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ELEMENTS

Natural Philofophy,

Confirm�dbyExpERiMENTs;

VOL.

Doflorof Laws and Philofophy, Profeflbrof Mathe^ maticks and Aftronomy at Leyden, and Fellow of thenbsp;Royal Society of London.

Tranflated into English By J. T. Defaguliers, LL. D. Fellow of the Royal Society,

Printed for J. Sen ex in Fketjlreet, W. Innys, in St. PauFs Churchyard; and J. Osborn andnbsp;T. L o N G M A N, in Pater-N^er-Row.

Sir Ifaac Nemton, K�'quot; PRESIDENT

ROTJL SOCIETY, amp;c.

N dedicating this Tranfla-tion to You, 1 do no more than what my learnednbsp;Friend, the ingenious Author, would have done, if Cuftomnbsp;and Gratitude had not obliged himnbsp;axnbsp;nbsp;nbsp;nbsp;to

iv nbsp;nbsp;nbsp;The Dedication.

to offer the firft Philofophical Work he has publiflied fince his beingnbsp;Profeflbr, to the Governors of thenbsp;Univerfity that gave him the Chair.nbsp;And as there are more Admirers ofnbsp;your wonderful Difcoveries, thannbsp;there are Mathematicians able tonbsp;underhand the two firft Books ofnbsp;your Trincipia', fo I hope You willnbsp;not be difpleafed, that both mynbsp;Author and myfelf have, by Extnbsp;periments, endeavoured to explainnbsp;fome of thofe Propofitions, whichnbsp;were implicitly believed by manynbsp;of your Readers; at the fame Timenbsp;that the greateft Part of your thirdnbsp;Book, and feveral of the Corollaries and Scholia in the other two,nbsp;gave them the higheft Satisfadionnbsp;that an inquifitive Mind is capablenbsp;of receiving. Mathematicians ofnbsp;the firft Rank, who want no fuchnbsp;Helps in reading your incomparable Works, take a frefh Pleafure innbsp;feeing thofe Experiments performed which you have made yourfelf;

The Dedication,

And though fome of ours may not always prove, but fometimes onlynbsp;illuftrate a Propofition ; yet, fuchnbsp;Mathematicians, as are of a communicative Temper, will be glad tonbsp;life them, as a new Set of Words,nbsp;to give Beginners fo clear a Notionnbsp;of the Syftem of the World, as tonbsp;encourage them to the Study of thenbsp;higher Geometry; whereby theynbsp;may know how to value your Solutions of the moft difficult Thano-mena, and learn from You, that anbsp;whole Science may be contained innbsp;a fingle Propofition.

For my own Part, fince � cannot enough acknowledge the Advantages which I have had by being admitted to your Converfation, and your generous Way of gratifying me for fuch Experiments asnbsp;1 have made by your Direction ;nbsp;therefore I fliall here forbear tonbsp;pay that Tribute which is due to

vj The Dedication,

You from all Lovers of Knowledge � and rather choofe to be thought Angular, than, by praifing, to offendnbsp;You, I ain.

Tour mofl Obedient^

yerjf Humble Sermnt

TRANSLATOR

READER.

HIS Book 'Will fufficientfy recommend it [elf; therefore, I need not fay anynbsp;thing in Traife of it, either on Aiycount of the ufeful Suhjeltnbsp;that it treats of, or the excellentnbsp;Method and familiar Way in whichnbsp;our Author has handled it: only Inbsp;thought proper to ohferve to thenbsp;Reader, that the Numbers in thenbsp;Margin expref fo many Tropofti-ons, which are referred to, as younbsp;go forward in the Book, to awdnbsp;Repetitions and Tautology. If whatnbsp;is printed in Italic Charalters henbsp;read hy it felf, it will appear to henbsp;a Compendium of the whole Book;nbsp;or the �DoHrinal Tart of it, without

the Experiments and *Demonfira^ tions. I have endeavoured to an-fu/er this End in my Englilh Tran-Jlation^ where you will find^ thatnbsp;whatever is in Italic Characters,nbsp;makes up the Senfe of the rejl ofnbsp;the Book; which alfo readily makesnbsp;Senfe hy it felf, though taken fromnbsp;Tlaces where it feems irregularlynbsp;difperfed.

Thefirfl Edition of thisTranJla-j tion had fome Errors of the *PreJ^, and Faults in the Tlates, whichnbsp;were occafioned ly the Hafle innbsp;which it was printed off, to pre-tvent the Injury that muji homenbsp;heen done to Dr. �s Gravefande, hy anbsp;Tranflation that fome Bookfellersnbsp;endeavoured to get out before mine*,nbsp;which was jo ill done, that no Body that had read the Latin Book,nbsp;would he able to know it again innbsp;their Englilh.

I have, therefore, in this Fourth Edition, carefully review'd and cor-reHed every Error, both in thenbsp;Book and Tlates,

THE

PREFACE.

: F we compare the Writings of different Philofopbers concerning Physics, we may eafily fee that they call different Sciences by the famenbsp;Name, tbo� they all profefs to explain the true Caufe of Natural ¹ Phenomena.nbsp;jlnd no Wonder if they difagree among themfelves,nbsp;fince even Mathematicians, who deal in Certainties, can hardly be kept from wrangling.

But that Diverfity of Opiniotts Jhonld not deter us from fearching after itruth j fince Labour andnbsp;Study will find it out j and the more we are innbsp;love with it, the lefs we are liable to Errors, excepting fuch as human Frailty renders unavoidable.

We mufi proceed cautionfly in Phyfics, fince that Science confiders the Works of the Supreme Wif-dom, and fets forth.

How the whole Univerfe Is govern�d by thofe Laws, and how the fame Laws run thro' allnbsp;the Works of Nature, and are confiantly obferv'dnbsp;with a wonderful Regularity.

We mufi take care not to admit Fibiion for 'fruth; for by that Means we Jhut out all further Examina-^nbsp;tion. No true Explanation of Pbanometia can fpringnbsp;out of a falfe Principle: And what a vaft Differencenbsp;there is betwixt learning the Fibiions of whimficalnbsp;Men, and examining the Works of the mofl wifenbsp;God! Since an Enquiry into Divine Wifdom, andnbsp;the Veneration infeparable from it, is to be the Scopenbsp;of a Pbilofopher, we need not enlarge upon the Vanity of reafoning upon fibiitious Hypothefes.

Nature herfelf is therefore attentively and in-cejfantly to he examined with indefatigable Pains. S'hat Way indeed our Progrefs will be but flow,nbsp;but then our Difcoveries will he certain; and oftentimes we fball even he able to determine the Limitsnbsp;of Human Underflanding.

What has led mofi People into Errors, is an immoderate Defire of Knowledge, and the Shame of confeffing our Ignorance. But Reafon fijould get thenbsp;better of that ill grounded Shame � fince there is anbsp;learned Ignorance that is the Fruit of Knowledge,nbsp;and which is much preferable to an ignorantnbsp;Learning.

Natural Philofophy is placed among thof ? Parts of Mathematicks, whofe Objebi is^iantitj in general.

Pangerct, omniparens legens violare Creator Noluit, astern ique operis fundamina fixit.

The P R E F A C E.

Mathematics are divided into Pure and Mixed. Pure Mathematics enquire into the General Properties of Figures and ahjira�ied Ideas. Mixednbsp;Mathematics examine Fhings themfelves., and willnbsp;have our Notions and Deduliions to agree bothnbsp;with Keafon and Experience.

Phyfics belong to mixed Mathematics. The Properties of Bodies, and the Laws of Nature, are the Foundations of Mathematical Reafoning, as allnbsp;that have examined the Scope of this Science willnbsp;freely confefs. But Philofophers do not equally a-gree upon what is to pafs for a Law of Nature^ andnbsp;what Method is to be followed in ^lefi of thofenbsp;Laws. I have therefore thought fit in this Prefacenbsp;to make good the Newtonian Method, which I havenbsp;followed in this IVork. What that Method is, Inbsp;have briefly fet down in the firft Chapter.

Phyfics do not meddle with the firft Foundation of Things. That the World was created byGon, is anbsp;Pofition wherein Reafon fo perfelily agrees withnbsp;Scripture, that the leaf: Examination of Naturenbsp;will fioew plain Footfteps of Supreme Wifdom. It isnbsp;confounding and overfetting all ourcleareft Notions,nbsp;to affert that the World may have taken its Rifenbsp;from fome general Laws of Motion, and that it imports not what is imagined concerning the firft Di-vifion of Matter. And that there can hardly benbsp;any thing fuppofed, from which the fame Effedbnbsp;may not be deduced by the fame Laws of Nature ^nbsp;and that for this Reafon : That fince Matter fuc-ceflively aflumes all the Forms it is capable of bynbsp;means of thofe Laws, if weconfider all thofe Formsnbsp;in Order, we muft at laftcome to that Form wherein this prefent World is framed �, fo that we havenbsp;no Reafon in this Cafe to fear any Error from anbsp;wrong Suppofition. This Ajfertion, I fay, overthrows ail our cleareft Notions, as has been fullynbsp;proved by many Learned Men^ and is indeed fonbsp;A znbsp;nbsp;nbsp;nbsp;m-

The PREFACE.

mreafonahle, and fo injurious to the Deity^ that it �will feem unworthy of an Anfwer to any one thatnbsp;does not know that it has been maintained by manynbsp;ancient and modern Philofophers, and fame of themnbsp;of the fir ft Kanky and far removed from any Sufpi^nbsp;cion of Jtheifm.

'then firft laying it down as an undoubted Truth, that God has created all Things, we muft after^nbsp;wards explain by what Laws every thing is governed-, and to mention only the Moon, we muftnbsp;explain, why

* The Silver Moon runs with unequal Pace,^ Which yet Aftronomers could never trace, gt;nbsp;Or fix in Numbers her uncertain Place : jnbsp;What Force her Apfides has forward driven.nbsp;And made her Nodes recede i�th� Starry Heaven.nbsp;What is her Pow�r to agitate the Sea,

Whofe various Tides her Prefence ftill obey; When thi�Ocean fwells,its topmoft Banks to lave.nbsp;Or ebbs from weedy Shores with broken Wave,nbsp;Leaving the Sands, the Sailor�s Terror, bare:

In Order to explain more fully which Way we trace out the Laws of Nature, we muft begin bynbsp;fame pre-vious and preparatory Reflexions.

What Subftances are, is one of the Things hidden from us. We know, for Inftance, fome of the

Paffibus baud sequis graditur; cur fubdita null; Haftenus Aftronomo numerorum fraena recufet.nbsp;Cur remeant Nodi, curq; Auges progrediuntur,

Viribus impellit, dum fraftis fiudibus ulvam Deferit, ac nautis fufpeiftas nudat arenas ;nbsp;Alternis vicibus fuprema ad littora pulfans.

The P R E F A C E.

Properties of Matter ; hut we are ahfolutely ignor rant, what Subject they are inherent in.

iVho dares affirm that there are not in Body many other Properties, which we have no Notionsnbsp;of? And who ever could certainly know, that, be-fides the Properties of Body which flow from thenbsp;EJfence of Matter, there are not others dependingnbsp;upon the free Power of God, and that extendednbsp;and folid Subfiance (for thus we define Bodyj is endowed with fame Properties without which it couldnbsp;^pcifi ? We are not, I own, to affirm or deny any thingnbsp;concerning what we do not know. But this Kule isnbsp;not followed by thofe, who reafon in Phyfical Matters, as if they had a compleat Knowledge of whatever belongs to Body, and who do not fcruple to affirm, that the few Properties of Body, which theynbsp;are acqtianted with, conftitute the very Effience ofnbsp;Body,

What do they mean by faying, that the Properties of Subfiance confiitute the very Subfiance ?

Can thofe things fiibfifi when joined together, that cannot fubfifi feparately ? Can Extenjion, Impenetrability, Motion, amp;c. be conceived without anbsp;Subjelf to which they belong? And have we anynbsp;Notion of that Subjebi ?

We mufi give up as uncertain what we find to be fo, and not he afhamed to confefs our Ignorance,nbsp;^ho'we need notfear being too bold in affirming, thatnbsp;a Sub jell altogether unknown to us may perhaps benbsp;endowed with fome unknown Properties. And thofenbsp;Men who at the fame �fime that they fay, conformably to this Axiom, 'That we mufi not reafon aboutnbsp;Things unknown, lay it down as a Foundation ofnbsp;their Keafonwgs, that nothing relating to Body isnbsp;unknown to us, are beholden to meer Chance, if theynbsp;are not mifiaken.

The Properties of Body cannot he known '3 We mufi therefore examine Body itfelf, and nicely

The P R E F A C E.

confider all its Properties j that we may he able to determine what natural Effedis do flow from thofenbsp;Properties.

Upon a farther Examination of Body^ we find there are fotne general Laws, according to whichnbsp;Bodies are moved. It ispaft Doubt, for Inftance,nbsp;That a Body once moved continues in Motion ;nbsp;that Readtion is always equal and contrary tonbsp;A�lion. And feveral other fuch Laws concerningnbsp;Body have been difcovered-, which can no Way henbsp;deduced from thofe Properties that are faid to C07i~nbsp;flitiite Body, and fince thofe Laws always holdnbsp;good, and upon all Qccafioits, they are to be lookednbsp;upon as getteral Laws of Nature. But then wenbsp;are at a Lofs to know, whether they flow from thenbsp;Eflence of Matter, or whether they are deduciblenbsp;from Properties, given by Go its to the Bodies, thenbsp;World confifls of �, but no Way effential to Body ; ornbsp;whatever finally thofe Effedis, which pafs for Lawsnbsp;of Nature, depend upon external Caufes, which evennbsp;our Ideas cannot attain to.

Who dares affirm any thing upon this Point concerning all, or any Laws of Nature, without incurring the Guilt ofKaJhnefs ? Befides, whoever examines the Phenomena of Nature will be fully fer-fuaded, that many of its Laws are not yet difcovered, and that 7nany Particulars are wanting towards the compleat Knowledge of others.

^he Study of Natural Philofophy is not however to be contemned, as built upon an unknown Foundation.nbsp;Fhe Sphere of htunan Knowledge is bounded within a narrow Compafs; and he, that denies his Af-fent to every thing but Evidence, wavers in Doubtnbsp;every Minute �, and looks upon many Fhings as unknown which the Generality of People never fo tnuchnbsp;as call in ^leftion. But rightly to diftinguijh Fhingsnbsp;known, from Fhings tmknown, is a Perfediion a-hove the Level of human Mind, Fhough many

The P R E F A C E.

Things in Nature are hidden from us; yet what is fet down in Phyjics, as a Science^ is undoubted.nbsp;From a few general Principles numberlefs particular Phxnomena or Effedls are explained^ andnbsp;deduced by Mathematical Demonftration. For,nbsp;the comparing of Motion, or in other Wordsofnbsp;^lantities, is the continual Theme �, and whoevernbsp;will go about that Work any other Way, than bynbsp;Mathematical Demonfirations, ivill be fure to fallnbsp;into Uncertainties at leaf, if not into Errors.

How much foever then may be unknown in Natural Philofophy, it fill remains a vaft, certain, and very ufeful Science. It corre�is an infinitenbsp;Number of Prejudices concerning natural Things,nbsp;and divine Wijdom i and, as we examine the Worksnbsp;of God continually, fets that Wifdom before ournbsp;Eyes 5 and there is a wide Difference, betwixtnbsp;knowing the divine Pozoer and Wifdom by a Meta-phyfical Argument, and beholding them with ournbsp;Eyes every Minute in their Effebis. It appearsnbsp;then fufficiently, what is the End of Phyfics, fromnbsp;what Laws of Nature the Phenomena are to benbsp;deduced, and wherefore, when we are once comenbsp;to the general Laws, we cannot penetrate any further into the Knowledge of Caufes. There remainsnbsp;only to difeourfe of the Method of fearching afternbsp;thofe Laws j and to prove that the three Newtoniannbsp;Laws, delivered in the firft Chapter of this Work,nbsp;ought to be followed.

The firft is. That we ought not to admit any more Caufes of Natural Things, than what arenbsp;true, and fufficient to explain their Phenomena.nbsp;The firft Part of this Rule plainly follows fromnbsp;what has been faid above. The other cannot benbsp;called in ^leftion by any that owns the Wifdom ofnbsp;the Creator. If one Caufe fuffices, it is needlefs tonbsp;f uperadd another j efpecially, if it be confidered,nbsp;that an Effsbi from a double Caufe is never exabl-A 4nbsp;nbsp;nbsp;nbsp;b

The P R � F A C E.

ly the fame with an Effe�i from a fingle one. There-, fore we are not to multiply Caujes^ 'till it appearsnbsp;one fingle Catife will not do the Bufinefs.

Jn order to prove the following Kules^ �we mufi premife fome general Reflediions.

We have already faid, that Mathematical De-monftrations have no Standard to he judged hy^ but their Conformity with our Ideas; and when thenbsp;^leftion is about Natural Things^ the firft Requi-Jite is.^ that our Ideas agree with thofe Things^nbsp;which cannot he proved by any Mathematical De-monftration. And yet as we have Occafion to rea-fon of Things themfelves every Moment.^ and ofnbsp;thofe Things nothhig can he prefent to our Mindsnbsp;hefides our Ideas, upon which our Reafonings immediately turn-, it follows, that God has efia-hlifhed fome Rules, by which we may judge of thenbsp;Agreement of our Ideas with the Things themfelves.

AllMathematicalReafonings turn upon the Com-parijon of ^lantities, and their Truth is evidenced by implying a Contradibiion in a contrary Propofi-tion. A rebiilineal Triangle, for Inftanee, whofenbsp;three Angles are not equal to two right ones, is anbsp;Thing impojftble. When the ^teftion is not aboutnbsp;the Comparifon of ^antities, a contrary Propofi-tion is not always impoffible. It is certain, for Infiance, that Peter is living, though it is as certainnbsp;that he might have died Tefterday. Now there being numberlefs Cafes of that Kind, where one maynbsp;affirm or deny with equal Certainty; there follows,nbsp;that there are many Reafonings very certain, tho'nbsp;altogether different from the Mathematical Ones.nbsp;And they evidently follow from the Efiablifhmentnbsp;of Things, and therefore from the pre-determinednbsp;Will of Go n. For by forcing Men upon the Necef-fity of pronouncing concerning the Truth or Falf-hood of a Propofitio.n j he plainly fhews they mufinbsp;affent to Agreements, which their quot;Judgments ne-Inbsp;nbsp;nbsp;nbsp;ceffarily

The P R E F A C E.

ceffarily acquiefce in-, and whoever reafons ether-wi/e, does not think worthily of God.

? return to Phyfics: IVe are in this Science to judge hy our Senfes, of the Agreement that therenbsp;is betwixt Ijhings and our Ideas. The Extenfionnbsp;and Solidity of Matter, for Inftance, afferted uponnbsp;that Ground, are pafi all Doubt. Here we examinenbsp;the Thing in general, without taking notice of thenbsp;Fallacy of our Senfes upon fome Occafions, andnbsp;which JVay Error is then to he avoided.

IVe cannot immediately judge of all Phyfical Matters by our Senfes. We have thenKecourfe to another jufi Way of Keafoning, though not Mathematical. It depends upon this Axiom(vi%.) Wemuft look upon as true, whatever being denied wouldnbsp;deftroy civil Society, and deprive us of thenbsp;Means of Living. From which Propofition the fe-cond and third Rules of the Newtonian Methodnbsp;moji evidently follow.

For who could live a Minute� sTime in Tranquil-if Man was to doubt the Truth of what paffes for certain, whatever Experiments have beennbsp;made about it j and if he did not depend upon feeing the likeEffebis produced by the fame Caufe P

The foUowingReafonings, for Example, are daily ly taken for granted as undoubtedly true, withoutnbsp;any previous Examination-, becaufe every Bodynbsp;fees that they cannot be called in ^lefiicn withoutnbsp;dejiroying the prefent Oeconomy of Nature.

A Building, this Day firm in all its Parts, will not of its felf run to Ruin to Morrow. Thus, by anbsp;Parity of Keafon, the Cohefion and Gravity of thenbsp;Parts of Bodies, which I never faw altered, nornbsp;heard of having been altered, without fome intervening external Caufe, will not be altered to Night,nbsp;becaufe the Caufe of Cohefion and Gravity will benbsp;the fame to Morrow as it is to Day. Who does

The Timber and Stones of any Country, which are fit for a Building, if brought over here, willnbsp;ferve in this Place, except what Changes^ maynbsp;arife from an external Caufe^ and I ihall no morenbsp;fear the Fall of my Building, than the Inhabitantsnbsp;of the Country, from whence thofe Material*!nbsp;were brought, would do, if they had built anbsp;Houfe with them, thus the Poiver which caufesnbsp;the Cohefion of Parts, and that which gives IVeifhtnbsp;to Bodies, are the fame in all Countries.

I have ufed fuch Kind of Food for fo many Years, therefore I will ufe it again to Day without Fear.

When I fee Hemlock, I conclude it to be poi-fonous, tho� I never made an Experiment of that very Hemlock I fee before my Eyes.

All tbefe Keafonings are grounded upon Analogy^ and there is noDouht., hut our Creator has in manynbsp;Cafes left us no otherfVayofKeafoning, and therefore it is a right Way.

Which being once proved, we may afterwards make ufe of the fame Method in other Mattersnbsp;where no abfokite Neceffity forces us to reafon at all.nbsp;When an Argument is good in one Cafel there isnbsp;no Reafon why we fooitld refufe our Ajfent to it innbsp;another. For who can conceive, that Fhin%s proved the fame Way are not equally certain d Befidesnbsp;thd� vee conclude in general, that this Method ofnbsp;Reafoning is right from the Meceffity of nfing itnbsp;yet it does not follow that particular Reafoningsnbsp;depend upon that NeceJJlty. I conclude from Analogy, that Food is not poifonous �, hit is that Argument only good, when I am hungry s'

In Pbyfics then we are to difcover the Laws of Mature hy the Phxnomena, then by Induliion provenbsp;them to he general Laws-, all the refi i^ to he handled

The P R E F A C E.

died Mathematically. Whoever will ferioiifly exa-�minet what Foundation this Method of Phyjics is huilt upon, will eafily di[cover this to he the onlynbsp;true one, and that all Hypothefes are to he laid a-fide.

So much for the Method of philofophifing. J have now a Word to fay of the Work it [elf, of which thisnbsp;is the firft ^ome.

��he whole Work is divided into four Books. Fhe firfi treats of Body in general, and the Motion ofnbsp;Solid Bodies. Fhe Second of Fluids. What belongsnbsp;to Light is handled in the F'hird. ^he fourth explains the Motions ofCeleJiial Bodies, and whatnbsp;has a Relation to them upon Earth. Fbe'two firfinbsp;Books are contained in this Fime.

In order to render the Study of Natural Philofo-phy as eafy and agreeable aspoffible, I have thought fit to illufirate every thing by Experiments, and tonbsp;fet the very Mathematical Conclufions before thenbsp;Reader�s Eyes by this Method.

He that fets forth the Elements of a Science, does not promife the learned World any thing newnbsp;in the main Fherefore I thought it needlefs, tonbsp;point out where what is here contained is to be found.nbsp;J have made my Property of whatever ferved mynbsp;Purpofe j and I thought giving Notice of it oncenbsp;for all, was fufficient to avoid the Sufpicionof Fheft.nbsp;I had rather lofe the Honour of a few Difcoveries,nbsp;di [perfed here and there in this Freatife, than robnbsp;any one of theirs. Let who will then take to him-felfwhat he- thinks bis own : J lay claim to nothing.

As to Machines which ferve for making the Experiments, I have taken Care to imitate feveralnbsp;from other Authors, have altered and improved o-thers, and added many new ones of my own Invention. And no Wonder I Jhould he forced to thatnbsp;Neceffity, having made Experi?nents upon manynbsp;2,nbsp;nbsp;nbsp;nbsp;Fbings

The P R E F A C E.

S'biamp;gs never tried perhaps hy any one before. For Mathematicians think Experiments fuperfluous,nbsp;vuhere Mathematical Demonjlrations will takenbsp;Place : But as all Mathematical Demonjlrationsnbsp;are ahftraSiedt I do not quejtion their becoming ea-jier^ when Experiments fet forth the Conclufions before our Eyes j following therein the Example ofnbsp;the Englifhj whofe IVay of teaching jjatural Phi-^nbsp;lofophy gave me Occafion to think of the Methodnbsp;J have followed in this Work. I ficall always glorynbsp;in treading in their Footjieps^ who^ with the Princenbsp;of Philofopbers for their Guide, have firji openednbsp;the Way to the Difcovery of Fruth in Philofophicalnbsp;Matters.

As to the Machines, I will fay thus much more by Way of Jdvertifement, Fhat moji of them havenbsp;been made hy a very ingenious Artiji of this Fown,nbsp;and no unskilful Philofopber, whofe Name is Johnnbsp;van Muflchenbroekj and who has a perfedi Knowledge of every thing that is here explained. Whichnbsp;Advertifement, J fuppofe, will not he diifpleafingnbsp;to thofe who may have a faney to get feme of thofenbsp;Machines made for themfelves.

THE

CONTENTS.

BOOK I.

Part I. Of BODT in General.

Chap. III. Of Extenfion^ Solidity.y and Vacuity. 5 Chap, IV. Of the Divifibility of Body ininfinitufn,nbsp;and of the Subtilty of the Particles of Matter. 7nbsp;Chap. V Concerning the Cohefion cf Parts j wherenbsp;we Jhall treat of Hardnefs, Softnefs, Fluiditynbsp;and Elajlicity.nbsp;nbsp;nbsp;nbsp;10

Part II.

Of the Mo tion of Solid Bodies.

Chap. VI, Of Motion in General where we fhall fpeak of Place and Time.nbsp;nbsp;nbsp;nbsp;19

Chap. VII; Of Motions compared together 21 Chap. VIII. How to compare the ASions of Powers.

XX The C o N T E N T S.

Chap. X. Of the Jingle IPuUey, Balanceand of the Center of Gravity.nbsp;nbsp;nbsp;nbsp;ay

BOOK

The C o N T E N T S.

Chap. II. Of the Adiions of Liquids againft theBot-toms a77d Sides of the Vejfels that contain �em. 138 Chap. III. Of Solidsnbsp;nbsp;nbsp;nbsp;immerfed in Liquids. 146

Of the AIR3 as an Elajlick 'Plaid.

Chap. XV. Several Experiments concerning the AiFs Gravity and its Spring.nbsp;nbsp;nbsp;nbsp;zzt

Mathematical

ELEMENTS

I^atural Philofophy,

EXPERIMENTS.

_B O O K I._

CHAP. r.

Orfinition T, and II-Natural Things are all Bodies: Jnd the JJJem- j llage or Syjfem of the^n all is called the Univerfe.

Mathematical Elements Book I.

i Natural Phaznomena are all Situations and all Motions of Natural Bodies^ not immediately depending upon the A^ion of an intelligent Being-, and whichnbsp;may be obferved by our Senfes.

We do not exclude, out of the Number of Natural Phsenomena, thofe which happen in our Bodies by our Will} for they are produced by the Motion of our Mufcles, and their Aftion dependsnbsp;upon another Motion: In thefe, there is only thatnbsp;Motion which arifes from the immediate Aftionnbsp;of the Mind, and is entirely unknown to us, whichnbsp;is not a Natural Phtenomenon.

All thefe Motions are performed by certain Rules, and always fubjeft to the fame Laws.

The Sun rifes and lets daily j and the Time of his Riling and Setting may always be determined,nbsp;according to the Time of the Year, and Latitudenbsp;of the Place. Plants of , the fame Kind, under thenbsp;fame Circumfiances, are always produced and grownbsp;in the fame Manner: And fo on in other Cafes.nbsp;Nay,- even in thofe things which appear to benbsp;wholly fortuitous and uncertain, certain Rules arenbsp;without doubt obferved.

^ Natural Philofophy explains Natural Pheeno-mena-, that is, gives an Account of their Caufes.

In enquiring after thofe Caufes, BODY in general is to be examined; and then the Rulesnbsp;which the Creator has eftablilhed, according tonbsp;which. Motions are to be perform�d. Thefe Rulesnbsp;are called Laws of Nature.

4 A Law of Nature then is, the Rule and Law, according to which Qod refolved that certain Motionsnbsp;Jhould always, that is, in all Cafes, be performed.

Every

Book 1. of Natural Thilo/bfhy.

Alfo in refpe�V to us, we call a Law of Nature, every EfFedl which in all Occafions is producednbsp;after the fame Manner; although its Canfe is unknown to us, and we do not fee that it flows fromnbsp;any Law known to^s.

For we make no Difl�rence between a thing which immediately depends upon the Will ofnbsp;God, and what it produces by the Intermediationnbsp;of a Caufe of which we have no Idea.

In order to find out the Laws of Nature, Sir Ifaac Neiuton's following Rules are to* be ob-ferved.

IVe are not to admit more Caufes of Natural f Things than fuch as are true, and fuffcient for explaining their Phcenotnena.

Such Qualities of Bodies, wfliole Virtue cannot 7 tgt;e increafed and diminilhed, and which belong tonbsp;all Bodies upon which Experiments may be made,nbsp;ffluft be looked upon as Coalities of all Bodies.

CHAP.

Mathematical Elements Book I.

What is meant by Extenfion, no Body is ignorant of. Its Idea is moll fimple, and almott always obvious to oUr Mindj frotii whence it is very intelligible, tho� we want Words to defcribenbsp;it.

Every Body has Extenfion; without Extenfion there is no fuch thing as Body. And yet all thatnbsp;has Extenfion is not always a Body; although itnbsp;is irapofiible to determine, how Body differs fromnbsp;mere Space, till the other Properties of Bodynbsp;lhall firft be afeertained.

P The Second Thing to be examined in Body is ^Solidity. . Body, having no Power to remove itfelf,nbsp;will confequently exclude every other Body fromnbsp;the Place pofTefled by it; and the moft fluid, asnbsp;well as the hardeft Bodies, have this Property.

10 nbsp;nbsp;nbsp;The Third Property of Body is Divifihilitynbsp;becaufe if a Body be extended, it is alfo divifible;nbsp;for you may always conceive one Extenfion leftnbsp;than another. From whence we fee, that therenbsp;are Parts in all Extenfions; which Parts in a Bodynbsp;may be feparated from each other; Becaufe,

11 nbsp;nbsp;nbsp;Body hath a Fourth Property, that is, that itnbsp;may be carried from one Place to another; whencenbsp;it is fiiid to be Moveable.

All Obftacles being removed, a Body yields to the leaft Blow: Neverthclefs there is a greaternbsp;Force required to move a Body with a greaternbsp;Celerity than with a lefs, as alfo to move anbsp;greater than a fmaller Body, .allowing their Velocity to be equal. There is alfo a greater Force

Book I, of natural �Philofofhy. nbsp;nbsp;nbsp;^

required in the fame Cafe to flop the Velocity of different Bodies in Motion. Hence it is, that Bodies at Reft, and Bodies in Motion, endeavour tonbsp;continue in their State.

This arifes from the Inaftivity of Matter, {Inertia) which in all Bodies is ever proportionable to their Quantity of Matter, becaufe it equally be-fongs to ail the Particles of Matter.

All Bodies have fome Figure; whence Figura- 11 lility (that is, to be of fome Shape or Figure) isnbsp;commonly efteemed one of the eflential Propertiesnbsp;of Body, though it feems rather to be derivednbsp;from other Properties.

If a Body be divided on every Side, and thofe Parts removed j what remains in the Middle isnbsp;terminated on �very Part, and confequently has anbsp;certain Figure. The fime Body is capable of havingnbsp;different Figures 5 becaufe it may be divided intonbsp;Parts, and thofe Parts placed in different Order innbsp;refpect to each other. Neither docs it imply anbsp;Contradiftion to fay, that a Body, that ftiouldnbsp;have no Figure, would be an infinite Body.

CHAP. IIL

Of Extenfor, Solidity^ and Vacuity.

Here the Qtieftion (fo often handled by ij the Learned) concerning a Vacuum^ is to benbsp;confidered ; namely. Whether there be an Exten-fion void of all Matter j for this Extenfion is called a Vacuum^ an Emptinefs or mere Space.

That there is really a Vacuum, is proved from Phaenomena: This Propofition therefore fhall benbsp;hereafter more fully treated of.

The Poflibility of a Vacuum appears from the bare Examination of Ideas. For whatever wenbsp;conceive to be poffible may exift.

^ nbsp;nbsp;nbsp;B 3nbsp;nbsp;nbsp;nbsp;The?

(?gt; nbsp;nbsp;nbsp;Mathematical Elements Book I,

The Queftion therefore amounts to this, vlz� Whether we have an Idea of an Extenfion that isnbsp;not folid ?

We acquire an Idea of Solidity by the Touch ; We feci that fome Bodies refill us; and indeednbsp;thofe Bodies refill us every Moment, that hindernbsp;us from defcending to the Icwell Places: Fromnbsp;which Refinance it appears, that a Body excludesnbsp;every ocher Body from the Place which itfelfnbsp;takes up-, that is, it appears that a Body is folid jnbsp;which Idea of Solidity we transfer to thofe morenbsp;fubtile Bodies, which, by reafon of the Smallnelsnbsp;of their Parts, efcape our Senfes; and we find bynbsp;Experience, that even thole refill other Bodies, asnbsp;well as the hardell.

14 Experiment7^ The x'^ir, in which we live, does almoll always efcape our Sight and Touch j yetnbsp;in a Syringe, that is clofe Ihut at the End, it refills the Pifton, fo that it can be pulh�d to thenbsp;Bottom of the Syringe by no Force.

The Idea of Solidity is not indeed contained in the Idea of Extenfion ; that only follows fromnbsp;Contadl, but this may be had without it j for ifnbsp;a Man had never touch�d a Body, he would havenbsp;no Notion of Solidity.

Let any one obferve an Image projefted in the Air, or reprefented between a Concave Mirrornbsp;and the Objcdlj fuch an Image does not refill,nbsp;and yet it feems to be a Body as denfe as the Ob-ic6l itfelf j for the Colours may appear more vividnbsp;in the Image than in the Objedl itfelf: If a Mannbsp;had never fecn any thing elfe but fuch Images, andnbsp;his own Body was like fuch an Image, could henbsp;have any Idea of Solidity ? It does not appear thatnbsp;he could i and yet he would certainly have an Ideanbsp;of Extenfion.

As we are here difputing of Ideas, we fhall not confider what the above-mentioned Image is �, itnbsp;is enough that there is fuch a Thing.

All the Difference between Space and Body does jj. not confilf in a Privation of Solidity.

That Space is infinite, and can be contained by no Limits, is plain to any one that attentivelynbsp;confiders it.

We plainly fee that Space has Parts, but they cannot be feparated from one another, being immoveable as Space itfelf.

The Idea of Space is very fimplc; that of Body is more complex, it may be moved, its Parts arenbsp;feparated, and what is finite is eafily conceived.

Solidity is by fome call�d Impenetrahility., and i6 they endeavour to deduce it from the Nature ofnbsp;Extenfion: For Example, one cannot add onenbsp;cubic Foot of Extenfion to another cubic Foot ofnbsp;Extenfion, without having two cubic Feet; fornbsp;each of them has all that is required to conftitutenbsp;that Magnitude j therefore one l-�art of Space excludes all others, and cannot admit them.

Anfwer. This is all true, becaufe the Parts of Space are immoveable; but it would be falfe, ifnbsp;it was not that it would imply a Contradidtion,nbsp;to fuppofe one Part of Space conveyed to anothernbsp;Place; And the Confequence follows only fromnbsp;the Immobility, not from the Impenetrability ornbsp;Solidity of the Parts of Space.

CHAP. IV.

Oj the DivjJihility of Body in infinitum ^ and of the Suhtilty of the Barticles of Matter.

The Extenfion of a Body implies its Divi- ij fibility 3 that is, one may confider Parts

8 nbsp;nbsp;nbsp;Mathematical Elements Book I.

Yet the Divifibility of Body differs from the Dtvifibility of Extenfionj for its Parts may benbsp;feparated from one another. Bur as this Property depends upon Extenfion, it muft be examinednbsp;under the Confideracion of Extenfion. And thennbsp;we may cafily transfer to Body what is demon-llrated.

not conceive any Part of its Extenfion ever fo final), but Hill there may be a fmaller.

Let there be a Line AD perpendicular to BE, {Plate I. Fig. l.) and another as GH at a fmallnbsp;IDirtance from A, alfo perpendicular to the famenbsp;Line, with the Centers C, C, C, and Di-Hanccs CA, C A, ��r. deferibe Circles cuttingnbsp;the Line GH in the Points lt;?, e, The greaternbsp;the Radius AC is, the lefs is the ParteG} thenbsp;Radius may be augmented in infinitum., and therefore the Part eG may be diminifhed in the famenbsp;Manner} and yet it can never be reduced to Nothing } becaufe the Circle can never coincide withnbsp;the Right Line BF.

Therefore the Parts of any Magnitude may be diminifh�d in infinitutn, and there is no End ofnbsp;fuch a Divifion.

The chief Obje�tions are, � That an Infinite cannot be contained by a Finite} � That it follows from a Divifibility in infinitum, that all Bodies are equal, or, that one Infinite is greater thannbsp;another.

But thefe are eafily anfwer�d} for to an Infinite may be attributed the Properties of a finite andnbsp;determined Quantity. Who has ever provednbsp;that there could not be an infinite Number ofnbsp;Parts infinitely fmall in a finite Quantity } or thatnbsp;all Infinites are equal? The contrary is demon-

llrated

Book I. of Natural Thilofofhy. nbsp;nbsp;nbsp;lt;

There are alfo other Objeftions propofed, fup-pofing that we affirm an aftual Divilion of a Body into an infinite Number of Parts feparated fromnbsp;one another. Wc neither defend nor conceivenbsp;fuch a Divifion: We have demonftrated, thatnbsp;however fmall a Body is, it may Itill be farther divided} and, upon that Account, we believe thatnbsp;we may call that a Divifion in infinitum^ becaufenbsp;what has no Limits is call�d infinite.

There are no fuch Things as Parts infinitely 20 fmall} but yet the Subtilty of the Particles of fe-veral Bodies is fuch, that they very much fui pafsnbsp;our Conception 5 and there are innumerable In-ftances in Nature of fuch Parts that are adluallynbsp;feparated from one another.

He fpeaks of a filken Thread 300 Yards long, 21 that weighed but two Grains and a Half.

He meafured Leaf-Gold, and found by weigh- 22 ing it, that fo fquare Inches weighed but onenbsp;Grain: If the Length of an Inch be divided intonbsp;200 Parts, the Eye may diftinguifii them all}nbsp;therefore there are in one fquare inch 40000 vi-fible Parts} and in one Grain of Gold there are twonbsp;Millions of fuch Parts} which vifible Parts nonbsp;one will deny to be farther divifible.

A whole Ounce of Silver may be gilt with eight Grains of Gold, which is afterwards drawnnbsp;into a Wire thirteen thoufand Foot long.

In odoriferous Bodies we can ftill perceive a greater Subtilty of Parts, and which are feparated from one another} feveral Bodies fcarce lofenbsp;any fenfible Part of their Weight in a long Time,nbsp;and yet continually fill a very large Space withnbsp;odoriferous Particles: Whoever will be at thenbsp;Pains to make Calculations concerning thofe fub-

Mathematical Elements Book I,

By Help of Microfcopes, fuch Objedls as would otherwife efcape our Sight, appear very large;nbsp;There arc fome fmall Animals fcarce vifible withnbsp;the bell Microfcopes; and yet thefe have all thenbsp;Parts neceflary for Life, Blood, and other Liquors: How wonderful mull the Subtilty ofnbsp;thofe Particles be, which make up fuch Fluids!

We cannot end this Chapter more aptly than by the following Eheorem, which is eafily deducednbsp;from what has been laid of the Subtilty of Matter.

2.(5 Any Particle of Matter, how fmall foever, and any finite Space, how large foever, being given; it is polfible that that fmall Sand, ornbsp;Particle of Matter, Ihall be diffufed thro� all thatnbsp;great Space, and fhall fill it in fuch Manner thatnbsp;there fhall be no Pore in it, whofe Diameter fhallnbsp;exceed a given Line.

Concerning the Cohejion of Earts'^ where we fhall treat of Hardnefs, Softnefs^ Fluidityynbsp;and Elafticity.

� A LL Bodies, that are perceived by our Sen-' jFX c-mfill of very fmall Parts, no one of which is indi'ifible initfelf; but all of them arenbsp;in refpecl to us; For all the Divifion we cannbsp;make, is only a Separation of Parts.

W hen a great l''orce is required to make fuch a Divifion, a Body is laid to be Hard.

If the Parts yield more eafily and fall in by being prefs�d, fuch a Body is faid to be Soft.

But

Book I. of Natural Thilofo^}^. nbsp;nbsp;nbsp;ii

But this great and lefler Force, in the common Signification, determine nothing 5 for a Body thatnbsp;is Hard, in refpe�l to one Man, feems Soft to another.

A Body is faid to be Hard, in a Philofophical Senle, 'when its Parts mutually cohere, and do not at z8nbsp;all yield in'wards,fo as not to be fubje� to any Motionnbsp;in refpe�t to each other, 'without breaking the Body.

A Body is faid to be in a Philofophical Senie, when its Parts yield inwards, and Jlh in upon one 2.pnbsp;another, even tho' it may reqiiire a Blow with anbsp;Hammer to do it.

A Body whofe Parts yield to any Imprejjion, and 3� hy yielding are eajily moved, in reJpeSt to each other,nbsp;is call'd a Fluid.

All thefe Things depend upon the Cohefion of Parts ; the clofer a Body is, the nearer it approaches to perfedl Hardnefs.

But the Hardnefs of the fmalleft Parts does not differ from their Solidity j it is an effential Property of a Body, which is no more to be explained,nbsp;than why a Body is extended, or a Mind thinks.

I do not know whether all Bodies confift of Parts that are equal and alike; And there are alfonbsp;feveralTkings very difficult, in Relation to the Caufenbsp;of the Cohefion of the fmall Parts of Bodies.

It is a particular Law of Cohefion, that all the 3 ^ Parts have an attraUive Force.

iz nbsp;nbsp;nbsp;Mathematical Elements Book I,

By the Word Attraction I underftand, any Force hy �which fwo Bodies tend to�ward each other j tho�nbsp;perhaps it may happen by Impulfe.

But that Attraction is fubjeft to thefe Lawsj Fhat it is very great, in the very ContaCt of thenbsp;Parts-, and that it fuddenly decreafes, infomuchnbsp;that it acts no more at the leaf fenjible Dijiance nay,nbsp;at a greater Diftance, it is changed into a repellentnbsp;Force, by �which the Particles fly from each other.

By Help of this Law, feveral Phenomena are very eafily explained j and that Attraction and Re-pulfion is fully proved by a vaft Number of chy-raical Experiments. That there is fuch a Thing,nbsp;appears alfo from the following experiments.

Experiment i.] We fee that in all Liquors all the Parts attra�t one another, from the Sphericalnbsp;Figure that the Drops always have ; and alfo be-caufe there is no Liquor whofe Parts are not flicking to one another, which is evidently true evennbsp;in Mercury itfelf.

^5 Experiment!.'^ But this mutual Attraftion of Particles is much better proved; becaufe in allnbsp;Liquids, two Drops, as A and B {Plate I. Fig. z.)nbsp;as foon as they touch one another ever fo little,nbsp;they immediately run into one larger Drop, asnbsp;F. All which Things, as they alfo happen innbsp;liquified Metals, it follows, that the Parts of whichnbsp;they are compounded do attraft one another, evennbsp;when they are disjoined by the Motion of thenbsp;Fire.

Thefe Appearances do not depend upon the Pref-fure of the Air, becaufe they alfo happen where there is no Air; neither do they depend upon thenbsp;Preflureof any other Matter equally from all Sides;nbsp;for thoiigh fuch a Prefiure is able to keep the

Drops

Book I. of Natural Th�lofophy. nbsp;nbsp;nbsp;13

In the Oval Drop acbcl, {Platei. Fi^. 3.) the ?4 Prefllires upon the Surfaces ad and cb are lefs thannbsp;the Prefllires upon the Surfaces ac, db-, for thenbsp;Drop is fuppos�d to be prefs�d equally from allnbsp;Parts, therefore the Pre/Ture is left in a left Space:

Yet the Drop can never become round, till thofe lefl�r Prellures overcome the greater, which is ab-furd.

On the contrary, in Attraftion, the greater the Number is of the Particles which attrad: one another between two Particles, the greater is the Forcenbsp;with which they are carried towards one another jnbsp;which produces a Motion in the Drop, till theDi-ftances between the oppofite Points in the Surfacenbsp;become every where equal j which can only happen in a Spherical Figure.

Several Bodies aft upon other extraneous Bodies by this Attraftion. I fhall give a few Examples,nbsp;in which the Effefts of it are mofl: remarkable.

Experiment 3.] immerge in Water the Ends of 3f fmall Glafs Tubes open at both Ends, in the Manner reprefented in Plate 1. Fig. 4. The Waternbsp;will fpontaneoufly afeend in them, and fo muchnbsp;the higher as the Diameter is lefs. It is not required that the Tubes be extremely fmall j for thenbsp;Experiments will fucceed in Tubes whofe Bore isnbsp;the lixth Part of an Inch. That this is not to benbsp;attributed to thePreffure of the Air, appears fromnbsp;the following Experiment.

Experiment 4 ] Having fixed the fmall Tubes A 35 to a Piece of Cork, and llifpended them with thenbsp;Brafs Wire AB, (Plate I. Fig. f.) pump out the Airnbsp;from the Recipient R, which rtands upon the Brafsnbsp;Plate of the Air-Pump i then by moving the Wire

14 nbsp;nbsp;nbsp;Mathematical Elements Book I.

AB, the Tubes may be immerfed in the Water which is contained in theGlafs CD, and the Water in this Cafe will rile up into the Tube juft asnbsp;ir did in the foregoing Experiment. How thenbsp;Wire may be moved, witho�t letting the Air intonbsp;the Recipient, will be explained hereafter.

97 Experiment f. ] A B C D are two Glals Plates, or Planes, {Plate I. Fig-6.) touching one another atnbsp;AB, but a little feparated at CD, by thruftingnbsp;a thin Plate of any Kind between them j they arenbsp;fiiftained by the wooden Frame HILM, in fuchnbsp;Manner, that the Side DC is always at the fimenbsp;Height j the Planes may be brought to make anynbsp;Angle with the Horizon, by railing the End A Bnbsp;where they are joined, the Cylinder N O likewilenbsp;fuftaining the Plane in any Pofition. The Screwnbsp;P makes fall the Cylinder at any Height.

A Drop of Water or Oil, G, is put between the Planes, lb as to touch both the Planes, whichnbsp;muft before-hand be made wet with the fime Liquor 5 this Drop is attrafted by both Planes, butnbsp;the Attraftion has a greater Effe6t upon the Drop,nbsp;where their Diftance is the lefs: that is, a greaternbsp;ate than at/, therefore the Drop is moved towardsnbsp;e-y that is, afeends, and moves upwards the fafter,nbsp;in Proportion as it is higher, theSurfices in whichnbsp;the Drop touches theGlalTes growing very much,nbsp;where the Diftances between the Planes isdimuiilh-ed. The Angle of Inclination of the Planes maynbsp;be fo increafed, that the Gravity of the Drop fhallnbsp;balance the Attrabfit'n, and then the Drop is atnbsp;reft} and in that Cafe, if you raife the End ABnbsp;of the Planes ftil! higher, the Drop will defeendnbsp;by its Gravity, which will then overcome the At-traftion.

Book I. of Natural Thilofophy.

Experiment 6. Let two Glafs Planes, ABCD, 38 {Platei. Pig-7-) touch one another at A B, andnbsp;at C D let them be a little feparated by the Inter-pofition of fome thin Plate, and then let theirnbsp;Ends be immers�d into Water tinged with Tome Colour, in fuch Manner, that the Sides A B and C Dnbsp;may be in a vertical Pofition, the Planes having beennbsp;moiftened with the Lme Liquor beforehand. Thenbsp;Water will rife between thofe Planes by their At-traftion, and rife higheft where the Planes arenbsp;nearell together j and as their Diftance continuallynbsp;decreafes from CD to AB, the Water rifes up tonbsp;different Heights in every Place, and makes thenbsp;Curve Line e f g.

Experiment 7.] Quickfilver unites itfelf to Tin 3P and Goldj alfo Water and Oil ftick to Wood andnbsp;clean Glafs.

We have Inftances of Repulfion between Water 40 and Oil, and generally between Water and all unctuous Bodiesj between Mrrrary and Iron j as alfo between the Particles of any Duft.

Experiment %If any greafed Body, lighter 4^ than Water, is laid upon theSurfiice of Water, ornbsp;Piece of Iron upon Mercurythe Surface of thenbsp;Fluid will be deprefled about the Bodies laid upon^nbsp;it, as it appears about the Ball A {Platei. Fig.S.)nbsp;And after the fifne Manner, where the Attractionnbsp;obtains, the Surface of the Liquor is higher aboutnbsp;the floating Bodies, as about the Ball B, and doesnbsp;not run to a Level by its Gravity; fo here wherenbsp;the repellent Force exerts itsACtion, Liquors, not-withflanding their Gravity, do not run down tonbsp;fill up the Cavities which are made round thenbsp;floating Bodies.

Upon this depend all the Pha:nomena of very 41 light Glafs Bubbles {Platei. Fig.nbsp;nbsp;nbsp;nbsp;which fwim

16 nbsp;nbsp;nbsp;Mathematical Elements BookL

upon Water i when they are clean, the Water rifes about them all roundj as at B j but, when theynbsp;are made greafy, the Water makes an Hollow allnbsp;round them, as at A j in the Glafs Veflel wherenbsp;the Experiments are made, the Water alfo Handsnbsp;higher all round next to the Glafs, as at C and D jnbsp;but when the Glafs is fo fill�d that the Water runsnbsp;down from all Parts, then, by the mutual A ttra�li-on of the Parts, of the Water, it Hands higher innbsp;the Middle than at the Sides, and forms the convex Surface ABC: {Plate I. Fig.9-) From thelenbsp;Principles only can the following Experiments benbsp;explain�d.

Experiment p, i o, 11, i z and 15. ] When a Glafs is not quite full of Water, a clean Glafs Bubblenbsp;always runs to the Side, and there flicks, provided it be not laid on coo far from it. The Bubblenbsp;is prels�d every Way by the Water, when it comesnbsp;to the Side of the Veflel} the fame Force, by whichnbsp;the Water is raifed there, does in part take offnbsp;that Preflure} fo the Preflure on the other Sidenbsp;overcomes, and the Bubble is moved towards thenbsp;Side of the Glafs.

When the Glafs is fo full as to be ready to run over, the Bubble goes off it felf from the Side tonbsp;the Middle of the Glafs, for the fame Reafon;nbsp;becaufe the Force, by which the Water is raifed innbsp;the Middle, does alfo diminifh the PrelTure uponnbsp;the Bubble towards the Middle.

Jufl the Reverfe happens when the Bubble is greafy, becaufe that Force, by which the Waternbsp;and the Bubble repel one another, is greateft wherenbsp;the Water is higheft.

Two clean Bubbles, or two greafy ones, run towards each other. As to clean Bubbles, we have jufl given the Reafon } when they are madenbsp;greafy, there is a Cavity round each of them }nbsp;Inbsp;nbsp;nbsp;nbsp;and

Book I. of Natural'Phtlofoj^hy, nbsp;nbsp;nbsp;i;

and where the Cavities join, the Preffure is dirai-nifh�d, and the Bubbles run that Way.

If one Bubble be clean, and the other greafy, they fly from one another, for the Reafons beforenbsp;given.

The Particles of any Salts attrafl: one another 43 with a very great Force, as appears by feveralnbsp;Experiments: The following will be fufficient tonbsp;prove, that that Attraftion exerts it felf at a verynbsp;fmall Dillance, and the repellent Force at anbsp;greater.

Experiment 14.3 Diflblve Salt in Water, and, when that Water is reduced into Vapour, thenbsp;fmall faline Particles will unite together andnbsp;form greater Lumps; which proves the Attra-ftion.

Thefe Particles are all equal, and of the flime Figure: Whence it follows, that the Icaft Parts,nbsp;of which they are form�d, had every where thenbsp;fame Situation in refpeft to each other j that is,nbsp;were every where diffufed in the Water at equalnbsp;Diilances; which cannot be, unlefs they all repelnbsp;one another with equal Forces.

�The Elaftkity of Bodies^ namely, that Property ^ whereby they return to their former Figure^ when itnbsp;has been alter'd by ^y Force^ is eafily deduced fromnbsp;what has been faid: For if a compaft Body benbsp;dented in without the Parts falling into that Dent,nbsp;the Body will return to its former Figure, fromnbsp;the mutual Attraftion of its Parts.

We fliall alfo in its proper Place (hew, that4f that Property of the Air, which is call�d its Ela-fticity^ arifes from the Force whereby its Parts repel one another.

And left any one fliould imagine, becaufe we don�t give the Caufe of the fard Attraflion and

t8 nbsp;nbsp;nbsp;Mathematical Elements Book �*

Repulfion, that they muft be look�d upon as occult ^alities .'We fay here with Sir Ifaac Newton^ � That we confider thofe Principles not as

* nbsp;nbsp;nbsp;occult Qualities, which are fuppofed to arifenbsp;� from the fpecifick Forms of Things j but as u-

* nbsp;nbsp;nbsp;niverfal Laws of Nature, by which the Thingsnbsp;� themfelves are form�d: For the Phsenomena ofnbsp;� Nature fliew that there are really fuch Principles,nbsp;� tho� it has not been yet explain�d what theirnbsp;� Caufes are. To affirm that the fcveral Species ofnbsp;� Things have occult fpecifick Qualities, by whichnbsp;� they ad: vvith a certain Force, is juft faying no^nbsp;� thing. But from two or three Phtenomena ofnbsp;� Nature to deduce general Principles of Motion,nbsp;� and then explain in what Manner the Propertiesnbsp;� and Adions of all things follow from thofenbsp;� Principles, would be a great Progrefs made in

part ii.

The Subjeft we are now entring upon has a large Scope in Phyficks; AU that happensnbsp;in natural Bodies belongs to Motion, and evennbsp;what has been faid of the Cohefion of Parts, hasnbsp;a Relation to it: For though the Parts are notnbsp;moved in their Cohellon itfelf, yet that Cohefionnbsp;Cannot be explain�d, nor can what is faid abciuc itnbsp;be confirm�d by Experiments, without Motion.

Motion is a 'Tranjlation from one Place to am-ther Place^ or a continual Change of Place. Every Body has an Idea of it j and Philofophers have innbsp;Vain laboured to find a Definition of the fimplenbsp;Idea, and proved with a great deal of Pains, thatnbsp;�ne may come to be ignorant of a Thing, Whichnbsp;t^therwife every Body knows.

Place is the Space taken up hy a Body; of which ^ay be find, what has juft been faid concerningnbsp;^lotion.

Mathematical Elements Book I.

yo Relative Place, which only can be diftinguifh�d by our Scnfes, h the Situation of a Body in refpeSinbsp;of other Bodies.

Whence arifes a and Abfolute Motion, and another Sort call�d a Relative Motion.

'True Time has no Relation to the Motion of Bodies, nor to the Succeffion of Ideas in an Intelligent Being, but by its Nature it always flowsnbsp;equally.

yz Relative Time is Part of the true Time mea-fured by the Motion of Bodies �, and this is the only Time that our Senfes perceive.

All Motion may become fwifter, as likewife a Body may move flower than it did before; andnbsp;it is very likely that there is no Motion of Bodiesnbsp;wholly equable; whence it follows, that relativenbsp;Time differs from true Time, which never flowsnbsp;fafter nor flower.

yj That AffeUion of Motion.^ by 'which a Body runs thro' a certain Space in a certain Time.^ is call'dnbsp;Celerity or Velocity; which is greater or lefs, according to Pat Bignefs of thsiX. Spaceto 'which itnbsp;is always proportionable.

yA The greater Force is imprefs�d upon a Body make it change its Place, the greater is its

The Direftion of Motion is in a Right Line., �which we fuppofe drawn towards the Place where thenbsp;moving Body tends.

CHAP. VII.

The Quantity, or Momentum of Motion, o follows the Proportion of the Caufe producingnbsp;the Motion.

Floe whole Effects of Motions, produced at the fame Fime, have the fame Relation to each other, as ^nbsp;the Momenta of thofe Motions.

If two equal Motions a�l in a contrary DireBion, (Sq ^hey dejlroy each other 5 and the one can never overcome the other.

. Bodies in Motion may differ in two Refpefts, Either in Refpeft to the Quantity of Matter innbsp;^^ch, or in Refpcft of the Space gone thro� innbsp;��ys fame Time, that is, in Refpcft of the Velo*

2 2 nbsp;nbsp;nbsp;MatbematicM Elements Book!.

j two 1'hings therefore, and only thefe two, are to be conlidered in comparing of Motions.

When the Velocities are equal, nothing is to be confidered but the Qtiantities of Matter j and ifnbsp;it be double in one Body, the Quantity of Motion in that Body will alio be double 3 becaufe fuchnbsp;a Body is made up of two Bodies, each equal tonbsp;the lead Body, and moved with the, fame Celeritynbsp;as the little one. The fame may be faid of all o-ther Relations between two Bodies 3 whencewede-duce this general Rule:

6t In equal Bodies that move with the fame Velocity, the ^.antity of Motion is as the ^entity of Matternbsp;in each.

When the Quantities of Matter are equal, the Velocities only are to be confidered: Andnbsp;then

63 In equal Bodies, the Momenta are as the Velo~ cities-. That is, as the Spaces gone thro� in thenbsp;�53. fame Time*. For the going thro� thofc Spacesnbsp;are the whole Effedts of the Motions produced innbsp;that Time, and are to one another as thofe Spaces 3nbsp;therefore the Momenta alfo arc in the fame Pro-portion.*

In order to determine the Relation bet ween two Motions, when the Velocities are unequal, and thenbsp;Bodies different in ^cantity of Matter-, you muftnbsp;find two ^lantities that are to one another as thenbsp;Maffes and as the Velocities. Multiplying the Velocities of each Body hy its Mafs or ^antity of Matter, the ProduSls will be to each other in the faidnbsp;Proportion.

When for Example, the Velocity is double, and the Mafs triple, a double (^rantity of Motion muft be tripled 3 therefore it will be fextu-ple: This is the Cafe when in one Body the Velocity is two, and four in another 3 and thenbsp;Mafs of the firfi; Five, the Mafs of the other being

Book I. of Natural Thilofophy. nbsp;nbsp;nbsp;45

ing Fifteen; multiplying each Mafs by its Velocity, the Produ�rs are 10 and 60, the laft of which is fix Times the firft.

This is called a Ratio, compounded of the Ratio of the Mafles and the Celerities.

A greater Body may move more llowly than a lefs, in fuch a Manner, that the Icfier Body maynbsp;have an equal Quantity of Motion with, or anbsp;greater than the other.

When the Velocity in the lejfer Body is to the Vdo-city in the greater^ as the Mafs of the greater to the Mafs of the lejfer y the ^antities of Motion arenbsp;equal in the tvco Bodies.

As much as the Quantity of Motion in the lefler Body is lefs, in Refpeft of its Mafs, fo muchnbsp;is it greater, in Refpeft of the Velocity: Whencenbsp;an Equality arifo. Likewife in that Cafe, thenbsp;Produfts of the Mafs of each Body by its Velocity are equal; and the Celerities are faid to be innbsp;an inverfe Ratio of the Mafies, or reciprocally asnbsp;the Mafles.

When fuch Momenta of Motion aEl contrarivcife-^ 66 they dejlroy each other. *nbsp;nbsp;nbsp;nbsp;*6q.

CHAP. VIII.

TH E Aftions of Powers, afting upon Bodies, may be compared together, in the famenbsp;Manner as the Quantities of Motion; and thenbsp;fame Rules ferve for both.

We Ihall hereafter fhew, that a Body once in Motion will continue in that Motion, tho� thenbsp;Caufe that firft gave it, ce.afeth: So that if a Bodynbsp;^ould be continually afted upon by any fower,nbsp;the Motion would become fwifte'r every Moment.'

Mathematical Elements Book I.

We do not here confider fuch an Aftion of Powers i but we take notice only of Powers thatnbsp;aft againft an Obftacle, in fuch Manner, that bynbsp;the Refiftance of the Obftacle the Aftion of thenbsp;Power is continually deftroyed, which is to benbsp;obferved gt; for in another Cafe, the following De-monftrations do not obtain. When therefore wenbsp;fpeak here of an Obftacle to be removed by anynbsp;Power, we fpeak of the greatefi Obj�ade that cannbsp;be moved by that Power j for otherwife the Obftacle would not deftroy the whole Aftion of thenbsp;Power.

The Aftions of Powers may differ from one another, both in refpeft of the Greatnefs of thenbsp;Obftacles, and in Refpeft of the Spaces run thro�nbsp;by the Obftacles; (that is, by the Points tonbsp;which the Powers are applied:) Thefe twonbsp;Things alone are to be confider�d in comparing ofnbsp;Powers.

The Obftacles, which may be removed by Powers, are to one another as the intenfities ofnbsp;*57-the Powers.*

as the Spaces run thro'. For they only differ in that Refpeft, becaufe the Obftacles are equal.

lt;58 IVhen the Spaces run thro' are equal.^ thofe Actions are as the Intenfities.

6p lEhen both the Spaces run thro'., and the Inten~ ^ fities, are different, the ASlions of the Powers are tonbsp;one another, in a Ratio compounded of the Intenfitiesnbsp;and the Spaces gone through.

70 When the Spaces gone thro� are in an inverfe Ratio of the Intenfities, the A�lions are equal.

Book I. of Natural Thilofophy.

Thefe Things may be demonftrated in the fame Manner, as what has been faid in the foregoingnbsp;Chapter.

General things concerning Gravity.

L L Bodies mar the Earthy if hinder'd by no 71 Obfackj are carried towards the Earth.

I�he Forceby which Bodies are carried towards 7 5 the Earth.^ is called Gravity.

Phat Force^ in RefpeSl to a Body aSled upon by it^ 74 is call'd the Weight of a Body.

'itloe Force of Gravity abls equally., and every Mo-1 f ment of Fime, near the Earth's Surface.

There is indeed a fmall Difference of Gravity in different Countries, which we fhall take Notice of hereafter, but it is too fmall to be confider-ed here, efpecially becaufe it is wholly infenfiblenbsp;in neighbouring Countries.

When the Defcent of a Body is hinder�d by an 76 Obllacle, it continually preffes that Obilacle e-qually, tending towards the Earth�s Center ; therefore it may be look�d upon as a Power adting upon an Obftaclc} and therefore what has in thenbsp;foregoing Chapter been demonftrated, concerningnbsp;Powers, does obtain here alfo.

Mathematical Elements Book L

77 Bodies �which defcend by the Force of Gravity^ (if all Refiftance be taken off) fall with the fame V?/o-city. Which is prov�d by an

Eyperiment.'] Pump out the Air from the tall Recipientnbsp;nbsp;nbsp;nbsp;Fig. i.) which is made

up of two Glaflhs, and is about three Foot high: Then from the Top of the Glafs within, by moving the Handle C D, let fall a Piece of Gold andnbsp;a very light Feather juft at the fame Time, andnbsp;they will always come down to the Brafs Plate ofnbsp;the Pump upon which the Receiver ftands, at thenbsp;Erne Inftant of Time.

-g For making the Experiment readily, the Top of the Recipient is cover�d with a Brafs Platenbsp;laid upon it. A thin Plate, bent into the Figurenbsp;E, is fixed to the covering Plate at e, by Help ofnbsp;two Screws H, that go thro� two lefler Plates,nbsp;one of which you fee gf and are joined to thenbsp;other Plate E.

The Ends of the Plate fpring together, and fo hold the Feather and the Gold, whilft the Receiver is exhaufting.

A Brafs Wire runs thro� the Cover, which, by Means of the Handle C D, may be turned roun^nbsp;without admitting the external Air j which wenbsp;{hall explain, when we come to fpeak of the Air-Pump.

The Wire goes thro� an Hole in the upper Part of the Plate e, and the End of the Wire, whichnbsp;comes down between the fpringing Plate, may benbsp;feen at L: It is fquare and hollow, that the Ovalnbsp;Plate I may be joined to it.

You muft obferve, that the fmall Diameter of the Oval be fmall enough for this little Plate to benbsp;contained between the Springs E, when their Endsnbsp;come together.

I nbsp;nbsp;nbsp;Now

Book I. �f Natural Tbilofofhy. nbsp;nbsp;nbsp;tj

Now when the Brafs Wire, and by it the Plate I i is turned round j by Reafon of the Differencenbsp;of Diameters in the Oval, the Ends of thenbsp;Springs open j and then the Bodies, that are fuf-pended, are let go at the fame Moment ofnbsp;Time.

The fame Phsenomenon is alfo deduced from another Experiment, which we fhall mentionnbsp;hereafter. *nbsp;nbsp;nbsp;nbsp;* i6o

Hence it follows, that Gravity in all Bodies^ 79 that is, their Weight, is proportionable to theirnbsp;^antity of Matter. *nbsp;nbsp;nbsp;nbsp;*62

Therefore all Bodies confift of Matter that is 80 equally heavy; and the Reafon that Bodies do notnbsp;appear equally heavy, is becaufe fome have moi'enbsp;Matter than others under the fame Bulk j that is,nbsp;in the fame Space.

When Weight is confidered as a Power, the 81 Intenfity of the Power is proportional to thenbsp;Quantity of Matter in the heavy Body) and thenbsp;Direflrion of the Power is towards the Center ofnbsp;the Earth.

CHAP. X.

Of the fngle Pulley , Balance, and cf the

Center of Gravity.

A Single Pulley is a little Wlosel moveable about its jJ.xis^ over which goes a drawing or running Rope^ dee {Plate I. Fig. 10.)

By this Engine, the Direction of the Power is changed, neither is it of any otherUfewhen fix�d 5nbsp;for in that Cafe, if the Force of Power apply'd to the 85nbsp;drawing Rope., as M, be equal to thcQbfacle P, it balances that Ohjlacle* ifor in that Cafe the Power can't *7�

Mathematical Elements Book I.

Weights are found, that is, the Quantities of *79 Matter in Bodies are compared together *, by anbsp;Balance, or a Pair of Scales, which is a well knownnbsp;Inftrument.

84 The Axis of a Balance is''that Line about 'which the Balance moves., or rather turns round.

8^ When we conGder the Length of the Brachia, ^ or of the Beam, then the Axis is to be lookednbsp;upon as a Point, and called the Center of the Balance.

We call Points of Sufpenfion, or Application, thofe Points where the Weights really are, or fromnbsp;which they hang freely j or the Scales in which thenbsp;Weights are placed.

Concerning this Machine, we are to obferve, 8^ float the Weight does equally prefs the Point ofnbsp;Sufpenfion, at whatever Height it hangs from it, andnbsp;in the fame Manner as if it was fixed at that verynbsp;Point.

For the Weight, at all Heights, equally ftretches *74(75 the Rope by which it hangs.* This is alfo provednbsp;by

Experiment i. ] In the Balance A B, the Weight P, by Means of the Rope BD {Plate'W. Fig. 2.3nbsp;is fufpended at different Heights: And the Pofi-tion of the Balance is not changed by it.

B�okl. of Natural Thilofiphy.

J�h� Atlion of a Weight to move a Balance h hy %% fo much greater^ as the Point prefed by the Weightnbsp;is more dif ant from the Center of the Balance-y andnbsp;that Jblion follows the Proportion of the Difiance ofnbsp;the [aid Point from that Center.

When the Balance moves about its Center, the Point B defcribes the Arc B^ {Plate II. Fig. y )nbsp;whilft the Point A defcribes the Arc A lt;3, whichnbsp;is the biggeft of the tv/o gt; therefore, in that Motion of the Balance, the A61:ion of the fame Weightnbsp;is different, according to the Point to which it isnbsp;applied, and it follows the Proportion of thenbsp;Space* gone through by that Point: At A there- *67,76nbsp;fore it is as A�, and at B as B^j but thofe Arcsnbsp;are to one another, as C A, to C B.

Experiment i.] The Brachia of the Balance A B {Plate W. Fig. ey.) are divided into equal Parts gt;nbsp;and one Ounce applied to the ninth Divifion fromnbsp;the Center, is as powerful as three Ounces at thenbsp;third i and two Ounces at the fixth Divifion actnbsp;as ftrongly as three at the fourth, ifc.

The Conftrudtion of a Balance, for this and foms other following Experiments, is plainnbsp;enough from the Figure, adding to it what isnbsp;faid at Numb. loz. Hence it follows, that thenbsp;Adtion of a Power to move a Balance is in anbsp;Ratio compounded of the Power it felf, and itsnbsp;Diftance from the Centerfor that Diftance is* 69nbsp;as the Space gone through in the Motion of thenbsp;Balance.

A Balance is faid to be in AEquilibrio, when the ^9 Actions of the Weights upon c^c/5'Brachium, to movenbsp;the Balance., are equal; fo that they mutually de-firoy each other y as appears by the foregoing Experiment.

Mathematical Blements Book L

When a Balance is in .^quilibrio, the Weights on each Side are (aid to (equiponderate.nbsp;po Unequal Weights can (equiponderate. For thisjnbsp;it is requifitc, that the Diftances from the Centernbsp;*89,70 be reciprocally as the Weights.* In that Cafe,, ifnbsp;each Weight be multiplied by its Diftance, thenbsp;Produ�s will be equal.

Experiment 3.] In the above-mentioned Balance {Plate II. Fig. 4.) one Ounce at the Diftance ofnbsp;the ninth Diviiion from the Center, acquiponde-rates with three Ounces at the third Divifion.nbsp;pi The Steel-yard, or Statera Romana whichnbsp;weighs every Thing with one Weight, is madenbsp;upon this Principle.

Experiments^.'] The Steel-yard AB {Plate W. Fig. f.) has two Brachia very unequalj a Scalenbsp;hangs at the fhorteft j the longeft is divided intonbsp;unequal Parts: Apply fuch a Weight to it, that,nbsp;at the firft Divifion, it ftiall tequiponderate withnbsp;one Ounce laid in the Scale j then the Body to benbsp;weighed is put into the Scale, and the above-mentioned Weight is to be moved along thenbsp;longeft Brachium, till you find the ^Equilibrium jnbsp;the Number of Divifions, betVv^een the Body andnbsp;the Center, ftiew the Number of Ounces that thenbsp;Body weighs, and the Subdivifions the Parts ofnbsp;an Ounce.

9Z Upon this Principle alfo is founded the deceitful Balance, which cheats by the Inequality of the Brachia,

Experiment ^.] Take two Scales of unequal Weights, in the Proportion of p to 10 {Plate\\\.nbsp;Fig. I.) and hang one of them at the tenth

Book I. of Natural Thilofofh^, nbsp;nbsp;nbsp;3

Divifion of the Balance above deferibed, and the other at the ninth Divifion, fo that there may benbsp;an ^Equilibrium; If then you take any Weights,nbsp;which are to one another as p to 10, and put thenbsp;firft in the firfl: Scale, and the other in the othernbsp;Scale, they will aequiponderate.

Several Wei^ts^ hanging at feveral Dijlances on 9quot;^ one Side^ may equiponderate with a fingle Weight onnbsp;the other Side. To do this, it is required, that thenbsp;Produ�l of that Weight, by itsDiftance from thenbsp;Center, be equal to the Sum of the Produfts ofnbsp;all the other Weights, each being multiplied bynbsp;its Diftance from the Center.

Experiment 6.'] Hang three Weights of an Ounce each, at the fecond, third and fifth Divi-fions from the Center, and they will mqtiiponde-rate with the Weight of one fingle Ounce applied at the tenth Divifion of the other Brachiumnbsp;{PlateW. Fig.6.) And the Weight of one Ouncenbsp;at the fixth Divifion, and another of three Ouncesnbsp;at the fourth Divifion, will tequiponderate with anbsp;Weight of two Ounces on the other Side at thenbsp;ninth Divifion.

Several M'^eights.^ unequal in Namher.^ on either Side., may equiponderate. In that Gale, if each ofnbsp;them be multiplied by its Diftance from the Center, the Sums of the Produfts on either Side willnbsp;be equal; and if thofe Sums are equal, there willnbsp;be an ^Equilibrium.

Experiment 7.] Hang on a Weight of two Ounces (Plate II. Fig. 7.) at the fifth Divifion,nbsp;and two others, each of one Ounce, at the fecondnbsp;and feventhj and on the other Side hang twonbsp;Weights, each alfo of one Ounce, at the ninthnbsp;and tenth Divifions j and thefe two will sequipon-dcrate with thofe three,

Mathematical Elements Book I.

The Center of Gravity is a Point in a Body^ about ivhich all the Parts of the Body {tiohatevernbsp;Pofition it is in) are in aequilibrio.nbsp;p8 When two or more Bodies are joined, whethernbsp;they are contiguous or fcparated, they have a common Center of Gravity.

Experiment The Body A III. Pig.i.) is fuftaincd and at reft, becaufe its Center of Gravity is fuftained by the Prop F.

When the Center of Gravity is not fuftained, the Body moves till that Center comes to be fuftained.

Experiment 9.] The Body A, laid upon the Table, will fall, and the Body B will not remainnbsp;in its Pofition, becaufc their Centers of Gravitynbsp;are not fuftained.

99 Hence may be known, why fome Bodies, laid upon inclined Planes, will roll off, whilft fomenbsp;only Aide off.

Experiment 10.] The Body A Aides, becaufe its Center of Gravity is fuftained by an inclinednbsp;Planej {PlateWY. 4.) that is, the Verticalnbsp;Line, which goes through that Center, c, cutsnbsp;the inclined Plane within the Body. But the Body B rolls, becaufe the Vertical Line, thro� itsnbsp;Center of Gravity, cuts the inclined Plane without the Body.

too From what has been faid it follow�s alfo, that a Body defeends, when its Center of Gravity de-feends, that is, is moved towards the Center ofnbsp;the Earth.

Book I. of Natural Thilofofhy, nbsp;nbsp;nbsp;33

Somenmes in that Cafe a Body feeins to alcend, aod oftentimes it does really afcendj but as allnbsp;Bodies defcend by Gravity, that is, their Centersnbsp;of Gravity defcend j it follows, that a Body Teemsnbsp;to afeend by its Gravity, and can really afeend.

6.) whofe Axis is made of two Cones, the Bafes of which join to the Wheel, when put betweennbsp;two Planes, whofe Sides D G, F H continued, makenbsp;the Angle FCD, which has the Bafe, (iuppofingnbsp;a Triangle made) higher than the Vertex, willnbsp;from H G, the lower Part of the Planes, roll towards FD, the higheft Part of the Planes.

By reafon of the greater Diftance between the Planes at FD,the Wheel A, whofe Axis is a Conenbsp;both Ways, defeends more between the Planes,nbsp;when it moves towards that Part, and is fo car-tied thither by its Gravity, provided that Defcentnbsp;be greater than the Afcent from the Inclination ofnbsp;the Angle FCD with the Horizon.

Experiment ii.] The Wooden Cylinder A {Plate III. Fig. f.) has within it, near the Side,

^ Leaden Cylinder j their common Center of Gravity is in a Scdlioa parallel to the Bafc,nbsp;which divides the Cylinder into two equal Parts,

Whatever Pofition this Cylinder be laid in, ir Will move until the above-mentioned Center of

If it be laid upon an inclined Plane, in the Portion deferibed in the Figure j the Center of Gra-Wy will defcend whilll the Body rifes along the I lane, if it be inclined in a fit Manner.

Mathe?natical Elements Book I.

The Body afcends by rolling towards the upper Part of the Plane, but care muft be taken, that, whilft it b endeavouring to roll up, it doesnbsp;not Hide down along the Plane j and thereforenbsp;you muft ufe a Rope, which goes in part roundnbsp;the Cylinder; one End of which is joined to thenbsp;Cylinder at/, and the other is fixed to the Planenbsp;at d.

From what has been faid of the Center of Gravity, it is farther deduced, that whatever Point of a Body or Machine fuftains the Center of Gravity of any Weight, that Point fuftaihs the wholenbsp;Weight: So that the whole Force,by which anynbsp;Body tends towards the Earth, is, as it were, collected to that Center.

Experiment 13.] If the Body AB [Plate III. Fig- 7.) whofe Center of Gravity is laid upon thenbsp;Brachiimi of a Balance, does in any Pofitionnbsp;eeqtiiponderate with any Weight P; it will innbsp;any other Pofition, as a a tequiponderatenbsp;with it, provided the Center of Gravity be ftillnbsp;at C.

loz That a Balance may be perfeft, it is required, I. That the Points of Sufpenfion of the Scalesnbsp;or Weights be exactly in the fame Line as thenbsp;Center of the Balance. 2. That they Be precifelynbsp;equidilhnt from that Point on either Side. 3. Thatnbsp;the Brachia of the Balance be as long as theynbsp;conveniently can. 4. That there be as little Fri-�lion as poffiblc in the Motion of the Beam andnbsp;Scales, f. And laftly, that the Center of Gravity of the Beam be placed a little below the Centernbsp;of Motion.

CHAP.

Book I. fff Natural Th�lofophf CHAP. XI.

Of the LEVER,

A Lever is called by Marhenaaticiaos, a?} inflexible RightUne^f made ufe of to raife Weights^ either 'weighing nothing itfelf or of fucb Weight asnbsp;may be balanced. {Plate lY. Fig.i.)

It is the firft of thofe that wc call Simple Machines (or Mechanical Powers) as being the moll firnple of all j and it ferves, when Weights are tonbsp;be raifed but to a fraall Height,

There are four other Simple Machines, which �vve lhall treat of in the three following Chapters.

2- The Power, by which it is to be raifed or fiiflained, which here is reprelented by the Weightnbsp;tho� commonly it is the Aftion of a Man.

?� Vhe. Fulcrum^ or Prop, by which the Lever is luftained, or rather on which it moves round,nbsp;quot;'bilfl; the faid Point F remains fixed.

I. Sometimes the Fulcrum is placed between ro4 t^he Weight and the Power. {Plate IV. Fig. i.)

?� And often alfo the Power a�ls between the Weight and the Fulcrum. {Plate IV. Fig. ?.)

The fame Rules ferve in all thefe Cales, which ^llow from what has been faid of the Balance*; *88nbsp;this fhews the Analogy between the Levernbsp;^quot;^d the Balance. The Lever of the firfl; Kind is,

Action of a Power and the Refijlance of the Weight increafe.^ in proportion to their Difisnee fromnbsp;D znbsp;nbsp;nbsp;nbsp;the

35 nbsp;nbsp;nbsp;z,2thematical Elenients Book I.

*amp;% the Fulcrum*-, and therefore, that a Power may be able to fuflain a IVeight^ it is required, that thenbsp;Diflance of the Point in the Lever, to �which it is applied, be to the Diflance of the PPeight, as the Weight

�90 to the Intenfity of the Po'wer* j 'which, if it be ever fo little increafed, 'will raife the Weight.

Experim. i, 2,, and 5.] This Rule is confirm�d by Experiments, in refped of the three Levers,nbsp;as it appears from the firft, fecond and third Figures of the fourth Plate; for there is an Equilibrium, when the Weights P, and the Weightsnbsp;M, which reprefent the Powers, and alfo the Di-ftanccs from the Fulcra, bear thofe Proportions tonbsp;each other, as the Numbers written in the Figuresnbsp;cxprefs. A Sight of the Figures fo plainly Ihewsnbsp;the Conftru�tion of the Machines wherewith thenbsp;Experiments are made, that a farther Explanationnbsp;would be needlefs.

Workmen alfo make ufe of a Lever to carry Weights; and there are feveral remarkable Cafesnbsp;of thofe Levers, the Demonflration of which maynbsp;be deduced from what has been faid.

106 nbsp;nbsp;nbsp;111 all Cafes this is generally to be obferved, thatnbsp;the Intenfity of the Po'wer, or the Intenfities of thenbsp;Po'wers taken all together (when there are more thannbsp;one) muft a5l as firongly as the Gravity of the Weightsnbsp;to be carried or fuflained.

107 nbsp;nbsp;nbsp;If a Weight is to be carried or fuflained bynbsp;two Powers, it muft be placed between the twonbsp;Powers, and the Dillances of the Powers on eachnbsp;Side of the Weight muft he in an inverfeRatio ofnbsp;the Intenfities of the Powers,

ExperimentThisPropofition is confirm�d by the Experiment of Fig. 4. which requires no fat'nbsp;ther Explanation.

P.npS�

Boolv I. of Natural ^htlofophy.

Experiment f.] Whefl two Weights are to be io8 fuftainedby one Power, that Power muft be placednbsp;between the Weights j and then what has beennbsp;faid before of the Powers,, mufl be applied to thenbsp;Weights. See Fig. i. Plate V.

Several Weights are often carried or fuftained by one or more Powers. In .which Cafes it is lopnbsp;to be obferved, that all IVeigbts., in whatever Po-fition^ have one common Center of Gravity, whichnbsp;Center is fuch,thatif, on either Side, each Weightnbsp;be multiplied by its Diftance from that Point,nbsp;the Sums of the Produfts on each Side will benbsp;equal.*nbsp;nbsp;nbsp;nbsp;_nbsp;nbsp;nbsp;nbsp;*55,99

Let the Powers alfo he difpofed in any Pojiiion, they have a common Center of Gravity, for theynbsp;may be reprefenced by Weights* j and here the *76nbsp;Intenlity of each Power is to be multiplied by itsnbsp;Diftance from the Center, and the Sums of the,nbsp;Produfts will then be equal on both Sides: Fhatnbsp;the Powers may be able to fuftain the Wei^ots, it isnbsp;required that t he Center of Gravity of the Powersnbsp;eand the Weights be the fame.

Experiments 6, and 7.] What has been fiid fuf-ficiently explains the Figures, (Plate V. Fig. 2, and 3.) where C denotes the Center of Gravitynbsp;common to the Weights and the Powers.

Experiment 8.] Wlnat has been faid is true, if no the Lever is drawn (Plate V. Fig. 4.) by Powersnbsp;on each Side i which we fee in the Lever of Fig.^^.nbsp;�which is drawn horizontally on each Sidej wherenbsp;the^Equilibrium only depends upon what has beennbsp;laid down in the Rules above-mentioned.

We ^may alfo make ufe of a compound Lever i ]� i for railing Weights: In which Cafe, inftead ofnbsp;^ Power, a fecond Lever applied to the firft,

38 nbsp;nbsp;nbsp;Mdthematkal Elements Book I-

and a third to ti;.it, and lo on as car as you will, and a Power is �plied t'o the kft Lever j andnbsp;then the Ratio of the Power to the Weight (whennbsp;it luftains it) is compounded of the Ratio's of thenbsp;Power to the Weights in each Lever^ when they arenbsp;ufed feparately.

E'tperiment 9.] The three Levers A, B, D, are fo difpofed (Plate IV. Fig. y.) that the Powernbsp;M fu'ftains the Weight P. In the Lever A, ifnbsp;it were ufed dngly, the Power would be to thenbsp;Weight as I to yj the Lever B, as i to 4; andnbsp;in the Lever D, as i to 6. The Ratio, compounded of all thele, is as i to no. For onenbsp;Ounce M does here fuftain the Weight P, ofnbsp;I zo Ounces. Obferve, that in the Motion ofnbsp;this Engine, the Spaces gone thro� by the Weightnbsp;and the Power are to one another, as i to 120 ;nbsp;that isj in the faid inverfe Ratio, which is requir�dnbsp;*70 to make an ^Equilibrium.*

CHAP. XII

T\ H E Lever, as was faid in the Beginning of the foregoing Chapter, ferves to raife Weightsnbsp;to a fmall Height j when they muft be raifed highernbsp;we life an ^xis in Peritrochio.

We call Axis in Peritrochio, a Wheel which turns together with its Axis. {Plate V. Fig. y.)

The Power in this Machine is applied to the Circumference of the Wheel, by whole Motionnbsp;a Rope that is tied to the Weight, is wound aboutnbsp;the Axis by which the Weight is raifed.

Book I. of Natural Thilofophy. nbsp;nbsp;nbsp;39

Let ah be the Wheel, {PlateY. Fig. 6.) de the Axis, p the Weight to be raifcd, ni the Power; as the Wheel is moved by the Power, thenbsp;Points b and d defcribe fimilar Arcs, wliich arenbsp;the Ways of the Power and the Weight, and arenbsp;to each other, ns ch to cd, that is, as the Diameter of the Wheel to the Diameter of the Axis;nbsp;whence the following Rule is deduced.

Fhe Power has the greater Force, the greater the 11 3 Wheel is, and its A5�on increafes in the fa'/ne Rationbsp;as the WheeVs Diameter. Fbe Weight rejifis fonbsp;much the lefs as the Diameter of the Axis is lefs, andnbsp;its Refiflance is diminijhed in the fame Ratio as thenbsp;Diameter of the Axis. And that there 'may be annbsp;ALqiiilibrum between the Weight and the Power, it isnbsp;always requifite that the Diameter of the BRjecl be,nbsp;to the Diameter of the Axis, in an inverfe Ratio ofnbsp;the Power to the Weigfot.*nbsp;nbsp;nbsp;nbsp;*7�

It is to be obferved, that you mufl; add the Diameter of the Rope to that ot the Axis.

Experiment i h] This Rule is varioufly confirm�d {PlateY. Fig.f.) by help of the Machine herenbsp;teprefented. When the Axis is the twelfth Parcnbsp;of the Diameter of the Wheel, half a Pouitd fu-ftains fix Pounds; and fo on.

The Power alfo may be applied to an Handle or Spoke; as at D, and then the Diflance of thenbsp;Point to which it is applied, reckon�d from thenbsp;Center, mull be look�d upon as the Wheel�s Se-midiametcr.

The Wheels, that have Teeth, work in the fame Manner as this Machine; they being, in rciped ofnbsp;the Axis in Peritrochio, what the compound Lc-Vor is, in refpeft of the fimple Lever.

If the Axis of the Wheel has Teeth alfo, it Cn-ves to move a Wheel, whofeCircumference has

40 nbsp;nbsp;nbsp;Mathematical Elements Book I.

agiin communicate Motion to a third Wheel, and fo on. In that Cafe, that the Power may fuftainnbsp;thenbsp;nbsp;nbsp;nbsp;its Ratio to the 1^'^eight is compounded of

the Ratio of the Diametcrr of the Axis of the lafi Wheels to the Diameter of the fifft-j and the Rationbsp;of the Revolutions of the laft Wheelf to the RevoIu~nbsp;tions of the firfl in the fame Pime.

The Demonftration of which Rule is alfo deduced from theComparifon of the Ways run thro� by the Weight and the Power. {PlateY. Fig.j.)

Experiment Let the Power reprefented by the Weight M be applied to the Wheel A B, andnbsp;the Weight P to the Axis of the Wheel FG, thenbsp;Diameter of that Axis is the eighth Part of thenbsp;Diameter of the Wheel AB, and this Wheelnbsp;goes round five Times, whilft the Wheel FGnbsp;goes round once: Therefore the Ratio of thenbsp;Power to the Weight is compounded of the Ratio�s of I to 8 i and i to f: Therefore it is thenbsp;Ratio of 1 to 40 5 half a Pound in this Gale fu-flaining 20 Pounds. .

CHAP. XIII.

Of theDgt;U LLE Y.

IN feveral Cafes, where the Axis in Peritrochio cannot conveniently be applied, Pullies muftnbsp;be made ufe of to raife Weights} a Machine madenbsp;by combining feveral of them, lies in a little Com-pafs, and is eafily carried about.

* 82 What a Pulley is, has been already explain�d.* If the Weight be fix�d to thePulley, fo that itnbsp;^ may be drawn up along with it, each End of thenbsp;drawing or running Rope fuftains half the Weight.nbsp;Therefore when one End is fix'either to a Hook^ ornbsp;any other Way^ the moving Force or Power appliednbsp;to the other End^ if it be equal to half the PTcight^nbsp;will keep the Weight in Mquilibrio,

Expe-

Book I. of Natural Th�lofophy.

Experiment 1.3 Make faft the Weight P of two Pounds to a Pulley {Eig.v. Plate'S^^.) yet lb thatnbsp;the Wheel or Sheave may not be hindered fromnbsp;turning round; let one Part of the Rope �� benbsp;tied to a Hook ; and the ether End rr/ go roundnbsp;the fix�d Pulley, to change the Direction, * thenbsp;Weight M of one Pound, fix�d to this End of thenbsp;Rope, will fuftain the Weight P.

Several Sheaves may be joined in any Manner, 116 and the Weight be fix�d to them; then if one Endnbsp;of the Rope be fix�d, and the Rope goes roundnbsp;all thofe Sheaves, and as many other fix�d ones asnbsp;is neceffary, a great Weight may be raifed by anbsp;fmall Power: In that Cafe the greater the Numbernbsp;of Sheaves fix�d in a moveable Pulley or of move-able Wheels^ (for the fix�d ones do not change thenbsp;A6tion of the Power *) fo much may the Povoernbsp;be lefis, which fufiains the tVeight; and a Powernbsp;which is to the Weighty as the Number One to twicenbsp;the Number of the Sheaves^ will fufiain tbePTeight.

The Reaion is, that the Number lall mentioned is the Number of the Ropes that fultain the Weight, and the Power is applied only to onenbsp;Rope.

N. B. The Workmen /�England call Block, the Box or Piece of Wood that has one or feveral Wheelsnbsp;in it; and thofe Wheels^ Sheaves or Shcevers.

Experiment 2..3 Hang the Weight Pof � Pounds to the Piece AB {Plate VI. Pig. z.) in whichnbsp;three Sheaves turn freely round. Let one End ofnbsp;the Rope be failen�d to an Hook, and let thenbsp;Rope go round thofe three Sheaves, and threenbsp;other fix�d ones; One Pound, fix�d to the othernbsp;End of the Rope, will make an ^Equilibrium.

Experiment 5, and 4.I] The different Make of the Pullies, or the different W ay of joining the

iiceration; the laft Sort is not very convenient for raifing Weights,nbsp;and therefore W ot kmen make ufe of unequalnbsp;Sheaves, joined together in the manner reprefentednbsp;inF;^. 3.5 till' the different Bigneis of the Sheavesnbsp;makes no Alteration. Oftentimes all the Sheavesnbsp;move round the fame Axis, as in the 4ch Fig. and

both thefe Cafes the fore.

When the End of the Rope, which in the foregoing Experiments was fix�d, is joined to the Weight, or to the moveable Wheels, then thenbsp;Ratio of the Power to the Weight is no longer,nbsp;as one to twice the Number of the Sheaves joinednbsp;to the Weight-, but this double Number muft benbsp;increafed by i y and then, where two Sheaves arenbsp;joined to the Weight, the Ratio will be as i to f jnbsp;for there are juft fo many Ropes, which fuftain the

tij Experiment f.j If feveral moveable Pullies with one Wheel in each, and each having its ownnbsp;particular Rope, be difpofed in the manner reprefented \nFlateF[\\. Fig. i. the Adfion of thenbsp;Power will be very much increafed j for everynbsp;Wheel doubles it, and therefore it is four Timesnbsp;greater with two Wheels, and eight Times greater with three, and fo on.

The Rule above-mentioned, (namely, that the Spaces gone thro� by the Power and Weight,nbsp;when they balance each other, are to one anothernbsp;inverfely as the Power to the Weight,) may benbsp;applied in all the Cafes above-mentioned.

Here we always fuppofe the Ropes parallel j we fhall hereafter fliew what Difference is made bynbsp;the Obliquity of the Ropes.

CHAP XIV.

Of the WEDGE and SCREW.

FRom what has been faid, it plainly appears, how great a Weight may be fuftained, ornbsp;even raifed, by a little Power 5 but thofe are notnbsp;the only Ways of producing the fame Ef��:. Me-chanicks are not confined only to thofe Methods jnbsp;the Actions of Powers may be increafed in allnbsp;Cafes: A very good Inftance of it appears in thenbsp;Wedge^ which is contrived for cleaving Wood,nbsp;and alfo ufeful in feveral other Cafes.

A Wedge A a Prifm of a [mail Height, iz-hofe Ba- 118 fes are eeq^uicrural �Triangles, as A, Plate VI. Fig. 6.

The Edge of the Wedge is a Right Line, 'oshich joins the Vertices of the Triangles-, as bf.

The Edge of the \fedge is applied for cleaving Woodj and the Power is the Blow of a Hammernbsp;Mallet, which drives the Wedjze into thenbsp;Wood.

When the whole Wedge, isdrivenin, theSpacc, gone thro� by means of the Blow or Blows, is thenbsp;Height of the Wedge, which therefore may benbsp;look�d upon as the Space gone through by thenbsp;Power j and the Space, which the Wood goes

44 nbsp;nbsp;nbsp;Mathematical Elements Book �.

thro� as it yields on each Side, is half the Bale of I ip the Wedge. Whence it follows, Thaf the Powernbsp;is to the Refinance of the Wood ( when its A6tion isnbsp;equal to it) as the half Bafe of the Wedge is to itsnbsp;Height.

What is here faid, of the Refinance of the Wood, may be applied to all the other Ufes of thenbsp;Wedge.

12,0 The two wooden Rules A A, A A, are kept up in a parallel and horizontal Situation by thenbsp;Feet BB, B B. (Plate VI. Ftg.j.)

Between them are moved the two Barrels, or Wooden Cylinders EE, which turn upon fmallnbsp;Steel Axes that come out behind the Rulers,nbsp;and have a fmall Return at their Ends, or the Bafesnbsp;are bigger than the Cylinders gt; each return is anbsp;little Convex on the Outfidc, that the Friftionnbsp;againft the Rulers CC, CC, may be the lefs. Innbsp;the Middle of each Ruler A A, there are twonbsp;Pullies i, which almofl touch one another,nbsp;and whofe upper Part is even with the Top of thenbsp;Rulers CC.

The Rope, which in its Middle carries the Weight P, goes round the Pullies d, asd eachnbsp;End of it is fixed to the Axis of one Cylinder E,nbsp;by means of a fmall Plate that has a Hole throughnbsp;which the Axis goes. The other Weight Pnbsp;hangs in the fame Manner upon fuch anothernbsp;Rope.

Therefore the Cylinders E E muil be carried towards one another in an horizontal Motionnbsp;(their Axes remaining parallel) by the Weightnbsp;P, if they are equal.

Let there be a Wedge made of two wooden Planes F F, which make any Angle at Pleafurenbsp;by Help of the Screw gg^

z nbsp;nbsp;nbsp;Experiment.'}

Book I. of Natural Thilofophy^ nbsp;nbsp;nbsp;45

Experiment^ The Cylinders EE are feparated bv letting down the Wedge FF-between them,nbsp;which is drawn down by the Weight M, andnbsp;you have an ^Equilibrium, when the Weight M,nbsp;together with the Weight of the Wedge, is to thenbsp;Weights P,P, as the half Bafe of the Wedge tonbsp;its Height.

The Force with which the Cylinders are carried towards one another, and which muft be overcome to feparate them, is here inftead of the Rc-fiftance of the Woodj the Force with which thenbsp;Wedge is driven or drawn between the Cylinders,nbsp;that is the Weight of the Wedge, together withnbsp;the Weight M, is here taken for a Blow with anbsp;Mallet i and fo the foregoing Rule is reduced tonbsp;Experiment, and confirmed by it.

The firft, which is called the Male Screw, or izr Outfide Screw, is a Cylinder cut in^ in a Helicalnbsp;Form^ as AB {Plate VI. Fig. 8.)

The fecond, which is called the Female Screvv, or Infide Screw, and fometimes the Nut, and isnbsp;different according to the different Ufes of thenbsp;Machine of which it is made a Part, is a folidnbsp;Body that contains an hollow Cylinder.^ wljofe Concave Sur face is cut in the fame Manner as the Malenbsp;Screw, fo that the Prominent Part of the one, maynbsp;fit the Cavity of the other-, as DE.

Thefe two Parts are to move one within another, when the Screw is applied to Ufe. It ferves chiefly to prefs together fuch Bodies as mull; benbsp;joined and firmly united j for in this Machine

4^ nbsp;nbsp;nbsp;Mathematical Elementsnbsp;nbsp;nbsp;nbsp;Book I.

the fmalleft Power may produce a very great Preffure. The Screw may alfo be uftd for raifingnbsp;Weights. In every Revolution of it, one Parcnbsp;remaining at reft,, the ocher is thruft out as far asnbsp;the Interval between two Threads. The Powernbsp;which moves the Screw is applied to an Handlenbsp;or Hand-Spike 5 and then the Power^ which acts asnbsp;llrongly as the Reflftance, is to the Refiflance asnbsp;the laid Difiance between two 'threads to the Periphery of a Circle^ run thro� by that Point of thenbsp;Handle to which the Power is applied. For thenbsp;Way gone through by that Pointer Plane, wherewith the Refiflance is overcome, has the famenbsp;Ratio to the Way of the Power.

Here we mull oblervc, that when the Power balances the Weight in any Machine whatever,nbsp;where no Friftion is fuppofed �, that, by increafingnbsp;the Power ever fo little, it will over-balance thenbsp;Weight. But when there is any Fridlion, thatnbsp;Friftion mull alfo be overcome by the Power;nbsp;and how much mull be added to it, to producenbsp;that Effedl, cannot be determined mathematically.nbsp;In the Machine lall mentioned, this Fiidlion isnbsp;very fenfible, and alfo of a great Ufe; for by itnbsp;the Machine is kept in its Pofition, and cannotnbsp;(either by the Aftion of the Bodies that arenbsp;prefibd, or the Gravity of the Weights) receivenbsp;a contrary Motion, fo as to be pufiied back to itsnbsp;firll Pofition.

CHAP. XV.

Of Compound Engines.

quot;HI TTITE have already Ihewn*, how a Machine fiiy V Vnbsp;nbsp;nbsp;nbsp;compounded of feveral Levers-fquot;,

or feveral Wheels; and that in fuch Machines the Power is to the Refiflance (when it counterba-Inbsp;nbsp;nbsp;nbsp;lances

Book I. of Natural Thtlo/ofhy. nbsp;nbsp;nbsp;47

lances it) in a Ratio compounded of all the Ratio's^ �which the Powers in each fmple Machine would havenbsp;nbsp;nbsp;nbsp;'

to theRefiflance^if they were feparately applied. This Rule alfo obtains in all other Machines.

Not only limple Machines of the fame Kind may be joined j but one may compound a Machine of feveral other Machines in diff�rent Manner; This will be plain enough by two Examples.

Experiment i.] Join the running Rope of the Pullies tothe Axle of the Wheel {PlateVl.Fig.f .)nbsp;and apply the Power to the Wheel: Now, as innbsp;this Cafe, the A�fion of the Power becomes fivenbsp;times greater by help of thefe Pullies, and thenbsp;Diameter of the Axis is but the third Part of thenbsp;Diameter of the Wheel j the Ratio of the Powernbsp;to the Weight is compounded of the Ratio�s ofnbsp;I to * y, and I to 3 j it is therefore as i to if j *116nbsp;and therefore one Pound M fuftains the Weight P tquot;3nbsp;of If Pounds.

The Axis in Peritrochio may be moved by a ^2,2 Screw: ,For this Purpofc the Wheel mull havenbsp;Teeth, and thofe Teeth mull Hand askew, ornbsp;be inclined, as you may fee in the Wheel A,

(Pig. 9.) which is carried round by the Screw B C. Such a Screw is called an endlefs ScreWjnbsp;and very much increafes the Action of the Power jnbsp;for there are fo many Revolutions of the Screw,nbsp;or of the Handle of it, required to turn thenbsp;Wheel once about as the Wheel has Teeth. Andnbsp;if another Wheel with Teeth be added to the firll,nbsp;the A�lion of the Power will Hill be much morenbsp;increafed.

Expermentzh] The Machine of PlateYl. Fig. 9. *^onfi;ls of- an cndlcfs Screw, which is moved bynbsp;the Handle DE. Here the Ratio of the Power

Mathematical Elements Book I.

ed of the Ratio of the Sernidiaraeter of the of the lall Wheel F,to the Length of the Handlenbsp;DE, and the Ratio of the Revolutions ol thatnbsp;Wheel to the Revolutions of the Handle or Screw.nbsp;The firft Ratio in this Machine is the Ratio ofnbsp;I to 303 the fecond is gathered from the Number of Teeth j the laft Wheel F has in its Cir

Wheel 7 j therefore the firft Wheel goes round five times, while the fecond W^heel goes roundnbsp;once: But this firft contains 3(5 Teeth j thereforenbsp;the Screw goes round fo many times, for onenbsp;Turn of the Wheel.* The Ratio compounded of thefe two is i to 180, which is the fecond Ratio fought J and the Ratio made up ofnbsp;that and the firfi (which is i to 30) is the Rationbsp;of I to 7400, which would be the Ratio of thenbsp;Power to the Weight, if there was no Friftioninbsp;but as it is pretty great in all thefe Engines, thenbsp;Power mu ft be pretty much increafed, to make itnbsp;raife the Weight; tho� ftill a very fmall Power applied to it, will raile a prodigious Weight. Thenbsp;Handle E D may be twice or three times as long.

farther increafe, the Adfion of the Power: And, in thatCaft, a fmall Hair will overcome the Forcenbsp;of the ftrongeft Man.

A great number of other compound Machines may be made, whofe Forces are in the famenbsp;Manner determined, by Computation, by thenbsp;Rule mentioned in the Beginning of this Chapter jnbsp;or alfo by comparing the Way gone through bynbsp;the Power with that gone through by the Weight,nbsp;or any other Obftacle} for their Ratio will be thenbsp;inverfe Ratio of the Power, and the Weight ornbsp;Refiftance.

Book I. of Natural ^hilofofhy. nbsp;nbsp;nbsp;49

The Forces, which adting contrariwife balance � one another, are always equal j if therefore the In-tenfity of a Power be Icfs than that of the Refinance, it mufl: run thro� a greater Space in the famenbsp;Timej and that muft always be in proportion asnbsp;its Intenfity is lefsj for the Forces can differ in nonbsp;other Refpect, neither can we compenfate any o-gt;nbsp;tber way for the Difference of Intenfity.

CHAP. XVI.

Of Sir Ifaac NewtonV Laws of Nature.

IN what we have faid of Machines, we have confider�d the Actions of Powers and Weightsnbsp;afting continually againfl: Obftacles and other Refinance; now we fiiallconfider Bodies lefttothem-fflves and continuing in Motion, or freely falling: ?nbsp;And here we mufl reafon from Phaenomena, (asnbsp;one mufl do in all Natural Philofbphy) and fromnbsp;them deduce the Laws of Nature.

Sir Ifaac Newton has laid down three, by'which ^ve think that every thing that relates to Motionnbsp;may be explained.

Bodies continue in their State of Reft^ or Mo- j 2,4 iion^ uniformly in a Right Line, except fo much as theynbsp;are forc'd to change that State by Forces imprefs'd.

We fee that Bodies, by their Nature, are in-a�live,and incapable of moving themfelves; wherefore unlefs they be moved by fome extrinfical A-gcnt, they mull neceffarily remain for ever at reft.

A Body alfo, being once in Motion, continues in Motion according to the fame Direflrion, innbsp;the fame Right Line, and with the fame Velocity,nbsp;as we fee by daily Experience; for we nevea;nbsp;fee any Change made in Motion, but froninbsp;fome Caufe. But (fince Motion is a continualnbsp;Change of Place) how the Motion in the fecond

Moment of Time fhould flow from the Motion in the firfl:, and what fhould be the Caufe of thenbsp;Continuation of Motion, appears wholly unknownnbsp;to me i but, as it is a certain Phtenomenon, wcnbsp;muft: look upon it as a Law of Nature.

12 � 'The Change of Motion is always proportionable to the moving Force imprefs'd^ and is always made according to the Right Line in which that Force isnbsp;imprefs'd.

If to a Body that is already in Motion, another Force be fuperadded to move it in the fame Dire-�fion, the Motion becomes quicker, and that in

When a new Force imprefs�d is contrary to the Body�s Motion, the Retardation follows the Proportion of the Impreflioni fo that a Force whichnbsp;is double or triple, ^c. produces a double ornbsp;triple Retardation. And generally all Forces produce Changes in Motion, according to their Di-re�tions and Quantities j other A6tions of Forcesnbsp;would imply a Contradiftion; This will appearnbsp;more clearly by fuch Experiments made upon oblique Forces, as we fliall mention in fome of thenbsp;following Chapters.

LAW III.

12.� A^ion is always equal, and contrary to Re-a�ion j that is, the Actions of two Bodies upon each other arenbsp;always equal, and in contrary DireStions.

Which Way foever one Body afts upon another, we fee that Body always fuffers an equal and contrary Re-afl:ion. If I prefs a Stone with my Finger, my Finger is equally prefs�d by the Stone.nbsp;A Horfe that draws a Cart forward, is as muchnbsp;drawn backward by the Cart; for the Geers ornbsp;Traces are equally flretched both Ways.

j nbsp;nbsp;nbsp;When

Book I. of Natural T'hilojo^hy, nbsp;nbsp;nbsp;jfi

When a Body ftrikes againft another, whatever the Stroke be, both fuf�er it equally; but the Im-preffions are contrary. This is clearly confirmednbsp;by the Experiments of the Congrefs of Bodies.

Experiment.Sufpend the Loadftone M, {Plate VII. Fig. z.) in fuch manner that it maynbsp;eafily be moved; then, bringing a Piece of Ironnbsp;within a fmall Diftance of it, the Loaddone willnbsp;come to the Iron; And if you pull back the iron,nbsp;before the Stone be come to it, the Stone willnbsp;follow the Iron; juft as the Iron goes towards the ' �nbsp;Stone and follows it, when the Iron is fufpend-cd and moveable, and the Loadftone brought tonbsp;the Iron.

When a Man fits in a Boat, and by a Rope pulls towards him another Boat, juft as big andnbsp;as much laden, both Boats will be equally moved,

^�ud meet in the Middle of the Diftance of the Places in which they were at firft. If one Boatnbsp;be greater than the other, or more laden, the Ve-Ipciries in each will be different, when they havenbsp;different Quantities of Matter; but the Quantitiesnbsp;of Motion on both Sides will be equal, abftradlingnbsp;from the Refiftance of the Watcr.

And this Law takes place generally in all the AQrions of Bodies upon one another.

CHAP. XVlf.i

the Acceleration and Keta]rdahon of heavy Bodies.

N accelerated Motion is that Motion^ �whofe i ij Velocity becomes greater every Moment.

Mathematical Elements Book 1.

The Force of Gravity a�ts continually upon all Bodies, in proportion to their Quantity of Matter * When a Body falls freely, the Impreffionnbsp;made upon it the firlf Moment is not deftroyednbsp;in the fecond Moment j therefore there is fuper-added to it the Impreffion made in the fecondnbsp;lip Moment, and fo on. ithe Motion then of a Bodynbsp;that falls freely.^ is accelerated, and that eqitally innbsp;equal Times-, becaufe the Force of Gravity a6lsnbsp;every Moment in the fame Manner,* and therefore communicates an equal Velocity to Bodies innbsp;equal Times. Whence that Celerity, which is acquired in the Fall, is always as the Time in whichnbsp;the Body has fallen. For Example: The Velocitynbsp;acquired in a certain Time will be double, ifnbsp;the Time be double j and triple, if the Time benbsp;triple, 0�c.

Let that Time be exprelTed by the Line AB (Plate VII. Fig. 3.) and let the Beginning of thenbsp;Time be A. In the Triangle ABE, the Linesnbsp;I �, zg, 3^, which, being parallel to the Bafe,nbsp;are drawn throiigli the Points, i, 2, 3, are tonbsp;one another as their Difhances from A, A i, At,nbsp;Ay, that is, as the Times which are expreffiednbsp;by thofe Diftances, and exprefs the Velocitiesnbsp;of a Body falling freely after thofe Times. If,nbsp;inftead of Mathematical Lines, others be takennbsp;with a very fmaU'Breadth equal to each of them,nbsp;the Proportion will not be changed thereby jnbsp;and thole fmall Surfaces will in the fame mannernbsp;denote the above-faid Velocities. In the leaftnbsp;Time the Velocity may be looked upon as equable, and therefore the Space gone through innbsp;*53 that Time is proportionable to the Velocity.* Innbsp;Inbsp;nbsp;nbsp;nbsp;each

Book I. of Natural ^hilofophy.

each of thole firtail Surfaces above-mentioned, if the Breadth of the Surface be called ; he Time,nbsp;the Surface itfelf will be the Space gone ��hro�.nbsp;The wholeTime A B confiits of thofe very ;r:,allnbsp;Times j and the Area of the Triangle ABE,nbsp;of the Sum of all thofe very little Surfaces, an-fwering to thofe fmall Parcs of Ti�cs: Thereforenbsp;that Area exprefles the Space gone through in thenbsp;Time A B. After the fame Manner the. Areanbsp;of the Triangle A i � reprefents the Space gonenbsp;thro� in the Time A i j thofe Triangles are limi-lar, and their Areas are to one another as thenbsp;Squares of the Sides A B, A i : That is, the Spaces^ gone through from the Beginning of the Fail-, arenbsp;to one another^ as the Squares of the Fimes duringnbsp;finhich the Body fell.

The Balance AB {Plate VW. Fig. which i kas but one Scale, is exaflily in T�quilibrio j whennbsp;a Weight is put into the Scale, an Iron, made innbsp;the Form of a Gnomon, keeps fail: the Brachiumnbsp;A, and the Balance is retained in a horizontalnbsp;Pofition.

Ac � there is a thin Spring � g fixed to the Gnomon, and which, when extended, reaches tonbsp;h where the End g is retained by help of the littlenbsp;Plate r, which is made fall Co the Brachium A.nbsp;Now by this Means the leafi: Motion of thenbsp;Balance becomes fenfiblci becaufe then the Springnbsp;f g-i being free, flies out, and returns to the Figure

fg-

At the End of the Brachium B, there is aPlclc, thro� which the String faftened to the Hook Dnbsp;pafles freely, that String is kept in a vertical Situation by hanging on the Weight N.

The Weight M has a Hole thro� it for the above-mentioned String to pafs freely thro�; innbsp;E 3nbsp;nbsp;nbsp;nbsp;making

5�4 nbsp;nbsp;nbsp;Mathematical Elements Book I.

making Experiments, the Weight M is raifed up along the String, and, when you let it go, it fallsnbsp;upon the lame Point of the Brachium B.

Experiment^ Put the one Pound Weight Pinto the Scale j then the Body M, falling from thenbsp;Height of three Inches, will move the Balance,nbsp;When P is a two Pound Weight, letM fall fromnbsp;twelve Inches, and the Balance will be moved.nbsp;If you lay on three Pounds in the Scale, the Bodynbsp;M muft be let fall from a Height of xj Inches,nbsp;to move the Balance and raife P. And in all thefenbsp;Cafes, if the Height from which M falls be takennbsp;but a little Icfs, the Balance, with the Weight innbsp;the Scale, will not be moved.

In this Experiment, the Weight which is laid upon the Scale, and raifed by the Blow of thenbsp;falling Body, is proportionable to the Stroke j thenbsp;Quantity of Motion in the Body follows the Proportion of the Stroke: And that Quantity (be-caufe we make ufe of the fame Body) is propor-*63 tionable to the Celerity j* and, lallly, the Celeri-Wjjo ty here is proportioned to the Time of the Fall*;nbsp;Therefore the Weights above-mentioned follownbsp;the fame Proportion of theTime. TheWeightsnbsp;here are as the natural Numbers i, z, 5, andnbsp;therefore the Times are in that Proportion: Butnbsp;the Spaces gone thro� in thole Times are as 3,nbsp;iz, 27, or as I, 4, p, which Numbers are thenbsp;Squares of the others.

115 Having divided the Time A B {Plate VII. Fig. 3.) into the equal Parts A r, i 2., 2. 3, 3B, thro�nbsp;the Divifions draw Lines parallel to the Bafej thenbsp;Spaces gone thro' in thofe Parts of Time, that is,nbsp;in the fir ft., fecond, and third Moment.^ amp;c. fiuppo-fing the Moments equal, are to one another as the

Book I. of Natural Thilofophi.

If the Body, after it has fallen, during the Time A B, fhould be no more accelerated, but with thenbsp;Celerity BE, acquired by that Fall, Ihould uniformly continue its Motion, during the equal Timenbsp;BC, the Space gone thro� by that Motion is ex-prefs�d by the Area BE DC, which is double thenbsp;Triangle ABE. And therefore,

A Body falling freely from any Height^ ivith that V�ocity which it has acquired by that Fall^ will innbsp;a Fime equal to the �Time of the Fall (by an equablenbsp;Motion') run thro' a Space double the faid Height.

The Motion of a Body thrown upwards is retarded in the fame Manner, as the Motion ofnbsp;^ falling Body is accelerated by the fecondnbsp;Law*: In this Cafe the Force of Gravity con-fpires with the Motion acquired; and in that itnbsp;afl:s contrary to it. But, as the Force of Gravitynbsp;is equal every Moment, the Motion of a Body thrown xnbsp;ttp is equally diminijhed or retarded in equal fimes.

The fame Force of Gravity generates Motion w the falling Body, and deftroys it in the rifingnbsp;Eodyj therefore the fame Forces are generatednbsp;and deftroyed in equal Times. A Body thrown upnbsp;rifes till it has loft all its Motion; and fo goes upnbsp;during the fame 1�ime^ that a Body falling could havenbsp;acquired a Vdocity equal to the Velocity with whichnbsp;the Body is thrown up.

If a Body be thrown up with the fame Velocity that it would acquire in falling down Line B C {Plate VII. Fig. f.) it wouldnbsp;^cend in a Time equal to the Time of thenbsp;Eall*, (and with an equable Motion) fo as to � gnbsp;Come up the Height C A, the double of B C*; but * 13^nbsp;^ in the fame Time, by the Force of Gravity, thenbsp;Body goes thro� a Space equal to the Space AB,

j6 nbsp;nbsp;nbsp;Mathematical Elements Book I.

the (ame Time, and act contrariwife, the lefler muft be fubltratted from the greater j thereforenbsp;the Body, after the End of the Afcent, will be atnbsp;I37B. Whence it follows, that a Body thrown upnbsp;will rife to the fame Height from which fallings itnbsp;13^ would acquire the Velocity with which it is thrown up.nbsp;And therefore, the Heightwhich Bodies thrown upnbsp;with different Velocities can rife to^ are to one am-� 131 they as the Squares of thofe Velocities*

CHAP. XVIII.

C B in Plate VII. Fig. 6. reprefent a Line parallel to the Horizon j AB makes with it thenbsp;oblique Angle ABC, and reprefents an inclinednbsp;Plane gt; and the Perpendicular A C is let fall fromnbsp;A, the upper Part of the Plane, to the Horizon.

Let two equal Bodies defeend by the Force of Gravity from the Point A, the one along thenbsp;Line AB, and the other along the Line ACjnbsp;when they are come to the Points B and C, theynbsp;have defeended equally, that is, they will be gotnbsp;each equally near to the Earth�s Center: Therefore the Forces with which they are impelled, asnbsp;they are direded towards the Earth�s Center,

Book I. of Natural Tbilofophy. nbsp;nbsp;nbsp;5:7

are equal; but the Intenfities of eqital Forces are reciprocally as the Spaces gone through*; and * 70nbsp;therefore here the Inten�ty of the Force, bynbsp;which the Body is impelled along an inclined Plane,nbsp;is to the Intenfity of it, by which it is diredtlynbsp;impelled towards the Center of the Earth, as ACnbsp;to AB. Therefore, A Body laid upon an inclinednbsp;Plane^ lofes Part of iti Gra'vity, and the IVeightnbsp;required to fuftain it^ is to the Weight of the Body,nbsp;as the Height of the Plane to its Length.

The Plane NOQL {P late Y\l. Fig.-j.) isplaced 145 in an horizontal Situation; the Plane A BIHnbsp;moves upon Hinges; and may be fixed at anynbsp;Height, by help of the Screw V, and Quadrant t.

The wooden Ruler EF has a Pulley C firftened at one end, and revolves.about the other; thenbsp;Head D, about whofe Center this Ruler moves,nbsp;may be fixed (in any Place of the Slitpr) to thenbsp;Plane N OQL, by a Screw under the Plane.

M is a wooden Cylinder, whofe Axis is of Steel, and whofe Bafes fomewhat exceed thenbsp;Cylinder; fo that, as it turns round along thenbsp;Plane A BIH, the Bafes only touch the Plane.

The Cylinder is fuftained by a String that goes over the Pulley G; which String is fixed tonbsp;a thin Brafs Ruler, bent in fuch Manner, that thenbsp;Axes of the Cylinder go thro� its Ends, and turnnbsp;in them.

In making Experiments, the Pulley is fo plac�d by the inclining Ruler EF, and moving the Headnbsp;D along the Slit rs ; that the String by which thenbsp;Cylinder is fuftained, is parallel to the inclinednbsp;Plane ABIH.

Experiment i.] Let the Plane ABIH be inclined in any Manner, the Weight of the Body M has the fame Ratio to the Weight P, as thenbsp;Length of the Plane A B to its Height A C; and

the

jS nbsp;nbsp;nbsp;Mathematical Elements Book I.

the Body M, in what Parc foever of the inclined Plane it be fee, will be fuftained upon it by thenbsp;Weight P.

As the Force, by which a Body is made to de-feend along an inclined Plane, arifes from Gravity, it is, of the fame Nature with itj and thereforenbsp;that'Force every Moment, and in all Parts of thenbsp;� 75 Plane, is equal*: For the fame Reafon the Motionnbsp;144 of a Body ^ freely rmningdown upon an inclined Plane^nbsp;is of the fame Nature with the Motion of a Bodynbsp;freely falling j and what has been fiid of the one,nbsp;may alfo be affirmed of the other. It is therefore

therefore the Propofitions of Numb. 130, 131, 133, 134, i3r, I3lt;^gt;nbsp;nbsp;nbsp;nbsp;maybe here

applied, if we fuppofe a Motion upon an inclined I4f Plane, inftead of a direct Afeenc or Defcent. ,nbsp;The Forces by osshich two Bodies defeend^ one ofnbsp;�which falls freely^ and the other runs down an inclined Plane^ if the faid Bodies begin to fall at the famenbsp;Time^ are always to one another in the fame

* nbsp;nbsp;nbsp;,29 Ratio as in the Beginning of the Fall*'j thereforenbsp;144 the Effedt of thofe Forces, that is, the Spaces gone

'� In the Plane AB (Plate VII. Fig. 8.) the Space gone through by a Body, whilft another falls freely down the Height of the Plane AC, is determined by drawing C G perpendicular to A B ;nbsp;for then the Length of the Plane A B is to itsnbsp;Height AC, as AC to AG. If a Circle be deferibednbsp;with the Diameter AC, the Point G will be innbsp;the Periphery of the Circle j becaufe an Angle innbsp;a Semicircle, as AGC, is always a Right Anglenbsp;and therefore a Point taken, as G, in any Inclination of the Plane, will always be in the Peripherynbsp;of the faid Circle; Whence it follows, that all the

Book I. of natural Thilofofhy. nbsp;nbsp;nbsp;59

Chords, as A G, are gone thro� by Bodies running along them, in the Time that a Body, fallingnbsp;freely, would run down the Diameter ACj andnbsp;therefore the Time of the Falls thro� thofe Chordsnbsp;are equal. Thro� the Point C there can be drawnnbsp;no Chord, as H C, but what a Chord, as Anbsp;may be drawn parallel to it, (that is, equally in- 147nbsp;dined) and equals therefore in a Semicircle, asnbsp;A H C, �whether a Body falls freely along the Diameter A C, or 'whether it falls down along any Chordsnbsp;as H C, it 'will in the fame fime come to the lowejinbsp;Point of the Semicircle.

The Time of the Fall along the whole Plane AB may be compared with the Time of the Dc-feent along the Height A C 5 which for that Timenbsp;is equal to the Time of the Fall along AG j andnbsp;fo the Squares of thofe Times are to one another,nbsp;as A B to A G.* But A B is to A C, as A Cnbsp;to A G i therefore the Squares of the Lines A Bnbsp;and A C are to one another, as A B to A G j andnbsp;therefore thofe Lines AB and AC are to one anothernbsp;as the Limes of the Fall along A B and A G, or A C jnbsp;ihat is^ the Limes in that Cafe are, as the Spacesnbsp;gone through.

the fame Cafe, the Velocities, at the End of 14p '^heDefcent,are equal-, for after equal Times, whennbsp;the Bodies are at G and C, the Velocities are innbsp;the fame Ratio as in the Beginning of the Fall*,nbsp;that is, as the Forces by which the Bodies arenbsp;tmpell�d, or as A C to A B.* When the Bo-oy defeends from G to B, the Velocity increafesnbsp;as the Time; and the Velocity at G is to the Ve-W at B, as AC to AB;* Therefore the*nbsp;Velocities at B and C have the fame Ratio tonbsp;the Velocity at G, and fo arc equal. Hence itnbsp;appears, that a Body acquires the fame Velocity, j -q

falling from a certain Height, 'whether it falls di-f telly do'wn^ or along an inclined Planeand fince

the

Mathematical Elements Book 1.

the Angle of Inclination caufes no Alteration, a Body may run down feveral Planes differently inclined^ and even along a Curve^ (which may benbsp;confider�d as made up of an innumerable Number ofnbsp;Planes differently inclined) and the Celerity acquirednbsp;will always he the fame., when the Height is equal.

^Experiment z.] In this Experiment it is to be obferv�d, that a Body hanging by a Thread, andnbsp;defcribing a Curve by its Fall, falls in the famenbsp;manner, as if it was to run down fuch a Curve cutnbsp;hollow in a folid Body without any Friftion.

Let the Body P, [Plate YW. Fig. p.) fufpended by a Thread, Tall from the Height AC, in thenbsp;Curve B C, and in the Curve D C, and in thenbsp;Curve E F G C, made up of Parts of differentnbsp;Circles, and in each Cafe it will, with the famenbsp;Force, ftrike againft the Body Q_, which is atnbsp;refti and always drives it to the fame Height.

Body that has acquired any Celerity in falling down along any Surface, whether Plane or Curve,nbsp;will rife up to the fame Height along another fimilarnbsp;Surface, with the fame Velocity, in the fame �ime.

A Body will,with the fame Celerity that it has acquired in falling down from a certain Height, rife up to the fame Height in any Curve whatever.

Experiment 3.] Let the Body P, [Plate VIII. Eig. I.) hanging by a Thread, fall from thenbsp;Height A C, along any Curve B C with the Celerity which it has thereby acquired, it will afcendnbsp;to the fame Height on the other Side, in thenbsp;Curves CD or CE, orCHGF.

CHAP.

CHAP. XIX.

Of the Ofcillation or Vibration of Pendulums.

A Heavy Body^ hanging by a [mail �Thread and i f 3 moveable 'with the �Thread about the Point tonbsp;labicb the Thread is fix'd, is call'd a Pendulum.

The Motion of a Pendulum is an ofcillatory or vibratory Motion. When the Weight, the Threadnbsp;being extended, is raifed upon one Side, it de-fcends by its Gravity, and, with the Celerity thatnbsp;jt has acquired, rifcs up to the fame Height on thenbsp;other Side} * and then it returns by its Gravity, *151nbsp;snd fo continues in its Vibrations.

We here fuppofe the Motion about the Point of Sufpenfion to be perfedlly free, and that there isnbsp;oo Refiftance of Air, which is very fmall in greatnbsp;Pendulums.

The Body P {PlateYWl: Fig. 2.) does, in its lyq Motion, defcribe the Arc PBF; if, infcad ofnbsp;that Motion, a Body fliould defcend along thenbsp;Chord PB, and again afcend along the Chordnbsp;�P, and fo Jhouldperform its Vibration in Chords-,nbsp;theDefcent would be made in that Time in whichnbsp;the Body by its Fall would go thro� the Lengthnbsp;of the Diameter A B} * that is, twice the Lengthnbsp;of the Pendulum. In an equal Time, it afcendsnbsp;slong the Chord B F} * therefore in the Times ofnbsp;'whole Vibration, the Body in falling might runnbsp;through four Diameters} * that is, eight Times thenbsp;Length of the Pendulum. And as the Defcent andnbsp;j^feent in any Chord is performed in the famenbsp;Time, all the Vibrations in Chords, whether greatnbsp;ot fmall, are likewife performed in the fame Time,nbsp;fmall Vibrations, the Arcs of a Circle do not

6z nbsp;nbsp;nbsp;Mathematical Elements Book I,

fenflblv differ from the Chords j and the Vibrations I�� of the fame Pendulum^ tho' unequal^ a,re performednbsp;in the fame �time^ as far as our Senfes can difiin-guifh.

Experiment Plate VIII. Fig. 4.] If the two equal Pendulums, CP and cp, are let fall from thenbsp;Points P and p in the fame Moment of Time,nbsp;they will at the fame Time come to B and andnbsp;then to F and �; and fo they will continue theirnbsp;Motion, in the Arcs PBF and � always in thenbsp;fame Time.

Here it is to be obfcrved, that tho� the Proportion I yy be true in all Pendulums, the Demon-ftration given is only to be applied to fhort Pendulums i in the longer Sore the Time of the De-fcent along a Chord differs fenfibly enough from its Defcent along an Arc j but in fmail Arcs thenbsp;Differences are equal, tho� the Chords be of different Lengths.

Let the Circle FB {Plate YIW. Fig, 3.) roll along the Line A D, till the Point B comes to Anbsp;in the fame Linej by fuch a Motion the Point Bnbsp;defcribes a Portion of the Curve BPA. Suchnbsp;another Curve B D may be defcribed in the famenbsp;Manner, and the whole Curve A B D is call�d anbsp;Cycloid. Let it be divided into two equal Partsnbsp;at B, and let the Parts B A and B D be fo difpofednbsp;as to have the Points A and D fall in together atnbsp;C i and let the Point B coincide with the Pointsnbsp;A and D in the Line AD. Let two Plates ofnbsp;Metal be bent according to thefe Curves, fo thatnbsp;the Thred of a Pendulum fufpended at C, maynbsp;on either Side apply itfelf to thofe Plates, and failnbsp;in with their Curves as the Pendulum vibrates.nbsp;Now if the Length of the Pendulum be C B, thenbsp;Bodv P in its Vibrations will defcribe the Cycloidnbsp;ABD.

Book I, of Natural Thilofo^hy. nbsp;nbsp;nbsp;6

It is a Property of this Curve, that in what- iflt;3 ever Point of it the Body P be placed, the Force,nbsp;with which it is carried by its Weight along thenbsp;Curve, is proportional to the Part of the Curvenbsp;'vhich is between that Point and the loweft Pointnbsp;of the Curve B. Whence it follows, that if twonbsp;Pendulums, as CP, be let fall in the fame Moment from different Heights, the Velocities, withnbsp;'vhich they begin to fall, are to one another, asnbsp;the Spaces to be run through before they come tonbsp;P: If therefore they fhould be adled upon bynbsp;thofe Forces alone with a Motion not accelerated,nbsp;they would come to B at the fame Moment ofnbsp;Timej * after the fame Manner by the Forces * 53nbsp;''^hich are acquired, the fecond Moment, theynbsp;^^fo come to B at the fame Time: The fame maynbsp;be fiid in Relation to the following Moments;

^nd the half Vibrations made up of all the For-together, however unequal they are, as alfb whole Vibrations are performed in the famenbsp;Time.

^hat the 1�ime of each Vibration is to the �fime of a quot;^^rtical Fall^ along the half Length of a Pendulum^nbsp;^tfhe Perpihery of a Circle to its Diameter. Innbsp;Curve the lower Part coincides with a fmallnbsp;ptc of a Circle, as to Senfe: And this is the truenbsp;veafon, why in a Circle the Times of fmall Vi-tations (however unequal thofe Vibrations be)nbsp;equal; and therefore alfo the Duration ofnbsp;hofe Vibratiqns has the above-mentioned Rationbsp;the Time of a vertical Fall. The Durationsnbsp;tnbsp;nbsp;nbsp;nbsp;Vibrations of unequal Pendulums may be com-

^ted together. When the Arcs are fimilar, the ^yiations, in Refpedl: of the Chords, are allbnbsp;^milar, and the Times of the Vibrations in thenbsp;^re, as the Times of the Vibrations alongnbsp;^Chords; but they are, as the Times of the

Mathematical Elements Book I.

�uperiment 2.] Two Pendulums CP, cp, (^PlateYWl. Fig-^-) whofeLengths are as 4 to i,nbsp;are let fall at the fame Time from the Points Pnbsp;and /), fo that in their Vibrations they defcribenbsp;hmilar Arcs} the longer Pendulum vibrates once,nbsp;whilft the fliorteft vibrates twice} and fo thenbsp;Squares of the Durations of the Vibrations arenbsp;as 4 to I, namely, as the Lengths of the Pendulums.

When the Vibrations are fmall, this Ratio al-fo holds, tho� the Pendulums fliould not vibrate in

159 phe Velocities of Pendulums in the lowefi Pointy when the Vibrations are unequal, are to one another,nbsp;as the Subtenfes of thofe Arcs, which the Body de-fcribes in its Defcent. So the Velocity of the Body P, {PlateYlll. Fig. z.) falling in the Arc PB,nbsp;is to its Velocity when it falls along D B, as thenbsp;Chord PB to the Chord DB: For if you drawnbsp;in a Circle the Lines P�, D d parallel to the Horizon, the Squares of the faid Chords are to onenbsp;another, as theLines � B, iB. The Squares of thenbsp;faid Velocities are alfo, as thofe Lines �B, dB�nbsp;therefore the Velocities are as the Chords.*

Concerning all that has hitherto been faid of Pendulums, it is to be obferved, that it is nonbsp;160 Matter how big the Weight of the Pendulum isgt;nbsp;or whether the Weights of two Pendulums benbsp;different in Magnitud^e or different Sorts of Bodies } fince Gravity is proportioned to th^nbsp;Quantity of Matter in all Bodies, all Bodies innbsp;the fame Circumftanccs are moved by Gravity

Book I. of Natural Thilofophy.

Experiment 3.] Take two equal or unequal Balls, the one of Lead, and the other of Ivory, hangnbsp;them up by Threads, that they may makePendulanbsp;of equal Lengthsj let them vibrate, and theirnbsp;equal Vibrations (or even all their unequal ones,nbsp;provided they be fmall Vibrations) are perform�dnbsp;in the fame Time.

Oftentimes inftead of a Thread, a fmall, but 161 ftiff, Iron-Rod is made ufe of, and fometimes alfo two or more Weights are fix�d to it, and it isnbsp;called a Compound Pendulum; in that Cafe thenbsp;Rules above-mentioned are not applicable j butnbsp;thofe Pendulums are reduced to fimple ones, bynbsp;determining in them fuch a Point, that, if all thenbsp;Weights were united in it, the Vibrations wouldnbsp;be of the fame Duration as thofe of the compoundnbsp;Pendulum. This Point is called the Center of Of-dilation.

The Center of Percufflon in a compound Pen- i6z dulum is aPoint, in which the whole Force of thenbsp;Pendulum is as it were collefted} fo that if thatnbsp;Point ftrikes againfi: an Obllacle, the Blow willnbsp;be greater than if any other Point of the Pendulum fnould ftrike againft it.

In a Vacuum, or a Medium that does not re-flfi? thefe two Centers coincide. They alfo ..coincide in the Air, as to Senfe, by reafon of the hnall Refifiance.

A Body of any Figure may be fufpended,. and '�ibrate about a Point, or rather an Axis and innbsp;foch a Body one may alfo determine the Center ofnbsp;Cfcillation.

t^Pben a Right Linefuch as is an Iron Wire^ vi- 16^ hrates about one End.^ the Center of Ofcillation is,

66 nbsp;nbsp;nbsp;Mathematical Elements Book I.

ExperimentThe flat Iron AB {Plate V\l\. Fig. 6.] muft be fo hung up, as to vibrate aboutnbsp;the End A 5 let the Ample Pendulum C P, whofcnbsp;I.cngth is equal to two third Parts of A B, be fuf-fercd todefcend at the fameTime as the Iron j andnbsp;the Vibrations of the Pendulum and the Iron willnbsp;be perform�d at the fame Time.

and this Property of Pendulums is of great Ufe in Clocks, to which an equable Motion is communicated by flxing on a Pendulum.

By carrying Clocks to different Places, it has appear�d that the Force of Gravity is not equalnbsp;in all Parts of the Earth, becaufe the Vibrationsnbsp;of the fame Pendulum, in divers Countries, havenbsp;been found unequal, in refpedl to Timej and thatnbsp;Difference of Gravity is meafured by Pendulums.

l d�4 Let there be two Pendulums., whofe Lengths are to one another, as the Forces of Gravity by whichnbsp;they are aEluated-, if they run out into Arailarnbsp;Arcs, in correfpondent Points, the Force will always have the fame Ratio to one another, and in-*nbsp;deed the Ratio of the Spaces to be gone thro�,nbsp;f becaufe Amilar Arcs are as the Lengths of Pendulums, ) which therefore will be run thro� in equal

If* they be reduced to the fame Length by changing one Pendulum, the Square of the Timenbsp;of the Vibration of the Pendulum, that isnbsp;changed, is to the Square of the Time of Vibration before the Change (that is, to the Square ofnbsp;the Time of the Vibration of the Pendulum that

Book I. of Natural Thilofofhy. nbsp;nbsp;nbsp;67

is not changed) as the Length of the Pendulum after the Change to its firft Length : * which * is*nbsp;Lengths are to one another, as the Force of Gravity in the Pendulum that is not changed, to thenbsp;Force of Gravity in the Pendulum that is changed.

And therefore the Squares ofthe Times of theVibra-tions in equal Pendulums are to one another, in-verfely, as the Forces of Gravity voith which tlse Pendula are a�ed upon: which therefore are to onenbsp;another^ direftly, as the Squares of the Vibrationsnbsp;perform�d in the fame �time.

But whence this Difference of Gravity arifes, fhall be explain�d hereafter, when we fpeak of thenbsp;Figure of the Earth.

CHAP. XX.

' Very Body that is at Reft, and hinder�d by ^ no Obftacle, may be pufh�d forward by anynbsp;other Body in Motion j and, when once it is putnbsp;in Motion, it will continue in it, till it is hinder�dnbsp;by fome external Caufc.* That Caufe is fomc- * 124nbsp;times a Stroke of another Body againft it, or anbsp;Stroke which itfelf gives another Body j or laftlynbsp;^ Stroke of both meeting.'

All Bodies, here taken notice of, are fuppofed jpherical j becaufe the Laws of Motion ought to

A Body is faid to impinge dircdily againft ano- 166 ^ner, or two Bodies to ftrike or impinge againftnbsp;another, when the Dire�ion of the Motion^ ornbsp;Motions.^ {if both are moved) goes thro' the Centersnbsp;V both Bodies.nbsp;nbsp;nbsp;nbsp;Fanbsp;nbsp;nbsp;nbsp;D e f i-

Mathematical Elements Book I.

In all other Cafes the Stroke is [aid to he Oblique. When eliiftic Bodies impinge againll one another, the Parts that are ftruck yield inwards,nbsp;and, by the Reftitution of the Pans, the Bodiesnbsp;repel one another, and are feparated froui onenbsp;another.

In Bodies that are perfedtly foft^ or perfectly hardy there is no fuch Afijion j and therefore, in a dire^nbsp;Stroke they are not feparated after the Blowy becaufenbsp;169 after their meetingy as well as beforOy they are movednbsp;in the fame Line j for nothing happens that cannbsp;change the Direftion.

I {hall in this Chapter fpeak of the Pcrcuffion of Bodies that are not elaftic, and here, as alfo innbsp;the whole following Chapter, I fhall fpeak of di-reft PercLif��on , and confirm the whole by Experiments made with the following Machine.

ABC is a vertical Plane of Wood almoft triangular, about 4 Foot and a half high, and 3 Foot wide at the Bottom. Plate IX.

In the upper Part there is a Slit st quite thro�it, which is horizontal, along which two fquarc Pinsnbsp;(and fometimes more) are moved; thefe Pins, having a Shank that goes through the Plane, may benbsp;made faft in any Part of the Slit by Screws whichnbsp;take the Shanks behind the Plane or flat Board, asnbsp;may be feen from the Figure of the Pins at V.''!

A little fquare Pipe of Iron X flips upon each Pin, and may be faften�d to it by a little Screw Cynbsp;in the upper Part of any Place of the wholenbsp;Length of the Pjn. Thefe little Pipes have Hooksnbsp;in the under Parts, thro� which fmall Threads ornbsp;Fiddle-ftrings run, and fuftain fuch Balls as P andnbsp;Thofe Strings go round the wooden Keys, /, hnbsp;by turning which, the Balls are rais�d or lower�d.

Book �. of Natural Thilojbfhy.

The Pin, from which any Ball hangs, is fix�d to fuch a Parc of che Slit rr, that its Center is di-

ftanc from the Line AD (which divides the Machine into two Parts vertically) jull ane Semidia-nieter of the Ball; and that is to be done for all the Balls by means of Marks in the Surface of thenbsp;Board.

The little Pipe and Hook, from which the Ball hangs, is fix�d to fuch a Part of the Pin, that thenbsp;Thread hangs but a little farther from the SurBicenbsp;of the Board than a Semidiameter of the Ball;nbsp;There are Divifions in the Pins, to determine thenbsp;Place of the Iron Pipes upon the Pins, accordingnbsp;to the Bignefs of the Balls.

When you ufe two Balls, the Line A D fepa-I'ates them, and in that Cafe (as alfo when leveral at once are made ufe of) if they are of differentnbsp;Bignefs, the great Ball always determines the Di-ftance of the little Ball from the Board ; and thenbsp;little Pipes are fix�d at fuch Divifions of the Pins,nbsp;that the Centers of all the Balls may be equallynbsp;lt;^iftant from the Board. The Keys I bring allnbsp;thofc Centers to the fame Heights; which is tonbsp;be obferv�d in all the Experiments.

There are two Brafs Rulers EG, EG, which Bide horizontally in the Board, whofe Surface isnbsp;hollow�d to receive them, fo that their Surfacenbsp;*ttay lie even with it. Behind each Ruler there isnbsp;^ Slit in the Board of about y Inches, to tranfmicnbsp;^ Screw coming from the Backfide of the Ruler,nbsp;^hich is fix�d behind a Nut in any Part of thenbsp;blit. In making Experiments, the End G ofnbsp;^ach Ruler is diflant from the Line A D, one Se-�^'diameter of the Ball, which hangs on the famenbsp;bide.

Thefe Kuiers are fo divided as to fhew equal ^t�gles, run thro� by the Threads which carry thenbsp;Balls,

7Cgt; nbsp;nbsp;nbsp;Mathematical Elements Book �-

To mcafure thofe Angles in making the Experiments, there are four Indices, two great ones M M, and two lels N N.

Thefe Indices, Aiding in a Groove, are moved along the Slits �r, (?r, and are faAen�d behind thenbsp;Board, where you pleafe, in the fame Line bynbsp;Screws. The longer Indices reach to the Edge ofnbsp;the Board, tho� the Slits want about 3 Inches ofnbsp;it.nbsp;nbsp;nbsp;nbsp;'

The feparated Figure M reprefencs the greater Indices, in which ah is a Plate, which Aides innbsp;the Groove of the Board} cd is the Index, per-

The other feparated Figure N reprefents one of the lefler Indices, whofe Length is equal to thenbsp;Semidiameter of the fmaller Balls, which are applied to the Machine, and whofe Diameter maynbsp;be about i Inch and a Half: Thefe Indices are putnbsp;among the great ones, becaufe they don�t hindernbsp;the Motion of the Balls: Sometimes the two fmallnbsp;ones are put in the fame Slit, when three Anglesnbsp;are to be meafured on one Side.

In that Cafe the Ball Q_is raifed up, or rifes after its Fall towards the Side of the Board B.nbsp;That the Index may be placed right for mea-furing that Angle, the end G of the Ruler EG,nbsp;which is on the Side B, muft be joined with thenbsp;End G of the other Ruler, placed as above-mentioned.

The three Iron Screws FFF ferve to let the Machine or Board truly vertical, fo as to bringnbsp;the Line A D perpendicular to the Horizon}nbsp;which may be eafily done by hanging on any onenbsp;of the Balls, and putting on one of the greatnbsp;Indices} fo that the Thread, cutting any Mark onnbsp;the Index, may hang parallel to the Line A D.

Book I. of Natural Thilofophy.

For making Experiments on Bodies that have no Elafticity, you muft ufe Balls of foft Clay,nbsp;made in the wooden Mould L.

This Mould confiftsof five Parts, four of which may be feen at H, H, H, H} thefe being join�d,nbsp;contain a fpherical Cavity of an Inch and a Halfnbsp;�iameter; with a Hole in the lower Part; therenbsp;is a Screw on the Outfide, by which they arenbsp;prefs�d together by means of the Ring I, that hasnbsp;a Screw on the Infidc.

L reprefents all the Parts join�d together j there is a Hole in which has a Communication withnbsp;the Infide of the Mould; thro� this Hole muft gonbsp;a Thread, which lies irregularly in the Clay, almoftnbsp;thro� it. Before you put the Clay into the Mould,nbsp;you muft anoint the Infide of it with Oil *, then,nbsp;when all the Parts have been join�d and prels�d together by ferewing on the Ring, take them afun-der again, and you will find a fmooth and roundnbsp;Clay Ball, to the Thread of which you may faftennbsp;another Thread, and immediately hang it upon thenbsp;iVTachine.

The Experiments, relating to elaftick Bodies, made with Ivory Balls. You muft have fixnbsp;final 1 ones, of an Inch and half Diameter. Be-fides thole, one of double the Weight, anothernbsp;three times the Weight, and a fourth of fournbsp;^imes the Weight.

In the I ith Experiment of the following Chapter, the fix equal Balls above-mentionednbsp;hung on to the Machine at the fime Time, fonbsp;to touch one another. And this is done (feenbsp;Z) by means of the Plate mn^ which is fix�dnbsp;the Machine by help of the Screws y y, whichnbsp;8*^ thro� the Slit st. This Plate contains four Pins,nbsp;�P�Ai/gt;,/gt;, in whofe Ends are Holes, thro�whichnbsp;the Threads pafs that carry the Balls. The Threadsnbsp;brought to a proper Length, and ftaid by thenbsp;F 4nbsp;nbsp;nbsp;nbsp;Keys

7i nbsp;nbsp;nbsp;Mathematical Elements Book 1.

Keys /, /, /, /. The two other Balls hang from the two Pins V, already defcribed.

To make the I�th Experiment of the next Chapter, there mull be three fuch Pins as V.

In this Machine the PerculTIon of the Balls, in the lowed Part of it, is always direftj and thenbsp;Balls (whether you let them go from differentnbsp;Heights, tlie fime Way or contrary Ways) will

and fo in that Cafe thePercuffion is always direft; the Celerities at the Bottom are mark�d by thenbsp;'59 Divifions of the Rulers EG, EG; ^ for in Arcsnbsp;no greater than fuch as the Balls defcribe in thisnbsp;Machine, the Ratio between the Arcs and Chordsnbsp;does not feiifibly differ. The Heights, fromnbsp;which the Balls are let fall, determine the Celerities before the Stroke; and the Heights, tonbsp;which the Balls rife, their Celerities after thenbsp;Stroke.

All that relates to the Percuffion of Bodies not eladic, may be referr�d to the four followingnbsp;Cafes.

* nbsp;nbsp;nbsp;�68 tion in the fame DireSlion as the firfl Motion ; * andnbsp;'^9 the ^mntity of Motion^ in the two Bodies^ will he the

For the Adtion of the Body in Motion, upon the other, communicates to it all the Motionnbsp;that it acquires; now the Re-adfion of this ladnbsp;in the fird retards its Motion; and as Adlion

of Motion, acquir�d by one Body, is equal to the Chiantity of Motion lod by the other; and fo thenbsp;Qi^iantity of Motion is not changed by the Stroke.

This

Book I. of Natural ThiloJb^hj. nbsp;nbsp;nbsp;73

This Quantity of Motion is found by multiplying the Mafs of the firft Body by its Velocity;

* and dividing that Quantity by the Mafs of both * 64 Bodies, you will have the Velocity after thenbsp;Stroke.nbsp;nbsp;nbsp;nbsp;/

For Example, take two equal Bodies, in each of which the Qtiantity of Matter may be ex-ptefs�d by One; let the Velocity of the movingnbsp;Body be Ten, the Quantity of Motion will alionbsp;be Ten, which muft be divided by Two, the Mafsnbsp;of both Bodies, and Five, the Quotient ot thenbsp;Divilion, will be the Celerity of the Bodies afternbsp;the Stroke.

Experiment i.] Take the two foft Clay Balls P and Q^, and hang them upon the Machine ofnbsp;Numb. 170. See Plate X. Fig. i.

Let fill the Ball P from the Height anfwcrable to the tenth Divilion of the Ruler EG, fo thatnbsp;it may Ifrike againft the Ball Q, which is at rell;nbsp;after the Stroke they will both move together,nbsp;and rife up on the other Side to the fifth Divifionnbsp;of the other Ruler EG: The reft of the Experiments in this Chapter are made with the fame Sortnbsp;of Balls.

Cafe 1.] If one Body ftrikes another that movea 172, the fame Waj, but flo'wer.y they quot;ooill both continuenbsp;their Motion in the fame Direction as before; and thenbsp;^antity of Motion.^ after the Stroke., will be thenbsp;fame as before.

In this Cafe the Celerity of the Bodies, after the Stroke, is determined by multiplying eachnbsp;Body by its Celerity, the Products of whichnbsp;i^ultiplications w'ill give the Qiiantitics of Motion in each Body; * by collecting them into one * 6 nbsp;3nbsp;nbsp;nbsp;nbsp;Sum,

74 nbsp;nbsp;nbsp;Mathematical Elements Book I.

Sum, you have the Quantity of the whole Motion gt; which if you divide by the Mafs of both Bodies, the Quotient will be the Celerity required.

Experiment 2,.] Take the equal Bodies P and Q_ {^Plate'X.. Fig. z.) and let them go towards thenbsp;fame Side, P with the Velocity lo, and Q^withnbsp;the Velocity 6 �, as the Mafs of each Body is i, the

* nbsp;nbsp;nbsp;Quantity of Motion in both together will be l�j*nbsp;which if you divide by 2, the Mafs of both Bodies, the Quotient will be 8; and the Experiment will Ihew the Velocity to be anfwerable tonbsp;this.

Cafe 3.] IFhen two Bodies.^ with equal ^anti-^ ties of Motion.) are carried tozvards contrary Sides^ the whole Motion will he deftroyed hy their meetingsnbsp;and the Bodies will be at reft.

* 168 The Bodies are not feparated after the Stroke, * and the Line in which they move cannot be

*169 changed i * but that they may continue to move in the fame Line, it is required that one Motionnbsp;fhould overcome the other, which implies a Con-

Experiment 5. Plate X. Fig. 3.] Let two equal Bodies P and Q^fall from contrary Sides withnbsp;equal Velocities, and as foon as they meet theynbsp;will be at reft.

17^ Cafe 4.] 'Two Bodies moved with different Felo-cities contrariwifc) after having (truck one another, will both together continue their Motion in the famenbsp;Diredlion, towards that Side where there is mojinbsp;Motion-) and the Quantity of Motion, after theirnbsp;meeting, is equal to their Difference of Motidn beforenbsp;the Stroke.

Book i. of Natural ^hilofofhy.

The greateft Motion overpowers j therefore � the Bodies mull be carried together the Way thatnbsp;the Motion is direftedj* and a Body, which *168nbsp;has a lefs Quantity of Motion, is carried in thenbsp;fame Line (but in a contrary Direftion) as before the Stroke 3 for this is required, that by thenbsp;Aftion of one Body, the whole Motion of thenbsp;other be deftroyed, which cannot be done, unlefsnbsp;that Body by the Re-adion lofes an equal Quantity of Motion} there remains therefore only thenbsp;Difference of the Motions.

Multiplying the Mafs of each Body by its Celerity, we have the Quantities of Motion} the lead of which muft be fubftradled from the greaternbsp;to have the Difference of the Motions} whichnbsp;Difference, if it be divided by the Mafs of bothnbsp;Bodies, will give the Celerity after the meetingnbsp;of the Bodies.

Esepsriment 4. Plats 10. Fig. 4.] Let the Body Qbe moved with the Celerity 14, and an equalnbsp;Body P in a contrary Direftion with the Celeritynbsp;6} after meeting, the Body Q^continues its Motion, and carries along with it the Body P withnbsp;the Celerity 4.

Becaufe of the Equality of the Bodies, the Quantities of Motion will alfo be 14 and 6}^nbsp;^nd their Difference is 8} which Number beingnbsp;divided by 2,, the Mafs of both Bodies, the Quotient 4 will be the Celerity after the Stroke.

call Relative Celerity, that 'with which one Pody is carried towards another.^ or ivith which twonbsp;Eodies are feparated} in Motions diredled the famenbsp;^ay, it is the Difference of the Celerities of thenbsp;Bodies} and, in contrary Motions, it is the Sumnbsp;�f the Celerities.

rS nbsp;nbsp;nbsp;Mathematical Elements Book t.

I ~j6 In the Congrefs of Bodies^ the Stroke is proportional to that Relative Celerity. For the Force of Bodies,nbsp;ftriking againd each other, is increafed or dimi-niihed, according to the Celerity with which twonbsp;Bodies come towards hnc another.

CHAP. XXT.

An Elaflic Body, whofe Figure is changed by any Force, will, when the Aftion ofnbsp;that Force ceafes, by its Eladicity or Spring, re-turn to its fird Figure.*

J Body has perfedl Eladicity, ivhen it returns to its firJi Figure.^ 'with the fame Force 'with 'which itnbsp;'was prefs'd in.

In that Cafe, the Stroke., arifing from the Refii-tution of the Spring., is equal to the Stroke by which the Figure of the Body was alter'd.

In this Chapter we fuppofe this Sort of Eladicity, tho� we know no Bodies perlcftly eladic; in different Bodies, the Force by which the Partsnbsp;return to their former Figure is very unequal,nbsp;for which Reafon we can give general Rules only,nbsp;concerning perfect Eladicity; the nearer Bodiesnbsp;approach this Eladicity, the more exactly willnbsp;their Motion agree with thefe Rules.

The Experiments, that we fhall mention in this Chapter, are to be made with the fame Machine * that the Experiments of the lad Chapternbsp;were made with; and here we are to ufe Ivorynbsp;Balls, fuch as are mention�d in the Defcriptionnbsp;of the Machine; for the Want of perfeft Eladicity, and the Reddance of the Air, do not makenbsp;a fenfiblc Error in the Experiments; which alfo,nbsp;.nbsp;nbsp;nbsp;nbsp;when

Book I. of Natural Thilofophy. nbsp;nbsp;nbsp;77

v^hen Necefficy requires, may be corre�led by de-tei mining the DifFerencc arifing from it.

Which Way foever two Bodies ftrike againft each other, the mutual Aftions of the one againftnbsp;the other are always equal.* By that Adionnbsp;the Parts of Bodies are punt�d inwards, and thatnbsp;tvith equal Force in both Bodies; by their Elafti-city alfo they return with equal Force to thenbsp;fird; Figure. The Adion of Bodies upon eachnbsp;other, from their Reftitution by their Spring, isnbsp;equal to the firft Adion from the Stroke ;* *'7^nbsp;whence it follows, that ihe Action of Bodies upon lypnbsp;each other is double in elaflic ones; that is^ double innbsp;refpedl of each Body conftder'dftngly^ becaufe of thenbsp;Equality of the Adrion in each. �TheChange therefore, which in that Cafe is produced in the Motion ofnbsp;each Body by the Stroke, is double that which thenbsp;Stroke would by the fame Motion produce in Bodiesnbsp;that have no Elaflicity, and as, in refped of thefcnbsp;Bodies, the Change (both in refped: to the ^an-tity of Motion, and in refped to the Celerity) isnbsp;determined in the foregoing Chapter; we may alfo determine what tlie Change will be in thofe,nbsp;that is, in claftic Bodies: In which the followingnbsp;�Rules are to be obferv�d.

RULE I.

IVljcn Bodies that are not elaflic flrike againji jgo each other^ if one Body acquires a certain ^tantitynbsp;of Motion.^ it would require twice as much., if thenbsp;Bodies were elaflic, and this double ^antity is tonbsp;be added to the firft Motion, in order to deterrainenbsp;the Motion after the Stroke.

TFhen two Bodies that are -not elaflic flrike ag.iinfl 181 each otherflf one Body lofes a certain ^antity of Mo-^^on, it would lofe twice as much, if the Bodies were

Mathematical Elements Book I.

elafiic; and that double ^antity muft be fubftraUed from the firfl Motion^ in order to determine the Motion after the Stroke.

What is faid of Motion muft alfo be underftood of Velocity i becaufe in the fame Body the Mo-*63 tion is proportionable to tite Velocity.*

Experiment i. Elate yi. Fig.'p.f Let the Body P, whofe Mafs is z, and Celerity p, ftrike againftnbsp;Q^, a Body at reft, whofe Mafs is i; after thenbsp;Stroke, moves with the Velocity iz, and Pnbsp;continues its Motion with the Velocity 3 j whichnbsp;agrees with the former Rules: For if the Bodiesnbsp;were not elaftic, the Celerities of both after meet-*171 ing would be �5* and fo the Body Q^would ac-*iSo quire 6 Degrees of Velocity, and bynbsp;nbsp;nbsp;nbsp;there

fore it muft acquire i z Degrees; the Body P lofing *iSi 3 Degrees of Velocity, by Rule II.* muft lofe lt;J,nbsp;which if you take from p, the former Velocity,nbsp;there remain 3 Degrees of Velocity.

Experiment z. Plate X. Fig.6.'\ If the Body P, whofe Mafs is z, and Velocity 8, ftrikes thenbsp;Body Q_, whofe Mafs is i, and which is carriednbsp;the fame Way with the Velocity f j after thenbsp;Stroke, the Body moves with the Velocity p,nbsp;and P with the Velocity 6} which again mightnbsp;have been determined by the foregoing Rules.

If the Bodies had not been elaftic, both would have moved after the Stroke with the Celeritynbsp;�173 7:* the Body Q.would acquire z Degrees of Celerity, which, by Rule I. muft be doubled, andnbsp;added to y, the firft Celerity, which gives us p:nbsp;The Body P loft one Degree of Velocity, and, bynbsp;Rule II. it muft lofe z 3 therefore it has 6 left.

RULE

Book I. of Natural Thilofo^hy.

When a Body lofes its whole Motion^ and acquires � Motion the contrary Way^ thoje two Motions mu ftnbsp;he colkhled into one Sum^ in order to have the Motionnbsp;that is lofi.

When the ^antity^ which is to he fuhfiraSled hy Rule II. exceeds the ^antity of Motion before thenbsp;Stroke^ from which it muft he fubftrabied, that wholenbsp;^antity of Motion is dejlroyed^ and what remains^nbsp;(that is, its Difference from that which it fhculdnbsp;have been fubftrafted from) gives the Motion thenbsp;contrary Way.

Experiment 3. PlateY.. Fig. j7\ Let the Body I* ftrike with the Velocity 12 againft another Bo-

Q� which is three times as heavy, and at reft, and it will return with the Velocity 6. In thisnbsp;Cafe, Bodies not claftic would move with the Celerity 3 j therefore the Body P would have loft pnbsp;Degrees of Velocity, but� by Rule II. * it muft *181nbsp;lofe 18} which if you fubftrad from the formernbsp;yelocity 12, you have 6 Degrees the contrarynbsp;^ay, by Rule III.* In this Manner may be de- *^^2nbsp;termined, by the following Experiments^ what isnbsp;laid down in the Rules.

Experment 4.. Plate X. H?-8.] Let the Body P be carried with the Velocity ip, the fame Waynbsp;that is, three times as heavy, and movenbsp;'virh the Velocity 35 after the Stroke the Bodynbsp;returns with the Velocity f.

Experiment y. Plate X. Fig p.] Let the two odies P and come tow-ards one another withnbsp;�^ualQ^iantities of Pvlotionj after the Stroke bothnbsp;^ftl return with the fame Celerities with whichnbsp;came upon each other.

Mathematical Elements Book L

Eitpcrimeut 6. Plate X. Fig. lo.] Let P with the Velocity y, and Q_of triple the Weight, withnbsp;the Velocity ii, move in contrary Direftionsj after the Stroke, Q_ continues its Motion with thenbsp;Celerity 3, and P returns with the Celerity ip.

Experiment 7. Plate X. Fig. 11.] Let the fame Bodies P and Q_be carried in contrary Directions,nbsp;P with the Celerity 16, and CL_with the Celeritynbsp;8 y both will be reflcdled after the Stroke, P withnbsp;20, and Q_with 4 Degrees of Velocity.

All the Cafes of the Percuffions of elallic Bodies /maydje determined by the Rules above-mention�d jnbsp;/ the following remarkable Obfervations are alfo deduced from them.

183 When the Bodies are equals and snove the fame IVay'y they continue their Motions^ interchangingnbsp;their Velocities j if they move contrari-wifey then theynbsp;are reflected from each othery likewife interchangingnbsp;their Velocities.

Cafe I. Plate'll. Fig. i.] Let the Bodies move the fame Way, and let A B be the Velocity ofnbsp;one Body, and B C the Velocity of the other jnbsp;the Velocities here are as the Quantities of Mo-,nbsp;tion.* Let the Line A C be divided into twonbsp;equal Parts at D, and let D ^ be equal to D Bjnbsp;A D or D C gives the Celerity of each Body af-*,.,2 the quot;Stroke, if they are not elallic; *fo thenbsp;Celerity B C is increafed by the Quantity DBinbsp;but as it mult be doubly increafed becaufe of thenbsp;*jgo Elallicity, ^ it will become hC: The Celeritynbsp;A B, in Bodies not elallic, is diminilBed by thenbsp;Qiiantity, DB; but it mull be diminillied bynbsp;double that Qjantity, for the Rcafon above-men-tioned5* and therefore it will now become Ah-

There-

Book t. of Natural Thilofophj. nbsp;nbsp;nbsp;8 i

Therefore the Velocities AB and BC are changed into the Velocities Kh and b C; but A B and ^C, as alfo BC and A^, are equal to one ano-theti

Experiment 8.] Let two equal Bodies P andCX, the firfl; with the Velocity lo, and the other withnbsp;the Velocity �, be carried the fame Way 5 theynbsp;will continue their Motion after the Stroke, interchanging their Velocities: Which alfo agrees withnbsp;the Computation by the foregoing Rules.

Cafe 1. Plate'SA. Fig. i.] Let AB be the Celerity of one Body, C B the Celerity of the other ; let the Difference A C be divided into two equalnbsp;Parts at D, and let A^ be equal to CB. Whennbsp;the Bodies are not elaftic, the Velocity of each ofnbsp;them, after the Stroke towards the fame Side, isnbsp;AD } * therefore the firfl Body has loft theVelo'^ *nbsp;city DB, and the other has loft the whole Velocity CB, and acquired DC the contrary Way jnbsp;therefore the whole Quantity loft is alfo DB; **i8^nbsp;if this Quantity be doubled, it will be b B, thenbsp;Quantity of Celerity loft by both Bodies; '* the � is*nbsp;I^ifference of that Velocity with the Velocity ofnbsp;each Body, does in each Body give a Velocity thenbsp;Contrary Way; * that Difference for the Motion * 18�nbsp;ABis^A, and for the Motion CB is Cb-, butnbsp;C b and A B, as alfo b A and C B are equal to eachnbsp;other.

Experiment p.] If the equal Bodies P, with the Celerity 10, and Q_with the Celerity f, are car-�quot;ied contrariwife, they will both be reflefted afternbsp;the Stroke, interchanging their Velocities.

IFhen an elaftic Body ftrikes another equal Wit ^b^t is at reft, that the Velocities may be changed,nbsp;the percutient Body -will be at reft after the Stroke^

Mathematical Elements Book I.

and the other 'will go on quot;with all the Velocity of the Percutient: Which is confirmed by

Experiment 10.1 Let the Body P, with the Velocity lo, ftrike the Body Q_which is at refij P will be at reft after the Stroke, and (^will go forward with the Velocity lo. And this ferves tonbsp;explain the following

Experiment ii.] i/, Let feveral equal Bodies P, Q, R, S, T, V, [^Plate XI. Fig. 3.) be placednbsp;in the fame Line, and touching one another j ifnbsp;the Body ftrike againft Q_with any Velocity, after the Stroke, P, Q^�R, S, and T, will remainnbsp;at reft, and V only be moved.

zdly.^ Let P and (^move with equal Velocities, fo that Q_may ftrike againft Rj after the Stroke

P, nbsp;nbsp;nbsp;Q,R, and S, will be at reft 5 but T and V willnbsp;move forward together.

^hly and laftly. If P,Q^,R and S be moved at once, fo that S ftrikes T, after the Stroke P and

Q. will be at reft, and R,S,T, V will move together. In general, let the Numbers of Balls benbsp;what it will, how many foever move on beforenbsp;the Stroke, fo many alfo will move off in thenbsp;fame Direftion after the Stroke.

In the firft Cafe, the Body P ( Plate XL Fig. 3.) f *84 ftrikes Q, and then is at reft, * Q^ftrikes R, andnbsp;is alfo at reft after the Stroke gt; and fo it happensnbsp;to the others, till at laft T ftrikes V, which, having' no Obftacle to ftop it, does alone continuenbsp;in Motion.

In the fccond Cafe, the Body Q_ (Plate XI. Fig. 4.) does in the fame Manner drive forwardnbsp;the Body Vj P immediately follows, and ftrikesnbsp;Q, which, on account of the firft Stroke, was

Book I. of Natural Thilofo^hy.

at reft, but now communicates its new Motion forward to T, (in the Manner above-mentioned )nbsp;which is not able to ftrike V, that is already innbsp;Motion i and as the Motions of P and Q^are e-qually fwift, and thofe Bodies follow one anothernbsp;very clofe, there is no fenfible Time betweennbsp;thole two Communications of Motion; which isnbsp;the Reafon that the Bodies V and T are movednbsp;equally fwift, and not leparated from each other.

��he relative or refpeStive Velocity^ v:ith 'which i8j* t'wo elafiic Bodies 'whatever recede from each othernbsp;after the Stroke^ is the fame as the refpehiive Vdo-city 'with 'which the Bodies came againft one another.

If the Bodies were not elaftic , they would jointly continue their Motion 5 * and in that Cafe,nbsp;by the Adfion of the Bodies upon each other, thenbsp;whole refpeftive Celerity, by which they comenbsp;to one another, isdeftroyedj the Adtion from thenbsp;Reftitution of the Spring is equal and contrary,

* and therefore it muft generate the fame refpe- ** 178 �iive Celerity, with which they recede from eachnbsp;pther. Let the Inequality of the Bodies be whatnbsp;It will, nothing is changed thereby, becaufe ofnbsp;the Equality of the firft and fecond Adlion uponnbsp;^achBody.*nbsp;nbsp;nbsp;nbsp;� 126

The ^antity of Adotion towards the fame Side^ 185 the fame W'ay.y is the fame after as before thenbsp;Stroke.

. For nbsp;nbsp;nbsp;Bodiesnbsp;nbsp;nbsp;nbsp;that are notnbsp;nbsp;nbsp;nbsp;elaftic, thisnbsp;nbsp;nbsp;nbsp;Propoli-

*�100 is nbsp;nbsp;nbsp;provednbsp;nbsp;nbsp;nbsp;in allnbsp;nbsp;nbsp;nbsp;Cafes jnbsp;nbsp;nbsp;nbsp;* for whennbsp;nbsp;nbsp;nbsp;the Mo- * tji

ttons do not confpire, the contrary Motion muft 172 0 fubftradted from the Motion one Way, in or- J73nbsp;or to determine the Motion that Way. By thenbsp;^oftitution on account of the Elafticiry, equalnbsp;Quantities are generated towards each Side, **nbsp;y which the Quantity of Motion towards onenbsp;*ue is not changed.nbsp;nbsp;nbsp;nbsp;Thefenbsp;nbsp;nbsp;nbsp;two laft Propofitions

Mathematkal Elements Book I.

187 JVhen a [mall Body flrikes againfi another greater Body^ which is at refl^ the great Body acquires anbsp;greater ^antity of Motion than the fmall one hadnbsp;before the Stroke.

The Quantity of Motion, acquired by the great Body, is double the Quantity which the little onenbsp;would lofe, if the Bodies had no Elafticity j butnbsp;in that Cafe, the little Body would lofe more than

Experiment \z. Plate'KI. Fig.�j.~\ Let the Body P, with the Velocity ly, (which Number alio docs here exprefsthe Quantity of Motion) ftrike the Body Q, which weighs four Times as much,nbsp;and is at reft j the Body Q^acquires the Velocitynbsp;6 j * that is, 24 Degrees of Motion. But thenbsp;Body P returns with p Degrees of Velocity j andnbsp;fo the Quantity of Motion, towards that Waynbsp;� 186 where P was firft direfted, continues to be if.*

The Motion is more increafed in the Body Q, by the Interpofition of a Body of mean Bignefsnbsp;between the Bodies P and Q^

188 Experiment PlaleXl. 8.] Let there be the two Bodies P and Q^before-mentioned, andnbsp;between them the Body R double the Body P j ifnbsp;the Body P, with 18 Degrees of Velocity, comesnbsp;upon R which is at reft, it will communicate to itnbsp;* 171 12 Degrees of Velocity; * with which if this Bodynbsp;^ 80 ftrikes upon Q_^ that is at reft, it will communicate to it the Celerity 8, that is, 3 2 Degrees ofnbsp;Motion; but in this Experiment, becaufe of thenbsp;double Percuftron, the Error arifing from thenbsp;Want of perfed: Elafticity, is more fenfible thannbsp;in the others, where there is but one Percuftioi'*�

Book I. of Natural ^hilofophy.

fhe greater the Number is of unequal Bodies .^ i8p which are interpofed between two Bodies^ if the Maffes always increafe from the firfi to the laft^ fo muchnbsp;the greater will the ^antity of Motion be in thenbsp;greatefl; and it will be the greatefl of all (the Number of Bodies interpofed remaining the fame) whennbsp;the Maffes of all the Bodies increafe in a Geometricalnbsp;Progreffion. ^

Tho� the Quantity of Motion, directed the fame Way in the Congrefs of Bodies, whethernbsp;elaftic or not elaftic, remains the fame, the Quantity of the Motion itfelf does not always remainnbsp;the fame, but is often diminith�d, * and alfo oftennbsp;increafed j * fo that there is no Reafon to fay, thatnbsp;there is always the fame Quantity of Motion innbsp;the World.

CHAP. XXII.

Of compound Motion^ and cblique ^ercujjton.

A Body in Motion may be a�ted upon by a new Force, and driven according to anothernbsp;Direftionj in that Cafe the Change of Motionnbsp;follows the Proportion and Diredrion of thatnbsp;Force: * And as the firft Motion is not deftroy�d * 125nbsp;by that Aftion, from thefe two Motions a thirdnbsp;arifes, according to a new Diredion.

het the Body P {Plate 'Kl. Fig. g.) he driven ipo by any Force., according to the Direction P C, and atnbsp;doe fame time let it be driven by another Force, according to the DireWionVB�-, and let the Celerities,nbsp;urifing from thofe Forces, be as thofe Lines PC,

PB. In order to determine what will happen, let the Parallelogram P B A C be corapleated, by draw-ttig the Lines BA, CA, parallel to the Linesnbsp;above-mention�d j let PA be the Diagonal of

Mathematical Elements Book 1.

that Parallelogram. Let the Body be fuppofed by the firft Force, that is, with the Celerity PC,nbsp;to deferibe the Line P C, and let that whole Linenbsp;be carried along in the Diredion, and with thenbsp;Celerity PB; when that Line is come to ha^ thenbsp;Body will be atj), fo that P b will be to PB, asnbsp;bp to ba, or AB, that is, it will be in the Diagonal PA, and fo always. When the Line PCnbsp;is at BA, the Body will be at Aj therefore/rownbsp;a Motion compounded of the two Motions ahonse-mentionedy there arifes a Motion along the Diagonalnbsp;PA, whofe Celerity is proportional to tha Length of

*53 the Diagonal-, ^ for the Diagonal will be run thro� m the fame Time by the Body P with anbsp;compound Motion,as the Line PB or PC wouldnbsp;have been gone thro� by the fame Body, by onlynbsp;one Motion} that is, aded upon by one of thenbsp;Forces.

To confirm'this Propofition experimentally, we muft make nfe of the following Machine.

}Sgt;i It confifts of two Boards, or wooden Planes, CDE, CDE, (Plate Xll. Fig. i.) of the Figurenbsp;of a right-angled Triangle, whole Side CD is innbsp;Length about 3 Foot and a half, and the Side DEnbsp;about I Foot and a half; thefe Boards are fixed fonbsp;as to move in a vertical Situation about the Hingesnbsp;A and B.

The Experiments upon this Machine are made with Ivory Balls of an Inch and a half Diameter. The Planes are fo joined, that if you conceive two other Planes to run parallel to them,nbsp;at the Diftance of a little more than a Semi-diameter of the Balls; the Lines, in which thefenbsp;imaginary Planes interfeft, fliall be the Axis ofnbsp;their Circumvolution; Which is brought to palsnbsp;by the Contrivance of the Hinges, (Fig. 2-)nbsp;whofe Pares b, b, are let into the Wood to be thenbsp;firmer. In the Center of the upper Hinge A,

Book I. of Natural ^hilofofhy.

there is a fmall Cylinder {Fig. 2.) in whofe Cafe there is a Hole, which meets another in thenbsp;Side, thro� which the Thread ih is to run 5 atnbsp;one End of this Thread a Ball, as P, hangs, andnbsp;the other End is joined to the Key 1. By Help ofnbsp;the Screws F,F,F,F,F, this Machine is fet perpendicular gt; fo as to have the Thread hi hang innbsp;the Axis of the Machine.

At ra, w, there are two Pins fix�d to the two Planes, from which Pins the Balls Q., Q_, hang,nbsp;at fuch a Diftance from the Planes, that they maynbsp;almoft touch them} fo that if you fuppofe anbsp;Line to pafs thro� the Centers of the Balls P andnbsp;Q^, it fhall be parallel to the Plane on that Side:nbsp;Befides, it is required, that, when thofe Ballsnbsp;hang at the fame Height, they lhall touch onenbsp;another.

The Threads which are tied to the Balls go thro� the Holes in the faid Pins, and arc fix�d tonbsp;the Keys //, fo that the Balls may be raifed ornbsp;let down cafily, and have all their Centers broughtnbsp;to the fame Horizontal Plane. There is a Brafsnbsp;Ruler, or graduated Limb R, bent up in the Arcnbsp;pf the Circle, fo as to have the Ball P rife along itnbsp;iti its Motion; and this Limb turns one of itsnbsp;Ends on a Center which is in the Axis of thenbsp;Machine. This Piece of Brafs ferves to ihew tonbsp;�vvhat a Height the Ball P afeends.

Each Ball Q_^, when it fwings, moves along the Plane to which it is applied; and the Height,nbsp;from which it is made to fall, is fliewn by an In-^cx fixed to the Plane; to which End there arenbsp;four Holes in each Plane containing equal An-8ks, in refpeft to the Motion of the Threads.

ftrikes upon the Ball P, and drives it to the Mine Height in the fame Direftion.*nbsp;nbsp;nbsp;nbsp;*

M(^thematical Elements Book 1.

Experiment 1.3 The Horizontal Seftion of this Machine is here repreiented, (P/a/gXlI. Fig. 3.nbsp;and 4.) the Body P may be driven by either ofnbsp;the Bodies Q^, with any Diredtion and Velocity.nbsp;If the Bodies QLand (l_are let fall at the fvmenbsp;time, the Body P has two Motions imprefs�d uponnbsp;*155 it at the fame time,* and therefore runs in thenbsp;Diagonal P p of the Parallelogram made in thenbsp;�15Q Manner above-mentioned,* to eiprefs thofe twonbsp;Motions, and runs up to an Height proportionablenbsp;to the Length of that Diagonal.

The Experiment anfwers very exaftly, whether the Balls Qjand Q^are let fall from the fame Height,nbsp;or from unequal Heights \ and whatever the Anglenbsp;be that is made by the two Planes, that is, by thenbsp;Diredtions of the Motions, whether the Angle benbsp;right, acute, or obtufe.

{Plate XI. Fig. 10.) may always be confider'd as aSled upon by two Motions-y and that as many Waysnbsp;as you pleafe: For you may draw as many different Parallelograms as you pleafe, as P B, A C,nbsp;pbac, pbacy whofe Diagonal is the Line above-mentioned 5 and, in every one of them, if therenbsp;be fuppofed two Forces adfing in the Diredlionsnbsp;PB and PC, from which theCelerities which thenbsp;Body would have are as the Sides P B and P C, anbsp;Motion will always be produced by the Adtion ofnbsp;them both at once, which will give a Celerity proportional to the Diagonal.

From thisR cfolution of Motion into two other Motions, may be determined the Motion of Bodies that ftrike one another obliquely.

and P with the Diredion and Celerity P A, ftrike againft it. When P is come to A, drawnbsp;thro� the Centers of both Bodies the Line D E,

Book I. of Natural ^hilofophy. nbsp;nbsp;nbsp;8^

and then PB perpendicular to it, and compleat the Parallelogram A B P C, the Motion along P Anbsp;is refolved into two others along P B and PC,

Or B A, C A: By the Motion in the Direftion C A, the Body P does not a6t upon the Body Q__}nbsp;the Aftion therefore arifes folely from the Motion in the Diredtion along B A, that is, the Body P, by the oblique Stroke along P A 'with the Celerity P Abatis upon the Body in the fame Man- tP?nbsp;�er, as if it Jhould firike it diredtly along B A 'withnbsp;the Celerity BA. And fo the Motion of the Body Q_from that A�tion, whether the Bodies benbsp;elaftic or not, is determined from what has beennbsp;faid of direft Percuftion.

The Motion of the Body P {Plate XI. Fig. 11. and iz.) after the Stroke, is deduced from thenbsp;fame Principle} the Motion along C A is notnbsp;changed} therefore by that Motion, with an equalnbsp;Celerity, the Body P is carried in the Direftionnbsp;A E. Now let A E be equal to C A} the Changenbsp;in the Motion BA is determined, in refpe�t ofnbsp;the Body P, in the fame Manner as the Motionnbsp;of Q^, by the two foregoing Chapters} let thenbsp;Celerity of that Motion be A D, in Fig. 11. whennbsp;the Body goes forward, and in Fig. iz. whennbsp;it returns back} from that Motion, and the Motion along A E, arifes a compound Motion in thenbsp;diagonal A/�, which, by its Situation and Length,nbsp;denotes the Diredtion and Celerity of the Bodynbsp;P after the Stroke.*

When Bodies are equal and elaftic^ the whole Motion along BA is deftroyed by the Percuffi-on,* and only the Motion along C A is left, andnbsp;the Body P is alfo carried in that Direftion. Innbsp;that Cafe, both the Bodies do always fy from eachnbsp;other in DireSlions that are at Right Angles 'with onenbsp;another'which JVay foever the Body P comes uponnbsp;t.he other Body,

Mathematical Elements Book �.

Experiment z. Plate XII. Fig.' f.] In the Machine deferibed Numb. ipi. let the Ball Q^and P hangi having fet the Planes at Right Angles;nbsp;let the Body Q^with any Direftion, and from anynbsp;Height, come down upon P, and ilrike againilnbsp;it: after the Stroke the Bodies will follow thenbsp;Direftions of the Planes, and rife to Heights,nbsp;which may be determined by what has been faidnbsp;hitherto.

ipy We may alfo the fame Way determine the Motion of two Bodies after the Stroke, when both Bodies are moved, which Way foever they comenbsp;upon one another. The chief Cafes are repre-fented in Plate XIII. and all of them are explainednbsp;exactly the fame Way.

Plate XIII. Fig. i, 2, 3, 4, y, and 6.] Let the Body P be moved with the Diredion and Celerity P A, and the Body Q^, with the Diredionnbsp;and Celerity j draw the Line B whichnbsp;goes through the Centers of both Bodies wherenbsp;they touch one another, and let CA and c a henbsp;drawn perpendicular to the Line above-mentioned, and let the Parallelograms P B A C andnbsp;(XP a e he compleated. The Motion of P is re-folved into two others, of which the Celeritiesnbsp;and Diredions are exprefs�d by C A, B A. Thenbsp;Motions, into which the Motion of Q^is refolved,nbsp;are exprefs�d by ea, ha-, by the Motions alongnbsp;C A and c a the Bodies do not ad upon one another j therefore thefe Motions are not changed,nbsp;and after the Stroke arc exprefs�d by A E and a e,nbsp;which are equal to AC and ac-, the Percuffion,nbsp;from the Motions in the Lines BA ba, is dired,nbsp;and determined in the foregoing Chapters: Letnbsp;the Body P move towards D, and its Celerity benbsp;AD, and the Body Q_move towards d with thenbsp;Celerity a d. After the Stroke therefore, the

Book I. of Natural ^hilofo^hy.

Motion of the Body P is compounded of the Motions along A E and A D, and moved in thenbsp;Diagonal kp. The Motion of the Body Q^,nbsp;after the Stroke, is compounded of the Motionsnbsp;along ac mAad, whence that Body is carriednbsp;in the Diagonal ae-, and the Lengths of thofenbsp;Diagonals exprcfs the Celerities of the Bodies after their Meeting. In the ift, id, and 3d Figuresnbsp;the Bodies are fuppofed not elafticj and innbsp;the 4th, yth, and 6th, the fame Bafes are put,nbsp;fuppofing the Bodies elaftic. There are fomenbsp;Letters wanting in the firft Figure, becaufe thenbsp;Points, which are marked with thofe Letters innbsp;the other Figures, do here coincide with othernbsp;Loints, and are not neceflary for determining thenbsp;Motions.

CHAP. xxm.

The Body P {Plate. XIII. Fig-J.) being driven in the Directions P B and P C, with t-'elerities proportionable to thofe Lines; fromnbsp;thence arifes a Motion along P, the Diagonalnbsp;of the Parallelogram P B A C, with a Celeritynbsp;that is denoted by that Diagonal; * if there be anbsp;third Force aCting along the Line Pa, fo that thenbsp;Celerity arifing from it be PA; by that Afeionnbsp;the Actions of both the faid Forces arc deftroyed,nbsp;^nd the Body comes to reft : If the aforefiidnbsp;�Actions continue, the Body will continue at reft;nbsp;^hich happens when the Body is drawn towardsnbsp;C, B and a, with the faid Forces pulling bynbsp;Threads. Whence it follows. That a Body 'uoillnbsp;at reft, 'which is drawa by three Powers, thatnbsp;to one another^ as the Sides of a Friangle tnade bynbsp;Fines parallel to the Direblions f the Powers.

Mathematical Elements Book I.

This Propoficion is confirmed experimentally, by the Machine reprefented in Plate XIII. Fig. 8.nbsp;It confifts of a round Board of about 8 Inchesnbsp;Diameter, which is in a horizontal Pofition, andnbsp;fullain�d by a Foot.- Round the Edge of it,nbsp;within the Thicknefs of the Wood, is a Groovenbsp;whereby Pullics are applied at Pleafure to anynbsp;Part of the Circumference j fgr each Pulley has anbsp;Brafs Plate perpendicular to it, which fits intonbsp;the Groove, when the Pulley is applied. See thenbsp;Pulley with the Plate reprefented by F.

The Board above-mention�d is a little hollowed in, in the upper Part, fo as to receive a lefs orbicular Board DF A, whofe Thicknefs is about anbsp;Quarter of an Inch, and its Surface rifes a littlenbsp;above the firft Board j fo that a Thread that runsnbsp;over any of the Pullies, being extended horizontally, may juft touch the faid Surface.

You muft have feveral of the lefier round Boards, for making different Experiments. Theynbsp;have Paper parted upon them on both Sides, thatnbsp;the Lines (to be mentioned hereafter) may be thenbsp;more eafity drawn upon them.

Experiment i.] Let C be the Center of the fmall Board, and let there be drawn upon it thenbsp;Triangle ABC, whofe Sides are to one anothernbsp;as z, 5, and 4: Let CE be parallel to the Sidenbsp;A B of the 'I'riangle, and let the Side A C be continued towards D.

^ Now if there be three Threads joined together at C, and rtretched over the Pullies fallened tonbsp;the greater Board, fo as to be in the Lines CD,nbsp;CE, and CB*, if to the Thread CD you hangnbsp;4 Pounds, to C E ^; and lartly, but z to thenbsp;Thread CF, the Threads will not be mov�d, andnbsp;the Knot remains over Cj but if it be mov�d outnbsp;of that Point, it will not be at reft.

Book I. of Natural �Philofo^hy,

In this Propofition any two Powers are balanced by a third, that is, a�i but as onej which afts contrariwife in the Direftion of that third.nbsp;Therefore the AUiom of two Powers may be reducednbsp;to the A�ion of one.

So, when a Point is drawn by four Powers, there will be an ^Equilibrium, if reducing twonbsp;Powers to one, this new Power, with the othernbsp;two remaining, be in the Pofition of Numb. ip6}nbsp;that is, if thofe remaining Powers being alfo reduced to one, the Power ariGng thence be equalnbsp;Vi^ith, and afts contrary to the new Power mentioned.

Experiment i. Plate XIII. Fig. p.^ The Point C is drawn by four Threads j towards B by thenbsp;Weight of I Pound, towards F by 3 Pounds,nbsp;towards E by 2, Pounds} and lallly, towardsnbsp;� by 4 Pounds} and this produces an .Equilibrium. Having drawn the Triangle CFlt;?, or thenbsp;Parallelogram CF 0 E, the abovclaid Powers,nbsp;drawing in C F and C E, are reduced to one thatnbsp;afts in the Direftion Ca] with the Force of 4nbsp;Pounds} and then the three Powers, drawing innbsp;the Lines CB, CD, Ca, give us the Cafe ofnbsp;Numb. ip(S: And therefore if the Powers, drawing along C B and C D, be reduced to one drawing along C A, it will aft in the fame Direftion,nbsp;but pull againft the Power pulling in C and benbsp;equal to it.

What is here faid of the four Powers, might be faid of five or more} for of five, if two benbsp;reduced to one, we come to the lall mentionednbsp;Cafe.

Experiment 3. Ptoe XIV. Fig. i.] The Point 2,00 C is drawn by f Powers, pulling by the Threadsnbsp;CA, CB, CD, CE, andCF} the Powers arenbsp;3nbsp;nbsp;nbsp;nbsp;to

94 nbsp;nbsp;nbsp;Mathematical Elements Book I.

are to one another as the Weight by which the Threads are drawn, and they have the famenbsp;Proportion to one another as the Numbers thatnbsp;you fee at the Pullies in the Figure, and younbsp;have an ^Equilibrium.

The Powers, drawing in CE and CF, may be reduced to one afting in CH; which brings us tonbsp;the Cafe of Numb. ip5. Laftly, thofe two newnbsp;Powers, drawing in C H and C G, are reduced tonbsp;one afting in Ca., which are equal to the fifthnbsp;drawing along CA, and pulls in the fame Line,nbsp;but contrariwife.

ioi Befides this, we deduced from the Propofition mentioned Numb. i6p, that the fame Thing maynbsp;be faid of the ASlion of the Po*wer^ which has beennbsp;fiid concerning Motion in the foregoing Chap-

�192 ter j * namely, that it may be refolved into the A5ti~ ons of two other Powers, and that in numberlefsnbsp;Manners, becaufe Triangles of numberlefs Kindsnbsp;may be made, tho� you keep one Side Hill the fame.nbsp;Thus in all Engines we can reduce a Power, thatnbsp;a6ts obliquely, to a diredt one and can determinenbsp;the Proportion between a direft and an obliquenbsp;one: Which will appear by the following Examples, that are confirmed by Experiments,

^2, Experiment 4. Plate^lY.Fig. 2 and 3.] To the Lever A B, whofe Brachia are equal, apply at Bnbsp;the Weight P of two Pounds, and at A a powernbsp;aiSing obliquely in the Dire�lion A D, and whichnbsp;is reprefented by the Weight M. If you imaginenbsp;a Line, as D E, parallel to the Lever in a horizontal pofition, and AE perpendicular to that Lever;nbsp;and if AD be to A E as 3 to 2, and the Weightnbsp;M be of three pounds, there will be an jEquili-b'ium.

Book I. of Natural Thilofofhy. nbsp;nbsp;nbsp;95

The Direction of the Motion of the Point A, by the Motion of the Lever, is perpendicular tonbsp;the Lever, therefore it a�ts in the Line EA produced. As the Diftance B A always remains thenbsp;fame, in the fecond Figure A is hindered fromnbsp;coming towards B, and as it were repelled in thenbsp;Diredlion B Aj in Fig. 3. the Point A is hinderednbsp;from receding from B, and fo A is as it were drawnnbsp;towards B. Befides, the Point A is by M drawnnbsp;towards Dj Therefore that Point is drawn by threenbsp;Powers, whofe Direftions are parallel to the Sidesnbsp;of the Triangle AEDj which therefore, to produce an Equilibrium, mult be to one another asnbsp;thofe Sides.

The Point A, by reafon of the Equality of the Diftances* of the Points A and B fromnbsp;the Fulcrum, moving along EA, is drawn withnbsp;the fame Force as P dcfcends, that is, with thenbsp;Force of two Pounds 5 the Force therefore alongnbsp;A D mull; be of three Pounds, becaufe the Sidesnbsp;A D and A E are to one another as 3 to 2.

The Side DE exprefles what the Fulcrum fuftains by the Force with which the Point Anbsp;tn Fig. 2. is pulhed towards B, or is drawnnbsp;from it m Fig. 3.

Experimentlt;) .Plate'KlY.Fig.il^P\ Letthe Weight 203 fixed to a Pulley, be fuitained by Powersnbsp;Applied on both Sides to the running Rope, butnbsp;drawing obliquely in the Diredlions C A and

B; thefe Powers are equal to one another, be-^aufe no Part of the Rope, that goes about the Fulley, can be at rell, unlels it be equally drawnnbsp;On both Sides. * The Weight P is as it were a *83nbsp;third Power, and fo the Point C is drawn by �

95 nbsp;nbsp;nbsp;Mathematical Elements Book L

three Powers j fuppofe the Line C E perpendicular to the Horizon, and the l,ine A E parallel to CB: If CE be to AE or AC (for thefe twonbsp;Lines are equal, becaufe of the above-itientioncdnbsp;Equality of the Powers drawing along C Bjnbsp;*196 CAj *) as � to f, the Weight P of 6 Poundsnbsp;will be fuftained by the Weight Q_and Q_ofnbsp;f Pounds eachj the Reafon of which is evidentnbsp;hy Prop. 196.

If one End of the running Rope is faftened to a Pin, the Weight P will be fuftained by only onenbsp;of the Weights

Experiment6.Plate'KYV.Fig. y.~\ If the Weight P be not joined to the Pulley, but fuftained bynbsp;the Rope CA and CB faftened �to it, it maynbsp;be fuftained by two unequal Powers. Draw thenbsp;Triangle C A E, as was done in the foregoingnbsp;Experiments, and let AE be ii, and CA iij,nbsp;and CE iz j you ftiall have an iEquilibrium, ifnbsp;the Weights Q_and Q_are to P, as the firft Numbers to the laft} the Reafon of which Experimentnbsp;is alfo evident from Numb. ip�.

^0^ Here we are to obferve, that from the given ^ Inclinations of the Threads G A and C B to thenbsp;Horizon, the Proportion of the Weight (^andnbsp;Q_to the Weight P may be determined by Triquot;nbsp;gonometrical Tables. If in the Triangle ACEnbsp;you conceive a Line Ae drawn thro� the Pointnbsp;A parallel to the Horizon, and that Line benbsp;taken for a Radius of a Circle, C A will be thenbsp;Secant, and eC the Tangent of the Angle whichnbsp;C A makes with the Horizon, and A E willnbsp;be the Secant, and eE the Tangent of the Anglenbsp;of Inclination of the Thread CB to the Horizon: VV'quot;hence it appears that the Weights Q,nbsp;are proportional to the faid Secants, and that the

Book I. of Natural Thilofo^hy, nbsp;nbsp;nbsp;97

On the Machine with which thefe laft Experiments are made, (the Make of which the Figure alone fufficiently exprefTes, efpecially if it be compared with the 4th Fig. of PlateW.) draw Linesnbsp;along which the Threads that go over the Pulliesnbsp;may be ftretched \ in the Middle of the Linesnbsp;write down the Numbers, which exprefs the Secants of the Angles which thofe Lines make withnbsp;the Horizon 5 and, at the Ends of the Lines,nbsp;write down the Numbers expreffing the Tangentsnbsp;of thefe Angles.

Now in every Cafe where there is an AEquili-brium, the Weights Q^and C^are as the Numbers in the Middle of the Lines along which the Threads are ftretched j and the Weight P as thenbsp;Sum of the Numbers at the Ends of thofe twonbsp;Lines.

Experimenty. Plate'KW. Fig.6.~\ For this Ex- toS perimenc we niuft make ufe of the Machine ofnbsp;Numb. 14^. FlateVW. Fig.y. The Body M, being laid upon an inclined Plane A B, is fuftainednbsp;by a Power drawing along MS; let M R be anbsp;Line perpendicular to the Horizon, and ASRnbsp;perpendicular to the Surface of the Plane; in e-very Cafe where the Weight P is to the Weightnbsp;of the Body M, as MS to MR, the Body willnbsp;be at reft.

The Body M by its own Weight is drawn in the Liredtions R M, by the inclined Plane it isnbsp;fuftained in aDirc�fion perpendicular to the Plane,nbsp;and fo that Experiment is reduced to the Propo-fition of Nimib. ip�.

Experiments. PlateXIV. Fig-y.J TheBrachia zoy of the Lever A C B are equal, and form fuch an

Mathematical Elements Book I.

Angle, that if A C be continued towards D, and BD be drawn perpendicular to CD, DC fliallnbsp;be the Half of BC or CA. At A hang onenbsp;Pound /), and at B the two Pound Weight P; thennbsp;fetting theBrachiumC A in a horizontal Pofition,nbsp;105 you will have an ^Equilibrium j * becaufe thenbsp;Weight P hangs, as it would do upon a ftreightnbsp;Lever hanging at the Point D.

Change the Weights, and let the greater hang at A, and the lelTer be laid upon the Brachiumnbsp;BC at B i {Plate^iX^. Fig. 8.) if by a verticalnbsp;Plane you hinder this laft Weight from falling off,nbsp;you will again have an iEquilibrium.

TheBrachia of the Lever are equal, and by the Motion of the Lever move equally ; therefore, bynbsp;the Force of the Weight P, the Weight p is as itnbsp;were drawn towards E, in the Dire�tion perpendicular to the Brachium B C gt; by the Aftion ofnbsp;the vertical Plane, that Weight is puih�d horizontally j and at laft is pufti�d vertically by the Forcenbsp;of Gravity. Therefore the Weight/) is drawn bynbsp;three Powers, which are to one another as thenbsp;196 Sides of the Triangle B E D,* Therefore thenbsp;Force tending towards the Earth, (that is, thenbsp;Weight p to the Force drawing towards E, namely,nbsp;the Weight P) is as B D to B E, or D C to C Bnbsp;or CAj that is, as i to 2. Which is alfo thenbsp;Ratio between the Weights/) and P. And herenbsp;therefore the Reafon of the Experiment is deduced from the often mentioned Propofition of Numb.nbsp;ipd} to which all other Cafes imaginable, relating to oblique Powers, alfo belong.

CHAP.

CHAP. XXIV.

A Body , moved by two Impreffions, has a zo8 Motion compounded of both �, * if a Body � 190nbsp;be projeBed^ or thrown in the Line AB, (PlateX.V.

Fig. I.) in the Time in which it could run the Length AB, it is by the Force of Gravity carried towards the Center of the Earth the Lengthnbsp;BF, and fo, by a Motion compounded of both,nbsp;it is moved in AF j and by that Motion thenbsp;fecond Moment it would run through F C, equalnbsp;to A F, if that fecond Moment it was not by thenbsp;Force of Gravity carried in CG, fo that thenbsp;Motion in the fecond Moment is in F G: Afternbsp;the lame Manner, the Motion in the third Moment is in G H, and the fourth Moment in HI jnbsp;but as Gravity afts continually, thofe Momentsnbsp;of Time are to be look�d upon as infinitely fmall, and fo you will every where have anbsp;Motion compounded in different Diredlionsj thatnbsp;is, an Infledtion of Direction in the Body�s Motion J in that Cafe therefore it md move in a Carvenbsp;Line.

Lhis Motion of a projedled Body., or Projc�tile, 109 may be confider�d more fimply in all Projcdlionsnbsp;''^hich we make i becaufe all Lines, which tend to-quot;^ards the Center of the Earth, may be look�dnbsp;Upon as parallel, and the Direction from that Motion is always the faraej when the proje�lile Mb-tion is made up of two Motions., the firft equable innbsp;^he Line of the ProjeStion^ and the fecond acceleratednbsp;towards the Earth.*

Let a Body be projected in the Line A D, pa-mllel to the Horizon j in equal Times, by that H znbsp;nbsp;nbsp;nbsp;Motion

too nbsp;nbsp;nbsp;Mathematical Elements Book I.

Motion, it will run thro� the equal Spaces A B, B C, C D; By Gravity it will, in a Motion per-ptndicular to the Horizon, be carried in the Di-reftion B F, C G, or D H, which here are ilip-pofed parallel; this Motion is accelerated, andnbsp;therefore if after the firft Moment the Body benbsp;at the Point F, after the fecond it will be at G,nbsp;after the third at Hj fo that if you call BFnbsp;* *31 one, CG will be four, and DH nine. * The Body will run in a Curve, which goes through allnbsp;the Points that may be determined in the famenbsp;Manner as F, G, H, and that Line is called a P�-rahola.

The Machine made ufe of, for proving thisPro-pofition experimentally, is made of three Parts, as may be feen inP/^/eXV. Fig.^. kh ii 6 Inchesnbsp;high, DE is exaftly of the fame Height: Thenbsp;Length H is of 12 Inches, fuppofing the Pointnbsp;H to be diftant i Inch from the End of the Cavity in which it is taken.

Let E A be hollowed circularly j or in any o-ther Curve i and let this hollow Channel be overlaid with a Plate of very fmooth Tin or Brals, that a Brals Ball may freely roll down itj butnbsp;Care mull: be taken that the lower Part of thenbsp;Curve, at A, fhall have a horizontal Dire�lion,nbsp;that the Ball may quit it in that Dire�iion.

When, to this firll Part of the Machine, you add the fecond B, (PtoeXV. Fig.^.) it reachesnbsp;to^, and^G is 8 Inches long: If upon this younbsp;lay the third Part C, this lall reaches to/, and/Fnbsp;is of 4 Inches.

The Diameter of the Ball P, which in making the Experiments is to be let fall along the Curvenbsp;E A, is of about half an Inch: neither mull a

■f

Book I. of Natural Thilofofhy.

Ball, or a bigger Machine than what is here mentioned, be made ufe ofj for the lefs the Bodies are, and the fwifter their Motion, the more innbsp;Proportion is the Motion retarded by the Air�snbsp;Refiftanccj as fhall be fhewn in its Place.

When the Ball P is let fall from E, running down the Curve EA, it acquires fuch a Degreenbsp;of Velocity, as appears to be always the fame innbsp;feyeral Trials; and with that Velocity and horizontal Direction, it continues its Motion.

ExperimentHaving joined together the three Parts of the Machine, as in Fig.^. let go the Ballnbsp;P from E, and it will llrikc the Point F. Takenbsp;away the leaftPart C, and let the Ball comedownnbsp;as before, and it will Itrike G. Latlly, take awaynbsp;the Part B, and the Ball, defeending as before, willnbsp;ftrike againfl; H.

If you ftick on a Piece of fofc Clay upon H and G, the Point of the Stroke will be exactlynbsp;mark�d i this will not do fo well in the Point F,nbsp;becaufe of the great Obliquity of the Motionnbsp;there; but, by repeating the Experiment, thenbsp;Point F will be well enough determined, by Sightnbsp;only.

The Propofition of Numb, r 34 may be experi- ; mentally confirmed by this Machine; for, its wenbsp;have already laid, the Ball running down E A willnbsp;ftrike the Point H.

Coming down E A, it acquires a Celerity which it could have acquired in falling in E D ; * with 'nbsp;that Celerity it is horizontally projedted from thenbsp;Point A, and it moves that Length h H equablynbsp;according to that Direftion, whilft by its Fall itnbsp;goes thro� A h equal to E D; but h H is doublenbsp;the Length A� or E D.

loz nbsp;nbsp;nbsp;Mathematical Elements Book I.

ziz What has been faid of the Curve, deferibed by a Body projefted horizontally, may alfo beappliednbsp;to any Projeftion.

Let the Body be projected in the DireSion A E, {Plate XV. Fig. f.) and let AB, BC, CD,nbsp;D E, be equal j the Body will go through thenbsp;Curve AFGHI, fo that BF, C G, D H, El,nbsp;will be to one another, as i, 4, p, and 16j innbsp;which Cafe the Curve is alfo called a Parabola.

Z13 Let AI be drawn horizontal, and the Curve a-bove-mentioned will cut it it I j A I is called the Amplitude of the Proje�tion.

The Motions of Bodies projefted with the fame Celerity, but different Dire�fions, may be compared together.

Let A L be the Height, to which a Body^ thrown up with a determinate Degree of Celerity,nbsp;may rife : Let the Body with the fame Celerity be thrown along A B, cutting in B the Semicircle deferibed in the Diameter ALj let ABnbsp;exprefs that Celerity, and M B be parallel to thenbsp;Horizon. The Motion in A B may be refolvednbsp;*192 into two others j * the firfl: along M B, a horizontal, and the fecond along A M, a vertical Line jnbsp;and it is only by that fecond Impreflion that thenbsp;Body afeends: The Height therefore, to whichnbsp;the Body afeends in that Cafe, is to the Height tonbsp;which it would afeend with the Celerity A B, asnbsp;�,38 the Square of A M to the Square of A B j* thatnbsp;is, as xA M to AL ; but this is the Height to whichnbsp;the Body afeends with the Celerity of the Proje-(flion; therefore alfo A M is the greateft Heightnbsp;to which the Body comes in that Projeftion. Innbsp;the Time of the Afcent in A M, the Body might

Book I. of Natural Thilofifhy.

by an uniform Motion, with the fame Celerity quot;With which it moves in A M, in a horizontalnbsp;Motion go thro� twice the Length of the Linenbsp;MB; and as the Time of the Fall is equal tonbsp;the Time of the Afcent,* the Amplitude AB isnbsp;four times the Length of the fame Line M B.nbsp;Now this Demonftration will ferve, whatever thenbsp;Inclination of the Direftion of AB is. Whencenbsp;we deduce,

I. �that the Amplitude is the greatefi, �l�th the -^4-fame Celerity^ uehen the Angle of the ProjeUion is a Half Right Angle. For then the Line mb^ being anbsp;Radius of the Semicircle, is the greatell of all.

z. Except this Cafe.^ there are always two Indi-'f^ations.y that give the fame Amplitudefor if thro�

B;, B/^ be drawn parallel to AL, cutting the Sc-tnicircle at and m b parallel to the Horizon, this Line will be equal toMB ; therefore the Amplitude of 'the Projeftion, in the Direftion A Z-,nbsp;will alfo be A I. In the fecond Parc of the following Book, all this will be confirm�d by Experiments.

If the Celerity be changed, and the Body pro- zip je�ied in the fame Direction, the Amplitude isnbsp;changed, in the fameRatio as the Diameter A f,;nbsp;that is, the Amplitudes., the Direblion remaining thenbsp;fume., are as the Heights to which Bodies, withnbsp;the fame Celerities, being thrown up, may afeend;nbsp;and therefore they are as the Squares of the Celerities.

CHAP.

CHAP. XXV.

A Body in Motion continues its Motion in a Right Line,* and does not recede fromnbsp;it, unlefs a new Impulfe a�s upon it j after fuchnbsp;an Impulfe the Motion is compound, and fonbsp;from the two there arifes a third Motion in anbsp;*190 Right Lincalfo. * If therefore a Body is movednbsp;in a Curve, it receives a new Impulfe every Moment i for a Curve cannot be reduced to Rightnbsp;Lines, unlefs you conceive it divided into Parts infinitely fmall. We have an Example of that Mo-*208 tion in the Projedtion of heavy Bodies j* and another in all Motions round a Point as a Center.nbsp;zi6 If a Body^ that is continually driven towards anbsp;Center., he projeSled in a Line that does not go thro'nbsp;that Center.^ it �will defcribe a Curve-, and., in allnbsp;the Points of it, it endeavours to recede from thatnbsp;Curve, according to the Direhlien of a Curvature jnbsp;that is, of a Pangent to the Curve fo that if thenbsp;Force driving towards the Center ibould immediately ceaie to aft, the Body would continue itsnbsp;Motion in a Right Line along the Tangent.

A Stone whirl�d round in a Sling defcribes a Curve, becaufe the Sling does every Moment, asnbsp;it were, draw it back towards the Hand; but, ifnbsp;you let the Stone go, it will fly out in the Tangentnbsp;of the Curve.

2,18 Phe Force 'with 'which a Body in the Cafe ahove-mentioned endeavours to fly from the Center, fuch as the Force by which the Sling in Motion isnbsp;Ihetch�d, is call'd a Centrifugal Force.

Book I. of Natural Thilofo^hy.

But the Force, by 'which a Body is dra'wn or im- zip fell'd to'wards that Center, is call'd a centripetalnbsp;Force.

In all Cafes, the centrifugal and centripetal Forces zzi are equal to one another-, for they aft in contrarynbsp;Direftions, and deftroy one another.

The whirl�d Sling is equally ftretch�d both Ways,* and the Stone endeavours to recede fromnbsp;the Hand with as much Force as it is drawn towards it.

Central Forces are of great Ufe in Natural Philofophy; for all the Planets move in Orbits,nbsp;and moft of them, if not all, turn upon theirnbsp;Axes.

I {hall chufe out the chief Propolitions relating to thefe Forces, and confirm them by Experiments; but firft we muft defcribe the Machinesnbsp;with which thefe Experiments are performed.

Plate XVI. Big. i. and 2.] A is a round Board 1 or Table of z Feet and a half Diameter; whofenbsp;vertical Seftion is feen in Fig. 2. in which a a re-prefents the Seftion of the Table itfelf, and ghnbsp;the Seftion of its turn�d Foot, which is joinednbsp;perpendicularly to its Center; this Foot or Supporter of It confifts of two Pieces feparatcd at D,nbsp;which are fix�d together by four fmall Irons,nbsp;whofe Ends are rivetted to Rings of the fiimenbsp;Metal.

The upper Part of the Foot has a Groove round it at cc; and has a cylindric Hole thro� it atnbsp;f g of three Quarters of an Inch Bore.

io6 nbsp;nbsp;nbsp;Mathematical Elements Book L

The Frame of the whole Machine, reprefented at C, is very folid j and one Side receives the Footnbsp;of the Table A, which palTes freely thro� an Holenbsp;in the upper Parc of the Frame Cj to which isnbsp;firmly joined the wooden Collar F of Fig. 5. whichnbsp;fits in the faid upper Part of the Fr^ie.

The Table with its Foot bears upon the croft Piece ED, which has a Plate of Iron to receivenbsp;the Brafs Center h. This tranfverfe Piece is fix�dnbsp;butjuft above the Feet of the Frame, that, whennbsp;the Table-Foot is let down upon it, the Groovenbsp;cc may be but juft above the wooden Collar jnbsp;eg the Top of which are ferewed down two Ironnbsp;Plates RR {Fig. 3.) by four Screws, fuch asnbsp;j r. In this Polition the Table will very freelynbsp;move horizontally about its Center j and, that itnbsp;may move the more eaftly, there is flipped on upon the Foot clofe to the Table (where it is notnbsp;round, but fix or eight Square) a fmall Wheel ornbsp;Pulley, whofe Se�ion you fee in hb.^ and whichnbsp;is joined to the Table by means of the Screws f/,nbsp;el. There muft be three other fuch Wheels whofenbsp;Circumferences, taking them at the Bottom ofnbsp;their Grooves, are to one another as one, two,nbsp;three i the leaft of all the Wheels is of about ynbsp;Inches Diameter.

Another Table B, made juft like the firft, is to be whirl�d about round its Center in the oppofitenbsp;Part of the Frame C. Tho� there is a fmall Dil-fcrence between them j for in this the lower Partnbsp;of the Foot has an Hole thro� it as well as thenbsp;upper, (fee i h Fig. 4.) yet it is turn�d freely,nbsp;having fix�d to its Bottom a Brafs Plate with anbsp;Hole in it to receive the little Pipe of anothernbsp;Plate M, whoft vertical Se6tion is feen at L,nbsp;and which is fixed to E D, the other croft Piecenbsp;of the Frame, at the Place m. This croft Piece

muft

Book I. of Natural Thilofophy. nbsp;nbsp;nbsp;107

mult be bored thro� to anfwer to the Hole of the PI ite, in fuch Manner that a Thread may gonbsp;from the Top of the Table quite thro� the wholenbsp;Foot and the Piece ED- Such a Wheel ^ b bnbsp;is made fad to the Table immediately under it,nbsp;and is in Bignefs juft equal to the leaft of thofenbsp;which are made to take olF and on, belongingnbsp;to the Table A.

The two Tables A and B may be whirl�d, very fwiftly, either feparately, or both together, by help of the great vertical Wheel Qj For performing of which, you muft make ufe of thenbsp;Machine of Fig, y. which is a wooden Plane, ornbsp;ftrong flat Board, to which is perpendicularlynbsp;fix�d a Parallelopiped, in whofe upper Surfacenbsp;arc vertically fixed the two Pullies v v, at thenbsp;Ends, and fideways at one End there is anothernbsp;Pulley, as r, which is horizontal. The Surface,nbsp;when the Machine is applied to the Frame, isnbsp;in the fame Plane with the fmall Wheels of thenbsp;Tables.

If B alone is to be whirl�d, the Piece of Fig. f. is to be fix�d to the Frame C, by help of twonbsp;Screws going thro� fuch Holes as x in the lowernbsp;Part of the Piece, which muft be fo fixed, in re-fpeft to the little Wheel of the Table, as is re-prefcnted in Fig. 6. where b reprefents that littlenbsp;Wheel: The Rope goes round the great Wheelnbsp;Qj and from its lower Part goes from d towards't',nbsp;goes round the Wheel and againit the Pulley ?,nbsp;towards r, and fo comes back to the upper Part ofnbsp;the Wheel

The feventh Figure reprefents the Pofition of the Machine or Piece of Fig. y. when both Tables are to be whirl�d round at once. A Sight ofnbsp;the Figure fliews the Way of the Rope, whichnbsp;fi'om V goes down to and fo to the lower Partnbsp;of the great Wheel.

io8 nbsp;nbsp;nbsp;Mathematical Elements Book I.

Befides this, in feveral Experiments you muft make ufe of long Boxes or Troughs IF, IF,nbsp;which are laid upon the Tables, and fix�d to themnbsp;with Screws i the Center of every one of thefenbsp;Boxes lies juft over the Center of the Tablenbsp;where there is a Hole equal to the Hole �,nbsp;{Fig. z.) and exaftly anfwering to it, in whichnbsp;Hole of the Box is thruft a wooden Cylinder N,nbsp;{Fig. z.) as may be feen at G; thro� the Middle ofnbsp;this Cylinder goes a little Glafs Tube of about anbsp;Quarter of an Inch Bore, whofe Ends are thicken�d at the Flame of a Lamp, fo as to make thenbsp;Hole fomething lefs and fmoothi that aThread ornbsp;fmall String may run up and down thro� it, without any fenfible Friftion.

One of the Troughs holds a Ball, tied to a Thread which goes thro� the above-mentionednbsp;Tube, and is alfo fattened by a Screw to thenbsp;Weight O, {Fig. 7.) which lies in D the Separation of the Foot. This Weight is laid upon thenbsp;lower Half of the Foot, from whence it is raifednbsp;up, as the Ball recedes from the Center of thenbsp;Trough.

This Weight is a round Plate of Lead, and of about z Inches Diameter; it has a Brals Cylindernbsp;fix�d to its Center, whofe upper Part, in order tonbsp;receive the String, is cleft into two Parts, whichnbsp;are drawn together by means of a Screw: thisnbsp;Plate of Lead with its Cylinder weighs half anbsp;Pound; and there muft be two fuch Weights.

There are feveral other Weights, fome of half, fome of a quarter of a Pound, reprefented by P,nbsp;{Fig. 7.) which are to be laid upon thefaid Weightnbsp;O; that one may at Pleafure vary the Weight tonbsp;be raifed by the Ball.

Book I. of Natural Thilofofhy,

Plane^ and fo defcribe a Circle-y if the centripetal Force^ by which the Body is every Moment drawn ornbsp;impelled towards the Center^ Jhould ceafe to aSt^ andnbsp;the Plane Jhould continue to move with the fame Celerity j the Body will begin to recede from the Centernbsp;{in Refpe6i of the Plane) in a Line which pajfesnbsp;thro' the Center.

Experiment i.] Take a Ball which is tied to a Thread, the other End of the Thread beingnbsp;faftened to the Center of one of the Tubes Anbsp;or B, and lay it on the Table, which muft benbsp;whirl�d fingly fo long, till the Ball is carriednbsp;round with it j here the Ball is at reft, in refpeftnbsp;of the Table, and in that Situation it is retainednbsp;only by the String faftened to the Center *, therefore it fuffers no Jmprcftion in that Plane, exceptnbsp;that by which the String is ftretched, that is,nbsp;whofe Dire�tion paffes thro� the Center of thenbsp;round Table; and fo, if it be left to itfelf, it cannot the firft Moment move in any other Direftionnbsp;in that Plane.

When a Body moves about a Center^ if in its Mo-tion it comes nearer to the Centerits Motion is accelerated ; but on the contrary retarded^ if it recedes from the Center.

Experiment 2.] Let the Trough F I, through whofe Center the Cylinder G, with its GlafsTube,nbsp;is fixed into the Center of the Table B, be faftened to the faid Table.

Let the Ball L, tied to a Thread, be laid in the Trough, and the Thread put thro� the Tube a-bove-mentioned, as alfo thro� the whole Foot ofnbsp;the Table, and the crofs Piece at Bottom that fu-ftains the Foot, and then with your Hand holdnbsp;the End of the Thread.

no nbsp;nbsp;nbsp;Mathematical Elements Book I.

Let the Table be turned round, and you will obfcrve, that, during that Motion, the B dl willnbsp;apply itlelf to one Side of the Trough, and isnbsp;carried round fo as to move with the lame Velocity as the Trough. Let the Thread be pull�d,nbsp;fo as to bring the Ball nearer the Center, andnbsp;it will immediately ftrike the oppofite Side of thenbsp;Trough, becaufe it moves falter than the Trough.nbsp;If you bring your Hand nearer to the Foot ofnbsp;the Table, lb as to git^e more String, the Ballnbsp;recedes from the Center, and ftrikes the firll: Sidenbsp;of the Trough, as moving more flowly than thenbsp;Trough.

This Acceleration when a Body approaches nearer to the Center, and Retardation when itnbsp;recedes from it, is determined by Geometricians: If a Body, for Example, which is drivennbsp;towards the Center C {Flats Fig. ii.) benbsp;moved in the Curve A E, it will move falter at Enbsp;and flower at A: Draw the Lines AC, BC, andnbsp;EC, DC, fo that the Areas ABC and DECnbsp;may be equal to one another, the Parts A B andnbsp;DE of the Curve are defcribed in equal Times bynbsp;the Body j and therefore, a Body that is retainednbsp;^ in a Cur've, by a Force tending towards a Center, isnbsp;faid to defcribe round that Center Areas proportionable to the Times.

zz6 The inverfe Propofition is alfo demonftrated, namely. That a Body which is moved in any Curvenbsp;in a Plane., and defcribes about any Point Areas proportionable to the Times, is turned out of the rightnbsp;Line, and urged by a Force tending to that Point.

227 The greater the ^antity of Matter in any Body is, the greater is its centrifugal Force-, which arilesnbsp;from a greater Quantity of Motion.

2z8 Liquors of different Denfities be included in a determinate Space, fo that the heavier cannot recede from the Center, unlefs the lighter

Book I. ef Natural Thikfophy. nbsp;nbsp;nbsp;in

Come towards it} and they be fo difpofed, that by their Weight the heavy Fluid comes to the Center j upon moving the Whole about that Center,nbsp;the light Fluid will come towards the Center, andnbsp;the heavy one fly from the Center.

If a Solid be included with a Liquid in a determinate Space, the fame may be faid, as was faid of the two Liquids: If it be lighter than the Liquid,nbsp;it will come towards the Center j if heavier, itnbsp;will recede from it. All which ariles from thenbsp;great Centrifugal Force in the heavier Body.

Experiment 5. Plate XVI. Fig. 8.] Take four Glals Tubes of about one Inch Diameter each,nbsp;and a Foot long, and having hermetically lealednbsp;them, let them be firmly tied to an inclined Plane.

In the firfl:, youmufthaveQuickfiivcr and Water} in the fecond. Oil of Tartar per deliquium., andnbsp;Spirit of Wine; and in the third Water with a Leaden Bullet} and laftly, in the fourth, Water withnbsp;a Piece of Cork} and all of them muft be aboutnbsp;Iialf empty.

This inclined Plane muft be faftened to the 'vhirling Table A or B {Plate XVI. Fig. i.) fonbsp;that the lower Parc of the Plane may come almoftnbsp;to the Center of the Table, by means of twonbsp;Screws, one of which goes thro� x {Fig. 8.) Leenbsp;the Table be whirled round, and immediatelynbsp;the lower Part of the Tubes will remain empty,nbsp;and the heavier Bodies will go to that Part of thenbsp;Tube which is fartheftfrora the Center} theCorknbsp;defeends and ftrikes to the lower Part of the Water, whilft the Leaden Bullet goes to the Top ofnbsp;the Tube.

Central Forces not only differ on account oi2,1^ the Quantity of Matter, but the Diftance docsnbsp;alfo caufe a Change, and likcwife the Celeritynbsp;'vith ^hich the Body is moved round; There is

112 nbsp;nbsp;nbsp;Mathematical Elements Book I.

nothing elfe that can make any Difference in chefe Forces: and thefe are all the Things to be con-fidered, when we compare them together.

2,20 The Periodical Time is the itim, in which a iQody going round a Center performs one whole Revolution ; that is, if it defcribes a Curve that returnsnbsp;into itfelf, the Time elapfed between its Departurenbsp;from, and Return to a Point: If the Curve doesnbsp;not return into itfelf, inftead of a Point we muftnbsp;take a Line paffing thro� its Center.

Zji The Periodical Time depends upon the Celerity of the Bodyj and therefore, in comparing central Forces, it muft be taken for the Velocity.

When the Periodical �Times are equal, and the Difiances from the Center are alfo equal-, the central Forces are as the ^antities of Matterdn the revolving Bodies.

ExperimentPlate'X.'Vl. Fig. i.] Of the three Wheels or Pullies bb, mentioned in the Deferip-tion of the Machines, apply the leaft r j the Tablenbsp;A j fo that if the two Tables A and B be whirlednbsp;at the fame Time by the Motion of the Wheel Q_,nbsp;they may run round in equal Times ^ to each ofnbsp;them fix the long Troughs IF, IFj and the Cylinders GG, that contain Glafs Tubes, muft benbsp;thruft thro� the Center Holes of the Troughsnbsp;quite into the Feet of the Tables.

Put a Ball L of half a Pound into the Trough of the Table B, and a Ball L of one Pound in thenbsp;Trough of the Table A: Threads tied to theB.allsnbsp;go thro� the little Tubes GG, and are faftened tonbsp;Weights placed in the Separations or Hollows ofnbsp;the Feet of the Tables, in fuch Manner, that thenbsp;Diftances of the Bails from the Center, when thenbsp;Threads are ftretched and the Weights not

Book I. of Natural Thilofo^hy. nbsp;nbsp;nbsp;i ij

raifed may be equal v now if the Weight in the Separation or Hollow of the Foot of A be onenbsp;Pound, that in the Separation of B muft be halfnbsp;a Pound j or if this lalt (hould be one Pound, thenbsp;other muft be two Pounds.

Let the Wheel Q_^be turn�d round fitfter and fafter, till by the centrifugal Force of the Ballsnbsp;the Weights above*mentioned be raifed, and bothnbsp;Weights will be lifted tip precifely at the fxmenbsp;Time; therefore Weights, that are as the Bodies,nbsp;tvill, ceteris paribus, be overcome by the centrifugal Force.

When the ^antities of Matter in the re�VoIving ^5^ l^odies are equal, and the periodical�times alfo equal,nbsp;the Forces are at the Dijiances from the Center.

Experiment f. Plate XVI. Fig. i.j This Expe-timent is made in the fame Manner as the foregoing; inftead of a Ball of half a Pound, put in the Trongh of the Table B a Ball equal to thenbsp;other that is of one Pound. Let the Diftancesnbsp;from the Center be taken, in any Proportion; ifnbsp;the Weights joined to the Balls in the famenbsp;Proportion, and the Wheel Q_be moved fafternbsp;and fafter, you will fee the two Weights rifenbsp;exadlly at the fxme Time. As, for Example, ifnbsp;the Diftance of the Ball upon A be of 12 Inches,nbsp;and the Weight joined to it of i Pound and anbsp;half; and the Diftance of the other Ball of 8 Inches,nbsp;and the Weight joined to it of i Pound, the Experiment will lucceed.

When the periodical times are equal, hut the Di- 2-35 fiances and the ^lantities of Matter in the revolvingnbsp;Bodies differ, the central Forces are in a Ratio corn-pounded of the Quantities of Matter and the D/-fiances-, which follows from the two laft Propoft-tions. To determine that compound Ratio, thenbsp;Quantity of Matter in each Body muft be multi-

Experiment 6.] If in the lafl: Experiment the Ball upon B be changed, and you place a Ballnbsp;of half a Pound at 8 Inches from the Center, andnbsp;you alfo change the Weight joined to it, and halfnbsp;a Pound be tiled inftead of a Pound j the Experiment will, alfo then fucceed, and the Weightsnbsp;will begin to rife at the fame time. If you multiply the half Pound Ball by its Diftance of 8nbsp;Inches, the Produdt is 4, and multiplying thenbsp;I Pound Ball by 12 Inches, its Diftance from thenbsp;Center, the Produft is ii-, which Produfts arenbsp;to one another as i to 3 j that is, as the Weightnbsp;of half a Pound to that of i Pound and a half,nbsp;which are in this Experiment both lifted up at thenbsp;fame Moment.

The Differences of central Forces arifing from the different Diftances from the Center, and thenbsp;different Quantities of Matter may compenfatenbsp;2,^one another; for fuppofmg the Quantities of Mat-ter in the revolving Bodies to be in an inverfe Rationbsp;of the Diftances from the Center^ the central Forcesnbsp;voill be equal, as much as one Force is greater idnbsp;refpe�: of the Qiiantity of Matter, fo much doesnbsp;the other exceed it by reafon of its greater DPnbsp;fiance.

Experiment 7.] Let a Ball of half a Pound be placed at the Diftance of 14 Inches, and a Bali ofnbsp;one Pound at the Diftance of 7 Inches; every thingnbsp;elfe being as in the foregoing Experiment; if thenbsp;Weights in the Foot or Spindle of each Table benbsp;alike, they will rife at the fame Moment.

There is a Cafe of this Propofition, �when t�W0 Bodies joined by a Ehread revolve about their commonnbsp;2nbsp;nbsp;nbsp;nbsp;Centcf

Book I, of Natural Th�lofophy. nbsp;nbsp;nbsp;115'

Center of Gravity. For the Diflances from that Center are in an inverfe Ratio of the Weights ofnbsp;the Bodies,* and therefore the central Forces are *95,90nbsp;equal. By the Force by which one Body endeavours to recede from theCenter, the other is drawnnbsp;towards it j and by reafon of the Equality of thenbsp;Forces, they retain one another., and continue theirnbsp;Motion-, if they revolve about any other Point,nbsp;they do not continue their Motion j and the Bodynbsp;'vhofe centrifugal Force overpowers, recedes fromnbsp;the Center, and carries the other Body alongnbsp;With it.

Experiment 6. Plate Fig. 10.] Let two Unequal Bodies P andQ_be joined by a Thread, innbsp;which you mull; mark tire Point C, which is thenbsp;Common Center of Gravity of thofe Bodies, whennbsp;the Thread is ftretch�d.

In this Experiment you mull ufe but one Table, and fix upon it a long Trough that reaches beyond the Diameter of the Table both Ways,

3nd whofe middle Point is over the Center of the Table. In this Trough you muft place the Bo-tlies above-mentioned, and the Thread that joinsnbsp;them being Ifretched, the Point C mull be put innbsp;the Middle of the Trough. When the Table isnbsp;Whirl�d round, the Bodies are carried round withnbsp;It, and remain at reft in it. If the Point C benbsp;temoved from the Middle of the Box, uponnbsp;'Whirling theTable, both Bodies will be carried tonbsp;that End of the Box which the Point C was placednbsp;ueareft to.

The Difference of the central Forces is alio determined from the Difference of the periodicalnbsp;Fime.

tVben the ^lantities of Matter in the Bodies 2,55 quot;thirled round, and the Difiances from the Center arenbsp;the central Forces are in an inverfe Ratio of

I z nbsp;nbsp;nbsp;the

xi6 nbsp;nbsp;nbsp;Mathematical Elements Book I.

the Squares of the periodical Times: That is, di-rc6tly as the Squares of the Revolution made in the lame Time.

Experiment 9. Plate quot;KVl. Fig. i-H To the Table A apply a Wheel or Pulley, fuch as b {Fig. 2.) whofe Circumference is double the Circumferencenbsp;of the Wheel which is fix�d to the Table B 5 fonbsp;that when tbe two Tables are whirl�d both together, B Ihall go round twice for A once j that is,nbsp;the periodical Time of that fhall be double thenbsp;periodical Time of this.

In each Trough I L, I L, lay a Ball of one Pound at equal Diftances from the Center. Thenbsp;Ball laid on the Table B muft be tied to twonbsp;Pounds in the Foot, and the other Ball upon Anbsp;is joined by its Thread to half a Pound in thenbsp;Foot. Upon whirling the Tables, both Weightsnbsp;will rife at the fame Time: Which Weights arenbsp;here as i to 4, the periodical Times being as 2 tonbsp;I, whofe Squares are reciprocally as i to 4.

However the central Forces differ from one another.^ they may, according to what has been faid, be compared to one another gt; for they are alwaysnbsp;in a Ratio compounded of the Ratio of the ^anti-ties of Matter in the revolving Bodies.^ * and thenbsp;Ratio of the Diftances from the Center, * and alfonbsp;an inverfe Ratio of the Squares of the periodicalnbsp;TmesF If you multiply the Quantity of Matternbsp;in each Body by its Diftance from the Center, andnbsp;divide the Produdt by the Square of the periodicalnbsp;Time, the Quotients of the Divifion will be tonbsp;one another in the faid compound Retio, that is,nbsp;as the central Forces.

Experiment 10.3 Every thing being prepared as in rhe former Experiment, fet a Ball of half anbsp;Pound, at the Diltance of 8 Inches from thenbsp;5nbsp;nbsp;nbsp;nbsp;Center

*•'. •

- •lt;.-.•

y, • gt;»■»• nbsp;nbsp;nbsp;^ •«

.- nbsp;nbsp;nbsp;■nbsp;nbsp;nbsp;nbsp;. Inbsp;nbsp;nbsp;nbsp;' ^

r,.:.- » nbsp;nbsp;nbsp;• ' ■’

Book I. of Natural f^hilofophy.

Center of the Table B, and let it by the Thread be joined to one Pound in the Foot, let anothernbsp;Ball of two Pound be placed at the Difiance ofnbsp;iz Inches from the Center of the Table A, andnbsp;joined with a Weight of 3 Quarters of a Pound inbsp;whirl the Tables, and the Weights will be raifednbsp;jufl at the fame Time.

Here the Bodies are as J to i j the Diflances as 8 to iZj the Squares of the periodical Timesnbsp;at I to 4i multiplying one Half by 8, and dividing the Produft by I, the Qiiotient of the Di-vifion is 4j multiplying i by iz, and dividingnbsp;the Produdl by 4, the Quotient is 3. Thereforenbsp;the central Forces are to one another as 4 to 3,nbsp;which Ratio alfo the Weights in the Feet have tonbsp;one another.

When the ^{untities of Matter are equal., the 2,28 Diflances themfelves mufl be divided by the Squaresnbsp;of the periodical Dmes^ to determine the Proportionnbsp;of the central Forces,

In that Cafe, if the Squares of the periodical Times be to one another as the Cubes of the Diflances.,nbsp;the Quotient of the Diviflons, as well as the central Porces,will be in an inverfe Ratio of the Squaresnbsp;of the Diflances.

Experiment ii.j Let the periodical Times of A and B be as i to 2, in the fame Manner as innbsp;the two laft Experiments. Take two equal Balls,nbsp;and let the Diflance from the Center on B be i onbsp;Inches, and the other Ball�s Diflance from thenbsp;' Center be of 16 Inches: To the Thread of thenbsp;firfl, faflen one Pound and a Quarter, and to thenbsp;Thread of the other faflen half a Pound in thenbsp;Hollow of the Foot A j whirling the Tables, thenbsp;Weights will rife the fame Moment.

In that Experiment the central Forces are as f to z, which you alfo find by Calculation.�^ *238

Mathematical Elements Book I.

This Ratio diflFsrs very little irom the inverfe Ratio of the Squares of the Diftances, which arenbsp;to one another as 200 to 5'i2i the Cubes of thenbsp;Diftances are alfo almoft as the Squares of the periodical Times: Thcfe Squares are as i to 4, andnbsp;thofe Cubes as izy to yi?-, which Ratios do notnbsp;much dift'er. If you take other Numbers, thefenbsp;Ratios will be exactly the fame, and the Experiment will fuccced in the fame Manner; but it isnbsp;not eai'v in the Experiment to vary the periodical Times or the Weights in what Ratio younbsp;pleafe.

When the Force, by which a Body is carried towards a Point, is not every where the fame, but is either increafed or diminifhed in Proportion tonbsp;the Diftance from the Center, feveral Curves willnbsp;thence arife in a certain Proportion.

If the Force decreafes in an inverfe Ratio of the Squares of the Diftances from that Point, thenbsp;Body will deferibe an EUipfis, which is an ovalnbsp;Curve, in which there are two Points called thenbsp;Foci^ and the Point towards which the Force isnbsp;diredled falls into one of them; So that in everynbsp;Revolution the Body once approaches to, and oncenbsp;recedes from it. The Circle alfo belongs to thatnbsp;Sort of Curves, and fo in that Cafe the Body maynbsp;alfo deferibe a Circle: The Body may alfo (bynbsp;fuppofing a greater Celerity in it) deferibe thenbsp;two remaining Conic Se�lions, wz. the Parabola^nbsp;or Hyperbola^ Curves, which do not return intonbsp;themfelves.

On the contrary, if the Force increafes with the Diftance, and that in the Ratio of the Diftancenbsp;itfelf, the Body will again deferibe an El-�ipf�-, but the Point, to which the Force is di-redted, is the Center of the Ellipfe, and the Bodynbsp;in each Revolution will twice approach to, andnbsp;again, twice recede from that Point. In this

Book I. of Natural Thilofophy.

Experment 12..] Hang up a leaden Ball with a long Thread ; if the Ball be drawn back from itsnbsp;Point of reft, it is alw.ays carried towards it by itsnbsp;Gravity, and from equal Sides with equal f'orcc, ifnbsp;the Diftance be equal. The Ball in its Motionnbsp;deferibes an Arc of a Circle, which Way foever itnbsp;falls when you let it go: If thofe Arcs are not verynbsp;great, they coincide with a Cyr/wV, and the Force,nbsp;with which the Ball inanyPointis carried tow ardsnbsp;the loweft Point, is as its Diftance from that Point;

Let the Ball be pull�d back from the loweft Point, and projected obliquely ; then it will dc-feribe an oval Curveabout that Point, which (whennbsp;the Ball does not runout to a great Diftance) willnbsp;hardly differ at all from an Ellipfe, becaufe of thenbsp;Proportion of the Forces, and becaufe in thatnbsp;Cafe the Ball does fenfibly move in the famenbsp;Place.

The Center of the Ellipfe is the Point in ''^hich the Ball is at reft when it is not projected ;nbsp;^nd in every Revolution the Ball does twice approach to, and twice recede from it. If the Ballnbsp;hangs over a Table fo as alraoft to touch it whennbsp;ft is at reft, and the Point, over which it is, benbsp;rnark�d upon the Table, the Experiment will become more fenfible if you draw an Oval upon thenbsp;Table with Chalk, by following the Body withnbsp;your Hand.

If the Proportion (mentioned Niinib. 241, and , 2-42.) of the Forces by which a Point is drivennbsp;t^owards a Center, be a little changed, the Bodynbsp;'vill no longer deferibe an Ellipfe; but fuch anbsp;Curve as may be reduced to an Ellipfe, by fup-

120 nbsp;nbsp;nbsp;Mathematical Elements Book I,

pofing the Plane in which the Body moves agitated by fome Motion, which therefore will make the Ellipfis moveable.

Experiment 13.] Every Thin^ being as in the former Experiment, let the BallBe fo thrown thatnbsp;it may run out to a greater Dillance j and then itnbsp;will deferibe a Curve which may be referr�d to thenbsp;moveable Oval: it will indeed twice in every Revolution come towards the Center, and twice recede from it j but the Place of the Points in whichnbsp;it is leaft, or molt diltanc, is changed every Revolution, and thefa Points are always carried thenbsp;fiime Way, their Motion confpiring with the Motion of the Ball.

r nbsp;nbsp;nbsp;CHAP. XXVI.

Of the Laws of Elaficity.

WE have already Ihewn what FJapicity is, and whence it arilesj * and what is itsnbsp;Effe�t in the Congrefs of Bodies, whether theynbsp;llrike one another directly or obliquely gt; whatnbsp;remains is to examine the Laws of Elapicity itfelf,nbsp;which we fhall do from Phtenomena.

All Bodies, in which we obferve Elafticiiyt conlill of fmall Threads or Filaments, or at leaftnbsp;may be conceived as conllfting of fuch Threads gt;nbsp;and it may be fuppoled that thofc 1'hreads laidnbsp;together make up the Bodyj therefore that wenbsp;may examine Elafliciiy in the Cafe which is thenbsp;leaft complex, we muft conftder Strings of mu-lical Inftruments, and fuch as are of Metal i fornbsp;Catgut-Strings have a fpiral Twift, and cannot benbsp;confider�d in the fame Manner as thofe Fibres ofnbsp;which Bodies are form�d.

^. The Elafticity of Fibres confifts in this, that they ^ can be extended., and taking away the Force by which

they

Book I. of Natural Thilofo^hy. nbsp;nbsp;nbsp;121

Fibres have no Elafticity, unlefs they are extended 2,^y with a certain Force; as it apppears in Strings thatnbsp;have their Ends fix�d without being ftretch�dj fornbsp;if you remove them a little from their Pofition,nbsp;they do not return to it: but what the Degree ofnbsp;Tenfion is, which gives Beginning to Elajlicity^nbsp;is not yet determined by Experiments.

JVhen a Fibre is extended with too much Force^ it t-iS lofes its Elafticity �, and this Degree of Tenfion isnbsp;alfo unknown this we know, that the Degree ofnbsp;Tenfion in Fibres, which conftitutes Elafticity, isnbsp;confined to certain Limits.

Hence appears the Dilference of Bodies that arc 2 j p elaftic, and fuch as are not foj why a Body lofesnbsp;its Elafticity, and how a Body dellitute of Elafticity acquires that Property. A Plate of Metal,nbsp;by repeated Blows of a Hammer, becomes elaltic,nbsp;and being heated, does again lofe that Virtue.

Between the Limits of Tenfion, that terminate Elafticity, there is a different Force required for different Degrees of Tenfion, in or to ftretchnbsp;Chords to certain Lengths: What this Proportionnbsp;is, muft be determined by Experiments, whichnbsp;mufl be made with Chords of Metal, as was faidnbsp;before. But as thefe Wires are fcarce: fenfiblynbsp;lengthen�d, the Proportions of the Lengtheningnbsp;cannot be diredtly mcafured j therefore they muffnbsp;be meafured by another Method.

Let AB {Plate XVII. Fig. i.) be a fmall Wire ftretch�d horizontally with a certain Force, whofenbsp;Ends are fix�d at A and B: Let it be bent by anbsp;Weight hanging in the Middle of it, fo that itnbsp;may come to the Pofition ACB.

Mathematical Elements Book I.

Z48 '�he Line C c drawn from the middle Point of a String or Chord after its Inflexion^ to the middlenbsp;Point of the fame when it was in its natural State,nbsp;is call'd the ScXgitfi. (^Arrozv) of the Chord.

Let ce be an Arc of a Circle deferibed about the Center B, with the Radius Bf. Half thenbsp;Chord or Wire by the Inflexion was ftrctch�d thenbsp;Length Ce, which Quantity has a certain Relationnbsp;to the Sagitta Ce.

The Weight alfo, by which theString is ftretch�d, has a certain Relation to the Force with whichnbsp;the Fibre is lengthen�d, that is drawn along BC;nbsp;and fo in fevcral Experiments by comparing thenbsp;Sagitta Ce, and the Weights with which thenbsp;Chords are infle�lcd, the Proportions of thenbsp;Lengthenings are determined j as will be fhewnnbsp;in the following Experiments.

249 The Machine for performing them is a vertical Board, about 3 Foot long, and i Foot high.nbsp;See Plate XVII. Fig. 2.

The Rulers of Wood mn, mn, are fix�d to the Board like a Moulding, and carry two Prifmsnbsp;H,H, made like a Wedge, which Hide along uponnbsp;the Rulers, and are fix�d any where upon them bynbsp;means of Screws, which hold them behind thenbsp;Board, their Shanks being moved backwards andnbsp;forwards, by means of a Slit in the Board.

Between A and B there are equal Divifions reckon�d from the Middle on either Side, in order to determine the Places where to fix thenbsp;Prifms.

At O there is a Groove, to hold the Pulley T in the Side of the Board ; which Pulley is repre-fented in Plate XIIL Fig. 8.

The Wire, with which the Experiments are made, is fix�d at one End of the Ruler m n, and

Book I. of Natural Thilofophy. nbsp;nbsp;nbsp;123

at the other End goes over the Pulley T, the Weight P llrctching it, and the Prifras, H, H,nbsp;fuftaining it in Points, which are equally diftancnbsp;from the Middle of the Machine.

There is a Brafs Plate de let into the Middle of the Board, and mark�d with very fmall Divi-fions, along which moves another Brafs Plate ornbsp;Indexwhich hangs upon the Wire, having anbsp;Hole thro� which it runs: This Index has a Scalenbsp;hanging from it, which, together with the Indexnbsp;fg^ weighs juit an Ounce. The Length ol thenbsp;Wire is determined in each Experiment by thenbsp;Diftancc of the Edge of the Prifm.*; H, H; lor innbsp;the fmall Inflexions made by hanging on Weightsnbsp;in C, concerning which alone Experiments arcnbsp;made, the String is not moved upon the Prifras,nbsp;nor is the Weight P raifed up, but only the Parcnbsp;AB is extended by thefe Inflections.

In the Inflexions of the String, the Sagittae are meafured by the Diviflons on the Plate ed-, fornbsp;the End g of the Index gc docs always defeendnbsp;equally with the Point C in every Inflexion.

Experiment i.] Let P be a two Pound Weight, zyo and letthe WirebeinfleXed at C with the Weightnbsp;of an Ounce, that is, with the Weight of thenbsp;Scale and Index fg-, and obferve the Diviflon ofnbsp;the Plate cd^ to which the End g of the Indexnbsp;f g defeends. Change the Weight P to q. Pound.s,nbsp;and alfo double the Weight by which the Stringnbsp;is infieXed, that g may defeend to the lame Diviflon, and this Weight will be two Ounces: Threenbsp;Ounces will give the fame Inflexions, when thenbsp;Weight P is of flx Pounds.

From this Experiment it follows, that the zyi fVdght^ by 'which a Fibre is increafeda certain Lengthnbsp;by its jlretching^ is in the different degree of Eepfmt^nbsp;the Eenfton itfelf-j if, for Example, tliere be

three

Mathematical Elements Book I.

three Fibres of the feme Kind, Length andThiek-nefs, whofe Tenfions are as i, z and 3; any Weights in the feme Proportion will equally tiretchnbsp;thofe Fibres.

�the kafi Lengthenings of the fame Fibres are to one another nearly as the Forces by �which the Fibresnbsp;are lengthened. As for Example, let a Fibre benbsp;feretched with the Weight of 100 Ounces, if itnbsp;be feparately lengthen�d with the Weights of inbsp;Ounce, 2 Ounces, and 3 Ounces, the Lengthenings will be nearly as I, z, and 35 that is, eachnbsp;Ounce fuperadded does equally lengthen the Fibre:nbsp;For the Tenlions by the Weights of too, loi,nbsp;and loz Ounces, by which the Fibre is ftretch�dnbsp;in each Cafe, when an Ounce is fuperadded, donbsp;not fenfibly differ from each other.

The Property of fubres may be applied to their Inflexion, and is of great Ufe. Let thenbsp;Wire AB [Platenbsp;nbsp;nbsp;nbsp;Fig. 3.) be fo infleded,

as to acquire the Pofitions A c B, A r B, and ACB, yet fo that in the greateft Inflexion thenbsp;Sagitta may not be ~ Inch long, fuppofing thenbsp;Wire z Feet and a Halfj In thofe Cafes thenbsp;I-engthenings of the String are very fmall, therefore they are in the Ratio' of the Forces thatnbsp;2^1 produce them, * and they ferve to exprefs themnbsp;let cD exprefs the Force by which a String isnbsp;ftretch�d when it is not inflefted, and with thenbsp;Center B deferibe the Circle Dd-, the Lines dc,nbsp;dc., dC, which are longer than rD by the Quantity by which the F�ibre was lengthened in everynbsp;Cafe, exprefs the whole Forces, by which thenbsp;Fibre is ftrctched in every Cafe. Bur here thenbsp;Arc Dd is hardly of one Degree, and D is always fer Enough diftant from the Point c, wherefore Dd may be looked upon as a Right Linenbsp;parallel to cC, and the Lines c d., cd, Cd havenbsp;the feme Ratio to the Lines f B, r B, C B. Therefore

Book I. of Natural Th�lof�;[gt;hy. nbsp;nbsp;nbsp;izy

fore the Point C is always drawn towards Br and A, by Forces proportionable to the Line CB ornbsp;CA, and the Force by which the Wire is in-flefted, whofe Direftion is along cC, is as thenbsp;double Sagitta, * or as the Sagitta itfclf. There- * 203nbsp;fore in all the leafi Inflexions of a Chords Muficalz�f^nbsp;String or Wire^ the Sagitta is increafed and dimi-nijhed in the fame Ratio as the Force with which thenbsp;Chord is inflected.

Experiment 2.. Plate XVII. Fig. 2.] Let the Wire AB, Idretched by any Weight, be infleftednbsp;by the Weight of i, 2, and 3 Ounces} the De-fcents of the Pointy, that is, the Sagittre them-felves are to one another as i, 2, and 3.

In Chords of the fame Kind., Thicknefs., and which 2f4 are equally frctched.^ but of different Lengths �, thenbsp;Lengthenings.) which are produced by fteperadding equalnbsp;IFeights^ are to one another as the Lengths of thenbsp;Chords. This is plain, becaufe the Chord isnbsp;equally ftretched in all its Parts} therefore thenbsp;Lengthening of a whole Chord is double thenbsp;Lengthening of half of it, or of a Chord of halfnbsp;the Length.

As to the Inflexion of thofe Chord?, let A B, ab.) {Plate'KVll. Fig. 4..) be Chords of the funenbsp;Kind and Thicknefs, but of different Length,nbsp;equally ftretched, and fo infledted, that A C Bnbsp;lhall be the Poficion of the firfl, and adb that ofnbsp;thelaft} and let the Triangles BCc, and bOdnbsp;be fimilar; cB is to D^, that is, the Lengths ofnbsp;the Chords are as CB to 0^^} therefore the Chordsnbsp;are lengthened in Proportion to their firfl Length,nbsp;and confequently they are drawn by equal Porcenbsp;in the Diredlions aef BC, AC: * But by *254nbsp;the Likenefs of the Triangles above-mentioned,nbsp;the Forces alfo adting along c C and D d are equalnbsp;to one another*, and the Sagitta cC^ Dr/, ^ 203

iz6 nbsp;nbsp;nbsp;Mathematical Elements Book 1.

the Lengths of the Chords-, which does therefore, casteris paribus ^ obtain in unequal and inflehlednbsp;Chords.

Experiment 3. Plate XVII. Fig. i.~\ Let the Chord AB be ilretched by any Weight, havingnbsp;fixed the Prifms H, H, at the fixth Divifion onnbsp;each Side; Now let it be infledted with anynbsp;Weight, fo that the Sagitta may be equal to fixnbsp;Divifions of the Plate ed. Let the Prifms benbsp;brought to and fixed at the fourth Div.ifion onnbsp;each Side, and the S-agitta will be equal to fournbsp;Divifions of the Plate gt; and fo on for any Pofitionnbsp;of the Prifms.

One may compare together Fibres of the fame Kind, but different Thicknefs; they may be looked upon as made up of feveral very fine Fibres ofnbsp;the fameThicknefs, whofc Number in the above-mentioned Fibres mult be taken in a Ratio of thenbsp;Solidity of thofe Fibres, that is, as the Squares ofnbsp;the Diameters, or as the Weight of the Fibresnbsp;when their Lengths are equal. Therefore thefenbsp;Fibres will be equally llretched by Forces that arenbsp;in the fime Ratio of the Squares of the Diameters; which Ratio alfo is required between thenbsp;Forces by which the Chords are infledted, thatnbsp;theSagittie may be equal in the given Fibres. Butnbsp;by diminifliing the Force by which the Fibre isnbsp;ffretched in the (ame Ratio as the Force by whichnbsp;* 250 it is infledted, ihcSagitta is not changed*. Thereof lt;5 fore, if the Forces by which the Fibres are Jiretchednbsp;be equal, and they are infleEled by equal Forces, evennbsp;in that Cafe alfo the Sagitta will be equal, howevernbsp;different the Fbicknefs be.

Experiment\. Plate ^V\\. Fig. 2.I Take any Chords of the fame Kind, and unequal Thick-

Book I. nbsp;nbsp;nbsp;of Natural Thilofophy.

nefs j and let them be feparately applied to the Machine, leaving the Prifms HH in the famenbsp;Place 5 if the)' be {Iretched by the fame Weightnbsp;P, and alfo be inflefted by the fame Weight L,nbsp;the Sagittae will be equal.

Let the Chord B (Plate XVII. Fig-1.) ftretch-1 yy ed any ho'-jo-y be fo inflehled as to acquire the Figurenbsp;A C B, then left to it felf.^ and by its Elafticicy itnbsp;will return to its firft Figure, and in that Cafe thenbsp;Motion of the Point C is accelerated j for whennbsp;the Chord is let go from the Pofition A C B, thenbsp;Point C is moved with the Force that is able tonbsp;retain it in that Pofition. This Motion is not de-ftroyed, but there is fuperadded to it, in all thenbsp;Points of the Sagitta, the Force by which thenbsp;Point C might be retained in them. The Celerity is the greateft of all at e, and by that Celerity the Point C is carried farther, and then returning, it 'will perform federal Vibrations., in whichnbsp;the Point C runs out but Htort Spaces j for whichnbsp;Caufe the Force, by which the Point C is actednbsp;Upon in all Diftances from r, is as the Diftancenbsp;* in each Point. Therefore the Motion agrees *nbsp;�with the Motion of a Body vibrating in a Cycloid,

lt;ind how unequal foever the Vibrations are, they are performed in the fame Lime. *nbsp;nbsp;nbsp;nbsp;*

If there be two equal and fimilar Chords, but un- 2 y8 ^yually flretched, unequal Forces are required tonbsp;infleft them equally j therefore the Vibrationsnbsp;are performed in unequal Times. One may compare their Motions with the Motions of the Pendulums which vibrate in Cycloids and deferi- � 2;nbsp;l^iug fimilp Cycloids by different Forces; whichnbsp;Porces are inverfly as the Squares of the Times ofnbsp;tbe Vibrations.* In Chords therefore likewfifc the �nbsp;^^uares of the Times of the Vibrations are to onenbsp;Another inverfly, as the Forces by which they are

Mathematical Elements Book 1.

Xfp Meloen the Chords are fimilar^ equally Jiretched^ hut of different Lengths^ their Motion muft benbsp;compared with that of Pendulums by . another Method } for as the Times of the Vibrations are to benbsp;confider�d, the Celerities alfo, with which thenbsp;Chords are moved, muft alfo be confidered: Andnbsp;in the Chords ACB, adh (Plate XVIJ. Fig. 4.)nbsp;whofe Sagittre are equal, and in which the Pointsnbsp;C and d may be conlidered as deferibing fimilarnbsp;Cycloids, the Celerities, with which thofe Pointsnbsp;are moved in correfpondent Points, are to each othernbsp;in an inverfe Ratio of the Squares of the Times of

JLet the Chords AC, ah, be divided into very fmall Parts, but each into an equal Number ofnbsp;Parts i the Ways to be run thro� by correfpondent Parts, fuppofing the Sagittae equal, will benbsp;equal, and thefe fmall Parts will perform fimilarnbsp;Vibrations 5 but the Particles of Matter in thenbsp;correfpondent Particles are as the whole Chords:nbsp;That therefore their Celerities may be determinednbsp;in correfpondent Points, the Forces with whichnbsp;the Chords are inflefted, when the Sagittae arenbsp;equal, muft be divided by the Quantity of Matter in the Chords, as it follows from Numb. 6^nbsp;It is therefore plain, that thofe Celerities are tonbsp;one another dire�ly as the Weights by which thenbsp;Chords are infiefted, and inverflyas the Quantitiesnbsp;of Matter in thofe Chords, that is, inverfly as theirnbsp;Lengths: But thofe Weights are alfo in an inverfe

* nbsp;nbsp;nbsp;25s Ratio of the Lengths of the Chords j* thereforenbsp;^53 the Celerities are in an inverfe duplicate Ratio of

Book I. of Matural Thilofofhy. nbsp;nbsp;nbsp;i zp

Length j and then, as was faid before, the Squares of the Times of the Vibrations will alfo be in thatnbsp;inverfe Ratio, ithe Lengths therefore of the Chordsnbsp;'Will be as the Limes of the Vibrations.

One may, in the fame Manner, compare the i6o Times of the Vibrations of Chords of different Tkick-nefs., fuppofing the Chords equal., and firctcbed withnbsp;equal pVeights\ the QiJantitics ol Matter are asnbsp;the Squares of the Diameters j therefore to determine the Celerities of the con efpondent Points,nbsp;the Weights, by which the Chords are inficcted,nbsp;are to be divided by thofe Squares, when thenbsp;Sagittie are equal 5 * the Celerities therefore are * 256nbsp;invcrfly as the Squares of the Diameters, andnbsp;therefore the Diameters are as the Times of thenbsp;Vibrations.

.dny Chords of the fame Kind being gi-xen, the z6l Durations of the Vibrations may be compared together; for they are in a Ratio compounded of thenbsp;inverfe Ratio of the f'quare Roots of the fVeights.,nbsp;by which the Chords are fret deedand the Ratio of* 258nbsp;the Lengths of the Chords, * and the Ratio of the * 259nbsp;Diameters. * If you multiply the Diameters by * 260nbsp;the Lengths, and divide the Produft by tlicnbsp;Square Root of the Weight that ftretches thenbsp;Chord, and go through the fame Operation for federal Chords; the Quotients of the Divifion willnbsp;be to one another as the Times of the Vibrations.

Elaftic Plates may be confidered as a Congeries, 261 or Bundle of Chords ; when the Plate is infleflred,

Ibme Fibres are lengthened, and there are unequal Lengthenings in feveral Points of the Plate 5 nownbsp;the Curve, which is formed by the infle�led Plate,nbsp;tnay be dil covered from what has been faid concerning Chords.

By comparing together the Inflexions of The z6i fttne Plate they are proportional to the Forces by

Mathematical Elements Book I.

'which the Plate is bent. Let A.B {Plate XVII. Fig. f.) be an elaliic Plate or Spring, whofe Endnbsp;A is fixed, and let it be inflefted by two Forces,nbsp;2.64 fo as to be brought into the Pofition ab and abynbsp;if the one be doubled, the other, bb and i B, willnbsp;be equal j and therefore in the Vibrations thenbsp;Motion of the Spring is accelerated in the famenbsp;Manner as the Motion of a Chord and the Motion of a Pendulum in a Cycloid quot;quot;j and the Vibrations of this Plate are performed in the famenbsp;Time.

Experiment F Plate'lK.VW. Fig. 6. The Spring A is made up of feveral elaftic Plates, and put into the Box B, and there moves on each Side j between the Rulers cd, od, two Strings are fixed tonbsp;the upper Part of the Spring, and run throughnbsp;the Hole e, 1?, in the Bottom of the Box. Ifnbsp;you hang half a Pound upon the Threads, itnbsp;will defeend half an Inch; add another halfnbsp;Pound, and it will defeend half an Inch more}nbsp;and fo on, �till the Spring can be comprelTed nonbsp;fitrther.

Each fmall Plate is bent in Proportion to the Weight; and the Motion of the Weight, on account of all the Inflexions together, follows thenbsp;fiime Proportion. The Experiment is made withnbsp;feveral Plates joined together; becaufe in various Inflexions the Diredrion of the Adlionnbsp;of the Weight on the Plates is not fenfiblynbsp;changed.

What has been fiid of the Inflexion of Plates, mav be applied to the curve Plate or Springnbsp;AGB {Plate XVII. Fig 7.) If it be prelTcd bynbsp;two Weights, fo as to acquire the Pofition acb-,nbsp;aeb., and the Weights are to each other as i tonbsp;2, the Diftances cc and rC will be equal**nbsp;Therefore the bending in of the Spring, or Spacesnbsp;5nbsp;nbsp;nbsp;nbsp;gone

Book I. of Natural Thilofophy. nbsp;nbsp;nbsp;131

gone through by the Point C, are as the Weights �with which the Plate is prefled. Which may alfonbsp;be applied to the bending in of feveral Plates joined together.

The Ball ACB {PlateXVU. % 8.) being z66 made of an elafticSnbftance, may be coafidered asnbsp;confiding of feveral Plates ; and the Ttitroceifionsnbsp;(orYieldings inward) of the Point G will be proportionable to the Forces with which the Body isnbsp;comprelTed.

Let the Point C of the Ball A C B E (Plate XVII. Fig. 9.) ftrike feveral times againft anynbsp;Liane, and let that Point go inwards to P, d, andnbsp;� j the Strokes will be to each other as the Linesnbsp;G�f, Cd, and CD. At the firft Stroke the Parcnbsp;becomes flat, the fecond Stroke ach is flattened, and the third ACB: As here we alwaysnbsp;confider the leafl: Arcs, the Arcs (that is, the Diameters of the plane Surfaces made by the Strokes,)nbsp;are to one another fenfibly as the Chords Ca,

C a, and C A j therefore the Surfaces are as the Squares of thofe Chords j in which Ratio alfo,nbsp;from the Nature of the Circle, are the Linesnbsp;Gr/, Cd, and CD, which are to each other asnbsp;the Strokes. Therefore � elafik Spheres.^ thenbsp;Planes made by the Strokes follow the Proportion ofnbsp;the Strokes.

Experiment 6.3 Take a flat Piece of blue Mar- 2.68 ble made fall in a hormontal Pofition, and a little

wet, fo as to make the Colour the more in-tenfe ; if you let an Tvory Ball fall upon this Plane, that Part of the Ball, which by being madenbsp;fiat applies itfelf to the Stone, leaves a verynbsp;found Spot in the Surface of it: Let the Ballnbsp;fall from the Height of 9 Inches, and the ^otnbsp;E: then let it fall from the Height of 3 Feetnbsp;''^bich is the Quadruple of the other, and thenbsp;K 4nbsp;nbsp;nbsp;nbsp;Spoc

Mathematical Elements Book I.

Spot will be F j laftly, let it fall from the Height of 6 Feet and p Inches, which is nine times thenbsp;firft, and the Spot will be G. In that Experiment, the Strokes of the Body againll; the Stonenbsp;*'3i are to each other as i, z, and 3:* In whichnbsp;Ratio alfo are the Spots E, F and Gj for if younbsp;draw the right-angled Triangles DAB, DBG,nbsp;in which the Sides DA, AB, BC, are equal tonbsp;one another, and to the Diameter of the Spot E,nbsp;the Line B D will be exactly equal to the Diameter of the Spot F, and the Line C D to the Diameter of the Spot G.

ELEMENTS

Natural Philofophy,

EXPERIMENTS.

BOOK II.

CHAP. I.

Of the Gravity of the L�arts of Fluids., and its Effed, in the Fluids themfelves.

� FLUID is a Body whole Parts I yield to any Force imprciTed, and bynbsp;yielding are very eafily moved onenbsp;amongft another.* Whence it follows, *nbsp;that Fluidity arifes from this., Fhat the 2,^.^nbsp;Parts do not flrongly cohere., and that the Motion isnbsp;^ot hinder'd by any Inequality in the Surface of thenbsp;Parts., as it happens in Powders.

�34 nbsp;nbsp;nbsp;Mathematical Elements Book 11.

But the Particles, of which Fluids confift, are of the fame Nature with the Particles of othernbsp;Bodies, and have the fame Properties �, for Liquidsnbsp;are often converted into Solids, when there is anbsp;more ftrong Cohelion �f them, as in Ice. On thenbsp;contrary, melted Metals give us an Inftance of anbsp;Solid changed into a Fluid.

tyo fluids agree in this with Solid Bodies^ viz. 'That they confift of heavy Particles^ and have their Gravity proportionahle to their ^antity of Matter, innbsp;any Pojition of the Parts. If in the Liquid itfelfnbsp;that Gravity be not fenfible, it is owing to this,nbsp;that the lower Parts fuftain the upper, and hindernbsp;them from defcending ; But it does not follownbsp;from thence, that the Gravity is taken away j be-caufe a Liquid contained in a VelTel will prefsnbsp;down the End of a Balance, which carries thenbsp;Veflcl, in Proportion to its Quantity. The following Experiment will alfo fhew, that the Gravity is preferved in any Part of the Liquid.

271 In this, as well as in other hydroftratical Experiments relating to the Gravity of Fluids, we ufe a very exaft Pair of Scales, differing from commonnbsp;Scales only in this, that each Scale has a Hooknbsp;VV under it, {Plate'^yill. Fig.ii) for fufpend-ing fuch Bodies as are to be immerfed in Liquids.nbsp;tyt The Balance itfelf hangs by a Line which goesnbsp;round two PuliiesTT, and is taftened to a Weightnbsp;P, (Plate XVIII. Fig. i.] that, by moving thenbsp;Weight, the Balance may be conveniently raifednbsp;and depreffed, and fufpend�d at any Height.

Experiment i.] Tmmerfe in Water the Phial D clofc ihut, and hanging by a Horfe-hair, nndnbsp;balance it with the Weight in the oppofite Scale jnbsp;then, without taking the Phial out of the Water,nbsp;oDcn it, and let it be filled with Water, and younbsp;'nbsp;nbsp;nbsp;nbsp;Will

Book II. of Natural�Thilofophy. nbsp;nbsp;nbsp;135*

^vi]l find, that the Water in the Phial will bring down the End of the Balance, although it has nonbsp;Communication with the external Water; If younbsp;teftore the. jEquiJibrmm, by putting more Weightnbsp;into the oppofite Scale, the Phial will remain fuf-pended in any Part of the Water.

From this Gravity it follows, that the Surface of t-jz ^ Fluid contained in a Veffel^ to keep it from flowingnbsp;out^ if it he not prefed from above^ or if it be equal-bprefed (for that makes no Alteration) wfllbe-lt;^ome plain^ or flat, and parallel to the Horizon. For,nbsp;as the Particles yield to any Force imprefled, theynbsp;'will be moved by Gravity, �till at laftnoneof themnbsp;Can defcend any lower.

fbe lower Farts fuftain the upper, and are pref- zyj fed by them gt; and this Prefure is in Proportion tonbsp;the incumbent Matter.y that is, to the Height of thenbsp;Liquid above the Particle that is prefed-, but, as thenbsp;Upper Surface of the Liquid is parallel to the Horizon, * all the Points of any Surface, which you�*' 272nbsp;may conceive within the Liquid parallel to thenbsp;Horizon, are equally prefs�d.

If therefore in a Part of fuch a Surface there is zyq � lefer Preffure than in the other PartSy the Liquid^

Which yields to any Impreffion there ^ will be mov�d, that is, will afcendy 'tall the Prefure betomes equal.

Experiment z. Plate XVIII. Fig. z.'] Take a Glafs Tube C open at both Ends, and floppingnbsp;nne End with your Finger, immerfe the other innbsp;Watery when the Tube is full of Air, the Water will rife in it but to a very fmall Height: Ifnbsp;you take away your Finger, that the Air that isnbsp;compreflcd may go out, the imaginary Surface (asnbsp;Mr. Boyle ufed to call it) that you conceive in thenbsp;Water, juft at the Bottom of the Tube, and parallel to the Horizon, is lefs prefled juft againft

3*5 nbsp;nbsp;nbsp;Mathematical Elements Book II.

the Hole of the Tube, fo that the Water will rife up into the Tube till it comes up to the famenbsp;Height with the external Water.

Zjp 'The PreJJ'urc upon the lower PartSy which arifes from the Gravity of the fuperincimhent Liquid, exerts itfelf every Way^ and every IVay equally.

Which follows from the Nature of a Liquid, for its Parrs yield to any impreflion, and are cafilynbsp;moved j therefore no Drop will remain in itsnbsp;Place, if, whilft it is preflcd by a fuperincumbentnbsp;Liquid, it is not equally prellcd on every Side:nbsp;But it cannot be moved on account of the neighbouring Drops, which arc prelTcd in the famenbsp;Manner, and with the fame Force, by the fuperincumbent Liquid 5 and therefore the firft or low-ell Drop is at rell, and equally prefled on all Sides,nbsp;that is, in all Directions.

Experiment Plate XVIII. Fig. z.] Let the Glafs Tubes A, B, D, be immerfed in Water, innbsp;the fame Manner as in the laft Experiment} and,nbsp;upon taking away the Finger, the Water will rife

all the Pubes to the lame Height as in the Tube C; In C the Preflure is diredlcd upwards,nbsp;in B dowmwards, in A fidewife, and in D obliquely} yet the Preflure is equal in each. If younbsp;pour in a greater Q^iantity of Liquid into thenbsp;Veflel, it will alfo rile equally in each Tube.

Communication^ whether equal or unequal^ whether flrait or obliquey anbsp;Fluid rifes to the fame Height} that is, all thenbsp;upper Surfaces are in the fame horizontal Plane}nbsp;which is ealtly deduced from what has been faid.

Plate

Hence it follows, that all the Particles of Liquids are preffed equally on all Sides, and therefore arenbsp;at ref-, and that they do not continually movenbsp;among themfelves, as feveral have fuppofed.

Book II. of Natural Thilofophy.

Plate XVIII. Fig. 3.] Let A be a VefTel, and B a vercical Tube, and D an inclined Tube; theynbsp;niuft communicate by Means of the Tube C, E:

Let there be a Liquid poured into them, and let f g h be a Surface parallel to the Horizon. If thenbsp;kleights �/ and ^ / be unequal, the Water willnbsp;^feend where that Difference is lealt. For the * ^74nbsp;Lme Reafon, unlefs the Preffures at g and h benbsp;^qual, the Water will not be at reft; but they arcnbsp;^'lual when I and n are in the fame horizontalnbsp;Liane: For fince the Preflure arifes from the Gra-''ity of the Parts, which tends towards the Centernbsp;of the Earth, the Height of the prefling Liquidnbsp;^uft be meafured according to that Direftion, that

it will he hm-, but the Obliquity of the Column kn caufes no Change, becaufe at the fame Depthnbsp;the Preflure every Way is equal.*nbsp;nbsp;nbsp;nbsp;* 275

Experiment 4. Plate'KNlll. F(^-4 ] Pour Water 278 into the Machine reprefented by Fig. 3. and afternbsp;�'tiy Agitation it will not reft, unleis all the Surfaces be in the fime horizontal Plane. The Glafsnbsp;VefTel A is joined to the Glafs Tubes B and D,nbsp;by help of the Brafs Tube CE.

All Liquids are not equally heavy, that is, have not the fame Qiiantity of Matter in the famenbsp;Space; but what has been faid will agree to everynbsp;Liquid by itfelf.

IVhen Liquids of different Gravities are contained t-Jp in the fame Vffel^ the heaviefi lies at the lotvejinbsp;Place., and is preffed hy the lighter, and that in Proportion to the Height ff the lighter.

Experiments;. Plate XVIII. Fig. f.] Take Water tinged with fome Colour, and pour it into the Glafs VefTel A to the Height o� h c-, im-nierge into it the Glafs Tube d E; the Water

Mathematical Elements Book H.

Now pour in IS a Liquid lighternbsp;than Water, and immediately the Water will rifenbsp;in the Tube; and fo much the higher as the Oilnbsp;is poured in, to a greater Height: Yet the Waternbsp;in the Tube does not rife to the fame Height asnbsp;the Oil in the Vedel �, becaufe, dnee Water is heavier, there is not required the fame Height ofnbsp;Water as there would be required of Oil to produce the lame Prcdlire.

If you have a Mind to try this Experiment with Mercury and Water, you will find a greaternbsp;Difference in their Heights, by reafon of theirnbsp;greater Difference of Gravity.

Experiment6.Platel^YlW. Fig.6.'] Let the End of a Tube be immerfed in Water, and pour Oilnbsp;into it. The Water in the Tube is deprefs�d asnbsp;far as yet the Height of the Oil de is greaternbsp;than the Height of the Water in the Velfel. Ifnbsp;the Tube be immerfed deeper, the Water will runnbsp;into it in greater Quantity j if you raife it up, thenbsp;Water will again go out at and the Water innbsp;the Tube de follow it, if it be raifed to fuch anbsp;Height, that the Predure of the Oil may overcome the Predure of the Water in the lower Partnbsp;of the Tube.

CHAP. II.

Of the JBlons of Liquids againfl the Bottoms and Sides of the Vejffels that contain them.

280 HE Bottom and Sides of a Heffel.^ which con-1 tain a Liquid.^ are prefjed by the Parts of the Liquid which immediately touch them j and be-V j26caufe Re-aftion is equal to Aftion,* thofe Partsnbsp;all fuffain an equal Predure. But, as the Predion of

Book II. of Natural �Phtlofophy.

l-iquicls is equal every Way, the Bottom and Sides are prcls�d as much as the neighbouringnbsp;Barts of the Liquids 5 therefore this Action in-treafes, in Proportion to the Height of the Liquid, * �nbsp;and is every Way equal at the fame Depth, de- 273nbsp;pending altogether upon the Height, and not atnbsp;all upon the Quantity of the Liquid. Therefore,nbsp;\vhen the Height of the Liquid, and Bignefs ofnbsp;*^he Bottom remain the fame, the Action upon thenbsp;Bottom is always equal, � however the Shape ofnbsp;the Body be changed. In every Cafe the Breffiire,nbsp;Biltalned by the Bottom, is equal to the Weightnbsp;nf a Column of Water, whofe Bafe is from thenbsp;Bottom itfelf, and the Height of the vertical Di-ftance of the upper Surface of the Water fromnbsp;^he Bottom itfelf.

P/ate XVIir. Fig- 7 and 8.] Take the hollow 2,gi Cylinder A, open at both Ends, and finely polilh-^d within, whofe Diameter and Height alfo arenbsp;^Sout three Inches and an half; the Ring E isnbsp;faftened to it by a Screw, fo as it may be fuflainednbsp;Sy a Trevet.

Let the Cylinder have a moveable brafs Bottom F, with which the brafs Ring G, having a Screw in the Infide, is joined: This Ring retainsnbsp;fixes a Leather Ring, broader than the Bottom,

^11 round by half an Inch; this Leather covers the external Surface of the brafs Ring when the Bottom is thruft into the Cylinder, and it hinders thenbsp;Water from going out when it is moved up andnbsp;down. This Leather muft be foaked in Oil, andnbsp;^fter a few Days it muft be taken out and foakednbsp;as long in Water; after which Preparation thenbsp;Eeather muft be well anointed with Oil and Wa-and moved feveral times up and down thenbsp;yhnder, and left in it in that Condition two ornbsp;�ttee Days. When you are going to ufe the Machine

Mathematical Elements Book II,

chine, you muft anoint the Leather again with Oil and Water, then the Bottom will move eafily,nbsp;and hold W ater. The Leather muft be neither toonbsp;thick nor too thin, which muft be left to the Judgment of the Workman.

The Bottom has in its Middle a fmall Brals Cylinder h i faftcned to it, by which the Motionnbsp;of the Bottom is direfted, for this Cylinder goesnbsp;thro the Hole m in the Plate which is laid upon the larger Cylinder yf, and let into it by a Cutnbsp;in the Edge. In the upper Surface of the Cylinder hi there is a Cavity which contains a Screw,nbsp;by which the Bottom is joined to the Brafs Wirenbsp;�/gt;, which js carried through the Tube D, thatnbsp;the Bottom may be faftened to the Brachium of anbsp;Balance by the help of this Wire.

Let the Cylinder A have the Cover C laid upon it} and, to hinder the Water from running out, the Mouth of the Cylinder muft be cover�dnbsp;with a Leather Ring, which is ttrongly prefled bynbsp;the Screw which joins the Leather Cover to thenbsp;Cylinder. To the Cover and Cylinder itfelf may benbsp;added a Handle, that the Cylinder may the morenbsp;cafily be fluit and open�d. The Cover has a Holenbsp;in the f.liddie, and the hollow Cylinder /, whichnbsp;has a Screw on the Outfide, is faften�d to it, thatnbsp;the Tube d may be joined to the Machine with anbsp;Leather upon the Screw, to hinder the Waternbsp;from coming out.

Experiment i. Plate XIX. Fig. i.j Having joined together all the Parts of the Machine iunbsp;the Manner juft mentioned, hang upon one Endnbsp;of the Beam of a Balance the Brals Wire whichnbsp;is fixed to the moveable Bottom, fo that thenbsp;Beam may be exa�i:ly horizontal when the Bottom is two Inches diftant from the Cover} then

Book 11. of Natural �Philofofhy.

puc fuch a Weight in the oppofite Scale as will make an ^Equilibrium with the Weight of thenbsp;Bottom only. Let the Tube be one Foot longjnbsp;and, the Beam of the Balanre being placed horizontal, pour Water into the Tube D, fo that itnbsp;may rife up to its upper Endj another Weight of

Pounds, being put into the upper Scale, will make an Equilibrium with the Water j and, ifnbsp;you diminiih or increafe this Weight, the Bottomnbsp;will move upwards or downwards. But you muftnbsp;obferve, that, in altering the Weight, you muftnbsp;put in, or take out a pretty confidcrable Weight jnbsp;for Example, half a Pound, becaufe of the Fri-ftion of the Bottom.

The Diameter of the Bottom is almoft 5 �? Inches, and theHeight of theTop of the Water,nbsp;in this E.xperiment, is 14 Inches; the Weight ofnbsp;a Pillar of Water of that Height, whofe Bafe isnbsp;equal to the Bottom, is q-f Pounds; and juft fonbsp;much does the Water prefs againft the Bottom;nbsp;tho� there be but a fmall Quantity of Water innbsp;the Machine.

Since only the Motion of the Bottom is to be obferved, the Machine is to be fix�d down, left itnbsp;Ihould be wholly raifed; which is done by layingnbsp;fuch Weights upon it as are reprefented by PP,nbsp;Plate Fig. i.

Experiment z. Plate NIX. Fig. z.'] Having taken away the Cover and the Tube, join thenbsp;Cylinder A to the Veflel DE, which has at thenbsp;Bottom a Ring with a Screw. Into this Machine pour Water upon the Bottom as high as innbsp;the foregoing Experiment; the reft of the Expe-timent is made in the fame Manner as the formei',nbsp;^nd the Succels is the fame; for the Prefllire isnbsp;Oot changed, tho� you alter the Figure of the

Mathematical Elements Book H.

VelTel and the Quantity of the Water, provided you keep the Water to the fame Height.

Experiment 5. Ptoe XIX. Fig. 3.] Hang the Cylindrick Veflel A to the End of a Balance,nbsp;which VelTel mull be filled in part by a woodennbsp;Cylinder De, which Cylinder is fixed any hownbsp;to the Piece of Wood B C, and neither touchesnbsp;the Sides nor the Bottom of the aforeliiid Veflel}nbsp;if you pour Water into this VelTel to any Height,nbsp;and make an ^Equilibrium by putting Weightsnbsp;in the oppollte Scale, that Weight will be thenbsp;Weight of the whole Water which would benbsp;contained in the Veflel, the Cylinder being takennbsp;away, fuppofing it filled to the fame Height as innbsp;the Experiment. And fo a fmall Quantity ofnbsp;Water, whpfe upper Surface is railed, lb as thenbsp;Preflure againft the Bottom be increafed, will fu-ftain a great Weight.

It will vifibly appear, that the katerai is equal to the vertical Preflure, making ufe of the following Machine.

282 Plate XIX. Fig. 4.]} The Veflel DB is a Pa* rallclopiped of Wood about a Foot and a halfnbsp;high} in the Side towards the Bottom there isnbsp;a Hole in which there is a Brafs Ring containingnbsp;a Screw, that the Cylinder A, mentioned in thenbsp;firft and fecond Experiments, may be fcrewed tonbsp;it. Here you mufl: take away the Trevet whichnbsp;fuftained the Cylinder in thofe Experiments, andnbsp;was fixed to the lower Ring by Screws. Nownbsp;the Motion of the Bottom, in the Cylinder isnbsp;horizontal. Two crofs Pieces of Wood are joinednbsp;to the Sides of this Machine, one of which isnbsp;leen in G H} along them the Ruler C C isnbsp;moved horizontally, which is wider in the Middle towards F, that by its Motion the Bottomnbsp;of the Cylinder may be thruH inwards, which

the

Book ir. of Natural '^hilofo^by.

the Ruler preffes a little below the Center. At CC Ropes^ as CE, are faften�d to this Ruler,nbsp;v-^hich are ftretch�d along the Pieces, as G H,nbsp;and going over Pullies at the Extremities ofnbsp;the faid Pieces, as T, have Weights joined tonbsp;them, as P.

Experiment Af\ Pour Water into the Veflel BD, fo that the Surface of the Water may be highernbsp;by 14 Inches than the Ruler CCj let the Weights,nbsp;as P, be of ^ Pounds and a Quarter each j fo thatnbsp;both taken together fhall amount to 4 Poundsnbsp;and a Half, the Preffure of the Water will fuftainnbsp;that Weight, and the Bottom in that Cafe willnbsp;be moved with the fame Eafe towards eithernbsp;Part.

The following Experiment proves that the Force, with which Water prefies upwards, isnbsp;equal to that with which it preffes downwardsnbsp;and Tideways.

Experiment . Plate'K'K. Fig.i7\ Tn the Middle of the upper Surface of the Block or Foot B therenbsp;is a Cylinder of about a Inches Diameter, on whichnbsp;you muft put the moveable Bottom of the Cylinder A, fo often mentioned j fo that, the Bottomnbsp;Remaining fixed, the Cylinder may be moved.nbsp;Fhe Cylinder muft have its cover on, and to itnbsp;^he Tube D, 3 Feet and a Plalf long, muft benbsp;faftenedj pour in Water, by which, the Bottomnbsp;^'ernaining fix�d, the Machine will be raifed; putnbsp;the Weights PP P, which all together weigh 9nbsp;Pounds, upon the Cover, and they, with thenbsp;Weight of the whole Machine, will be fuftainednbsp;by the Water in the Tube} but the V/eight ofnbsp;Machine is more than 3 Pounds and a Half.

. The Force, which afts againft the Cover, ^ equal to the Weight of a Pillar of Water,

Mathematical Elements Book II.

whof� Bafe is the Cover, excepting the Hole to which the Tube is fix�d, and whofe Height is thenbsp;Height of the Water-Tube above the inward Surface of the Covxrj which agrees with this Experiment.

If you apply the fame Tube to a greater Machine, the A�lion .againft the Cover will increafe in the fame Ratio as the Cover j fo that a prodigious Weight may be fuitained, and even raifednbsp;by a fmall Quantity of Water.

Plate XX. Fig. z.'\ Take two round Boards AB, AB, of If Inches Diameter, and join themnbsp;together with a Piece of Leather, fo that theynbsp;may make a Cylindrick Vefiel fomething like a-Pair of Bellows, fo that it may contain Water.

which is fix�d a brafs Cylinder that has a Screw, whereby the Tube D is fix�d to it, which is asnbsp;long as the Tube uled in the farmer Experiment.

Experiment d.] Pour Water into this Bellows thro� the Tube, and the Water in the Tube willnbsp;fuftain the Weights P, P, P, P, P, P, all whichnbsp;together weigh more than 2fo Pounds. Thefcnbsp;Weights will even be raifed by continuing to pournbsp;Water into the Tube.

Though thefe are Paradoxes, they follow from the Nature of Liquidity; every Drop which isnbsp;at reft, endeavours to recede every Wav withnbsp;�275 equal Force; *if therefore it be prefled on onenbsp;Side, it endeavours to recede that Way with thenbsp;fame Force, becaufe Aftion and Re-a6tion arenbsp;equal, and with that very Force itfelf will preftnbsp;every Way. In the firft Experiment, the Waternbsp;which touches the Bottom, and correfpopds with

the

Book II. of Natural Thilofophy.

the Tube, fuftains the Weight of the Column of contain�d in the Tube, and reaching quitenbsp;to the Bottom, and preffes the Bottom with fuchnbsp;a Force, that it ads with the fame Force upon thenbsp;Water next to it} and fince that Water cannotnbsp;flow out againft the Bottom, the Water next tonbsp;that is alfo prefs�d with the fame Force. Thenbsp;fame may be faid of the Water next to that; andnbsp;fo in all Parts of the .Bottom there is a PrcfTurenbsp;equal to that which lies under the Water in thenbsp;Tube; and therefore the Bottom in this Cafe isnbsp;as much prefs�d as if a Pillar of Waters, of thenbsp;fame Height as the Water in the Tube, and of anbsp;Bafe equal to the Bottom, fhould lie upon it.

In the fecond Experiment, fuppofe that th� Cylinder A fhould be continued, fo as to reach upnbsp;to the Surface of Watery by this Means thenbsp;externalwould be feparated from Waternbsp;contain�d in this Cylinder, and then no Water butnbsp;this interior Water would prefs upon the Bottom,nbsp;and the Bottom would fuftain it all. The Waternbsp;in the Cylinder prefles againft the Sides of thenbsp;Cylinder, and the external Water prefles upon thenbsp;external Surface of the Cylinder, and the outwardnbsp;Surface is prefs�d exactly in the fame manner asnbsp;the inward, and the PrefTures againft oppofitenbsp;Points are precifely equal; fo that if the Surfacenbsp;gt;vas taken away, thefe Prefiures would deftroy onenbsp;another; therefore it is no matter, whether therenbsp;be fuch a Surface or not, fo that -taking it awaynbsp;(that is, taking away the Continuation of the Cylinder) the Aftion againft the Bottom is no waynbsp;alter�d.

The third Experiment is alfo illuftrated by �vvhat has been faid; for the Weight placed iiinbsp;*he Balance is not only fuftain�d by the Water

1^6 nbsp;nbsp;nbsp;Mathematical Elements Book H.

in the Veflel, but alfo by the A�tion of the inferior Surface in the Cylinder De againll the Water.

Tho� all that has been faid depends upon the Gravity of Liquids, their A�lions muft be di-ftinguiiTi�d from their Gravity, which laft is al-* 270 ways proportionable to the Quantity of Matterquot;^*

CHAP. III.

Of Solids immerfed in Liquids.

WE have often faid, that the different Gravity of Bodies, whether Solids or Liquids, arifes from this, that they contain a greater or lefsnbsp;Quantity of Matter in an equal Space.

7�he ^lantity ef Matter in a Body being confi' der'd in relation to its Bulk^ that is, in relation tonbsp;the Space pofTefs�d by it, is call'd the Denfity ofnbsp;the Body.

A Body is faid to have double, or triple, f* the Denfity of another Body, when, fuppofingnbsp;their Bulks equal, it contains a double, or triple�nbsp;6?c. Qiiantity of Matter.

zSf Body is faid to be Homogeneous.^ uohen it ts every where of the fame Denfity.

5 nbsp;nbsp;nbsp;^ The

Book II. of Natural ^h'llofophy.

The fpecifick Gravity is faid to be double, wheu under the fiime Bulk the Weight is double.

Therefore the fpecifick Gravities and Denfities of 2,88 Bodies^ in homogeneous Bodies^ are in the fame Ratio ; and they are to one another as the ff'^eight 's ofnbsp;equal Bodies^ in refpetl to their Bulk.

If homogeneous Bodies are of the faine fF'eight.y 2.8p their Bulks will be fo much lefe as their Denfitiesnbsp;are greater, and under thefarrie Weight the Bulknbsp;is diminiflied in the ftme Ratio in which theDen-fity is increafed j therefore in that Cafe the Bulksnbsp;are inverfly as the Denfities.

When a Solid is immer fed in a Liquid., it is prefs'd by the Liquid on all Sides^ and that Prejjure increa-fes in Proportion to thi Height of the Liquid abovenbsp;the Solid-, as it follows from what has been faid innbsp;the foregoing Chapter) and which may alfo benbsp;proved by a diredt Experiment.

Experiment i. PlateNK. Fig.'^f] Tie a Lea!thcr ^3g S to the End of a Glafs Tube ^m, and fillnbsp;tt with Mercury) you may alfo make ufe of anbsp;bladder; let this Bag be immerfed in Water., butnbsp;^ that the End B of the Tube may be abovenbsp;^he Water-, by the PrcfTure of the Water againftnbsp;the Surface of the Bag, the Mercury in the Tubenbsp;^ill rife to m-, and the Afcent of the Mercurynbsp;follows the Proportion of the Height of the Wa~nbsp;above the Bag.

When a Body is immerfed in a Liquid to a Efeat Depth, the Preflure againft the upper Partnbsp;iffcrs very little from the Preflure againft the un-Part) whence Bodies very deeply immerfed, are^nbsp;u werei equally prefs'd on all Sides-, which Pfef-may be fuftained by foft Bodies, without anynbsp;hange of Figure, and by very brittle Bodies,nbsp;^^mout their breaking.

Mathematical Elements Book II.

Experiment z. PlateXlX. Fig. j-.] Take a Piece of foft Wax of an irregular Figure, with an Egg,nbsp;and iaclofe it in a Bladder full of Water, and thenbsp;Bladder being exa�tly iTiut mud: be put into anbsp;brafs Box Aj let it be covered with a woodennbsp;Cover O, fo that it may be fuftained by the Bladder, lay on a Weight P of 70 or 8 o Pounds, andnbsp;the Egg will not be broken, nor the Figure of thenbsp;Wax any way changed.

zpi Body fpecifically heavier than a Liquid.^ being itnmerfed in a Liquid in any Depth., will defcend.nbsp;The inferior Part of the Body prefles the Surfacenbsp;of a Liquid which it touches, and this Preflure isnbsp;equal to the Weight of a Column made up ofnbsp;the Body itfelf and the fuperincumbent Liquid,nbsp;and with this Force the Body is carried downwards. The Weight of a like Column, but whichnbsp;confift wholly of a Liquid, is the Force by whichnbsp;* 290 the Body is prefs'^d upwards by the Liquid.* Butnbsp;^75 when the Solid is fuppofed fpecifically lighternbsp;than the Liquid, this Force is lefs than that, andnbsp;therefore is overcome by it.

2,p5 It is proved by the fame Reafoning, Ehat a Solid fpecifically lighter than a Liquid., and immer fed into it, : mufi afeend to the higheft Surface of thenbsp;Liquid.

But fuppofe a Solid of the fame fpecifick Gravity ^ with the Liquid, it will neither afeend nor defcend,nbsp;but remain fufpended in the Liquid at any Height,nbsp;and the Liquidwill fuflain the whole Body �, in whichnbsp;Cafe, by rcafon of the E,quality of the fpecificknbsp;Gravities, the Liquid fuftains a Weight equal tonbsp;the Weight of the Quantity of the Liquid, whichnbsp;would fill the Space taken up by the Body. Butnbsp;a Liquid adts in the fame manner upon all equalnbsp;Solids immerfed to the fame Depth, and will

Book l�. of Natural �Philofophy.

A If^eight^ 'which keeps a Body immer fed in a Li- 2py qv.id^ is called its refpeftive Gravity.

And this refpeSiive Gravity is the Excefs of the 2,p� fpecifick Gravity of a Solid above the fpecificknbsp;Gravity of a Liquid-, for fince a Solid immer fed innbsp;a Liquid lofes that Part of its Weight which isnbsp;fuftained by the Liquid, it lofes the Weight of thenbsp;^antity ff the Liquid, 'which could fill the Spacenbsp;taken up by the Body.

Experiment Plate'KXl. Fig. i.] Hang the hollow Brafs Cylinder E to the Balance above-mentioned ; * hang the folid Cylinder G of the�lame Metal by a Horfe-hair to a Hook fix�d to thenbsp;Bottom, which, if it be put into the other Cylinder E, will cxaftly fill it; fo that E, whennbsp;it is full of Water, will contain fiich a CLiantitynbsp;of Water as will fill the Place taken up by C-, putnbsp;a Weight in the oppofite Scale to make an iEquEnbsp;librium; let the Balance defeend, that the Cylindernbsp;C may be immerfed into the Water contained innbsp;the Veflel D; by that Means the ^Equilibrium isnbsp;deftroyed, becaufe C is partly fuftained by thenbsp;Water; but is reilored, if E be fill'd with Water.

Hence it follows, that all equal Solids, but of different fpecifick Gravity, when they are immerfednbsp;into the fame Liquid, they lofe equal Parts of theirnbsp;td^eight. The laft mentioned Experiment willnbsp;fucceed in the fame Mannhr with a Cylinder ofnbsp;any other Metal, and by pouring in the famenbsp;Quantity of Water, that is, fo much as will fillnbsp;the Veflel E, the ^Equilibrium Will always be-teftored,

f 50 nbsp;nbsp;nbsp;Mathematical Elements Book II.

zp9 Moreover, from what has been faid it follows, that however the Denjities of equal Bodies differnbsp;among themfelves^ if they be immer fed in the famenbsp;Liquids^ the If eight which they lofe is in the Rationbsp;cf their Bulks^ for the Spaces which they take upnbsp;in a Liquid are in the fame Ratio.

Therefore Bodies of the fame Weight, but of different Denfities, lofe an equal Part of theirnbsp;Weight, when they are immerfed in the fame Liquid, becaufe of the Inequality of their Bulks.

Experiment c[..Plate 'KX. Fig.Let two Pieces of Metal of the fame Weight, the one of Gold,nbsp;and the other of Lead, g, be fufpended withnbsp;the Hook V V of the Balance above-mentioned

librij.imj let the Balance defeend, and the Bodies be immerged in the Water contained in thenbsp;Veilcl F F, and the iLquilibrium will be dc-Broyed. When a Solid, fpecifically heavier thannbsp;a Liquid, is fufpended in a Ihquid, the Liquidnbsp;afts on every Side againft that Solid, in Propor-

againfl: it; therefore thofe Aftions are the fame as if the Space taken up by the Solid were fill�d withnbsp;the Liquid gt; therefore it is no Matterin refpePl ofnbsp;the Gravity of the Liquidy whether a Solidy fpecifi^nbsp;(ally heavier than the Liquidy be fufpended in ity ornbsp;a ^antity of the Liquid be poured iriy which takeynbsp;up a Space equal to the Solid.

Experiment y. Plate XXI. Pig. 2.] Take the VcfTel A containing Watery hang it to one End ofnbsp;the Balance, immerge into it the Brafs Cylindernbsp;C, which is fulliaincd by a Horfe-hair, left itnbsp;fhould touch the Bottom of the Veifel, puttingnbsp;a Weight into thp oppofitc Scale, and you willnbsp;have an Aiquilibriura j take the Cylinder Court

.. nbsp;nbsp;nbsp;of

Book II. of Natural Thilofo^hy.

of the Water, the ^Equilibrium will bedeftroyed: and it will be rellored again by pouring in Waternbsp;as much as can be contained in the hollow Cylinder E, which will be exa�lly filled by the following Cylinder C.

By comparing together the Numb, zpj and 501 500, as alfo the third and Mth.Experiments., whichnbsp;confirm them, it appears, that a Liquid acquiresnbsp;the fVeight istbich the immerjed Solid lofes. Thenbsp;Force of Gravity is always proportionable to thenbsp;Qtiantity of Matter, and is not changed by thenbsp;Immerfion of a Solid into a Liquid} whereforenbsp;the Sum of the Weight of the Solid, and ofnbsp;the Liquid, do not differ before and after the Immerfion.

Experiment 6. PlateNN.. Fig.p.~\ Hang the Solid C to the Balance, and make an ^Equilibrium, by putting into the oppofite Scale B the Weightsnbsp;P and p, of which p equal to the Weight whichnbsp;the Body C lofes in Water. Take the Vcflcl E,nbsp;which contains Water, and is fufpended to thenbsp;Balance EF, and, putting a Weight into the oppofite Scale, make an ^Equilibrium here alfo} letnbsp;the Balance defeend with the Body C, that it maynbsp;be immerfed in the Water contain�d in D, by thisnbsp;Means you will dellroy the ^Equilibrium in bothnbsp;Balances, which will be rellored by raking out ofnbsp;the Scale B the Weight /gt;, and putting it into thenbsp;Scale of the Brachium F.

A Body fpecifically heavier than a Liquid, and 304 which defeends in it, is carried downwards withnbsp;a greater Force than it is prefled upwards, as hasnbsp;been explained before*} thfi Difference of which * 29^nbsp;Forces is the refpedive Gravity of the Body.

The firfl Force in part confifts of the Weight of the Liquid incumbent over the Body, and thenbsp;Body may be immerfed to fuch a Depth, that

L 4 nbsp;nbsp;nbsp;that

Mathematical Elements Book l�.

that Weight fhall be equal to the above-mentioned fpecifick Gravity: If in that Cafe you take awaynbsp;305 the fuperincumbent Liquid, the Body will be fu-ftained by the Preflure of the Liquid under it. Ifnbsp;the Body immerfed to a greater Depth, and thenbsp;Liquid be alfo hindered from palling upon the upper Surface of the Body (becaufe the Preflure bynbsp;which a Body is pulhed up, incrcafes as the Depthnbsp;^ 290 to which it is immerfed) * the Body then will benbsp;carried upwards with greater Force than downwards by Gravity; wherefore, if it could movenbsp;freely, it would afeend.

linder C, which is open at both Ends, apply at Bottom the Plate of Lead F, a Quarter of annbsp;Inch thick �, if it fits fo exactly to the Cylinder asnbsp;to let no Watcr flip by, and the Plate be held upnbsp;by a Thread faftened to the Hook V in the Center of the Plate, until it be immerfed with thenbsp;Cylinders to the Depth of about 3 Inches, thenbsp;Lead will be fuftained by the Water, as appearsnbsp;by letting go the Thread. If you immerge it tonbsp;a greater Depth, it will flick clofer to the Cylinder but if to a lc(s, it will fall off.

If this Experiment was made with a Plate of Gold, it ought to be immerfed to a greater Depth inbsp;for the fpecifick Gravity of Gold is to the fpecifick Gravity of Water as ip to i j and thereforenbsp;its refpeftive Gravity is to that of Water as 18 tonbsp;*296 I.* Therefore to have a Pillar of Water equalnbsp;in refpeftive Gravity to the Plate of Gold, thatnbsp;Pillar muft be above 18 times its Heightj andnbsp;therefore the Height of the Water, above thenbsp;upper Surface of the Plate of Gold, muft be atnbsp;leaft equal to as many times its Thicknefs.

Book II. of Natural�Philofo^hy. nbsp;nbsp;nbsp;\$3.

Experiment PlateXNl. Fig.'^.~] Take a Cylinder A with a moveable Bottom, that has alio a Cover with the Tube D joined to it, as was before deferibedj * immerge it in Water, andwhenit * 381nbsp;comes to be a Foot under Water, the Bottom willnbsp;rife, although it weighs a Pound and a Quarter,nbsp;and has Pa Pound Weight ferewed to it at Bottom.

If the fame Solid be immerfed into Liquids of 3�4 different Denfity, it will lofe different Parts of itsnbsp;Weight: * And therefore, when two Bodies of * 297nbsp;the fame Denfity and Weight are immerfed in Liquors of different Denfity, they will lofe theirnbsp;^Equilibrium.

Experhnent Ptoe XXL 4.) Take two flat Pieces gg^ of the fame Metal and equal, andnbsp;hang them upon the Hooks VV of the Scalesnbsp;A and B j then by theDefcent of the Balance im-merge them in the Liquids contained in the Vef-fels F F, the one in Water, the other in Oil ofnbsp;Turpentine, and the ^Equilibrium will be deftroy-fd, the Piece which was immerfed in Oil becoming lighter.

^ Solid Hooter than a Liquid^ and immerfed in it^ jOf afeends and remains at the upper Part of the Liquid, * fo as to be immerfed only in Part} but the * 293nbsp;greater is its fpccifick Gravity, the more it defends, and the Body will not be at reji till the im-^lerfcd Part takes up fuch a Space in the Liquid^ thatnbsp;the Bulk of the Liquid^ which would fill that Space^

Jhall-weigh as much as the whole Body. For in another Cafe the Solid does^not aft with the fame Force againfl; the neighbouring Parts of the Liquid, as the Liquid would aft, if it fhould take upnbsp;the Place of the Body} therefore in this Cafe ftonenbsp;the Liquid and the Body can be at reft. *nbsp;nbsp;nbsp;nbsp;* 27^

Mathematical Elements Book II.

It follows from this Propofition, that the immer fed Parts of the Bodies^ fwimming on the Surface of the fame Liquor^ are to one another^ as the Weights of the Bodies. Therefore if, by fuperad-ding a Weight, the Gravity of the Body is changed,nbsp;the iminerfed Part is increafed in the fwne Proportion, and the Parts.^ which defend into the Liquidnbsp;by laying on of different IVeights.^ are to one anothernbsp;as thofe IVeights.nbsp;nbsp;nbsp;nbsp;r

Experiment lo. Plate XXII. Fig. i.] Take a Vefl'el A containing Water in itj letC be a hollownbsp;Cylinder of any Metal gt; lay upon it the Weightnbsp;�gt;, that it may defeend into the Water with its Parcnbsp;bd-, adding the Weight of one Pound, meafurenbsp;how far it will defeend; then adding another equalnbsp;Weight, you will find that it will defeend equallynbsp;every time.

508 In Numb. 30Z and 303, confirmed by the Experiments 7 and 8, it appeared how a Body, fpe-cifically heavier than a Liquid, may be made to fwim j by the fame Method a Body, fpecificallynbsp;lighter than a Liquid, may be retained at the Bottom: In that Cafe thePrefllire of thefuper-incum-bent Water is taken offj but here you mufl: takenbsp;off the Preffure of the inferior Water wherebynbsp;the Body is pulhed upwards.

Experiment \\. Plate'^^W. Fig.z.'] Upon the Foot D, which is fixed at the Bottom of thenbsp;Vefi'el A, there is a Brafs Plate be exadlly flatnbsp;and polillicd j there is another Brafs Plate bctnbsp;the former, faflened to a large Piece of Cork E,nbsp;fo that together with the Cork it lltall make up anbsp;Body fpecifically lighter than Water: Lay this Platenbsp;upon the ocher, fo that they may fit, and keep thenbsp;Cork down with a Stick while you pour in Water^ leaving the Cork, it will not afeend until, by

of Natural Thilofophy.

moving it out of its Place, the Plates are fepara-ted, fo that the Water may exert its Preflure a-gainft the Plate joined with the Cork, and puilt it upwards together with the Cork,

CHAP. IV.

Since the Denfity of Bodies is in Proportion to pp their Gravity, by comparing the Weights ofnbsp;equal Bodies we difeover their Denfities. * If �nbsp;therefore any Vejfel he exactly filled with a Li(yuid^nbsp;and that Liq^uid be weighed; and if yon make thenbsp;fame E,xperiment with other Liquids^ their IVeigbtsnbsp;will be as their Denfities. But, as this Method isnbsp;liable to feveral Difficulties in Praftice, I ffiall notnbsp;fpend any Time in explaining it here.

IVhen thr^refiures of two Liquids are equaf the 31 o Quantities of Matter, in Columns that have equalnbsp;Bafes, do not differ j * wherefore the Bulks, that * 273nbsp;is, the Heights of the Columns, are inverfiy as thenbsp;Denfities-, * whence may be deduced the Method *nbsp;of comparing them together.

Experiment I. PlateXXll. F/ff. 3.] Pour Mercury into a curve Tube A, fo as to fill the lower Part of the Tube from b 10 c-, pour in Water innbsp;one Leg from b to c-, in the other Leg pour innbsp;Oil of Turpentine, till both the Surfiices of thenbsp;Mercury be in the fame horizontal Line, andnbsp;the Height of the Oil be cd: Thefe Heights willnbsp;be as 87 to 100, which is the inverfe Ratio thatnbsp;the Denfity of f;he Water has to the Denfity ofnbsp;Oil of Turpentine} and therefore thefe Denfitiesnbsp;fli'e to each other qs 100 to 87.

1^6 nbsp;nbsp;nbsp;Mathematical Elements Book II.

? The Mercury is poured in, left the Liquids ftiould be mix�d in the Bottom of the Tube.

The Denfities of Liquids are alfo compared together, by immerging a Solid into them �, for if a 311 Solidf lighter than the Liquids to be compared together^ be immerfed fuccejjively into different Liquids,nbsp;the immerfed Parts �usill be inverfely as the Denfitiesnbsp;of the Liquids-, for, becaufe the fame Solid is madenbsp;ufe of^ the Portions of the different Liquors, whichnbsp;in every Cafe would fill the Space taken up by thenbsp;^305 immerfed Part, are of the fame Weight;* therefore the Bulks of thofe Portions, that is, the immerfed Parts themfelves, are inverfely as the Den-*289 fities.*

Plate'KX.ll. Fig. 4-'] Take theGlafs A, which is a hollow Ball that has a Tube divided into equalnbsp;Parcs i at the Bottom of the great Ball there isnbsp;a fmall one, Part of which is fill�d with Mercury, or very fmall Shot, whole Weight fervesnbsp;to make the Tube defeend vertically in Liquids,nbsp;and ftand in that Pofition ; Care muft be takennbsp;not to have too much Weight in tfie little Ball,nbsp;for the whole Glafs muft be lighte^ than the Liquids to be compared together. The Hydrometer,nbsp;(for fo it is called) defeends to different Depths innbsp;different Liquids; and thofe Denfities, as we havenbsp;already faid, are inverfely as the Paits immerfed,nbsp;which therefore are to be compared together. Ticnbsp;a Thread to the Hydrometer, and weigh it together with the Thread; the Weight (if it be likenbsp;mine) will be f73 Grains; if put into Water, icnbsp;will defeend to b; therefore a Bulk of Water, equalnbsp;to the immerfed Part of the Hydrometer, weighsnbsp;r73 Grains,* and may be expreffed by that Number. Faften the Thread above-mentioned to thenbsp;Hook V of the Scale A of the Balance repre-fented in Plate XVIIL Fig. i. the Hydrometer

Book II. of Natural�Philofofhy. nbsp;nbsp;nbsp;iS7

remains iramerfed j put zo Grains in the Scale B, ^ and let the Weight P be moved gently to raifenbsp;the Balance, (by which the Tube will be drawn anbsp;little way out of the W^ater) till there be annbsp;./Equilibrium, and the Surface of the Water thennbsp;will be given with the Point d-, the Water fuilainsnbsp;the Weight of the whole Machine, except 20nbsp;Grainsgt; that is, it fuftains yy; Grains; and thenbsp;Weight of the fame Bulk of Water^whxch is nownbsp;immerfed, weighs juft fo many Grains, and isnbsp;exprefled by that Number; wherefore one maynbsp;call the Bulk of the Parcs db of the Tube 205nbsp;if the Space db he divided into 10 equal Parts,nbsp;and you continue the Divilions upwards beyond

and downwards below each Divillon may be called 2; and by obferving the Diviiion tonbsp;which the Inftrument defeends in a Liquid, younbsp;will have the Bulk of the immerfed Part; fo, ifnbsp;the whole Tube ftands out above the M^ater, thenbsp;immerfed Bulk will be yqp; if it rifes to thenbsp;Upper Divifions, the immerfed Bulk will be yyp gt;nbsp;and the Denfities of the Liquids, in which thisnbsp;happens, will be inverfly as thofe Numbers, thatnbsp;is, as yyp to yqp, and only the intermediatenbsp;Denfities may be compar�d by this Inftrument}nbsp;if the Ball was lefs in Proportion to the Tube,nbsp;it would ferve for comparing together Liquidsnbsp;whole Denfities differ more than this. Whennbsp;feveral Liquids are compared together, thenbsp;Numbers which exprefs the Bulk of the iramerfed Parts are the Denominators of Fraftions,nbsp;which have i for their Numerator; and thefenbsp;Fraftions exprefs the Ratio of the Denfities;nbsp;for they are to one another inverfly as the Denominators.

Experiment 2.] Let the Denfities of Waters, containing different Quantities of Salts, be to be

Mathe?Mtkal Elements Book II.

compared, the Hydrometer defcends in the one to the Divifion if it be iramerfed in another,nbsp;it only defcends to the Divifion c, their Denficicsnbsp;will be to one another as to as may be ea-fily deduced from what has been laid.

This Method is alfo liable to feveral Difficulties befidesthis, that it is difficult to compare togethernbsp;Liquors very different in Denfity by the fame Hydrometer.

813 The beft Method of all is, to make ufe of a Solid heavier than the Liquids. When the famenbsp;Body is immerfed in different Liquids, the Weights,nbsp;'which it lofes in the Liquids, are to each other asnbsp;*297 the Denfities of thofe Liquids. * Here you muftnbsp;ufe a hydroftatical Balance, * and befidcs a folidnbsp;Piece �f Glafs, as C, which may hang to onenbsp;of the Scales by a Horfe-hair, Plate XXI. Fig^nbsp;y, and 6, you mult have a Weight, as P, whichnbsp;aequiponderates with the Glafs C, when it isnbsp;immerfed in Water, as is reprefented in Fig. 4-The Difference between the Weight P, and thenbsp;Weight of the Glafs C, When it is taken out ofnbsp;the Water, is the Weight which the Body hasnbsp;loft, when weigh�d in Water: This muft be ob-ferved, that it may fervc in all the Experiments;nbsp;in our Balance it weighs jiz Grains. Sufpend thenbsp;Body in any other Liquid, unlefs it be of thenbsp;fame Denfity as Water, the ^Equilibrium will notnbsp;be preferved : Let it be reftored by putting Grainnbsp;Weights in either of the Scales j if they be put into the Balance A, add them to the above-mentioned Difference of yzz Grains j if the Weightsnbsp;be put into B, fubftraft them from that Number}nbsp;and by that Means in each of thofe Cafes, as itnbsp;appears, the Weight loft by a Body is determined,nbsp;that is, the Weight which expreffes the Denfitynbsp;of the Liquid.

Book II. of KatiiralThilofofhy.

Experiment Letthe WeightC,which hangs from the Scale A, be immeifed in Oil of Turpentine, whilft the Weight P hangs from thenbsp;Scale B; put P4 Grains in the Scale B, and younbsp;will have an Equilibrium. Then immerge thenbsp;fame Weight in Milk, that the Balance may return to an Equilibrium., a Weight of zi Grainsnbsp;mud be put in the Scale A. Subllrafting the firftnbsp;Number from 722., and adding the fecond to k,nbsp;you will have 628 and 744, exprelEng the Den-lities of the above-mentioned Liquids^ whilft 72anbsp;ftiews the Denfity of the Water itfelf.

CHAP. V.

IN all homogeneous and equal Bodies, the Den-fities are as the Weights;* in unequal Bodies *288 of the fame Weight, the Denfities are inverfty asnbsp;the Bulks;* if therefore both the Bulks and Weghts *289nbsp;tliffer, the Ratio of the Denfities is compoundednbsp;of the dire�t Ratio of the Weights, and the in-'^erfe Ratio of the Bulks; and therefore, dividing , jnbsp;the Weights by the Bulks, you have the Denfities-, �nbsp;that is, you will have Numbers that are to each o-iher as thofe Denfities.

The Weight of allBodibs may be compared by , Means of the Balance; the Bulks are found by m~nbsp;Merging Bodies in the fame Liquid-, for the Weights,nbsp;�t^hich they lofe, are as the Bulks.*nbsp;nbsp;nbsp;nbsp;*299

Plate XXL Fig. 7.] Here alfo the hydroftati-ojil Balance is to be ufed,*as likewile theGlafi Vef-M D, in which the Bodies to be compared are to replaced; you muft alfo have fuch a Weight asnbsp;ts reprefented by P in FE 6. that is, equal to the

Mathematical Elements Book l�.

Weight of D5 and laftly, the Weight p {Fig.^.) equal to the Weight which the Glais D lofes,nbsp;�when it hangs in the Water.

With a Horfe-hair you muft fallen the Glafi D in the Place of the Body C (Fig.4.) in the Scalenbsp;A, and hanging the Weight P in the Scale B, younbsp;will have an ^Equilibrium. The Body, whofenbsp;Denfity is required, is placed in the Glals D (asnbsp;we have faid before) and weighed, the Verfelnbsp;and Body being immerfed in Water; putting thenbsp;Weight p into the Scale B, the .Equilibrium isnbsp;reftored in refpeft to the Glafs D j you muft addnbsp;as much Weight befides as is: required to make annbsp;Equilibrium, and that will be the Weight loft bynbsp;the Body weighed j by this Weight therefore younbsp;muft divide the Weight of the Body itfelf,to havenbsp;*315 the Denfity.^

Experiment i.] APieee of Gold, weighing 137 Grains, loft in Water y-~ Grains. A Piece ofnbsp;Silver, weighing 248 Grains, loft in Water 24nbsp;Grains. Therefore their Denfities are as tonbsp;lof, that is, nearly as ii to 6. By faftening a Body, whofe Denfity is required, and is heavier than anbsp;Liquid, to a Body lighter than the Liquid, thenbsp;Denfity is alfo defcovered.

Plate XXII. Fig. y.] Take the Machine A, like the Machine deferibed in the preceding Chapter,* and let it have fixed to its Bottom the Ringnbsp;DE, and to its Top the Ring FG; then thenbsp;Ball will, by its own Weight, be in part immerg�dnbsp;in Water. This Machine cannot be apply�d tonbsp;Ufe, unl�fs you know by fome other Method,nbsp;how much of its Weight any Body lofes in PV%-ter-, therefore we lay down as known, that lopWnbsp;Grains of Lead weigh \w Water \nViX. lOO'Grains gt;nbsp;therefore lay juft as many Grains upon the Ringnbsp;znbsp;nbsp;nbsp;nbsp;D E,

Book �I. of Na/nral Thilafophy.

DE as will make the Machine, when immerg�d in Water, defcend to a-, lay what Number ofnbsp;Grains you pleafe upon the Ring F G, for Example, Eight, and the Machine defcends to cnbsp;the Space a c muft be divided into Eight Parts,nbsp;and the Divifions nauft be continued upwards andnbsp;downwards; if you make a the hundredth Divifi-on, c will be the hundred and eighth Divifion, andnbsp;the loweft of all in this Figure will be the 97th.nbsp;�Tis plain, that if the Propofition of Nnm. 307 benbsp;compar�d with the aforefaid Preparation', the Divifion, to which the Machine defcends in Water,nbsp;fhews the Weight of the Grains which prefs downnbsp;the Machine i therefore laying a Body upon thenbsp;Ring D E, itsWeight inWater will be determinednbsp;by fubftra�ling this Weight from its Weight outnbsp;of the Water, you will have the Weight loft innbsp;the Water j by which if the Weight out of thenbsp;Water be divided, the Denfity is difcover�d, asnbsp;has been faid in the Beginning of this Chapter.

Experiment 2.] Lay a Piece of Brafs, weighing, for Example, 100 Grains, on the Ring DE by which the Ball of the Machine is not immerg�d i lay any Weight, for Example, of 17nbsp;Grains, on the Ring F G, and the Machine de-feends to b, that is, to the 105th Divifion ^ whichnbsp;proves, that the Machine is prefs�d down by fonbsp;many Grains j from this Number of Grains fub-ftratft the 17 laft mentioned, the remaining 88nbsp;are the Weight of the Piece of Brafs in W ater,nbsp;which therefore lofes 12 Grains. If again, thenbsp;Weight 100 Grains be divided by the 12 Grains,nbsp;you have 8 7, expreffing the Denfity of the Brafs.nbsp;The Denfities of any other Bodies may be foundnbsp;after the fame Manner.

M nbsp;nbsp;nbsp;CHAP.

CHAP. VI.

319 nbsp;nbsp;nbsp;A LL Bodies moved in Fluids fuffer a Refiji-jLjL. ance, which arifes from two Catifes. �Fhenbsp;firfi is the Cohefion of the Farts of the Liquid. Anbsp;Body in its Motion, feparating the Parts of a Liquid, muft overcome the Force with which thofenbsp;Parts cohere, and thereby its Motion is retarded.

12 'The fecond is the Inertia, or Inadfivity of Matter, that belongs to all Bodies, which is the Rea-fon, that a certain Force is required to remove the Particles from their Places, in order to let the Bodynbsp;pafs. The Body adls upon the Parts to removenbsp;them, and they diminilh its Motion by Re-aftion.

320 nbsp;nbsp;nbsp;The Retardation from the firft Caufe, that is,nbsp;the Cohefion of Parts, is always the fame in thenbsp;fame Space, the fame Body remaining, be thenbsp;Velocity of the Body what it will. The famenbsp;Cohefion is to be overcome in every Cafe; therefore this Refinance increafes as the Space runnbsp;through, in which Ratio the Velocity alfo in-

321 nbsp;nbsp;nbsp;Fbe Refifiance arifingfrom the Inertia, or Inactivity of Matter, when the fame Body moves throughnbsp;different Liquids with the fame Velocities^ followsnbsp;the Proportion of the Matter to be removed in thenbsp;fame Time, which is as the Denfity of the Liquid.

3 22 When the fame Body moves thro' the fame Liquid with different Velocities, this Refifiance increafes in Proportion to the Number of Particles ftruck in an equal Time, which Number is as thenbsp;Space run through in that Time, that is, as thenbsp;Velocity. But this Refinance does farther increafenbsp;in Proportion to the Force with which the Body

Book II. of Natural Thilofophy.

the Velocity of the Body. And therefore, if the Velocity is triple, the Refiftance is triple fromnbsp;a triple Number of Parts to be removed out ofnbsp;their Places. It is alfo triple from a Blow threenbsp;times ftronger againft every Particle j thereforenbsp;the whole Refiftance is ninefold, that is, as thenbsp;Square of the Velocity.

A Body therefore moved in a Liquid is refifted323 partly in a Ratio of the Velocity, and partly innbsp;a duplicate Ratio of it. The Refiftance from thenbsp;Cobejion of Parts in Liquid/, except glutinous ones,nbsp;is not very fenfible in refpedt of the other Refiftance j which as it increafes in a Ratio of the Squarenbsp;of the Velocities, * but the firft in a Ratio of the *nbsp;Velocity itfelf: *By how much the Velocity in-* 210nbsp;creafes, by fo much more do thefe Refiftancesnbsp;differ amongft themfelves; wherefore, in fjoifter 324nbsp;Motions the Refiftance alone is to be confidered,nbsp;which is as the Square of the Velocity.

I lhall not now treat of tenacious or glutinous Liquids, nor of flow Motions, in which the Refiftance, arilmg from the Cohefion of the Parts,nbsp;muft be confidered.

If a Liquid be included in a VeflTel of a prifmati- 3 25 cal Figure, and there be moved along in it withnbsp;equal Velocity, and a Diredtion parallel to the Sidesnbsp;of the Prifm, two Bodies, the one fpherical and thenbsp;other cylmdric, fo that/be Diameter of the Bafe ofnbsp;this laft he equal to the Diameter of the Sphere,nbsp;andJhe Cylinder be moved in the DireClion of itsnbsp;Axis, thefe Bodies will fuffer the fame Refiftance.

To demonftrate this, fuppofe the Bodies at reft, and that the Liquid moves in the Veflel with thenbsp;fame Velocity that the Bodies had; by this the re- 'nbsp;lative Motion of the Bodies and the Liquid is notnbsp;changed, therefore the Atftions of the Bodies onnbsp;'he Liquid, and of the Liquid on the Bodies, arenbsp;*iot changed. The Retardation which the Liquor

Mathematical Elements Book 11.

rufrefs in palling by the Body, arifes only front this. That in that Place it is reduced to a narrowernbsp;Space, but the Capacity of the Vefl�l is equallynbsp;diminilhed by each Body; therefore each Bodynbsp;produces an equal Retardation. And becaufe Action and Re-aftion are equal to one another, thenbsp;Liquids a�l: equally upon each Body; whereforenbsp;alfo each Body is equally retarded, when the Bodies are moved, and the Liquid is at reft.

326 nbsp;nbsp;nbsp;^his Demoiiflration will alfo obtain, tho� thenbsp;Vefl�l be fuppofed much bigger j and it will donbsp;in an infinite Liquid comprefled; therefore itnbsp;may be referred to Bodies deeply imtnerfed. Herenbsp;we fpeak of a continuous Liquid, and whofenbsp;Parts cannot be reduced into a lefler Space bynbsp;PreflTure; otherwife there will be an Accumulation before the Body, and a Relaxation behind jnbsp;and fo much the more, the more blunt the Body

327 nbsp;nbsp;nbsp;isj which alfo caufes a greater Irregularity in thenbsp;Motion of the Liquid, and a greater Retardationnbsp;in the Motion of the Body.

IVheu a Body is moved in any Liquid along the Surface^ the Liquid is raifed before the Body, andnbsp;depreflTed behind �, and thefe Elevations and De-preflions are greater, the more blunt the Body is,nbsp;and by that Means it is more retarded for therenbsp;is alfo a greater Irregularity in the Motion of the

32.8 Liquid in this Cafe, which ftill more increafes the Retardation of the Body, ithis is alfo true, ifnbsp;the Body be not immerfed deep yet in that Cafenbsp;the Irregularity of the Motion of the Liquid isnbsp;the chief Caufe of the Retardation.

Therefore, to take away thefe Irregularities, we muft confider Bodies as deeply immerfed, and givenbsp;Rules relating to them; by which the Retardations in feveral Cafes may be compared together.nbsp;We fuppofe the Bodies fpherical, tho� the De-monftrations will ferve for all fimilar Bodies movednbsp;in the fame manner,

Book II. of Natural�Philofo^hy. nbsp;nbsp;nbsp;165

Here you tnuft obferve, thzX the Kefiftajic, is to hediftingtdjhed from the Retardation j the Refift-ance produces the Retardation. When we/peak ofnbsp;the fame Body, the one may he taken for the other,nbsp;becaufe they are in the fame Proportion j but, fup-pofing the Bodies different, the fame Refiftancenbsp;often generates different Retardations. From the 330nbsp;Refiftance arifes a Motion contrary to the Motionnbsp;of the Body j the Retardation is the Celerity, andnbsp;the Refiftance itfelf is the ^lantity of Motion.

Let the Bodies be equal, but of different Denfi- 331 ties, and moved thro^ the fame Liquid with equalnbsp;Velocity, the Liquid afts in the fame Manner upon both i therefore they fuffer the fame Refiftance, but different Retardations �, and they arenbsp;to one another as the Celerities, which may be generated by the fame Forces in the Bodies propo-fed; * that is, they are inverfly as the Quantities * 33nbsp;of Matter in thofe Bodies, * or inverfty as the * enbsp;Denfities.

Now, fuppofing Bodies of the fame Denfity, but 331 tmeqtial, moved equally faft thro' the fame Fluid,nbsp;the Refiftances increafe according to their Super-fices, that is, as the Squares of their Diameters;nbsp;the Quantities are increafed in Proportion to thenbsp;Cubes of the Diameter the Refiftances are thenbsp;Quantities of Motion, the Retardations are thenbsp;Celerities arifing from them j * dividing the*nbsp;Quantities of Motion by the Quantities of Matter, you will have the telerities i * therefore thenbsp;Retardations are diredtly as the Squares of thenbsp;Diameters, and inverfly as the Cubes of the Diameters, that is, inverfly, as the Diameters them-felves.

If the Bodies are equal, move equally �^^;r�z�, 332 and are of the fame Denfity, but are moved thro'nbsp;different Liquids, their Retardations are as thequot; 339nbsp;Renfijties of thofe Liquids,*

Mathematical Elements Book II,

334 nbsp;nbsp;nbsp;iVhen Bodies^ equally denfe and equals are carriednbsp;thro� the fame Liquid with different Velocities, the

* 3Z4 Ketardations are as the Squares of the Velocities. * 319 From what has been faid, the Ketardations of

335 nbsp;nbsp;nbsp;any Motions may be compared together, for theynbsp;'' m are firfi.^ as the Squares of the Velocities ; * fe-

condly, as the Denfities of the Liquids thro� which 353 the Bodies are moved j thirdly, inverjly, as theDi-quot; ameters of thofe Bodies^ lafily, inverfly, as thenbsp;331 Denfities of the Bodies them [elves. *

The Numbers in the Ratio, compounded of thofe Ratio�s, exprefs the Proportion of the Retardations. Multiplying the Square of the Velocity by the Denfity of the Liquid, and dividingnbsp;the Product by the Produ�t of the Diameter ofnbsp;the Body multiplied into its Denfity, and working thus for feveral Motions, the Quotients of thenbsp;Divifions will ftill have the fame compound Rationbsp;to one another.

Thefe Retardations may alfo be compared together, by comparing the Refiftance with the

336 nbsp;nbsp;nbsp;Gravity. It is demonftrated, that the Kefiftoncenbsp;of a Cylinder, which moves in the Dire61 ion of itsnbsp;Axis (to which the Refiftance of a Sphere of the

^^5 fame Diameter is equal *) isequal to the Weightlt;ff a Cylinder made of that Liquid, thro� which thenbsp;Body is moved, having its Bafe equal to the Body'snbsp;Bafe, and its Height equal to half the Height, fromnbsp;�which a Body fallingm'Wamp;c\io may acquire the Velocity with which the faid Cylinder is moved thronbsp;the Liquid. From the given Celerity of the Bodynbsp;moved, the Height of the Liquid Cylinder isnbsp;found, as alfo the Weight of it from the knownnbsp;fpecifick Gravity of the Liquid and Diameter ofnbsp;the Body. Let a Ball, for Example, of 3 Inchesnbsp;Diameter be moved in Water with that Celeritynbsp;with which it would go thro* 16 Foot in a Second : From what has been faid of falling

Book IL of Natural Thilofophy. nbsp;nbsp;nbsp;1�7

Bodies and Pendulums, * as alfo by Experiments made on Pendulums, it has been found that thisnbsp;is the Celerity which a Body acquires in fallingnbsp;from a Height of 4 Footj therefore the Weightnbsp;of a Cylinder of Water, of 3 Inches Diameter,nbsp;and 2 Foot high, that is, a Weight of about 6nbsp;Pounds and 3 Ounces, is equal to the Refiftancenbsp;of the aforefaid Ball.

Let the Refiftance fo difcovered he divided by the U''�eigbt of the Body-, which determines its (Quantity of Matter, and you, will have the Retardation*

By which Rule the Proportion of the feveral Retardations is difcovered, * and found to be the * fame as is given by the foregoing Rule.

Having confidered the Retardations of diredl Motions, we pafs on to the Motion of Pendulums.

The Arc defcribed by a Pendulum ofcillating in Vacuo, with a Celerity that it has acquired by defending, is equal to the Arc which is defcribednbsp;by the Defcent; * the fame does not happen in a *nbsp;Liquid, and there is a greater Difference betweennbsp;thofe Arcs, the greater the Refinance is j that is,nbsp;if you fpeak of the fame Liquid and Pendulum,nbsp;the greater the Arc is which is defcribed in thenbsp;Defcent.

Let theRefifiance of the Liquid be in Proportion to the Velocity, and two Pendulums^ entirely alike,nbsp;ofcillating in a Cycloid, perform unequal Vibrations^nbsp;and begin to fall the fame Moment; they begin tonbsp;move by Forces that are as the Arcs to be defcribed �,* *nbsp;if thofe Impreflions alone, which are made thenbsp;firft Moment, be confidered, after a given Time,nbsp;the Celerities will be in the fame Ratio as in thenbsp;Beginning; for the Retardations, which are asnbsp;the Velocities themfelves, * cannot change their *nbsp;Proportions, for the Ratio between (Quantitiesnbsp;is not changed by the Addition and Subftradlionnbsp;of the (Quantities in the fame Ratio. Therefore

Mathematical Elements Book I�.

in equal Times, however the Celerities of Bodies are changed in their Motion by the Re-fiftance, Spaces which are gone thro� are as the �* 53 Forces in the Beginning; * that is, as the Arcs tonbsp;be defcribed by the Defcent; therefore after anynbsp;Time the Bodies are in the correfpondent Pointsnbsp;of thofe Arcs. But in thefe Points the Forces arenbsp;15� generated in the fame Ratio as in the Beginning,*nbsp;and the Proportion of the Celerities, which is notnbsp;varied by the Refiftance, fuffers no Change fromnbsp;the Gravity. In the Afcent, Gravity retards thenbsp;Motion of the Body, but in correfpondent Pointsnbsp;its A�lions are in the fame Ratio as in Defcents,nbsp;And therefore every where in correfpondent Pointsnbsp;the Celerities are in the fame Ratio. But as innbsp;the fame Moments the Bodies are in thefe correfpondent Points, it follows that the Motion ofnbsp;both is deftroyed in the fame Moment, that is,nbsp;they finijh their Vibrations in the fame 'Time. Thenbsp;Spaces, run thro� in the Time of one Vibration,nbsp;are as the Forces by which they are run thro�;nbsp;340 that is, the Arcs of the whole Vibrations are asnbsp;the Arcs defcribed by the Defcent., whofe Doublenbsp;are the Arcs to be defcribed in Vacuo. The De-2,^ife�is of the Arcs to be defcribed in Liquids^ fromnbsp;the Arcs to be defcribed in Vacuo, are the Differences of Quantities in the fame Ratio, and are asnbsp;the Arcs defcribed by the Defcent.

542 Since there is the fame Proportion between thofe different Arcs, it follows, that the Celerities.^nbsp;in the correfpondent Points of the Arcs defcribed.,nbsp;are every where as the Arcs defcribed by the Defcent ; for thefe correfpondent Points are alfo thenbsp;correfpondent Points of the Arcs to be defcribednbsp;in Vacuo, in which we have demonftrated thatnbsp;this Proportion holds.

343 Now let the Kefiftance increafe in the Duplicate Ratio of the Velocity, and let the Pendulum perform

Book II. of Natural Thilofo^hy. nbsp;nbsp;nbsp;i6^

unequal Vibration^ the greatefi will laft the longefiy � becaufe the Refiftance increafes more than in thenbsp;Cafe Numb. 239.

Yet the Celerities.^ fnppofing the Arcs not very 344 unequal in the correfpondent Points of the Arcs de~nbsp;fcribed, are every where nearly in the fame Ratio,nbsp;and indeed in the Ratio of the Arcs defcribed bynbsp;the Defcent. If the Refiftance was in the Rationbsp;of the Celerity, this Proportion would obtain j * but now it is difturbed by reafon of*

� a greater Refiftance in a greater Vibration, by which the Motion in this is more diminilh-ed. But it is more accelerated by two Caufes,nbsp;ift, this greater Vibration lafts longer, * and the * 341nbsp;Jl Body ftays longer in a certain Space, than in thenbsp;correfpondent Space in a lefs Vibration, and isnbsp;accelerated during a longer Time. 2dly, thenbsp;Defeft of the Arc defcribed here, from an Arc tonbsp;be defcribed in Vacuo, is greater in Proportion,nbsp;in a greater Vibration, becaufe in this the Refiftance differs more from the Refiftance in a lefsnbsp;Vibration than in Numb. 241. Therefore the correfpondent Points, keeping the fame Proportion,nbsp;are more diftant from the loweft Point in the greater than in the lefler Arc, as long as the Body de-fcends in it: therefore in Proportion it has a greater Acceleration, becaufe the Force which a�tscon-4nbsp;nbsp;nbsp;nbsp;tinually on the Body, is as its Diftance from the

lower Point gt; * therefor^ there is a Compenfation, * and the Proportion above-mentioned is reftored.

In the Afcent of the Body, the Duration of the !nbsp;nbsp;nbsp;nbsp;Retardation concurs with the Refiftance to di-

nuing) than in the lefiTer, and the Gravity in Proportion produces a lefs Retardation j and

1^0 Mathematical Elements Book II.

therefore now (the Proportion continuing) the Difference of the Diftance of the correfpondentnbsp;Points from the loweft Point is increafed, fo thatnbsp;a Compenfation is given from this alone.

The Refiftances which are as the Squares of the Celerities, and therefore every where in correfpondent Points, as the Squares of the Arcsnbsp;defcribed by the Defcent, in which Ratio alfo thenbsp;� 329 Retardations are; * but, as each of them keep thenbsp;fame Proportion incorrefponding Points, the Sumsnbsp;of them all will be in the fame Proportion j thatnbsp;is, the whole Retardations, which are the Defeftsnbsp;of the Arcs defcribed in the Liquid from the Arcsnbsp;to be defcribed in Vacuo ; or, what is the fame,nbsp;the Differences hetwen the Arcs defcribed in thenbsp;345 Defcent and the next Afcent. Therefore thefe Differences, if the Vibrations are not very unequal^nbsp;are nearly as the Squares of the Arcs defcribed bynbsp;the Defcent. Which is alfo confirmed by Experiments in greater Vibrations j for in thefe the Proportion of Refiftance, which we treat of here, ob-� 324 tains.

2^6 Fill the wooden Veffel A B FCD (Plate�KX.ll. Fig. 6.) 3 Foot long, i Foot wide, and i Footnbsp;high, with IVater ; hang up the Pendulum V pnbsp;by a Hook V hanging over the Middle of thenbsp;Veflel. This Pendulum is made of an Iron Wirenbsp;7 or 8 Foot long, and a Leaden Ball p of thenbsp;Diameter of an Inch and a Half; when the Pendulum is at reft, the Ball is diftant 3 Inches fromnbsp;the Bottom of theVeflel. At P there is a greater Ball of Lead, of 3 Inches Diameter, joined tonbsp;the Iron Wire, that the Ball P may be the lefsnbsp;retarded in Water.

A-crofs the Top of the Veffel, upon the Brim of it, may be moved a Board about 5 Inchesnbsp;high, to which muft be applied the divided Brafsnbsp;Rulers EG, EG, and the Indices M, M, fornbsp;2nbsp;nbsp;nbsp;nbsp;meafuring

Book II. of Natural Thilofo^hy. 171

meafuring the Angles defcribed by the Pendulum in the Defcent, or Afcent, by the Method givennbsp;Ntmb. 170. Page 70.

Experiment.'] Let the Rulers EG, EG, be fo difpofed, that the Ends G, G, may be over-againft the Pendulum when it is at reft, and in fuchnbsp;Manner that between their Ends there may be anbsp;Diftance equal to the Diameter of the Wire tonbsp;which the Bodies Pp are fij^ed. Let one Indexnbsp;be applied to the i6th DiviCon of the Ruler, andnbsp;another to the 14th Divifion of the other Ruler jnbsp;let the Pendulum fall from that Divifion, andnbsp;it will rife almoft to this. If, inftead of thefenbsp;Divifions, you take 20 and 16 s, the Experiment will fucceed in the fame Manner, asnbsp;alfo when you apply the Indices to the Divifions 24 and 19 2- Take care that the Water benbsp;perfedfly at reft.

In this Experiment the Arcs defcribed in the Defcent are to one another as 4, 5, and 6, whofenbsp;Squares are 26, 35, 26 j the Difl�rence of thofenbsp;Arcs from the Arcs defcribed in the Afcent, arenbsp;2, 3f, 4ij which Numbers are to one anothernbsp;as the aforefaid Squares, as appears by multiplying them by 8.

Body freely defcendingin a Liquid is accelerat- 347 ed by the refpe�iive Gravity of the Body whichnbsp;continually afts upon it^ yet not equally, as in anbsp;Vacuum, * the Refiftance of the Liquid occafions * 129,nbsp;a Retardation, that is, a Diminution of Acceleration, which Diminution increafes with the Velocity of the Body. For there is a certain Velocity 348nbsp;which is the great eft that a Body can acquire hyfal-ling ; for ?� its Velocity be fuch, thoXthe Kefijlancenbsp;arifingfrom it becomes equal to the reftpeSlive Weight

the Body, its Motion can be no longer accelerated i for the Motion which is continually generated

Mathematkal Elements Book II.

nerated by the reCpeftive Gravity, will be de-ftroyed by the Refiftance, and the Body forced to go on equably ; ^he Body continually comesnbsp;nearer and nearer to this greateft Celerity^ but cannbsp;never attain to it.

When the Denfities of a Liquid and a Body are given, you have the refpeftive Weight of thenbsp;Body ; and by the knowing the Diameter of thenbsp;Body, you may find out from what Height anbsp;Body falling in Vacuo, can acquire fuch a Velocity, that the Refiftance in a Liquid ihall be e-f jjcqual to that refpediVe Weight, * which will benbsp;that greateft Velocity above-mentioned.

If the Body be a Sphere, it is known that a Sphere is equal to a Cylinder of the fame Diameter, whofe Height is two third Parts of that Di-^yoameter , which Height is to be increafed in thenbsp;Ratio in which the refpelt;ftive Weight of the Body exceeds the Weight of the Liquid, in ordernbsp;to have the Height of the Cylinder of the Liquid,nbsp;whofe Weight is equal to the refpeSiive Weight ofnbsp;the Body but, if you double this Height, you willnbsp;have a Height, from which a Body, falling in Vacuo, acquires fuch a Velocity as generates a Re^i-* 3 36fiflance equal to this refpehiive Weight, * andwhicbnbsp;therefore is the greateft Velocity which a Body cannbsp;�* 3 8 acquire,falling in a Liquid from an infinite Height.*nbsp;Lead is eleven times heavier than Water, wherefore its refpe�tive Weight is to the Weight ofnbsp;Water as 10 to i j therefore a leaden Ball, as itnbsp;appears from what has been faid, cannot acquirenbsp;a greater Velocity, falling in Water, than it wouldnbsp;acquire in falling in Vacuo, from an Height ofnbsp;13 � of its Diameters.

3 � I A Body lighter than a Liquid,and afcendinginit by the ASlion of the Liquid, is moved exadily by thenbsp;fame Laws as a heavier Body falling in the Liquid.nbsp;Where-ever you place the Body, it is fuftaiped by

Book II. nbsp;nbsp;nbsp;of Natural Thilofophy.nbsp;nbsp;nbsp;nbsp;173

the Liquid, and carried up with a Force equal to the Diference of the Weight of the Quantity ofnbsp;the Liquid, of the fame Bulk as the Body, fromnbsp;the Weight of the Body, as appears by comparing Numb. 293 with Numb. 292 j therefore younbsp;have the Force that continually adls equably upon the Body, by which, not only the Adlion ofnbsp;the Gravity of the Body is deftroyed, fo that itnbsp;is not to be confidered in this Cafe, but by whichnbsp;alfo the Body is carried upwards by a Motionnbsp;equably accelerated, in the fame Manner as a Body, heavier than a Liquid, defcends by its re-fpeftive Gravity j but the Equability of the Acceleration is deftroyed in the fame Manner by thenbsp;Refiftance, in the Afcent of a Body lighter thannbsp;the Liquid, as it is deftroyed in the Defcent of anbsp;Body heavier than the Liquid.

When d' Body, fpecifijcally heavier than a Liquid, 352 is thrown up in it, it is retarded upon a doublenbsp;Account, on account of the Gravity of the Body, and on account of the Refiftance of the Liquid i therefore a Body rifes to a left Height than 353nbsp;it would rife in Vacuo with the fame Celerity.

But the Defelfs of the Height in a Liquid, from the Heights to which a Body would rife in Vacuonbsp;with the fame Celerities, have greater Proportionnbsp;to each other than the Heights themfelves j and,

, in lefs Heights, the Defeats are nearly as the �Slquares of the Heights in Vacuo.

P A R TII. Of the Motion of Fluids.

CHAP. VII.

Of the Celerity of a Fluid arijing from the Trejfure of the fuperincumbent Fluid.

An inferior Fluid is prefled by a fuperior, and that equally every Way j * becaufenbsp;A�lion is equal to Re-adion, it endeavours tonbsp;recede every Way with equal Force ; therefore ifnbsp;you take off the Preffure on one Side, the Liquid willnbsp;354 towards that Side-, and which Way [oever thenbsp;Preffure he taken away, it will move with the famenbsp;Celerity which will be confirm�d by the �xperi-ments to be mention�d in the following Chapter.

At the fame Depth the Celerity is alfo every where the fame, by reafon of the Equality of thenbsp;* Preflure, * but when the Depth is changed, thenbsp;a/zCelerity is alfo changed.

Yet the Velocity does not follow the fame Proportion as the Depth; tho� the Preflure, from which the Velocity arifes, does increafe in thenbsp;* 273 fame Ratio as the Depth. * The Quantity of Motion, which is produced in the Liquid, is the Ef-fed of the whole Preflure; and this Quantitynbsp;�* 59increafes as the Preflure; * but the Ratio of thenbsp;Quantity of Motion is compounded of the Rationbsp;ot the Velocity and the Quantity of the Matternbsp;64moved. * Here the Matter moved is the Water,nbsp;which goes out of the Hole, whofe Quantity,nbsp;the Time remaining the fame, increafes with the

Book 11. of Natural Thilofofhy. nbsp;nbsp;nbsp;175'

Celerity; it will be double, if the Celerity be doubled, in which Cafe the Quantity of Motion 355nbsp;is quadruple; that is, increafed as the Square ofnbsp;the Celerity^ which obtains in any Celerity: Therefore that Square increafes as the Preflure; that is,nbsp;is the Height of the Liquid above the Hole fromnbsp;�which the Water [pouts.

P/^?reXXlV. Fig. I.] Fill with Water the Pa-356 rallelopiped A B, which is 15 Inches long, andnbsp;as wide, and 2 Foot high; it muft be fo placednbsp;that its Bottom may be raifed about 8 Inches abovenbsp;-the horizontal Bottom of a hollow Trough C D,nbsp;whofe Length is almoft 4 Foot, and Breadth anbsp;Foot and a half, and Depth 5 or 6 Inches,

At E, near the Bottom of the Veflel AB, there is fix�d a Brafs Tube horizontally, above halfnbsp;an Inch in Bore; the fore Part of it is ihut bynbsp;a Plate, in the Middle of which there is a Hole,nbsp;whofe Diameter is equal to h Inch; that Hole isnbsp;Ihut with a Cover that fcrews on upon the forenbsp;Part of the Tube.

The Celerities with which the Water flows out from E, when you have open�d the Hole,nbsp;are compared together by Help of this Machine.nbsp;Let it move, for Example, in the Line E L, andnbsp;at L let it come to the Bottom of the Veflel CD;nbsp;this Motion may be refolved into two Motions;nbsp;the one horizontal along E I, in the Diredfionnbsp;that the Water has in going out of the Hole, andnbsp;the other vertical along 1 L; the firft is equable, and the Water^ with the Celerity with whichnbsp;it goes out, runs thro� the Space E I, in thenbsp;fame Time that in falling it runs thro� IL; *nbsp;�whatever the Celerity be, I L is not changed,nbsp;hecaufe E I and the Bottom of the VeflTel arenbsp;horizontal, therefore the Time is not changed,nbsp;in which feveral fuch Lines as E I may be run

Mathematical Elements Book It

thro�, and therefore they are as the Celerities with �* 53 which the Water goes out: * if you meafure thenbsp;Diftance to which the Water fpouts, you will havenbsp;the Line E I.

Experiment 1 Let there be Water in the Vef-fel A B, up to the Height of five Inches above the Hole at Ej let the Diftance be meafurednbsp;to which the Water fpouts �, if more Water benbsp;poured in, to the Height of twenty Inches, it willnbsp;fpout to a double Diftance. The Squares of thenbsp;Diftances are here as the Height of the Water,nbsp;in which Ratio alfo are the Squares of the Celerities.

357 The Velocity of a Liquid^ at any Depth, is the fame as that which a Body, falling from a Height equal tonbsp;the Depth, would acquire �, for the Velocity of a Liquid increafes, when the Depth of the Hole belownbsp;the Surface of the Liquid increafes, in the famenbsp;Ratio as the Celerity of the falling Body increafes, when the Space gone thro� by the Fall in-* 355 creafes*; and in the Beginning thefe Velocitiesnbsp;*^^are equal j for in a Liquid the upper Parts, asnbsp;^ well as in a Body at the Beginning of the Fall,nbsp;endeavour to defcend by Gravity only.

Experiment 2, Plate XXIV. Fig. i.] This is performed with the fame Machine as was ufed innbsp;the former Experiment the Veffel A B is fillednbsp;with Water, and a Tube with a Hole, like thenbsp;Tube E, is placed at F^ fo that the Height ofnbsp;the upper Surface of the Water, above thenbsp;Bottom of the VelTel C D, is divided by thatnbsp;Hole into two equal Parts; the Water from thatnbsp;Hole will fpout to M, fo that the horizontalnbsp;Diftance from the Point M to the Hole will benbsp;double the Height of the Hole, above the Bottomnbsp;cf the VelTel C D i therefore the Water, by an

Book n. of Natural Thilofofhy. 177

equable Motion, and with the Celerity with which it goes out, runs thro� double the Spacenbsp;of that Height, in the Time in which a Body cannbsp;fall from F to the Bottom of the Veflel C D, andnbsp;therefore it moves with the Celerity which a Bodynbsp;can acquire in falling from that Height * j but thisnbsp;Height is equal to the Height of the Surface ofnbsp;the Water above the Hole.

CHAP. VIII.

Of ff outing Liquids,

Liquid, [pouting vertically out of a Hole, a-riles up with that Celerity, with which it

vjould come up to the upper Surface of the Liquid, yet it never comes up to tbqt Height ^; and that for *nbsp;feveral Caufes. i. The Celerity, by which the Liquid afcends, is diminifhed every Moment, andnbsp;the Column of the fpouting Liquid confifts ofnbsp;Parts, which are moved to different Heights bynbsp;different Celerities ; all the Parts of a Column,nbsp;which is every where of the fame Thicknefs, arenbsp;necelfarily moved by the fame Celerity the faidnbsp;Column every where will be broader every Moment, as the Celerity of the Liquid is diminifhed �,nbsp;which arifes from the Impulfe of the Liquid following, and which from the Nature of a Liquidnbsp;yields to every ImprefHon, and is eafily movednbsp;every Way i by that Impreffioh the Motion isnbsp;retarded every where. 2. This Motion is alfonbsp;diminilhed by the Liquid, becaufe, when it hathnbsp;loft all its Motion, it hangs in the upper Part ofnbsp;the Column, and is fuftained for a Moment by thenbsp;Liquid that follows, before it flows off on thenbsp;Sides, which retards the Liquid that follows it,nbsp;and that Retardation is communicated to thenbsp;whole Column. 3. By the Friftion againft the

N nbsp;nbsp;nbsp;�ide$

Mathematical Elements Book II.

Sides of the Hole, the Celerity of the Liquid is diminifbed j which Fridionis increafed, when thenbsp;Liquid is brought through Pipes and Codes.nbsp;4 Laftly, the Air�s Refiftance flops the Motionnbsp;of Liquids.

The fecond is corre�ted by fomewhat inclining the Direction of the Liquid, as is felf-evident jnbsp;359nbsp;nbsp;nbsp;nbsp;is the Reafon why a Liquid rifes higher^

Experiment!. XXIV. Fig. i. ] To the ' 356 Machine above-deferibed, * by help of a Screwnbsp;at N, join the Curve Tube N O, from whichnbsp;the Water, thro� a fmall Hole, fpouts up vertically by turning the Tube a little, which isnbsp;eallly done by reafon of the Screw at N, thenbsp;Dire�bion of the fpouting Water will be inclined,nbsp;and it will afeend higher. But by this Inclinationnbsp;the Beauty of a Jet is deftroyed.

As to the third Caufe of the Retardation, �tis to be obferved, that there is a greater Fri�cion, innbsp;Proportion, in fmall Holes, than in great ones;nbsp;the Celerity being increafed, the Friction alfonbsp;j5Qis thereby increafed j and therefore the Holes arenbsp;to he increafed according to the Heights of the fpouting Water.

The Ends of the Pipes, from which the Water Ipouts, have commonly the Figure of a truncated Cone, as is reprefented at P, (Elate XXIV.nbsp;Fig. 3:) in which End the Water fuffers a greatnbsp;deal of Friction, and is moved irregularly, andnbsp;fpouts up with that Irregularity. This may be a-361 mended by covering the End of the Tube quot;with aflat,nbsp;fmo0th,andpolifloedPlate,fixed to it, which has a Holenbsp;inft-, �ox�iamp;n the Water fpouts higher,andhQOOjfcnbsp;.nbsp;nbsp;nbsp;nbsp;it

Boo/c II. of Natural Th'ilofofhy. nbsp;nbsp;nbsp;179

Experiment 2. Plate XXIV. Fig. 3.] Take the Tube above-mentioned P, as alfo the Cylindernbsp;Q, (hut up at one End with a bored Plate j letnbsp;thofe be fcrewed on, one after another, to the Endnbsp;O of the Tube N O QFig. i.) the Water remaining at the fame Height in the VefTel A B, letnbsp;it fpout from the Tube P, and the Cylinder Q,nbsp;and the Experiments will fully confirm what wenbsp;have faid.

Fbe Pipes which bring the Water from a Refer-^^62 voir, niuft be very wide, in Proportion to the fpout'nbsp;ing Hole, that the Water may move (lowly in thefenbsp;Pipes, and have no fenfible Fridlion.

For the fame Rtamp;ionthe Water-way, orPaf-363 fage, of the Cocks muji be very largCi

Experiment 3. Plate XXIV. Fig. i.] To the VefTel A B, at the fame Height as the Tube F,nbsp;fix the Cock H, the Pipe of the Cock mull benbsp;(hut up in the fame Manner as the Tube F, andnbsp;bored with a Hole of the f�me Size �, the Waterway of this Cock is a Quarter of an Inch. Thenbsp;Water which goes thro� this Cock is broughtnbsp;thro� a narrower Space than that which movesnbsp;thro� the Tube F �, therefore this laft is morenbsp;tranfparent, and fpouts to a greater Diftance.

Fhe Refifiance of the Air has a fenfible EffeBupon 364 the Motion of Liquids �, (for itfelf may be reckonednbsp;amongft Liquids, as w ill be faid in the Secondnbsp;Partj) therefore we may here apply what hasnbsp;been faid of the Afcent of any Body in a Liquidjnbsp;and in fmall Heights, the DefeCis of the Heightsnbsp;from the Heights in Vacuo are in the Ratio of thenbsp;Squares of thofe Heights * j that is, abftradl-*nbsp;5ng from the other Caufes of Retardation, theynbsp;N 2nbsp;nbsp;nbsp;nbsp;are

Mathematical Elements Book I�.

are in the Ratio of the Square of the Height of the Liquid above the Hole. Befides this Refiftancenbsp;there is alfo another, not to be over-looked, whichnbsp;is the Adtion of the Air againft the fpouting Liquid. It enclofes the whole Column of the fpouting Liquid, and refills that Part of its Motion,nbsp;whereby it fpreads itfelf fide-wife, as it becomesnbsp;wider, and there is required a greater Force of thenbsp;Liquid that comes after, than if this Refiftancenbsp;was taken away j therefore the Air refifts by itsnbsp;lateral PreflTure. The Refiftance from the Strokenbsp;of the Liquid againft the Air encreafes with thenbsp;impingent Surface, that is, if the Celerities remainnbsp;the fame, encreafes with the Hole ; in which Ratio alfo, the Quantity of the Matter moved encreafes, and upon this Account it is no matter ofnbsp;what Bignefs the Hole is.

The lateral Preflure follows the Proportion of the Surface of the Column j the Matter moved,nbsp;which (the Celerity being the fame) is in the famenbsp;Ratio as the Quantity of Motion, * follows thenbsp;Proportion of the whole Column, that is, of thenbsp;Square of its Surface and therefore, if the Holenbsp;be encreafed, the Quantity of Motion encreafesnbsp;fafter than the Caufe retarding it j and for thatnbsp;365 Reafon in the greateft Heights of fpouting Liquids,nbsp;that the lateral PrefiTure (which exerts a greaternbsp;Adlion when it ads the longer) may be the better overcome, greater Holes are required whichnbsp;we have alfo fhewed before to be required in thenbsp;quot;*360 fame Cafe from another Caufe : * where, as well asnbsp;here, we fuppofed the greater Holes only neceflarynbsp;for the greateft Heights, though the Demonftrati-ons prove that thefe Holes, which are very necef-fary in the greateft Heights, are in general to benbsp;preferred to others.

Great Holes alfo hinder the Motion ; for then there is a greater Surface which is preflTednbsp;inbsp;nbsp;nbsp;nbsp;upon

Book 11. of Natural Thilofofhy. i8i

upon by the higheft Part of the Liquid, which has loft all its Motion, and hangs on a longernbsp;Time, before it runs off down the Sides. Fromnbsp;thefe two contrary Effects joined together, innbsp;Heights there is a certain Meafure of the Hole, thro*nbsp;which the Liquid will rife to the greateft Heightnbsp;pofible. Yet one cannot give Rules to determinenbsp;the Diameter of the Hole, becaufe the Bignefs ofnbsp;the Pipes of Condu�l and their Inflexions requirenbsp;it different, fo that there may be a Variation tonbsp;Infinity.

In refpedl to the greateft Heights, it is to be 367 obferved, that the Bignefs of the Hole, and alfothenbsp;Height to which the Liquid can afeend, have theirnbsp;Limits, which they cannot exceed. Not only thenbsp;Liquid -which is direftly againft the Hole runsnbsp;out, but, that there may be a conftant Supply,nbsp;the neighbouring Liquid continually comes towards the Hole with an oblique Motion, andnbsp;in going out it fpouts with a compound Motion,nbsp;and the Motion of the Liquid, fpouting vertically,nbsp;is difturbed ; the greater the Hole is, the greaternbsp;is the Difturbance arifing from that Caufe, andnbsp;in fpouting Waters the Holes fhould never exceednbsp;an Inch and a Quarter. When the Celerity ofnbsp;the Liquid is too great, it ftrikes agajnft thenbsp;Air with fo much Force, that it is difperfed intonbsp;Drops i in which Cafe, by diminilhing the Celerity, the Height to which the Liquor fpoutsnbsp;will be increafed, and there is a Height which isnbsp;the greateft to which a Liquor can afeend, whichnbsp;Height in fpouting Water fcarce exceeds 109 Feet.

Liquids, which fpout obliquely, are not retarded from fo manyCaufes, nor fo rnuch,asthofe that fpoutnbsp;vertically.ThQ fecond Caufe of Retardation, above-mentioned, * has no Place here, and the Effedl of* 35�nbsp;the firft is lefs. As for the reft, one may applynbsp;^ere w'hat has been faid of Solids oblique pro-N 3nbsp;nbsp;nbsp;nbsp;jedled

Mathematical Elements Book II.

369jefted in Cha^. XXIV. Book I. from Numh. 212. quite to the End of the Chapter. And a Liquidnbsp;may he conjide�fed as an innumerable ^lantity ofnbsp;Solids^ following one another.^ and running the famenbsp;Way. In the Motion of the Liquid, the Waynbsp;gone thro� may be perceived by our Scnfes, andnbsp;what has been faid of Solids, obliquely projeded,nbsp;may be reduced to Experiments by the Help ofnbsp;Liquids i for doing which we muft make ufe ofnbsp;Quickfilver, becaufe of the great Specifiek Gravity of this Liquid in refped of others : Butnbsp;thefe Experiments are to be made by a particularnbsp;Machine contrived as follows.

gyo Blate XXIII. Fig. i.] The wooden Trough A, B, C, D, E, F, H, is four Foot and an halfnbsp;broad, eight or ten Inches long, and fix or fevennbsp;Inches high j the Bottom is made of a Boardnbsp;hollowed in half an Inch, to contain the Mercury the better.

In the End H, of the Side E, F, H, you have a Board H I fix Inches wide, and two Footnbsp;high, which has in it a Slit 0 t. By this meansnbsp;you may fix to any Height upon the Board thenbsp;wooden Parallelopiped s, which has a Screw fixednbsp;in its hinder Part.

The fecond Figure reprefents this Parallelopiped at S: There is faftened to it a cylindric Veflelnbsp;of Box Wood, which has a Groove round it tonbsp;receive two Brafs Plates, one of which may benbsp;feen at �^? j their Ends are joined together bynbsp;the Screw G, fo as to make the Box VeflTel immoveable, till it is loofened by unferewing g,nbsp;which will allow it to move about its Axis.

In the Bottom of this Veflfel there is a cylindric Cavity a b, a. Quarter of an Inch Diameter; this communicates with a like Cavity h c, whichnbsp;terminates in the Middle of the greater Cavity

�whofe Diameter is above half an Inch, that it may receive the truncated Cone of Box H, whichnbsp;is joined to the Cylinder I L, Fig. 3.

The truncated Cone H exactly fills the Cavity c and is held fall in it by help of the Screw R, that goes thro�the Brafs Plate Q^O, but fonbsp;that this truncated Cone may turn upon itsnbsp;Axis.

In this Cone, as well as in the Cylinder I L, there is a Cavity h i /, of the fame Diameter asnbsp;the Cavity b c, and anfwering to it. This Cylinder I L has a Glafs Tube N cemented to it.

The Tube is a Foot and a half long, one End of which is feen at N M (Hg. 5.) which is cemented alfq to the Box Cylinder L I, which isnbsp;hollowed at / i h, w ith a round Hole in the Formnbsp;of a Gnomon, or the Carpenter�s Square, ztb cnbsp;the Cavity is greater, to receive the truncated Conenbsp;C D, that exaflly fills it, and is moveable aboutnbsp;its Axis by the Help of the Handle E A.

The Cavity h i anfwers to the Cavity d e, w hich communicates with � g-, this Part of thenbsp;Box has driven upon it an Iron Ferril B Q, innbsp;which is drilled a very fmall Hole g, which, whennbsp;the Parts of the Machine are joined together,nbsp;communicates wdth the Cavity of the Box P

To prevent the Tube from breaking the Ends L L of the Box Cylinders (PVg. 3. and 5.) togethernbsp;with the Tube, are joined clofe to a ftreight andnbsp;ftiff Piece of Wood � (Fig. i.) whofe lower Endnbsp;ni has an Iron Plate fixed to it (as may be feen innbsp;Fig. 6.) whofe End LPBQis bent in the Figurenbsp;of a double Gnomon ^ w'hen the End L (Fig. 5.)nbsp;of the Box Cylinder is applied to the End of thenbsp;Piece M N I of Fig. S to I of Fig. 6. andnbsp;the Screw Q, apnlied to 0, preffes the Cylindernbsp;N 4nbsp;nbsp;nbsp;nbsp;' B D

Mathematical Elements Book 11.

L 1

All the Parts of the Machine may be feen joined together at Fig. i. Quickfilver being poured into the Vefl�l /gt;, fpouts out of the Holeg, Fig.$.nbsp;When the Mercury is at the fame Height innbsp;the Box, and you do not vary the Inclination ofnbsp;the Piece ?/ ;�, the Mercury fpouts with the famenbsp;Celerity in any Diredtion; W the Inclinationnbsp;of the Dire�lion may be varied by moving thenbsp;Handle e (E A, in Fig. 5.) the Angle that thenbsp;Diredlion, in which the Mercury goes out of thenbsp;Hole, makes with the Horizon, may be meafurednbsp;by Help of the Quadrant lt;7, along with the Indexnbsp;� h is moveable, which by its Weight is alwaysnbsp;kept in a vertical Pofition. This Quadrant maynbsp;be feen in Fig. 7. with its Index F H : It has twonbsp;Rings behind, to receive the Handle E A, Fig-S-When thisHandle is vertical,the Index hangs againftnbsp;the 45th Degree, and the Dire�lion of the Motionnbsp;of the Mercury, which fpouts out then, makes anbsp;half Right Angle with the Horizon.

In Fig. I. the Jets of Mercury in their feveral Directions are reprefented; They become thenbsp;more vifible by help of a wooden Plane G painted black, along which the Mercury in its Motionnbsp;does almoftllide : Upon this Plane mull be drawnnbsp;the Ways which a Body (according to what is faidnbsp;in Numb. 212) runs thro�, when it moves withnbsp;the fame Celerity according to Direftions whichnbsp;tnake different Angles with the Horizon. Alfonbsp;the Semicircle A L of Plate XV. Fig. 5. rnuft benbsp;drawn upon this Plane, tho� it could not be reprefented in this Figure.

There are feveral other fuch Planes, in which the fame Things are drawn, but fo as to repre-fent the Ways of Proje�tiles, ^c. according tqnbsp;different Celerities.

This

This Plane ftands upright near the Middle of the Trough, and bears agaiml the Side E F H,nbsp;fo as to move backwards and forwards accordingnbsp;to the Length of the Trough.

The Celerity of the fpouting Mercury is varied, as you change the Inclination of the Piece n m;nbsp;and, by lowering the Veflei p, the Hole throughnbsp;which the Mercury fpouts, is fet to the Dire�tionnbsp;of the Lines drawn on the Plane.

The Mercury will ftop its fpouting, when the Cavity a b {Fig. 2.) is flopped with the Pin D Enbsp;of Fig, 4.

Experiment Plate XXIII. Fig. i.] The Parts 371 of this Machine being joined and fixed together,nbsp;as in the Manner above described, incline the Piecenbsp;n m, till the Height to which the Mercury is tonbsp;fpout, when it afeends to a Direflion almofl vertical, is nearly equal to the Diameter of the Semicircle deferibed on the Plane G. Let the Vef-fel P be fixed at fuch a Height, and the Plane Gnbsp;be fo placed, that the Axis of the Circumvolutionnbsp;of the Cylinder B D {Fig. 5.) fhall anfwer to thenbsp;loweft Point of the Semicircle above-mentioned.nbsp;Which Way foever the Inclination of the Diredi-on of the Jet (that is, of the Projection) be, itsnbsp;Amplitude will always be the Quadruple of thenbsp;Line B M in the Semicircle A B L {Plate XV.

Fig. 5.) There is indeed a fmall Difference, which chiefly arifes from the Refiflance of the Air, andnbsp;muft be obferved in the following Experiments.

Experiment 5.) The Machine being difpofed as 372 in the foregoing Experiment, if the Mercury fpoutsnbsp;in two Directions, and the Inclination of onenbsp;of them exceeds a half Right-Angle as much asnbsp;the ether is under it, the Mercury will cut thenbsp;Horizontal Line which is drawn from the lower

Point

18(5 nbsp;nbsp;nbsp;Mathemaftcal Elements Book II.

373 nbsp;nbsp;nbsp;Experiment 6.] Every thing being difpofed asnbsp;before, if the Way for any Dire�lion of Motionnbsp;be drawn on the Plane, and the Index � g agreesnbsp;with that Divifion of the Quadrant which denotes that Inclination, the Mercury in its Motionnbsp;will follow the Line drawn to reprefent itsnbsp;Way. If you draw the Ways for feveral Angles, by the Motion of the Handle a e, you willnbsp;bring the Mercury to fpout in Jets that go alongnbsp;thefe very Lines.

374 nbsp;nbsp;nbsp;Experiment 7.] Let there be another Plane asnbsp;G, in which all the Lines above-mentioned arenbsp;drawn from another Celerity of the Mercurial Jet,nbsp;and the Experiments will fucceed in the famenbsp;Manner.

By the fame Method, as we do by a Semicircle (determine the Diftance to which Bodies obliquely projedfed will fall, one may find the Diftance to which the Liquor coming out of the Side of a Veflel fpouts, when the Veffel is fetnbsp;upon a Horizontal Plane; which Diftance is different according to the different Height of thenbsp;Hole, the upper Surface of the Liquid remaining the fame.

375 nbsp;nbsp;nbsp;P/ate XXIV. Fig. 4.] Let AB he the Height ofnbsp;a Veffel filled with a Liquid �, fuppofe this Heightnbsp;cut into two equal Parts at G j with the Centernbsp;C and Diftance E A deferibe a Semicircle j letnbsp;there he a Hole at E, laftly, draw E D perpendicular to AB, and terminated in the Circumferencenbsp;of the Semieircle at D. Let the Liquid fpout fromnbsp;� to F in the Hori':e,Qntal Planeand the Difiancenbsp;B F tciil he double the Perpendicular ED.

Book II. of Natural Thilofofhy.

Which will be demonftrated, if we confider, that the Liquid, with an equable Motion withnbsp;the Celerity that it has coming out of the Hole,nbsp;wouid (in the Time that a Body can fall from Enbsp;to B) run thro� the Space B F

In all Motion, the Time remaining the fame, the Space gone thro� is as the Celerity * � the �nbsp;Celerity remaining the fame, it is as the Time :nbsp;Therefore if you change the Time and the Celerity, the Space gone thro� will be in a Rationbsp;compounded of the Celerity and the Time andnbsp;multiplying the Time by it, you will have thenbsp;Space gone thro�; that is,if you apply this Operation to different Motions, you will have fuchnbsp;Quantities as will exprefs the Proportion of thenbsp;Spaces gon� thro�- If you compute the Squaresnbsp;of the Celerities and the Time, you will havenbsp;the Proportion of the Squares of the Spaces gonenbsp;thro�. A � here exprelfes the Square of the Celerity ; * E B the Square of the Time; * therefore the Product of thofe Lines expreffes thenbsp;Square of the Space gone thro� E F. But thatnbsp;Produd is the Square of the Line E D, whichnbsp;therefore, changing the Hole, increafes and di-minilhes in the fame Ratio as the Diftance B F.nbsp;Suppofe the Hole in the Center C ; E G, thenbsp;Diftance to which the Liquor fpouts, is equal tonbsp;BA, * arid it is double the Perpendicular, whichnbsp;from C may be drawn to A B in the Semicircle ;nbsp;which therefore obtains in all Holes, and E Dnbsp;will be the Half of B F.

Hence it follows, that a Liquid, [pouting from 376 a Hole in the Center C, �will go to the greatefi J)i-Jiance pofjible.

'Experiment 8. Elate XXIV. Fig. i.quot;! Here we muft make ufe of the Machine, defcribed in thenbsp;foregoing Chapter *. Let the Water fpout from � .56

Mathematical Elements Book 11.

the Hole F, as in Experiment 2. Chap. VII. Let it fpout at the fame Time from E, and alfo from G,nbsp;where there is a Tube like thofe which are madenbsp;faft at F and E j the Hole G is lefs than F, butnbsp;the Hole E is farther diEant from the Surface ofnbsp;the Water, the Water comes from neither of themnbsp;to the Diftance to which it comesj when fpout-ing from F.

Plate XXXIV. Fig. 4.] From what has been faid, it follows, That the Water [pouts to the famenbsp;Diftance from the Holes Ee equally dift ant from thenbsp;Center C, becaufe in that Cafe the Perpendicularsnbsp;E D and E d are equal.

Experimeftt 9. Plate XXIV. Fig. i.] From F let there be drawn a horizontal Line which goesnbsp;thro� H ; if H G and H E are equal, the Waternbsp;will go from each Hole G and E to L.

CHAP. IX.

Of a Liquid flowing out ofVejfels., and the Irregularities in that Motion.

378 HE Quantity of a Liquid, which in a given J ^ime ^ows from a given Hole, increafes innbsp;Proportion to the Velocity of the Liquid goingnbsp;out j this depends upon the Height of the Liquidnbsp;above the Hole, and it is no matter to whatnbsp;354 Part the Motion of the Liquor is dire�tedj * therefore the Squares of the ^lantities flowing out arenbsp;in the Ratio of the Heights 0/ the Liquid above

In the Time in which a Body falling freely goes through the Height of the Liquid abovenbsp;the Hole, a Column of the Liquid flows outnbsp;equal in Length to twice that Height 3 * the

Book II. nbsp;nbsp;nbsp;of Natural Thilofophy,

given ; if the Height of the Liquid above the Hole is known, the whole Column is known ;nbsp;the Time alfo is eafily determined by Experiments of Pendulums; * but having found what' 157nbsp;Quantity flows out in a known Time, one maynbsp;know what Quantity will flow out in a givennbsp;Time.

Here you muft obferve, that the Reliftance of the Air, and the Fri�lion of the Liquid againftnbsp;the Sides of the Hole, hinders the Motion of thenbsp;Liquid, and that the Rule above-mentioned doesnbsp;not exadly obtain, and that there always flownbsp;out a lefs Qiaantity than what there is determined by it. Yet making Experiments with Water, it is plain, that the Quantities, which flownbsp;from the fame Hole in equal Times, fenfibly keepnbsp;the Proportion of the Squares of the Heights ofnbsp;the Water above the Hole, in Heights not exceeding fifty Foot.

In VeflTels which are not fupplied by the flowing in of the Liquid, the Celerity of the Liquid flowing out is continually changed; to whichnbsp;Regard muft be had, when you compare together the Times in which different Veffels arenbsp;emptied.

Here we confider cylindric Veffels ; and what is here faid may be applied to any Veffels that arenbsp;of the fame Bignefs from Top to Bottom; wenbsp;fuppofe the Liquid to flow out from a Hole in thenbsp;Bottom.

5tbe �times in 'which cylindric Vejfels of the fame Diameter and Height are emptied, the Liquidfloiv-ing from miequal Holes, are to each other inverjlynbsp;as tbofe Holes.

If we fuppofe that thefe Veffels are divided into very fmall equal Parts, by Planes parallel tonbsp;their Bafe ; and that the Divifions of each Veffelnbsp;do not differ from one another, when we confider

Mathematical- Elements Book II.

the (malleft Parts, one may conceive that the Celerity is not changed in the Evacuation of onenbsp;Part. The Quantity of a Liquid which flowsnbsp;from a Hole, if the Celerity is not changed, in-creafes with the Hole and with the Time j thatnbsp;is, in a Ratio compounded of the Time and ofnbsp;the Hole. The correfpondent Parts in the Vet-fels are emptied with equal Celerities, and thenbsp;aforefaid compounded Ratio obtains here ; Thenbsp;fame Parts alfo, that is, the Quantities of thenbsp;Liquid which flow out, are equal j whereforenbsp;the Difference of the Times is recompenfed bynbsp;the Difference of the Holes ^ that is, the Timesnbsp;are in the fame, but inverfe Ratio, as the Holes.nbsp;Now as this happens in all the correfpondentnbsp;Parts, it muft alfo be referred to the Times ofnbsp;the whole Evacuations of the Veflels.

380 nbsp;nbsp;nbsp;When the VeJJels are cylindric, unequal., and equally high, they are emptied thro' equal Holes, in Timesnbsp;that are to one another as the Bafes of the Cyliitders.

Let the Veflels again be fuppofed to be divided into very fmall Parts, and equal in Number innbsp;each VeflTel �, the Liquid of the correfpondentnbsp;Parts flows thro� equal Holes, and with equalnbsp;Celerity : Therefore the Quantities that flow outnbsp;are as the Times �, and confequently the correfpondent Parts themfelves are in that Ratio ofnbsp;the Times, which are as the Bafes of the Cylinders : But the Times of the whole Evacuationsnbsp;are as the Times in which the correfpondent Partsnbsp;are evacuated.

381 nbsp;nbsp;nbsp;Laftly, Let there he two cylindric Vejfels, ivhofcnbsp;Bafes are equal, hut their Heights, for Example, asnbsp;I to 4, and let them he evacuated thro� equal Holes.

Let thefe alfo be conceived to be divided into very fmall Parts, by Planes parallel to the Bafe ;nbsp;and let the Number in thofe Parts be equal innbsp;each Veflel j thofe Parts will be to one another

Book II. of Natural Thilofophy.

as the Veflels, that is, as i to4. ^We can confider every Part as evacuated by an equable Motionnbsp;becaufe the Parts are very fmall �, the Celeritiesnbsp;in the correfpondent Parts are every where, as inbsp;to 2, * becaufe the Heights of thofe Parts above*nbsp;the Bafes are as the Heights of the Veflels, whichnbsp;are as the Squares of thofe Numbers. Whencenbsp;it follows, that the Times, in which correfpondent Parts are evacuated, are to one another,nbsp;as I to 2 j becaufe in twice the Time with anbsp;double Celerity, a quadruple Quantity is evacuated. But as the Times are in the fame Rationbsp;for each correfpondent Part, the Times in w'hichnbsp;the whole Veflels are evacuated are alfo, as i tonbsp;2. If the Veffels are, as i to 9, the Times willnbsp;be, by a like Demonftration, as i to 3 ; andnbsp;generally the Times are as the Celerities in whichnbsp;correfpondent Parts are evacuatad, the Squaresnbsp;of whofe Celerities are as the Heights of the Vef-fels^ * in which Ratio alfo are the Squares of thenbsp;^imes.

Experiment i. Plate XXIV. Fig. 2. ] Let there be three thin cylindric VeflTels of Metal A, C, B,nbsp;having equal Diameters, and whofe Heights are,nbsp;as I, 3, and 4 ^ let each of them have a Lip in thenbsp;Top to let the Water run out when it comes tonbsp;a certain Height, which Lip muft be reckonednbsp;the Top of the Veflel; in the Bottoms of thenbsp;VeflTels A and B, which are as i and 4, let therenbsp;be equal Holes, and let them be filled with Water ; let the Holes be opened in the fame Moment j if the Water running out of B be receivednbsp;in the Veflel C, it will be filled in the fame Timenbsp;that A is evacuated. C contains three Quarters ofnbsp;the Veflel B; the Quarter which is left will alfonbsp;be evacuated in the fame Time as the Veflel A,

Maihematkat Elements Book IT,

which is evident to Senfe j therefore A is emptied twice, whilft B is emptied once.

382 nbsp;nbsp;nbsp;The 'Times in which any cylindric Vejjels are eva-quot;^liocuated^ are in a Katio compounded of the Bafes, *nbsp;*379 of the inverfe Ratio of the Holes^ * and of thefquarenbsp;quot;�b*� Roots of the Heights.*

383 nbsp;nbsp;nbsp;The cylindric Vcffel may be fo divided, that thenbsp;Parts intercepted between the Divifions Jhall benbsp;emptied in equal Times, which will happen, if thenbsp;jyiftances of the Divifions from the Bafe be as thenbsp;Squares of the natural Numbers �, for the Times ofnbsp;the Evacuations of the Veflels, whofe Heights are

*3*� in that Proportion, are as the natural Numbers,* and the Differences of the Times are equal.

The Time in which a cylindric Veffelis emptied, is as the Celerity with which the Liquid be-�*3?I gins to run out j * therefore the Celerity, while ^5 5 the Liquid defcends in the Veffel, is diminifhednbsp;in the fame Ratio as the Time of the Evacuationnbsp;of the Liquid remaining in the Veffel, and thenbsp;Motion of a Liquid running out of a cylindric Veffel,nbsp;is equally retarded in'equal Times.

and out of another Veffel of the fame Height (and in which the Liquid is always fupplied foasto be kept atnbsp;the fame Height) in the Time in which the Cylindernbsp;is emptied,there runs out twice as much Water fromnbsp;the other Veffel as from the Cylinder. For,becaufeofnbsp;the equal Height of the Veflels, the Celerities in thenbsp;Beginning are equal the Celerity of the Liquid,nbsp;which comes out of the Veffel that is alwaysnbsp;kept full, is equable ; the Celerity of the Liquid,nbsp;which runs out of the Cylinder, is equably retard-^384 ed. ^ Therefore, whilft the Cylinder is emptying,nbsp;there will flow twice as much Water out of thisnbsp;Veffel as out of the Cylinder ; For if two Bodiesnbsp;are driven with the fame Celerity, and the firft

goes with an equable Motion, and the fecond with a Motion equally retarded, and they move tillnbsp;they have loft all that Motion, the firft in thatnbsp;Time will run double the Space of the fecond; ^nbsp;* here the Liquor that runs out may be looked ,nbsp;upon as the Space gone through, becaufe the *nbsp;Holes are equal.

Befides the Irregularities from FriHion^ and the Kefijlance of the Air^ there are feveral others ari-fing from the Cohefion of Parts, even in Liquorsnbsp;that are not glutinous. I ftiall here only fpeak ofnbsp;Water. We obferve in Relation to it, that tho�nbsp;it be driven by the fame Force in any Diredfion,nbsp;the Height of the Water above the Hole re- *nbsp;maining the fame, yet it will defcend the morenbsp;fwiftly in a vertical Diredfion j the Water in falling, is continually accelerated in its Motion, itnbsp;coheres with the following, and accelerates that,nbsp;and increafes the Velocity of the Water flowingnbsp;out of the Velfel.

Plate XXIV. Fig. 2.] For this Reafon the ilib- 387 tion out of a Vejfel, that has a Fiihe fixed to its undernbsp;Side, is alfo accelerated. Let E be fuch a Veflelnbsp;equal and fimilar to the Veflel A, and which,nbsp;together with the Tube, makes up the Heightnbsp;of the Veflel Bj let the Tube have the Holes atnbsp;both Ends, equal to the Holes at the Bottoms ofnbsp;the Veflels A and B, fill the Water in the Vef-fels A, E, and B. In the Beginning of the Motion, the Water flows from the Veflel E and Bnbsp;with equal Celerity, becaufe the Heights of thenbsp;Water above the Holes, from which the Waternbsp;goes out, are equals but the Celerity, in the Vef-fei E, is immediately diminifhed, becaufe therenbsp;cannot run a greater Quantity of Water out ofnbsp;the Tube than what comes in at the upper Holenbsp;of the Tube, into which Hole no more Water

O nbsp;nbsp;nbsp;can

Mathematical Elements Book II.

can run in, than what can flow out at the Vefl�l A. Since the Parts of the Water cohere, the Water, which runs out, accelerates that which runsnbsp;into the Tube, and this laft retards that whichnbsp;runs out j and fo the Quantity of Water, whichnbsp;in a certain Time runs out of the Vefl�l E, is anbsp;mean Quantity between the Quantities of Waternbsp;that can run out at the fame Time from the Vef-fels A and B.

Experiment 2.] The Veflels A E and B being made of fome thin Metal, in the Proportions a-bove-mentioned, fill with Water A and E ; havingnbsp;opened the Holes at the fame Inftant of Time,nbsp;the Water of the Surface at E will defcend fafternbsp;than that at A : On the contrary, if you makenbsp;ufe of the VefTels E and B, it will defcend fafternbsp;in the laft than in the firft.

388 Let the upper Hole of the itiihe, hy which it communicates with the Veffel^ remain as before j and the lower Hole be opened wider ; then a greaternbsp;^lantity of Water will flow out, and the Waternbsp;which goes into the Tube will be more accelerated ; this Hole may be made fufficiently widernbsp;without altering the Length of the Tube, infomuchnbsp;that a greater Quantity of Water (hall flow out fromnbsp;it than from the VefTel B. In that Cafe thro�thenbsp;upper Hole of the Tube, at a fmall Depth belownbsp;the Surface of the Water, there flows out a greater Quantity of Water, than from an equal Holenbsp;four Times the Depth. The fame may be donenbsp;by applying a longer Tube, without widening itsnbsp;lower Hole.

Experitnent 3. Plate XXIV. Fig. 2.] Take the VeflTel F no Way different from the VefTel E, butnbsp;in having the lower Hole of its Tube bigger jnbsp;take alfo the above-mentioned Vefl�l B. The

Book II. of Natural Thikfophy.

Diameters of the Hole in the Bottom of this, and of the upper Hole of the Tube which isnbsp;joined to F are of four Lines (of an Inch)nbsp;the lower Hole of this Tube is of five Lines.

Let the Veflels be filled with Water ; and let the Water begin to run out of both at the famenbsp;Moment; the Surface of the Water in F will de-fcend fafter than that of B. The VeflelBis a-bout i6 Inches high.

CHAP. X.

Of the Running of Rivers. Definition I.

TH E Water that runs hy its own Gravityin 3^9 a Channel open above., as A E, is called a River (Jlate XXV. Fig. i.)

A River is faid to remain in the fame State, 39� or to be in a permanent State, when it flows tmi-fornily, fo as to he always at the fame Height in thenbsp;fame Place.

A Plane, which cutting a Kiver is perpendicular to the Bottom, as p on q, is called the Sedlion of anbsp;River.

When a River is terminated by flat Sides parallel to each other, and perpendicular to the Horizon, and the Bottom alfo is a Plane either horizontal or inclined, the Seftion of the River with thefe three Planes makes Right Angles, and is anbsp;Parallelogram.

In every River that is in a permanent St ate,the 39^ fame ^lantity of Water flows in the fame Fime thro�nbsp;every Sehlion. For unlefs there be in every Place

Mathematical Elements Book II.

as great a Supply of Water, as what runs from it, the River will not remain in the fame State.nbsp;And this Demonftration will not hold good, whatever be the Irregularity of the Bed or Channel,nbsp;from which, in another Refped:, feveral Changesnbsp;in the Motion of the River arifej as, for Example,nbsp;a greater Fridtion in Proportion to the greaternbsp;Inequality of the Channel.

The Irregularities in the Motion of a River may be infinitely varied, and Rules cannot benbsp;given to fettle them ; Therefore fetting afide allnbsp;Irregularities, we muft examine the Courfe ofnbsp;Rivers ; for, unlefs the Laws of Motion be knownnbsp;in that Cafe, we have no certain foundation fornbsp;determining any Thing.

Therefore, we fuppofe the Water to run in a regular Channel, without any fenfible Fridtion,nbsp;and that the Channel is terminated with planenbsp;Sides, that are parallel to one another and vertical ; and alfo that the Bottom is a Plane, andnbsp;inclined to the Horizon.

I-et A E be the Channel, into which the Water runs from a greater Receptacle or Head ; and letnbsp;the Water always remain in the fame Height atnbsp;the Head, fothat the River may be in a permanentnbsp;State. The Water defcends along an inclinednbsp;gt; plane, and is accelerated; * whereby, becaufenbsp;the fame Quantity of Water flows through everynbsp;' ,Q,Sedlion, * Height of the IVater^ as you re-393 cede from the Head of the River, is continuallynbsp;diminifloed, and the Surface of the Water willnbsp;acquire the Figure i q s.

To determine the Velocity of the Water in different Places, let us fuppofe the Hollow ofnbsp;the Channel A D CBto be (hut up with a Plane;nbsp;if there be a Hole made in the Plane, the Water will fpout the fafter through the Hole, as thenbsp;Hole is more dillant from the Surface of the

Book II. of Natural T?hiloJhfhy.

Water iamp;z; and the Water will have the fame Celerity that a Body, falling from the Surface ofnbsp;the Water to the Depth of the Hole below it,nbsp;would aquire; * which arifes from the Preflure *nbsp;of the fuperincumbent Water. There is the famenbsp;Preifure, that is, the fame moving Force, whennbsp;the Obftacle at A G is taken away; then everynbsp;Particle of Water enters into the Channel, withnbsp;the Celerity that a Body would acquire in fallingnbsp;from the Surface of the Water to the Depth ofnbsp;that Particle. This Particle is moved along innbsp;an inclined Plane in the Channel, with an accelerated Motion i and that in the fame Manner,nbsp;as if, in falling vertically, it had continued its Motion to the fame Depth below the Surface of thenbsp;VVater in the Head of the River.* So, if you*nbsp;draw the horizontal Line i r, the Particle at rnbsp;will have the fame Celerity as a Body falling thenbsp;Length i C, and running down Cr, can acquire ; which is the Celerity acquired by thenbsp;Body in falling down t r. 1 herefore the Celerity of a Particle may be every where meafured,nbsp;drawing from it a Perpendicular to the horizontal Plane, which is conceived to run along thenbsp;Surface of the Water in the Head of the River ;nbsp;and the Velocity, which a Body acquires in falling down that Perpendicular, will be the Celerity of the Particle ^ which is greater, the longernbsp;the Perpendicular is. Prom any Point, asr, drawnbsp;rs perpendicular to the Bottom of the River,nbsp;which will meafure the Height or Depth of thenbsp;River. Since r j is inclined to the Horizon, ifnbsp;from the feveral Points of that Line you drawnbsp;Perpendiculars to i ?, they will be the Ihorter,nbsp;the more diftant they are from r, and the Ihort-eft of them all will he f v : Therefore the Celerities of the Particles in the Line r s are fo muchnbsp;the lefs, the nearer they are to the Surface of the

ip8 nbsp;nbsp;nbsp;Mathematical Elements Book II.

395 nbsp;nbsp;nbsp;But yet the Celerities of thofe Waters^ as the River runs on, continually approach nearer and nearer to an Equality. For the Squares of thofe Celerities are as r ? to ^ t?, the Difference of whichnbsp;Lines, as you recede from the Head of the River,

393 is continually leflened, becaufeof the Height r s,* which is alfo continually diminifhed as the Linesnbsp;themfekTs are lengthened. Now as this obtainsnbsp;in the Squares, it will much more obtain in thenbsp;Celerities themfelves, whofe Difference thereforenbsp;is diminifhed as they increafe.

396 nbsp;nbsp;nbsp;If the Inclination of the Bottom be changed atnbsp;the Head of the River, fo as to become y z, andnbsp;a greater ^lantity of Water flows into the Channel,nbsp;it will be higher every �where in the River, hut thenbsp;Celerity of the Water is no where changed. For thisnbsp;Celerity does not depend upon the Height of thenbsp;Water in the River, but, as has been demonftrated,nbsp;from the Diftance of the moved Particle from thenbsp;horizontal Plane of the Surface at the Head continued over the faid Particle ; which Diftance isnbsp;meafuredby the Perpendicular rt or s v butthefenbsp;Lines are not changed by the Afflux of Water,nbsp;provided that the Water remains at the, famenbsp;Height in the Bafon or Head.

397 nbsp;nbsp;nbsp;Let the tipper Part of the Channel he flopped tiphynbsp;an Ohflacle, as'Pi,wbich defcends a little Way belownbsp;the Surface of the Water j the whole Water whichnbsp;comes cannot run through, therefore it muft rifenbsp;up : But the Celerity of the Water below this

'* 596 Cataracft is not increafed j * and the Water that comes on is continually heaped up, fo that atnbsp;laft it muft rife fo as to flow over the Obftaclenbsp;or the Banks of the River. But if the Banks benbsp;raifed and the Ohflacle be continued, the Height ofnbsp;fhe Water would rife above the Line i t-, but, before

Book IL of Nat Ural Thilofo^hj. nbsp;nbsp;nbsp;159

that, the Celerity of the Water cannot be in-creafed: In which Cafe the Height of all the Water in the Head will be increafed ; for as wenbsp;fuppofe the River in a permanent State, therenbsp;muft continually be as great a Supply of Waternbsp;to the Head, as there runs from it down thenbsp;Channel j but, if lefs Water runs dowm, thenbsp;Height muft neceflariJy be increafed in the Head,nbsp;till the Celerity of the Water flowing under thenbsp;Obftacle be fo much increafed, that the famenbsp;Quantity of Water (hall run under the Obftacle,nbsp;as ufed to run in the open Channel before.

All thefe Things, as we have already faid, if we abftra�t from all the Irregularities, are true jnbsp;and, the lefs the Irregularities are, the more willnbsp;the true Motions agree with what we have faid :nbsp;concerning which, before we can make any Judgment, we muft be able to compare the Velocities of Water by Experiments, and fo determinenbsp;the Velocities themfelves, as to know the Spaces gone through in a certain Time.nbsp;nbsp;nbsp;nbsp;^

Plate XXV. Fig. 2.j Let ACE be a Quadrant divided into Degrees, * with a Thread in*39s the Center, that has at the other End a Ball Pnbsp;hanging, which is heavier than the Water.

Let the Ball hang wdthin the running Water,399 whilft you hold the Side CA of the Quadrantnbsp;in a vertical Pofinon ; the Ball by the Motionnbsp;of the Water will be fo far fuftained, that thenbsp;Thread PC will make the Angle PC A, withnbsp;the Side C A, which will ferve to determine thenbsp;Celerity of the Water running, againft the Ball.

The Ball, being at reft in the Water, is drawn by three Powers ; by its Gravity it endeavours tonbsp;defcend vertically � by the Adfion of the Liquidnbsp;it is carried in the Diretftion of the Motion ofnbsp;the Water i and laftly, it is drawn by the Threadnbsp;O 4nbsp;nbsp;nbsp;nbsp;along

ioo nbsp;nbsp;nbsp;Mathematical Elements Book II.

along PC. Draw the Triangle EFG, in which E F reprefcnts the vertical Line j let F G makenbsp;with that Line the Angle EFG, equal to thenbsp;Angle which the Dire�lion of the Motion of thenbsp;River makes with the vertical Line j laftly, let thenbsp;Angle GEF be equal to the Angle PC A. Thenbsp;Sides of the Triangle E F G are parallel to the Di-re�tions of the three Powers above-mentioned ;nbsp;therefore the Powers are to one another, as thofe

Gravity of the Ball, F G will exprefsthe Adlion of the Water on the Ball. If you make feveral Experiments in different Places with the fame Ball, younbsp;muft draw fuch Triangles, the Side F remaining,nbsp;( which denotes the refpeifive Gravity of thenbsp;Ball that never changes) the Sides that are asFG,nbsp;will have the fame Proportion as the Aftionsnbsp;of the Water on the Ball. But thefe are as thenbsp;Squares of the Velocities of the Waters in the

* nbsp;nbsp;nbsp;Places in which the Experiments are made* 5 fornbsp;there is no Difference in refpedf of the Adfion ofnbsp;the Water on the Ball, whether the Ball be moved and the Water at reft, or, on the contrary,nbsp;the W^ater be moved and the Ball at reft.

The Atftion of the Water againft the Ball may be compared with the Weight, for it is to thenbsp;refpeftive Gravity of the Bali, as F G to E F.

400 But this Adtion is equal to the Refiftance w'hich a Body fuffers, when it is moved throughnbsp;quiefcent Water with the fame Celerity withnbsp;which the flowing Water does now ftrike againftnbsp;the Body which is at reft : By knowing thenbsp;Vs'eight, which is equal to the Refiftance, wenbsp;know what Space could be run through in a given Time, with the Celerity with which the

' 3=6 Body moves; * therefore we fhall alfo here know what Space the W ater can go through in anbsp;knpwn Time, and fo likcwjfe what Quantity of

Water

Book II. of Natural Thtlofofhy. nbsp;nbsp;nbsp;zoi

Water flows in a given Time through a Place given in the Se�lion of the River.

Here it is to be obferved, that the Determination of the Velocity of the Water will not be exa�tly fettled, if the Experiment he made towards the Surface of the Water, becaufe therenbsp;the Adion of the Water upon the Globe is irregular. *nbsp;nbsp;nbsp;nbsp;_nbsp;nbsp;nbsp;nbsp;*

This Celerity may be determined by immer-4,01 ging in Water a Body which is but a little lighternbsp;than Water, and which, fwimming at the Surface,nbsp;does not float fo high above it, as to be affe�ted bynbsp;the Motion of the Wind ; as the fpecifick Gravities of the Water and the Body fcarce differ at all,nbsp;and this Body may be looked upon as wholly im-merfcd, it will move with the fame Celerity asnbsp;the Water ; and you may, by Help of a Pendulum, meafure the Time in which a Body runsnbsp;through a certain Space that was meafured before.nbsp;When the Surface of the Water is agitated by thenbsp;Wind, the Experiment will not fucceed well, becaufe of the Motion of the Waves, which caufenbsp;an Irregularity in the Motion of the Body.

CHAP. II.

Of the Motion of the Waves.

The Surface of the ftagnant Water is plain,402 and parallel to the Horizon * j if it be- * 27^nbsp;comes hollow at A (Plate XXV. Fig. 3.) uponnbsp;any Account whatever, this Cavity is furround-ed with the Elevation B B j this raifed Waternbsp;defcends by its Gravity, and, with the Celeritynbsp;acquired in defcending, it forms a new Cavity,nbsp;by which Motions the Water afcends at the Sides

Book II.

of this Cavity, and fills the Cavity A, whilft there is a new Elevation towards C j and, whennbsp;this laft is deprefifed, the Water rifes anew towards the fame Part j whence there arifes a Motion in the Surface of the Water, and a Cavity,nbsp;which carries an Elevation before it, is movednbsp;from A towards C.

403 nbsp;nbsp;nbsp;Sr7j/j Cavity^ with the Elevation next to it^ isnbsp;called a Wave.

404 nbsp;nbsp;nbsp;The Breadth of a Wave is the Space taken upnbsp;hy a Wave in the Surface of the Water.) and rnea-Jiired according to the Direliion of the Waversnbsp;Motion.

The Cavity, as A, is encompalfed every Way with an Elevation, and the Motion above-mentioned expands itfelf every Way; therefore the

Elate XXV. Fig. 4.] Let A B be an Obflacle, againfl: which the Wave, whofe Beginning is atnbsp;C, does run ; we muft examine, what Changenbsp;the Wave fuffers in any Point, as E, when it isnbsp;come to the Obftacle in that Point. In all Placesnbsp;through which the Wave runs, whilft it goes forward its whole Breadth, the Wave is raifed , thennbsp;a Cavity is formed, whi()h is again filled up,nbsp;w:hich Change while the/ Surface of the Waternbsp;undergoes, its Particles go and come through anbsp;fmall Space. The Dirediion of this Motion isnbsp;along C E, and the Celerity may be reprefentednbsp;by that Line j let this Motion be conceived tonbsp;be refolved in two other Motions along G Enbsp;and D E, w hofe Celerities are refpe�livcly re-*nbsp;nbsp;nbsp;nbsp;^kofe Lines *. By the Motion a-

Book II. of Natural Thilofofhy. zo}

long D E the Particles do not aft againft theOb-ftacle, and after the Stroke continue the Motion in that Direction, with the fame Celerity �, andnbsp;this Motion is here reprefented by K F, fuppo-fing E F and ED to be equal to one another, bynbsp;the Motion along G E, the Particles ftrike direft-ly againft the Obftacle, and this Motion is de-ftroyedj * for though thefe Particles are elaftick,quot;* '6*nbsp;yet, as in the Motion of the Waves they runnbsp;thro� but in a fmall Space, going backward andnbsp;forward, they are moved fo flowly, that the Figure of the Particles cannot be changed by thenbsp;Blow, and fo they are fubjeft to the Laws ofnbsp;Bodies perfectly hard. But there is a Refleftionnbsp;of the Particles from another Caufe, the Waternbsp;which cannot go forward beyond the Obftacle,nbsp;and is pufhed on by that which follows it, yieldsnbsp;that Way where there is the leaft Refiftance, thatnbsp;is, afcends: And this Elevation greater than innbsp;other Places is caufcd by the Motion along G E,nbsp;becaufe �tis by that Motion alone that the Particles come againft the Obftacle. The Water, bynbsp;its Defcent, acquires the fame Velocity withnbsp;which it was raifed; and the Particles of Waternbsp;are repelled from the Obftacle with the famenbsp;Force in the Direftion E G, as that with whichnbsp;they came againft the Obftacle. From this Motion and the Alotion above-mentioned along EF,nbsp;arifes a Motion along E H, w'hofe Celerity isnbsp;exprefled by the Line E H, which is equal tonbsp;the Line C E j and by the Reflexion the Celerity of the Wave is not changed, but it returns a-long E H in the fame Manner, as if, taking awaynbsp;the Obftacle, it had moved along 'Eh. If fromnbsp;the Point C, CD be drawn perpendicular to thenbsp;Obftacle, and then produced, fo that Dc fliallnbsp;be equal to C D, the Line H E continued willnbsp;Inbsp;nbsp;nbsp;nbsp;go

Mathematical Elements Book II.

go through and, as this Demonftration holds good in all the Points of the Obflacle, it follows

406 nbsp;nbsp;nbsp;that the reJle�iedlVave has the fame Figure on thatnbsp;Side of the Obfiacle^ as it would- have had beyondnbsp;the Line A B, if it had not run againft the Objla-

407 nbsp;nbsp;nbsp;cle. If the Obftacle be inclined to the Horizonynbsp;the Water rifes and defcends upon it, and fuffersnbsp;a Fridfion, whereby the Reflexion of the iVave isnbsp;diflurbed, and often wholly deflroyed. This is thenbsp;Reafon why very often the Banks of Rivers donbsp;not refle�t the W aves.

408 nbsp;nbsp;nbsp;When there is a Hole, as I, in an Obfiacle, asnbsp;B L, the Part of the Wave, which goes throughnbsp;the Hole, continues its Motion dire�ly, and expands itfelf towards Q Q, and there is a newnbsp;Wave formedy which moves in a Setnicircky whofenbsp;Center is the Hole. For the railed Part of the Wave,nbsp;which firft goes through the Hole, immediatelynbsp;flows down a little at the Sides, and then by de-fcending, makes a Cavity, which is furroundednbsp;with an Elevation on every Part beyond the Hole,nbsp;which moves every Way in the fame Manner, asnbsp;was faid concerning the Generation of the firft

409 nbsp;nbsp;nbsp;In the fame Manner a Wave, to which an Obftacle, as A O, is oppofed, continues to movenbsp;between O N 5 but expands itfelf towards R innbsp;a Part of a Circle, whofe Center is not very farnbsp;from O.

10 Hence we may eafily deduce what muft be the Motion of a Wave behind an Obftacle, as M N.

411 Waves are often produced by the Motion of a tremulous Bodyy whichalfoexpandthemfel-ves circularly y tho� the Body goes and comes in a Right Line; for the Water, which is raifed by the Agitation,nbsp;defcending, forms a Cavity, which is every wherenbsp;furrounded with a Rifing.

Book II. of Natural Thilofofhy. nbsp;nbsp;nbsp;aof

Different Waves do notdiftnrh one another, vihen^iz they move according to different DireSiions. Thenbsp;Reafon of which Effe�l is, that whatever Figurenbsp;the Surface of the Water has acquired by thenbsp;Motion of the Waves, there may in that be annbsp;Elevation and a Depreffion, as alfo fuch a Motionnbsp;as is required in the Motion of a Wave.

Whoever has attentively confidered the Motion of the Waves, w'ill find that all thefe Things a-gree with Experiments.

To determine the Celerity of the Waves, ano-4J3 ther Motion, analogous to their Motion, is tonbsp;be examined. Let there be a Liquid in the recurve cylindric ��'ube E H (flateNXN. Fig. 5.)nbsp;and let the Liquid in the Leg E F be highernbsp;than in the other Leg by the Diftance / E ; whichnbsp;Difference is to be divided into equal Parts atnbsp;i. The Liquid by its Gravity defcends in the Legnbsp;E F, whilft it afcends equally in the Leg E FI jnbsp;and fo, when the Surface of the Liquid is come tonbsp;/, it is at the fame Height in both Legs, and thatnbsp;is the only Pofition in which the Liquid can be atnbsp;reft : But, by the Celerity acquired by defcending,nbsp;it continues its Motion, and afcends higher in thenbsp;Tube G H, and in E F it is deprefled quite tonbsp;/, except fo much as it is hindered by the Fri-ftion againft the Sides of the Tube. The Lquidnbsp;in the Tube G H, which is higher, alfo defcendsnbsp;by its Gravity i and fo the Liquid in the Tubenbsp;rifes and falls, till it has loft all its Motion by thenbsp;Friftion.

The Quantity of the Matter to be moved is the whole Liquid in the Tube j the moving Forcenbsp;is the W^eight of the Pillar IE, whofe Height is al-w^ays double the Diftance E? �, which Diftancenbsp;therefore increafes and diminifhes in the famenbsp;Ratio with the moving Force- But the Diftancenbsp;E i is the Space to be run through by the Liquid,

Mathematical Elements Book II.

that from the Pofition E H it may come to the Pofition of Reft ; which Space therefore is alwaysnbsp;as the Force which continually a�ls upon thenbsp;Liquid : But we have demonftrated that it is upon this Account, that all the Vibrations of anbsp;Pendulum, ofcillatinginto a Cycloid, are perform-

�whatever he the Inequality of the Jgitations, the Liquid always goes^ or comes^ in the fame ��ime.

414 ^ime in which a Liquid^ thus agitated, a-fcends, or defends, is the fime in which a Pendulum vibrates, whofe Length, that is, the Diftance between the Center of Ofcillation and Sufpenfion,nbsp;is equal to Half the Length of the Liquid in thenbsp;��'uhe, or to Half the Sum of the Lines E F, F G,nbsp;GH. This Length is to be meafured in the Axisnbsp;of the Tube.

Plate XXV. Fig. 6.] Let fuch a Pendulum vibrate in a Cycloid, in the Manner explainednbsp;above (Page 62 and 63.) Let the Pendulum BCnbsp;and the Arc A D be of the fame Length j fornbsp;the Arc C A is equal to the Arc A D, andnbsp;the Thread, by which the Pendulum is fuf-pended, applies to it, when the Body fufpend-ed is at A 5 in that Point the Dire�lion of thenbsp;Curve is perpendicular to the Horizon, and thenbsp;Body endeavours to defcend with all its Weightnbsp;along the Curve : But this Weight is to the Forcenbsp;a�ling upon the Body, when it is at P, as A D,

fuch a Pofition, that i E (Fig. y.) be equal to P, D ; the VVeight of the whole Matter tonbsp;be moved, that is, of the whole Liquid, isnbsp;to the Weight / E (which is the Force aftingnbsp;upon the Liquid in that Pofition) as thenbsp;Length of the Liquid in the Tube to the Linenbsp;I E, in which Ratio alfo the Halves of thofe

Book II. ef Natural Thilofofhy. nbsp;nbsp;nbsp;xo7

Quantities are, that is, P C to P D {Ng. 6.) Therefore in the Pendulum the Weight of thenbsp;Matter to be moved is to the Force ading uponnbsp;it, at P, as in the Tube, the Weight of the Matter to be moved is to the Force ailing upon itnbsp;in the Pofition E H. Therefore the pendulousnbsp;Body and the Liquid, in this Cafe, are aded upon by equd Forces, and this always obtainsnbsp;where the Spaces to be run through by the Liquid in Agitation, and by a Body in Vibration,nbsp;are equal; therefore in this Cafe the Agitationnbsp;and the Vibration are performed in the famenbsp;Time, and not only in this Cafe, but always. * *nbsp;But, as the fmall Vibrations in a Circle do notnbsp;differ from the Vibrations in a Cycloid, the De-monftration will agree with them.

Experiment.'] Take a cylindric recurve Tube as E F G H ; let the Length of the Legs b� onenbsp;Foot, and the Bore of the Cylinder half an Inch ;nbsp;pour Mercury into this Tube, and having madenbsp;a Pendulum, whofe Length is equal to Half thenbsp;Length of the Cylinder of Mercury in the Tube ;nbsp;if the Mercury be agitated in the Tube, it willnbsp;afcend and defcend in the fame Time as the Pendulum will go and come.

Elate XXV. Fig. 7.3 To determine the Celerity of the Waves from what has been faid, we muftnbsp;confider feveral equal Waves that follow one another immediately, as A, B, C, D, E, F, whichnbsp;move from A towards F, the Wave A runs itsnbsp;Breadth, when the Cavity A is come to C; whichnbsp;cannot be, unlefs the Water at C afcends to thenbsp;Height of the Top of the Waves, and again de-fcends to the Depth C ; in which Moment thenbsp;Water is not agitated fen�bly belo\y the Linenbsp;F /� j therefore this Motion agrees with the Motion

Mathematical Elements Book II.

tion in the Tube above-mentioned, and the Water afcends and defcends j that is, the Wave goes through its Breadth, whilft a Pendulum of thenbsp;Length of half B C performs two Ofcillations,

?414* or whilft a Pendulum of the Length BCD, that is four times as long as the firft, performs

Therefore the Celerity of a Wave depends upon the Length of a Line B, C, D, which is greater, according as the Breadth of the Waves is greater, and as the Water defcends deeper in thenbsp;Motion of the Waves.

415 In the broadeft Waves, which do not rife high, fuch a Line as B C D does not much differ fromnbsp;the Breadth of the Wave; and in that Cafe alVavenbsp;runs through its Breadth^ wbtlfia Pendulum^ equalnbsp;to that Breadth, ofcillates once. In every equablenbsp;Motion, the Space gone through increafes withnbsp;the Time and the Celerity; wherefore multiplying the Time by the Celerity, you have the Spacenbsp;gone thro�; whence it follows, that the Celeritiesnbsp;of the Waves are as the Square Roots of theirnbsp;Breadths : For as the Times in which they go

fame Ratio is required in their Celerities, that the Produffts of the Times by their Celeritiesnbsp;may be as the Breadth of the Waves, which arenbsp;the Spaces gone through.

All thefe Things muft be only looked upon as nearly true, becaufe the Motion of the Wavesnbsp;differs fomething from the Motion in the Tube ;nbsp;which Error is in part taken off, becaufe thenbsp;Length of the Pendulum is meafured along thenbsp;inclined Lines BC and CD.

�r

Part III. Of the Air, as an Ulaftic Fluid.

CHAP. XII.

WE have often fpoken of the Airj and as we live in, and are always encompafl�d bynbsp;It, We muft have Regard to its Effedl in feveralnbsp;Experiments, as we have faid in other Parts ofnbsp;this Treatife ; But now we fhall conlider itsnbsp;Properties lingly.

The Air is corporeal, heavy, its Parts yield to 417 any Force imprefled, and are very eafily movednbsp;one amongft another ; it prefles in Proportion tonbsp;its Height, and the Preflure every Way is equal;

All the Air which the Earth is encoinpaffed with, 418 confidered together, is clt;^//eithe Atmofphere of thenbsp;Earth, or fimply, the Attnofphere.

The Height of the Air above the Surface of the 415 Earth, js called the Height of the Atmofphere.

rio nbsp;nbsp;nbsp;Mathematical Elements Book 1�.

421 nbsp;nbsp;nbsp;^bat it yields to any ImpreJJlon, and has itsnbsp;Parts eafily moved., is not doubted by any one.

422 nbsp;nbsp;nbsp;That it is heavy, is proved by its preffing upon the Surface of other Fluids, and fuftainingnbsp;them in Tubes.

Experiment i. Plate XXVI. Fig. i.] Take a Glafs Tube A B, about three Foot long, of aboutnbsp;i Inch Bore ; if you flop up the End A, and letnbsp;the Tube be filled with Mercury, and let thenbsp;other End be immerfed in a Veflel full of Mercury, the Mercury will be fuftained attheFIeightnbsp;of about 29 Inches. This is occafioned by thenbsp;Preflure of the Air on the Surface of the Mercury in the Veflel, which cannot prefs equally innbsp;every Part of it, unlefs in the Tube where no Airnbsp;is, there be a Column of Mercury, which preflesnbsp;' J74equally with outward Air*.

Experiment 2. Plate XXVI. Fig. i.] That this Preflure may not be changed when the Tubenbsp;is inclined, it is required that the Mercury Ihouldnbsp;' 277 keep the fame perpendicular Height * ; If therefore there be tw'o Veflels containing Mercury, innbsp;which Tubes in the Manner above-mentioned arenbsp;immerfed, of which E D is inclined to thenbsp;Horizon, the Mercury is fufliained at the Heightsnbsp;h fundi g,io ihut f and gare in the fame Horizontalnbsp;Lines j fuppofing the Surfaces of the Mercury innbsp;the Veflels to lie in the fame Plane.

Experiment 3. Plate XXVI. Fig. 2.] The fame Preflure of Air fuftains the Water of the Glafs U)nbsp;which is yet immerged in Water and filled with it,nbsp;and then is pulled out all but the Orifice, whichnbsp;ftill remains immerfed.

423 Water would be fuftained in the fame Manner,nbsp;th�� the Height fliould be 32 Foot j for Quick-

Book II. of Natural Thilofoploy. nbsp;nbsp;nbsp;iii

filver 14 times heavier than Water, and a Pillar of Water a little more than 32 Foot high prelTesnbsp;equally with a Column of Mercury 29 Inchesnbsp;high, which Preflure is equal to the Preflure ofnbsp;theAtmofphere.

1'hat the Prejptre of the Air depends upon its 424 Height, may be eafily deduced from what has beennbsp;faid; but it is immediately proved by carryingnbsp;the Tube with the Mercury above-mentioned tonbsp;a higher Place; for, when you carry this Machinenbsp;up a Hill, for 100 Foot that you rife perpendicularly, the Mercury defcends a Quarter of annbsp;Inch.

That Air prejfes equally every way, appears from 425 this, that the Preflure is fuftained by fofc Bodiesnbsp;without any Change of Figure, and brittle Bo*nbsp;dies without their breaking, though this Preflurenbsp;be equal to the PreflTure of a Pillar of Mercurynbsp;29 Inches high, or a Height of Water of 32nbsp;Foot * i any Body may fee that nothing can pre- *nbsp;ferve thefe Bodies unchanged, but the equal Pref-fure on all Parts; but it is plain that the Air doesnbsp;prefs in that Manner. * If you take away the Air *nbsp;on one Side, the PreflTure is fenfible on the oppo-fite Side.

Exper. 4. Plate XXVI. Fig. 3.] Hang a Glafs Tube to one of the Scales of a Balance A B,nbsp;which is fhut at D, and 3 Foot long ; fill thisnbsp;Tube with Mercury, and let the End E be im-merfed in the Mercury that is contained in thenbsp;Veflel U. The Mercury by the Air�s Preflurenbsp;is fuftained at the Height/ in the Tube, and thenbsp;upper Part of the 1 ube � D is left void of Air ;nbsp;to make an jpquilibrium, you muft put into thenbsp;oppofjte Scale a Weight equal to the Weight ofnbsp;the Tube and th� Mercury contained in it. I'henbsp;Mercury in the Tiibe cannot prefs the Balance jnbsp;P 2nbsp;nbsp;nbsp;nbsp;for

a 12 Mathematical Elements Book II.

for its A�ion againft the Sides of the Tube is horizontal j but the Air adts upon the uppernbsp;Part of the Tube, and the Column that is fu-ftained by the Tube is sequiponderate withnbsp;the Column of Mercury that is contained in thenbsp;Tube : If letting the Mercury run out, you fuf-fer the Air to come in, then nothing but thenbsp;Tube weighs down the Scale ; which proves,nbsp;that the Adlion againft the inferior Surface ofnbsp;the upper Part of the Tube deftroys the Adlionnbsp;on the exterior Surface, and that the Air pref-fes upwards and downwards with the famenbsp;Force,

0/ the Air�s Elajlicity, or Spring.

WE have fhewn, the Air has the Properties of other Liquids j but befides, it has another Property, which is, that it can takenbsp;up a greater or lefler Space, according as it isnbsp;comprefled with a different Force; and, as foonnbsp;as that Force is diminilhed, it expands itfelf.nbsp;426 By reafon of the Analogy of this Effed with thenbsp;Elafticity of Bodies, this Property of the Air isnbsp;called its Elafticity.

A27 That the Air may be compreftid^ appears from an ,4 Experiment already mentioned. *

Experiment x. Plate XXVI. Fig. 4.] Take the Tube A B clofe at the End A, and pour Mercury into it, fo that there may be fome Air leftnbsp;in the Tube, which, when in the State of the external

Book II. of Natural ^htlofophy. 213

ternal Air, will take up the Space A / j if the End B of the Tube be immerfed into Mercury in anbsp;VefTel, the Mercury in the Tube will defcend tonbsp;g, and there remain. The Height i g differs verynbsp;much from the Height of the Mercury in thenbsp;firft Experiment of the foregoing Chapter, whichnbsp;does not arife from the Weight of the Air in thenbsp;Tube; for its Weight is too little to producenbsp;any fenfible Difference in the Height of thenbsp;Mercury : The Expanfion of the Air caufes thisnbsp;Effea.

From this Experiment we deduce this Rule, That the Air dilates itfelf in fuch a Manner^ thatnbsp;the Space taken up by it is always inverfely as thenbsp;Force by which it is comprejfed.

The Force, by which the common or external Air is comprefled, is the Weight of the wholenbsp;Atmofphere, which is equal to a Pillar of Mercury of the Height b f Fig. i. therefore thecom-prefling Force may be expreffed by that Height;nbsp;the Space taken up by the Air in the Tube,nbsp;when it is comprelfed with fuch a Force, isnbsp;A /.

But in the laft Experiment, the PreflTure of the Atmofphere exerts two Effeas; it fuftains thenbsp;Pillar of the Mercury /g, and it reduces the Airnbsp;in the Tube to the Space g A ; if the Force, bynbsp;which the Mercury is fuftained at the Height g /,nbsp;be fubftraaed from the PreflTure of the whole Atmofphere ; that is, if the Height g i be takennbsp;from the Height hf, {Fig. i.) there remains thenbsp;Force by which the Air is compreffed in the upper Part of the Tube; but this Difference of thenbsp;Heights of the Mercury h f, and g ?, is always tonbsp;hf as A/to Ag, that is, their Forces are inverfely as the Spaces.

Mathematical Elements Book If.

Experiment 2. Plate XXVI. Fig. 5.] Take a curve Tube A BCD, open at A, and fhut at D ;nbsp;let the Part B C be filled with Mercury, fo thatnbsp;the Part C D may contain Air of the fame Statenbsp;or Tenor as the external Air; therefore, thenbsp;comprefiing Force is the Column of Mercury,nbsp;whofe Height is h E Fig. i. and by this Heightnbsp;tnuft this Force be exprefled, as in the foregoing Experiment; but the Space taken upnbsp;by the Air is CD. Pour Mercury into the Tubenbsp;AB, that it may rife up to g, the Air willnbsp;be reduced to the Spacer D : Now the comprefling Force a�ts as ftrongly as a Column ofnbsp;Mercury of the Height /g, and alfo the Pref-fure of the external Air upon the Surface g ofnbsp;the Mercury ; this Force is exprefled by the Sumnbsp;of the Heights/g in this Figure, and in Fig. i.nbsp;This Sum is always to h f (Fig. i. ) as CDnbsp;to c D; and again the Forces are inverfely asnbsp;the Spaces.

430 Fhe Elajiicity of the Air is as its Denfity; for this laft is inverfely as the Space taken up by thenbsp;* 4^9 Air * ; therefore, as the Force comprefled thenbsp;419nbsp;nbsp;nbsp;nbsp;which is equal to that by which the Air

Hence it follows, that the Air in which we live is reduced to the Denfity which it has nearnbsp;the Earth, by the PreflTure of the fuperincumbentnbsp;Air, and thSt it is more or lefs comprefled, according to the greater or lefs Weight of the At-mofphere ; for which Reafon alfo the Air is lefsnbsp;denfe at the Top of a Mountain than a Valley,nbsp;as being coinpreifed by a lefs Weight.

How far this Property of expanding itfelf is extended, we do not certainly know ; and it isnbsp;very probable that it can be determined by no

Book II. of Natural Thilofopby. nbsp;nbsp;nbsp;a

Experiments. Neverthelefs, if you compare the following Experiments with the Experiment ofnbsp;the Air comprelTed in a Pump*, it will appearnbsp;that the Air may take up twenty thoufand Timesnbsp;more Space in one Cafe than in the other.

Experiment 3. Plate XXVI- Fig. 6.] Let the Glafs A B, about fourteen Inches high, be exafllynbsp;filled with Water i it has a brafs Cap fixednbsp;to it at the End B, by which it is to be fcrewednbsp;to the Pump that is reprefented in Plate XXIX.nbsp;Fig. 6. by drawing out the Piiton of the Pump,nbsp;the Water defcends into it by its Gravity j andnbsp;the Place in the upper Part of the Veflel is voidnbsp;both of Air and Water. The Air bubbles in thenbsp;Water, which are now comprefled, becaufe thenbsp;Air does not a�t upon the Surface of the Water,nbsp;expand themfelves, and rife up to the Surface ofnbsp;the Water; in that Motion the Bubbles are accelerated, fo as not to be feen diftin�fly near thenbsp;Surface, upon Account of their very fwift Motion ; they alfo grow bigger as they afcend, andnbsp;if you compare the Diameter of a Bubble at Bnbsp;with its Diameter, when it is come almoft up tonbsp;the Surface of the Water, but fo far from it asnbsp;to be feen diftinftly, its Diameter is at leaft fournbsp;Times as great as before.

The upper Part of the Glafs, as was faid before, is entirely void of Air; for the fmall Quantity of Air, w'hich is continually going out of the Water, is not to be taken notice of here jnbsp;therefore the Air-bubbles near B, which is aboutnbsp;a Foot below the Surface of the Water, are com-prelTed only by the fuperincumbent Water; whichnbsp;PrefTure is to the Preflitre of the Atmofpherenbsp;nearly as one to thirty-two ; * in which Rationbsp;alfo is the Space taken up by the Air, when it isnbsp;comprefled by the whole Atmofphere, to thenbsp;P 4nbsp;nbsp;nbsp;nbsp;Space

Mathematical Elements Book II.

^ Space taken up in the Bubbles above-mention-429 ed; * their Diameter in their Afcent, as has been faid before, becomes quadruple j that is, thenbsp;Bubble becomes 64 times bigger than it was jnbsp;and fo the Space taken up by the Air, in this laftnbsp;Cafe, is to the Space taken up by the Air, whennbsp;compreffed by the Atmofphere, as 64 times 32nbsp;( that is 2048 ) to I. The Air comprefled bynbsp;the Atnaofphere is reduced to a Space 10 timesnbsp;Jefs in a Forcing pump ; and fo the Denlity ofnbsp;the Air above-mentioned is to the Deiifity of thisnbsp;Air, as i to 20480. Extra�fing the Cube Rootsnbsp;pf thefe Numbers, we (hall find that the Diftan-ces between the Center of the Particles, in thefenbsp;two Cafes, are, as i to 27.

Hence we conclude, that the Particles of Air are not of the fame Nature with ocher elafticnbsp;Bodies, for the fingle Particles cannot expandnbsp;themfelves every way into 27 times the Space,nbsp;and fo be increafed 2000 times, preferving theignbsp;Surface free from every Inequality or Angles fornbsp;in every Expanfion or Compreflion, the Parts arenbsp;cafily moved one amongft another ; but, as thenbsp;4ji Air may be dilated much more than in this Experiment, it follows, that the Ah confifts of Parthnbsp;(jles quot;which do not touch one another^ and that ref elnbsp;each other. We have fliewn, that in feveral Cafesnbsp;there arc Particles endowed with fuch a Proper-*40 ty j * and it is plain enough, that it obtains here;nbsp;but wc are entirely ignorant of thp Caufe of thisnbsp;Force, and it mult be looked upon as a Law ofnbsp;Nature, as is plain from what has been faid between Numb. 4 and Numb. 5.

43 ^ ^he Force., by which the Particles of the Air fly from each other, increafes in the fame Ratio as theDi-fiance in which the Centers of the Particles are di-pinifhed; that is, that Force is inverfely as this Di-ftance. To demonflrate which;, let us confider two

Book II. of Natural Thilofofhy. nbsp;nbsp;nbsp;7.jj

equal Cubes A and B (flate XXVI. Fig. *],) containing unequal Quantities of Air^ let the Di-ftances between the Center of the Particles be as 2 to I, the Numbers of the Particles will be innbsp;the fame, but inverfe Ratio, in the Lines de andnbsp;h i ; the Numbers of the Particles adingupon thenbsp;Surfaces d g and b m are, as i to 4, namely, as thenbsp;Squares of the Numbers of the Particles in equalnbsp;J^ines, and as the Cubes of thofe Numbers, thatnbsp;js, as I to 8, fo are the Quantities of Air contained in the Cubes j in which Ratio alfo arenbsp;the Forces comprefling the Air in the Cubes. ^ *nbsp;The Forces ading upon the equal Surfaces dgnbsp;and h m are as the Forces by which the Air isnbsp;comprefled ^ ^ they are alfo in a Ratio compounded of the Numbers of the Particles ading, andnbsp;the Adion of the Angle Particles ; therefore, thisnbsp;compound Ratio is the Ratio of i to 8: Thenbsp;firft of the compounding Ratio�s, as has beennbsp;faid, is that of i to 4 j wherefore, neceffarily thenbsp;fecond is that of i to 2, which is the inverfe Rationbsp;of the Diftances of the Particles. And this De-monftration is general for by i and 8 we exprefsnbsp;any Cubes whatever �, by i and 4, the Squares ofnbsp;the Cube Roots i and lalily, by i and 2, the Rootsnbsp;of thofe Cubes. This Demonftration proves thatnbsp;the Adion, which the Particles continually fuffernbsp;from all Sides, is increafed between the Ratio innbsp;which the Diftance of the Centers of the Particles is diminilhed, whether the Adion is to benbsp;referred only to neighbouring Particles, or alfonbsp;to thofe which are more diftant. In the firft Cafenbsp;the repellant Force itfelf, which every Particlenbsp;is endowed with, is as the Adion above-mentioned, that is, inverfely as the Diftance between thenbsp;Centers of the Particles.

In the fecond Cafe, the repellent Force is equal at all Diftances i for then the Adion againft

Mathematical Elements Book II.

each Particle depends upon their Number in the fame Line, which Number is inverfely as thenbsp;Diftance between the Number of the Particles.nbsp;Then alfo, fuppofing the Air of the fame Denfi-ty, the Elafticity will be the greater, where thenbsp;Quantity of the Air will be the greater j but, asnbsp;this does not agree with Experiments, therefore,nbsp;the firft Caufe muft be true.

43 3 ^he EffeSts of the Elafticity of the Air are like thofe of its Gravity-) and included Air adls by itsnbsp;Ea-fticity, juft as Air not included does by itsnbsp;Weight.

The Air which is loaded by the Weight of the whole Atmofphere, prefling every Way from thenbsp;�Very Nature of Liquids, and the Force which itnbsp;exerts, does no Way depend upon the Elafticity,nbsp;becaufe, whether you fuppofe Elafticity, or not,nbsp;that Force which arifes from the Weight of thenbsp;Atmofphere, and is equal to it, can be no Waynbsp;changed j but, as the Air is elaftic, it is reducednbsp;to fuch a Space by the Weight of the Atmofphere,nbsp;that the Elafticity, which re-acfts againft thenbsp;^*^^compreffing Weight, is equal to that Weight*.

But the Elafticity increafes and diminifhes as the .'^432. Diftance of the Particles diminifhes or increafes *,nbsp;and it is no Matter, whether the Air be retainednbsp;in a certain Space by the Weight of the Atmo-fphere, or any other Way; for in either Cafenbsp;it will endeavour to expand itfelf with the famenbsp;.Force, and prefs every Way. Therefore, if thenbsp;Air near the Earth be included in any Veflel,nbsp;without altering its Denlity, the Preffure of thenbsp;included Air will be equal to the Weight of thenbsp;whole Atmofphere,

Expriment a^.Elate'Ei.'^ElW. Hg 3.] Take the Tube mentioned in the firft Experiment of thenbsp;laft Chapter, immerge it in Mercury included in

Book II. of Natural Thilofofhy. nbsp;nbsp;nbsp;2,i�)

the Glafs D C, fo that the Air, preffing upon the Surface of the Mercury contained in the Veflel U,nbsp;may have no Communication with the externalnbsp;Air j the Mercury in the Tube is fuftained atnbsp;the fame Height by the Elafticity of the Air, asnbsp;it was fuftained in the open Air.

The Tenor of the Air continuing the fame, 434 what we have faid will always obtain j but thisnbsp;Tenor or Temper of the Air is not always thenbsp;fame j the repellent Force of the Particles is oftennbsp;increafed, or diminifted, though the Dijiance between their Centers is not changed: I lhall fpeak ofnbsp;this Alteration in the following Book : FJoe Elajii-city increafes by Heat, and diminijhes hy Cold.

CHAP. XIV.

Of the Air-Ttmf.

The Elafticity of the Air is the Founda-435 tion of the Conftitution of a Machine, bynbsp;which the Air may be drawn out of any Veflel.nbsp;This Machine is called an Air-pump, which isnbsp;^ade feveral Ways: The chief Part in all of themnbsp;Is a Barrel, or hollow Cylinder of Metal, borednbsp;fmooth, and polifhed in the Infide j in this Barrel muft move a Pifton, that fills its Bore fo ex-^ftly as to let no Air flip by. This Pifton is thruftnbsp;down clofe to the Bottom of the Barrel, and thennbsp;raifed up in fuch a Manner as to exclude all thenbsp;Air from the Cavity of the Cylinder or Barrel ^ ifnbsp;rhis Cavity communicates with any Veflel, bynbsp;Cleans of a Pipe at the Bottom of the Barrel, thenbsp;Air in the Veflel will expand it felf, and Part of itnbsp;will enter into the Barrel, fo that the Air in thenbsp;Barrel, and in the Veflel, will have the fameDen-Shut up the Communication between thenbsp;\ eflTel and Barrel, and letting the Air out of the

Barrel,

Z20 nbsp;nbsp;nbsp;Mathematical Elements Book II.

Barrel, apply the Pifton clofe to the Bottom. If you raife the Pifton a fecond time, and open thenbsp;Communication between the Barrel and Veflelnbsp;above-mentioned, the Denfity of the Air in thenbsp;VefTel will again be diminilhed; and repeatingnbsp;the Motion of the Pifton, the Air in the VefTelnbsp;will be reduced to the leaft Denfity. Yet all thenbsp;Air can never be exhaufted by this Method j fornbsp;at every Stroke the Air does fo expand itfelf, asnbsp;to have the fame Denfity in the Barrel as in thenbsp;VefTel i in which laft therefore, there is always anbsp;little Air left.

436 All Air-Pumps have in common the Parts a-bove defcribed, but they differ in feveral other Things. AVgt;y?, the Communication between thenbsp;Receiver to be exhaufted, and the Cylinder ornbsp;Barrel, is opened and fhut� different Ways. Secondly^ there are different Ways of getting thenbsp;Air out of the Cylinder or Barrel, when the Pifton is brought to the Bottom. ��'hirdly^ the Pi*nbsp;ftons differ in different Pumps. Fourthly^ the Po-fition of the Cylinder is not the fame in all Pumps.nbsp;Fifthly^ there are different Contrivances for moving the Pifton.

There are often two Barrels, in one of which the Pillon is raifed, when it is depreffed in thenbsp;other.

Our Pump is here reprefented in Plate XXVII. Fig. I. the other Side of it is reprefented m Platenbsp;XXX. Fig. 2. I fhall defer the particular Defcrip-tion of it to another Place, and only here mention fome Things in general. This Pump has twonbsp;Brafs Barrels C, C, of 2 Inches Diameter, andnbsp;about 5 Inches high.

In thefe Barrels the Piftons move, one of which defcends, while the other rifes, which Motion is communicated to them by the Wheel R,nbsp;which is moved by the Handle M M, Fig. 2.

Book II. nbsp;nbsp;nbsp;of Natural Thilofophy.nbsp;nbsp;nbsp;nbsp;ixf

fixed to the Axis a. The angular Motion of the Wheel is the eighth Part of a Circle, by whichnbsp;in a lefs Wheel there is produced an angular Motion of 120 Degrees. This lefifer Wheel is fixednbsp;to a third Wheel, by means of which the Piftonsnbsp;are immediately moved they make a Stroke of 3nbsp;Inches and a half.

The Contrivance of the Pifton is much the fame as in the Pumps which they ufe in Englafid j tho�nbsp;we think that we have made ours more perfect,nbsp;by fome Alterations in them.

The Glafs is to be exhaufted, or fet upon the round Plate L L; they communicate with thenbsp;Barrels, by means of a Pipe, one End of whichnbsp;is at D, and which foldered to the lower Side ofnbsp;the Plate, the Continuation of this Tube is feennbsp;at E Ej there are two Cocks in it, E,E, betweennbsp;the Cocks is fixed the Pipe /, /, which communicates with the Cylinders C, C.

When the Air is exhaufted, one of the Cocks above-mentioned ferves to Ihut the Communication between the Receiver (fo the Glaflcs arenbsp;called from which the Air is to be pumped out)nbsp;and the Barrels; the other Cock ferves to let thenbsp;Air in again, and to cut off the Communicationnbsp;with the mercurial Gage.

The mercurial Gage could not be convenient]y43g reprefented in this Figure; it ferves to determinenbsp;what Quantity of Air is drawn out of the Receiver, as alfo what Quantity of Air remains innbsp;it; it is likewife of ufe for meafuring the folidnbsp;Contents of the Receivers, which ought tonbsp;be exaftly known in feveral Experiments; ournbsp;Gage differs from the common Gages in feveralnbsp;Refpefls.

A little Cylinder, with a Screw upon it, is often fcrewed into the Plate at D, for applyingnbsp;a Globe to be e.xhaufted to the Pump.

Mathematical Elements Book IL

In the Middle of the Plate L L there is a Holej which is Ihut up with a Screw j but fometimes itnbsp;ferves for joining feveral Machines to the Plate.

439 nbsp;nbsp;nbsp;By this Means alfo there is often applied to thenbsp;Pump a cylindric Box, full of Leathers foaked innbsp;Wax, thro� the Center of which a brafs Wire paffes, which may be moved by the Help of a Handle, fo as to communicate Motion into a Place voidnbsp;of Air j the Box has a Cover, which enters intonbsp;it with'a Screw, for preffing the Leathers together, and to prevent theentring in, or efcapingout

440 nbsp;nbsp;nbsp;of the Air ; fuch a Box, or Collar of Leathers, isnbsp;often joined to the Cover which is laid over thenbsp;Recipients, as may be feen in Fig. 2. Plate XXVIII.nbsp;and in Fig. 2. Plate XXIII.

441 nbsp;nbsp;nbsp;/ When the Receivers are laid upon the Platenbsp;^/jL, L, or when the Receivers are floped with Co-

/ vers, or when the Screws are joined to the Ma-/ chine, and in general, when the Air is to be hin-/ dered from running in, we make ufe of Wax, which is foftned by mixing as much Oil andnbsp;Water to it as is found necellary.

CHAP. XV.

Several Experiments concerning the Air�s Gravity., and its Spring.

W be weighed like other Bodies, and fo its Denfity may be compared with that of other Bo-

weighed, when it is full of Air, and again, when the Air is exhaufted, the Difference between theirnbsp;Weights is the Weight of the Air 5 which Methodnbsp;has this Inconvenience, that fuch a fmall Difference of Weight cannot eafily bedifcovered, whennbsp;a Balance, tho� ever fo nice, isloaden with a greatnbsp;Weight; therefore we muft make ufe of the following Method.

Book It. of Natural Thilojbphy, 1x3

Experiment 1. Plate'KX.Ylll. Fig. 2.] Having442 exhaufted the Air out of the Glafs Ball, whofenbsp;folid Contents are 283 Inches, and having tiednbsp;fuch a Weight to it, that it may be almoft equalnbsp;in fpecifick Gravity to Water, let it be immerfednbsp;into the Water continued in the Veflel D E, andnbsp;let it be faftned by a Thread to the Hook ofnbsp;the Scale of the Balance A B, above defcribed i * * i/rnbsp;raife the Balance till you make an ^Equilibriumnbsp;with a very fmall Weight ^ if by opening thenbsp;Cock you let the Air into the Globe, a Weight I,nbsp;of about 100 Grains, will be required in the op-pofite Scale to reftore the jEquilibrium, fome-times more, or fometimes lefs, according to thenbsp;different Tenor of the Air, which here near thenbsp;Earth is varied according to the different Weightnbsp;of the Atmofphere, and according to the Difference of Heat and Cold.

Bodies immerfed in Liquids are fuftained by them, and the more or lefs according to thenbsp;greater or lefs Bulk of the Body, and the Weight * 199nbsp;loft in that Cafe is determined from the knowrnnbsp;Denfity of the Liquid ^ * by the foregoing Expe-* ^97nbsp;riment, therefore, it may be known, how muchnbsp;Bodies gravitate lefs in Air than in a Vacuum.

Hence alfo may be deduced, that Bodies tbat^i are in FEqiiilihrio in the Air, if their Bulks arenbsp;unequal, will lofe their jEquilihrium in a Vacuum : Which is confirmed by the following E:tpe-riment.

Experiment 2. Plate XXVIII. Fig. 3. ] Ini the Scales of the Balance a b, lay a Piece of Wax, c,nbsp;and a Weight of Metal, p, and you will have annbsp;-Equilibrium. Hang up the Balance in a Glafsnbsp;Receiver, and, having exhaufted the Air,, the

ii4 Mathematical Elements Book IJ.

Wax will preponderate, its Bulk being greater than the Bulk of the Body �gt;, it muft be morenbsp;fuftained by the Air j and, therefore, when younbsp;let the Air into the Receiver again, the JEq�ili-brium is reftored.

The Elafticity or Spring of the Air, which has been proved in Chap. XIII. becomes more fenfl-ble by the following Experiment.

444 nbsp;nbsp;nbsp;Experiment 3.] Tie up a Bladder very clofe,nbsp;with a fmall Quantity of Air in it j put a Receivernbsp;over it, and pump out the Air, whereby thenbsp;Preflure upon the external Surface of the Bladder is diminilhed, and immediately the Air included in the Bladder will expand itfelf, and fwellnbsp;it out. We have proved, that the Spring of thenbsp;Air is equal to the Weight of the whole Atmo-fphere j the following Experiment will make itnbsp;� vifible.

445 nbsp;nbsp;nbsp;Experiment 4. Elate XXVIII. Fig. i.] Take anbsp;Bladder tied up very clofe, and not quite full ofnbsp;Air, and put it in a Brafs Box A, whofe Diameternbsp;is three Inches and an half; fo that the Cover,nbsp;which is of Wood, and does not exactly fit thenbsp;Box, may be fuftained by the Bladder j you muftnbsp;put the Lead Weights P, P, upon the Cover:nbsp;They have a Hole in the Middle for a woodennbsp;Cylinder E, which is fixed to the Cover, to gonbsp;thro�: When you pump out the Air, the Bladder

I is fwelled, as in the foregoing Experiment, and by ' that means the Weights are raifed. You may wfenbsp;feveral Weights according to the Bignefs of yournbsp;Glafs Receiver i and tho� they Ihould amount tonbsp;60 or 70 Pounds, they would be eafily raifed-The Gravity of the Air, its Preflure that arifesnbsp;from the Gravity, as alfo its Elafticity, produce

Book 11. of Natural Thilofophy.

very different Effects ^ fome of which I fhall fe:-kdt, and confirm by Experiments.

Experiment 5. Elate XXVII. Fig. 3.I To the446 Hole in the Middle of the under Side of the Air-Pump LL, fcrew on a fmall hollow brafs Cylinder which has a Hole through it, and an open Glafsnbsp;Tube A B cemented to it at Bottom, whofe lower End B muft be immerged into Mercury. Letnbsp;Mercury be fuftained in the Tube eg^ that isnbsp;clofe at e, and void of Air in the Manner be-fore-faid. Set the Vef��l U with the Tube up- *4^�nbsp;on the Plate L L, and cover it with a tall Glafsnbsp;DC, fo as to cut off all Communication between the external Air and the Veffel U, as alfonbsp;the Cavity of the Tube A B. The Air in thisnbsp;Tube does, by its Elafticity, hinder the Mercurynbsp;from rifing up in the Tube, by the Prelfureof thenbsp;external Air. The Air alfo, that is included innbsp;the Receiver D C, does by its Spring fuftain thenbsp;Mercury in the Tube ge.* Pump the Air out *43?nbsp;of D C; as the Denficy diminithes, the Elafticitynbsp;does alfo decreafe, * and the Force by which the *43*nbsp;Mercury is fuftained in the Tube ge becomesnbsp;lefsj therefore, the Mercury defcends. At thenbsp;fame time the Preffure of the external Air overcomes the Refiftance in the Tube A B, and thenbsp;Mercury afcends in the Tube, The Diminutionnbsp;of the Spring in the Tube A B, and in the Veffel B C, is the fame, and the Effe�f of the Diminution the fame in both Cafes j therefore, thenbsp;Mercury defcends as much in the Tube e g as itnbsp;rifes in the Tube AB, which agrees with thenbsp;Experiment. By this Method the Mercury isnbsp;raifed up to �, while the Tube g e becomes al-moft wholly empty j when you let in the Airnbsp;again, the Mercury rifes in the Tube g e, as itnbsp;is deprelTed in the Tube A B.

Mathematical Elements Book II,

447 Experiment 6. XXVIII. 4.] Take the little Pump or Syringe A, and its Pifton beingnbsp;thruft ciofe to the Bottom, let the Tube whichnbsp;is joined to the Syringe be immerged in Water ;nbsp;when you raife up the Pifton, the Water will follow it, and fill up the Cavity between the Bottomnbsp;of the Pump and the Pifton which Effect arifesnbsp;from the Preffure of the external Air.

44^ Experiment 7. Plate XXVIII. Fig. 5.] Join the Glafs Tube ^cto the Syringe A, which is fere wednbsp;to the Cover of a Glafs Receiver, fo that thenbsp;End c of the Tube may defeend below the Surface of the Water in the Veflel U �, thruft downnbsp;the Pifton to the Bottom of the Syringe, andnbsp;let all the Air be pumped out of the Receiver,nbsp;if then you pull up the Pifton, the Water willnbsp;not rife.

449 nbsp;nbsp;nbsp;Experiment 8. Plate XXVIII. Fig. 6.] Thenbsp;Force, by which the Air pref��s upon Bodies, often breaks them, when the Preflure is not equalnbsp;every Way. Let the Brafs Cylinder A be coverednbsp;with a flat Piece of Glafs j when you pump thenbsp;Air out of this Cylinder, the Plate of Glafs willnbsp;be broken into a great many little Pieces by thenbsp;Preffure of the external Air.

450 nbsp;nbsp;nbsp;Experiment 9. Plate XXIX. Fig. i.] Take anbsp;Syringe A of an Inch Diameter pufh down thenbsp;Pifton to the Bottom of it, and fhut up the Holenbsp;at the Bottom of the Syringe, and hang on anbsp;Weight P of 10 Pounds to the lower End of thenbsp;Syringe j if you hold the Handle B of the Piftonnbsp;in your Hand, the Syringe will not defeend j fotquot;nbsp;it cannot defeend, unlefs the Weight hanging at

Book II. of Natural Thilofophy.

it overcomes the Preflure of the Air and Fridlion of the Pifton ^ but the Preflure of the Air alonenbsp;does here exceed lo Pounds.

Experiment lo. Plate XXIX. Fig. 2.] The Sy-451 ringe defcends^ by the Weight �gt; alone in a Vacuum, which is but juft fufficient to overcome thenbsp;Fri�tion of the Pifton.

Experiment ii. Plate XXIX. 3.] We fee more fenfible Effect of the Preflure of the Air,nbsp;when the two Segments of a Sphere, H and I, arenbsp;joined together. Let the Brim or Edge of each ofnbsp;them be well polifhed, fo that they may fit together, and, when they are applied clofe, put a littlenbsp;Wax between, to exclude the Air. There is anbsp;Cock in the Segment H, by which the two Segjnbsp;ments, when joined together, may be applied tonbsp;the Air-Pump, and which muft be fhut, whennbsp;you have exhaufted the Air. The Segments arenbsp;iufpended by the Ring A, and, by the Help of thenbsp;Ring Q, you may hang to them the Weights thatnbsp;are laid upon the great wooden Scale T. If thenbsp;Diameter of the Segments be three Inches and annbsp;Half, a Weight of about 140 Pounds will be required to pull them afunder.

Experiment 12. Plate XX.X. Fig. i.] Let the453 Segments be joined together and exhaufted, as innbsp;the former Experiment i if they be fufpended innbsp;a Vacuum, with a little Weight P hanging on,nbsp;which is juft able to overcome the Cohefion ofnbsp;the Wax, they will be feparated in this Experiment. There muft be faftned to the Plate L L,nbsp;the little brafs Box, or Collar of Leathers, * thro� �nbsp;which a brafs Wire, that has the Weight hangingnbsp;to it, flips. Left the Receiver Ihould be broke bynbsp;the Fall of the lower Segment, you muft put un-Q znbsp;nbsp;nbsp;nbsp;der

12 8 nbsp;nbsp;nbsp;Mathematical Elements Book II.

der it the hollow wooden Cylinder M, to let it fall into. In this Figure the Segments are fuf-pended to the Cover of the Receiver which is tonbsp;be exaufted j they may alfo be fufpended fromnbsp;a Pillar faftened to the Cylinder M. That thenbsp;Hemifpheres may not be feparated without Difficulty, it is not required that they fhould benbsp;empty of Air; as great a Force will be requirednbsp;to feparate them as in the nth Experiment jnbsp;when having included them in a Veflel, and applied them clofe together (fo as to leave themnbsp;full of common Air, and (hut the Cock, that thenbsp;Air between them may not be changed the Airnbsp;ontheOutfideof them in the Veflel, that containsnbsp;them, is reduced to a double Denfity j which tonbsp;confirm by an Experiment, we muft firft deferibenbsp;the Machine with which we make Experiments innbsp;comprefled Air.

45^ Ftate XXXT. Fig. 5.] Around brafs Plate N is laid upon a Board a a, about 15 Inches long,nbsp;and 10 wide j the Diameter of this Plate is a-bout 5 or 6 Inches, as you may fee by its feparated Figure at N, in P/ale XXIX. Fig. 4. andnbsp;has fixed to it here a Cylinder P, which is notnbsp;perforated, and goes through the Board a a. Upon this Plate you muft put a Glafs UU, about 10nbsp;Inches high, that is terminated at each End in anbsp;cylindric Form, and the cylindric Parts havenbsp;brafs Rings, or Ferrels, upon them.

The Vefiel muft have upon it the Cover D. The Pillars G S, CS, are faftened to the Board aa^nbsp;and go thro� the Wood by which the Covernbsp;D is firmly joined to the Glafs, as alfo the Glafsnbsp;to the Plate N, by the Force of the Screws ��.nbsp;It is very necelTary to prefs all thefe Parts clofenbsp;together, having firft fpread Wax upon the uppernbsp;and lower Edges of the Glafs.

The

Book II, of Natural T^hilofiphy.

The Cover is reprefented in the feparated Fi-p,ure D, Plate XXIX. Fig. 4 ] There is fixed to it the Collar of Leathers : * and, left the Piece �nbsp;of Wood de fhould be applied to too fmall a Surface, the Cover is made in the Shape of a roundnbsp;open Box.

There is a perforated brafs Wire C, which goes through the Collar of Leathers, to which a Cocknbsp;B is joined.

PLate XXIX. Fig. 6.] In order to comprefs the 455 Air in this Veftel, fcrewing on the Syringe Anbsp;B to the Cock Blaft mentioned, which Syringe hasnbsp;joined to it another Cock, in the Key of which,nbsp;befides the ufual Hole, there is another obliquenbsp;Hole which goes to �, and by which when younbsp;flrut the Communication between the Glafs andnbsp;the Syringe, the Syringe has a Communicationnbsp;with the common Air, and is filled with it, whennbsp;you raife up the Pifton. When you open thenbsp;Communication between the Glafs and the Syringe, by pufhingdown the Pifton, you force thenbsp;Air which was contained in the Syringe into thenbsp;Glafs j and, by often repeating this Operation,nbsp;you at laft bring it to the Denfity required.

Experiment 13. Plate XXIX. Fig. 4 Now, to 456 feparate thefe Segments or Cups in compreflednbsp;Air, the Segment I is joined to the Plate N,nbsp;by Means of the Pillar M L, which has Screwsnbsp;at M and L. The other Segment H, bynbsp;Screws at F and E, is joined to the Wire C.

As for the reft, you muft obferve w'hat has been faid in the Defeription of the Machine j and thenbsp;Air muft be comprelTed in the Veftel, fo as tonbsp;have twice the Denfity of that which is compref-fed only by the Attnofphere. At P the Ring

0.3 nbsp;nbsp;nbsp;is

Z30 nbsp;nbsp;nbsp;Mathematical Elements Book II.

is joined to the Plate N, as alfo the Ring A to the Cock B. Invert the Machine, as in Pm- S,nbsp;and fufpend it by the Ring Q. The Scale T, onnbsp;which the Weights are laid, hangs upon the Ringnbsp;Aj and, until the Weights, laid on, come tonbsp;be about 140 Pounds, the Segments will not benbsp;fepai'ated. Three Screws X, X, hinder the Scalenbsp;T from defcending too low in feparating the Segments,

457 Experiment 14. PlateX.X.'X., Fig. 2.] Apply the Tube A B to the under Side of the Air-Pumpnbsp;Plate LL, which Tube has a Cock in its uppernbsp;Part, and is joined to the fmall Tube which Handsnbsp;above the Plate. Put on the Receiver R, whichnbsp;covers the prominent Tube. The End B of thenbsp;Tube A B muft be immerfed in the Water contained in the VefTel U, and having exhauftednbsp;the VefTel R, you muft open the Cock; the Water will fpout up into the Receiver wdth a greatnbsp;Force, for the fame Reafons as the Water is fuftainednbsp;'' 4^3 at the Height of 32 Foot in a Pipe void of Air. *

^58 Experiment 15. Plate XXX. Fig.s.] The Air�s Elafticity produces the fame Effe�t. Let therenbsp;be a brafs Cylinder U exactly ftiut. There muftnbsp;be a Hole in the Bottom to pour in Water, whichnbsp;afterwards you fhut up with a Screw. To thenbsp;upper Part of the Vefl'el there is foldered a Pipe,nbsp;which goes down almoft to the Bottom; and tonbsp;the other End of it, that Hands above the Vefiel,nbsp;a Cock is joined (^See Fig. 4.) This VefTel is tonbsp;be ferewed on to the lower Part of L L, the Air-Pump Plate, and from it a Pipe goes quite thro�nbsp;the Plate, and ftands up above it, which is covered by the Receiver R. After you have pumped out the Air, the VefTel U being about twonbsp;Thirds full of Water, when you open the Cock,

Book II. of Natural Thiloffhy. nbsp;nbsp;nbsp;^3I

the Water will violently fpout up into the Receiver, by the Force of the Spring of the Air contained in the upper Part of the Veffbl U. Here the Air prefles upon the Surface of the Water ;nbsp;when you open the Cock, the Preflure in thenbsp;Tube becomes lefs, therefore the Water muft gonbsp;into the Tube.

Experiment i6. P/^?eXXX. 4-1 inthe459 open Air, the Water will violently fpout out ofnbsp;the VefTel �, if, having filled it two Thirds fullnbsp;of Water, the Air be compreffed in the uppernbsp;Part of it; which is done by help of the Syringenbsp;above-mentioned *�

Experiment 17. P/lt;JteXXXI. Fig. i. j Invert the 460 Glafs R, and immerge it in the Water containednbsp;in U, the Air keeps out the Water at whatevernbsp;Depth it be immerged ; yet the deeper the Glafs isnbsp;put down, the lefs Space the Air is reduced into.nbsp;Upon this Principle are made the Machines 111461nbsp;which divers go down into the Sea. They arenbsp;made like Bells, and defeend by their own Gravity ; the Water does not rife up to the Diver innbsp;the Bell ; frelh Air is fent down continually bynbsp;Bladders tied to a Rope, which he draw's down tonbsp;him ; the Air, heated by his Refpiration, rifes tonbsp;the upper Part of the Bell, and is there driven outnbsp;through a Cock, by the PrelTure of the Water, thatnbsp;pufhes up, and comprelfes the Air in the lowernbsp;Part of the Bell; which Preffure overcomes thenbsp;Force with which the Water endeavours to defeend thro� the Cock; for the Prelfure of Liquidsnbsp;is increafed in Proportion to their Depths. *

Experiment 18. Pt�eXXXI. 2.] Take lit-463 tie Figures of Glafs that are made hollow, of annbsp;Inch and half long, reprefenting Men, whichnbsp;may be had at the Glafs-Blowers; thefe little

Mathematical Elements Book II,

Images have a fmall Hole in one of their Feet, and are lighter than Water. Immerge them into the Water contained in the Glafs A B. Thisnbsp;Glafs is about a Foot, or 15 Inches high, and covered with a Bladder, which is tied fall over thenbsp;Top. A fmall Quantity of Air is to be left between the Bladder and Surface of the Water. Ifnbsp;the Veffel be preffed w'ith the Finger, the Airnbsp;above-mentioned is reduced to a lefs Space, andnbsp;the Surface of the Water is more comprelTed jnbsp;the Water which is more comprefled, enters innbsp;the little Men through the E.ole at their Feet, andnbsp;comprefles the Air in their Bodies more than itnbsp;was. The little Images, becoming heavier, bynbsp;this Means defcend towards the Bottom of thenbsp;Veffel, and that fafter or flower, according to thenbsp;Bignefs of the Hole, and alfo according as thenbsp;fpecifick Gravity of the Images comes nearer tonbsp;thefpecifick Gravity of the Water. Taking awaynbsp;your Finger, the Air in the little Men, beingnbsp;lefs comprefled, expands itfelf, and drives outnbsp;the Water j fo the Images rife up again to thenbsp;Surface of the Water.

^63 Experiment 19. Plate XXXI. Fig. 3.] Animals cannot live without Air. If any Animal be included in the Receiver U, and the Air be drawnnbsp;out, the Animal will immediately be in Convul-fions, and will fall down dead, unlefs the Air benbsp;fuddenly re-admitted. Some Animals will live innbsp;a Vacuum longer than others.

^64 Experime72t 20. Plate XXXI. Fig. 4.] Some Fifhes alfo cannot live without Air j but in othersnbsp;you fee no fuch Change, but the Swelling of theirnbsp;Eyes. What Experiments you make upon Fifhes,nbsp;muft be made in the Glafs Receiver U, which isnbsp;fet upon the Plate of the Air-Pump, and to the

Hole,

Book II. nbsp;nbsp;nbsp;of Natural Th�lofofhy.nbsp;nbsp;nbsp;nbsp;a33

Hole, through which the Air is drawn out, you muft fcrew on a Pipe, which comes up almoft tonbsp;the upper Part of the Glafs U ^ pour in Water,nbsp;and then put a Cover over the Glafs Receiver U,nbsp;andexhauft the Air out of the upper Part of it.nbsp;Having taken away the Preffure of the Air fromnbsp;the Surface of the Water, the Air in the Filh�s Bodynbsp;expands itfelf, by which Means the Fifh, becoming lighter, cannot defcend in the Water.

Experiment 21. Plate XXXI. Fig. 5.] Experi-/|65 ments are made upon Animals in comprefled Air,nbsp;by Help of the Machine above defcribed.* Innbsp;that Cafe Animals do not foon die, becaufe thenbsp;VeflTels in the Body are not broken ; yet if theynbsp;continue long in that condenfed Air, it muft benbsp;hurtful to them^ nay, and in a greater Compref-fion of the Air (for which a Veftel of Metal is required) they will die in a little Time.

Experiment 22.] Several Liquors contain Air. 466 If you put them under a Glafs Receiver, andnbsp;draw out the Air, then the Air contained in thenbsp;Liquors will expand itfelf and go out. In thatnbsp;Cafe very often the Adlion of the Particles of thenbsp;Liquid upon one another, is changed, and a Ferrnbsp;mentation arifes.

The Tgt;efcription of feveral Machines^ and the Exf lunation of their EfeEts.

LE T one End a, of the Curve Tube a S be immerged in Water, whilftthe other Endnbsp;h defeends below the Surface of the Water. Ifnbsp;hy fucking, or any other Way, the Air be taken

out of this Tube, the Water will run through b. This Inftrument is called a Syphon.

468 nbsp;nbsp;nbsp;This EfFedt arifesfrom the PreflTure of the Air,nbsp;which drives on the Water in the Syphon, by itsnbsp;Preffure upon the Surface of the Water in thenbsp;Veflel. The Air does alfo prefs againft the Water that goes out of the Orifice b, and fuftains it.nbsp;Thefe Preflures are equal, and ad contrariwifeinnbsp;the upper Part of the Syphon, with a Force e-qual to the Weight of the Atmofphere, takingnbsp;away the Weight of the Pillars of Water whichnbsp;are fuftained by the PreflTure. The Pillar of thenbsp;Water in the LegS^, is longer than the oppofitenbsp;Pillar of Water; therefore, the PreflTure of thenbsp;Air is more dirainilhed on the Side b S, and thenbsp;oppofite Frelfure overcoming it, the Water flowsnbsp;towards b.

469 nbsp;nbsp;nbsp;Experiment 2. Plate XXXI. Fig, 6.] The Syphon above-mentioned has this Inconveniency,nbsp;that, if once it ceafeth to work, the Water willnbsp;not run again, unlefs the Air be drawn out of thenbsp;Tube afrefh. This may be correded by makingnbsp;a Syphon in the Figure a S b, whofe Legs arenbsp;equal, and turned up again; For if the Syphon benbsp;filled with, and one Leg be immerfed in Water,nbsp;fo that the Surface of the Water may be abovenbsp;the Orifice, then the Water will run out throughnbsp;the other Legj for the Reafon given in the Explication of the former Experiment. Since thenbsp;Tegs are returned upwards, the Syphon will notnbsp;be emptied, when the running out of the Waternbsp;ceafesj and fo the Syphon, being once filled, isnbsp;always ready to work its Effed. The Water runsnbsp;backward or forward through it, according as it isnbsp;higher on one Side or the other,

Book II. of Natural Thilofophy.

Plate XXXII. Fig. 2,] Upon the fame Princi-4'70 pie as the foregoing Machines, is contrived thenbsp;Syphon for raifing Water into a Ciftern, TheEf-fe�t of this Syphon becomes vifible by the Helpnbsp;of a Machine made up of two hollow Glafs Ballsnbsp;Hand I, which are joined together by the Tubenbsp;C D E, the Ball I communicates with the Waternbsp;to be raifed by means of the Tube A B, whichnbsp;comes up almoft to the Top of the Eallj to thenbsp;Ball H at the lower Fart is joined the Tube F G,nbsp;as long as the whole Tube A B.

The Ball H rauft be filled with Water through a Hole by a Funnel, and then the Hole muft benbsp;lliut up clofe.

In fuch Machines as are applied to Ufe, for raifing Water out of aRefervoir that contains it,nbsp;the Water is brought away in the Veflel H, andnbsp;the Communication between the VefiTel and thenbsp;Refervoir is fhut up with a Cock.

Experiment �}.I, Opening the Cock G, the Water will tun out that Way, and the Water will afcend through the Tube A B up into the VelTelnbsp;I; which being filled, the Water is fulfered tonbsp;run away to the Place where you would have it;nbsp;and, by repeating the Operation, the Elevation ofnbsp;the Water continues.

Opening the Cock G, the Air prefTes 3galnftthe47l Watergoing out of the Tube F G; the Air alfonbsp;prelTes upon the Water in the E.efervoir, and fu-ftains that vvhichis in the Tube AB. ThefePref-fures are equal, and if you take from them the Columns of Water which they fuftain, you will havenbsp;the Forces by which they aft upon the Air, contained in the upper Part of the Veflels and the Tubenbsp;CDE. The Pillar F G, becaufe there is fuperaddednbsp;to it the Height of the Water in the Veflel H, does

Mathematical Elements Book IT.

always overcome the Column in the Tube A B, as being longer, therefore the PrelTure at G isnbsp;lefs diminiflied than the other, and fo overcomenbsp;by it j and therefore the Water muft rife in thenbsp;Tube AB, and defcend down F G.

4'72 To render the Effedt of common Pumps vifible. Jet there be a little Pump made of Glafs in the following Manner; AB (P/^reXXXII. F/g. 3.)niuftnbsp;be a Cylinder of Glafs, and about an Inch and anbsp;half Diameter. In the Bottom of it join a Tubenbsp;of any Length, as C D. Let the upper Part ofnbsp;it be ihut with a leaden Ball, fo that the Waternbsp;may not be able to defcend out of the Cylinder,nbsp;but may eafily rife into it, by railing up the Ball,nbsp;which we make ufe of here inftead of a Valve.nbsp;The Pifton is moved in the Cylinder A B, which,nbsp;being furrounded with Leather, exadtly fills itsnbsp;Cavity; There is a Hole in the Pifton, which like-wife is flopped with a Ball of Lead inftead of anbsp;Valve ; fo that the Water may rife, but not defcend through the Pifton.

Experiment 4.] Pufh down the Pifton to the Bottom; pour Water upon it to hinder the Paflfagenbsp;of the Air; if the End of the Tube CD be im-merfed into Water, and the Pifton be raifed, the

*447 Water will afeendupinto the Cylinder A B *from which it cannot defcend; wherefore, it comes upnbsp;through the pifton, when it is pulhed down. Ifnbsp;you raife the pifton again, the Cylinder is againnbsp;filled with other Water, and the firft Water isnbsp;raifed up into the wooden Cylinder which is joinednbsp;into the glafs one, from which it runs out thro�nbsp;the Tube G.

4173 Since the Effe�fs of all the Machines, deferibed in this Chapter,' depend upon the Freflure of thenbsp;Atmofphere, the Water will not rife in thefe Ma-chines much higher than 32 FootA

There

Book II. of NaturalThilofofhy. nbsp;nbsp;nbsp;Z37

There are feveral little artificial Fountains, that are called the Fountains of Hero; I fliall herenbsp;give the Conftruftion of one of them.

Plate XXXII. Fig. 4.] Let there be two equal 474 elliptical Veffels AB and C D, exactly Ihut on allnbsp;Sides, made of one Sort of Metal.

In each of them there is a Separation pafling through the Center of the Ellipfe, which dividesnbsp;the whole Vefiel into two equal Parts.

pendicular to the Axis of the Ellipfe, the Separation cfgb of the other Veffel muft be inclined tonbsp;that Axis.

There is a Brim raifed round about the upper Part of the Veflel A C B, to make a Bafon.

Four Tubes are joined to thefe Veffels. The firft 0 p goes through the Cavity B of the Veflelnbsp;A B, without having any Communication withnbsp;it, and defcends almoft to the Bottom of the Cavity D ; the fecond J ? is foldered to the uppernbsp;Part of the Cavity D, and afcends to the uppernbsp;Part of the Cavity B, but not quite fo high as tonbsp;touch the upper Plate of it. The third q r reachesnbsp;from the lower Part of the Cavity B, almofl: tonbsp;the Bottom of the Cavity C; the 4th, X �, is madenbsp;faft to the upper Part of the Cavity C, and reachesnbsp;almoll to the upper Part of the Cavity A.

Laftly, there is a Tube zy,' which, going thro� the upper Plate, is foldered to it, and reachesnbsp;down lb deep in the Cavity A, that its End %nbsp;is but a little Way off of the Bottom.

There are Cocks joined to every one of the Cavities ; orelfe they have other Holes that are Ihut up with Screws that have Leathers on themi thenbsp;chief Ufe of them is to let out the Water verynbsp;clean from the Cavities, leil they Ihould grownbsp;rufty, when the Machine is not in Ufe.

Mathematical Elements Book 1�

Experiment 5.] Pour in Water through the Tube 0 �), fo as to fill the Cavity D j if you continue to pour in Water, it will rife up through thenbsp;Tube s t^ and then defcend through q r into thenbsp;Cavity C, which is alfo filled, the Air afcendingnbsp;up through X and going out through zy.

Turn the Machine upfide down, opening the Cocks of the Cavity C and D, the Water will defcend into the Cavities B and A. Having againnbsp;fhut the Cocks, as alfo the Hole y of the 1 ubenbsp;zy, fet the Adachine again the right Side upwards,nbsp;and pour in Water again through the Tube ap,nbsp;till the upper Surface of the Machine be coverednbsp;with Water. Now, if the Hole y be opened, thenbsp;Water will fpout up to almoft twice the Heightnbsp;of the Alachine, and the Afotion of the Waternbsp;will continue, till the Cavity A be emptied of itsnbsp;Water. The Height of the fpouting Water willnbsp;continually diminifh, and at laft it will not benbsp;double the Diftance of the Veflels.

^gt;75 The EfFedi; of this Adachine is to be attributed to the Compreffion of Air in the Velfels. Thenbsp;PreiTure of the Atixiofphere at 0 andy, as alfo innbsp;the VelTels, is equal, and thefe Preflures deftroynbsp;oneanothcr^ therefore, they are not tobe confider-ed in the Examin.ation of the Alachine. When atnbsp;laft the Water is poured into the Tube op, it isnbsp;fuftained in it by the Preffure of the Air contained in the Cavity D, and adting upon the Surfacenbsp;of the Water which ftands at a finall Height in thatnbsp;Cavity ; which Air, therefore, is comprefted bynbsp;the Weightof the Water, whofe Height is p 0. Wenbsp;fpeak of the Prefiure, by which the Prefllirc of thenbsp;Atmofphere is overcome. The Air in the uppernbsp;Part of the Cavity B communicates with thenbsp;Air above-mentioned by the Tube.rf, and is c-qually comprefted, and a�fs with the lame Force

upon

Book II. of Natural hilofojgt;hy.

This Prefllire is to be added to the Preflure ari-fing from the Height of the Water, in order to have the Force by which the Air is comprefled in thenbsp;Cavity C, as alfo in the upper Part of the Cavitynbsp;A, by reafon of the Communication through thenbsp;Tube X It. The Preflure, therefore, upon the Surface of the Water in that Cavity A, is equal to anbsp;Pillar of Water, whofe Height is almoft doublenbsp;the Weight of the whole Machine. And thereforenbsp;it fpouts as if it was prefled by fuch a whole Column, that is, to a Height not much wantingnbsp;from the Height of that whole Column. *nbsp;nbsp;nbsp;nbsp;* 358

The Height is continually diminifhed, for the Columns of Water, which comprefs the Air, continually become Ihorter, becaufe the Water af-cends in the Cavities G and D, and its Height isnbsp;diminiflied in the Cavity B. In the fame Timenbsp;the Cavity A is continually evacuated, and thenbsp;Water afcends through a greater Space, before itnbsp;comes to y ^ therefore, it is driven to a lefs Heightnbsp;above y.

CHAP. XVII.

Of the undulatory Motion of the Air, iz'here VL�e Jhall treat of Sound.

IF the Air he agitated in any Manner, the Par-4.176 tides moved recede from their Place, and^^�nbsp;drive the neighbouring Particles in a lefs Space �nbsp;and as the Air is dilated in one Place, it is com-preflTed in the Place next to it ^ the compreflednbsp;Air, by the Reftitution of the Spring, not onlynbsp;returns to its firft State, but is alfo dilated by thenbsp;Motion acquired by the Particles.

The Air, being firft dilated by that Motion, is reftored to its firft State, and the Air is cotnpref-

fed towards other Parts. This again obtains, �when the Air lafb comprefled expands irfelf, bynbsp;which a Compreffion of Air is again produced ^nbsp;therefore,/ro?// any Agitatioti there arijes a Motionnbsp;analogous to the Motion of a Wave on the Surface of

*^01 Water * The Air is compreiTed in that Manner with a Dilation following, fo as to make what is

�*703 called a Wave of Air : * Comprel��d Air always dilates itfelf every Way, and the. Motion of theje

478 nbsp;nbsp;nbsp;Waves is the Motion of a Sphere expanding itfelfnbsp;in the fame Manner as the Waves move circularly

,478 Whilfi a Wave moves in the Air^ vcberever it paffes^ the Particles are removed from their place^nbsp;and return to it, running through a very ffsort Spacenbsp;in going and coming.

XXXIII. Fig. I.] Now, to explain the Laws of this Motion, let us conceive Particlesnbsp;of Air to be placed at equal Diftances, and to benbsp;in a right Line, as a, h, c, d, amp;c and f. Let thenbsp;Wave be fuppofed to move along that Line. Nownbsp;let us fuppofe it to be come forward along thatnbsp;Line, as far as between h and p, and that thenbsp;Air is dilated between b and h, but compreflednbsp;between h and p, as all this is reprefented in Line i.

479 nbsp;nbsp;nbsp;The greateji Denfity is at m, which is the Middle between h and p; and the greatef Dilation between b and h, is in the Middle e.

480 nbsp;nbsp;nbsp;Wherever the neighbouring Particles are not e-qually diftant, the Motion, arifing from Elafiicity,nbsp;caufes the lefs diflant Particles to move towardsnbsp;thofe that are niofi diftant j * and this Motion a-lone, abftra�fing from all other Motion acquired,nbsp;is to be examined.

Between band e there is a Motionb towards e, that is, confpiring with the Motion of the Waves-,nbsp;there is alfo fuch a Motion between mandp.

But

Book It. of I^aturai Thilojophy. 241

But there is a contrary Motion between e andsfi 'z and, it is direbbed from m towards e.

At m and e, where the Directions of the Mo-483 tions are changed, no .Adtion arifes front the Ela-Jiicity^ becaufe the neighbouring Parts are placednbsp;at equal Diftances among themfelves.

In the Places and p, the Difference of the 484 neighbouring Parts is the greateft of all j and therefore, there is the greateft A�iion of the Elafticity.

From this it follows, that a Particle, according to its different Place in a Wave, fuffers a different Action from the Flafticity by which its Motion is generated, accelerated, diminifhed or de-llroyed j therefore, the Direction of the Motionnbsp;of a Particle cannot be determined from the Action of the above-mentioned DireCtion only, andnbsp;does not always agree with that DireCtion, andnbsp;the Motion of the fingle Particles is changed everynbsp;Moment.

All the Particles between h and p are removed according to the Order of the Letters. The Particles between h and p continue theif Motion,nbsp;and the reft between h and b return towards asnbsp;will be faid hereafter.

Thefe continue in the Motion by which they return, until, by the Aftion of the Elafticityjnbsp;whofe Diredfion is changed in the Point e, thenbsp;Motion acquired anew be deftroyed in whichnbsp;Cafe a Particle,, as b, returns to reft, and its firfl:nbsp;State. In the following Moment the Particle enbsp;comes to reft in its firft State, but p comes forwardnbsp;to gf, as in the Line 2, and fucceffively in equalnbsp;Moments, the Wave has all the Pofitions whichnbsp;are here reprefented in the Lines i, 2, 3, ye.485nbsp;and 13; and, whilft the Wave, from the Pofitionnbsp;in the Line i, comes to the Pofition in the Linenbsp;13, it runs throquot; its whole Breadth. The Particlenbsp;Pj in that Motion, goes and returns^ and the Mo-Rnbsp;nbsp;nbsp;nbsp;tion

Mathematical Elements Book II.

tion of it is made fenfible in the Figure, and, as it plainly appears, this Particle goes jucceffivelynbsp;through all the Situations of the Particles j in thenbsp;Waves all the Particles lingly are agitated by thenbsp;like Motion.

486 sThe Motion of any Particle, 3S /gt;, in its going backward and forward, is analagous to the Motionnbsp;of a �Vibrating Pendulum, whilfi it performs twonbsp;Ofcillations, that is, does once go forward andnbsp;backward. A Pendulum defcends in its Ofcilla-tion, and the Motion acquired confpires with thenbsp;Motion of Gravity, and is accelerated by it, until it comes down to the loweft Part of the Arcnbsp;to be defcribed, that is, the Middle of the Waynbsp;to be run through ; the Pendulum goes on bynbsp;the Motion acquired, which is deftroyed by thenbsp;Adtion of Gravity, whofe Diredlion changes innbsp;this Point, whilft the Body afcends up the othernbsp;Part of the Arc to be defcribed : This Body returns by the fame Laws.

The Particle p is moved by the Elafticity, and this Motion is accelerated by the Adlion of thenbsp;Elafticity, until it comes to the Situation of thenbsp;�^481 Particle m, the Line i, * which Situation is feennbsp;in Line 4, in which the Particle p is, in the Middle Point of the Space, to be run through by thenbsp;Motion backward and forward. By the Motionnbsp;*4*1 acquired, though Gravity atfts againfl: it, * itnbsp;perfeveres in its Motion, until, by the Acftion ofnbsp;the faid Elafticity, the Motion be wholly deftroyed, which happens, when it has gone through anbsp;Space equal to that in which it was generated jnbsp;then the Particle is in the Pofition which is feennbsp;in Line 7, which anfwers to the Situation of thenbsp;Particle h in Line i. Then, by the Elafticity, thenbsp;Particle returns, and is accelerated, until it hasnbsp;acquired the Situation of the Particle e, innbsp;*4* I Line i, * as in Line lo, that is, until, as in

Book II. of Natural Thilofophy. nbsp;nbsp;nbsp;2-43

Line 4, it comes again to the Point that is in the. Middle of the Way to be run through. The Particle continues in its Return, until by the Actionnbsp;of the Elafticity, whofe Direction is again changed *, the whole Motion is deftroyed 3 and then *483nbsp;the Particle returns to its firft Pofition, as in thenbsp;Line 13, and there, not being agitated by anynbsp;new Motion, it remains at reft. Therefore, theCifgt;*\nbsp;Motion of the tremulous Body, hy which the Air isnbsp;agitated, ceafing, there are no new Waves generated, and the Number of the Waves is the fame asnbsp;the Number of the Agitations of that Body.

If, after two Vibrations of a Pendulum, the Adtion of Gravity ftiould ceafe, as in the Air,nbsp;after the going and returning of a Particle, thenbsp;Adlion of the Elafticity on that Particle ceafes,nbsp;the Motion of a Particle of Air would whollynbsp;agree with the Adfion of a Pendulum. In thenbsp;middle Point of the Arc, which is to be run thro�nbsp;in the Ofcillation, there is no Adtion of Gravity,nbsp;and its Diredlion is changed j in the middlenbsp;Point of the Space to be gone through by thenbsp;Particle p in its going and coming, in which it isnbsp;in the 4th and loth Line, the Situation of thisnbsp;Particle agrees with the Situation of the Particlesnbsp;in and e in Line i, in which Points there is nonbsp;Adfion of Elafticity, and its Diredlion is chang-ged*. In a Pendulum, the more a Body ofcilla-*4g5nbsp;ting is diftant from the loweft Point or Middle ofnbsp;the Arc to be defcribed, by fo much greater is thenbsp;Force of Gravity adting upon itj the more alfonbsp;the Particle f is diftant from the Space to be runnbsp;through, the more is the Adlion of the Elafticitynbsp;Upon it j and in the Lines i, 7, and 13, the Particle is moft diftant from the Point above-mentioned, and its Situation there agrees with thenbsp;Points, b, h and p in the Linei, in which thenbsp;Adlion of the Elafticity is greateft of all *�nbsp;nbsp;nbsp;nbsp;�

Mathematical Elements Book II,

According to which Law, fince this Adiion of Elafticity increafes with the increafed Diftance ofnbsp;the often-mentioned middle Point, it is determined from the very Law of the Elafticity of thenbsp;Air, whofe Particles drive one another away with anbsp;Force which is inverfely as the Diftance betweennbsp;�^43, the Centers of the Particles*; and it is demon-ftrated, that the Adfion of Elafticity upon fuchnbsp;a Particle, as p, is increafed or diminifhed in Proportion to the Diftance of the middle Point ofnbsp;the Space to be run through; And, therefore, aJfonbsp;in that Part there is an Analogy between the Motion of a Particle and the Motion of a Pendulumnbsp;ofcilating in a Cycloid *.

If the Breadth of a Wave remaining, the Particles run out thro� a greater Space, the Compref-fton and Dilatation of the Air in the Wave will be greater, and there will be a greater Adfion of Elafticity, and that greater in the fame Ratio in whichnbsp;the Space gone through in the going and comingnbsp;is increafed: And the Motion of a Particle,as pinnbsp;this Cafe, differs from the Motion in the foregoingnbsp;Cafe, as the unequal Ofcillations of different Pendulums differ ^ which, as they are performed innbsp;*156 equal Times *, the fame will alfo obtain here.

Therefore, a Particle, as /), if the Breadth of the Wave continues, the fame goes and comes innbsp;the fame Time, through whatever Space it be carried out of its Place ; that is, the Wave will go its

488 nbsp;nbsp;nbsp;Breadth in the fame Time j therefore, a^ equalnbsp;Waves, �whether the Air he more or lefs agitated,nbsp;are equally jwift.

489 nbsp;nbsp;nbsp;Now let us examine unequal Waves ; let themnbsp;he as A to B, and let the Space gone through bynbsp;the Particles in the Motion of each of them, innbsp;going and coming, be in the fame Ratio �, in thatnbsp;Cafe the Compreftions and Dilatations in corre*

fpondent Places will be equal; the AtWom^ therefore, from the Elafticity, don�t differ in corre-fpondent Diftances from the middle Point of the Spaces to be run thro� by the Particles, in theirnbsp;going and coming. Therefore, thofe Motions arenbsp;analogous to the Motions of two Pendulums, whofenbsp;Lengths are as A and B, and which run thro� fi-milar Arcs; for, in the correfpondent Points ofnbsp;thofe Arcs, the Adion of Gravity is the fame.

In Pendulums the Adtion of Gravity increafes as the increafed Quantity of Matter; and whatever be this Quantity, the Motion is equally fwifr,nbsp;when the Gravity is not changed : On the contrary, the Adlion of Elafticity is determined innbsp;the Motion of Waves, and depends upon theDi-ftance between the Particles and the Velocity,nbsp;which is generated from it; the Elafticity remaining the fame, is inverfly as the Quantity of Matter to be moved.^ In the Ifavss above-mentioned,nbsp;the Quantities of Matter are as the Breadth of^^onbsp;the Waves a and b, and the Velocities generatednbsp;by the Elafticity are, therefore, in correfpondentnbsp;Points as b to a. Therefore, thefe Motions arenbsp;analogous to the Motions of Pendulums deferibingnbsp;fimilar Aers, and moved with different Forces ofnbsp;Gravity, which are to one another B A; for,nbsp;in correfpondent Points of fimilar Arcs, the Celerities arifing from different Gravities areas thofenbsp;Gravities.

Now to compare the Motion of Waves with the Motion of Pendulums, we muft confider Pendulums differing in Length, and on v.'hich different Forces of Gravity adf*, and we have Ihewnnbsp;what thefe Caufes produce fingly in the Dura- 490nbsp;tion of the Vibrations.* Both thefe are to be ^158nbsp;joined together, and the Squares of the Times ��5nbsp;of the Olcillation of pendulums, whofe Motions

Mathematical Elements Book II.

are analogous to the Motion of the above-men-tioned Waves, are as the Length A and B, * and inverfely, as the Gravities B and A j * that is,nbsp;16} again dire�fly, as A and Bj the Ratio of whichnbsp;Ratio�s is a Ratio, compounded of the Squares ofnbsp;the Quantities A and B. Therefore, the Times ofnbsp;the Ofcillations are as A and B, and the Timesnbsp;are in the fame Ratio in which the Particles ofnbsp;the Waves go and comej that is, the Waves runnbsp;through their Breadths, which are as A to B jnbsp;which Times are, therefore, as the Spaces gonenbsp;thro� by the Waves, and, therefore, the Motionsnbsp;are equally fwift. If the Space be changed thro�nbsp;which the Particles go and come, the Velocitynbsp;4S8 of the Waves is not changed5 * wherefore, thenbsp;Proportion which we have put down for a De-monftration, between the Spaces gone thro� bynbsp;the Particles in their going and coming, may benbsp;negledted, and the Propofition will be generally

491 nbsp;nbsp;nbsp;true, that IVaves^ �whether equal or any �way unequal, move �with the fame Velocity.

492 nbsp;nbsp;nbsp;This Rule will hold good, if the State ofnbsp;the Air is not changed but the Elafiicity remaining the fame, the Denfity of the Air often variesnbsp;and� the Elafticity may be changed, the Denfitynbsp;remaining the famcj laftJy, both are often liablenbsp;to be changed.

In the firft Cafe, fupppfing both the Waves to be equal, and alfo the Spaces thro�which the Particles go and come, the Celerities arifing fromnbsp;the Elafticity, which is always the fame, are in-^6? verfely as the Denfities 5 * but this Variation ofnbsp;Celerity anfwers in the Motions of equal Pendu-*49; lums, with the Variation of the Gravity *, innbsp;which Cafes the Squares of the Celerities of thenbsp;*165 Vibrations are as the Gravities themfelves *jnbsp;therefore,nbsp;nbsp;nbsp;nbsp;their Squares of the Celerities are

inverfely

inverfely as the Denfities. The Suppofition of the Equality of the Waves, and the Spaces gone thro�nbsp;by the Particles, does not hinder this Demonftra-tion from being univerfal *.nbsp;nbsp;nbsp;nbsp;�4gg

When the Denjity remains the fame, hut the E- 9� lafticity is changed, the Celerity arifmg from it va- 493nbsp;ries in the fame Ratio as the Elafticity j w here-fore, from the Demonftration of the foregoing Fro-pofition in this Cafe, the Squares of the Celeritiesnbsp;of the Heaves are as the Degrees of the Elaflicity.

Jfthe Elafticity and theDenfity differ, the Squares 494 of the Velocities of the Waives will be in a Rationbsp;compounded of thequot; dire�l Ratio of the Elafticitynbsp;and the inverfe Ratio of the Denfty *.nbsp;nbsp;nbsp;nbsp;*492-

Jf the Denfty and the Elafticity increafe or de- 49J creafe in the fame Ratio, the inverfe Ratio of thenbsp;Denfity will deftroy the direft Ratio of the Elafticity, and the Celerity of the Waves will not henbsp;changed.

This laft Cafe happens in the Compreflion of496 the Air*. Therefore,/row the changed Height of *430nbsp;the Pillar cf Mercury, which is fnftained in a Tubenbsp;void of Air by the Preffure of the Atmofphere *, �'412nbsp;which (hews, that the Weight, by which the Airnbsp;is comprefied near the Earth, is changed, we muftnbsp;not judge the Celerity of the Waves to he changed.

For th� fame Reafon, the Waves are moved with the fame Celerity in the Top of a Mountain as innbsp;a Valley, unlefs there be a Change of the Elafti-eity itfclf, by reafon of the Cold, which is almoftnbsp;always more intenfe on the Top of a Mountainnbsp;than'in a Valley; and this would occafion thenbsp;Waves to move flower *.nbsp;nbsp;nbsp;nbsp;�

It is plain alfo, that the Waves move fafter in 498 Summer than in Winter*.nbsp;nbsp;nbsp;nbsp;^4?4

The Celerity of the Waves is compared to 499 the Celerity which a Body acquires in failing,nbsp;by determining, from the known Xieight of thenbsp;R 4nbsp;nbsp;nbsp;nbsp;Mer-

Mathematical Elements Book 11,

Mercury, which weighs equally with the Pref-'*42afure of the Atmofphere % and alfo the Denfity of the Air*i the Height of the Jtmoj 'phere, Jup-^nbsp;pofing it every where equally denje with the Airnbsp;near the Earth j the Velocity of the Waves will benbsp;the fame as a Body could acquire in falling fromnbsp;half that Height. Which Velocity, from whatnbsp;has been faid, may be eafily difcovered by Expe-rin^ents made upon Pendulums*.

If the Weight, by which the Air is compreffed, be diminifh�d, the Air expands itfelf in the famenbsp;*42 Ratio*; and fuppofing the Atmofphere everynbsp;where of the fame Denfity, its Height does notnbsp;vary, which agrees with what has been faid, thatnbsp;the Velocity of the Waves is the fame in diferentnbsp;^496Compreirions of the Atmofphere*.nbsp;ypo The Motion of the Air, which we confider innbsp;this Computation, arifes from Elafticity alone;nbsp;and the Computation would be exa�t, if the Particles themfelves had not a fenfible Proportion tonbsp;the Interftices between them; but if we fuppofenbsp;here that they bear a fenfible Proportion to them,nbsp;the Motion of the Waves will be fwifter; for itnbsp;is propagated through folid Bodies in an Inftant,nbsp;which muft alfo be referred to heterogeneous Cor-puf:-les fwimming in the J\ir,

5� � iThe Motion of Waves in the Air produces Sound; of which, before we fpeak, we muft lay downnbsp;fomething in general relating to Senfation.

502 So ftricl is the Union of the Body and the Mind, that fome Motions in the Body do, as itnbsp;were, cohere with certain Ideas in the Mind, andnbsp;they cannot be feparated from each other. Fromnbsp;the Motion of the Body are new Ideas every Moment excited in the Mind, and fuch are the Ideasnbsp;of all fenfible Cbjeds; yet we find nothing com-fTion between the Motion in the Body and the

Idea

Book II. of Natural Thilofofhy. nbsp;nbsp;nbsp;149

Idea in the Mind. We cannot perceive what Connexion is here, nor that any Connexion isnbsp;poffible. There are an infinite Number of Thingsnbsp;hidden from us, of which we have not fo muchnbsp;as an idea.

The undulatory Motion of the Air agitates 3:03 the ^tympanum, or Drum of the Ear, by whichnbsp;Means a Motion is communicated to the Airnbsp;contained in that Organ, which, being carried tonbsp;the auditory Nerve, excites in the Mind the ideanbsp;of Sound.

The Strufture of the Ear, both internal and external, is wonderful �, but here we treat of thenbsp;Motion of the Air ^ that it is the Vehicle of Sound,nbsp;is proved by the following Experiment.

Experiment i. Plate XXXIII. Fig. 2.I Take the Leaden Plate O, which has two cylindric Pillars of the fame Metal C,C, fixed to it; join anbsp;little Bell A to the brafs Wire B D, and let it benbsp;tied with Strings to the Pillars C,C; lay the Platenbsp;O upon the brafs Plate of the Air-Pump, putting between a little CuOiion of Cotton, or Raw-Silk ; fet a Receiver on over all this Apparatus:nbsp;Cover the Receiver with a Plate that has the Collar of Leathers ferewed to it, through which thenbsp;brafs Wire DE can flip up and down to the *440nbsp;brafs Wire you muft fallen the Plate e f, fo that,nbsp;by turning the W ire round, the Bell A may benbsp;agitated. Pump out the Air from the Receiver,nbsp;and (baking the Bell in the Manner before deferi-bed, you will net hear the Sound. By turningnbsp;the Wire D �, the Bell will move backward andnbsp;forward feveral times; but we are only to obfervenbsp;that Motion in which the Plate e � doth not touchnbsp;the Wire hd. Letting in the Air, the Soundnbsp;will be heard as before.

2,50 Mathematical Elements Book II.

505 From this alone, that the Air is the Vehicle of Sound, and that Sound is moved thro� it without the Air�s being carried from one Place to another, it evidently follows, that in Sound therenbsp;is an undulatory Motion of the Air, and thatnbsp;Sound arifes from the tremulous Motion of Bodies. That this obtains in Cords, or Strings ofnbsp;mufical Inftruments, no Body doubts, fince, bynbsp;giving them a tremulous Agitation, they producenbsp;a Sound. In great Bells, and feveral other Bodies, this tremulous Motion is very fenfiblej butnbsp;it will become vifible by the following Experimentnbsp;made upon a founding Glafs Beil.

Experin?ent 2.] Let the Glafs Bell CC be fixed with Plaifter, or Cement, to a wooden Screw, bynbsp;Means of which it may be made very faft to thenbsp;tranfverfe Piece of Wood AB 5 this Wood muftnbsp;be fuftained by two wooden Pillars SS, to whichnbsp;it is firmly join�d with Screws and Nuts. Therenbsp;is a Pin with a Screw upon it, that goes throughnbsp;one of the Pillars, jull even with the Mouth ofnbsp;the Bell j fo, by fcrewing it forwards or backwards, you may fet it nearer to, or farther from,nbsp;the Edge of the Bell. If this Diftance be verynbsp;fmall, and the Bell be ftruck, it will, by its tremulous Motion, ftrike feveral times againft thenbsp;Pin with its Edge.

Hence we deduce, that a Body that is firiick, continues to give a Sound forne E�ime after thenbsp;Blow; the agitated Fibre will continue his Vibra-*2�5tion fome Time, on Account of the Elafticitynbsp;we often fee, as in Experiment i, that a Bodynbsp;gives a Sound, tho� the Air, agitated by jt, has nonbsp;Communication with the outward Air j whence itnbsp;^'^7 follows, that hy the Agitation of the Air, the Fibresnbsp;1nbsp;nbsp;nbsp;nbsp;of

Book II. of Natural Thilofo^hy, 2,5-1

of which Bodies confift^ are moved which Motion is transferred into the external Air.

This Tranflation of the Sound, by the tremulous Motion of the Fibres, is very remarkable j and how the Communication of this Motion extends itfelf, will appear by a fingle Experiment,

Experiment 3. Elate XXXIII. Fig. 2.] This Experiment differs from the firft only in this;nbsp;that if, inftead of tying the Bell to the leadennbsp;Machine C O C, it be foftened to the Ends ofnbsp;a brafs Plate bent in the Figure of a double Gnomon, which is made faft by a Screw to the Platenbsp;of the Air-Pump, and the Air be pumped out,nbsp;and the Bell fliak�d in the fame Manner as in thenbsp;firft Experiment; you will find but very little Difference between the Sound that is made, when the;

The tremulous Motion of the Parts of the Bell is communicated to the brafs Wire hd.^ fo as tonbsp;move the Strings by which the Bel! is fufpended,nbsp;and this Motion is transferred to the bent brafsnbsp;Plate; the Screw, with which this Plate is joinednbsp;to the brafs Plate of the Air-Pump, touches thenbsp;Plate, and communicates a tremulous Motion tonbsp;it, by which the Air is agitated, and the Soundnbsp;of the Bell is heard.

Fhe Celerity of the Sound is the fame as theCe- 508 lerity of the IVaves.^ which ftrike the Ear 5 and tonbsp;this muft be referred what has been faid of their *491nbsp;Celerity. * In refpedt to Numb. 499, it is to be 491.nbsp;obferved, that the Celerity of Sound can no way '^^5.nbsp;be determined by Calculation; for the Proper-tion between the Diameters of the Particles and 49f',nbsp;in the Interftices between them, is not known,

Mathematical Elements B�ok l�.

509 nbsp;nbsp;nbsp;If a Flalh of Fire goes off at Night with anbsp;Noife, and a Speilator ftands at any known Di-ftance from the Fire, who, with a Ihort Pendulum, meafures the Time between feeing the Light,nbsp;and hearing the Sound, he will have the Celeritynbsp;of the Sound; for the Motion of Light, at leaft,nbsp;thro� the Space as fuch an Experiment can benbsp;made in, is inftantaneous.

510 nbsp;nbsp;nbsp;By fuch an Experiment made in France^ it appear�d, that Sound run 1800nbsp;nbsp;nbsp;nbsp;Feet in a Se

511 nbsp;nbsp;nbsp;If at the fame Time in which the Velocity ofnbsp;the Sound is determined by this Method, therenbsp;be made the two Experiments above-named

�*4^9of Sound by the Elafticity of the Air*, and by comparing it with the Velocity immediately mention�d, you will have the Acceleration of thenbsp;Sound, from the Thicknefs of the Particles, and

�'491 Fhe Celerity of the Sound is equable ; yet in going through a greater Space, it is fometimes ae~

513 The Celerity of the Sound does not ninch differ, whether it goes with the Wind, or againft the Wind.nbsp;By the Wind a certain Quantity of Air is carriednbsp;from one Place to another j the Sound is accelerated as long as it moves through that Part ofnbsp;the Air, if the Direction of the Sound be thenbsp;fame with the Direction of the Wind ; but asnbsp;Sound moves very fvvife, in a very fliort Time it

Book II. of Natural Thilofofhy. nbsp;nbsp;nbsp;25-3

will run through the Air which is agitated by the Wind, and the Acceleration does not laltnbsp;long, which, indeed, is not very great j for thenbsp;molt violent Winds, which are ftrong enough tonbsp;root up Trees, and blow down Houfes, havenbsp;their Celerity to the Celerity of the Sound, butnbsp;about as one to 33 ; by the fame Argument it isnbsp;proved, that no fenfible Retardation is occafionednbsp;by the Wind, when the Sound moves againftnbsp;it.

The Space which the Particles run through, as they come and go, ftiay be increafed and diminilh-ed by the VV^ind j therefore^ the Sound may he heardnbsp;at a greater or [mailer Dljiance, according to thenbsp;Direi�ion of the Wind.

The Intenfity of the Sound depends upon the Strokes of the Air on the auditory Nerve, andnbsp;thefe Strokes are as the Quantities of Motion innbsp;the Air.

Whence it follows, that, cceteris paribus, the Intenfity of the Sound is as the Space run throughnbsp;by the Particles in their going and coming*. *488

All Things remaining as before, if the W'eight s?, by which the Air is comprelfed be changed, the*^^�^gnbsp;Celerity does not vary *, but the Denfity ischan-*4,9�nbsp;ged in the fame Ratio as the Weight*.nbsp;nbsp;nbsp;nbsp;2^9

Therefore, Cieteris paribus, the Intenfity of the Sound is as the Weight by which the Air is com-preffed *, that is, this Intenfity increafes and de-creafes, as the Pillar of Mercury, which is in JE~nbsp;qiiilibrio with the Weight of the Atmofpherc.

Experiment 4. Plate XXXIII. Fig. 3.] Shake the Bell A in comprelfed Air * exadly in the *454.nbsp;Manner as it was lhaked in Vacuo, in Experim. 1,nbsp;and the Sound will be increafed j which will againnbsp;be diminilhed, if opening the Bell you let thenbsp;Air return to its firft State.

Mathematical Elements Book II.

As the Intenfity of the Sound in comprefled Air, included in a Veflel, is greater, fo the Fibres, of which the Glafs V V is made, are agitated, and a greater Agitation is communicatednbsp;to the external Air.

517 nbsp;nbsp;nbsp;If a� Things remain as before.^ hut the Ela^icitynbsp;he increafed^ the Denfity is diminilhed in the fame

*430 Ratio as the Elafticity is increafed * j but the Celerity increafes as the Square Root of the Ela-

*493 fticity * j therefore, the Intenfity of the Sound is di-redly as the Square Koot of the Elafticity^ and in-

*64 verfely as the Elafticity itfelf but the Ratio, compounded of thefe, is the inverfe Ratio of the

518 nbsp;nbsp;nbsp;above-mentioned Square Root of the Elafticity.nbsp;The Intenfity of the Sound is diminilhed �, there-

519 nbsp;nbsp;nbsp;fore, as its Velocity is increafed, and in Summer,nbsp;cseteris paribus, the Intenfity of Sound is lefs thannbsp;in Winter 5 yet, in Summer Bodies do more eafilynbsp;tranfmit Sound, becaufe their Parts cohere lefsnbsp;ftriftly, as will be explained at a proper Time, andnbsp;they do more eafily acquire a tremulous Motion.

Experiment 5. Plate XXXIII. Fig. 5.] Hang up the Bell Aina Glafs, and opening the Cock,nbsp;that the Air in the Glafs may have Communication with the external Air, let the Glafs be Iha-ked, and the Diftance be determined when thenbsp;Sound can be heard j warm the Glafs, and repeatnbsp;the Experiment, and the Sound will be heard at anbsp;greater Diftance-

520 nbsp;nbsp;nbsp;Fbe Intenfity of Sound, confidered in general, isnbsp;in a compound Ratio ofthe Space run thro' by the Par-

* lie the Weight compreffmg the Air * j and laftly, of the inverfe Ratio of the Square Root of the Elafticity .*

521 nbsp;nbsp;nbsp;There is alfo a Difference in Sound, from thenbsp;Number of the Vibrations of the Fibres of the

Body

Book II. of Natural�Philofofhy.

Body which produce the Sound, that is, of the Number of the Waves produced in a certainnbsp;Time, according to the different Number of thenbsp;Percuffions in the Ear, the Mind receiving a different Senfation.

A Mufical Tone depends upon this Number of jr22 Vibrations, which is faid to be the more acntefnbsp;according as the Returns in the Air are more frequent j and more praw, the lefs the Number ofnbsp;the Waves is j aad the Degrees of the Sharpnefs ofnbsp;the different Sounds are to one another^ as thenbsp;Number of the Waves which are produced in thenbsp;Air at the fame Time.nbsp;nbsp;nbsp;nbsp;S^3

A Tom does not depend upon the Intenfity of the 5 24 Sounds and an agitated Cord gives the fame Sound.^nbsp;whether it vibrates through a greater or a lefsnbsp;Space.*

Concords arife from the Agreement between the S 2^ different Motions of the Air, which affedi the Auditory Nerves at the fame Time.

If two tremulous Bodies perform their Vibra-526 tions in the fame Time, there will be no Differencenbsp;between their Tones and this Agreement, whichnbsp;is the moft perfe�t of all, is called Unifon.

If the Vibrations are as i to 2, this Confonance3 527 or Agreement, is called OBave, or Diapafon.

Suppofing the Vibrations as 2 to 3, that is, if528 the fecond Vibration of one Body always agreesnbsp;with the third of another, fuch a Confonance isnbsp;called a Fifth, or Diapente.

Vibrations, which are as 3 to 4, give a Confo-529 nance, which is called a Fourth, or Diatejfaron.

And Sefquiditonus is a Confonance, from a Con-531 courfe of the fifth Vibration of one Body withnbsp;fhe fixth of another.

Mathematical Elements Book I�.

A Confonance from the Agitation of Cords, if they be of the fame Kind, is cafily determined, bynbsp;knowing their Dimenfions and Tenfion,

532 nbsp;nbsp;nbsp;C�eteris paribus, if the Lengths of two Cordsnbsp;are as the Number of Returns in a Confonance^nbsp;you will have the Confonance between the Sounds

533 nbsp;nbsp;nbsp;1�he fatne obtains, ?�, casteris paribus, the Dia-meters have the aforefaid Proportion. *

534 nbsp;nbsp;nbsp;Add alfo, if cECteris paribus, the Proportionofnbsp;the Vibration in a Confonance be given between the

535 nbsp;nbsp;nbsp;Jnd generally.^ fnppofing any Cord of the famenbsp;Kind.y if the Ratio he compounded of the direbi Ratio of the Lengths and of the Diameters, and thenbsp;inverfe Ratio of the Square Roots of the 'Tenfonsnbsp;he the Ratio between the Numbers of Vibrationsnbsp;performed in the fame 1�ime in any Confonancenbsp;whatever, you will have that Confonance by the

All thefe have been experimentally tried by Muficians ^ they have obferved a very remarkablenbsp;Phsenomenon relating to thefe Cords, whofe different Cafes very well deferved to be explained.

536 nbsp;nbsp;nbsp;Let any Mufical Strings be fo extended, as to perform their Vibration in equal itimes �, if you give Motion to the one, the other will alfo move. Everynbsp;Wave of the Air, arifing from the tremulous Motion of the firft String, ftrikes the fecond String,nbsp;and gives it a little Motion �, the String, from thenbsp;leaft Motion, goes backward and forward feveral

*257 Times and is moved by the Stroke of the firft Wave, whilft the fecond Wave comes forward,nbsp;whofe Motion confpires with the Motion of the

�*i5^ String *, and accelerates it. What is faid of the fecond Wave muft alfo be referred to thenbsp;other Waves that follow, and there will be an

Acce-

From the fame Demonftration it follows, that53? an agitated String �will communicate Motion tonbsp;another, which performs two or three Vibrations,nbsp;�whilft thefirji performs but one.

Now, if the agitated String performs feveral Vibrations, whilft the String tW is to be movednbsp;by the Air can perform but one, from the foregoing Demonftration it would follow, that it multnbsp;communicate a particular Motion to it. To dif-cover which, it is to be obferved, that the Duration of the Vibration and the Length of thenbsp;String are reciprocal j fo that every Thing elfenbsp;continuing as before, the determined Length cannbsp;no way be feparated from the unchanged Duration of the Vibration. If therefore any Stringnbsp;be ftruck with feveral Strokes, by which, Motionnbsp;is communicated to it, and the Strokes are morenbsp;frequent than what is agreeable to the Length ofnbsp;the String j that Part of it, whofe Length agreesnbsp;with the Time of the communicated Vibration,nbsp;will be agitated as much, and there will be, as itnbsp;Were, an undulatory Motion communicated tonbsp;the String j the Length of the Waves in the Stringnbsp;will depend upon the Duration of the communicated Vibration, that is, upon the Time betweennbsp;the Strokes.

^�ake two Strings, in fuch Proportion, that one may vibrate twice whilft the other vibrates butnbsp;once, and let the firft String be put in Motion;nbsp;the Duration of the Vibrations, which are communicated to the laft String with the Motion ofnbsp;the Air, agrees with a String of half its Length *, * ,59nbsp;and fuch is the Length of the Waves in it: Therefore, by the communicative Motion, the String isnbsp;divide^ into two equal Parts, and the Middle

Mathematical Elements Book II.

Point is at reft. This is confirmed by an Experiment, if you lay a Piece of Paper upon the String to which the Motion is communicated} for it willnbsp;remain at reft, if you lay it upon the Middle ofnbsp;the String, but any other Part of it will be af-fe�ted with a tremulous Motion.

539 nbsp;nbsp;nbsp;Jf the Stringwhicb is put into Motion.^ in order to caufe Motion in another, performs three Vibrations, whiljl the String, to be moved, performsnbsp;hut one, the lafi will be divided into three Partsnbsp;by the communicated Motion, and there will be twonbsp;Points of Keft ^ which may be confirmed by thenbsp;fame Experiment above-mentioned. All othernbsp;Cafes that have communicated Motion, which arenbsp;obferved by Muficians, are eafily deduced fromnbsp;what has been faid.

to their Reflexion in Air, the Elafticity in this 4,0 Caufe producing the fame Effed: as the Preffurenbsp;of the raifed Water in that.

541 nbsp;nbsp;nbsp;Fhm the Reflexion of the Sound there often a-rifes a Repetition of it, which is called an Eccho.nbsp;If different Parts of the fame Wave, expanding

ces, fo that being refiedled they concur together, the Motion of the Air will be ftronger there, andnbsp;the Sound will be heard. Fhe fame Sound is oftennbsp;repeated different times from the different Partsnbsp;of the fame Way refleded to different Diftanpes,nbsp;and Ibme of which alfo fucceffively concur at the

542 nbsp;nbsp;nbsp;fame Place. Such a Repetition fometimes happens from the Reflexion being repeated.

541 Fhe Soufid is often increafed by Reflexion in a Fiibe : Themoftperfeft Figure of all that can be given to fuch a Tube, is that of a Parabola, revolvingnbsp;about a Line a Quarter of an Inch diftant from

the

Book IL of Natural Thilofo}gt;hy.

the Axis. For if any one fpeaks in fucha Tube, fetting his Mouth in the Axis of the Machine,nbsp;and in the Focus of the Parabola, the Waves willnbsp;be fo reflefted, that every one of their Parts willnbsp;acquire a Motion parallel to the Axis of the Machine, whereby the Force of the Wave, and alfonbsp;of the Sound, will be very much increafed. Therenbsp;muft be a Mouth-Piece, to lit the Lips, fixed tonbsp;the End of the Tube.

FINIS.

Books printed/�r J. S e n e x, W. I n n v s* J. O SBORN and T. Lon GMAN.

An Introdudlion to Natural Philofophy; or Philofo-phical Lefturcs read in the Univerfity of Oxford, A. 1700. To which are added. The Demonftrations ofnbsp;Monjkur Huu/gemh Theorems, eonceming the Centrifugal Force and Circular Motion. By Keill, M. D.nbsp;Savilian Profeffor of Aftronomy, F. R. S. 8�, 1720.

The Religious Philofopher ^ or the right Ufe of con-tomplating the Works of the Creator, ijl. In the wonderful Structure of animal Bodies, and in particular Man. zdly. In the no lefs wonderful and wife Formationnbsp;of the Elements, their various E�Ee�ls upon animal andnbsp;vegetable Bodies. And, ^dly. In the moft amazingnbsp;Strufture of the Heavens, with all its Furniture. Defign-ed for the Conviflion of Atheids and Infidels. By thatnbsp;learned Mathematician, Dr. Nietmentyt. To which isnbsp;prefixed a Letter to the Tranflator, by the Reverendnbsp;y. fDefaguliers, LL. D. F. R. S. The Fourth E-dition, adorned with Cuts. 5 Vols. 8�, 1750.

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The Hiflory of Ttmttrhec, known by the Name of ^amerhyne the Great, Emperor of the Moguls and ^ar~nbsp;farS} being an hiftorical Journal of his Contmefts innbsp;and Europe. Written in Perfian by Cherejed dire Mi,nbsp;Native ofnbsp;nbsp;nbsp;nbsp;his Contemporary. Tranflated mto French

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Optice, five de Reflexionibus, Refra�lionibuslnflexio-nibus amp; Coloribus Lucis. Libri tres. Authore Ifaaco Nenvlon, Equite Aurato. Latini reddidit Samuel Clarke,nbsp;S. T. P. Editio fecunda audior. 8�, 1719.

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_ A new Syftem of Arithmetick, Theorical and Practical : Wherein the Science of Numbers is demonftrated in a regular Courfc, from its Firfi Principles, through allnbsp;the Parts and Branches thereof 5 either known to thenbsp;Ancients, or owing to the Improvements of the Moderns. The Pradlice and Application to the Affairs ofnbsp;Life and Commerce being alfo fully Explained j fo asnbsp;to make the Whole a Compleat Syttem of Theory, fornbsp;the Purpofes of Men of Science, and of Praflice fornbsp;Men of Bufinefs. By Alexander Malcolm, A. M. Teacher of the Mathematicks at Aberdeen.

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Three Effays in Artificial Philofophy j or, Univerfal Chemeftry; Viz. I. An Effay for the farther Applicationnbsp;and Advancement of Chemiftry in England. II. Annbsp;Eflay for the Improvement of Diftillation, in the Handsnbsp;of the Malt-Stiller, Re�iifier, Compounder, and Apothecary. III. An Effay for Concentrating Wines, and othernbsp;Fermented Liquors j or taking the fuperfiuous Water outnbsp;pf them to advantage. By �Peter Sha^, M. D. 8�, 1731.

Philofophical Principles of Univerfal Chemiftjyr: or, the Foundation of a fcientifical Manner of inquiring inrnbsp;to, and preparing the natural and artificial Bodies fornbsp;the Ufes of Life, noth in the fmallerway of Experiment,nbsp;and the larger way of Bufinefs. Defigned as a generalnbsp;Introduftion to the Knowledge and PraiSice of artificialnbsp;philofophy, or genuine Chemiftry in all its Branches,nbsp;Drawn ftomtha Collegium ^/enenfe of Dr. George Ernefinbsp;Stahl. By Peter Shaw, M, D.

A new Method of Chemeftry 5 including the Theory and Praffice of that Art: Laid down on Mechanicalnbsp;Principles, and accommodated to the Ufes ofLife. Thenbsp;whole making a clear and rational Syftem of Chemicalnbsp;Philofophy, To which is prefixed, a Critical Hiftory ofnbsp;Chemiftry and Chemifts, from the Origin of the Art tonbsp;the prefentTime. Written by the very learned HI Scer-haave, Profeflbr of Chemiftry, Botany, and Medicine innbsp;the Univerfity of Leyden, and Member of the Royalnbsp;Academy of Sciences at Paris. Tranllated from thenbsp;printed Edition, collated with the beft Manufeript Copies.nbsp;By P. Shaw, M. D. and E. Chambers, Gent. Withnbsp;Additional Notes and Sculptures 5 and a copious Indexnbsp;to the Whole.

The Wifdom of God manifefted in the Worksofthe Creation 3 In two Parts. To which are added, Anfwersnbsp;to fome Objeflions. By ^ohn Ray, late Fellow of thenbsp;Royal Society. The Eighth Edition, 8�,J7ai.

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ELEMENTS

Natural Philofophy,

Confirm�dbyExpERiMENTs;

VOL.

Sir Ifaac Nemton, K�'quot; PRESIDENT

ROTJL SOCIETY, amp;c.

iv nbsp;nbsp;nbsp;The Dedication.

The Dedication,

vj The Dedication,

Tour mofl Obedient^

yerjf Humble Sermnt

TRANSLATOR

READER.

THE

PREFACE.

The P R E F A C E.

.f

The PREFACE.

The P R E F A C E.

The P R E F A C E.

The P R E F A C E.

The P R � F A C E.

The P R E F A C E.

The P R E F A C E.

The P R E F A C E.

THE

CONTENTS.

BOOK I.

Part I. Of BODT in General.

Part II.

Of the Mo tion of Solid Bodies.

XX The C o N T E N T S.

BOOK

The C o N T E N T S.

Of the AIR3 as an Elajlick 'Plaid.

Mathematical

ELEMENTS

I^atural Philofophy,

EXPERIMENTS.

_B O O K I._

CHAP. r.

Mathematical Elements Book I.

Every

Book 1. of Natural Thilo/bfhy.

CHAP.

Mathematical Elements Book I.

Book I, of natural �Philofofhy. nbsp;nbsp;nbsp;^

CHAP. IIL

Of Extenfor, Solidity^ and Vacuity.

^ nbsp;nbsp;nbsp;B 3nbsp;nbsp;nbsp;nbsp;The?

(?gt; nbsp;nbsp;nbsp;Mathematical Elements Book I,

As

CHAP. IV.

8 nbsp;nbsp;nbsp;Mathematical Elements Book I.

llrated

Book I. of Natural Thilofofhy. nbsp;nbsp;nbsp;lt;

Mathematical Elements Book I,

But

Book I. of Natural Thilofo^}^. nbsp;nbsp;nbsp;ii

iz nbsp;nbsp;nbsp;Mathematical Elements Book I,

Drops

Book I. of Natural Th�lofophy. nbsp;nbsp;nbsp;13

14 nbsp;nbsp;nbsp;Mathematical Elements Book I.

Book I. of Natural Thilofophy.

16 nbsp;nbsp;nbsp;Mathematical Elements BookL

Book I. of Natural'Phtlofoj^hy, nbsp;nbsp;nbsp;i;

t8 nbsp;nbsp;nbsp;Mathematical Elements Book �*

part ii.

Mathematical Elements Book I.

CHAP. VII.