefore evident that the narrower the ba^�^nbsp;is, the eafi�ra b�dy may be moved, and, on the connbsp;frary, the broader the bafe is, and the nearer thnbsp;fine of diredlion is to the middle of it,, the

firmly

firmly does the body ftand. Hence it appears that^ a ball, or a circular plane figure Handing upright,nbsp;fuch as a wheel, is moved upon a plane with greaternbsp;facility than any other figure; for the leaft change'nbsp;of pofition is fufficient to remove the line of direction of a fpherical or circular body, out of the bafe.�nbsp;Hence alfo it is that bodies w'ith narrow terminations, fuch as an egg or a Hick, amp;c. cannot benbsp;niade to ftand upright upon a plane,- at leaft: notnbsp;without the utmoft difficulty.nbsp;nbsp;nbsp;nbsp;^

The application of the properties of the centre of gravity to animal ceconomy is eafy and evident.nbsp;If the line of dire�tion falls within the bafe of ournbsp;feet, we remain ereft; and the fheadieft, when thatnbsp;line falls in the middle of that bafe; otherw'Ife w-enbsp;inftantly fall to the ground.

On account of the great importance which the prefervation and management of that centre is tonbsp;animal motion, the infinite wifdom of the Creatornbsp;has implanted in all animals a natural propenfitynbsp;H balance themfelves in almoft every circumftance.nbsp;hdany animals acquire the habit of keeping them-fclves upon their-legs within a few hours after th�irnbsp;hirth, and fuch is particularly the cafe with calves.

It is wonderful to refledt, and to obferve, how a ^hild begins to try and improve his liability. Henbsp;Ecnerally places his feet at a confiderable diftaneenbsp;hrom each other, by which means he enlarges thenbsp;hafe,,and diminifhes the danger of a lateral fallnbsp;he �ndeavours to ftand'quite ercdl,and with his body

immoveabje, fo as to prevent as much as poffibic a fall backwards or forwards;�and when he liftsnbsp;up one foot, he inftantly replaces it upon thenbsp;ground, finding his inability to reft upon fo fmallnbsp;a bafe as that of one foot. Farther advanced i0nbsp;years, he adopts farther methods of preferving and.nbsp;ufing the centre of gravity, and that without thenbsp;leaft knowledge of the mechanical principle uponnbsp;which he ads. Thus a man naturally bends hisnbsp;body when he rifes from a chair, in order to thrownbsp;the centre of gravity forwards. He leans forwardsnbsp;when he carries a burden on his back, in order tonbsp;let the line of diredion (which in that cafe de-fcends from the common centre of gravity of hisnbsp;body and burden) fall within the bafe of his feet.nbsp;For the fame reafon he leans backwards when benbsp;carries a burden before him j and leans on one fidenbsp;when he carries fomething heavy on his oppofit�nbsp;fide.

Human art improved by conftant exercife experience, goes far beyond thofe common ufesnbsp;the centre of gravity, and line of diredion.nbsp;fee, for inftance, men who can balance theo^'nbsp;felves fo well as to remain ered with one f��^nbsp;upon a very narrow ftand, or upon a rope, and eveOnbsp;with their heads downwards and their feet upp�^nbsp;moft.�Their art entirely confifts in quickly counnbsp;terpoifing their bodies, the moment that the bn�nbsp;of diredion begins to go out of the narrow bn ^nbsp;upon which they reft. Thus, if they find

felves falling towards the right, they ftretch out the left arm or the left leg, and vice verfa; for thoughnbsp;the weight of the arm be much lefs than the weightnbsp;of the body j yet by being extended farther fromnbsp;the fulcrum, its momentum may be rendered equalnbsp;to that of the rell of the body, which lies vaftlynbsp;nearer to the fulcrum, or to the line of direction.�nbsp;See chap. VI.

Xiiis explanation llkewife fhews the great ufe of a long horizontal pole in the hands of a rope-dancer ; for as the extremities of the pole, whichnbsp;are generally loaded with leaden weights, lie farnbsp;from the rope, which is the fulcrum �, when thenbsp;pole is moved a little one way, the momentum ofnbsp;that extremity of it which lies that way, is in-creafed confiderably, and fo as to counterpoife thenbsp;body of the man, when he finds himfelf going thenbsp;other way.

Notwithftanding the ufe of the centre of gravity which mankind acquires naturally or merely by experience; yet in many cafes people ait feen tonbsp;adt contrary to the laws of nature ; and the confe-quenccs are fometimes quite fata). Thus we frequently find that when a boat or carriage is oversetting, the perfons in it rife fuddenly from theirnbsp;Scats; by which means they remove the centre ofnbsp;gravity of the whole higher up, and thereby accelerate the fall, (exadly like the cafe which hasnbsp;been reprefented in fig. 9, Plate viii.) which theynbsp;probably prevent, either by remaining onnbsp;Y 4nbsp;nbsp;nbsp;nbsp;their

their feats, or rather by lowering themfelves down � as clofe as they could to the bottom of the boat ornbsp;carriage.

The natural application of the mechanical laws might be inftanced in almoft every occurrence ofnbsp;life ; for whatever moves muft move conformablynbsp;to thofe laws. But, to avoid prolixity, we tlrall onlynbsp;mention a few more particular inftances; whencenbsp;the attentive reader may eafily learn how to applynbsp;the foregoing doctrine of motion to a variety of occurrences.

The man, or the horfe, that runs in a circular-path, naturally leans towards the centre of the circle, or towards the concave part of the curvilinear pathway j and that in order to counteraftnbsp;the effedt of the centrifugal force, which wouldnbsp;otherwife throw him out of the perpendicular*nbsp;And the fwifter he runs, the more he leans towardsnbsp;the concave fide �, the centrifugal force encreafmgnbsp;with the velocity.

When a man is to hold a great weight in his hand, he naturally places the hand near the body �nbsp;for if he extend his* arm, the momentum of thenbsp;weight which is placed at the end of it, as if dnbsp;were at the end of a long lever, becomes too gre^-tnbsp;for his power; confidering that the arm becomes anbsp;lever where the power and the fulcrum lie near onenbsp;extremity, viz. near the Ihoulder, and the weightnbsp;lies at the oppofite extremity.

Perfons who are accuftorned to ufe a hammer generally hold'it by the lowed part of the handle,nbsp;for the purpofe of rendering the ftroke as powerful as pofllble; for in that cafe the head of the hani-mer, by being fartheft from the ceotre of motion,nbsp;moves with the greateft velocity, and, of courfe,nbsp;flrikes with the greateft momentum. But fuch,nbsp;perfons as are not fufficiently accuftorned to ufe anbsp;hammer, generally place their hand near the headnbsp;of the inftrument, by which means they rendernbsp;the ftroke of very little effeft.

The like obfervation might be made with re--fpeft to the ufe of almoft all other tools and in-ftruments, including thofe which are commonly in ufe, fuch as fciflars, knives, razors, amp;c. And itnbsp;it is by the different management of fuch inftru-ments that a mechanical hand is diftinguilhednbsp;from an unmechanical or clumfy one; and that anbsp;perfon poffeffed of ufeful experience, ufeful habitsnbsp;and ufeful knowledge, is diftinguiflied from one ofnbsp;the contrary defcription j excepting indeed whennbsp;the aukward management of tools, amp;c. is wilfullynbsp;adopted, under the refined idea, and for the purpofenbsp;of fhewing, that a perfon has never been under thenbsp;difgraceful neceffity of handling any mechanicalnbsp;inftrument.

Of the different machines of luxury or convenience, that are in ufe amongft civilized nations, none have been more generally adopted, and morenbsp;univerfally ufed, thah wheel carriages j and yet it

Js very remarkable, that, notwithftanding the prc-fent greatly improved ftate of mechanical knowledge, thofe machines are by no means conilrudted or ufed in the moft advantageous manner pof-fible.

Could the furfaces of bodies be rendered pcr-fedtly fmooth, flat, and deftitute of adhefion, it would be as eafy to drag a body upon a plane, asnbsp;it would be to move it upon wheels ; but as this isnbsp;far from being the cafe, the advantage which arifesnbsp;from the ufe of wheel carriages, is too evident tonbsp;need any particular demonftration; and, in fadt, wenbsp;almoft every day obferve, that a Angle horfe is ablenbsp;to carry upon a cart fuch a load as ten horfesnbsp;would perhaps have not ftrength fufficient to movenbsp;on the bare ground.

When a heavy body is dragged upon the ground, the fridtion is very great, becaufe the ground ftandsnbsp;Hill and the body moves upon it, fo that allnbsp;the inequalities of the ground, the accumulationnbsp;of dirt, and ftones, the finking of the ground,nbsp;amp;c. form fo many obftables to the moving body,nbsp;which obflacles muft be overcome by the powernbsp;which is applied to draw it. But when the bodynbsp;is carried upon wheels, as upon a cart, or waggon,nbsp;amp;c. the furface of the rims of the wheels does notnbsp;rub on, but is fucceffively applied to the ground,nbsp;(agreeably to W'hat we have faid above with refpedtnbsp;to rollers; fee page 279) and the obflacles arifingnbsp;from finking, from ftones, fand, amp;c. offer an oblique

lique oppofition to the wlieels, which is overcomtf vaftly eafier than a diraft oppofition, and the morenbsp;fothe larger the wheels are in diameter; fo that bynbsp;the ufe of large wheels the friftion againft tli^nbsp;ground is almoft entirely removed. There is how-�nbsp;ever another fort of friftion introduced by the ufenbsp;of wheels, viz. the friftion of the wheel upon thenbsp;axle; but this friftion, when the parts are properlynbsp;fliaped, and oiled or greafed, is not very material,nbsp;efpecially when the wheels are large; for whe^ anbsp;wheel turns upon an axle, the force neceflary tonbsp;overcome the friftion is diminiflied in the ratio ofnbsp;the diameter of the wheel to the diameter of thenbsp;axle.

A wheel carriage is drawn with the lead power, when the line of draught paffes through the centrenbsp;of gravity of the carriage, and in a direftion parallel to the plane on which it moves. It thereforenbsp;follows that the height of the carriage fliould benbsp;regulated by the nature of the power which is tonbsp;draw the carriage; viz. whether it is to be drawnnbsp;by high or low horfes, by bullocks, amp;c. It muflnbsp;however be obferved, that, from the make of hisnbsp;body, a horfe draws with the greateft advantagenbsp;when the traces, or the lhaft, makes a fmall anglenbsp;with the plane which paffes through the axles ofnbsp;the wheels. But this angle muft not exceed a fewnbsp;degrees, otherwife part of the power of the horfenbsp;js employed in lifting up the fore wheels from the

ground; confequently the power is not entirely employed in drawing the carriage forward.

Four-wheeled carriages are almoft always made with the two fore wheels fmaller than the hindnbsp;wheels. The fore wheels are made fmaller thannbsp;the others for the conveniency of turning, as theynbsp;require lefs room for that purpofe. The fmallnefsnbsp;of their fize does alfo prevent their rubbing againfhnbsp;the traces; but, thofe objedts excepted, fmall wheelsnbsp;are by no means fo advantageous as thofe of a larger diameter, as has been already mentioned, and asnbsp;will.be confirmed by the following illuftration.

In fig. 2. Plate IX. let the hollows BGC, and DFO be equally large, and equally deep in thenbsp;ground. It is evident that the large wheel A willnbsp;not go fo far into the hollowing, as the fmall wheelnbsp;R. Befides, even fuppofing that .they defcendnbsp;equally deep into thofe hollowings, the large wheel,nbsp;by the power adting far above the impediment, maynbsp;be eafily drawn out of it; whereas the fmall wheelnbsp;can hardly be drawn out by means of an horizontal draught, unlefs indeed when the ground givesnbsp;way before it, which is not always to be expedted.

The idea of the two large wheels helping to drive the fore fmall ones, is a vulgar error, -which has notnbsp;the lead; foundation in truth. The abfurdity ofnbsp;this idea might be proved various ways, but bynbsp;none more fatisfaClorily than by the following experiment.

Take a real carriage, or the model of a large one, having two large and two fmall wheels. Fallen anbsp;rope at each of its ends, but equally high from thenbsp;ground; then extending one of thofe ropes horizontally, let if go over a pulley, which muft benbsp;' placed at fome diftance from the carriage, and tienbsp;as much weight to the defeending extremity of thenbsp;rope, as may be juft fufficient to move the carriage.nbsp;This done, difeharge this rope; turn the carriagenbsp;with its other end towards the pulley, and, in Ihort,nbsp;repeat the experiment with the other end of thenbsp;carriage foremoft. It will be found that precifelynbsp;the fame weight will be required to draw the carriage, and to draw it with equal velocity, whethernbsp;the large or the fmall wheels be placed foremoft.

The figure, or rather the breadth of the rims of the wheels, influences confiderably the motion ofnbsp;the carriage. Upon a fmooth and hard road nonbsp;advantage is derived from the ufe of broad wheels;nbsp;but upon a foft road the broad wheels are muchnbsp;more advantageous than narrow ones; the latternbsp;cutting and finking into the ground; on whichnbsp;account they muft be confidered as always goingnbsp;up hill, befides their fufiering a great deal of friction againft the fides of the ruts that are made bynbsp;themfelves; whereas the broad wheels producenbsp;nearly the fame effed as a garden-roller; that is,nbsp;they fmooth and harden the road, befides theftnbsp;moving with great freedom. It muft however benbsp;obferved, that upon fand, as alfo upon ftiff clayey

roads, lefs force is required to draw a cart with narrow, than one with broad, wheels. Upon fand the broad wheels form their own obftacles, by drivingnbsp;and accumulating the fand before them. Uponnbsp;davey roads they gather up the clay upon theirnbsp;furfaces, and become in a great meafure cloggednbsp;by it.

Some perlbna imagine that the broad wheels, by touching the ground in a great many more pointsnbsp;than narrow wheels, muft meet with proportion-ably greater obftruftion. But it fhould be confi-dered, that though the broad wheels touch thenbsp;ground with a larger furface, yet they prefs uponnbsp;it no more than narrow wheels do. Let, fornbsp;inftance, two carts be equal in every refpedl andnbsp;equally loaded, excepting that the wheels of onenbsp;of them be 3 inches broad, whilft thofe of thenbsp;other be 12 inches in breadth. It is evident thatnbsp;the latter wheels reft upon the ground with a fur-face which is equal to four times the furface uponnbsp;which the former wheels reft. But fince an equalnbsp;weight is fupported by the wheels of both carts,nbsp;every three inches breadth on the furface of thenbsp;broad wheels fuftains a quarter of that weight jnbsp;whereas the three inches breadth of the narrownbsp;wheels fuftain the whole weight; fo that the broadnbsp;wheels touch the ground as much lighter as theynbsp;are broader than the narrow wheels. It is for thenbsp;fame reafon that if a heavy body in the form of *nbsp;parallelepipedon, viz, like a brick, be dragged upo^*nbsp;*nbsp;nbsp;nbsp;nbsp;a

a plane furface, the fame power will be required to draw it along, whether its broad or its narrow fidenbsp;be laid upon the plane.

� If the wheels were always to go upon fmooth and level ground, the beft way v.'ould be tonbsp;make the fpokes perpendicular to the naves;nbsp;� that is, to ftand at right angles to the axles;nbsp;� becaufe they would then bear the weight of thenbsp;** load perpendicularly, which is the ftrongeft waynbsp;� for wood. But becaufe the ground is generallynbsp;� uneven, one wheel often falls into a cavity or rutnbsp;� when the other does not and it bears muchnbsp;� more of the weight than the other does; innbsp;� which cafe, concave or dilhing wheels are beft,nbsp;� becaufe when one falls into a rut, and the othernbsp;quot; keeps upon high ground, the fpokes becomenbsp;� perpendicular in the rut, and therefore have thenbsp;quot; greateft ftrength when the obliquity of the loadnbsp;quot; throws moft of its weight upon them; whilftnbsp;� thofe on the high ground have lefs weight tonbsp;bear, and therefore need not be at their fullnbsp;� ftrength. So that the ufual wayof making thenbsp;� wheels concave is by much the beft.�

� The axles of the wheels ought to be perfectly � ftraight, that the rims of the wheels may benbsp;� parallel to each other; for then they will movenbsp;� cafieft, becaufe they will be at liberty to go onnbsp;� ftraight forwards. But in the ufual way of prac-tice, the axles arc bent downward at their ends jnbsp;�' which brings the fides of the wheels next the

� ground nearer to one another, than their oppo-� fite or higher fides are : and this not only makes � the wheels drag fidewife as they go along, andnbsp;� gives the load a much greater power of cruthingnbsp;� them than when they are parallel to each other,nbsp;� but alfo endangers the overturning of the car-� riagewhen any wheel falls into a hole or rut, or

when the carriage goes in a road which has one � fide lower than the other, as along the fide of anbsp;quot; hill¹ for on that conftruftion the carriagenbsp;ftands upon a narrower bafe, than when the rims ofnbsp;the wheels are parallel to each other.

Upon level ground a carriage with four equal wheels may be drawn by the fame power withnbsp;equal facility, whether the load be placed on anynbsp;particular part of the carriage, or it be. fpreadnbsp;equally all over it.

Upon a two wheeled carriage the mofl: advantageous difpofition of the load is, when the centre of gravity of the weight coincides with the middle ofnbsp;the axle, or with a perpendicular line which paflhsnbsp;through that middle.

A carriage having the two hind wheels large, and the two fore wheels fmall, when going upon an horizontal plane, flrould have the principal part ofnbsp;the load laid towards its hind part; but when going upon uneven roads, up and down hill, andnbsp;when the load cannot be eafily fliifted, the beft

Do�irine of Motion. way is to ky the load principally in the middle, ornbsp;to fpread it equally all over the carriage�

The common practice of carriers, who place the principal part of the load upon the fore axle of thenbsp;waggon, is evidently very abfurd; for by thatnbsp;means they prefs that axle with greater force uponnbsp;the wheelSj and the fore wheels deeper into thenbsp;ground, in confequence of which thofe wheels,nbsp;which turn oftener round than the large wheels,nbsp;will not only wear much fooner, but require anbsp;greater power to draw them along, and efpeciallynbsp;over any obftacle.

The lower the centre of gravity of the load is fituated, the lefs apt is the carriage to be overturned. The following obfervations are of Profef-for Anderfon of Glafgow.

� In Glafgow and its neighbourhood, a tingle � horfe, on a level turnpike road, draws 25 cwt.

� in a cart which weighs about 10 cwt. having ' � wheels fix feet high, and its axle paffing throughnbsp;quot; the centre of gravity of the,load and cart, bur,

quot; in a common cart, he draws only the half of that load. Two horfes yoked in a line,nbsp;in a common cart eafily draw 30 cwt. uponnbsp;� an even road. And fix horfes, yoked two a-bread:, draw 80 cwt. in a common waggon,�

� Six horfes, in fix carts, with high wheels, can draw 150 cwt. on a level road �, and fix horfes,

� in three common carts, with two horfes in each, . � can draw, upon an uneven road, 90 cwt. that

� is 10 cwt. more than they can do in a waggon � � the weight, tear and wear, and the eafe in draw-� ing a waggon, or three carts, being, it is faid,nbsp;� nearly equal; and the price of the three cartsnbsp;� being lefs than that of the waggon.� ¹

XT 7HATEVER body is impelled by any power, and is afterwards left to proceed by itfclf, isnbsp;called a projeSlile, which denomination is derivednbsp;from a Latin word, the meaning of which is tonbsp;throw, to hurl. Thus the bullets which arenbsp;thrown out of fire arms. Hones that are thrown bynbsp;the hand, or by a fling, or by any other projedtingnbsp;inftrument, amp;c. are called projeEiiles.

It has been already (hewn (in chap. IX.) that projeftiles, unlefs they be thrown perpendicularlynbsp;upwards or downwards, mufl; defcribe a curve line inbsp;becaufe they are adted upon by two forces, one ofnbsp;which, viz. the impelling force, produces an

equable motion ; whilft the other, viz. the attraction of the Earth, produces an accelerated motion.

It has likewife been Ibewn that proje�liles de-fcribe fuch curves as are called by the mathematicians parabolas ; or rather that they would defcribe fuch curves, if they were not influenced by certain fludluating circumftances, which caufe thenbsp;paths of projedliles to deviate more or lefs fromnbsp;true parabolic curves.

Thus much might have fufEced with relpedt to the motion of projedtiles. But the great ufenbsp;which is made of them both in peace and in war,nbsp;obliges us to confider this branch of the dodlrine ofnbsp;motion in a more particular manner, and to derivenbsp;from the theory fuch rules as may be of ufe in thenbsp;pradtical management of projedtiles.

There are three caules, which force the projectile to deviate from the parabolic path: viz. ift, the force of gravity�s not adling in diredlions perpendicular to the horizon i 2dly, The decreafe ofnbsp;the force of gravity according to the fquares of thenbsp;diftances from the centre of the Earth; and jdly,nbsp;the refitlance of the air.

The effedts which are produced by the firfl and fecond of thofe caufes, are too fmall and trifling jnbsp;for the centre of the Earth is at fo great a dif-tance from the furface, that both the height andnbsp;the diftance to which we are able to throw projectiles, are exceedingly fmall in proportion to it.

z 2 nbsp;nbsp;nbsp;But

But the refiftance of the air offers a very confider-* able oppofition to the motion of projedliles, andnbsp;its aflion is fo very fluftuating, that, not with-(landing the endeavours of feveral able philofophersnbsp;and mathematicians, the deviation of projedlilesnbsp;from the parabolic path has not yet been fubjeftednbsp;to any determined and pradlical rules.

After the preceding remarks it will be ealify al-* lowed, that the only method we can follow Is to laynbsp;down the theory on the fuppofition that the pro-jediles move in parabolas, and then to fubjoin anbsp;concife (latemcnt of the refult of the principal experiments, which have been made for the purpofenbsp;of (hewing the deviation of the real path of anbsp;projeftile from the parabolic curve.

When a body is projected obliquely from any kind of engine, fuch as a ball from the canquot;nbsp;non A, fig. 3, Plate IX. in the direclion AC,nbsp;-the force of gravity, adling upon it in di-redlions nearly perpendicular to the horizontalnbsp;plane AB, forces it to deviate from the ftraight di-reftion A C, and to deferibe the parabola ADBgt;nbsp;lafily falling upon the horizontal plane at B Jnbsp;whence it evidently follows, that a ball or any othernbsp;projedlile cannot move even for a moment in ^nbsp;ftraight line; but that it muft deviate more ornbsp;lels from the ftraight line of its initial direftioo�nbsp;and muft immediately begin to incline towards th�nbsp;ground ; excepting however fome particular cales,

tion round its axis, or when its fhape is fomewhat oblong and bent; for in thofe cafes it may deviatenbsp;not only tideways, but even upwards for a fliortnbsp;time.

The diftance AB between the mouth of the projefting engine and the place where the flrot fallsnbsp;upon the horizontal plane, is called the range ofnbsp;the jhot, or the amplitude of projection; D E is thenbsp;height of its path, or of the parabola ADB. Thenbsp;angle CAB, which the dire�tion of the projection,nbsp;or of the cannon, makes with the horizontalnbsp;plane, is called the angle of elevation. The timenbsp;during which the fliot performs the path ADB, isnbsp;called the time of flight, and the force with whichnbsp;it ftrikes an obje�l at B, is its momentwn.

It will be found demonftrated in the note that thofe particulars, viz. the range, the height, thenbsp;angle of elevation, amp;c. bear a certain determinatenbsp;proportion to each other, fo that when two ofnbsp;them are known, the others may thereby be foundnbsp;Out. It is demonftrated likewife, that, cceteris paribus, the greateft range or greateft diftance tonbsp;which a fliot may be thrown Upon an horizontalnbsp;plane, takes place when the angle of- elevation isnbsp;equal to half a right angle, or 45 degrees (i.); we

(I.) Propofuion I. A body which is projedled in a direction not perpendicular, but oblique, to the horizon, will de-flrihe a parabola j and its velocity in any point of that parabo-

3\'i nbsp;nbsp;nbsp;Of ProjeSiles,

fhall therefore proceed in this place to fhew a practical method of determining the moll ufeful of thofe particulars, when the angle of elevation, andnbsp;the greateft range the cannon is capable of, arenbsp;known.�This greateft range, viz. the diftance tonbsp;which the cannon when charged with the ufualnbsp;quantity of powder, and elevated to an angle ofnbsp;45'�. is capable of throwing the (hot, muft be af-certained by means of adual experiment andnbsp;menfuration in every piece of artillery, efpeciallynbsp;with large cannons, and mortars j fince the baljsnbsp;from thofe pieces deflecft much more from thenbsp;ftraight diredtion.�The angle of elevation is af-certained by means of a graduated circular inftru-ment, and a plummet, or a level.

Let

la IS the fame as it would acquire ly defcending perpendicularlj through a fpace equal to the fourth part of the parameter be~nbsp;longing to that point as a vertex', fuppofng that the force ofnbsp;gravity is uniform, and that it a�ls in directions perpendicularnbsp;to the horiooontal plane 5 alfo that the air offers no refiflancenbsp;to the motion of prcjeCiiles.

Let a body be projected from A, fig. 5. Plate IX-the diredtion AE, and let AE reprefent the fpace, through which the projedting force alone would carry it with annbsp;equable motion in the time T. Alfo let AB reprefent thenbsp;fpace through which the force of gravity alone would caufenbsp;it to defcend in the tame time T. Complete the parallelogram ABEC, and it is evident that the body, being ttn-pelled both by the projedling force, �and by the fores

Of ProjeSiiles� nbsp;nbsp;nbsp;343

Let the greatefl horizontal range of a cannon, or mortar, be 6750 yards, and let the actual anglenbsp;of elevation be 25� j the other particulars maynbsp;be found by delineating this cafe upon paper; fornbsp;which purpofe the inftruraents that are generally

gravity, muft, at the end of the time T, be found at C. Now AE is as the time T, becaufe it reprefents the fpacenbsp;defcribed uniformly; but AB is as the fquare of the timenbsp;T; therefore AE or its equal BC, is as the fquare of AB.nbsp;And the fame reafoning may be applied to any other contemporary diftances, as AH, AF, or FG, AF. But AEnbsp;is a tangent to the curve at the point A, AF is a diameternbsp;at the point A, and B C, FG, amp;c. being parallel tp thenbsp;tangent AE, are ordinates to the diameter AF ; and fincenbsp;the fquares of thofe ordinates have been demonftrated to benbsp;as the refpe�tlve abfciflas AB, AF, amp;c.; therefore thenbsp;curve ACGD is the parabola.

The velocity of the proje�file at any point, as A, in the curve is fuch, that the fpace AE would be defcribed uniformly by it, in the fame time that the body would employ in defcending perpendicularly by the force of gravitynbsp;from A to B. Alfo (fee p. 66.) the velocity acquired bynbsp;the perpendicular defcent AB is fuch as would carry thenbsp;body equably through twice AB in the fame time, (that is,nbsp;in the fame time that AE is defcribed;) therefore the velocity which is acquired by the perpendicular defcent AB,nbsp;is to the velocity with which AE is defcribed, as twice ABnbsp;is to AE. But the velocity acquired by the perpendicularnbsp;defcent through AB, is to the velocity acquired by the perpendicular defcent through a quarter of the parameter belonging to the vertex A of the parabola ACG, alfo as

344 nbsp;nbsp;nbsp;�nbsp;nbsp;nbsp;nbsp;Pt'ojeSfiks.

put in a common cafe of drawing inftruments, arc quite fufficient, viz. a pair of compafles, a rulernbsp;with a fcale of equal parts, and a protradlor.

Draw an indefinite right line AK, fig. 4. Plate IX. to reprefent an horizontal plane, palling

fore, fince the like reafoning may be applied to any other point of the parabolic path, we conclude that, univerfally)nbsp;the velocity of the projeflile in any point of its path is thenbsp;fame as would be acquired by a perpendicular defdentnbsp;through a fpace equal to the fourth part of the parameternbsp;belonging to that point as a vertex.

Corollary i. It is evident that the projediile muft move in the plane of the two forces, viz. in the plane which paiTe*nbsp;through AE, AB, and is, of courfe, perpendicular to thenbsp;horizon.

Cor. 2. It follows from the laws of compound motion, that the projedtile will deferibe the arch AC, in theEamenbsp;time in which it would defeend by the force of gravity from.nbsp;A to B, or in which it would deferibe uniformly the fpace

Of TrojeBiles. nbsp;nbsp;nbsp;345

through the point of projeftion A. Make AB perpendicular to it, and equal to twice the greateftnbsp;horizontal range, viz. equal to 13500 j^ards 5 whichnbsp;is done by making it equal to 13500 divifions of

Cor. 4. Either in the fame, or in different parabolas, the parameters belonging to different points are to eachnbsp;other as the fquares of the velocities of the projeiSile atnbsp;thofe points (fee p. 65); whence it follows, that at thenbsp;vertex of the parabola the velocity or the momentum ofnbsp;the projedile is the leaft, and at equal diftances from thatnbsp;vertex the velocities or the momentumsare equal.

Propofition II. The initial velocity being given^ to find the direSlion in which a body mujl be prujeSied in order to hitnbsp;a given point.

Let A, fig. 6, Plate IX, be the proje�ling point, and C the objedf, or point which is required to be hit.

The velocity of projection being given, the parameter of the parabola which muff pafs through the point C will eafilynbsp;be found by means of the preceding propofition i viz. bynbsp;finding the fpace, through which a body muft fall fromnbsp;reft, in order to acquire the given velocity ; for that fpacenbsp;is equal to the fourth part of the parameter belonging to thenbsp;point A.

Join AC, draw the horizontal line AL, and at A ere�l AP perpendicular to the horizontal line AL, and equal tonbsp;the above-mentioned parameter. Divide AP into two equalnbsp;parts at G, and through G draw an indefinite right linenbsp;KGH parallel to the horizontal line AL. Through A

the fcale of equal parts, for thofe parts muft re-prefent yards. Upon AB, as a diameter, defcribe the femicircle AFB. At A, by means of the pro-traftor, draw the line of projedtion AF, making annbsp;angle of 25� with the horizontal line AK.nbsp;Inbsp;nbsp;nbsp;nbsp;.nbsp;nbsp;nbsp;nbsp;Through

draw AK perpendicular to the diretSlion AC of the obje�l, which AK will meet KH in a point K. With the centrenbsp;K and radius KA draw the circular arch PHEA. Throughnbsp;the point or objeA C draw BCI perpendicular to the horizon, and if this perpendicular meets the circular arch, asnbsp;at E and I, draw AE, AI; and either of thofe directionsnbsp;will anfwer the defired purpofe.

Join PI and PE; and the triangles PAE, E AC are fimi-lar ; the angle PAE being equal to the angle AEC (Eucl. p. 29, B. I.) and the Angle APE equal to the angle EACnbsp;(Eucl. p. 32, B. III.) Hence PA; AE : ; AE ; ECi

AIC are alfo fimilar; the angle PAI being equal to the angle AIC (Eucl. prop. 29, B. I.) and the angle API equalnbsp;toIAC (Eucl. p. 32, B. III.) Hence PA: AI:: Al-

rameter belonging to the point A of the parabola, which is to be dcfcribed by the projeclile, amp;c. the faid parabola (bynbsp;cor. 3 of the preceding prop.) muft pafs through thenbsp;point C.

Corollary I. The angular diftance CAP between the object and the zenith, is divided into two equal parts bynbsp;the line AH ; for KH being equal to KA, the angle AHKnbsp;is equal to tiie apgle HAKj and likewife equal (on account

Oj FrcjeBiles. nbsp;nbsp;nbsp;347

Through the point F, where AF cuts the femi-, circle, draw OF perpendicular to the horizon AK.nbsp;Divdde AO into two equal parts, and at the pointnbsp;of divifion C ereft CD perpendicular to the horizontal line A K; alfo make C D equal to a quarternbsp;of the parallelifm of KH, AB) to the Angle HAB. Butnbsp;K AC is equal to GAB ; for they are both right angles; therefore, fubtradting the angle GAC from both, there remainsnbsp;KAG equal to CAB; confequently GAH is equal to HAC.

Cor. 2. 7'he two directions AI, AE are equidiftant from the direftion AH; for KH being perpendicular tonbsp;PA and to IE, the arch AH is equal to HP. and EH i$nbsp;equal to HI.

Cor. 3. When the directions AI, AE coincide with AH, then the diftance AC is the greatefl diftance to whichnbsp;the projectile can be thrown upon the plane AO with thenbsp;given velocity of projection. Hence it appears that whennbsp;the objeCt C is placed upon the horizontal plane, as in fig,nbsp;7, Plate IX. where AC coincides with AB, AK coincidesnbsp;with AG, PHA becomes a femicircle, and HAC half anbsp;right angle; then the greatefl: diftance to which the pro-je�ile can be thrown, takes place, viz. when the angle ofnbsp;elevation HAB is half a right angle.

Cot. 4' The velocity of projection being known, the greatefl: diftance AL, to which the projectile can benbsp;thrown upon the horizontal plane, or greatefl: range, isnbsp;likewife known ; it being equal to half the parameter A P ;nbsp;for AL is equal to the radius KH, or AK, which is thenbsp;half of AP.

Cor. 5. When the point of projection A, and objeCt C, are both upon the fame horizontal plane, as in fig. 7 gt;

then

34^ nbsp;nbsp;nbsp;0/ ProjeBiles.

ter of OF; then the path of the Ihot is repret�nt-ed by a curve line, which pafles through the points A,D,0. Take the diftance AO in your compafles,nbsp;and, applying it to the fame fcale of equal partsnbsp;as was ufed before, you will find it equal to 5170,nbsp;which reprefent yards. If you apply the diflance

�ienthediftance ACof the objedt is as the fine of twice the angle of elevation CAE, or CAI; for (Eucl. p. 32, B.nbsp;III.) CAE is equal to APE, and likewife equal (Eucl. p.nbsp;20, B.III.) to half AKE, vvhofe fine is h N; and EN isnbsp;equal to AC. EN is likewife the fine of double thenbsp;angle CAI; for CAI = API � ^ AKI; and IS =: NE,nbsp;is the fine of the angle AKI.

Cor. 6. If AE be the diredlioh of the projeAile, the greateft height of the parabolic path above the horizon, isnbsp;equal to a quarter of AC, and is as the verfed fine of twicenbsp;the angle of elevation CAE. For divide AC into twonbsp;equal parts at T, and eredl TR perpendicular to it. Divide TR into two equal parts at V; then TV is equal tonbsp;half TR, and to a quarter of EC. It is evident that ACnbsp;is an ordinate to the axis TR of the parabolic path;nbsp;and that V muft be the vertex of that parabola; fornbsp;the dircdlion AE being a tangent to the parabolic curve,nbsp;the part VT of the axis is (by conics) equal to the par'^nbsp;4'R. Farther, CE is equal to AN, which is the verfeonbsp;fine of the angle AKE, viz. of twice, the angle of elevation EAC, or IAC ; therefore TV* is equal to a quarter ofnbsp;EC, and is as the verfed fine of twice the angle of elevation CAE.

Proje�iiles, nbsp;nbsp;nbsp;349

C D to the fame fcale, it will be found equal to 603 divifions, or 3�ards; which fliews that withnbsp;the angle of elevation equal to 25% the cannon innbsp;qqeftion will throw the (hot to the horizontal di{-tance of 5170 yards, and that the vertex, or great-

dicular to the horizon, is equal to a quarter of the para-i meter AP; for in that cafe AE, EC, and AP coincide;nbsp;confequently a quarter of CE becomes the fame thing asnbsp;a quarter of AP.

Cor. 8. In the cafe of the preceding corollary, the time of the projedlile�s remaining in the air, is the fame that anbsp;body would employ in defcending from P to A, merely bynbsp;the force of gravity; for the projedlile will be as long innbsp;afcending, as in defcending along a quarter of AP; viz�nbsp;it will employ twice the time which is required to defeendnbsp;aloiig one quarter of PA, and which is equal to the timenbsp;that is required to defeend from P to A; the (paces de-feribed by defcending bodies being as the fquares of th�nbsp;times.

Cor. 9. The time of flight when a body is projefted. in any direflion, as for inftance AE, is as the fine of thenbsp;angle of elevation EAC; for it is as the chord AE, ornbsp;as half AE, which is the fine of the angle APE, or ofnbsp;half AKE, which is equal to the angle of elevation CAE.

From the abovementioned two propofitions, with their corollaries, the moft ufeful properties of projeiStiles havenbsp;been derived, and are concifely exprefied together with thenbsp;refults of experiments, in the following pra6lical rules,nbsp;which every gunner fhould imprefs in his mind ; for whennbsp;the greateft range that a piece of artillery is capable of withnbsp;the ufual charge of powder, is known, and which muft be

3^0 nbsp;nbsp;nbsp;Prejgfiiles�

yards.

By the like means the range anfwering to any other angle of elevation may be afcertained �, and

learned from aftual experiment; thofe rules will anfwer all the neceflary cafes in gunnery; excepting the obftruc-tion which arifes from the reliftance of the air.

tion and the objedl are not both upon the fame horizontal plane) for the heft elevation take the compl^' ment of half the angular diftance between the objectnbsp;and the zenith.

If, on the other hand, the diftance of the ob-jeft, and the cannon�s greateft range being known, the angle of elevation neccflfary to hit that objectnbsp;be required; you muft proceed by the reverfe ofnbsp;the preceding method. Let, for inftance, thenbsp;diftance of the object from the cannon be 5170nbsp;yards, and the cannon�s greateft range, 6750 yards.nbsp;Draw a right line AO, fig. 4. Plate IX. equal tonbsp;^170 divifions of the fcale of equal parts. Makenbsp;AB perpendicular to AO, and equal to twice thenbsp;cannon�s greateft range; viz. to 13500 divifionsnbsp;of the fame fcale. Upon AB, as a diameter, de-fcribe the femicircle AFB. At O ere�t the linenbsp;OG perpendicular to the horizontal line AO, andnbsp;the perpendicular OG will meet the femicirclenbsp;either in one, or in two points, or not at all.nbsp;Should this line meet the femicircle in one point,nbsp;it muft be at Y, its middle, and then the requirednbsp;angle of elevation is YAO, viz. of 45�. If OG

fines; but there is an inflrument in ufe, called the gunner�� s callipers, which anfwers every purpofe relative to the application of thofe rules; as it contains a graduated circle,nbsp;and is fufceptible of the application of a plummet, amp;c. Anbsp;table of fines and verfed fines, together with many othernbsp;tables and meafures are likewife marked upon it.�See anbsp;Very good defcription, and account of the various ufes ofnbsp;the gunner�s callipers, in Robertfon�s Treatife on the Ufenbsp;�f Mathem. Inftrum.

meets th� femicircle in two points, as at F and G, which is the cafe in the prefent inftance; thennbsp;either th� dire�tion AF or AG will anfwer thenbsp;purpofe ; and if the angles which thofe diredtionsnbsp;make with ,the horizon AOj be meafured by meansnbsp;of the protradtor, the former will be found equalnbsp;to 25% and the latter to 65�. But when the perpendicular OG does not meet the femicircle, then wcnbsp;muft conclude that the given power, or force of thenbsp;cannon in queflion, is not fufficient - to throw thenbsp;fhot to the propofed diftanc�.

It follows from the foregoing conftrudtion, that the higher thediredtion line cuts the femicircle,nbsp;the longer is tire horizontal range j not exceedingnbsp;however the middle point Y j for S Y, which isnbsp;equal to the radius of the circle, is longer than anynbsp;other line that can be drawn in the femicirlc AYB#nbsp;parallel to the horizon ; hence the horizontal rangenbsp;is the greateft when the angle of elevation is 45�'nbsp;It is alfo evident, that at equal diftances from thenbsp;point Y, as at F and G, the horizontal range will benbsp;the.fame; the height of the path only being different. When the angle of elevation is 90quot;, thennbsp;the line of diredtion, becoming perpendicular tonbsp;the horizon, coincides with the line AB, and thenbsp;horizontal range becomes equal to nothing, viz.nbsp;the fhot will afeend to a quarter of AB, and willnbsp;then defeend again to A.

fliot�s remaining in the air, is equal to the time that a body would employ in defcending perpendicularly, by the force of gravity, from a heightnbsp;equal to the horizontal range, which may be foundnbsp;by the rules given in chap. V.�In order to findnbsp;the time of flight, at any other inclination, as fornbsp;inftance, the inclination OAF; fay as AY (mea-.nbsp;fured on the fcale of equal parts) is to AF (alfonbsp;meafured on the fame fcale), fo is the time that anbsp;body would employ in defcending perpendicularlynbsp;through a fpace equal to the greateft horizontalnbsp;range, to the time in queftion, which will be knownnbsp;by the common rule of three.

Thus much may fuffice with refpedt to the fup-pofed parabolic paths of projeftiles. We (hall now fubjoin a (hort account of the refult of fuch experiments as have been made for the purpofe of determining how far the parabolic theory may be depended upon.

In the common praftice of direfting cannons, an arbitrary allowance is made for the deviation of thenbsp;(hot from the ftraight line. Praeflice indeed renders fome gunners very expert; but their pradticalnbsp;accuracy cannot be reduced to certain rules; viz.nbsp;fuch as may be of ufe to other perfons.

Mortars, which throw the (hots in general with lefs velocity than cannons, have heretofore beennbsp;directed by means of a graduated inftrumentj preferring, of the two diredions which produce thenbsp;fame horizontal range, that which may be thoughtnbsp;preferable according to the nature of the objed.

But of late another method has been found more advantageous in praftice.�The mortar is fteadilynbsp;^xed upon its bed or carriage, at an elevation ofnbsp;45�; but it is loaded with more or lefs powdernbsp;according as the flrell is required to go farther ornbsp;nearer; not exceeding, however, the greateft horizontal range the piece is capable of; the errors ofnbsp;amplitude having been found to be lefs with annbsp;elevation of 45�, than with any other elevation;nbsp;and the horizontal ranges having been found to be,nbsp;cceteris paribus, nearly as the charges of powder;nbsp;viz. half the weight of the full charge will throw thenbsp;fhell nearly to half the greateft horizontal range; anbsp;quarter of the weight of the full charge will thrownbsp;the fhell to a quarter of the greateft range, amp;c.

It has been found that a 24 pounder (viz. a cannon whofe ball weighs 24 pounds) whennbsp;charged with 16 pounds of gunpowder, and elevatednbsp;to an angle of 45�, will generally range its fliot uponnbsp;an horizontal plane 20230 feet, which is not abovenbsp;one fifth of the range afligned by the theory, viz¹nbsp;of what it ought to be, if the air could be removed.nbsp;-�The oppofition which the ftiot meets with fromnbsp;the air in this cafe has been eftimated by the ing^¹nbsp;nious Mr. Robins, equivalent to 400 pounds.¹nbsp;Hence it appears that the path of a flgt;ot is farnbsp;different from the parabola.

� The refiftance of the air varies principally according to its fiuftuating qualities, viz. temperature, gravity, amp;e. according to the fliape of the fhot, and according to the velocity with which thenbsp;fliot is impelled, viz. the initial or incipient velocity.�When that velocity is fmall, the refiftance ofnbsp;the air is very trifling; but when the initial velocitynbsp;is very confiderable, the refiftance of the air becomes fo great as to render the theory quite inapplicable to practice. The refiftance fometlmesnbsp;amounts to 20 or 30 times the weight of the fhot,nbsp;and the horizontal range frequently is much lefsnbsp;than the tenth part of what it ought to be, according to the parabolic theory.

It has been found that with the fame angle of elevation, the horizontal ranges are in proportionnbsp;to one another as the fquare roots of the initialnbsp;velocities, and that the times of flight arc as thenbsp;ranges; whereas, according to the theory, the timesnbsp;ought to be as the velocities, and the ranges as thenbsp;fquares of the initial velocities.

Mr. Robins likewife found that very little advantage was gained by projefting a body with a velocity greater than 1200 feet per fecond. Whennbsp;a 24 pound tbot is projeded with the velocity ofnbsp;2000 feet in a fecond, it will meet with fo greatnbsp;an oppofition from the air, that when it has advanced not more than 1500 feet, viz. in about onenbsp;fecond, its velocity will be reduced to that of aboutnbsp;AA 2nbsp;nbsp;nbsp;nbsp;i20q

1200 feet per fecond. In confcquence of this quick redudlion of velocity, Mr. R. concluded that anbsp;certain projeftile velocity at the fame angle, mightnbsp;carry a fliot farther than a greater velocity; fornbsp;the body projefted with the greater velocity, whennbsp;its velocity becomes equal to that of the other pro-jedtion, has a lefs angle of elevation, on which account it may go not fo far from that point, fo as tonbsp;make the whole diftance fliorter.

No gun to carry far fhould be charged with powder whofe weight exceeds one fixth, or at moftnbsp;one fifth part of the weight of the fhot; for innbsp;field-pieces that quantity of powder will impel thenbsp;(hot with the initial velocity of about 1200 feetnbsp;per fecond. In a battering piece, when the objedlnbsp;is near, the weight of the powder fhould be aboutnbsp;one third part of the weight of the fliot.

When the initial velocity is greater than about 1100, or 1200 feet per fecond, the refiftance of thenbsp;air feems to be three times greater than it ought, if itnbsp;varied only as the fquare of the velocity, � The ve-� locity at which the variation of the law' of refiftquot;nbsp;� ance takes place, is nearly the fame as that withnbsp;� which found moves. Indeed if the treble re-� fiftanc^ in the greater velocities is owing to anbsp;� vacuum being left' behind the refitted body, 't isnbsp;� not unreafonable to fuppofe that the celeritynbsp;� of found is the very laft degree of celerity with

� can in fome fort avoid the preflure of the at-� mofphere on its hinder parts. It may perhaps � confirm this conje�lure to obferve, that if anbsp;� bullet, moving with the velocity qf found, doesnbsp;� really leave a vacuum behind it, the preflure ofnbsp;� the atmofphere on its fore part is a force aboutnbsp;� three times as great as its refiftance, computednbsp;� by the laws obferved for flow motions.*�

A fliot, befides its being drawn downwards from the line of diredtion, is fornetimes defledled fide-way; which takes place when the fhot by rubbingnbsp;againft one fide of the cavity of the piece, acquiresnbsp;a rotatory motion round its axis, and proceedsnbsp;through the air with that motion; for in that cafenbsp;the fide of the fhot, which in its courfe throughnbsp;the air turns forwards, meets with greater refiftancenbsp;than the oppofite fide, whofe motion coincides withnbsp;that of the air. It is eafy to conceive that whennbsp;the axis of rotation happens to be parallel to thenbsp;horizon, then the rotation will contribute to thenbsp;flrot�s defledion, not fideways, but upwards ornbsp;downwards.

* Robins�s Eflays on Gunnery,�The reader may derive confiderable information frotn Dr. Hutton�s Paper on thenbsp;Force of fired Gunpowder, and the initial Velocities of Cannon Ball, amp;c. in the 68th vol. of the Phil. Trans. And fromnbsp;the Chev. de Borde�s Memoirs in the Hift. of the Acad, ofnbsp;Scienc. for 1769.

the

the difficulty which attends the pradtical management of projedtiles, that even with the very fame piece of ordnance, like thots, equal weights ofnbsp;the fame fort of gunpowder, and the fame anglenbsp;of elevation, the places on which the fliots ftrikenbsp;the horizontal plane frequently differ by feveralnbsp;yards from each other.

- v 'quot;'^ ït,.iL

*1%, .,-r^ r '-^.r

gt;.-■ nbsp;nbsp;nbsp;, v ^

.^-C:

V.. r '\

STiCNTING

UNIVERSiTniTSMUSEUM

Q ilt;D nbsp;nbsp;nbsp;\ O�]gt;

ELEMENTS

PHILOSOPHY.

PREFACE.

A 4 nbsp;nbsp;nbsp;It

By

T. C.

CONTENTS.

VOLUME

CONTENTS.

b2 nbsp;nbsp;nbsp;VOLUME

m nbsp;nbsp;nbsp;CONTENTS.

VOLUME III.

Chap. VIII. Natural phenomena relative to light - 301

CONTENTS.

VOLUME

PART

CONTENTS.

PART V.

Containing a few unconnected Subje�ts, p. 315.

DIRECTIONS /or the BooUinder,

nri

VOL. I. nbsp;nbsp;nbsp;C

mid its Properties. nbsp;nbsp;nbsp;19

c 2 nbsp;nbsp;nbsp;fions

20 nbsp;nbsp;nbsp;Of the General

Properties of Matter, nbsp;nbsp;nbsp;29

50 nbsp;nbsp;nbsp;Of the General

Properties of Matter. nbsp;nbsp;nbsp;31

Properties of Matter.

Whft

the

fupp. ft

g 2 nbsp;nbsp;nbsp;LafWy.1

I ffi^U

he

Propofition.

|;5g

Ci^ntripeial, and Centrifugal, Forces, i^c. 6g

time

Ceniripeta!, and Cenirifugal, Forces, nbsp;nbsp;nbsp;69

Ceniripelal, and Centrifugal, Forces, tele. 71

sbi

'yz nbsp;nbsp;nbsp;Of the Motton arijing from

. Centnpeta�i ond Centrifugal, Forces, 73

� nbsp;nbsp;nbsp;In

yS nbsp;nbsp;nbsp;The Method of nfcertaintng the

h g f e d nbsp;nbsp;nbsp;�

Centre of Gravity, i^c, nbsp;nbsp;nbsp;79

!� If two bodies be conneBed together by means of

an

So 7he Method cf. ajcertaining the

afi inflexible line or rod, as A and B, fig^ 2 2. Plate �/ and the line or rod be fupported by a prop, or (as It is

commonly'

iiifi

In �which Value muft, by means of the equation of the figure,nbsp;he expreffed in terms wherein x is the only variable letter.

Sz nbsp;nbsp;nbsp;The Method of afcertaining the

In this fituation no one of the bodies will have more tendency towards the earth than the other ;

for

Centre of Gravity., ^c. nbsp;nbsp;nbsp;S3

7, nbsp;nbsp;nbsp;lot

G 2 nbsp;nbsp;nbsp;yy ~

34- nbsp;nbsp;nbsp;�^he Method of ajcertahihig the

__ .3

In the lecond place, if the two bodies, togethegt;^

SS nbsp;nbsp;nbsp;^he Method of ajceriaining the

V. If tzvo bodies be carried tozvards contrary partSy

Front�

Perciitient Bodies, amp;c. nbsp;nbsp;nbsp;95

Since

PercHtknt BodieSflSc. nbsp;nbsp;nbsp;97

At

no nbsp;nbsp;nbsp;^he Theory of