■ nbsp;nbsp;nbsp;-nbsp;nbsp;nbsp;nbsp;■ , 'quot; _nbsp;nbsp;nbsp;nbsp;■ .-' . X ''nbsp;nbsp;nbsp;nbsp;.-'

,'(■'■.■ '•' . ' ■ ''^ nbsp;nbsp;nbsp;■ .jy' ^

ELEMENTS

Natural Philofophy,

Confirm�d by Experiments;

INTRODUCTION

Sir I s A A c N E w T 0 K�s Philofophy.

Do6lor of Laws and Philofophy, ProfelTorof Mathe-maticks and Aftronomy zx. Leyden, and Fellow of the Royal Society of London.

L 0 ND 0 N:

Printed for J. Sen ex in Fleet-Jlreet, W. Innys in St.nbsp;nbsp;nbsp;nbsp;Church-yard; and J. Osborn and

THOM AS

Lord PARKER,

Baron of ^Macclesfield^

Lord High Chancellor of Great Britain^ See,

My Lord,

F whilft Your Lordfiiip�s Hours are taken up with an.nbsp;Employment of the great-eft Fatigue as well as thenbsp;higheft Honour, there can be anynbsp;Time allowed for Recreation, yournbsp;a %nbsp;nbsp;nbsp;nbsp;Lord-

iv DEDICAT 10 N.

Lordfliip makes that your Diverfion, which few can attain to without painful Application and laborious Study,nbsp;To become a skilful Mathematiciannbsp;and a found Philofopher, and at thenbsp;fame Time fhine in other Parts ofnbsp;Learning, requires a great and exten-five Genius; But to lead a Life of Bu-finefs, and be eminent in the Law,nbsp;where Reputation is only got by con-ftant Practice, as well as Brightnefs ofnbsp;Parts ; and yet, in thofe few Minutesnbsp;of Leifure that are allowed to breathenbsp;in, and are, as it were, ftolen from Sleep,nbsp;to play with all the Intricacies ofnbsp;Lines and Numbers, to view and un-derftand the Syftem of the World,nbsp;the Proportion, Symmetry, and Harmony of its feveral Parts 5 to be acquainted with all the Experiments ofnbsp;Confequence that have been made,nbsp;and be able to contrive new onesnbsp;as ufeful as inftrudive; was only re-

DEDICJTIO N. V

ferved to Men of uncommon Capacities, and to none in fo eminent a Degree as to your Lordfliip.

The Honour I have had of being admitted into your Lordihip�s Con-verfation, has given me an Opportunity to know which of the Mathematical Sciences are chiefly lik�d by you:nbsp;And as AJironomy and Optics feem tonbsp;have the Preference, I thought thisnbsp;Tranflation of Dr. �sGravesande�snbsp;Second Volume would not be unacceptable.

Here you have the Principles of the common Optics reduced into anbsp;fmall Compafs, and confirmed by newnbsp;Experiments of the Author�s own Invention : A fine Application of the Action of Ele�lric Bodies to difcover thenbsp;Nature of Fire: And Sir Isaac New-toh�s Do�lrine of Lj^t and Colours

vj D EDICJTION.

proved by the moft conliderable of his Experiments, which Dr. j Gravefandenbsp;performs with an Apparatus verynbsp;ingenioufly contrived, and nicely ex-prefled by curious Figures. The laftnbsp;Part of this Volume not only leads anbsp;Beginner gradually on from the moftnbsp;fimple to underftand the moft difficultnbsp;Thitnomena of AJironomy; but givesnbsp;fuch a phylical Account of the Celefti-al Motions as muft be fully fatisfac-tory to the beft Geometricians. Therenbsp;your Lordftiip will fee with Pleafure,nbsp;that there are Profeflbrs abroad whonbsp;underftand the Principia; andnbsp;have fo juft a Value for that excellentnbsp;Book, as to tcke Pains to propagate the wonderful Truths which itnbsp;demonftrates, fo as to make themnbsp;plain to fuch Phllofcphers as are notnbsp;yet able to go through all the difficultnbsp;Propofitions from whence they arenbsp;deduced. But 1 will detain your Lord*-

fliip

DEDICATION, vij

Ihip no longer, either from the important Bufinefs of your Station, or the pleafing Truths of the Author,nbsp;than while I beg Leave to fubfcribenbsp;myfel^

PREFACE.

Spoke of the Method of Reafoning te he iifed in Natural Philojophy, innbsp;the firjt Chapter of the firfi Volume,nbsp;and in the Preface endeavoured tonbsp;vindicate the Goodnefs of the Method I have followed, ^tere arefe-veral remarkable Specimens of thisnbsp;Method in the prefent Volume, which evidently Jloewnbsp;Sir Isaac Newton�s great Superiority of Geniusnbsp;above all other Philofophers.

Before him, Nattiralijls were in the Dark in mm-herlefs �Things relating to Light, and efpecially to Co^ lours. For Jnftance, whoever fufpeSted before, that.nbsp;the Opacity of Bodies depended upon their Interfiices,nbsp;fo that a Body becomes trmfparent, when thefe Interfiices are filled with a Medium of the fame Denfitynbsp;as the Particles of the Body itfelf.

X nbsp;nbsp;nbsp;The PREFACE.

His Account of the 'Planetary Syftem, and partku~ larly of the Motion of the Moon, is not lefs worthynbsp;of eternal Praifes, being likely to carry Afironomy tonbsp;a greater Pitch of PerfeSlion than the niceft Obferva-,.nbsp;tions alone could poffibly do. For if a Manis acquaint-'nbsp;ed with the Laws that govern the Syjiem cf the World,nbsp;he will be able to make a better Ufe of his Obfervations,nbsp;and to compute the Motions of the heavenly Bodiesnbsp;more exa^ly than if he had nothing but Obfervationsnbsp;to direbi him.

It was my Defgn in thefe two Books, to give my Reader a general Notion of the chief Fhitsgs difcoverednbsp;by Sir Isaac Newton in Natural Philofophy, andnbsp;thereby to encourage him to the Study of the more ab-ftrufe, and, at the fame Fime, more fublime Parts ofnbsp;Mathematics., after be has learned the firft Principlesnbsp;cf Geometry, to fit him for the reading of thefe plainnbsp;Elements.. He will, as it tvere, go to the Fountainnbsp;Head, when he reads the Writmgs of our great Philo-fopber, which will reveal fuch Fhings to him, as werenbsp;unknown to the profoiindeft Philofophers before him jnbsp;and zvhich, though pnblified, are fiill a Secret to albnbsp;but thorough Mathematicians.

I have only a few Words more to fay to the Englijh Reader, concerning the two. EngLJb Franflations ofnbsp;this IVorkf.

As tedious and difiaftefnl as an Authors Complaints generally prove, they cannot, however, be difallowed,nbsp;when they are grounded upon fuch an Injury done tonbsp;the Author, as it is his Reader's Interefi to he w-forined of.

Soon after the Publication cf the firfi 'Volume of thefe �lemcntSj it was tranfiated into Englifh, and printed

The PREFACE,

ed in London, with the Name of a Celebrated Profef' for of Mathematics, eminent for his Writings, in-fcribed on the Title Page, as if he bad looked overnbsp;and correBed this Tranflation. But whoever examines any one Page in the Book, will immediatelynbsp;difcover the Wrong done to that Learned Gentleman,nbsp;and the Abiife made of bis Name j J�nce be will every where perceive manifeft Signs of the Tranflator'snbsp;Ignorance of the very Principles of Phyfics and Mathematics, not to mention his Negligence with regardnbsp;to the correBing of the Additions at the End of thenbsp;Book, to the Numbers in the Margin, and the Di-finBion of the Propofitions.

Dr. Desaguliers, whofe Knowledge of Pbu lofophy, and Skill in making Experiments, are fonbsp;well known, began to tran/late the fame Work aboutnbsp;the fame Time as the other, or rather before. Butnbsp;this obliging him to make more than ordinary Hafte,nbsp;he could not himfelf wholly avoid the ufiial Confequen-^es of too much Precipitation, amp;c.

THE

Second Volume.

BOOK III.

Chap. II. ithat Fire adheres to Bodies^and is contained in them 3 �where voe JJfall alfo [peak of EleSiricity.nbsp;nbsp;nbsp;nbsp;p. 2.

Chap. III. Concerning the Motion of Fire, where wejhall fpeak of Heat and Light.nbsp;nbsp;nbsp;nbsp;p. 13.

BOOK III.

The CONTENTS.

Chap. VI. Concerning the Kefra�iion of Lights and its Laws.nbsp;nbsp;nbsp;nbsp;Page.nbsp;nbsp;nbsp;nbsp;26.

Chap. VII, Of the Kefradion of Light, when Mediums are feparatcd by a plane Surface. p. 3 9.

Chap VIII. Concer7iing the KefraBion of Light, zvhen a fpherical Surface feparates the Mediums,p. 44.

Chap. IX. Concerning the Motion of Light through a more denfe Medium. Where we fhall take Noticenbsp;of the Properties of Lenfes.nbsp;nbsp;nbsp;nbsp;p. 53.

Chap. XVIII. Concerning the different Refrangi~ hility of the Sun�s Rays.nbsp;nbsp;nbsp;nbsp;p. 108;

BOOK IV.

Chap. III. Of the phenomena or Appearances of the Sun from the Motion of the Earth in its Orhitap. 164.

Chap. IV. Of the Ehanomena of the inferior Planets, arifing from the Earth's and their own Motions innbsp;their Orhits.nbsp;nbsp;nbsp;nbsp;p.nbsp;nbsp;nbsp;nbsp;167.

Chap. V. Concerning the Phenomena of the fuperi-or Planets, arifng from their Motions and the Motions of the Earth in their refpeSlive Orhits. p. 171.

Chap. VI.- Concerning the Phxnomena of the Satellites, from their Motions in their Orbits. Where we fhall fpeak of the Eclipfesof the Sun and Moon I 'l^.

Chap. VIII. Of the Ph(snomena which relate to the Surface of the Earth,and its particular P arts. p. i Sjr.

BOOK IV.

Part II. The Phyfieal Caufes of the Celeftiai

Motions.

The C o N T E N T S.

Chap. XII. Of the Celejlial Matter j where a Vacuum is proved. nbsp;nbsp;nbsp;Page 216.

Chap. XV. I�he Phyfical Explanation of the whole Planetary Syftem.nbsp;nbsp;nbsp;nbsp;p.nbsp;nbsp;nbsp;nbsp;233.

Chap. XVI. The Phyfical Explication of the Moon's Motion.nbsp;nbsp;nbsp;nbsp;p.nbsp;nbsp;nbsp;nbsp;241.

Chap. XVIII. The Phyfical Explanation of the Motion of the Jxis of the Earth. nbsp;nbsp;nbsp;p. 270.

Chap. XIX. nbsp;nbsp;nbsp;Concerningnbsp;nbsp;nbsp;nbsp;the Tides.nbsp;nbsp;nbsp;nbsp;p.nbsp;nbsp;nbsp;nbsp;272.

Chap. XX. nbsp;nbsp;nbsp;Of thenbsp;nbsp;nbsp;nbsp;Moon�snbsp;nbsp;nbsp;nbsp;Denfity and Figure.

Mathe-

Natural Philofophy,

EXPERIMENTS.

H O we know feveral Properties of Five, yet we are ignorant of anbsp;great many things relating to it.

I ihall not invent Hypothefcs, but reafon from Experiments,nbsp;and leave untouched what isnbsp;not fully known.

Fire eajilypenetrates thro' all Bodies.^ however denfe and hard they are. For we have never yet known �nbsp;VoL. II.nbsp;nbsp;nbsp;nbsp;Bnbsp;nbsp;nbsp;nbsp;any

%, nbsp;nbsp;nbsp;Mathematical Elements Book III,

any Body, that by the Application of Fire has not been heated in all its Points j that is, innbsp;every Part.

5'46 Fire unites itfelt to Bodies j for when they are brought to the Fire, they grow hot, as we faidnbsp;before ; and in that Cafe they expand or fwell :nbsp;Which Expaniion is alfo obferved in fuch Bodies,nbsp;f48nbsp;nbsp;nbsp;nbsp;Parts do not cohere, in which Cafe they

alfo acquire a great Degree of Elafiicity^ as is obferved in Air and Vapours.nbsp;f49 That Fire is attrafled by Bodies at a certainDi-fancefrorn them^ will be Ihewn in the followingnbsp;*^6ii Part of this Book. *

If any Bodies are violently moved againft one another, they will grow hot by fuch a Friction,nbsp;and that to a great Degree j which fhews thatnbsp;all Bodies contain Fire in them : for, by rubbing,nbsp;Fire may be put in Motion, and feparated fromnbsp;Body, but can by no means be generated that way.

Having laid down thefe general Heads, we mull examine feveral Things more particularly.

CHAP. II.

Fhat Fire adherers to Bodies, and is contain�d in them; where we Jhall alfo fpeak of EleBricity.

AS we have already faid, one may prove that Fire is contain�d in all Bodies, be-caule there are no Bodies but what may be heated by Attrition; and that it coheres firmly withnbsp;the Parts of Bodies, appears in Smoke and Vapours : for Smoke and Vapours are made up ofnbsp;Parts feparated froin Bodies, and agitated (fome-2.nbsp;nbsp;nbsp;nbsp;times

Book III. of Natural Thilofophy. . 5

There are, befides, fevcral remarkable PhfE-nomena arifing from Fire, contained in Bodies, fome of which welhall here mention} amongftnbsp;which, there are fuch as relate very much tonbsp;Ele�tricity, for which Reafon, we mull alfo treatnbsp;of the Phtenomena of Electricity.

Electricity is that Property of Bodies^ by which ff i {when they are heated by Attrition) they attraSl-ynbsp;and repel lighter Bodies at a fenfible Diflance.

Experiment i.] Take two Pieces of Rock yfz Cryftal and rub them together, and immediatelynbsp;they will appear luminous all over, tho� they donbsp;not acquire any fenfible Heat by that Attrition.nbsp;Light (as well as Heat) is a Proof that there isnbsp;Fire in a Body. The greateil Light is in thofenbsp;Points, where the Bodies touch one another.

Experiment z.] Take a Glafs Tube ly or 18 ffj Inches long, and one Inch in Diameter, and rubnbsp;it with a Linnen or Woollen Cloth, and it willnbsp;emit Light in the Dark.

Experiment 3. Plate I. Fig. i.(] This Tube, yy4. heated by rubbing, has a very fenfible Electricity}nbsp;for if light Bodies, fuch as Pieces of leaf Gold,nbsp;and Soot, be laid upon a Plane, and the Tubenbsp;be brought near them, they will be put in Motion, attracted, repelled, and driven feveralWaysnbsp;by the Tube, The Tube aCts at dilferent Di-llances, according to the different State of the Air}nbsp;fometimes at the Diftance of one Foot} whennbsp;the Air is full of Vapours the EffeCl is dimi-niihed.

4 nbsp;nbsp;nbsp;Mathematical Elements Book III.

j-j-y There is one Thing remarkable, and very hard to explain in this Experiment, concerning thenbsp;Direction of the Attrition 5 when you rub thenbsp;Tube, one End of it his held in one Hand, whilftnbsp;it is rubb�d with the other j which, if it be donenbsp;from the Hand that holds towards the other Endnbsp;of the Tube, the Effc6l will not befenfiblej butnbsp;if you rtib from the free End of the Tube towards tire End held in the Hand, the contrarynbsp;will happen. And this happens indifferently,nbsp;whether you hold the open or the fhut End ofnbsp;the Tnbe in your Hand.

yy� In the following Experiments {Platei. Fig. z.) Glafs Globes are fwifely whirl�d about} to'perform which, there muff be a cylindric Neck atnbsp;each End of every Globe} but only one of thefenbsp;Necks is to be open, and both are to haveBrafsnbsp;Ferrels, fuch as are reprefented atG} a CockEnbsp;mtift be ferew�d on to the Ferrel at the open Neck,nbsp;and the oppofite Brafs Ferrel muft have a littlenbsp;Wheel r of -about an Inch and a half Diameter,nbsp;join�d to it, with a fmall Brafs Axis ftandingout:nbsp;There is juft fuch another Piece of Brafs ferew�dnbsp;on to the Cock, fo that it may be taken on andnbsp;off at Flcafarc. Thefe Axes go about a Quarter of an Inch into the Pillars S S, that fupportnbsp;the Globe, and are Centers for it to whirl uponnbsp;v/hen it turns about its Axis.

The Pillars SS ffand upon a flrong horizontal Board of about an Inch and a half thick, framednbsp;into three other Boards, as may be feen in thenbsp;Figure; In that which Hands foreright, there isnbsp;an Hole � that you may come at the lower Partnbsp;of the Pillar, to make it fait with a Nut or Screw.nbsp;The other Pillar S is likewife faffen�d by a Screwnbsp;applied under the horizontal Board, and movednbsp;forwards and backwards in a Slit of 4 or y Inchesnbsp;before it be made faft, in order to take thenbsp;Inbsp;nbsp;nbsp;nbsp;Glafs

Book Iir of Natural �Philofofhy. nbsp;nbsp;nbsp;^

Glafs Globes in and out, and to take in fuch as are larger or fmaller than others, according tonbsp;the different Experiments to be made.

There is a great Wheel R which is turn�d round by means of the Handle M, and thereby gives anbsp;very fwift whirling Motion to the Globe G.

In the Side of the upper Board, there is a Slit, along which the Fully t may be moved, by meansnbsp;of the Screw c, in order to keep ftretch�d thenbsp;Rope that goes round the Wheels R and r.

Experiment 4 amp; f. Plate II. Fig. i.] Apply a Glafs Globe of about 8 or p Inches Diameter,nbsp;to the Machine above-mention�d, and let it benbsp;briskly whirl�d in a dark Place, tlie Hand all thenbsp;while being held againft it, to give it Attrition.

If the Globe be exhaufted of its Air, it will appear all luminous within, but moltly fo wherenbsp;the Hand touches the Glafs.

But if the Globe has Air in it, and being f y8 whirl�d in the fame Manner, the Hand be applied to it, no Light appears, either in the innernbsp;or outer Surface of the Glafs j but Bodies at anbsp;fmall Diftance from the Glafs (as for Example,nbsp;at a Quarter of an Inch, or nearer) become luminous j and fo only thofe Parts of the Hand heldnbsp;againll the Glafs, which terminate, or rathernbsp;environ the Parts that immediately touch thenbsp;Globe, are luminous.

Experiment 6. Plate\. Fig.zP\ Take the Globe pyp made ufe of in the foregoing Experiments, andnbsp;put it in between the Pillars, to whirl it as before; Then take a Brafs Wire rr ^ circularlynbsp;bent in the upper Part, and fix it fo that its curvenbsp;Part may be about 4 Inches off of the Globe,nbsp;with fmall Threads hanging from it, which being extended cowards the Center of the Globenbsp;B 3nbsp;nbsp;nbsp;nbsp;come

5 Mathematical Elements Book III.

Whirl the Globe, and apply the Hand, and immediately the Threads will be moved irregularly by the Agitation of the Air j but when thenbsp;Glafs is heated by the Attrition, all the Threadsnbsp;are direfted towards the Center of the Globe, asnbsp;may be feen in the Figure : And if the Hand benbsp;applied a little on one Side, or nearer the Polenbsp;of the Globe, the Threads will be direfted towards that Point of the Axis which is under thenbsp;Hand.

f6i Experiment j. Platei. Fig.Take another Globe like the former j only differing in this,nbsp;that the open�d Neck mufl have a larger Openingnbsp;than that of the Globe G, fo that you may putnbsp;into it a round flat Piece of Wood o, that has anbsp;Brafs Wire for its Axis. In order to fix thisnbsp;Piece of Wood in the Middle of the Globe, itsnbsp;Axis mufl: be firmly fcrew�d to the Middle ofnbsp;the Cover that is join�d to the open Neck of thenbsp;Globe at h: and the Cock E is alfo join�d to thenbsp;Middle of the Cover on the outfide.

To the wooden Circle fmall Threads are fii-ften�d, which, being extended, almofl; touch the inner Surface of the Globe.

Turn the great Wheel fo as to whirl the Globe, and rub it till it becomes warm, as was donenbsp;in the former Experiments gt; if you ceafe tonbsp;whirl the Globe, and the Hand be taken off,nbsp;the Threads will immediately ftretch themfelvesnbsp;out like Radii from the Center towards the Surface of the Globe, yet they hardly remain onenbsp;Moment at refl; j for tbo the Globe be exaftlynbsp;fhut, thefe Threads will be put in Motion, as

Book in. of Natural ^hilojbphy. nbsp;nbsp;nbsp;f

appears by blowing againft the Globe, tho you ftand two Foot off, or farther, from the Globe.

If you bring your Finger towards the Globe^ tho you do not touch it, the Threads next to itnbsp;will be attracted by the Finger, and directed towards it j nay, and foraetimes they fly from it.

If you apply the whole Hand to the Globe, the Threads will be mov�d violently and irregularly.

And if all the Air be drawn our, as in the f6i foregoing Experiment, the whole Effect ceafesnbsp;and the Threads (both before and after thenbsp;Fridtion) onlv hang down by their Gravity.

f63

* nbsp;nbsp;nbsp;554.'

559.

561.

* nbsp;nbsp;nbsp;554-

If we attend to all the foregoing Experiments, the following Conclufions feem to be naturallynbsp;deduced from them, which we do not give outnbsp;as certain, but very probable j for we muit always diftinguifh Certainty from Probability.

7quot;hat Gla/s contains in it^ and has^ about its Surface^ a certain Atmofphere^ which is excited by Fri-tiion, * and put into a vibratory Afotion-, foritat-trafts and repels light Bodies j *the fmalleft Parts of theGlafs are agitated by the Attrition, and bynbsp;reafon of their Elafticity, their Motion is vibratory, which is communicated to the Atmofpherenbsp;above-mention�d j and therefore that Atmofpherenbsp;exerts its A�tion the ffarthcr, the greater Agitation the Parts of the Glafs receive when a greaternbsp;Attrition is given to the Glals.

Floe Fire^ contained in the Glafs^ is �xpeWdhy f64 the Action of this Atmofphere \ at leafl: it is mov�dnbsp;with it. For when light Bodies are put in Motion at a Diftance from the Glafs the Bodies alfo * 559,-beeome lucid at a Diftance. *nbsp;nbsp;nbsp;nbsp;55^-

It is plain, that this Atmofphere asidFire is more y6y eafily mov'd in a Place void of Air : For if thenbsp;Globe be exhaufted of its Air, there can be perceiv�d no Light, nor any Effeft ofElcdtricity onnbsp;the outfide of the Globe *, but the inflde of the �

Mathematical Elements Book III.

Globe appears very luminous, and this Fire is perceiv�d to be in greater Qiiantity in this Experiment, than in that immediately after men-

which feems to overthrow what we faid of the more caly Morion of the Atmofphere of Glafsnbsp;in Facuo: But yet it is not probable that it fhou�dnbsp;move no whither in this Cafe. On the contrary,nbsp;it feems to go the fame Way as the Fire, andnbsp;to move that way where there is the leaft Refinance ; and that the ceafing of the Adtion of E-ledlricity is to be attributed to the Want of Air,nbsp;by means of which, the 'threads are moved by the

But laying afide Conjedlures, tho they have a great many Experiments for their Foundation,nbsp;let us return to other Things relating to Fire; asnbsp;feveral Experiments are to be made inVaciio^ wenbsp;muft deferibe the following Machine contriv�d tonbsp;give Attrition to Bodies in Facuo.nbsp;f�y Plate \\.�ig.zd\ Ler M be the Air-Pump de-

' on which the Glafs Receivers are fet ; on each Side of the Plate there is a wooden Pillar, AD,nbsp;AD, which Pillars ftand on the Board that carries the Air-Pump Plate, having their Bafeuponnbsp;it, and a Part below the Bafe, and going thro�nbsp;the Board with an Hole in it, to receive twonbsp;crofs Pieces of Wood or Wedges, that makenbsp;them fall: during theExperimenr, but fo that thenbsp;ftid Pillars may be taken aWay afterwards.

The Glaft Receiver, in which the Experiments are to be made, is about lt;? Inches high, and 6

Book III. of Natural�Philofophy. nbsp;nbsp;nbsp;9

in Diameter, with a Cover * that has annex�d to * 44� it a Box or Collar full of oil�d Leathers thro�nbsp;which a Brafs Axis paffes j and left in thenbsp;whirling round of the Axis the lower End ftiou�dnbsp;move, Part of the Box is made fquare, fo as ex-a6tly to fit a fquare Hole in the Horizontalnbsp;Board E E that preffes on the Cover, and isnbsp;made faft by bringing down the Nuts or Screwsnbsp;B B at each End, -without bearing upon thenbsp;Shoulders of the Pillars.

Towards the upper Part of the Pillars, there is a narrower horizontal Board F F prefs�d downnbsp;alfo and fattened with Nuts, which has fix�d tonbsp;it, on the under Side, a Piece of Brafs with annbsp;Hole in it for the upper End of the above-mentioned Axis to turn in, whilft the biggeftPartofnbsp;this Axis, being in the middle of theGlafs, hasnbsp;an outfide Screw upon it, with the two Wing-Nuts r/, moveable upon it, in order to fattennbsp;feveral Bodies upon the Axis.

The Brafs Spring �� is joined to the Piece r, which fcrews it down to the Air-Pump Plate atnbsp;the Hole where the Air is drawn out 5 but therenbsp;muft always be made a fmall Hole or Paffage innbsp;the Piece c, for the Air to be drawn out at.

Having exhaufted the Receiver, the Axis above-mentioned (byreafon of the oiled Leathersnbsp;in the Collar P) may be turned round withoutnbsp;admitting any Air j but to give it a quick Motion, there is a little Brafs Wheel r of about 2.nbsp;Inches Diameter, with Points in its Groove, thatnbsp;the Rope, tliat turns it round, may not flip.

The great Wheel R, of about 3 Foot Diameter with its Frame, is brought up clofe to the Air-Pump, and made faft to its Foot by a Screwx

The Rope, that goes about the leffer Wheel r, is brought down over Ptillies obliquely placed

Mathematical �l�ments Book lit

in the upper {�art of the Pillars C, C, and being over them, comes down to the great Wheelnbsp;which it goes round; Co that by turning the vertical Wheel R, the Brafs Axis, above deferibed,nbsp;is carried round very fwiftly, whereby a Motionnbsp;is communicated to Bodies l^acua for leveralnbsp;Experiments.

Experiment 8.] Take a Glals Globe of three Inches Diameter, or of two Inches and a halfnbsp;Diameter, with an Hole on each Side, where itnbsp;may alfo have cylindric Necks. The Axis above-mention�d mull; go thro� thofe Holes, in order tonbsp;give the Globe a whirling Motion; Pieces ofnbsp;Cork mull: be put on each Side of the Globe tonbsp;cover its Holes, and made fall: with the Nutsnbsp;c/, in the Manner that they appear in theFigure,nbsp;to hold fall; together the Plates or little Wheels,nbsp;thro� which the Axis goes.

This Globe, thus fix�d, will be fwiftly mov�d inVacuo with its Axis, by turning the WheelR:nbsp;To caufe Attrition, a Piece of Woollen Clothnbsp;mull; be tied on to each Side of the Braft Springnbsp;/', �, which by its Elafticity prefles the Globenbsp;hard on each Side.

Making the Experiment in the dark, the Globe will appear luminous; and if the Motion be continued till the Globe grows hot by the Attrition,nbsp;the Light will indeed be increafed, but \villnbsp;appear fix�d in the Places where the Attrition isnbsp;made.

It follows from that Experiment, that the Fire, contained in Glafs, does not nvant Air to make itnbsp;�-jifible, for it grows hot, and ihines when bothnbsp;the internal and external Air are taken out.

yyo Experiment p.] Take a round Piece of Wood of a, or a 4- Inches Diameter, and about i an

Book III. of Natural Thilofofhy. Il

Inch thick, and let it have feveral Hollows cut in its Edge, that it may be encompafs�d roundnbsp;with .Beads of Plaifter, which may be madefaftnbsp;to it with Threads going thro� them. I.et thisnbsp;Piece of Wood, thus prepar�d, be made fail uponnbsp;the Axis above-mention�d, in the fame Mannernbsp;as the little Globe had been fatten�d then givenbsp;Attrition in Vacuo^ as in that Experiment, andnbsp;Light will be thereby produced in the dark.

Experiment lo.] That Quickfilver contains yyi Five is plain, from Experiments made upon it,nbsp;in Vacuo. For if Mercury well clean�d be ttiak�dnbsp;about in an exhaufted Gkfs, it will appear luminous.

If you put Mercury into a Glatt Globe, the fji Globe may be whirl�d round, as m the formernbsp;Experiments, which will be delightful to fee, ifnbsp;the Glafs be mov�d flowly. If Mercury has nonbsp;Tin mix�d with it, it may be clean�d with boiling hot Vinegar.

The Globe, that has the Mercury in it, may yy 3 be exhautted by fcvewing on to it a Pipe aboutnbsp;z Foot long, the lower End of which mutt benbsp;fcrew�d on to the Hole iu the Middle of thenbsp;Plate of the Air-Pump.

Then, if you cover both that Hole and the Hole thro� which the Air is drawn out with onenbsp;of the Receivers above-mention�d, the Air willnbsp;be eattly drawn out of the Globe.

Take a Plate O, and hav- ^74 on to it the Pipe E E, whofenbsp;other End is fcrcw�d to the Air-Pump, putnbsp;the Glafs Receiver R upon it, and pump outnbsp;the Air from it. The Pipe, thro� which thenbsp;Air is drawn out, muftftand up beyond the Platenbsp;in the Receiver about 4 or y Inches, be bent,nbsp;and have a fmall Hole in it 5 which mutt be taken

1 i Mathematical Elements Book IIL

The middle of this Plate, has a Brafs PipeBj whofe lower End, that is without an Hole,nbsp;comes down almoft to the Bottom of the Veflelnbsp;V, that contains very clean Mercury : There isnbsp;an Hole in the Side of the Pipe, exaftly (hut bynbsp;the Pin A. The upper End of this Pipe has anbsp;very fmall Hole in it, and comes up thro� thenbsp;Plate into the Receiver.

The Height of the Receiver R is about i6 Inches, and its Diameter 4 Inches gt; pump thenbsp;Air out of it, and then open the Hole in thenbsp;Side of the Pipe B, and the Mercury by thenbsp;Prellure of the external Air will go up to thenbsp;Tube, and will fpout very ftrongly into the Receiver, ftriking againft the Top of it.

The Experiment muft be made in a dark Place, and the Mercury will appear luminous.

fjf Experiment 11.] Chymifts make a certain Preparation of Urine, which they czW Pbofphorus of Urine �, which muft be kept in Water. If it benbsp;roll�d up into fmall cylindrical Pieces like a Pencil, and you write with it upon a Paper, whennbsp;you hold the Paper in a dark Place, you will feenbsp;Letters of Fire. The Phofphorus itfelf, whennbsp;you take it out of the Water, will grow hotnbsp;and fmoke 5 all which ftiews that the Phofphorus contains a great deal of Fire.

In this Experiment, we may ohferve that Water aPts upon the Fire contained in Phofphorus j fornbsp;it keeps it in^ fo that it cannot get out fo long asnbsp;the Phofphorus is cover�d with Water j but asnbsp;foon as the Water is taken away, the Heat andnbsp;Smoke immediately drew that the Fire iftiiesnbsp;out of the Phofphorus. The Air does., in famenbsp;^ meafure, keep in the Fire contain'd in hot Water j

Book III. of Natural Thilofophy.

Experiment iz-] Take two Veflels that are e-qual and alike, and put an equal Quantity of boiling Water into each of them. Let one ofnbsp;them be fet upon the Air-Pump Plate under anbsp;Receiver, and pump out the Air j when you arenbsp;pumping, the Water boilsviolently by the Aftionnbsp;of the Fire that goes out of it ^ and then foonnbsp;becomes only lukewarm, whilfc the Water in thenbsp;other Veflel, which remain�d in the open Air,nbsp;has fcarce loft any of its Heat.

One may obferve Comething like this in fhi- r7� ning Wood for fome Wood that is rotted innbsp;the Ground, fhines when it is taken out j thenbsp;Earth that encompafs�d the Wood kept in thenbsp;Fire, which goes out when that is remov�d, andnbsp;the Wood continues to ftiine for fome Days. Innbsp;an exhaufted Receiver, the Light is foon de-ftroy�d, and is not reftor�d by the Re-admifllonnbsp;of the Air.

But it cannot be eafily determin�d how Fire is confined in a Body by the ambient Bodies, nornbsp;is it eafy to find cut what Aftion produces thisnbsp;EfFeft} it is hardly probable that Preflure is herenbsp;much concerned, fince Fire does eafily penetratenbsp;all Bodies by its Subtilty.

CHAP. III.

Mathematical Elements Book kf�.

Light are to be attributed to the different Mo-tions of fi'ire. Heat^ in a hot Body^ is the Agitation of the Parts of the Body and the Fire contained in it^ by which Agitation, a Motion is producednbsp;in our Bodies, which excites the Idea of Heatnbsp;ySo in our Mind. Heat^ in refpeSl of us^ is nothingnbsp;hut that Idea^ and in the hot Body is nothing hutnbsp;Motion.-^ Here we muff call to mind what hasnbsp;been faid of the Senfations in general (N. yoz.)nbsp;which alfo may be referr�d to Light.

When Fire enters our Eyes in Right Lines^ hy the Motion that it communicates to theYihxes in thenbsp;Bottom of the Eye^ it excites the Idea of Light ofnbsp;which Motion of the Fibres we fhall fpeakmorenbsp;* 716 particularly hereafter. * A rehlilinear Motion isnbsp;ySz the Motion of Light, as it appears from its being eafily flopped by an Obftacle. On the contrary, fuch a Motion is not requir'd in Heat: andnbsp;that an irregular Motion is more for it^ may bpnbsp;proved, becaufe the Rays, that come directlynbsp;from the Sun to the Top of a Mountain, produce no fenfible Heat j whilft in the Valley,nbsp;where the Rays are agitated with an irregularnbsp;Motion by feveral Reflexions, there is often produced a very intenfe Heat.

A Body is faid to he Lucid, 'when it emits Ldght, that is, when it gives Fire a Motion innbsp;right Lines.

5'83 That there is not always Light, 'where there is Fire, is beyond all doubt gt; for we daily fee hotnbsp;Bodies that do not fliine.

f84 But 'whether there be any Body without Heat, cannot be determin'd. Hear, in all Bodies, is anbsp;Motion that may be infinitely diminilh�dj andnbsp;there may be fuch a Motion, tho it be not fen

Book III. of Natural TbHofo^hy. !gt;

any thing of Heat for certainly there are a great many lucid Bodies, in which we can perceive no fenfible Heat.* Concerning which it * 571nbsp;may be obferv�d, that no Heat j.s fenfible to us,nbsp;unlefs the Body, that a�s upon our Organs ofnbsp;Senfe, has a greater Degree of Heat than that ofnbsp;our Organs. Which fltews us, that the Judgment of our Senfes, concerning Heat, is whollynbsp;Uncertain.

When the fmalleft Parts of which aqy Body is made up are agitated, either by Attrition, ornbsp;the Aftion of Fire applied externally to it, ornbsp;any other Way ; the Fire is feparated from thenbsp;fmall Particles and agitated in the Body �, thennbsp;alfo the Fire and the Particles of the Body a�tnbsp;upon one another by their Attra�fion, as isnbsp;prov�d by the Experiments which we Ihall hereafter mention.* By this Adtion fome Parts are *6unbsp;feparated from the Body, and carried off fromnbsp;it by the Motion of the Fire. And this is thenbsp;Caufe why hard Bodies are often fet on Fire by a ygynbsp;'violent Attrition.

Hence we deduce, that the burning of Bodies ygg is a Separation of their Parts by the mutual Abiionnbsp;of the Fire., and thofe Parts on each other-, fame ofnbsp;thofe Parts, carried off by the Motion of the Fire, j-87nbsp;tnake Flame and Smoke.

We fee befides, that a Body, that is biirn�d by the Application of Fire, is not only diifolv�dnbsp;by the Adlion of the external Fire, but alfo bynbsp;the Action of the Fire contained in the Bodynbsp;itfelfi and that the Heat is encreafcd by the Application of new Fire, and the Augmentation ofnbsp;the Agitation of the Fire which the Body itfeltnbsp;contains j and therefore that the Heat is not in f88nbsp;proportion to the ^lantity of Fire.

As to the Motion of Light, it is plain, as we faicl *, that it is perforrn�d in right Lines j but * 581

Mathematical Elements Book III.

whether it be fucceffive or inftantaneous, is di-fputed j that is, whether at the fame Moment that a Body begins to fhine, the Light is fenfi-ble at any Diftance, or whether the Light goesnbsp;on fucceflively to Places more and morediftant.nbsp;ygp It feems clearly to follow from feveral Aftro-nomical Oblervations, that that Motion is fuc-ceffive, and Philofophers did not long doubt ofnbsp;it j but by fome later Oblervations, the Conclu-fions, drawn from the former, are weaken�d, andnbsp;we are obliged to confefs that the Motion ofnbsp;Light has fomething unknown to us.

To fay that a Motion from one Place to another is not fucceffive, implies a Conttadidlion, j-po it can fcarce he doubted, that Light moves fromnbsp;one Place to another.

For we obferve that Fire is carried off in Vapours and Smoke ; in which Cafe, Fire carries with it the Bodies to which it adheres, and yetnbsp;is often mov�d very fwift : If the Subtilty ofnbsp;Fire be confider�d, it will eafily be found that itnbsp;is immenfly retarded by the Bodies to which itnbsp;adheres ; and, that, as foon as it is freed fromnbsp;them, it is mov�d with a very great Velocity.

There are feveral other Things very well worth obfcrving, in refpeft to Light and Heat,nbsp;but a great many of them are hard to explain.nbsp;In natural Philofophy, when we arc ignorantnbsp;of the Caufes, we muft only mention the Ff-fedts.

We (ce feveral heated Bodies that become lucid, if their Heat be increafed-, fuch are Metals; Theynbsp;emit Fire by the Agitation of their Parts, butnbsp;if the Motion of the Parts be increafed. Part ofnbsp;the Fire is mov�d in right Lines, and the Bodynbsp;ffiines.

Thus

Book III. of Natural Thzlofo_phy. 17

Thus if Smoke be made hotter by applying ypi Flame to it, it is chang�d into Flame j that

Experiment PlateW. Fig. 3.] Having blown out the Candle C, which fends out the Smokenbsp;F } let another lighted Candle A be brought to

it, nbsp;nbsp;nbsp;and the Smoke of the firft will be chang�dnbsp;into Flame, and that fucccffively quite to thenbsp;Candle C, which is thus lighted by the Candle A, tho it be 7 or 8 Inches from it.

We have faid that Air a�ts upon Fire ; one * 577 of its, EfFefts upon it, that in many Cafes is notnbsp;to be overlook�d, fhews itfelfinRefpecl of Light.

For the Prefence of the Air is often necejfary for the fP5 Production of Light or Prefervation of Fire., whichnbsp;may be oblerv�d in the burning of all thole Bodies that go out when the Air is taken awaynbsp;from them.

Experiment z.~\ Put a lighted Candle under the Receiver of the Air-Pump j exhauft the Receiver, and the Candle will immediately go out.

Plate IL Fig.z. Experiment 7, Take a round Plate of Steel of about 3 Inches Diameter, withnbsp;an Hole in the Middle to receive the Axis whichnbsp;is to be mov�d in Vacuo *, and let it be prefs�d * 567nbsp;and fix�d between two round Pieces of Woodnbsp;by the Help of the Screws d, d ; let Pieces ofnbsp;Flint be fatten�d to the Spring/, / fo that bynbsp;the Motion of the Axis the Steel Plate may rubnbsp;nimbly againft the Flints. As long as there isnbsp;any Air in the Receiver the Attrition will produce Sparks of Fire gt; when the Air is drawnnbsp;out, tho you continue the Attrition, you cannbsp;perceive no Light; but upon re-admitcing thenbsp;Air, it becomes again fenfible.

i8 Mathematical Elements Book III,

^94 We obferve alfo on the contrary, that �very often the Abfence of Air is neceffary for Lights as might be obferv�d in the Experiments alreadynbsp;*557 mention�d*.nbsp;nbsp;nbsp;nbsp;laftly, 'Y'asiS. taking ana ay the

57 � Air^ the fame Light^ �which may be feen in the open Air^ is fometimes increafed.

* 575 Experiment 4.] With t\\QPhofphorus * abovc-mention�d write or draw Figures upon Paper, and they will fhine in the dark, as we faid before 1 but the Paper will become much brighternbsp;in Vacuo.

CHAP. IV.

Of the Dilatation occajion�d hy Beat.

AL L Bodies are dilated by the Action of Fire*; but that Dilatation changes as thenbsp;Heat changes j fo that it feems to depend rathernbsp;upon the Motion than the ^antity of the Fire j fornbsp;Bodies are expanded as well by rubbing as bynbsp;applying Fire to them externally.

Experiment l. Plate III. Fig. i Take a fquare Iron Bar about 3 Foot long and i of an Inchnbsp;thick, and having laid it upon a Board betweennbsp;two firm Obftacles or Standards O, O, between one of the Obftacles and one End of thenbsp;B.ar, thruftdn an IronRuler about 4 or pinchesnbsp;long, (made Wedge-Fafhion, being ~ of an Inchnbsp;broader at R than at r, and divided into equalnbsp;Parcs) as far as it will go j and obferve thenbsp;Divifton that is againft the End of the Bar,nbsp;which muft be a little oblique, that it may thenbsp;better apply it felf to the Ruler.

Then let the Bar be heated, either by Attrition or Fire j then let it again be put between 3nbsp;nbsp;nbsp;nbsp;�hc

Book III. of Natural Thtlofo^hj.

As the Ruler will not go in fo far as the Divi-fion that was obferv�d before, it is plain that the Bar is lengthen�d by Heat.

�xperiment z. Plate II. Fig. 4.] That Gquids.^ f97 as well as Solids.^ are dilated by Heat nlay be prov�dnbsp;ih the following Manner. Take a Glafs BallGnbsp;that has a fmall Tube E D for its Neck, andnbsp;fill it with fome Liquor up to what Height younbsp;pleale in the Tube, and the Liquor will afcendnbsp;in the Tubcj when you heat the Ball j a fmallnbsp;Degree of Heat will produce this Effeft, evennbsp;tho you ufc Mercury, the denfeft of all Fluids.

The Experiment will be itiade in the fame Manner, but better, if inftead of a Ball you ufe an hollow Cylinder C with the long Tube B Anbsp;join�d to itj for the whole Idquor will be foonernbsp;heated in 'a Cylinder than in a Ball.

If the Tube ED or B A be divided into equal ypS Parts i or the Tube with its Ball or Cylindernbsp;be fitted to a Frame on which equal Divifionsnbsp;are mark�d, the Heat may in fome Sort be mea-fured by this Machine : For the Liquor rifes ornbsp;fubfides in the Tube, as the Heat of the neighbouring Bodies is increas�d or diminifb�d. Suchnbsp;Inftruments are call�d 'Thermometers.^ and are ofnbsp;Common Ufe. They do indeed fhew a Changenbsp;in the Heat, but it is uncertain whether theynbsp;fliew the Degree of Heat j that is, it is notnbsp;known what Relation there is between thenbsp;Change of Expanfion and the Change of Hear,nbsp;fo as to enable us to compare together the Degrees of Heat, by comparing the Degrees ofnbsp;Dilatation.

If the BallG or Cylinder C be fuddenly heated, 500' the Liquor in the Tube will immediately defcend,nbsp;but immediately after rife. By rcafon of the fud-G 2.nbsp;nbsp;nbsp;nbsp;den

to Mathematical Elements Book III.

den Heat the Glafs it felf grows hot fooner than the Liquor contain�d in it j and therefore whennbsp;the Glafs is dilated by the Heat, the Cavitynbsp;grows bigger, fo that the Liquor defcendsj butnbsp;the Heat being prefently communicated to it, itnbsp;rifes again by its Expanfion.

From the Expanfion of Bodies it is evident 6oi that the Particles of which Bodies confift^ from thenbsp;ASlionof theFire^ acquire a repellent Force^ by whichnbsp;they endeavour to fly from each other, and whichnbsp;a61:s contrary to that Force by which the Particles come to each other *. As long as this lafl;nbsp;Force is ftronger than the other, the Particlesnbsp;cohere more or lefs, according to the differentnbsp;Degree of Heat. When the repellent Force isnbsp;almofl: equal to the attraftive, the Particles,nbsp;which were before firmly join�d, fcareely cohere,nbsp;yield to any Impreffion, and are eafily mov�dnbsp;one among another j whence we fee that a So-602. lid Body is cleaned into a Liquid by Heat, whichnbsp;may be obferv�d in all Bodies that are liquifiednbsp;by Heat, and return to their firft State upon thenbsp;Diminution of the Heat. It is a Queftion,

603 nbsp;nbsp;nbsp;FFhether all Fluidity is not owing to Heat ? whichnbsp;cannot he determin'd, becaufe we know of no Body that is entirely without Fire in it it is certain that Heat is not only the Caufe of Fluiditynbsp;in Metals, Wax, and fuch like Bodies, but thatnbsp;feveral Bodies, which are commonly reckon�dnbsp;Fluid, are only fo by realbn of their Heat; thus

604 nbsp;nbsp;nbsp;Water is melted Ice j for when Part of the Heatnbsp;of the Water is gone, it grows fix�d.

605quot; Heat may be fo increas�d that in fome Bodies the attrafting Force is overcome by the repelling Force j in which Cafe the Particles fly from each other; that is, acquire an elafticnbsp;*431 Force, as the Particles of Air have*, whichnbsp;Elallicity is increafed even in the Air by Heat.

■ nbsp;nbsp;nbsp;■■'5.»v''	gt;t

		t
	’■	»
	^ quot;	1
» t- (V' ’'quot; •gt;*«»'■		1
-«t^gt;»-v nbsp;nbsp;nbsp;^ lt;,•,	■-.iw#

Book III. of Natural ^hilofophy. ai

Experiment 3. Plate III. Fig. z.'] Take an hollow Brafs Ball E, of about 4 Inches Diameter, with an HandleMj This Ball has aPipeT withnbsp;an Hole jof hardly the twentieth Part of an Inch.

Let this Ball be heated, and the Air in it will expand itfelf ^ and come out thro� the Pipe ; * 597nbsp;then, immerging the Ball in cold Water, thenbsp;Air will be again condens�d by the Cold, andnbsp;the Water will go into the Ball, by thePreflurenbsp;of the Atmofphere upon its Surface.

If this Ball, thus partly fill�d with Water, be laid upon the Fire, the Moment that the Waternbsp;is chang�d into Vapours, thofe Vapours will gonbsp;out thro� T j but if the Heat be increas�d, fo asnbsp;to make the Water boil violently, the Vapours,nbsp;comprefs�d inthe upper Part of the Ball, will, bynbsp;their Elafiicity, endeavour to recede from eachnbsp;other every Way, and rufh violently out of thenbsp;Pipe. Such an Inftrument is call�d an606

Experiment^.'] The following Experiment607 Ihews a more fenfible Effe�l of the Elaificity ofnbsp;V apours.

Platelll. 3.3 Take the Ball E, which is alfo of 4 Inches Diameter, made of Brafs ornbsp;Copper, but thicker than the former, and letnbsp;it be placed upon a little light Cart, fuchnbsp;as is reprefented in the Figure. In its uppernbsp;Part, it has a fquare Pipe T. In the middle ofnbsp;this Pipe, there is a Separation, and the hindnbsp;Part of the Pipe comnaunicatcs with the Ball.nbsp;There is an Hole of about 1 of an Inch, in thenbsp;middle of the Separation of the Pipe, which Pipenbsp;is open forwards. This Hole is Ihut up with anbsp;long Plate, that goes thro� two Holes in the

%% Mathematical Element:^ Book III,

Sides of the Pipe, and applies irielf to the above-mention�d Separation. As the Plate is made Wedge-wife, if you ftrike upon it with a Hammer, It will exactly (hut che Hole.

Take this Bail off of the Carr, and open the Hole, and, having heated the Ball, immerge itnbsp;in Water, to let it be in Part fill�d, as in the lafbnbsp;Experiment. Then, having fhut the Hole, fetnbsp;the Ball again upon the Fire, till the Water boilsnbsp;violently ; and then if you make it fall; to thenbsp;Carr, and open the Hole, the Vapour will flynbsp;out violently one Way, and the Cart be carriednbsp;with the Ball the contrary Way,

608 The Vapour, being very ftrongly comprefs�d, endeavours to recede equally every Way, andnbsp;therefore oppofitePreffiires deftroy one another 5nbsp;but when the Hole is open�d, the Vapour, whichnbsp;goes out, does not prefs; therefore the Prefllirenbsp;one Way being taken off, that which afts in anbsp;contrary Diredion prevails, and the Cart isnbsp;mov�d along.

dop A Sky-Rocket rifes up into the Air, becaufe the Gun-powder, being fer on Fire, acquires annbsp;Elatticky, and its Parts endeavour to recede everynbsp;W''ay: As the Pipe, or Cafe of it, is open at onenbsp;End, it is lefs prefs�d that Way, and confequentlynbsp;at the other End, the Preflure prevails, and carries up tlje Rocket.

BOOK

Part II.

Concerning the Inflexion, Refra�lion, and Reflexion of Light.

Concerning the Inflexion of the Rays of Light.

1A VIN G premis�d, in the foregoing Parr, what regards Fire in general let us again farther examine the Properties of Light andnbsp;the Phaenomenaarilmg from them.

The Things obferv�d concerning Light are very wonderful, yet mofl; of them are explain�dnbsp;by a few Laws of Nature.

Light moves in right Lines % can be inter- * jgj cepted by an Obftacle which wholly flops fonbsp;mucii Light a^ comes upon it, but flops no more.

Any Light confidefd according to the DireEtion of its: Motion^ if it be all carried in the fame Di-reclion.^ is call'd^ a Ray of Light.

Mathematical Elements Book III.

Fire, as has been faid, is attracted by Bo-* 549 dies ; the notable Effedts of which Attraftion may be obferv�d in the burning of Bodies theynbsp;�II are alfo fenfible in Light j for �when Light pajfesnbsp;near Bodies^ it is turn'd out of the fir ait TVay jnbsp;which may be dillinctly perceiv�d by the following Experiment.

Experiment i. Plate HI. Fig. 4.] Take a little Board T about 6 Inches long, and as high, withnbsp;an Hollow c c in its Surfltce, in which twonbsp;Plates of Steel Aide, each of which has an Edgenbsp;like a Knife gt; the two Edges may be broughtnbsp;together in the. middle of the Board, and benbsp;join�d together exaftly. At the Place wherenbsp;they meet, there is a fquare Hole in the Board,nbsp;of almoft an Inch, that a Beam of Light, let intonbsp;a dark Room thro� an Hole of a Quarter of annbsp;Inch in Diameter, may pafs thro� the faid fquarenbsp;Hole, fo as to come to the Plates.

If the Diftance between the faid Edges be of about the lothPart of an Inch, and the Lightnbsp;be made to pafs between them, the Board beingnbsp;fet at the Dillance of 3 Foot from the Window,nbsp;if the Light falls upon the Paper A, at f Footnbsp;beyond the Board, there will appear, on each Sidenbsp;of the tranfmitted Beam, a Light like that of thenbsp;Tail of a Comet 5 which proves that the Lightnbsp;is infieSled as it pafles by the Edges of the Plates,nbsp;as is plain from the Figure.

If you bring the Plates nearer (as for Example) within the hundredth Part of an Inch, in-Aead of the Light above-mentioned, you will, on each Side of the Light of the Beam upon thenbsp;Paper, fee three colour�d Fringes, parallel to the

Book III. of Natural Vhilofophy.

Edges of the Plates i which will ftill appear more diftin�t, if the Hole in the Window be madenbsp;lefs. I {hall hereafter explain whence thefe Colours arife. Now it is enough to deduce fromnbsp;this Experiment, that Light is attra�ledby'So�iQSnbsp;that infledl its Raysj for if there was not a Motion towards the Body, the whole Beam wouldnbsp;have continu�d in its diredt Motion.

But the Action of Bodies^ by which they a�l up- dii on the Light to attract it^ exerts itfelf at a fenftblenbsp;Diftance j as is plain from the Experiment.

Experiment z. PlateWl. Fig. Everything elfe being as in the former Experiment, if thenbsp;Plates be brought nearer together within tvvnbsp;Part of an Inch, no Light will appear upon thenbsp;Paper between the Fringes above-mention�d �, fonbsp;that in this Cafe, all the Light, which paffes between the Edges, is infledled on either Side, fonbsp;as to produce the Fringes above-mention�d.nbsp;Which plainly {hews, that Steel a�ls upon Lightnbsp;at leafl at the Diftance of -sFs- Part of an Inch.

It is alfo proved, that Fhat Action is increas'd lt;5 j ^ as the Dijiance is diminijh'd.

Experiment 7,. PlatelW. Fig. 4.] Things being ftill in the fame Difpofition, lelfen the Diftancenbsp;between the Plates j and the Fringes will vanifhnbsp;fucceffively, till the Plates being join�d together,nbsp;no Light palfes between them. The Fringes,nbsp;which vanifh firfl, arc thofe that are producednbsp;by the Rays which are leaft infleftedj and thofenbsp;vanifh laft, which are produced by the Rays thatnbsp;are moft inflefted j that is, when the Edges comenbsp;towards one another, the Shadow between thenbsp;Fringes, made by each Edge, is continually increas�d, till, at laft, the whole Light on eachnbsp;Side vanifhes. Whence it plainly follows, that

2-6 Mathematical Elements Book III.

the Rays are more inflcdled, the nearer they pais by the Edges j that is, the Attraftion is increafed,nbsp;as the Diiiancc is diminiih�d.

�14 But this Attraction has iomething particular, for the Attraction of one Edge is increafed^ as thenbsp;other is btougjst near it. Which is plainly feen innbsp;this Experiment ; for as the Edges are broughtnbsp;towards one another, the Infledicn of the Raysnbsp;continually becomes greater.

CHAP. VI.

When a Ray of Light goes out of one Medium into another, it is often turn�d out of the right Line,

Definition IT.

- , In order to produce RefraCtion.y the Mediums ntufl be of different Dcnfities, and the Ray tntijl makenbsp;an obli'jue Angle vjith the Surface thaU feparatesnbsp;the Medhms.

g j 0 Refraction arifes from thiSy that the Rays are ^ more attracted by a denfcy than by a rare Mediumnbsp;from which Attraftion, that we have prov�d innbsp;the foregoing Chapter, all that relates to Re-fraCtion may be deduced.

619 Let E. F be the Separation of the Mediums [PlateW. Fig, i.j and letXbeinthedCTifer, andnbsp;Z in the rarer Medium. All the Particles have

6i I in Light. Let the Diftance, at * which the Pai �� tides exert their Addon, be th.at which is comprehended between the Lines E F and G H.

Book III. of Natural Thilofophy. 17

Therefore, the Light, which comes between thofe Lines, will be attrafted by the Medium X.

At the Diftanceof the LineGH, only the extreme Parts of the Medium X aft upon the Light j at a lefs Diftar.ce, both they and other Parts adtnbsp;fo as to increafe the attradling Force, when thenbsp;Diftance is diminifli�d, as has been before ob-ferved*. In the denfer Medium X, let the Linenbsp;I L be fuppofed at the fame Diftance from E Fnbsp;as G H is in the Medium Z. Let the Light enter into the Medium X, and it will, on all Sides,nbsp;be attrafted by the Particles of the Medium,nbsp;whofe Didances from the Light are lefs thannbsp;the Diftaiace between E F and G H j for thenbsp;Light is fuppos�d to be attradedat that Diftancenbsp;by the Particles of the Medium X.

As long as the Light is between the Lines EF and IL, the attrading Force is theftrongeftnbsp;towards 1 L, becaufe there are more Particlesnbsp;that draw that Wayj but as the Number of Particles, that ad the contrary Way, increafes, thatnbsp;is, as the Diftance from E F increafes, the Forcenbsp;towards I L is diminifh�d, till, in the very Linenbsp;I L, the Light be equally attracted every Way jnbsp;^vhich alfo obtains every where in the Mediumnbsp;X, beyond I L.

Suppofe a Ray of Light to come in the Line and fall obliquely on theSurfitce, which divides the Mediums, or rather on the Surfacenbsp;G H, where the Adion begins, by which thenbsp;Light is driven towards the MediumX: Whennbsp;the Ray comes to it is turn�d out of the rightnbsp;Line, by the Force by which it is attraded bynbsp;the Medium X % that is, by which it is driven to-vvards it in a Diredion perpendicular to its Surface. And, indeed, the Ray of Light is bentnbsp;out of the right Idne in every Point, as long as itnbsp;is between the Lines G H and I L, between

Mathematical Elements Book III.

which the faid Attraction aCts j and therefore between thole Lines, it defcribes the Curve ab^nbsp;*2o2 in the fame Manner that ProjeCtiles move*. Beyond the Line I L, the AClion which bends thenbsp;Ray ceafes ; therefore it goes on then in a rightnbsp;Line, according to the Direction of the Curvenbsp;in the Point b.

The Diftance, between the Line G H and I L, is very fmall ; therefore, when we confi-der Refraction, we take no notice of the bendednbsp;Part of the Ray, which we look upon as if itnbsp;was made up of two llrait Lines AC, C B,nbsp;meeting at C, namely, on the Surface whichnbsp;divides the Mediums.

6zJ. tliisCafe, Wloen Light goes out of a rare into a denfe Medium, the Angle of Ref ration is lefs thannbsp;the jingle of Incidence, by reafon of the Inflexionnbsp;of the Ray j for thefe Angles wou�d be equal, ifthenbsp;Ray AC continu�d to move llrait on in the Linenbsp;CD. The Ray CB does alfo come nearer to the

Book III. of Natural Thilofophy,

Perpendicular CMj and thereioxethe Refradlioti is faid to be made towards the Perpendicular.

On the contrary, /� the Ray goes out of a denfer gty into a rarer Medium.^ it will recede from the Perpendicular, becaufe the denfer Medium attradsnbsp;the Ray in the fame Manner, whether it goes outnbsp;of a rarer into a denfer, or out of a denfer into anbsp;rarer Medium. Therefore if B C be the Ray ofnbsp;Incidence, C A will be the refradled Ray; that is.

Let the Ray come from either Side, audit willmove SiS inthe fame Lines. Therefore, if there be two Rays, 6i.jnbsp;and one comes out of a denfer Medium into a rarer,nbsp;and the other out of a rarer into a denfer, and thenbsp;An^e of Refraction of the one be equal to the Anglenbsp;of Incidence of the other, the two remaining Anglesnbsp;of Incidence and RefraClion will be equal to eachnbsp;other.

Whence it follows, xh^xThe Direction of the 6i^ Ray is not chanfd, if it moves thro' a Medium ter-mmed by two Surfaces parallel to each other; for asnbsp;much as it is turn�d towards any Side at its Entrance, fo much cxaftly is it turn�d the othernbsp;Way as it goes out of the ftid Medium.

ff a Ray falls perpendicularly on the Surface that �zp feparates two Mediums, it will not be turn'd out ofnbsp;the right Line by the Attrafliion of the denfernbsp;Medium j becaufe in that Cafe it adls in the Di-reftion of the Ray.

PlatelV. Fig. z.] To confirm what we have , faid by Experiments, take a Trough or open �nbsp;Box P, about a Foot long, 4 Inches wide, andnbsp;of the fame Depth. Its two Sides abed, abed,nbsp;are Planes of Glafs parallel to each other, and thisnbsp;Box mull: be fill�d with Water two Inches andnbsp;half high.

The Experiments rnufl. be made in a dark Place j the Light muft be let, in thro� a Slit in thenbsp;moveable round Plate O, which is fix�d to the

Mathematical Elements Book Ilf.

Window-fhut, and is eafily turn�d round, that the Slit may be inclin�d at Pleafure : The Slit isnbsp;4 Inches long and �? Inch wide.

The Light, that is let in, ftrikes upon th� Looking-GlalsS, and the Slit isfo inclin�d, andnbsp;Speculum difpofedj that the Beam, that comesnbsp;thro� the Slit, may be horizontally reflefted bynbsp;the Looking-Glafs, the Beam being in a verticalnbsp;Pofition, that it may go thro� a vertical Slit innbsp;the Board T, of the fame Dimeillions as that innbsp;the Plate O. This Board is us�d to diminifh thenbsp;Breadth of the Beam, which is continually iii-creafing, byreafonof the Light that comes fromnbsp;the Sides of the Sun.

Experiment i.] Things being difpos�d, as we have Lid in the Defcription of the Machine jnbsp;let the Beam fall perpendicularly on the Surfacenbsp;abed, and it will pafs perpendicularly thro� thenbsp;Water and the upper Part of the Trough, andnbsp;not be turn�d out of the Way at all, either goingnbsp;in or coming out j by which is confirm�d whac-has been faid (Nquot;. 6zp.)

Experiment z,] Now let the Beam fall obliquely oa the Surface abed,nbsp;nbsp;nbsp;nbsp;the upper Part of

the Beam will continue its Motion ftrait along but the lower Part of it, in the Water, will benbsp;bent towards/, coming towards the Perpendicular gt; which confirms (N*. 6Z4.)

Experiment Every Thing being, as in the former Experiments, the Beam, which, at/, goesnbsp;out of Water into Air, is turn�d out of its Waynbsp;frontwards the Perpendicular, and in fuch Manner, as to move in the fame Direftion as the incident Ray that goes into the Water at g j for it:

Book III. of Natural Thilofo^hy.

goes on parallel to the Ray � h continued ; which confirms (N*. 62f, �2.�, and 618.

In what has been faid hitherto, we have only Confider�d the Attradionof the denfer Medium,nbsp;becaufe it overpowers 5 but we rnuil not overlook the A�l:ion of the rarer Medium, becaufe itnbsp;diminifhesthe Aftionof the denfer, which Adfiotlnbsp;becomes fo much the lefs on the Rays of Light,nbsp;as the Mediums differ lefs in Denfity. Thereforenbsp;there is no RefraUion where the Denftties of Me-6�^tnbsp;Hums are equal ; and the RefraUion is greaiefinbsp;where the Denfities of Mediums differ moft.

The Laws of Refra�tion are deduc�d from the Acceleration which the Attraftion generates jnbsp;therefore this Acceleration muff be examin�d.

Plate 4. Fig. i.] The Actradlion a�l;s between the Planes reprefented by thefe Lines, but nonbsp;fiutherLnbsp;nbsp;nbsp;nbsp;_nbsp;nbsp;nbsp;nbsp;*5,g;

Fhe Direction of this AClion is perpendicular ^'�655 the Surface that feparates the Mediums.^ and con-fequently to the Surface I L } and its Form is unequal at different Dijiances from the Surface*. Bat * ^gt;9-it is equal at equal Diftances., becaufe both Mediums are fuppofed homogeneous, and alike in allnbsp;their Parts.

The Motion of the Ray AC may berefolved into two other Motions, according to the Dbnbsp;regions A O and O C % of which the firft is * 192,nbsp;parallel to the Surface E E, and the fecondper-pendicular to that Surface : The Celerities ofnbsp;which'Motions will be refpeflively proportionalnbsp;to thofe Lines A O and O C, whilil A G denotesnbsp;the Celerity of the Ray itfclinbsp;nbsp;nbsp;nbsp;* 192;,

3^ Mathematical Elements Book III.

634 quot;�beMotion^ according to the Diretlion AO., is not chan^d by the Attradlion perpendicular to the Surfacenbsp;I L, only the Motion according to OC is accelerated.

Keeping the Line AC, that is the Celerity of the Ray, its Inclination may be chang�d, whereby alfo the Celerity in the Diredtion O C is alfonbsp;chang�d i which Celerity wholly vaniflies, if thenbsp;Angle A a Ght very fmall. In which Cafe, ifnbsp;after Light has enter�d into a denfer Medium,nbsp;its Motion be refolv�d into two, fo that thenbsp;Diredion of the one be perpendicular to the Surface I L, its Celerity mull: be wholly attributednbsp;to the Attradlion fo often mention�d. For, as itnbsp;enters into the Space of Attradion, a Motion isnbsp;generated in that Diredion j and, as it goes thro�nbsp;that Space, in which a new Adion does everynbsp;where ad upon the Light in the fame Diredion,nbsp;it is continually accelerated. Which Acceleration obtains in every Pallage of Light thro� thenbsp;Space of Atinadion gt; but it is different, accordingnbsp;to the different Celerity with which Light comesnbsp;perpendicular to the Surface which parts thenbsp;Mediums.

If the Attradion was equable thro� the whole Space of Attradion, the Laws concerning thenbsp;faid Attradion might be determin�d (as wasfaidnbsp;of the Acceleration of heavy Bodies after N� 130.)nbsp;by Help of the redangular Triangle P Q_R,nbsp;{PlaieW. Fig.^.) in which the Lines parallel tonbsp;the Bafe reprefent the Celerities, whilft Portionsnbsp;of the Area of the Triangle reprefent the Spacesnbsp;gone thro�.

But here we have always the fame Space run thro�, namely the Breadth of the Space of Attradion, becaiife we only confider the Motionnbsp;which is perpendicular to the Surface, which f�-parates the Mediums} therefore that Space isnbsp;always reprefented by equal Parts of the Area ofnbsp;the Triangle P QR. Let V d c reprefent fuch anbsp;Inbsp;nbsp;nbsp;nbsp;Part

Book III. of Natural *PhUofophy, 5I�

Parc when the Celerity is equal to 0. The Light enters the Space ofAttraftion in theabove-men-tion�d perpendicular Direction, that is, when annbsp;incident Ray makes a very Itnall Angle with thenbsp;Surface that feparates the Mediums j ^ c in thatnbsp;Cafe will denote the Celerity acquir�d by thenbsp;Attra�tion, and with which the Light goes outnbsp;of the Space of Attraction.

But if the Light goes perpendicularly into the Space of Attraction with the Celerity that isex-prefs�d by f it will go out of that Space withnbsp;the Celerity h fuppofing the Areas ^ d c andnbsp;fgih equal to one another, as appears fromnbsp;what has been faid. The Triangle ? dC, Vfgynbsp;V hi are Similar, and therefore their Areas are tonbsp;one another, as the Squares of the homologousnbsp;Sides d c, � ^, hi-, but the Sum of the Areasnbsp;Vdc, Ffg, is equal to the Area P ^2 i (by reafonnbsp;of the equal Areas P d c and fg i h j) thereforenbsp;alfo the Sum of the Squares of the Lines d c andnbsp;� i;is equal to the Square of: the Line hi-, whencenbsp;it follows, that with the three Lines above-mentioned one may form a reCtangular Triangle,nbsp;whofe Hypothenufe will be h i.

Therefore, In a re^angular'�riangle^ one Side of 6^6 �which is the Celerity 'with which Light enters thenbsp;Space of AttraUion perpendicularly, and the othernbsp;Side the Celerity acquir'd in going thro' that Space,

�when the Celerity with which the Light enters into it is equal to o -, the Hypothenufe of the Lriangknbsp;will be the Celerity with which the Light goes perpendicularly out of the Space of AttraSlion on the othernbsp;Side: Which univerfally obtains, which way fo-ever the Attraction is chang�d, in the Space ofnbsp;Attraction, according to the different Diftance ofnbsp;the Planes that terminate that Space. Which tonbsp;prove,

Let us fuppofe the Space of Attraction to be divided into two Parts, whether equal or un-

Mathematical Elements Book III.

equal, by a Plane parallel to the Surfaces wich which it is terminated. Let us fuppofe befides,nbsp;that the Attraction is not the fame in thofe twonbsp;Parts, but yet that it docs not vary in one ofnbsp;�them. Thefe Parts are to be confider�d as twonbsp;different Spaces of Attraction. Let A {PlateYSl.nbsp;Fig. 4.) be the Celerity which the Light acquires in going thro� the firft Part of the Space,nbsp;when it enters the Space with the Celerity O.nbsp;Let B be the Celerity acquir�d in going thro� thenbsp;fecond Part of the Space, when the fime Lightnbsp;enters that Part with the Celerity O. It is to benbsp;obferv�d that this Demonftration relates to thenbsp;Motion, perpendieular to the Surface, whichnbsp;feparates the Mediums,

Let the Light enter the firftPart of the Space above-mention�d with the Celerity O, it will comenbsp;to the fecond with the Celerity A j if thereforenbsp;a right angled Triangle be form�d with the Sidesnbsp;A and B, the Hypothenufe E D will exprefs thenbsp;Celerity with which the Light will go out of

* nbsp;nbsp;nbsp;the Space of Attraction *.

If the Light enters the Space of Attraction with the Celerity F G, let the reCtangular Triangle H F G be drawn with the Sides FG andnbsp;A i the Hypothenufe H G will be the Celerity,nbsp;with which the Light goes out of the firftPart of

�525 the Space of Attraction*, and enters into the fecond j and if you draw the reCtangular Triangle H G I, whofe Perpendicular is equal to thenbsp;Line B, you will have the Hypothenufe I G tonbsp;denote the Celerity with which the Light goesnbsp;out, and continues its Motion after it has run

Now we muft demonftrate that the Celerity I G is alfo the Hypothenufe of the reCtangularnbsp;Triangle N M L, whofe Side M Ij is equal to thenbsp;Celerity F G, with which the Light enters the

Book III. of Natttral ^hilofofhy, 55:

Space of Attraction i and whofe other SideL N is equal to the Line ED, which is the Celeritynbsp;that the Light acquires in going thro� the wholenbsp;Breadth of the Space of Attraction, when it hasnbsp;enter�d it with the Celerity O j which being alfonbsp;demonitrated in that Cafe, in which two different Forces of Attraction aCt, it is plain thatnbsp;the Propofition of N� 636 is thereby prov�d.

But it is plain from the Confideration of the rectangular Triangles, that the Lines I G, andnbsp;N M, are equal. The Square of the Line N Mnbsp;is equal to the Squares of the Lines N L andnbsp;L M or F G : N L is equal to the Line E D,nbsp;whofe Square is equal to the Squares of the Linesnbsp;EC and CD, or of the Lines A and B, whichnbsp;are equal to F H and HI: therefore the Squarenbsp;of the Hypothenufc N M is equal to the threenbsp;Squares of the Lines FG, FH, and HI. Tonbsp;which three Squares, the Square oftheLinesGInbsp;is equal �, as having been prov�d equal to thenbsp;Squares of the Lines H I and H G, which lafl:nbsp;is equal to the Squares of the Lines H F andnbsp;FG.

If the Space of Attraction be divided into ever fo many Spaces by plain parallel Surfaces,nbsp;which terminate that Space, and the differentnbsp;Parcs have different Forces of Attraction, thenbsp;fameDecnonltration will fervej and the Numbernbsp;of Divifions may be made any how in infinitum jnbsp;which Cafe obtains in the RefraCtion whichnbsp;Light fuffers going out of any Medium into ano-^,._nbsp;iher of different Denfity*: to which RefiraBion * 5^^nbsp;therefore the Rule of N�, 6^6 may he applied.

Plate IV. Pig-fl} Let Z be the rarer, and X the denfer Medium, and let them be feparatednbsp;by the Plane E let a Ray of Light AC fallnbsp;obliquely on the Surface E F j let A C denotenbsp;the Celerity of the Light in the Medium Z, and

D z nbsp;nbsp;nbsp;let

2^ Mathematical Elements Book IIIJ

let that Line A G be fettled j that is, let it be always the fame,'whatever the Inclination of thenbsp;Ray is. With the Center C and theSemidame-ter C A, draw a Circle j and let N G M benbsp;drawn perpendicular toEFj from A, draw AOnbsp;perpendicular to N C, and A Q^to EF.

Let us conceive the Motion along A C re-folv�d into two others, the one along A O, and

will denote the Ray�s Celerity perpendicular to the Surface E F, which Celerity alone is in-f 634 creafed by the Attra�lion of the Medium *.

Let CP be the Celerity which Light acquires, in paffing perpendicularly thro� theS^pace of At-tradion of the Medium X j fuppofing the Celerity of Light at its Entrance to be O, the Hypo-thenufe O P of the rcdangular Triangle PCOnbsp;will be the Celerity of the Ray A C in thenbsp;Medium X, according to a Diredion perpendi-

Light in the Diredion A O or Q_C, parallel to *634 the Surface EF, isnot chang�d^. Therefore letnbsp;CV be equal to A O or Ci.C, and VB perpendicular to E F, equal to the Hypothenufe P O,nbsp;and draw C B} the Motion along C B is compounded of the T wo, and this Line by its Situation determines the Diredion, and by its Lengthnbsp;*190 the Velocity of the Light inthe Medium'K* �, whichnbsp;lt;5^8 Celerity is not chang'd by the different Inclination ofnbsp;the Ray AC. For the Square of the Line C B isnbsp;equal to the Square of the Line B V or P O,nbsp;the Square of the Line C V or A O j' butnbsp;the Square of the Line P O is equal to thenbsp;Squares of the Line PC and CO: Therefore thenbsp;Square of the Line C B is equal to the threenbsp;Squares of the Lines PC, CO and AOj whichnbsp;two laft, if join�d, will give us the Square of thenbsp;Semidiameter AG orC N j that is, CB is equal

Book III. of Natural ^hilofofhy'.

to P N, whofe Square is alfo equal to the Squares of the Lines P C and C N, and which undergoes no Change from the different Inclination ofnbsp;the Ray A C.

The Line CB does inT cut the Circle, which is defcrib�d with the Semidiameter C A j fromnbsp;the Points B and T, draw B S and T R perpendicular to C M; by reafon of the Similar Triangles CBS, CTR, B C will be to T C ornbsp;CA, as BS to TRi which Lines therefore, bynbsp;reafon that B C and C A are fettled, will havenbsp;the fame Ratio to one another, whatever be thenbsp;Angle of Incidence. T R is the Sine of thenbsp;Angle of Refraftion T C R, and B S which isnbsp;equal to C V, that is equal to A O, is the Sinenbsp;of the Angle of Incidence A C O.

Therefore, in every Inclination of the Incident �jp iSijy, there is a fettled and confiant Ratio betweennbsp;the Sines of the jingles of Incidence and Refraction.

Now fince B C and C A which are as the Sines above-mention�d, do alfo denote the Celerities of the Light in the Mediums X and Z, itnbsp;follows that thofe Sines are inverfy^ as the Celerities in the Mediums.

If the Medium Z be Air, and X Water, the lt;5^(5; aforefaid Sines are as 4 to 3 j and the Celerity ofnbsp;the Light in Air to its Celerity in Water, as 5nbsp;to 4. But if, Z being Hill taken for Air, X benbsp;Glafs, the Sines will be as 17 to 11. One Experiment determines this for all Mediums.

The Ratio between the Sines of any Angles is the inverfe Ratio of the Secants of the Complements, as it appears in this Figure, fuppofing anbsp;Circle drawn with the Semidiameter C Qjor C Vjnbsp;for then AC (which is equal to CT) andCBnbsp;will be the Secants of the Angle ACQ2nbsp;BCV Complements to the Angles of Incidencenbsp;and Refraction, and inverfly, as B S (which isnbsp;D 3nbsp;nbsp;nbsp;nbsp;equal

Mathematical Elements Book III.

equal to A O) and T R, -which are the Sines of Incidence and Refra�tion in the greaternbsp;Circle.

And this Proportion of the Secants gives us a Method of eafily reducing to Experimentnbsp;the Prop. ofN�. 63P.

Experiment IV. Plate IV. Eig-6l] In this Experiment the Light is to be let into the dark Chamber thro� a Slit, in the fame Manner asnbsp;in the former Experiments, and by means of anbsp;Looking-Glafs refle�ted thro� the vertical Slit innbsp;the Board T.

Take a Trough P, nearly of the fame Size as that which was us�d in the former Experiments,nbsp;but which has only one End of Glals, namely,nbsp;the little Side abed. And let it be half fill�dnbsp;�with Water.

The vertical Beam of Light being made to fall obliquely on the Glafs End of the Trough,nbsp;the Part of it, which is above the^ater, goesnbsp;dircdlly forwards, and at h falls upon one of thenbsp;long Sides of the Trough j but that Part of thenbsp;Beam, which is refradted in the Water, goes a-longgij and ftrikes againil the fame Side of thenbsp;Trough at i. Whatever the Angle be which thenbsp;Beam makes with the End or Side abed, thenbsp;Lines � ^ and i will always be to each other,nbsp;as 3 to 4 i as is very eafily ihewn in feveral In-clin-dtions, if you have two Scales, each withnbsp;fmail Divifions on them; which Divifions in thenbsp;one muft be to the Divifions in the other, as 3nbsp;to 4 : For the greater Line^ i will always contain as many of the great Divifions, as the leflcrnbsp;Line � h will contain of the fmail ones. Thenbsp;Angle, which �/d makes with the Plane abed,nbsp;is theComplement of the Angle of Incidence to

Book HL nbsp;nbsp;nbsp;of Natural VhiloJofhy.

a right Angle, and the Angle made by the Line g i with the (ame Plane is the Complement of thenbsp;Angle of Refraftion to a right Angle j thereforenbsp;g b and g i are the Secants of the Complementsnbsp;of the Angles of Incidence and Refraftion, whichnbsp;have a conftant Ratio to one another, as was tonbsp;be confirm�d by this Experiment.

We have hitherto confider�d a Ray of Light �4Z, going out of a rarer into a denfer Medium, butnbsp;the fame conftant Proportion of the Sines mention�d, N�. 639, holds good alfo in the contrarynbsp;Motion of the Rays j the Angles A C N, M C Bnbsp;{Plate IV. Fig.^) are not chang�d, whether thenbsp;Incident Ray be A C or BC*. In that Cafe, if�quot;nbsp;BC be the Celerity of the incident Ray, C Anbsp;will be the Celerity of the refrafted Ray gt; fornbsp;the Motion of the Ray, going out of X into Z,nbsp;is retarded in the fame Manner by the Attractionnbsp;towards the Medium X, as it is accelerated innbsp;the contrary Motion.

CHAP. VII.

RAys proceeding from the fame Point, as a Center, and continually receding from each 0-ther, are /aid to be divergent.

40 Mathematical Elements Book III.

645quot; '^oe Pointy from which the divergent Rays prO' ceed^ is call'd the Radiant Point.

646 Itbe more the Rays diverge, fuppofing them at equal Difiances from each other, the lefs is the ra-

^Aj diant Point difiant from them; and fo on the con-' trary, the Rays are often mov�d by Refraction, as if they came from a radiant Point, thotheydonbsp;not really proceed from fuch a Point j that is,nbsp;if the Rays fhould be continued or producednbsp;back the Way from whence they come, theynbsp;would meet in one Point. In that Cafe ajfo, thenbsp;Rays are faid to be divergent.

648 Rays which concur in one Point, or would concur if they were continued, are faid to he convergent �,

as (being intercepted or turned out of the PRay before their Concourfe) would have concurred, being continued, is called the imaginary Focus. Whichnbsp;Name is alfo given to that Point from whichnbsp;thofe divergent Rays are conceived to flow,nbsp;*647 which do not proceed from the radiant Point*.

6yz The more the Rays converge, fuppofing theirDi-fiance from each other to be the fame,the lefs Difiance from them is the Focus, whether real or imaginary.

Book III. of Natural Thilofoiphy. 41

If parallel Rays pafs out of any Medium into lt;?- �j'5 nother of different Denfity^ theywill alfo he parallelnbsp;after RefraUion: Becaufe they are all equally inflected i for, in all this Chapter, we fpeak ofnbsp;Mediums feparated by a Plain Surface.

Let X and Z he two Mediums^ the lafi more lt;5 �4 rare, and the other more denfe^ feparated by thenbsp;Plane E S, (Plate V. Fig. i.) from the Point R C,nbsp;let there proceed the divergent Rays PC, R 0, R w,nbsp;and enter into the denfer Medium: Let one of themnbsp;beRC, perpendicular to the Surface ESjnbsp;laft is not turn�d out of the Way *, but con-tinues its Motion along C G. The Rays R 0,

R^, are refraCted towards Perpendiculars, which are conceiv�d to fall upon the Surface E S in thenbsp;Points 0 and �*. Thefe Rays are moved in the den- *nbsp;ftr Medium.) as if they all proceeded from the imaginary Focus r, which is farther diflant from the Surface thanR..) if the Rays are not too much fcattered-,nbsp;which yet is not to be underftood Mathematically j for, by a Point, we underftand a fmallnbsp;Space, Etch as is otherwife call�d a Phyficalnbsp;Point.

To demonftrate this Propofition, we mull confider, that the Angle R 0 C, is the Complement of the Angle of Incidence to a right Angle } and, that the Angle r 0 Cis alfo the Complement of the Angle of RefraClion to a rightnbsp;Angle i and therefore that the Lines Ro, rc, arenbsp;the Secants of the Complements of the Anglesnbsp;of Incidence and RefraCtion, fuppofing the Semidiameter to be 0 C j between which Secants,nbsp;there is a conllant Proportion *. In the fmall * 641nbsp;Divergence R 0 and R C, as alfo r 0 and r C, 639.nbsp;they do not fenfibly differ, and, between RCnbsp;and r C, the Ratio is always conftant j that is,nbsp;r is fixed as well as R, tho the Inclination of

4^ Mathematical Elements Book III.

the Ray be chang�d ; And therefore R� is re-fradled along n A, as if it had proceeded from r.

If the Rays are too much difpers'thisDemon-ftration will not ferve, and the Place of Con-courfe r cannot be taken for a Point : In this Cafe a little Circle mufi be imagin'd there, intonbsp;�which all the Rays concur^ which will be the greater^nbsp;the greater the Angle- is that the divergent Raysnbsp;make.

much difperfed^ but fall very obliquely on the Surface ES, they will berefra^ed, as if they proceeded fromnbsp;a Point not very remote from the Point r : As isnbsp;plain from what has been fa id.

The Rays., fuch as A B lt;?, GC, which come converging from a denfer Medium X into a rarerh.,nbsp;concur fooner than they wou�d do, if they ihou�dnbsp;* 625 continue their Motion in a denier Medium %nbsp;that is, become more convergent.^ and the real Focus is left diftant than the imaginary one. Innbsp;this Figure, the imaginary Focus is r, and thenbsp;�525 teal Focus R *. This Propoiltion therefore isnbsp;properly the Inverfe of the Propoiltion of N�.(5f4.nbsp;� 626 and therefore * both are prov�d by the fame Experiment.

Experiment i. PlateN. Fig. zf] Thro� the Ball G, which is moveable inthe Window-ihut, andnbsp;has an Hole going thro� the Middle of it, let anbsp;cylindric Beam of the Sun come into the darknbsp;Room, and be reflcdled horizontally by the Looking-Glafs S �, let it then go through the Convex Lens of Glafs that is fix�d in the Board ornbsp;Stand T, and the Rays will meet together at R,nbsp;and beyond R will move as if they proceedednbsp;from that Point j which therefore is the radiantnbsp;Point.

Convex

Book III. of Natural Thilofofhy. 43

Convex Lenfes of Glafs are very common j we fliall hereafter mention their Properties*, it be- *691nbsp;ing needlefs to do it in this Place : Now we ^97nbsp;only want a radiant Point, and it is enough tonbsp;ihevv how to make it.

Take a Trough Box, P, whofe Side abcdis of Glafs, and let it be fill�d with Water ; Thenbsp;Rays, which diverge from the Point R, becomenbsp;lefs diverging when they go into the Water.

When convergent Rays,^ as H D, Ip, Ly [Plate V. Fig. I.) having their imaginary Focus at/, gonbsp;from a rarer Medium Z into a denfer X, they become lefs convergent *, and concur in the Focus F, *6z^nbsp;which is more diftant from the Surface ES *, *652nbsp;as appears by applying here the Demonftrationnbsp;given in N�. 6y4.

Rays proceeding from the Point F, and goingout �yp of a denfer Medium into a rarer.y become more diverging, and move as if they came from/, whichnbsp;Propofition is the Inyerfe of the foregoing, andnbsp;is confirm�d by the fame Experiment *.nbsp;nbsp;nbsp;nbsp;� ^26

Experiment 2. PlateY. Fig. 3.] Take the fame Box B, as was ufed in the former Experiment jnbsp;but here let in two Beams of the Sun into thenbsp;dark Room thro� two Holes in the moveablenbsp;Plate in the Window-Ihut; let them be bothnbsp;reflefted horizontally, and tranfmitted thro� fi-milar convex Lenfes } thereby the Rays of thenbsp;Beams will become convergent, having their Focus�s at the fiimeDiftancej but if the convergingnbsp;Rays are made to run into the Water in the Boxnbsp;thro� the Side abed, they will be colle�ted at anbsp;greater Diftance j which will plainly appear bynbsp;comparing together the Situation of the Pointsnbsp;F in the Air, and � in the Water.

CHAP.

CHAP. Vlir.

Concerning the RefraBion of Lights when a fpherkal Surface feparates the Mediums.

PlateYl.Fig.i:] T E T X and Z be Me-I 1 diums difFering in Den-fity, the kfcthe rarer, and the firft the denfer j let them be feparated by the fpherical Surfacenbsp;E S, whole Center is C, and whofe Convexitynbsp;is towards the rarer Medium.

660 To begin, byexamining themoftfimpleCafej Let us fuppofe parallel Rays, as B O and A n, going out of a rarer into a denfer Medium, and fallingnbsp;upon a convex Surface, fuch as we have juft de-ferib�d ; let one of them be B O, which, beingnbsp;continu�d, goes thro� the Center, and falls perpendicularly upon the Surface E S j and there-? 629 fore is not turn�d out of the right Line *. Allnbsp;the Rays, 'which are not toonbsp;nbsp;nbsp;nbsp;from that Ray,

come nearer to it by the Refraftion of the denfer Medium, and are colleSled into one Point F : As for Example ; Let A zz be a Ray which isnbsp;refrafled along zz F j thro� the Point zz draw tonbsp;the Center C the Semidiameter C zz, and let itnbsp;be continu�d to p j as this Line is perpendicularnbsp;to the Surface which feparates the Mediums,nbsp;the Angle of Incidence is A zzp, which is equalnbsp;to the Angle zz C O j the Angle of Refra�lion isnbsp;C zz F. If the Arc zz O be a very fmall one,nbsp;thele Angles are as their Sines, whofe Ratio isnbsp;* ^39 conftant *. Therefore thefe Angles zz C O andnbsp;CzzFare increas�d and diminifli�d in the famenbsp;Ratio, as long as their Difference, which the Angle zz FO, which confcquently follows thePro-protion of the Arc zz O, which is the Meafurenbsp;of the Angle zz C O j as long as the Arc n O

, r

H T X^

1 i	*r C nbsp;nbsp;nbsp;^nbsp;nbsp;nbsp;nbsp;E	/ / 1 / W /f s
77		7/
/	X nbsp;nbsp;nbsp;c
i		1/ v
/ / A. E t

Book III. of Natural Thilofo^hy, 45

does not exceed i f Degrees, the Angle � F O is increafed and diminilhed fenfibly in the famenbsp;Ratio, as the faid Arc j and therefore all thenbsp;Rays, between A n and B O, do by their Re-fradlion meet fenfibly in the Point F.

Experiment i. Plate VI. Fig. z.] Let a cylin-dric Beam of the Sun, of an Inch in Diameter, made up of innumerable parallel Rays (which arenbsp;fo on Account of the Sun�s immenfe Diftance)nbsp;be let into the dark Room, as in the firfi: Experiment of the foregoing Chapter, and be refledednbsp;horizontally by the Looking-Glafs S.

Fill with Water the Trough P, which is a-bout 5 Inches high, and as wide, and one Foot long. Let it have a Glafs V made fait in onenbsp;of its Sides; This Glafs mull be a Portion of anbsp;Sphere, thin, and every whereof the lame Thick-nefs, fuch as the Cryllal of a Pocket-Watch.

The Convex Part of the Glafs V mull be outwards, that the Water next to it in the Trough may put on a fpherical Surface. If the Raysnbsp;above mention�d go into the Trough thro� thisnbsp;Glafs, becaufe the Glafs is thin, and has its Surfaces parallel, there is no fenfible Change in thenbsp;Motion of the Light by the Refraftion of thenbsp;Glafs, and the Light enters into the Water, innbsp;the fiime Manner, as if there was no Glafs. Letnbsp;the Trough be fo difpos�d, that one of the Raysnbsp;may pafs thro� the Center of the fpherical Surface, and the others will come nearer and nearernbsp;to it, and at laft concur with it atF.

Plate'Yl. Fig. 3] Again let X be the denfer Medium, and Z the rarer, and let them be fepa-rated by the fpherical Surface ES, whofe Centernbsp;is at C, and whofe Convexity is towards the rarernbsp;Medium; From the radiant Point R kt^ Rays pro- j

Mathematical Elements Book III.

ceed^ and enter into the denfer Medium thro' the Sur~ face above-mention'd gt; fo that of thofe Rays, thatnbsp;which is exprefs�d by R O, being continued,nbsp;may pafs thro� the Center 5 this Ray is notnbsp;refrafted as it goes into the Water, and all thenbsp;reft of the Rays come towards it by the Refra-�lion, and when they are not too divergent are col-letled into one Point as F, in the fame Manner asnbsp;was faid of parallel Rays j with this Difference,nbsp;that the Focus F in that Cafe is more diftant.nbsp;The fame Demonftration will alfo ferve here, asnbsp;relates to parallel Rays, which is built uponnbsp;this Foundation, that the Angle of the Incidencenbsp;incrcafes in the fame Ratio as the Arc n O,nbsp;which does alfo obtain here, when the faid Arcnbsp;docs not exceed if Degrees. Let R� be a Raynbsp;of Light, and from the Center C thro� n drawnbsp;C np : The Angle R np will be the Angle ofnbsp;Incidence ; let it be divided into two Parts bynbsp;the Line nq, parallel to the Line R O C j thenbsp;Part p n qis equal to the Angle � C O, which isnbsp;meafur�d by the Arc n O, and which thereforenbsp;follows the fame Proportion as that Arc; andnbsp;which alfo the Angle % RO (if it be very fmall)nbsp;does follow, and is equal to the fecond Part ofnbsp;the Angle of Incidence, which alfo does whollynbsp;increafe and diminifh in the fame Ratio, as thenbsp;Arc � O j for the Ratio which holds in refpeftnbsp;of every Part, taken fingly, will alfo hold in re-fpe6t of the Whole.

66z The fame Demonftration may be applied to any diverging or converging Rays, which in anynbsp;Cafe are refrafted in paffing thro� a fphcrical Surface j and which (as appears by this Demonftration) when they diverge but little, have theirnbsp;Focus real, or imaginary, or run parallel. It isnbsp;enough to have obferv�d this in general.

Book III. of Natural Thilofo;^hy.

The Focus F of the Rays, that come fromFL,goes farther off 'when R is brought nearer �, and jo onnbsp;the contrary. For the radiant Point being broughtnbsp;nearer, if the Point n remains the fame, thenbsp;Angle of Incidence is increas�d j and as it in-creafes, fo does alfo the Angle of Rcfradionnbsp;F � C, and n F interfefts R C at a greater Di-ftance.

Etcperiment z. Plate VI. F/^.4.] This Experiment differs from the foregoing only in this, that a Cylindric Beam of the Sun, reflefted horizontally, muft betranfmitted thro� the Convex Lensnbsp;in the Board T, as was done in the Experimentsnbsp;of the foregoing Chapters, to form the radiantnbsp;Point R, from which the Rays, going forward,nbsp;diverging, arecollcdled in the Water at a greaternbsp;Diftance, than if they had been parallel.

As you move the Board T, the Point R alfo changes its Place : If this Point is farther offnbsp;from the Surface that feparates the Mediums, Fnbsp;falls nearer to it; On the contrary, if R be nearer, F is farther off.

The radiant Point may be brought fo near to the 64^ Surface above-mention�d, that the Focus will recede to an infinite Diftance ; that is, that the refracted Rays 'will run parallel.

Experiment'^. PlateVl. Fig.'fi] Things being difpos�d as in the former Experiment 5 by removing the Board T, let R be brought nearer tonbsp;the Trough 5 and it may eafily befo difpos�d, asnbsp;to make the rcfradfed Rays become parallel.

Experiment A.. PlateYW. Fig. i.] Now if the ��f Experiment be repeated, bringing nearer to thenbsp;Trough the radiant P oint R, the refracted Raysnbsp;'willbecome dl'vergent-, but they'will diverge lefs than

'48 Mathematical Elements Book III.

666 incident Rays. If the Rays^ which out of a rarer enter. into a denfer Medium thro' a convexnbsp;Surface, he converging and directed towards the Center of that Jpherical Surface, they will fuffer no Re-

* nbsp;nbsp;nbsp;629 fra�ionnbsp;nbsp;nbsp;nbsp;But if they be dindled towards another

^ Focus of thefe converging Rays will always be between the Center of the Surface which fepa-rates the Mediums (to which all the Perpendiculars are directed) and the Point to which thenbsp;incident Rays tend. That is, if the imaginarynbsp;Focus of the incident Rays be at a Ufs Difiancenbsp;than the Center, the refraSted Rays will be lefsnbsp;converging : But if this imaginary Focus be beyondnbsp;the Center, the refradted Rays will be more converging.

Experment^.PlateVW.Fig.z.'] Every Thing being in the fame Manner, as in the former Experiments, it is eafy to confirm thefe Propofi-tions by Experiments} for the Board T may benbsp;fo difpos�d, that the convergent Rays frail enternbsp;the Water, fo as to have their imaginary Focusnbsp;at any Diftance in it.

608 From what has been laid hitherto it is ealy to determine what happens in a contrary Motion

from a denier into a rarer Medium, the convex Surface remaining towards the rarer Medium.

670 Fhe radiant Point may be fo placed, as that the Focus frail fly out to an infinite Diftance j that

r- �' � �:. � nbsp;nbsp;nbsp;^^-.-.-'�.lt;5^,1*-nbsp;nbsp;nbsp;nbsp;., -..,nbsp;nbsp;nbsp;nbsp;.'frr.v iV;-J.,,

|pf?|P|||jjtfip^|||||||j||||^|^|^|||^^^^^^^|{i[i[|M|i[|i|||j||||[[|[||j|[|ummi|m|||j^ nbsp;nbsp;nbsp;........

! nbsp;nbsp;nbsp;:; l ��:::-;::-s;. |.- laa. r.;........... : : -::=::;! r::a;::aqs ;;;;.M!i;.S!...-.--� :; B,;:...;.!;i..::.:;:- } nbsp;nbsp;nbsp;quot;i: ::::sa:j!:!:!::i-i:!|i:i (aquot;-p.:quot;ll!li.:i- -

Book III. of Natural Thilofophy. 49

If the radiant Point he brought nearer^ the re- 671 fraked Rays will diverge; they will diverge lefsnbsp;than the incident Rays, if the radiant Point henbsp;more difiant from the Surface than the Center*. * 667

But if the radiant Point be between the Surface 6-yz and the Center, the refradied Rays will be morenbsp;divergent *.nbsp;nbsp;nbsp;nbsp;* ^^7

If the Rays are convergent, they become more con- 675 vergent in every Cafe, which follows from the Refraction being made fromwards the Perpendicular*, and which may be alfo deduc�d from N�. *625

PlateYll. Fig. 3.] LetuS2.faxn fuppofe the Rays to go out of the rarer Medium Z into the denfer X,nbsp;and that the Hollow or Concave of the fphericalnbsp;Surface ES, which feparates the Mediums, is towards the rarer Medium. If theRays be parallel,nbsp;as BOj An, EO, which goes thro� the Centernbsp;C of the Surface ES; will not be refracted; butnbsp;A is refraflred towards the Perpendicular Cpnbsp;along n G *, andj being continued towards Z, * 6z^nbsp;interfeas BCO at/, which is alfo true inrefpe6tnbsp;of the Rays between B O and An*-, thofe Rays * 66anbsp;therefore become divergent, having their imaginarynbsp;Focus/ in the rarer Medium.

Experiment 6. Plate VII. Fig. 4.3 This Experiment differs from the firft of this Chapter, only in this, that the Glafs V has its Concavity towards the Air, the Trough P being in all othernbsp;Refpefts the fame; in this Cafe the Rays of thenbsp;cylindric Beam diverge in the Water.

If Rays come from a radiant Point in C B, which is beyond C {PlateYW. Fig.t,) the Angle of Incidence A � C is diminiih�d, and therefore alfonbsp;the Angle of Refradion Gnp grows lefs; that is,nbsp;the refraSled Rays become more diverging, and thenbsp;Vol.ilnbsp;nbsp;nbsp;nbsp;Enbsp;nbsp;nbsp;nbsp;imagi-

Mathematical Elements Book III.

the radiant Point coming nearer, at length it coincides with the imaginary Focus at C j for in

and O, the imaginary Focus is farther from O than the radiant Point, for it is always betweennbsp;that Point and C, by reafon of the Angles of Re-

.EscpermentjP\ Things being, as in the former Experiment i ifyouufe the Board, with the Convex Lens in it, to form a radiant Point, the Experiments proving thefe Propofitions may benbsp;eafily made.

^77 If the Rays are convergings and the Point of Con-courfe be in the denfer MediumSs near the Surface which feparates the Medium^ the refraSled Raysnbsp;will alfo converges but lefs than the incident Rays.

If the imaginary Focus of the incident Rays recedes more and more from O, that is, if they converge lefss therefrattedRays will alfo converge lefs -y untily by the receding of the imaginary, FocuSy thenbsp;Rays become parallel.

g_Q If the imaginary Focus recedes yet furthers ^ refraSled Rays become divergent.

Rays which from a denfer go into a rarer Mediums the concave Surface being towards this lajiy are almoft fubjedt to the fame Laws.

Parallel Rays by Refraction become divergent'^. If they proceed from a radiant Pointy they be-

Experiment 8. Plate VII. Fig. fHere the Board mufl: be fo plac�d in refpeft of the Trough,nbsp;that the Rays may enter the Water converging;nbsp;and the Phsenomena above-mention�d may benbsp;feen, according as you remove the faid Board.

Book III. of Natural Thilofophy.

yind they diverge Jiill more and more^ astheradi- 6Sz ant Point is brought nearer

Converging Rays^ which tend to the Center 685 the fpherieal Surface^ undergo no Change *.nbsp;nbsp;nbsp;nbsp;* 629

If they converge more orlefs^ the imaginary Fo- 084 cus of the incident Rays is always between the Center of the Surfacef which feparates the Mediums^nbsp;and the Focus of the refrahed R.ays^ * which may *675,nbsp;recede in infinitum, fo as to make the refracted 676nbsp;Rays become parallel *.nbsp;nbsp;nbsp;nbsp;* 674

We have hitherto confider�d fuch Rays as are but little inclin�d to the Surface which feparatesnbsp;the Mediums j for we have mention�d fuch incident Rays as diverge but little, and one ofnbsp;which is perpendicular to the Surface that feparates the Mediums. Fhe fame Propofitions hold 686nbsp;good in oblique Rays, yet in that Cafe all the Raysnbsp;are infleSled, which does not happen fo in directnbsp;ones} for the Ray which is perpendicular tonbsp;the Surface is not inlledted. Oblique Raysnbsp;alfo undergo a greater Refraftion, that is,nbsp;they are more inflebled either to or from each other, 68 Jnbsp;than diredt ones, fuppofing the Circuraftancesnbsp;the fame.

PlateYlll. Fig. i.] Let Z be a rarer, andXa denfer Medium, E S the Surface feparating thenbsp;Mediums, and having its Center at C j and thenbsp;parallel Rays h.n,Pgt;m will come together at F.

PlateYlVL. Fig. 2.] If the Surface be turn�d fo as to have its Concavity towards the rarer Medium, the parallel Rays A n and B m will havenbsp;their imaginary Focus at f-, but its Dillance fromnbsp;the Stirfiice E S, as alfo that of the above-raen-tion�d Foci F and � in xhamp; Figure i. is lefs, thannbsp;if the Rays were diredt.

Mathematical Elements Book III.

688 All the Points of a lucid Body are radiant Points^ and have each their -particular Focus^ which fervesnbsp;to explain the following Experiment, made tonbsp;confirm what has been faidof the oblique Rays.

Experiment p. Plate VIII. Fig. 5.] Take the Trough P mention�d in the firft Experiment, andnbsp;fill it with Water; let theGlafs Vbe all cover�d,nbsp;but a circular Part in the Middle of about halfnbsp;an Inch 5 have in the Trough a moveable whitenbsp;Plane T. If theCairdle A befet at the Diftancenbsp;of 3 or 4 Foot from the Trough, let the Planenbsp;Tbe mov�d backward and forward in the Water,nbsp;and when it is come to the Dillance of the Focinbsp;of the Flame A, that Flame will appear exactlynbsp;reprefented on that Plane, all the Foei formingnbsp;a Picture. And this holds good, whether thenbsp;Rays from the Candle fall obliquely or dire�llynbsp;upon the Glafs V, only that when the Rays arenbsp;oblique (the Diftance of the Candle A remaining the fame) the Diftance of the Plane T fromnbsp;V muft be lefs. In this Cafe alfo the Candle andnbsp;the Glafs V will not be in the fame right Linenbsp;as the Reprefentation of it, as it happens whennbsp;the Rays are direft, by which thePropofitionofnbsp;N�. 686. is confirm�d.

The Candle is reprefented invertedy becaufe the Rays, which proceed from different Points, in-terfe�: one another as they go thro� V, as maynbsp;be plainly feen by looking at the firft Figure.nbsp;For which Reafon, if there are two Candles, asnbsp;A and B, the Reprefentation of the laft will benbsp;at by and that of the firft at a.

6po All the Changes that happen in Eighty which we have taken Notice of in this Chapter, are Jonbsp;much the more fenfibhy as the Surface feparatingnbsp;the Mediums is more curve j that is, a Part of anbsp;lejffer Sphere.

CHAP.

Book III. of Matural T^hilofo^hy. 53 CHAP. IX.

Concerning the Motion of Light thro� a more denfe Medium. Where we Jhall take notice of the Properties of Lenfes.

The Ufe of Glafles is common, they are more denfe than Air, and the Rays out ofnbsp;Air pafs thro� the Glafs into Air again. According to the feveral Surfaces that terminate thenbsp;Glais, Light undergoes different Changes as itnbsp;moves in it ; which to determine, the Glaffes,nbsp;or any Mediums encompafs�d with a rarer Medium, and terminated with different Surfaces,nbsp;muft be examin�d. If we confider only fane andnbsp;fphencal Sur faces, there are fix: Sorts.

The Medium may be plane or flat on both Sides. 2. Plane on the one Side and convex onnbsp;the other. 3. Convex on both Sides. 4. Planenbsp;on one Side and concave on the other, y. Concave on both Sides. 6. Laflly, It may be terminated with a concave Surface on one Side, andnbsp;a convex one on the other.

If the Glafs be made ufe of, and is not very (jpa thick, Glaffes, whofe Figure is mention�d in thenbsp;lafl five Cafes, are call�d Lenfes of Glafs.

In the fecond and third Cafe, a Lens is faid to be convex ; and if we diftinguifh thofe twonbsp;Cafes, in the fecond it is call�d plano-convex.

And fo in the fourth Cafe, it is faid to be plano-concave ; tho both this Cafe, and the fifth, is generally referr�d to concave Lenfes. But anbsp;concavo-convex Lens is referr�d to the concavenbsp;or convex Lenfes, according as the one or thenbsp;other Surface is predominant j and that is faidnbsp;E 3nbsp;nbsp;nbsp;nbsp;to.

6p5 In every Lens^ or Medium^ terminated in any Manner above deferibed^ a right Line^ vohich isnbsp;perpendicular to the two Surfaces., is call'd thenbsp;Axis. When both Surfaces are fpherical, thenbsp;Axes goes thro� both their Centers but if onenbsp;of them be plane, it falls perpendicularly uponnbsp;that, and goes thro the Center of the other.

6p4 In the Paflage of Light thro� a Medium, terminated by two plane Surfaces, the BireUion of

�pf It is the Property of all Sorts of convex Lenfes^ that the Rays in their Paffage thro' them are inflected towards one another j fo much the more, as

6^6 the Convexity is greater: And fo concave ones, that the Rays are defledled from one another, accordingnbsp;as the Concavity is greater. For the Diredtion of

but, by inflefting one or both Surfaces, another Direction is given to the Rays: There are morenbsp;infledted towards the Axis of the Lens, by reafonnbsp;ofthe Convexity of the Surface of the Glafs, and,nbsp;by making the Surface concave, they are defledtednbsp;from the Axis; as is plain in every Cafe, by comparing the Infledlion in the plane Surface that isnbsp;perpendicular to the Axis, with theinfledtion innbsp;the fpherical Surface at any Diftance from the Axis. And the Difference of their Inflexions, that is,nbsp;the Change of the Diredtion of the Rays, in-creafes, as their Diftance from the Axes does ;nbsp;and it is to be obferv�d in every Diredlion of thenbsp;Rays, as well in oblique Rays, as in diredt: butnbsp;the Changes are greater in oblique Rays, becaulenbsp;the Angles of Incidence are greater. From which

That

Book Iir. of Natural fhilofophy. 5-5-

That diverging Rays either diverge left, or run 6pp parallel, or laflly, converge j in 'which Cafe, thenbsp;radiant Point receding, the Focus comes nearer, and,nbsp;fo on the contrary: But this is. the Cafe, when thenbsp;radiant Point is farther diftant from the Lensnbsp;than the Focus of parallel Rays.

Laftly, That converging Rays converge more, 700 voben the Light emerges out of the Lens.

Lhe fame Things are obfervable in oblique Rays gt;701 concerning'wbich it is to be noted, that the Difiancesnbsp;of the Foci of the emerging Rays, are lefs than innbsp;the dire�1, and the other Changes more fenftble^\ *697

All thefe frmc Things may be deduced from examining the double Refradlion in the Entrancenbsp;and Emerllon of Light. And this double Refra-dtion is vifible in every Cafe, by the followingnbsp;Experiments j by which the aforefaid Propertiesnbsp;of convex Lenfes are confirm�d.

PlateYlll. Fig.^Sf..'] Takefevcral Boxes likeP, with Water in them, and thro� which Light is 'nbsp;tranfmitted thro� the Glafies V and V, whichnbsp;are placed in the oppofite Sides of the Box,nbsp;and are diftant from one another about onenbsp;Inch. Thefe Gl.aftes are thin j in the Box whichnbsp;reprefents a convex Lens, on each Side there isnbsp;placed one, like that of the firft Experiment ofnbsp;the former Chapter, which arc fo difpofed as tonbsp;have their Convexities without the Box. Whennbsp;a plano-comex Lens is to be reprefented, on onenbsp;Side there is a plane Glafs. A concavc-comtxnbsp;Lens is reprefented by two fpherical Glafies, thatnbsp;are Portions of different Spheres, and the Convexity of the Portion of the greater Sphere, mullnbsp;be turn�d towards the Infide of the Box.

Whilft the Light paffes thro� thefe Boxes, the Changes of the Light are vifible to the Eye innbsp;E 4nbsp;nbsp;nbsp;nbsp;hsi

Mathematical Elements Book III.

its Entrance into, and Emerfion out of a denfer Medium, and by the Affillance of thefe, allnbsp;Things, relating to convex Lenfes, are clearlynbsp;demonftrable.

Experiment i.] Ex. gr. Let P be one of the fore-mentioned Boxes, with the fphericalGlafTesnbsp;V, V, the Convexities being placed outwards �,nbsp;let it be fill�d with Water} in a dark Room let anbsp;cylindricBeamof theSunbe horizontally refle�l-ed from the Looking-Glafs S} let this Beam enter the Box} the parallel Rays, of which it isnbsp;form�d, will be infleflred towards one another,nbsp;and will converge } at their Emerfion on the o-therSide, they will converge more, and concurnbsp;in F. Experiments may be made of the incidentnbsp;Rays, which diverge or converge, by ufing thenbsp;Board with the convex Lens, as in the Experi-rpents of the former Chapter.

Every Point of a lucid Body, as was faid bc*^ *688 fore, is a radiant Point, * and being placed at anbsp;due Diftancefrom a convex Lens, every one hasnbsp;* 698 its Focus. *

Egt;(feriment i.] Let a lighted Candle be removed from a convex Lens beyond the Focus of parallel Rays} at the oppofite Part of the Lens,nbsp;upon a white Plane, by the Foci of the Pointsnbsp;of the Flame, it will be reprelented } and thisnbsp;Reprefentarion will be inverted, by reafon ofnbsp;the Interfeftion of the Rays in their Paflagenbsp;thro� the Glafs.

Convex Lenfes are alfo burning Glajfes., bccaufe � 698 they colledl the Rays of * the Sun, which, upon the Account of the immenfe Diftance of thenbsp;Sun, are efteem�d as parallel. But Rays unitednbsp;in a Focus, (by reafon that the Fire, that wasnbsp;before difperfed, is now collected, and by rea-Vnbsp;nbsp;nbsp;nbsp;fon

Book III. of Natural �T^hilofophy.

fon of the Motion of the Fire according to various Dire�lions,) do burn vehementljr.

Experiment 3.] Take a convex Lens of any Magnitude, ler it be fo expofed to the Rays ofnbsp;the Sun, that the Axis of the Lens may be innbsp;the Direftion of the Rays j if any combulHblenbsp;Body be placed in the Focus of the Sun�s Rays,nbsp;it will burn.

When, by reafon of the Magnitude of the Lens, the Rays are not exadfly enough collefted,nbsp;before they come to the Focus, they mull benbsp;tranfmitted thro� another convex Lens that isnbsp;Icfs, by which they will be reduced into a fmallernbsp;Compafs, fo as to burn more violently,

As for concave Lenfes, and their Properties, they may be eafily deduced from * what has * 696nbsp;been faid.

Converging Rays either converge the lefs., or be- yo6 come parallel.^ or (as it happens in fuch as converge lefs) go cut from the Lens diverging.

All which Things happen to oblique as well 7Q7 as dircdt Rays, but more fenGbly in the firft. *nbsp;nbsp;nbsp;nbsp;* 697

PlateYK. Fig. i.] Boxes, to reprefcnt thcEf- 708 fefts of concave Lenfes, are made in the famenbsp;Manner asthofe thatreprefent theEffcftsof convex ones.* The Difference is only in the Pofi- *702nbsp;tion of the Glaffes: In the firft Box the Concavities of the two fpherical Glafles V, V, areout-^vards. In the fecond, inftead of one of thofenbsp;Glaffes, you have a plane Glafs. In the third,nbsp;you have two fpherical Glaffes, but Scdlions ofnbsp;different Spheres gt; the Sedlion of the greaternbsp;Sphere has its Convexity outwards, and thenbsp;fher its Convexity inwards.

Mathematical Elements Book �II.�

Experiment 4.] Parallel Rays of the Sun, re-flefted horizontally in a dark Chamber by the Looking-Glafs S, muft be tranfmitted thro� thenbsp;Box P, which is full of Water, and reprefents anbsp;Lens concave on both Sides : As they go intonbsp;the Box they will begin to diverge, and as theynbsp;go out they will diverge more.

The remaining Experiments, relating to concave Lenfes, are made in the fame Manner, as has been faid, in refpe�t of convex Lenfes.

CHAP.

TIHE Properties and Laws of theRefra�i-on of Light, that we have explain�d, are ot wonderful Ufe in reprefenting Objefts to ournbsp;Mind.

By thefe I.aws, the Obje�s are beautifully painted in their proper Colours in the Bottomnbsp;of the Eye; and this Pifture, as I (hall fay here-*716 after, * is the Occafion of the Ideas which arenbsp;excited in our Minds concerning the Thingsnbsp;that we fee.

How this Piflrurc is form�d in the Eye, cannot be explain�d, without examining a new Property of Light j namely, its Divifibilicy, which is paft our Comprehenfion.

yop -ri Body that is not lucid,, and intercepts the Light, is faid to be opaque.

-jQ Several wcaong all the opaque Bodies, when ' exa�tly polifh�d (except perfe�tly black Bodies, ifnbsp;anbsp;nbsp;nbsp;nbsp;there

Book III. nbsp;nbsp;nbsp;of Natural ^hUofophy.nbsp;nbsp;nbsp;nbsp;59

there are any fuch) have the Property of dividing Light j ^or ihty refleSl the Lights fo that the Rapnbsp;from every Point, being ftruck back, are divided,nbsp;�ind recede every Way, fo that all the fingle Pointsnbsp;of the Body become as it vtiere radiant Points, fromnbsp;'vhich Light goes every Way.

Whence vve deduce a Method of painting Ob- j\ r je�ts upon a white Plane j for all the Points ofnbsp;the enlighten�d Body, from which the Rays comenbsp;Upon a convex. Lens, have their Focus of thenbsp;other Side of the Lens.*' The Foci of diftant * 699nbsp;Objefts, tho not exaftly, are fenfibly at thenbsp;fame Diftance from the Lens ; thofe Objeftsnbsp;tnay by thefe Foci be reprefented in the famenbsp;Place j which Reprefentation is inverted, (bynbsp;reafon of the Interfedtion of the Rays as they gonbsp;thro� the Glafs,) and fenfible in a dark Place, innbsp;which Light comes in no Way but thro� thenbsp;Lens, and only that Light by which theObjedtsnbsp;are reprefented.

This will do wherever the Lens is placed, and in Refpedt of all the Points of Objefts, enlighten�d by Rays of Light, from which right Linesnbsp;without Interruption may be drawn to the Lens jnbsp;in this Manner the above-mention�d Divifibilitynbsp;in Light may be proved, and the Aptnefs thatnbsp;Bodies, that refleft Light, have to divide it.

Experiment i. Plate\yi.Fig.z7\ Make an Hole in a dark Place, over-againft feveral Objedts thatnbsp;are at lead; fo Foot off or farther. Let the Holenbsp;be V, and let it have a convex Lens in it thatnbsp;colleds parallel Rays at the Diftance of about 4nbsp;or y Feet j if a white Plane be placed behindnbsp;the Lens a little farther from it than that Diftance, all the Objedts above-raention�d will benbsp;painted upon it in very beautiful Colours. It isnbsp;Co be obferv�d, that the Lens muft be placed in

6o Mathematical Elements Book III.

a Pofition parallel to the Plane j and that by moving the Lens or the Plane, the Diftance mullnbsp;be found at which the Obje�ls are reprefentednbsp;molt exadtly.

This Reprefentation of Objefts has great Affinity with that by which the Objefts, that we fee, are reprefented in the Bottom of the Eye, asnbsp;will appear from the Make of the Eye.

7^^ The Figure of the Eye, when taken out ofthe Head, is nearly fpherical; only the Fore-part isnbsp;fomething more convex than the reft.

The Part A A, which is moft convex, is tran-fparent, and call�d thcFunka Cornea.

The whole Covering of the Eye, except the Cornea., is call�d the Sclerotica., B A AB.

That Part of the Sclerotica, which is next to the Cornea, is call�d the Adnata, otTVhite of the Eye.

Behind the Cornea, on the Infide, is a Coat call�d l\iGUvea, which has in its Middle an Holenbsp;fp, call�d the Pupil.

The Uvea is made up of concentric circular Fibres, interfc�ted at right Angles by ftrait Fibres. If the firftare fwell�d, the laft are relax�d,nbsp;and the Pupil is leflened or contraflred j and anbsp;contrary Motion of the Fibres increafes or widens it.

In theMiddleof theEye, but nearer theForc-parr, there is a tranfparcnt foft Body C C, like a convex Lens, whoft Hind-part is moreconvexnbsp;than the Foic-parr. It is call�d the Cryjlallinenbsp;Humour. Its Axis coincides with the Axis ofnbsp;the Eye, that goes thro� the Centers of the Pupil and the whole Eye.

This cryftaliine Humour is fuftain�d by fmall Fibres or Threads, which are fix�d to all thenbsp;Points of its Circumference, and iikewife to thenbsp;%nbsp;nbsp;nbsp;nbsp;Infide

Book III. of Natural Thilofophy. 6i

Infideof the Eye: They are infledted in the Form of an Arc, and every one of them is a Mufcle gt;nbsp;they are call�d the Ligamenta CHiana^ and two ofnbsp;them are reprefented by / C, IC. They all cohere to one another, and, together with thenbsp;Cryftalline, make a Separation m the Eye, andnbsp;divide it into two Cavities, one forwards^ p,nbsp;and the other backwards S S.

The Cavity, that is forwards, is fill�d with a Liquor like Water, call�d the Aqueous Humour.

The hind Cavity is fill�d with a tranfparent Humour, nearly of the fame Denfity as the Aqueous Humour, but not fo fluid, call the Vitreousnbsp;Humour.

The hind Surface of the Eye within is lined with a Coat call�d the Choroides, which is againnbsp;cover�d with a thin Membrane cs\\'d the Retinu.

At the back Parc of the Bulb of the Eye, a little on one fide, is theOptic NerveNN fojoin�d to the Eye, that the Eye icfelf is, as it were, annbsp;Expanfionofthe Optic Nerve 5 for the expandednbsp;Coats of the Nerve form the Choroides and Scle-gt;nbsp;rotica, and the Fibres, which make up the Retina, concurring, makethe Marrow of the Nerve.

The Eye is moved in the Head by fevcral Mufcles inferred in the Sclerotica j but we ihallnbsp;not treat of them here: As we only confider thenbsp;Eye with refpe�t to the Motion of Light, wenbsp;purpofely forbear to take Notice of any Thingnbsp;elfe.

Rays that proceed from any Point and enter the Eye thro' the PupU.^ go out of a rarer into a denfernbsp;Medium thro� a fpherical Surface; and thereforenbsp;if that Point he at adue Dijiance from the Eye., thenbsp;Rays after Refrahlion 'willconverge * in the fame *nbsp;Manner as in the Experiment of N�.663. {Platenbsp;^l.Fig. 4.) in which the Glafs V reprelencs thenbsp;tranfparent Cornea of the Eye, whilll the Water

Mathematical Elements Book III.

in the Trough isinfteadof the aqueous Humour j and therefore, fuppofing only the Cornea and thenbsp;yiq. aqueous Humour^ there isoill be in the Eye an in-�'710, nserted Picture of the Objetls. *

Experiment z. Plate IX. Fig. 4.3 Let P be the Trough above-mention�d full of Water, and feenbsp;in a dark Place, which has an Hole about halfnbsp;an Inch wide, fo difpofed towards enlighten�d Ob-je�fs at a certain Diftance, that one may fee themnbsp;thro� the Holej let the Glafs V of the Troughnbsp;be applied to this Hole, and upon the whitenbsp;Plane T you will have an inverted Pi�ture of thenbsp;Objefts : By moving the Plane backwards andnbsp;forwards, you will find the Place where the Pi-�lure is moft diftinft.

If the abovemention�d Pifture, which we have imitated in this Experiment, was to be made innbsp;the Eye, it would be at too great a Diftancenbsp;from the Cornea, and beyond the Bottom of thenbsp;Eye j and therefore this Diftance is diminifti�dnbsp;71 f by Help of the Cryfialline Humourwhich is den-fer, but encompafs�d with rarer Mediums 5 fornbsp;the Rays, converging in the aqueous Humour,nbsp;pafs thro� the cryftalline into the vitreous Humour 5 that is, out of a rare Medium thro� anbsp;denier, (which is terminated by two fphericalnbsp;convex Surfaces) into a rare Medium again j bynbsp;* 700 which Motion the Rays converge ftill more; *nbsp;and therefore they concur fooner, and the Pitlurtnbsp;above-mentioned falls within the Eye.

prefented in the Bottom of the Eye, are painted upon the Retina; and by the Motion of Light thefinall Fibres, of which the Retina is made up, are agitated;nbsp;by which Agitation, the\d^f}amp;oftheObjePls,paintednbsp;in the Eye, are excited in the Mind. The Connexion between the Ideas and the Motions by which

Book III. of Natural Thilofophy. 6^

they are excited, is unknown to us, * as we faid * 502 before ; In determining the Caufes of Senfations,nbsp;we can go no farther than the Agitation of thenbsp;Nerves.

The more exa�l the PiSlure above deferibed is, the-j\-j quot;tiiore difiinSt �mill the Objects appear. When the-jx%nbsp;Rays, coming from the fame Point, are not exactlynbsp;united upon the Retina, its Pifture is not a Point,nbsp;but a Spot, whicli is confounded with thenbsp;Pidrures of the neighbouring Points 5 in whichnbsp;Cafe the Vifion is confufed.

But when, according to the different Difanceofji^ the radiant Point, its Focus is brought nearer, ornbsp;removed farther off, * it is neceffary that thereJloould * 665nbsp;be a Change in the Eye,nbsp;nbsp;nbsp;nbsp;the Place, in which the

Pifture is exad, Ihould fall ihort of, or beyond the Retina, and fo the Vifion Jhouldhe confufed.

But it is very difficult to determine what this Change is, and Philofophers are divided in theirnbsp;Opinions about it; I fhall only obferve in general, that it is not very probable that the Figure ofnbsp;the -whole Eye is changed, in order to put back or 7^^nbsp;bring forward the Retina �, and therefore we muffnbsp;exped to find this Change within the Eye.

For if the Figure of the Eye was changed, as this Change mull be equally neceflary in all Animals, the Eyes of all Animals would undergo thenbsp;fame Changes; for the fame natural Effeds cannotnbsp;have different Caufes. Now in the Whale thenbsp;Sclerotica is too hard to be fubjeft to any Altera- �nbsp;tion of Figure. Befides, if there was fuch anbsp;Change in the whole Eye, it would arife fromnbsp;the external Preffure of the Mufcles, whichnbsp;Would be different in different Pofitions of thenbsp;Eye, and only regular in one Situation of it.

If now we examine the Eye within, it will appear neceffary that there fliould be a Changenbsp;in the Cryftaliinei which by changing its Place

^4 Mathematical Elements Book Hi.

or Figure in the Eye, will produce the defired EfFeft gt; for the Rays that fall upon the Retina^nbsp;before they are united, will be made to unitenbsp;juft upon the Retina, if the Cryftalline becomes

721 That the Pofition of the CryftalUne Humour is eafily changed^ and that it is brought nearer to^ ornbsp;farther from the Retina^ its Axis remaining thenbsp;fame, is plain, becaufe the ciliary Ligaments arenbsp;mufcular : When thefe Mufcles are fwell�d, andnbsp;become flrorter, the Hollow which their Inflection makes at C/, C/, becomes lefs^ by whichnbsp;means the vitreous Humour is comprefs�d, andnbsp;therefore it prefles upon the Cryltalline, andnbsp;pulhes it forwards farther from the Retina jnbsp;which is neceflary when we look at near Ob-

^99 From an Experiment, that we fhall hereafter it has been demonftrated, that there isnbsp;another Change in the Eye that a�ls contrarynbsp;to this i and we fhall fhew what is the Occa-fion of it. The fecond Change is alfo to be re-622 ferr�d to the Cryftalline j which (ywhen it isnbsp;drawn by the ciliary Ligaments^ to make it re*nbsp;cede from the Bottom of the Eye) becomes alfonbsp;flatter^ and therefore it mufi recede farther thannbsp;if its Figure was unchangeable j that is, the Changenbsp;. becomes more fenfible gt; which we fhall fhew tonbsp;*730 be of Ufe.*

Thefe Changes in the Eye have their Limits, for which Reafon alfo Objebls appear only diftinfinbsp;724 within certain Limits^ which are different Diftancesnbsp;jiy accordingto the Difference of People's Eyes �, and verynbsp;often in the fame Man^ both Eyes have not the famenbsp;Limits i which is almoft of the lame Ufe, as ifnbsp;the Limits of both Eyes were more diftindfrom

Book III. of Natural Thilofophyi 6$

one another; For one may lee anObje�ldiftin�l-ly enough with only one Eye. In romeFerfons alfo the neareft Limit of one Eye is farther offnbsp;than the fartheft of the other : In which Cafenbsp;near Obje�ls and diftant Objedls are feen di-ftin�l:ly, but the intermediate ones appear coh'nbsp;fufed.

The Pi�l:urein the Bottoni of the Eye, as has jz6 been faid, * is inverted; whence a Queftion a- * 7t4nbsp;rifes, why we fee Objefts ere�t ? To which wenbsp;anfwer by asking another Queftion ; Whethernbsp;it is more eafy to conceive the Connexion between an Idea in the Mind, and anere�tFigurejnbsp;than an inverted one ? Weconfefs, that we havenbsp;no Notion of that Connexion in either Cafe :

But Experience teaches us, that there is a Connexion between an inverted Pi�ture in the Eye, and the Idea of an ere�t Obje�t; and furthernbsp;than this we do not know.

If we look at the fame Objefl: with both Eyes, -jtj it will appear fingle; but this happens only whennbsp;the Objeft is painted in correfpondent Pointsnbsp;of each Retina; which probably happens fromnbsp;the meeting of the Optic Nerves. For it isob-ferv�d in all Animals, which fee the fame Objeftnbsp;with both Eyes, that the Optic Nerves meetnbsp;and feparate again before they go to the Brain ;nbsp;but in Animals which fee different Objects withnbsp;each Eye, the Optic Nerves go feparately fromnbsp;the Eyes to the Brain,

Only one Point at a Time can he feen difiin^lylt;i 'J�2.% namely, that which is reprefented in the Axis ofnbsp;the Eye ; if we look at one Point with bothnbsp;Eyes, we muft fo direft the Eyes, that theirnbsp;Axes continued lhall meet in that Point; whichnbsp;happens when we have our Eyes intent upon anynbsp;Point.

55 Mathematical Elements Book III.

By this Direftion of the Axes we judge of the Diftance of Obje�bs j for the Situation ofnbsp;the Eyes alter according as the Axes make a different Angle, which Angle depends upon thenbsp;Dillance of the Objedt: Whence it happens,nbsp;that without perceiving when we do it, by Ulenbsp;we get a Habit of judging of theihiftance ofnbsp;yip Objects by the Dire�ion of the Axes gt; wtiich is fen-Hble to us, becaufe it depends upon the Motioirnbsp;of the Eye, that we feel. Therefore we maynbsp;fee the Ufe of having two Eyes placed at a certain Diftance from one another j as long as thisnbsp;Diftance of the Eyes hears a fenfible Proportion tonbsp;the Diftance of the Objetls^ �wecan judge of it prettynbsp;certainly.

tVe can alfo judge of leffer Diftances with one Eye alone 'j bccaufe in the Variation of a fmallnbsp;723 Diltance, the Change in the Eye is fenfible. ^nbsp;731 In great Diftances.^ if we look at known Ob-jedts, we judge from the apparent Magnitude andnbsp;the Colour.

731 It is impoffihle to judge of very great Diftances, except the fame Objeks befeenfrom different Places.

The apparent Magnitude of an ObjeSi depends � upon the Bignefs of the Pidure in the Bottomnbsp;of the Eye, which Pi�lure dependsnbsp;nbsp;nbsp;nbsp;the Angle

under which an Objebt is feen, that is, the Angle which is form�d by Lines drawn from the Extremities of the Obje�t to the Eye.

734 This apparent Magnitude is to be diftinguilh�d from the Magnitude which our Mind attributes tonbsp;the OhjeB that we fee, which laft is founded uponnbsp;the Judgment, whofe Foundation is not in thenbsp;Appearance alone. Every Body knows that thenbsp;Obje�t appears the left according as it is morenbsp;diftantj whence it happens, ih'i.t, accordingto thenbsp;greater Diftance of the Object, if it be known, thenbsp;apparent Magnitude of the Obje�t is increas'd

�ook Illi of Natural ^hilcfophy� 6j

in the Judgment that our Mind makes of itj' which we do without any Attention to it. Therefore the fame Object at the fame Diftance appears of a different Magnitude, if we judge differently of the Diftance.

We have a remarkable Example of this in re-fpe�tof the Sun and Moon^ which appear greater ' when near the Horizon, than at a greater Height}nbsp;tho, as is known to Aftronomers, the Pidtureofnbsp;the Sun in the Bottom of the Eye is the fame innbsp;both Cafes, and that of the Moon is lefs when itnbsp;appears nearer the Horizon j we cannot judge ofnbsp;the Diftance in either Cafe, * but it appearsnbsp;greater near the Horizon, by reafon of the Inter-pofition of the Fields and Part of the Heavens.

If we fee the Bodies above-mention�d thro� a Tube, this apparent Diftance vanilbes, as alfonbsp;the Magnitude which is deduc�d from it. Fromnbsp;our Childhood upwards, and fo continually,

We join the Idea of Diftance with the Incrcafe of apparent Magnitude, (which is neceflary fornbsp;tnaking a true Judgment concerning the Magni*nbsp;tude) whereby the Ideas are fo clofely join�d,nbsp;that they cannot be feparated, not even in thofcnbsp;Cafes, in which we know that they lead us intonbsp;Error. Logicians teach us, how many Errorsnbsp;are to be attributed to Ideas fo join�d.

C H A P. XI.

are as it were radiant Points * therefore *713 ^ Point appears in that Place from whence diverging 714nbsp;Rays are emitted.nbsp;nbsp;nbsp;nbsp;7 * S

68 Mathematical Elements Book III.

737 If Rays infletied any how enter the Eye diverging^ the vifible Point will he in the imaginary Focus of thenbsp;Rays} for the Rays enter the Eye exa�fly in thenbsp;fame Manner that Rays would do that came directly from that Focus j and to have them unitenbsp;upon the Retina, the fame Situation of the Cry-ftalline is requifite; fo that, in refpe�t to a Spe-61:ator, it is no Matter whether thofe Rays, afternbsp;Rcfraftion, or thefe diredlly enter the Eye ^ andnbsp;� there will be the fame Motion in the Eye, whennbsp;it fits itfelf for diftindl Vifions.*

A Point appears the more enlightenedy the greater Number of Rays coming from it enter the Eye.

719 hfhenObjeSls are feen thro� a plane Glafs terminated with parallel Surf aces.) they appear to he nearer than when feen with the naked Eye. Let h. {Platenbsp;X. Fig. I.) be a vifible Point} the Rays, goingnbsp;from it andentring the Eye, are between A ^ andnbsp;A b } thefe, being refrafted in the Glafs V V,nbsp;move along ^ c, and go out thro� c d^ c d.,

is, they interfe6b at i�, which is nearer than A ; therefore the imaginary Focus of the Rays whichnbsp;enter the Eye is at in which the Point A ap-

740 Fhat Point alfo appears to be more enlightened.^ when feen thro' the Glafs above-mentioned. For allnbsp;the Rays between A b and A b enter the Pupilnbsp;between d and^j but as the Lines Ab, Ab, arenbsp;parallel to the Lines c d, c d, and thefe are between thofe, Ab and Ab being continued wouldnbsp;fill beyond d and d-, therefore if the Glafs wasnbsp;taken away, the Rays, which now enter the Pupil, would take up a greater Space, and therefore would not all enter the Eye.

' . nbsp;nbsp;nbsp;Phis

Book Iir. of Natural Thilofophy. 6^

Plate X. Fig. 2.] Floe apparent Magnitude of an 741 ObjeSi IS incrcajcd by the Interpofition of a planenbsp;Glafs-j ihe Objeft: AE is feeii by the naked Eyenbsp;under the Angle A^^Ej but if you ufe theGiafs

V nbsp;nbsp;nbsp;V, by reafon of the Refraftion thro� A. b c dnbsp;and Fjb c it will be fecn under the Angleer/r,nbsp;which is greater than the lalV. But yet the Objebl 742nbsp;is not greater in Proportion to the increafed apparentnbsp;Magnitude 5 * for it appears to be at a lei's Di-ftance.^

Fhe Increafe of apparent Magnitude is fo mucit the greater, as is the Difference of the Anglesnbsp;A r/E and c d c �, whofe Difference incrcafcs asnbsp;the fntcrfeftions of the Lines A d with b r, andnbsp;E d with b come nearer to the Points ^ andnbsp;which obtainsar theObjebl is brought neixYcv toihcnbsp;Glafs j and therefore it is the greatell of all,nbsp;when the Objedt touches the Glafs ; whichnbsp;fhews that Objefl-s, even in the Glafs itfelf,nbsp;muff appear magnified.

And in general, the Eye being placed in a rarer 744 Medium^ the Objeil that is feen in a denfer Mediumnbsp;appears bigger.^ and is alfo brought nearer by thenbsp;Refraction.* This is every Day confirm�d by *659nbsp;Experience, when we look at Objefts in thenbsp;W ater.

V nbsp;nbsp;nbsp;V, {Plate X. Fig. 3.) and the Rays A b, A bynbsp;^tedy cdy will goout more diverging, as if theynbsp;came from aj * therefore that Point appears to be * 699nbsp;at a greater Diflance.* Ft appears alfo more en~ *737nbsp;iightened j for the Rays come nearer to each otlier 74'^nbsp;as they go thro� theGlals, * and are alfo reduced * ^95nbsp;into a lels Space, wherefore a greater Number ofnbsp;them muff enter the Pupil.

The apparent Magnitude of an Objeft is alfo 747 increafed j that is, the Objebi feen thro' a convey:

JO Mathematical Elements Book III.

pears from a Sight of the Figure j the Objeft A E, feen with the naked Eye, is feen under thenbsp;Angle A hquot; E, {Plate X. Fig and f.) but nownbsp;it is feenunder the Angle c dc, which is greater gt;nbsp;(in Fig-lt;\.) the Rays A E which are conrnbsp;vergent, converge more as they go out of the

* nbsp;nbsp;nbsp;734 creas�d j * and therefore the Magnitude^ that wenbsp;748 attribute to an ObjeSi.^ does not follow the fame Pro~

portion as the apparent Magnitude-j for which Rea-fon we fhall not delay Time in Demonftrations about it : But we fliall obferve in general,

That the Angley under which an ObjeEl is feen thro' a convex GlafSy diminijhes as the Eye recedesnbsp;from the Glafs j whilfi the ObjeSl is not more difiantnbsp;from the Glafs than the Point in which parallel Raysnbsp;are colleSled: Bat if the Object is farther offy thenbsp;apparent Magnitude is increafed as the Eye recedes.nbsp;7fo After the fame Manner, if the Eye he betweennbsp;the Glafs and the Focus of parallel Rayiy the Anglenbsp;abovemention'd is diminified as theObjedl is farthernbsp;removed gt; the Eye being placed at a greater Diftanceynbsp;the fame Angle is increafed as the Object is farthernbsp;removed-, in which laft Cafe the Objeft may benbsp;fo far removed, as not to be vifible beyond thpnbsp;754 Glafs, as will be faid anon.*

Alfo in thofe Cafes in which ObjefVs are vifible, they do not always appear diftinflly.

quired that the Rays that proceed from a Pointfcould * -36 enter the Eye divergingy * a-nd that the imaginarynbsp;*727 Focus of thofe RaySy in refpect of the Spedtatorynbsp;quot;H Jhould he within the Limits of diftindl Fifton.*

If the Objeft be removed beyond the Focus ^ ^ ** of parallel Rays, the Rays flowing from a Point

Book III. of Maturan^hilojb^hy. 71

of the Object, enter the Eye converging j * which 699 Cafe is impoffible to the naked Eye : In this thenbsp;Vifion is always confufed, and the Eye difpofcsnbsp;itfelf fo as to have the Vifion the leal! confufednbsp;that may be j from this Difpofition we judge ofnbsp;the Diftance, as we do in every Cafe in whichnbsp;We judge of it with only one Eye.*nbsp;nbsp;nbsp;nbsp;* 730

But this Diftance does not always appear to be the lame; whence may be deduced what isnbsp;faid of the Change of the Figure of theCryftal-line.* For if, fuppofing the Cry ftalline moveable, *722nbsp;Its Figure be unchangeable, in every Situationnbsp;of the Objefl: and the Eye, in which the Rays,nbsp;coming converging from a Point, enter the Eye,nbsp;there will be the leaft Confufion, if theCryftal-line comes back towards the Retina as far as maynbsp;be j fo that in every Cafe there would be thenbsp;fame Difpofition of the Eye, and the fame Judgment concerning the Diftance; which, as hasnbsp;been faid, is contrary to Experience : But if itnbsp;becomes flatter as it recedes from the Retina,nbsp;there will be Changes in the Eye, which agreenbsp;with the feveral Judgments made of the Diftance in different Situations of the Obieft andnbsp;the Eye.

If in the laft Cafe^ in which Rays^ coming from a Roint converge^ the Eye be fo removed^ that the Raysnbsp;^nite before they come to the Eye^ in all the Pointsnbsp;tti which the Rays concur, there will be radiantnbsp;�*oints; which are the Foci of all the Points ofnbsp;fhc Objedt, by which the Objedt is reprefentednbsp;inverted upon a white Plane; ^ and which are * 7i�nbsp;''ifible Points in refpedt of the Eye, to which ^nbsp;the Rays can come after their Interfedtion. * In 73^nbsp;that Cafe the ObjeSl appears inverted^ becaufe wenbsp;tJon�t fee the Objedt itfelf, but its Reprefenta-tion behind the Glafs, which we have faid tonbsp;be inverted.*nbsp;nbsp;nbsp;nbsp;�7�i

F 4 nbsp;nbsp;nbsp;We

7^ Mathematical Elements Book III,

the naked Eye, in which the Rays, coming from a Point, enter the Eye converging; and thereforenbsp;fuch Vifion is always confufed, becaufetheCon-llru�tion of the Eye cannot adapt itfelfto an iinpoG-fible Cafe : Yet fornetiaies, but rarely, even innbsp;that Cafe, theObje�lsarefeendiilin�tly �, which,nbsp;as it is occafioned by fuch a Deleft of the Eyenbsp;as takes away all diftinft Vifion from the nakednbsp;Eye, I did not think prope* to take Notice ofnbsp;fuch Exceptions of the general Rule,

75'� quot;the Fault of the Eyes of a great many old Men is, that they can fee none butdiftant Objeftsdi-llinftly, thofe that are near appearing confufednbsp;to them ; which Defeft is correSted by the Intersnbsp;fofition of a convex Lens. The Rays, which flownbsp;from a Point which is near, concur beyond thenbsp;Retina �, palfing thro� a convex Glals, they willnbsp;diverge lels as they enter the Eye, and fo concurnbsp;fooner in the Eye j that is, come to the Eye asnbsp;if they flow�d from a remote Point, fuch as isnbsp;feen diftinftly by an old Man.

7f7 Ehro' a concave Lens ObjeSis appear to be nearer^ lefs enlighten'd., and lefs.

Plate'K. Fig. 61] The Rays A^, A^, and all that are between, going thro� a concave Lens

* nbsp;nbsp;nbsp;if they came frorn a Point (!?, which is lefsdiflant,*nbsp;Yvhere the Point A appears to be.*

By making the Rays to diverge more, they are carried farther afunder and therefore fewernbsp;of them enter the Eye, which diminiflies the

Plate'K.. Fig.jL] The apparent Magnitude is alfo diminilli�d, bccaufe the Rays A 1^, E^, bynbsp;which we fee the Extremities of the Objeft,

� i^rV; it xf-ift^Tex nbsp;nbsp;nbsp;'� i

'samp;u,

ti��i;^ eas�c nbsp;nbsp;nbsp;.., ,.;

Sr; to

JooklII. of Natural Thilofophy. 73

Jnder which it isfeen by the naked Eye ; Therefore upon account of the Diminution, both of the Diftanceand the Angle above-mention�d, thenbsp;Objefl: appears diminilh�d*nbsp;nbsp;nbsp;nbsp;* 734

A concave Lens is of Ufe tothofe who fee no Ob- jfS je^s diJlinSlly^ but fuch as are near-y fuch are call�dnbsp;Myopes �, thro� this Lens remote Points appearnbsp;to be near, * and the Rays, which did concur * 757nbsp;before they came to the Retina, now enter thenbsp;Eye more diverging, and meet upon the Retina.

There are Glafl�s that have one Surface plane, and the other Side is made of feveral plane Surfaces that make Angles one with another, (likenbsp;a Diamond) thro� thelb the Rays that flow fiomnbsp;one Point fiiffer different Reflations \ and bynbsp;every Surface are made to enter the Eye in a different Diretion, as if they came from differentnbsp;Points ; That is, the fame Point forms feveralnbsp;iinaginary Focij and therefore appears multiplied j for it is feen in feveral imaginary Foci * : * 737nbsp;which as it happens in refpet of every Point ofnbsp;the Objet, thro' fuch a Lens^ which is a Poly- 7fPnbsp;hedron (or multiplyina; Glafs) the Objebt appearsnbsp;multiplied.

CHAP. XIL

WE have fhewn of what Ufe Glaffes, terminated with fpherical Surfaces, arc for correting the Defets of the Eyes of old Mennbsp;and of the Short-fighted *. How they ferve * 756nbsp;fordifcoveringthefmalleflObjets, and bringing 75^nbsp;the moll diflant (as it were) to the very. Eye, isnbsp;what we are now to conflder,

Mathematical Elements Book W.

* 747 the Objelt;9:s*i which magnifying depends upon the Refraftion of the Rays as they go thro� anbsp;convex Lens j whence it follows, that it is in-creafed, if, in the fame Circumftances, the Re-fraftion be increafed j which EfFe6t may be produc�d, by augmenting the Convexity of the Lens}nbsp;which will be the more convex, as the Surfacesnbsp;that terminate it are Segments of a lefs Sphere jnbsp;which can only be had in very fmall Glafles.

761 By a Microfeope fmall ObjeEls are niajily magnified j by which means. Things, which wou�d be invisible to the naked Eye, are very di-ftinftly feen.

7�Z. ^he Space feen thro' the Microfeope^ that �, the Circle in isohich Objedts are vifible thro' the Microfnbsp;cropey is call�d the Field of the Microfeope.

Experiment i. Plate IX. Fig. f.] If we look at the fmall Obje�t AE thro� the Microfeope V,nbsp;it will appear magnified at e

There are alfo compounded Microfeopes, made up of two or three Lenfes j what Foundation theynbsp;depend upon,will be fufficiently fliewn by explaining one of thofe which is made up of two Lenfes.

Plate XL Fig. 2.] Take a fmall Lens that is very convex, asV V, and let rheObjeftAE benbsp;plac�d at fuch a Diflance from it, that all itsnbsp;* 710 Points fliall have their Force beyond the Lensjnbsp;^99 let the Obje�t be brought fo near, that the Focinbsp;� may be remov�d to a e and you will therenbsp;havetheReprefentationof the Obied:) very much

s nbsp;nbsp;nbsp;3

Book III. of Natural Thilofo^hy. 75 enlarg�d, which will be fcnfible if you receive itnbsp;there upon a white Planh *.nbsp;nbsp;nbsp;nbsp;*711

Experiment 2. Plate XI. Fig. i.] The Lens a-hovemention�d muft be made lafl: in the End of a Tube at V} and the other End of the Tube,nbsp;�which is wider, muft be cover�d with a very thinnbsp;Paper C C} the Objedt A E muft be fo plac�d,nbsp;that the Foci of the Points of that Obeft lliallnbsp;Igt;e juft at the Diftance of the Paper; If then thenbsp;Objeft be well enlighten�d, you will have itsRe-Prefentation inverted, vifible thro� the Paper atnbsp;By moving the Objeft, you will have thenbsp;truePofition that brings the Reprefentation uponnbsp;^he Paper to be diftinft.

PlateXl. Fig. 2.] All the Points of the Representation a e are radiant Points, and therefore ''iGble*i which, being feen thro�a large Micro-*725nbsp;Scrope Y V, fhews the large Reprelentation a e 754nbsp;at ae*} that is, the Rays coming from the Ob- *761nbsp;jeft A E, after the Refraftions thro� both thenbsp;Lenfes'V�V, VV, will enter the Eye, as if theynbsp;Came from an Object at a e.

Therefore thro' fuch a compounded Microfeope the Objedt appears inverted.^ and much moremagni- � ^nbsp;fied than thro' a finf,e Microfeope.

In this Microfeope the fmalleft Lens., which is tiext to one Object., is called t�eObje�t-Glal^ andnbsp;the other the Eye Glafs.

This laft ought not to be too fmall j for the Points of the Reprefentation a b, tho they be radiant Points, do not emit Light every Wayjnbsp;poly the Rays, which paft thro�the Obje�l-Glaft,nbsp;jotefe�l one another in the ieveral Points of thenbsp;Reprefentation ^ 5 which Reprefentation there-fore will not be viftble, unleft the Rays that gonbsp;^hrq� the Obje�l-Glaft do alfq go thrp� theEye-

Glaft.

y6 Mathematical Elements Book III.

jSj Glafs. '�he Field therfforc (or the Space thaC the Microfcope can take in) dependi upon thenbsp;Magnitude of this Lens.

Fhe Eye alfo muft be To plac�d, that all the Rays that come to the Rye-Glaft, going thro�nbsp;it, fhall come to the Eye j which is done bynbsp;placing the Eye at the Point in which all thenbsp;Rays., which come from the Center of the OhjeP.-GlaJ's^ and pafs thro� the Eye-Glafs., are collected.

Objefts appear bright enough thro� Micro-fcopes, bccaule they are very near the Glafs j and fo the fame Number of Rays pafs thro�a fmallnbsp;Lens aswou�d not pafs at a greater Diftance, un-Icfs thro� a great Hole: Tet often., where Objectsnbsp;are the mofi magnified., they mufl be enlighten'd bynbsp;Rays collected thro' a convex Lens and thrown uponnbsp;' them.

769 That which we now treat of is call�d the AflronomicalTbccaufe it isnotfo fit fornbsp;feeing Obje�ts upon Earth j for it reprefents themnbsp;inverted: But Aftronomers do not much regardnbsp;the Pofition of the Appearance of the Objea.

This Telefcope conuftsof two convex Lenfes, the one an Objedl-Glals, which is plac�d nextnbsp;to the Objedt, and the other an Eye-Glafs, plac�dnbsp;next to the Eye. By Help of the firft, diftantnbsp;Objefts are reprefented at a certain Diftance be-7'�� hind the Lens *, as near Objects are in the compound Microfcope. If this Reprefentation benbsp;obferv�d thro� an Eye-Glafs, it will appear enlarg�d and inverted, as we have faid concerningnbsp;the Microfcope,

^ It

Book III. of Natural Thilofoj^hy. 77

It is plain alfo, that in this Cafe, as well as in 770 the Microfcope, the Field depends upon the Breadthnbsp;of the Eye-Glafs; as alfo, that the Place of the Eye 771nbsp;'muft, be determined in the fame Manner for the��ele-fcope as for the Microfcope *: For the Aftronomic � 766nbsp;Telefcope differs from the compounded Microfcope only in this, that in the Microfcope thenbsp;Lenfes are more convex, and therefore lefs proper for looking at diftant Obje�ts, efpecially innbsp;refpe�tof Objeft-GIaffes. In the Microfcope, thenbsp;Obje�t-Glafs is more convex than the Eye-Glalsjnbsp;but the contrary obtains in the Telefcope,

Telefcopes cannot be too long for oblerving the Stars: But, if they are above zo Foot long,nbsp;they are of no Ufe forffeing Objefts upon Earth jnbsp;becaufe of the conftani Trembling of the Air,nbsp;which is too fenflble inGlaffes chat magnify verynbsp;much.

A (hoxtJftronomic Telefcope will ferve to feeOb-yjz. jeSls upon Earthy by adding toil two convex Lenfes,^nbsp;which are alfo call�d Eye-Glaffes ; the threenbsp;Eye-Glaffes are alike, and left convex than innbsp;the Aftronomic Telefcope, the Objeft-Glals remaining the fame.

Plate XI. Fig. ? �] Take an Objeft-Glaft V V, -77 ^ which reprefents a diftant ObjeA at e athen^nbsp;take befid?s three Eye-Glaffes D D, D D, D D;nbsp;the firft muft be fo plac�d, that the Rays, comingnbsp;from the Point of the Reprefentation e lt;?, ftiallnbsp;become parallel when they have pafs�d the Lens * j * 699nbsp;in that Cafe the Rays, which come from the middle Point of the Objcdt-Glaft, will be colle�tcdnbsp;at Gj the Second Lens muft befo plac�d, thatnbsp;thefcRays which are colledted at G (where theynbsp;interfe�t one another, and move as if they camenbsp;from that Point) may go out parallel after theynbsp;have pafs�d thro� it which being perform�d, * 669nbsp;the Rays coming from the Objc�l:-Glafs to

Mathematical Elefnents Book lit.

and there interfering and forming that Point of the Rdprefentation of the Obje�t, being refrarednbsp;thro� the firft: Lens, pafs by G parallel to onenbsp;another j thro� the Second Lens they are refraftednbsp;'* 693 in the Direction D e, and collected at e *, fo asnbsp;to make it the Point of a new Reprefentation.nbsp;In the fame Manner th� Point of a new Reprefentation correfponds to the Point a of thcfecondnbsp;Reprefentation; which being alfo true concerning the intermediate Points, there will be foundnbsp;an erect Reprefentation of the Obje�t at a e.

Experiment '^.Plate'Kl. Fig. 4.] Let three little Boards D, D, D, with Eye-Glafles in them, that colle�t parallel Rays at theDiftanceof aboutnbsp;� Inches be moveable upon a Plane between twonbsp;Rulers, in fuch Manner that the three Glaflesnbsp;may be in the fame Line as the Hole V, thro�nbsp;which alone the Light enters into the Room,nbsp;and in which there is an Obje6t-Glafs, which isnbsp;fix�d in a IhortTube, in order to exclude all thenbsp;fide Light.

This Obje�t Lens is fuch as is able, at the Diftance of 3 Feet from V, to reprefent difiantnbsp;Objedfs inverted at F ; which Reprefentationnbsp;will be vifible, if you let the Rays fall upon anbsp;* 711 white Plane in that Place*. Five Inches farthernbsp;from F, you muft place the firft Eye-Glafs, andnbsp;10 Inches from that, you muft place the Second}nbsp;at/, which is five Inches from it, you will havenbsp;an ere�l Reprefentation of the fame Objefts,nbsp;which will alfo be vifible, if receiv�d upon anbsp;white Plane.

Phte^l. Fig.'^i] If the Reprefentation a e be 774feen thro� a third Eye-Glafs, fuppofing the Eyenbsp;at 0, in which the parallel Rays a D and a Enbsp;are collefted, the Object appears magnified^ broughtnbsp;near, and ere�t-, for it is feen und^er the Angle

DoD,

Book III. of Natural Th�lofophy, 79�

D 0 D, when with the naked Eye it wou�d appear Under a very fmall Angle; it will alfo appear near,nbsp;becaufe, tho it be feen beyond a e, yet the Di-flance, at which it appears, has no fenfible Relation to the Diftance of a very tdiftant Object.

'ExperimentPlate XI. Fig. 4.] Suppofing e-very Thing as in the foregoing Experiment gt; let there be a third Eye-Glafs plac�d 10 Inches fromnbsp;the Second, and f Inches from that a little Board,nbsp;or Eye-Stop, with an Hole O j if the Eye benbsp;plac�d at O, the Obje�t, as has been faid, willnbsp;�ippear ere�t, magnified, and near. If the Boardnbsp;O be difplac�d, that is, be brought nearer or remov�d farther off, the Field of the Telefcope isnbsp;tliminifh�d j bccaufe there is but one Situation ofnbsp;the Eye, in which all the Rays which pafs thro�nbsp;the Eye-GlafTes can come to the Eye.

We are to take notice, that the Eye-GIafles, made Ufe of here, are not convex enough inre-fpeft of the Objeft Glafs V j but they are bellnbsp;for making the 3d Experiment.

./ill the Points of the Object do alfo appear more enlighten'd: For the Rays which, coming fromnbsp;any Point of the feveral Points of the Objedt Glafs,nbsp;do interfeft one another in a Point of the Repre-fentation, by reafon of the fmall Dillance of thenbsp;Eye-Glafs from that Reprefentation, are but little difpers�d before they come to the Eye; fonbsp;that they all go into it. And therefore the Illu-tuination, given by the Telefcope, is to thatnbsp;u^hich is given by the naked Eye, as the Surfacenbsp;uf the Objeft Glafs, thro� which the Rays pafs,nbsp;to the Surface of the Pupil. *nbsp;nbsp;nbsp;nbsp;* 7^8

One may alfo �with two Lenfes make Telefcopcs.^ yy6 '^hich fall JhewObjedts ere�l.yenlighten'd.y andmag-r-ified. Thefe muft be fhorter ; for if they benbsp;above one Foot long, they become aimoftufelels,

Mathematical Elements Book III.

becaufe their Field will be fo ftnall j that is, they will take in fo little of an Objeft.

Plate XI. Fig. y,] Let V V be art Obje�t-Glafs, the inverted Reprefentation of a dill ant * 7� Objeft will be zte a*: The Rays are fo intercepted by the concave Lens D D, that fuch ofnbsp;them, as come from the Center ofjthe Lens Vquot; V,nbsp;are inflefted as if they proceeded from the Pointnbsp;by the fame Refraction in the Rays whichnbsp;concurr�d atlt;J, become diverging*} having theirnbsp;imaginary Focus at a j which alfo happens innbsp;refpedl to all the Points of the Reprefentationnbsp;e a, and, inftead of it, you have an imaginarynbsp;Reprefentation which is eredt at a e j that is,nbsp;the Rays enter the Eye as if they came from thenbsp;Objedt plac�d at ae.

jjj The Rays, in all refpedts, go out diverging from the Eye-Glals j and therefore theEyemufinbsp;be brought as near as pojjible to the Eye-Glafs.

77^ In this Felefcope., the Field depends upon the Big-^ nefs of theObjeSlGlafs �t for the Rays, which fromnbsp;aPoint come obliquelytotheCenterofthisLens,nbsp;very often do not enter the Eye gt; whilft othernbsp;Rays from the fame Point, which pafs thro� thenbsp;Lens near its Circumference, do come to thenbsp;Eye.

CHAP. XIII.

WE have Ihewn that Light is refledted from opaque Bodies, and that every Pointnbsp;*710 refledls it every Way *. The Occafion of thisnbsp;is the Inequality of the Surtaces, which arenbsp;made up of an innumerable Quantity of fmallnbsp;Planes, which, in all fenfible Points, are diredled

Book III. o/* Natuf'al ^hilofo^hy^ 8 i

every Way ; which will be eafily conceiv�d, if we imagine a Surface cover�d with an innumerable Quantity of fmall Hemifpheres. Thatnbsp;this is true, we deduce from the Reflexion ofnbsp;Light from a polilh�d Surface j that is, from anbsp;Surface whofe Inequalities are taken offj which 77Pnbsp;in all its Points reflefts the Light only one Way,nbsp;which holds in Curve as well as in plane Surfaces : Nay, from Surfaces that are not at allnbsp;polifli�d, the Light is moftiy reflected that Way,

^nd it would be all refle�ted if they were polic�d, as daily Experience fhews.

Plate XII. Fig. 2.] Let A C be a Ray of Light coming obliquely upon a plane Surface jnbsp;let C O be perpendicular to this Surface, andnbsp;the Rays be refledbed along CB.

The refle�ed Ray.^ together mth the incident 7gi ^^y-i in the fame Plane which is perpendicular tonbsp;^he refleSiing Plane.

^efleSled, is direSled perpendicularly to a Plane, ^hich is fuppofed like in all Points.

Experiment i. Plate'X.ll, Fig. 1.'] Take a plane Lqoking-glafs S, which may be fet in any Portion by Means of a Ball and Socket join�d tonbsp;Foot that fuftains it j thro� a Hole, in thenbsp;Flate of Metal L, that is in the Window, let innbsp;^ Sun Beam of about a Quarter of an Inch Diame-

Mathematical Elements Book III.

ter into the Room; the Glaft muft be fo difpo-fed that the Beam may come thro� an oblique cylindric Cati^ity (of the fame Bignefs as thenbsp;Beam) made in a little upright Board T; Ifnbsp;you turn this Board Side for Side, the reflectednbsp;Ray will go thro� the fame Cavity. This holdsnbsp;good, whatever the Inclination of the Cavitynbsp;be, as may be demonftrated by ufing differentnbsp;Boards.

y8y Plate'Kll. Fig. zi] If the refletied Ray becomes the incident Ray ; that is, if the Light comesnbsp;along the Line B C, it will return in the Linenbsp;C A, that is, the firft that was the incident Raynbsp;will become the reflected Ray-y as appears from thenbsp;Equality of the Angles B C O, OCA.

From this Equality of the Angles of Incidence and Reflexion, we farther deduce, that

786 nbsp;nbsp;nbsp;Lighty after it has fallen upon a Body^ recedesnbsp;from it with the fame Force that it came upon it.nbsp;Let the Motion along A C be refolv�d into two

parallel to the reflefting Plane, and O C perpendicular to it. Let A O be continued; the Motion in that Diredion is not alter�d from the Aftion of the Plane; Therefore let A O andnbsp;O B be equal; if the Light recedes from thenbsp;Plane with the fame Force, with which it camenbsp;upon it, the Motion occaflon�d by the Repul-flon is reprcfented by C O, and in that Cafe

787 nbsp;nbsp;nbsp;As to the Reflexion of Light, it is to be ob-ferv�d, thatXfgfo does not run againji the folidPartnbsp;of BodieSy when it is refleSledby them ; but that it isnbsp;reflebled in thofe PlaceSy where it cou'd very freely

Book lit. of Natural ^hilofo^hy,

pafs. I (hall prove this by feveral Experiments, by which many other wonderful Properties ofnbsp;Reflexion are difcover�d.

It is a common Experiment obferv�d by every 788 Body, that when Light is mov�d thro� any Medium, as for Example, Glafsj Water or Air,nbsp;it does not undergo a fenfible and regular Reflexion i but that it is reflcfted therej where twonbsp;Mediums of different Denfity are feparated ; fonbsp;it is refleftcd in the Surface of Water or Glafs.

Coil�d Light in fuch Qpantity ftrike againft the Particles, juft where the Mediums are feparated, whereas it moves thro� both the Mediumsnbsp;for a great Space without ftriking againft any �nbsp;fuch Particles i Are there more of thofc Particlesnbsp;near the Surface than elfewhere ? Light alfo is morenbsp;abundantly refleSied in a denfer Medium^ 'when itnbsp;comes againft the Surface of a rarer j than �when onnbsp;the contrary^ moving in a rarer Medium^ it ftrikesnbsp;againft the Surface of a denfer.

Experiment i. Plate XII. Fig. 5.3 In a dark Place in which the Light enters thro� an Holenbsp;in the Plane L, let there be placed a triangularnbsp;Prifm of Glafs A B j let the Light enter thenbsp;Prifm thro� one Side j if it comes to the nextnbsp;Side making an Angle of Incidence greater thannbsp;40 Degrees, it is wholly reflefted, and does notnbsp;at all penetrate into the Air j but Light movingnbsp;in Air is never wholly refleded by the Glafs.

But if the Reflexion be made by the ftriking of Light againft the folid Parts of Bodies, therenbsp;muft be more fuch Parts in Air than in Glafs jnbsp;for if Light was refle�ted from the Glafs icfelfnbsp;into the Air, the Light would never come tonbsp;the Separation of the Mediums j that the Lightnbsp;oan alfo go out of Glafs in the very Places^nbsp;Where it is reflededj is prov�d by following Ex-G anbsp;nbsp;nbsp;nbsp;periments-

^4 Mathematical Elements Book III, periments. Therefore in the Neighbourhood ofnbsp;the Glafs there muft be Co many Parts in thenbsp;Air, that there may be no Way for the Paflagenbsp;of the Light, to caufe it to be wholly refleftednbsp;into the Glafs gt; but it is plain that there are nonbsp;fuch Parts, becaufe Light comes thro� the Airnbsp;in all Direftions quite to the Glafs. Even innbsp;the fame Place of the Surface which feparatesnbsp;the Glafs and Air, the Light which comes fromnbsp;one Side is reflefted, whilfl that which comesnbsp;from the other Side is tranfmitted. Which clear-fo proves that the Light is reflefted in the verynbsp;Place where it can go thro�.

I being as in the former Experiment, if the Obliquity of the Light be chang�d, Part of it will pals thro� into the Air.

Who wou�d conceive that Light, which palTes from Glafs into Air, and does not run againftnbsp;the folid Parts, Ihou�d all of it (by a little in*nbsp;creafing the Obliquity) run againft thofe Parts gt;nbsp;when in each Medium, as has been already faid,nbsp;there are Pores and Paflages in all Direftions?

791 Experiment 4. Plate XII. Fig. 5.] Take a Glafs triangular Prifm A B, moveable about its Axis jnbsp;which is made fo by fixing brals Plates to itsnbsp;Ends, with brafs Wires perpendicularly fix�d tonbsp;them : The Prifm muft be fo laid upon thenbsp;Trough P, as that the faid Wires may bear upon the Brims of it, which are made a little hollow to receive them, yet fo as to let them turn,nbsp;that the Prifm may move freely about its Axis:nbsp;Let it be fo plac�d as to refle�t the Light in thenbsp;fame Manner, as in the focond Experiment. Letnbsp;the Trough be fill�d with Water up to thenbsp;Prifm i then the Light, which, ftriking againft

Book III. of Natural ^hilofoj^hy. 85quot;

the Air, was wholly reflefted, now running a-gainft the Water, does partly enter into it, and is only reflected in part.

Which Experiment does not at all agree with a Reflexion made by a Stroke upon the folidnbsp;Parts.

In the third Part of this Book we fliall alfo ihew, that thin Plates, which refie�t Light, willnbsp;tranfmit it, if they become thicker *.nbsp;nbsp;nbsp;nbsp;*894

The fourth Experiment alfo proves that the 7P2. ^efleEling Power is fo much greater^ as the Mediums^nbsp;which are feparated by a refle�ing Surface^ differnbsp;'cnore in Denfity gt; for Glafs and Air differ more innbsp;Denfity than Glafs and Water.

In this Experiment we alfo fee that Reflection 7^*5 is made by the fame Power by which the Rays arenbsp;f'efraSled, which produces different Effects in dffe^nbsp;rent Circumftances.

A Ray, which goes out of a denfer into a rarer Medium, by the Attraction of the former, ^nbsp;is made to recede from the Perpendicular * j if ^^5nbsp;the Obliquity of the incident Ray be increas�d,nbsp;the Obliquity of the refraCted Ray will alfo in'nbsp;creafe, till it comes at laft to move in the verynbsp;Surface which parts the Mediums. And thisnbsp;obtains, when the Sine of the Angle of Incidence is to the whole Sine, as the Sine of Incidence, in the denfer Medium, is to the Sine ofnbsp;Refraction in the rarer j for in that Cafe thenbsp;of Refraction is a right one. If the Obliquity of the incident Ray be more increas�d,

It is plain that the Ray cannot penetrate into the rarer Medium : This is the Cafe in whichnbsp;the Light is wholly reflected j which Reflexionnbsp;depends upon that Attraction by which the Raysnbsp;are refraCted. For when the Ray is mov�d thro�nbsp;the Space of Attraction, it is bent towards thenbsp;denfer Medium * j if it be in the denfer Medi- *618nbsp;G 3nbsp;nbsp;nbsp;nbsp;mn,

$6 Mathematical Elements Book III.

uin, and fo bent, that, before it has gone thro� the whole Space of Attraftion, the Tangent tonbsp;the Curve be parallel to the Surface that fepa-rates the Mediums, the Curve, being continued,nbsp;turns back again j and therefore the Ray is re-flefted by the Attradlion of the denfer Medium,nbsp;and this Continuation of the Curve is fimilarnbsp;and equal to the firft Part, and makes the Angle of Reflexion equal to the Angle of InciTnbsp;dence becaufe the Light returns thro� the famenbsp;Part of the Sphere of Attraftion j and the famenbsp;attrafting Force a�lsupon the Light in correfpon-dent Points of the two Parts of the Curve. 'Thusnbsp;a Projeftile, in its Afcent andDefcent, deferibesnbsp;flmilar Curves.

Yet that all Reflexion does not defend upon that 7P4 Attradtion^ in the fame Manner^ is evident j for innbsp;that Cafe, in which the Refra�ion is made, Partnbsp;of the Light is refle�ledj for the Light does notnbsp;wholly penetrate out of the rarer into a denfernbsp;Medium ; for even in that Cafe, in which thenbsp;Attraftion is the moll oppos�d to the Reflexionnbsp;that is poflible, yet Ibme Rays are refledfed.

Yet it cannot be doubted but that, in every Cafe^ Reflexion has Relation to the refradling Poiver.nbsp;^ 79lt;Snbsp;nbsp;nbsp;nbsp;Where Light pafles without Refradiion^ there it is

notrefledled* but where the Reflradlion is greatefl, �631, the Refleifion isalfo flrong�fl*-y which is true,nbsp;792. not only when Light, moving in a denfer Medium, ftrikes againft a rarer, as in the fourth Ex-perimentj but the fameThing is obferv�d,whennbsp;Light flrikes againft the denfer Medium: Thusnbsp;fuppofing the Light to move in Air, the Surfacenbsp;of Glafs reflects more ftrongly than that of Water gt; and that of a Diamond yet more ftrongly.nbsp;If Gl.�fs and a Diamond be immers�d in Water} the refrafting Power is left in the Separation of thofe Bodies with Water, than where

thofc

Book III. of Natural'^hilofophy. 87

thofe Bodies touch the Air*. Theie Bodies alfo *631 refled Light lefs flrongly in Water than innbsp;Air. From this Relation of therefleding and re-frading Powers, we deduce, thatL/^^/ is drivennbsp;hack at a certain Difiance from the Bodies^ in thenbsp;fame Manner that the refrading Power docs alfonbsp;ad at fome Diftance from the Body: This Pro-pofition is confirm�d from what has been demon-ftrated concerning Reflexion, which does notnbsp;depend upon a Stroke made againfl: the folidnbsp;Parts of Bodies j and this is fully made out, ifnbsp;we confider, that poUfio'd Bodies reflehl the Light /pSnbsp;regularly (which we obferve in Looking-Glafles)nbsp;iho there a great many Scratches in their Surfaces:

For as they are polifh�d with the Powder of Emery and Putty, tho their Parts are very fmall,nbsp;yet they leave very great Scratches on the Surface in Refpcd of the Papticks of Light i whencenbsp;in the Surface itfelf the Reflexion muft needs benbsp;irregular j but if we conceive the Reflexion tonbsp;be made at fome Diftance from the Surface, thenbsp;Irregularities are diminiftt�d, and almoft whollynbsp;taken off, as is eafily underftood by any one thatnbsp;confiders it with Attention. .

CHAP. XIV.

of ^lane Mirrors.

1 j of a Plane Mirror or^p^ Looking-Glafs} A a radiant Point. Let the Planenbsp;of the Glafs be continued, and from the Radiant Anbsp;let a Perpendicular A C fall upon it j if this Line benbsp;continued, and C a be made equal toC A, a will benbsp;the imaginary Focus of therefledledRays that proceednbsp;from A. Let A^ be the incident Rayj there-flededRayj which continue beyond the Glafs jnbsp;G 4nbsp;nbsp;nbsp;nbsp;becaufc

88 Mathematical Elements Book III-the Angles of Incidence and Reflexion are equal *784 to one another*, their Complements alfo, whichnbsp;are the Angles AbC, f b d 3xe equalgt; to this isnbsp;equal its oppofite and vertical bC: TheTrian-gles A bC^abC^ which are tedlangular, have thenbsp;SideC^ common, and the Angles Qba^ C^A,nbsp;equal i therefore they agree in all Refpefts, andCnbsp;A and Qa are equal to one another; Which De-monftration may be applied to all other Raysnbsp;which flow from A, in whatever Plane, perpendicular to the Glafs, they be conceiv�d to be.nbsp;Therefore wherever the Eye is, if the refle�tednbsp;Rays come to it, they will enter the Eye as ifnbsp;they came from a j and the Point A will appearnbsp;in that Place * gt; but the Jppearance of that Pointnbsp;*737 wi// have the fame Pofition in Refpebi of the Mirror^

801 nbsp;nbsp;nbsp;je�t, it will appear, that the ObjeB will appearnbsp;behind the Glafs in the fame Pofition that it hasnbsp;before the Glafs.

Of Spherical Mirrors.

ooz I'l ^ der�d^ as made of innumerable fmall Planes j and a Plane, which touches the Spherenbsp;in any Point, is as it were a Continuation of fuchnbsp;a fmall Plane.

^ooklll. of Natural Thilofo^hy. 89

A Ray comingufon any fpherkal Mirror ^together 804 '^ith its refieSied Ray-, is in a Planc^ 'which^ beingnbsp;lt;:ontinued^ goes thro' the Center of the Sphere for *782nbsp;filch a Plane is perpendicular to the Surface ofnbsp;the Sphere. A Line which is drawn thro' the 804nbsp;Center of the Sphere and Point of Incidence^ beingnbsp;�continued makes equal Angles with the incident andnbsp;^'cfleSied Rays *} for that Line is perpendicular to *78^nbsp;the Surface, and thofe Angles are Angles of Inci-lienee and Reflexion; Therefore the Ray that goes 8o5nbsp;thro' the Center^ or which, being continued, wou'dnbsp;go thro' the Center, whenrefledted returns upon itjelf.

Plate XII. Fig. p.] Let be a Portionofthe convex Mirror; A a radiant Point; let Ab, Ad,

A c, be incident Rays ; the reflefted Rays will be bf, dg, chi if from the radiant Point A, a 807nbsp;Tangent be drawn to the Mirror, the refledled Raynbsp;�will be a Continuation of the incident, or rather the Reflexion of the Rays terminated in thenbsp;Point of Contabl.

If b f, dg, ch, the Rays that are refleAedfrom'�o'� the convex Mirror, he continued, with all the intermediate ones, by their Inrerfedtions they willnbsp;form the Curve a a, which all thefo Rays touch,nbsp;and the neighbouring Rays interfedl in the Periphery of the Curve j fo that they always enternbsp;the Eye as if they came from a Point of the Periphery i in which therefore the Point A does alwaysnbsp;appear *, as long as the refledled Rays can come to * 737nbsp;the Eye, and the Eye is mov�d in a Plane whichnbsp;goes thro� the Center of the Sphere j but whennbsp;the Eye is mov�d out of that Plane the radiantnbsp;appears in another Curve, becaufe there arefuchnbsp;Curves in every Plane, which may be conceiv�dnbsp;to pafs thro� A and C.

Since all thefe Curves, and each of them 809 wholly arc behind the Giafs, all the QhjeHs alfonbsp;appear behind the Surface ofthe Giafs.

cgt;o Mathematical Elements Book III.

Point A be mov�d about the Mirror, the whole Curve a a \5 carried with the fame Motion jnbsp;which proves (in Refpeft to the ere�t inverfenbsp;Situation) that the Points of the Reprefenta-tion have the fame Relation to each other, asnbsp;the Points of the Objeft itfelf.

As the Point A is farther remov�d from the Glafs, the whole Curve does alfo recede by anbsp;contrary Motion j but fuppofing A at an infinitenbsp;Diftance, that Point of the Curve, which is thenbsp;fartheft remov�d from the Surface, willbediftantnbsp;from a Quarter of the Diameter \ whence itnbsp;gi I follows, that theObjeElsappear diminijh'd^htcuu�enbsp;all the Reprefentations are compris�d in fmallnbsp;Limits.

81 z V ^he Eye be inov'd^ the Appearance of the Object is alfo mov'df and its Figure chanfd: For all the Points are mov�d in their own Curves, andnbsp;that unequally, according to the different Situation of the Eye in Relpeft of each Curve jnbsp;whence of Neceffity the Figure muft be chang�d.

Experiment i ^ Plate XII. Fig. 6.] If any one fees his Face in a fpherical convex Mirror,nbsp;Handing at A, he will fee his Face at a erefl',nbsp;diminilb�d and unlike j by the Motion of thenbsp;Eye one may obferve the other Things mention�d in Refpe�t to fix�d Objefts.nbsp;g j, Let b d {Plate'K.Wl.Fig. ihe a concave Mir-^ ror.) and a Portion of a Sphere whofe Center isnbsp;C j letparallel Rays fall upon the Surface of the Mirror., one of which, Ci, is fiippos�d to pafs thro�nbsp;the Center; this Ray by Reflexion returns uponnbsp;* 806 itfelf*, and the Rays next to it, being reflefted, become converging, and concur with it in the Focusnbsp;F, which is the middle Point between^and C. Letnbsp;Ai- be a Ray very little diftant front Qd-j drawnbsp;2,nbsp;nbsp;nbsp;nbsp;tkc

Book HI, of NaturalT^hilofij^hy. 91

the Semidiameter Cb i the Angle of Incidence �vvill be A^C, to which the Angle of Reflexion

is equal, as alfo the alternate Angle *805 ^ C F j therefore ^ F C is an Ifofceles Triangle,nbsp;and the Sides F C and F b are equal ; becaufenbsp;b dh veryfmall, Fd and Fi^ do not fenfiblydiffer j therefore F C and F d are equal 5 whichnbsp;Hemonftration will ferve for all the Rays thatnbsp;are but very little diftant fromC d.

If parallel Rays are farther diftant from C lt;/, they do not meet at F ; yet they will all comenbsp;together into a little Circle, if the Diameter ofnbsp;the Mirror does not exceed the fifth or fixthnbsp;Part of the Diameter of the Sphere of which itnbsp;is a Portion.

Burning Mirrors are made upon this Founda- 814 tion, which colled: the Sun�s parallel Rays intonbsp;a Focus.

Experiment t. Plate XIII. Fig. z.] Let there be a concave Mirror S, made of Metal, or ofnbsp;Clafs quickfilver�d behind ; let it ftand upon thenbsp;tv^ooden Foot P, whofe upper Part is bor�d fonbsp;as to receive a Cylinder of Wood made fall to anbsp;tranfverfe Piece A A, which ferves for turningnbsp;the Glafs round with a horizontal Motion ; andnbsp;the Mirror itfelf mufl move upon two Ends ofnbsp;an Axis between the Pillars A B, A B, Co as tonbsp;he inclin�d in any Angle, and the Screws B, B,

Having expos�d theBurning-GIafi to the Sun-Beams in fuch a Manner, that the Ray, which Comes upon the Middle of the Glafs, is perpendicular to its Surface j fince all the others arenbsp;parallel to it, they are colleded in a Focus, atnbsp;a Quarter of a Diameter�s Diftance from thenbsp;Glals, and there burn violently.

92- Mathematical Elements Book III-

If the Diameter of the Surface of the Mirror (as it is in mine) be of about if Inches, andnbsp;the Focus is 18 Inches diftant from it. Woodnbsp;will immediately be in a Flame, and thin Platesnbsp;of Lead prefently melt.

If we confider the Rays that are at fome Di-ftance from C and parallel to it, thofe of them that are neareft one another being refle-fted, will interfed: before they come to C^, and

SI f in that Cafe, that is, where parallel incident Rays fall obliquely on the Glafs^ being a little differ s'd bynbsp;Reflexion^ they are collecled in a Point.

816 If the Focus, inwhich parallel Rays are colleSled by a concave Mirror, becomes the radiant Point, thenbsp;Rays, which are but little difperfed, are reflo-

From thefe Properties of a concave Mirror we deduce the Method of rcprefenting Objedsnbsp;in a dark Place, much like what was beforenbsp;(hewn in Refped of a Convex Lens*.

PlateXMl. Fig.7,r\ Let there be an Hole F thro� the Wallj let a b be a concave Mirror, fonbsp;plac�d as to colled the parallel Rays that arenbsp;perpendicular to the Wall at F: The Rays, com-

�8i6ing from F in that Diredion, are refleded and fuch are the Rays, which, being refledednbsp;from the external Objeds, interfed one anothernbsp;at F.

Let A F be Rays coming from a Point of a diftant Objed j thefe Rays are by the Mirrornbsp;refleded perpendicularly to the Wall �, and be-caufe Rays coming from a diftant Point, andnbsp;palTing thro� a fmall Hole, may be looked uponnbsp;as parallel, thefe Rays will, after Reflexion, benbsp;colleded into one Point at a, at the Diftance of

* nbsp;nbsp;nbsp;the Wall*, that is in its Surface-, where therefore the Point will be rcprelented. In the famenbsp;Manner the Rays which come from a Point

Book III. of Natural Thilofophy. 93

thro� B F, are colleflred at ^ j which, as. it is true with Regard to all the Points of an Ob-je6t, will give the Reprefentation of it upon thenbsp;Wall; and if the Wall be white, and the Ob-jc�l enlighten�d by the Rays of the Sun, thenbsp;Pi�lure will appear in very lively Colours.

Experiment 3. Plate XIII. Fig. 4.] In a dark Place cover the Backfide of the Window-fhutnbsp;with white Paper, an Hole being made in thenbsp;Middle of it little more than half an Inch Diameter, fo as to anfwer to an Hole behind it in thenbsp;Window-lhut, ovcr-againft which, ataDiftancenbsp;no lefs than of yo Foot, there are feveral Ob-jefts enlighten�d bj the Sun : Let a concavenbsp;Mirror whofe Surface is i y Inches wide, andnbsp;which colledts parallel Rays at the Diftance ofnbsp;18 Inches, be placed at that Diftance from thenbsp;Window, in fuch Manner, that a Line paffingnbsp;thro� the Center of the Hole, and the Center ofnbsp;the Surface of the Mirror, be perpendicular tonbsp;the Plane of the Paper and the Surface of thenbsp;Glafs. Then the Objc6ts will be reprefentednbsp;Upon the Paper in a Circle concentric with thenbsp;Hole, and whofe Diameter is equal to the Diameter of the Mirror. You muft join to thenbsp;Hole, on the Outfide of the Room, an hollow,nbsp;truncated Cone, to exclude the Light which doesnbsp;not come from the Objeds to be reprefented.

httbe {Plate'Klll.Fig. be a concave Mir-ror; C the Center of its Concavity; A a radiant Point farther dijlant than C the Center of the Glafs �, A^, Kd, A ^ incident vthoicrefle-amp;ed ones bf, dg, eh., with their intermediatenbsp;ones by their mutual Interfeblions form the Curve a a.,nbsp;vohich they touch ; therefore the Point A appearsnbsp;in that Curve* t and if the Eye be mov�d in the * 808nbsp;Plane of the Curve, the Appearance will change

Mathematical Elements Book III.

Place in that Curve. But in all the Planes which may be conceived to pafs thro� C Aj there isnbsp;Tuch a Curve, and they all concur in the Linenbsp;819 C A, namely, at the Point a. Therefore in thatnbsp;Point the refletied Rays are the mofi abundantlynbsp;colleSied^ which therefore is call�d the Focus of

* nbsp;nbsp;nbsp;the Rays coming from A. On the contrary, Anbsp;^ ^ will be the Focus, fuppofing the Radiant at a*-

In this Figure there is only Part of the Curve drawn, which is produced by one Part of thenbsp;Line AC; fuch another Part muft be conceiv�dnbsp;on the other Side, and both join in the Focusnbsp;of the radiant Point.

811 As the Radiant comes forward, the Curve recedes from the Mirror, and moves towards thenbsp;Radiant, till they concur at the Center C i innbsp;which, if the Radiant he placed., all the refietled

Szz If the Radiant is yet brought nearer to the Mir-* ror, fo as to be between the Center and the Glafs,nbsp;the Curve will recede farther, and be then beyondnbsp;the Center j and in the Curve, that Point willnbsp;recede moft of all, in which all the Curves concur which are conceiv�d in feveral Planes, that is,nbsp;the Focus of the radiant Point: And that Focusnbsp;813 will be at an infinite Diftance when the Radiantnbsp;is dijiant from the Mirror juft the fourth Part of the

extended in infinitum., and the two Parts sf!\�ch. concur in theFocusof the Radiant^jr^ feparatedinbsp;this feparated Part is feen at a a : {Plate XIII-Fig.6.) If the Radiant be brought ft til nearer., thenbsp;824 Curve Parts decline from one another, becaulb thenbsp;Rays, fuch as h,b, and thofe nearer it, beingre-

Book l�l. of Natural ^htlofophy. 9^

flcfted, do not touch the Curve, but become divergent j that is, thofc reflefted Rays, being continued beyond the Glafs, will interfedt one another, and form a new Curve behind the Glafs^ which hasnbsp;t'Wo Legs, one of which is feen azaa-, they concur in the Line C A continued, namely, atlt;?,and,nbsp;Receding from the Glafs, are ftretched out in infinitum. And there is alfo, on each Side of the radiant Point, a Point in the Surface as d, whichnbsp;Separates the Rays that form the Curves a a andnbsp;and the Ray A d, being reflefted in d g,nbsp;touches neither of the Curves, if it be infinitelynbsp;continued toward each Part g, g, tho it is continually coming nearer to each Curve. If thenbsp;^hole Sphere wascompleated, inRefpe�l of thenbsp;oppofite Part of the Sphere, the Radiant wouldnbsp;be beyond the Center, and the refledted Raysnbsp;''vould form theCurve which we have mention�d ^nbsp;before, * by which the feparated Legs, as a a,

If the Mirror be enlightened by a lucid Body, the 8 if , Rays which come from all the Points of the Objebl,nbsp;being refteSied, will form Curves, but are chieflynbsp;Collected in the Foci of thefe Points� �, * therefore if*f^9nbsp;thefe Pod are in the Surface of a white Plane, therenbsp;quot;pill be upon it a Reprefentation of the lucid Body, asnbsp;*n the fecond Experiment of Chap. IX. And thatnbsp;Reprefentation will be inverted gt; for the Line,nbsp;'^hich joins the radiant Point with its Focus, goesnbsp;�hro� the Center of the Sphere}* in which there- * 819nbsp;fore all fuch Lines interfe61: one another} and thisnbsp;Rterleftion is between the radiant Point and thenbsp;Pocus, * in which the Point is reprefented. * 818

But as the lucid Body is brought nearer to the Mirror, the Appearance recedes farther, * and m. * 820nbsp;this Cafe becomes bigger.

Mathematical Elements Book III.

Experme?it 4.] Hold a lighted Candle between the Mirror and the Center of the Sphere ofnbsp;which it is a Portion ; yet fo that it may benbsp;more diftant from the Mirror than that Center:nbsp;If then there be a white Plane perpendicular tonbsp;the Line that paffes thro� the Candle and thenbsp;Center of the Mirror, and this Plane be heldnbsp;beyond the Center, you will have upon it an inverted Reprefentation of the Candle j the proper Place will be found by moving the Planenbsp;forwards and backwards ; as likewife by thisnbsp;Proportion, viz. As the Difference of the Di-ffances of the Candle, from the Center of thenbsp;Sphere and from the Mirror, is to the fourthnbsp;Part of the Diameter of the Sphere, fo is thenbsp;Dillance of the Candle from the Glafs, to thenbsp;Diftance requir�d. As the Candle is broughtnbsp;nearer to the Speculum, the Plane muft be mov�dnbsp;farther off, and the Reprefentation will grownbsp;bigger.

02,7 Plate'KWl. Fig. y.] ObjePis, placed beyond the Center.^ appear between the Glafs and the Center., fornbsp;*o^'onbsp;nbsp;nbsp;nbsp;Points appear in a Curve as at lt;� j * the

verted : for they are reduced into a narrow Space j and as the Point A moves downwards, its Reprefentation will be carried upwards j for thenbsp;Line a a keeps the fame Situation in refpe�t ofnbsp;AC a^ as it is carried round the Center C.

Experiment f. Plate XIII. Fig. 7.] Left the Reprefentation of the Obje�t fliou�d be lefs vivid, the Mirror is to be included in a Box. Ifnbsp;you have a Mirror, whofe Surface is about 8nbsp;Inches wide, and which is a Portion of a Spherenbsp;of one Foot and a half in Diameter j fhut it upnbsp;in a BoxP, in whofe Fore-Part there is an Hole

�^ook III. of Natural Thilofophy.

of about 6 Inches Diameter, and from which theGlafs'is diftant above 6 or 7 Inches j and .nbsp;kt this Opening be turn�d from the Light. Nownbsp;if any Perfon, as A j beholds himfelf at the Di-ftance of about z Foot from the Glafs5 his Facenbsp;�'vill appear inverted in the Box towards thenbsp;Hole j and if the Beholder comes nearer, he willnbsp;fee a Head coming out of the Hole.

The Reprefentationof aPoint^ placed in theCen- 8zp' of a SpberCy coincides with the radiant Point it-Mf') and is as it were [wallowed up by it^.nbsp;nbsp;nbsp;nbsp;*821

if the Eye be plac'd in that Center^ no Objeblcan 850 ^0 feen by it-, for then only the Rays, that flownbsp;from the Eye, will be reflected to it *.nbsp;nbsp;nbsp;nbsp;* 806

. If the Objefi be between the Center and the Pointy 851 which parallel Rays are colleEled after Reflexion^nbsp;the Objebl will alfo appear without the Glafs, at anbsp;greater Diflance from the Glafs than the ObjePl it-filf*-, the Reprefentation is inverted^ which is pro- *

'�cd in the fame Manner ar in N*. 818 5 and tttagnified, beai�fe it is farther remov�d from thenbsp;Center, than the Objeft itfelf is diflant from it jnbsp;for the Reprelentation recedes from the Centernbsp;infinitum whilft the Objeft goes thro� thenbsp;fourth Part of the Diameter of the Sphere.

If the ObjeSl be lefs diftant from the Mirror than g, j the fourth Part of the Diameter of the Sphere, ac-oording to the different Situation of th� Eye, theOb-Jebl appears either before or behind the Glafs.

Plate XIII. Fig. lt;5.] If th� Eye be fo placM, fhat thofe Rays may come to it which form thenbsp;Curve a a, as towards/, it will fee the Appearance of the Objefts towards the Glafs * magni- * 824.nbsp;rrodj becaufe the Curves, as-a a, which belongnbsp;to fevefal Points, kre diverging;

If thofe Rays come to the Eye which form 8 ^ ^ the Curve a a, the Objedl will appear withoutnbsp;the Glafs ; And in both Cafes the Reprefentationisnbsp;VoL�ILnbsp;nbsp;nbsp;nbsp;Hnbsp;nbsp;nbsp;nbsp;ereit%

^8 Mathematical Elements Book IIL

enSt; for as the Point A afcends or defcends, the Curves a a and a a, in which it is repre-fented, are agitated with the fame Motion.

854 If the Eye be in the Point, in which the reflected Rays, that belong to each Curve, inter-fe6l one another, as in �, the jdytpearance of the Object wtll be double.

It is plain in every Cafe of the Appearance, that the Points have not the fame Relation tonbsp;one another, as the Points of the Objefl: �, and

^3r therefore that a Concave Minor never reprefents the Objebis exaMy t But the moft irregular Re-prefentation of all is that which is in fuch Lines,nbsp;as a a.

8^6 There are alfo cylindric Mirrors convex and concave 5 thefe in one Refpe�t are plane, and innbsp;another Refpeftfpherical j and therefore the Re-prefentation of the Objeds is very irregular gt;nbsp;which Irregularity, fince it depends upon a regular Figure, may be fo determin�d, that Pi-dures may be drawn, which, thotruly irregular,nbsp;will in fuch a Glafs appear regular by Refic':.nbsp;dion, in a determinate Situation of the Eye.

CHAP. xvr.

of the Magic Lantern.

There are feveral Machines made by the Combination of Mirrors and Lenfes,nbsp;which afford ufeful and pleafant Appearances jnbsp;whofe Explanation may be eafily deduc�d fromnbsp;what has been (aid.

Among many other I fhall only chufe to explain one, in which Figures, that are paintednbsp;upon fmall Pieces of Glafs, arc reprefented mon-ftroufly large upon a white Plane. This is a

Book l�i. of Natural Thilofofhy, 991

Phsenomenon wonderful enough to deferve a particular Explanation. The Inftrument thatnbsp;performs this is call�d a Magic Lantern^ whichnbsp;Optie Writers have not altogether pafs�d by,nbsp;but yet have not fufficiently explain�d.

Plate'KIY. Fig. i.] Let there be a wooden 83/ Box aboutaFootandanhalflong, 14 Inches high,nbsp;and as wide j there muft be a concave Mirror S,nbsp;of 8 Inches Diameter, and a Portion of a Spherenbsp;of 18 Inches ; This Mirror is fix�d to a Footnbsp;which moves upon Rulers, along the Length ofnbsp;the Box.

There is alfo in this Box a Lamp L, furtain�d by a wooden Foot which is moveable long-wifenbsp;between two Rulers, in the Side of the Box.

The Pipe of the Lamp Hands forward in fuch Manner, that the Genrer of the Flame is over-sgainft the Center of the Surface of the Mirror j this Flame is made up of four little Flames,-which, by touching one another, make one fquarenbsp;Flame, two Inches wide.

In the Top, or upper Plane of the Box, there is an oblong Hole, which has a Cover that Aidesnbsp;to two Grooves, or between two Rulers ornbsp;Ledg es: Thro� this Cover pafies the Chimneynbsp;C, which (as you fee in the third Figure) Handsnbsp;Up about one half Foot above the Box. Thenbsp;Chimney is moveable with the Cover, whilHnbsp;the Opening remains Alut j that the Chimneynbsp;J^iay be always over the Lamp.

In one of the little Sides of the Box, whicjr is overagainH the Mirror, there muH be a roundnbsp;Hole about y Inches widej which muH have innbsp;it a Convex .Lens of Glafs of the fame Bignefsnbsp;V, convex on both Sides, which are Portions ofnbsp;a Sphere of one Foot Diameter : The Axis ofnbsp;this Glafs, being continued to the Surface of thenbsp;Mirror, will be perpendicular to it, and fallnbsp;H anbsp;nbsp;nbsp;nbsp;wpoa

100 Mathematical Elements Book III.

upon its Center, as likewife to the Plane of the Flame, thro� whofe middle Point it alfo palles.

This Hole is fhut and open�d by a Plane moveable in a Groove, which is mov�d by anbsp;Cylinder that Hands out of the Box at E.

To this Hole without the Box anfwers the Tube T, whofe Length and Diameter is of a-bout 6 Inches, at the End of which there is anbsp;Ring, in which the fecondTube t moves, of a-bout 4 Inches Diameter, and for 6 Inches long.

In the lefler Tube there are two Lenfesgt; the firft in that End which is thruft into the Tubenbsp;T, and it is of the fame Convexity as the Glalsnbsp;V, and three Inches and a half Diameter. Thenbsp;fecond Lens is three Inches from the firft, andnbsp;flatter, being terminated by Portions of a Spherenbsp;of four Foot Diameter. Between thefe Lenfes,nbsp;at the Diftance of about one Inch from the fecond, there is plac�d a wooden circular Stop, ornbsp;Aperture, which fhuts up all the Tube, exceptnbsp;an Hole of an Inch and a Quarter Diameter innbsp;the Middle of the Wood.

The Objefts, that are to be reprefented, are to be painted upon a flat thin Piece of Glafs,nbsp;which muft be mov�d without the Box over-againft the Glafs V, between it and the Tubenbsp;T, the Pifture being in an inverted Pofition. Ifnbsp;thefe Pidlures are round, they may be of finchesnbsp;Diameter : That they may be moved eafily, theynbsp;are put into flat Boards, three in one Board.nbsp;The Pifture alfo may be painted upon longnbsp;Glafles, which may be fucceflively made to Aidenbsp;before the Glafs V.

P/a/e IV. F/g. 3.] This whole Box Hands up* on a Frame or Foot, made fo that it may benbsp;fix�d at different Heights. There areflat Piecesnbsp;of Wood fix�d to the Box, at Bottom, which Aidenbsp;in Grooves in the Frame ; each of them has a

Slit

Book III. of Natural Thilofophy. loi

Slit in it j fo that the Box may be made faft at any defir�d Height, by the Help of Screwsnbsp;join�d to the Frame, and moveable in the Slits.

The whole Machine is placed at the Diflance of ly, zo, or 30 Feet from a white Plane,nbsp;'''^hich Diftance muft be different according tonbsp;the Bignefs of the Plane ; for this Diftance maynbsp;he equal to the Length of the Plane; Thenbsp;Box muft be juft at fuch an Height that thenbsp;Tubes, in the Side of the Box, may be exaftlynbsp;oppoftte to the Middle of the Plane.

The Lamp being lighted, the Box muft be ftiut, and the Figures which are painted uponnbsp;theGlafs will be reprefented upon a white Plane.

By moving backwards and forwards the Tube that has the two Lenfes in it, you will find thenbsp;proper Pofition of the Glafies requir�d to givenbsp;a. diftinft Reprefentation. As for the Difpoft-tion of the feveral Parts of the Machine, whichnbsp;immediately ferve for exhibiting this Appearance, we fhall here more particularly explain.

PlateXlV. Fig. z.] The Parts in this Figure 838 are fhewn feparately j SS is the Mirror, 11 thenbsp;Blame which confifts of four Flames in the Linenbsp;^ ^ gt; V V is the Glafs V of the firft Figure jnbsp;00 is a Pi6lure painted upon a flat thin Piecenbsp;of Glafs �, a a the biggeft Lens �, d d the flatteftnbsp;Lens} bb the wooden Stop between the Lenfes,;nbsp;� the Aperture or Hole in the Middle of thenbsp;Wooden Circle.

Thefc Things being difpos�d as has been al-gt; ready explain�d, and as may be feen in the Figure, tlie Rays which proceed frortva Point ofnbsp;the Picture � O, by going thro� the Lens a a,nbsp;l^ocome lefs diverging, and fall upon the Lensnbsp;^d.y as if they came from a Point more remote^; * 669nbsp;from this Lens they go out more converging*, * ^99nbsp;are colleded upon the Surface of the whitenbsp;H 5nbsp;nbsp;nbsp;nbsp;Plane,

�02 Mathematical Elements Book III,

Plane, where they exhibit the Point of the Pi-fture that is painted on the Glafs *. This Picture is illuminated both by the Rays that proceed from the Flame I /, and by the Rays re-? flcdted by the Mirror S S.

For the Perfeftioh of this Machine it is requir�d, i/, That the Figure O O be enlighten�d as much as poffible ; zri'T)', That it be equallynbsp;enlighten�d in all its Points ; 3^/^', That all thenbsp;Light, by which every Point of the Pifturc isnbsp;enUghten�d, go thro� the Lenfes a a and d d tonbsp;th� white Plane, and ferve to make the Re-pr�fentation gt; ^hly^ That no other I-ight butnbsp;that go out of the Box, left the Reprefentationnbsp;fhou�d be leis lively, by reafon of extraneousnbsp;Light.

The firft Requifite depends upon the Bignefs of the Flame and of the Mirror, and of itsnbsp;Concavity j the more concave it is, the nearernbsp;it is to be brought to the Flame, and then thenbsp;more Rays will be intercepted and reflefted jnbsp;yet Care raiifl; be taken that the Mirror (whichnbsp;may be made of very good Glafs) be not toonbsp;much heated.

When the Flame and Mirror are fo contriv�d, that the Pidture is the moft enlighten�d that itnbsp;can poffibly be, and every where equally enlighten�d, the Flame and Mirror muft be fonbsp;plac�d, that the inverted Image of the Flamenbsp;^g^glhall fall juft upon the Pifture*. Now, as thenbsp;* Reprefentation of the Flame can be increas�dnbsp;j, 026 diminifh�d *, the Mirror and Flame muftnbsp;be fo difpoS�d, that the Reprefentation of thenbsp;Flame (hall cover the whole Pidlure upon thenbsp;Glafs, but fo as not to exceed it : For then thenbsp;Pidlure is as much enlighten�d by the refledtednbsp;Light as it can be, and all its Points are equallynbsp;illuininated 3 the diredt Light alfo docs pretty

near

Book III. of Natural Thtlofophy. 105

near equally fall upon all the Points of the Figure } this Light wou�d indeed be increas�d by btinging the Flame nearer j but the reflcftednbsp;Light wou�d be diminiflied, and the Diminutionnbsp;of this laft wou�d be greater than the Increafenbsp;of the other.

The Glafs V V ferves to infle�: the Light, by which the Pifture O O is illuminated, before it comes to the Pidure j by which Inflexion all the Light comes to the Lens a,a^ andnbsp;ferves for the Reprefentation on the white Plane.

All the Light, that is of Ufe for this Reprefentation, goes thro� the Hole � j and the Rays, coming from different Points, interfeft one another there j wherefore the Pidlure upon thenbsp;Glafs, which is placed inverted, is reprefentednbsp;erect upon the white Plane j by the Ring bb allnbsp;the Rays, which do not fcrve to form the Reprefentation, are intercepted, left theyftiou�d enternbsp;the Room, and make the Picture lefs diftin�t.

This Ring or Aperture alfo intercepts thofcRays by which one Point is more enlighten�d thannbsp;another, whereby the Light, which (from whatnbsp;has been faid) is equally enough diffus�d, is yetnbsp;Kiade more equal. But unlefs the Stop or A-perture b b he juft where the Rays interfc�t, itnbsp;does a great deal of Mifchief.

BOOK

f �4 Mathematical Elements Book III,

CHAP. XVII.

ODIES that tranfinit the Light are ca/W Tranfparent. Such arcnbsp;all Mediums*,nbsp;nbsp;nbsp;nbsp;2. Facuum.

Lhere is no Body to hofe fmallefi Parts are not tranfparent: No onenbsp;who is' ufed to Microfcopes willnbsp;doubt of this. There ^re fome Parts of Metals,nbsp;which, tho very frnall, do not tranfmit the Light;nbsp;But if they be difTolv�d in Menflrutms �, that is,nbsp;if they be divided into much lefs Parts, they become tranfparent. One may alfo prove by a verynbsp;eafy Experiment, that Light can go thro� fevc-;nbsp;ral opaque Bodies.

.-. . .. nbsp;nbsp;nbsp;■-.: m ■

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... -.mtF y»‘-‘‘ ■ nbsp;nbsp;nbsp;-tV. —'•nbsp;nbsp;nbsp;nbsp;.s-

Book III. of Natural Thilofo^hy. lOf

Experiment i.] In a dark Room, in which 'he Sim�s Light comes in thro� an Hole, let thatnbsp;Hole be cover�d with a thin Plate of an opaquenbsp;Body, and the Light will go thro� it; a Piecenbsp;of Wood, of theThicknefs of the tenth Part ofnbsp;Inch, does not intercept all the Light. Butnbsp;the perfcft Tranfparency of Parts in opaque Bodies is not prov�d by this Experiment, for thatnbsp;Pranfparency is only in very fmall Parts.

Opacity does not (as is commonly iniagin�d) 842 ^^appen in Bodies^ becatife the Wty-) thro' which thenbsp;Eight might pafs^ is flopp'd by Particles of Matter gt;nbsp;Light pafles thro� all the fmallell Parts ofnbsp;Bodies; neither is fuch an Interception of Lightnbsp;of any Ufe to caufe Opacity ; It is requir�d fornbsp;Opacity that the Light Ihou�d be refledfed andnbsp;defle�tcd from a right Line, for which there isnbsp;only requir�d the Separation of two Mediums *nbsp;nbsp;nbsp;nbsp;_nbsp;nbsp;nbsp;nbsp;*631

Let us conceive a Body, conlllling of very 79^ fmall Parts, perfedlly tranfparenc (fuch as are thenbsp;Particles of which Bodies conflft and fepa- *84,nbsp;tated from one another by Pores-, and thatthofenbsp;Interftices are either void, or fill�d with a Medium whofe Denfity is diferent from that of thenbsp;Particles j if Light enters fuch a Body, it willnbsp;every Moment fall upon a Stirfice dividing Mediums differing in Denfity; therefore it will undergo innumerable Refledlions and Refractionsnbsp;|n that Body*, fo as not to be able to get thro� * 631nbsp;'t- Therefore we fee that Opacity depends upon, 79^nbsp;the Pores j for if you fill the Pores with a Medium onbsp;of the fame Denfity as the Particles of the Bodiesnbsp;themfehes^ the Light will undergo no Reflectionnbsp;pr Refraftion in the Body, but pafs directlynbsp;tfixo�nbsp;nbsp;nbsp;nbsp;the Body will be tranfparcnt.nbsp;nbsp;nbsp;nbsp;* 841

Tho697

�o6 Mathematical Elements Book III.

Tho we cannot make Experiments, whereby to fill the Pores of the Body with a Medium ex-aftly of the fame Denfity as the Particles, yetnbsp;the following Experiments will clearly enoughnbsp;prove Sir Jfaac Newton's Dodlrine concerningnbsp;Opacity.

Experiment z.] Paper becomes more tranfpa-rcnt, when tnoillen�d with Water j for it fill* the Pores, and differs lefs in Denfity from the Particles than Air does. Oil has the fame Effect.

Experiment 3.] Take a Piece of Glals two Inches thick ; and take feveral Plates of thenbsp;lame Sort of Glafs laid upon one another, yetnbsp;fo as not to be quite two Inches thick gt; andnbsp;you will find that thefe will be le.fs tranfparcncnbsp;than the folid Piece, becaufe of the Airbetweennbsp;the Plates, which does not get into the folidnbsp;Piece where all the Parts cohere.

Experiment 4] Take twelve Plates of the fame Glafs, as near as may be of the famenbsp;Thicknefs j let fix and fix of them be laid together i if you take the leaft tranfparent of tholenbsp;two Particles, and, having dipp�d it in Water,nbsp;take it out again, it will become more tranfparentnbsp;than the other j becaufe the Water, which innbsp;that Cafe fills the Interftices between the Planes,nbsp;differs lefs in Denfity from the Glals than Airnbsp;does.

What has been faid of Opacity is farther confirm�d, and put out of all Doubt by innumerable 844 E-xperiments, by which Bodies perfeSlly tranfparentnbsp;become opaque^ by the Separation of their Parts^nbsp;without the Intervention of any opaque Body.

Book III. of Natural Thtlofophy. nbsp;nbsp;nbsp;107

Experiment f.] Let any perfeftly tranfparent Liquid, that may be chang�d into Froth, benbsp;^ak�d, till it be full of Bubbles j it will imme-^'!itely become opaque, by reafon of the Inter-ftices that are fill�d with Air.

Experiment 6Turpentine and Water are ti'anfparenc Bodies, but, when mix�d, they become opaque.

Experiment 7.] Water and Oyl, by being mix�d together, become opaque, tho, fingly, they arenbsp;tranfparent.

Experiment 8.3 Tho Glafs be tranfparent, yet if it be reduced to Powder it becomes o-paquej as it alfo does when it is crack�d.

We clearly fee in all thefe Cafes that Opacity is produc�d, becaufe there is a Medium of diffe-*'cnt Denfity between the tranfparent Parcs jnbsp;'^hich may alfo beoblerv�d in the Clouds, whichnbsp;^gt;�6 epaque on account of the Air interpos�d between the Particles of the Water.

If we add to this what is faid in the zzd Chapter following, concerning the Colours ofnbsp;thin Plates, we fhall have new Experiments, bynbsp;'^hich alone it is fully prov�d that Bodies intercept the Light, becaufe they confift of verynbsp;thin Particles, encompafs�d with a Medium differing from them in Denfity.

5nd the reft of the Light, by innumerable Di-''ifions which it undergoes in the Refledtions and ^efraftions above-mention�d, is extinguifh�d innbsp;the Body j fuch are black Bodies j if there werenbsp;^ty Bodies perfetlly blacky they 'wou'd refleSi no ^4rnbsp;for all Bodies, when no Light falls uponnbsp;5nbsp;nbsp;nbsp;nbsp;them,

io8 Mathematical Elements Bpok III.

Other opaque Bodies appear to have various Colours } fome tranfparent Bodies are alfo ting�dnbsp;with Colours ; And whence thofe anfe we aramp;nbsp;now to examine.

CHAR XVIIL

Bodies appear differently colour�d, tho� they be enlighten�d by the fame Rays ofnbsp;the Sun which are reflefted by them. Befidesnbsp;thefe there are feveral Phenomena of Light, relating to Colours, not to be overlook�d.

In thefe we are to obferve three Things, i/. The Rays are to be examin�d, zdly. Theirnbsp;Refledion is to be obferv�d.nbsp;nbsp;nbsp;nbsp;We are to

847 As to the Rays, the firft Property, to be taken Notice of here, is, that in the fame Cir-cumftance all Sorts of Rays do not undergo the fame Refradtion.

848 ^he Rays, which undergo this different Ref ration, are faid to have a different Refrangibility, and thofe are faid to be moft refrangible, whichnbsp;are mofl inflected by Refrailion.

Thofe Rays are faid to be Homogeneous, which do not differ from one another in Refrangibility.

Book III. of Natural Thilojb^^hy. 109

And thofe are call'd Heterogeneous, which^ un-der the fame Circimftances^ are not equally infleUed h Refra�ion.

Plate XV. Fig. i.] Between A B and C D let there be a Sun Beam made up of an innumerable Quantity of parallel Rays j thefe are notnbsp;equally refradled j for if they fall obliquely onnbsp;the Surface B D of the denfer Medium, fomenbsp;�f them are refradted between B E and D G,nbsp;and are mov�d according to that Direction innbsp;the denfer Medium �, others are more infledted,nbsp;and diredt their Motion between B F and D Hnbsp;according to the Diredtion of thofe Lines j andnbsp;indeed no Diredtion can be conceiv�d betweennbsp;the Mediums, along which the Rays do not movenbsp;in every Point between B and D: So that the Icafbnbsp;Beam whatever is, by Refradtion, divided into annbsp;innumerable Quantity of Rays j bccaufe everynbsp;Sun-Beam, however fmall, is heterogeneous, andnbsp;confifts of an innumerable Quantity of Rays refrangible in all Degrees of Refrangibility.

The above-mention�d parallel Rays falling Syi npon a plane Surface, -by Refradtion, are mov�dnbsp;between B E and D H which Lines divergenbsp;from each other, and, oeing continued, are morenbsp;^nd more feparated ; fo that the Rays above-ttiention�d are dilpcrs�d by Refradtion. ^n N�.

We have conftder'd the Rays as homogeneal.^ alfo every where in the Foregoing Part; the Difference of Refrangibility is fmall enough in thenbsp;Rays of the Sun, not to have been worth ob-ferving in the foregoing Propofitions. Befides,

�vvc were firft to examine what happens in ho-Oiogeneal Rays j and what mufi; be chang�d in *^he Propofitions, upon account of the differentnbsp;Refrangibility, will eafily appear.

, Mathematical Elements Book Ilk

That this Refrangibility of the Rays may be made vifible, the abovc-mention�d Divergencenbsp;muft be increas�d j which will be perform�d, ifnbsp;the Rays above-mention�d fall upon the Surfacenbsp;E H, which terminates a denfer Medium, andnbsp;feparates it from the rarer Medium, and whichnbsp;makes any Angle with the Surface B D, and isnbsp;fo inclin�d to it, that the Rays, which are morenbsp;refrangible, fall more obliquely upon itthanthof�nbsp;that are lels refrangible j fo that the former, goingnbsp;into a rarer Medium, are more turn�d out of thenbsp;Way, and diverge more from the others upon anbsp;double Account, that is, both upon the Accountnbsp;of a greater Refraqgibility, and of a greater Inclination. The Rays between B E and D Gjnbsp;which are the leaft refrangible, being refradtednbsp;alecondTime, continue their Motion betweennbsp;EI and G L, the others between F M and H N:nbsp;In which Cafe, if thefe Rays fall upon a Planenbsp;at the Diftance of ip or zo Foot, thofe thatnbsp;are moft refrangible will be feparated from tholenbsp;that are leaft refrangible, and the whole intermediate Space is enlighten�d with Rays endow�dnbsp;with a mean Refrangibility.

Experiment t. PlateXV. Fig.%1] Let there be an Hole in the Plate of a Metal in the Window-Ihut, of about a Quarter of an Inch, thro� whichnbsp;a Sun-Beam enters the darken�d Room ; Letnbsp;this Beam fall upon a triangular Prifm of Glalsnbsp;A A, in fuch Manner, that it may be wholly re-fledted by the lower Surface (fee the Exper- ofnbsp;N�. /Sp.) By the two Refra�iions which thenbsp;Light undergoes, Rays that are differently refrangible do not diverge, and, being refleded,nbsp;come upon a fecond Prifm B B, which is likCquot;'nbsp;wife of Glafs, and triangular, in the fame Manner, as if they had come dircdly from the Sun-�nbsp;nbsp;nbsp;nbsp;Thefe

Book III. of Natural�Fhilofophy, m

Thefc two Prifms are moveable about their Axes, as we before explain�d * j the fiill A A * 197nbsp;is laid upon a Piece of Wood faftcn�d to it, yetnbsp;fo as not to hinder it from turning about itsnbsp;Axis j this Piece of Wood is fatten�d toaThrce-legg�d Staff with a Ball and Socket, fuch as isnbsp;Us�d in practical Geometry. The fecond Prifmnbsp;B B is laid on a Frame or Stand S, in whofeop-pofite Sides there are feveral Slits that anfwernbsp;one another ; by thefe the Prifm is futtain�d atnbsp;different Heights, but always horizontally, thenbsp;End of the Axis being plac�d in corrcfpondentnbsp;Slits.

The Light, that comes to the Prifm B B, moves perpendicular to its Axis, and paffes thro�,nbsp;�ts is demonttrated in Fig. i. in which B D andnbsp;E H reprefent the Sides of the Prifm ; the Lightnbsp;is alfo equally inclin�d to each Side ; which willnbsp;happen by moving the Prifm about its Axis; fornbsp;the Light of the Sun (as you move the Prifm)nbsp;will afcend, and then defcend again ; and thenbsp;Situation requir�d is that in which the Light isnbsp;highett of all : Now both the Prifms are to benbsp;fo difpos�d, that in this Cafe the Light may gonbsp;horizontally out of the Prifm B B. This hori-2tontal Beam, at the Diftance of i y or zo Foot,nbsp;mutt fall upon the Board T which is cover�dnbsp;quot;With white Paper, and has fuch a Foot, thatnbsp;Upon it, it may be rais�d and fix�d at differentnbsp;Heights. The Rays come diverging to thenbsp;Eaper, and upon it form the Oblong Imagenbsp;B. V, terminated at the Sides by parallel Lines,nbsp;hut (emicircular at R and V.

If the Rays of the Sun, that go thro� the mund Hole, fall upon a Plane at a certain Di-ftance, they will form a bright round Spot, fonbsp;much the greater, as the Plane is more diftantnbsp;from the Hole j which arifesfrom the Rays that

iti Mathematical Elements Book Ilf.

come from the Sides of the Sun, which making an Angle with thofe that come from its Center to the Hole, and inteifelt;5ting them in thenbsp;Hole, gives the Image of the Sun upon thenbsp;Plane.

If the Rays did not go thro� the Prifm B B, and fall upon the Plane at the Diftance of thenbsp;Board Tj the Image of the Sun wou�d have itsnbsp;Diameter equal to the Breadth of the Imagenbsp;V R j which Breadth^ is not alter�d by the Re-8f 5 fraftion, becaufe the Rays enter the Prifm perpendicular to its Axis, and, in Refpeft of thenbsp;Breadth of the Image, are not inclin�d to it. Butnbsp;as, in another Refpeft, the Image of the Sun isnbsp;oblong.^ it follows plainly from thence, that allnbsp;the Rays have not undergone the fame Re-fmftion } for homogeneous Rays, tho refracted,nbsp;will give a round Image of the Sun. The leaft refrangible Rays go to R, and the moft refrangible to V} and the whole Image R V is terminated with Semicircles at R and V, becaufe thenbsp;whole Imagenbsp;nbsp;nbsp;nbsp;of Circular Images. NovV

between R and V there are innumerable Qtian-tities of circular Images, made by Rays of alt poffible Refrangibilities ; otherwife the Imagenbsp;R V wou�d not be terminated with rectilinearnbsp;Sides.

8f4 In moft Experiments, we have faid that the Light is let into the darkened Room thro� a Slitnbsp;or an Hole, which we leave to the Contrivancenbsp;of the Workman} our Method was the following.

Plate'XY. Fig. q.j We made an Hole 4 Inches fquareinthe Window-ftiut, which on three Sidesnbsp;had Ledges of Wood A B, BC, CD, forabbet-ted, as to make Grooves to keep in the fquarenbsp;flat Piece of Wood (^L, which is fix Inchesnbsp;long, and fix Inches wide. It may be drawn out,

Book III. of Natural Thilofo^hy.

and feveral Sorts of them ferve for feveral Experiments. That which we ufe in the following Experiment has a Hollow and a Hole in thenbsp;Middle behind to contain a convex Lens, whichnbsp;js the Objeft Glals of a Telefcopc of i6 or zonbsp;Foot j the round Hole in the Middle mark�d �nbsp;is above half an Inch in Diameter, thro� whichnbsp;the Rays of the Sun, paffing thro� theGlafs, enter the Room. In the Fore-part alfo the Boardnbsp;Ois hollow�d, but not in the Middle thisnbsp;Hollow contains the brals round Plate L, whichnbsp;towards its Circumference has an Hol� at �,nbsp;''^hich Hole is equal to the Hole in the Boardnbsp;Q_itfelf, and agrees with it. Together withLnbsp;there is another concentric Plate, which is le%nbsp;3nd moveable about the Center gt; this Plate hasnbsp;feveral unequal Holes, which fucceffively comenbsp;to � as the Plate is turn�d round j fo that yoiinbsp;may at Pleafure let the Light enter a darknbsp;Place thro� a bigger or a lelTer Hole } whichnbsp;in many Experiments, that are made withoutnbsp;the Lens above-mention�d, is of good UfCinbsp;yhamp;Pin join�d with the Plate, ferves to turnnbsp;it by.

Experiment z.] Let in a Sun-Beam thro� th� Lens above-mention�d, thro� an Hole a Quarter

an Inch wide, into the darken�d Room j by the Rays thus let in, at the Diftance where pa-tallel Rays are colleded, the Sun is very exaftlynbsp;teprefented, and its Image terminated with ex-^ly defcrib�d Bounds. For the Rays that comenbsp;from the feveral Points of the Sun, which, bynbsp;teafon of its immenfe Diflance, may be look�dnbsp;tipon as parallel, are collefted in one Point atnbsp;that Diftance.

Plate XV. Fig. a.] Now if with thefe Rays you make the Experiment above-mention�dj the

VoL. II. nbsp;nbsp;nbsp;Inbsp;nbsp;nbsp;nbsp;feveral

114 Mathematical Elements Book IIL

feveral Images made by the homogeneal Rays, fuppofing the Board at a juft Diftance, are ex-aftly terminated j and therefore the oblong I-mage R V, which is made of all thefe Images,nbsp;is likewife regularly terminated.

This Experiment will fucceed in the fame Manner, if the Rays pafs thro� a Prifm made ofnbsp;any Body which is denier than Air.

Experiment 3. Plate XV. Fig. 2.] Let a triangular Prifm be made of Wood and two glafs Planes, and fill�d with Water} fuch as is re-prefented at B B {Plate XVII. Fig. 3.) If younbsp;make ule of this Machine inftead of the Prifmnbsp;B B in this Fig. the Experiment will fucceed innbsp;the fame Manner j and, in palfing thro�Water,nbsp;heterogeneous Rays in the lame Manner are fe-parated by Refra^ions.

Experiment 4.] If any Perfon, Handing i y or 20 Foot olF, looks at the Hole thro� which thenbsp;Light is let in, it will appear round j but if beforenbsp;his Eyes he hold a triangular glals Prifm, ornbsp;the watery one mention�d in the laft Experiment,nbsp;fo that the Rays coming from the Hole (afternbsp;fuch Refra�bions as the Light fulFers in the foregoing Experiment) reach the Eyes, the Holenbsp;will appear oblong. The Situation of the Prifmnbsp;will be found, if holding it horizontally withnbsp;one Edge (or the refracting Angle) upwards, itnbsp;be a little mov�d about its Axis, fo as to makenbsp;the Image of the Hole to afeend and defeend }nbsp;and the Prifm be held fall in that Pofition innbsp;which the Hole appears moll deprels�d.

This Experiment, as well as the foregoing ones, proves the different Refrangibility of thenbsp;Rays} for by the homogeneal Rays of eachnbsp;Refraction, the Hole appears in the imaginarynbsp;Foci of the Rays coming from the feveral Points

Book III. of Natural Thilofoj^hy. 115'

oftheHoIe*, which Image is round; theRays,*737 which undergo a different Refraction, enter thenbsp;Eyes in different Direfbions ; and you have fevc-ral linages, all which form the oblong Image,nbsp;which is really feen.

But that this diffeYent Refrangibility does notde-pend upon the refraPling Medium^ but the different Confiitution of the Rays themfelves, is prov�d, bc-caufe thofe Rays, which in one Cafe undergonbsp;the greateft Refraftion, are in any Refradfionnbsp;turn�d out of the Way more than any.

Experiment f. Plate XV. Fig. 4.] All Things being difpos�d, as in the firft Experiment, at anynbsp;Diftance from the Prifm B B } let the oblongnbsp;Image of the Sun fall upon the Vertical Prifmnbsp;C C, which is alfo moveable about its Axis j asnbsp;the Rays go thro� k, let it be turn�d about itsnbsp;Axis, and left fix�d where the Rays are leak ofnbsp;all turn�d out of the Way by Refra�lion thro�nbsp;the Prifm : In that Cafe the Rays are refraftednbsp;in the fame Manner thro� this Prifm as thro� thenbsp;firfl:, but they are not difpos�d the fame Wa)^,nbsp;for that Wou�d form a fquare Image. But herenbsp;the Image, keeping the fame Breadth, is inclin�dnbsp;at R V, the Rays at V being moftly turn�d outnbsp;of the Way, as in the R.efradrion thro� the firftnbsp;Prifm B B.

�the Demonf rat ion.^ before given % of the con-6 flant and fettled Ratio between the Sines of the An- * 639nbsp;gles of Incidence and RefraSlion.) may be referred tonbsp;any homogeneous Rays but confidering the different Refrangibility of the Rays, this Proportionnbsp;Varies, as it follows from the Experiments ofnbsp;this Chapter.

But the Refrangibility in all the, different Sorts of Rays is every M-^ay unchangeable.^ as will appearnbsp;by the Experiments to be mention�d hereafter.

CHAP. XIX.

JL along with the different Colours and every Sort of Rays, as they are more or lefs infledled bynbsp;Refraction, have a particular Colour of their own,nbsp;and which is wholly unchangeable.

In Refpe�t of the Colours, the fame Thing is to be obferv�d, as has been taken Notice of innbsp;502 regard to other Senfations * j the Colours arenbsp;^ Ideas which have nothing common with thenbsp;Rays, by which they are excited: Therefore wenbsp;muft define what we mean by colour�d Rays andnbsp;colour�d Objefts.

jin Object is faid to be of fuch a Colour, whofe Idea is excited in the Mind by the Rays reflectednbsp;from the Object.

Homogeneous Rays, which, flriking upon the Re-tina, excite in our Mind an Idea of any Colour, are call'd Rays of that Colour.

We have faid that the Rays excite an Idea, by which we underftand that the Rays put thenbsp;Fibres into a trembling Motion, and that Motion occafions an Idea in the Mind.

Plate XVI. Fig. i.] It is very evident from the Experiments of the foregoing Chapter, thatnbsp;Rays of different Refrangibility have a different Colour ; for thofe Experiments fhew thatnbsp;the Image of the Sun, which is oblong, is

,J ..' ., nbsp;nbsp;nbsp;.

. -.-^'••* • ' ^^‘•v »i

• - nbsp;nbsp;nbsp;- •i'tPtv ‘ •■ -t'*' ■•■■-••

Book III. of Natural Thilofophy. 117

tinged with different Colours. The Rays^ which 861 are leaji turn�d out of the Way by Refra�lion^ arenbsp;Ted j the rcfi of the Colours are in the following Order y namely Orange, Yellow, Green, Blue, Purple,nbsp;andFiolet, oivsh 'ich laft Colours are the Rays thatnbsp;are moft refrafted The oblong Inlage of thenbsp;Sun above-mention�d, as has been faid *, isnbsp;made up of an innumerable Qtiantity of roundnbsp;Images j if their Diameters bediminiih�d (whichnbsp;is done by intercepting the Sun�s Rays, fo thatnbsp;only thofe that come from the Center of thenbsp;Sun fhall pafs thro� the Prifm) the Centers ofnbsp;the particular Images which make up the oblong Image will not be chang�d j and thereforenbsp;the Length of the Image a h, between parallelnbsp;Lines, is not chang�d : And this alone wou�dnbsp;remain the fame, if the Breadth of the Image,nbsp;fliou�d be infinitely diminifh�d : And thereforenbsp;this Length alone is to be confider�d in determining the Limits of the Colours in the Imagenbsp;itfelf 5 thefe Colours are reprefented in this Figure by the Letters a, b, c, d, e,f,g, h, and thenbsp;Number which is fet down againft each Colournbsp;denotes the Space taken up by it in the Image,nbsp;the whole Lenth of the Image being dividednbsp;into 360 Parts.

If the Breadth of the oblong Image of the Sun be diminijlo'd, the heterogeneous Colours are more fepara-ted in the Image �, becaule there are fewer particular Images confounded in the feveral Points,nbsp;where Rays of different Refrangibilitics differnbsp;little from each other.

The Colour of any Ray, as alfo its Refrangibility, o , cannot be changed by any RefraStions, or Reflexions, ^nbsp;or mingling of the Rays together.

In this Chapter I lliall fpeak of Reflexion and Refraftion, and of the Mixture of Rays innbsp;the following one.

I�8 Mathematical Elements Book IIT.

8164 That the Refrangibility is not chang�d, by Re-fraftion, is prov�d by the ph Experiment of the foregoing Chapter j which may alfo be referr�dnbsp;to Colour ; but is more clearly evinc�d by thenbsp;following Experiment. Concerning whiph it isnbsp;to be obl�rv�d (as may be alfo faid of what follows) that the Experiments are to be made withnbsp;Prifms of clear Glafs, free from Veins j for theynbsp;occafion the Eight to move irregularly i and thenbsp;Rays are not duly feparated by theRefraftions.

Experiment i. Plate XVI. z.J Every Thing being as in the fit ft Experiment of the foregoingnbsp;Chapter, you muft make the Experiment withnbsp;a Sun-Beam, going thro� an Hole of half annbsp;Inch Diameter: The Frame or Stand S, in whichnbsp;the Prifm is, muft be fuch, that the little flatnbsp;Board t, to be us�d in this Experiment, maynbsp;move between its Sides j this Board has an Holenbsp;F of T of an Inch Bore, thro� which the Lightnbsp;refrafted by the Prifm is tranfmitted, wherebynbsp;the Rays in the oblong Image are better feparatednbsp;from one another, the Rays being moftly intercepted 5 this oblong Image R V, at the Diftancenbsp;of 10 or I z Feet from the Prifm, falls upon thenbsp;Board r of a Stand like the former ; in whichnbsp;Board alfo there is a fmall Hole F like that innbsp;the firft 5 thro� this the Rays pafs upon a fecondnbsp;Prifm laid upon this Stand, and are refrafted innbsp;the fame Manner as in the firft. By moving thenbsp;firft Prifm a little about its Axis, the Image RVnbsp;afeends or defeends, whereby the different Raysnbsp;are fucceffively tranfmitted thro� the Hole ^ in every Cafe, the Rays refra�led thro� the fecond Prifm,nbsp;and ftriking upon the flat Board T which is cover�d with white Paper, are not difpers�d at H 5nbsp;but the Image is round, if the Rays fall perpendicularly upon the Paper, and alfo of the fame

Colour

Book III. of Natural�PhiloJbjgt;hy. 119

Colour as the Rays falling upon the fecond Prifm. Yet the Image H is fo much the morenbsp;lifted up, as the Rays, by Refraftion thro� thenbsp;firft Prifm, are more turn�d out of the Way j thatnbsp;is, thofe, that are moft refrangible in the one Cafe,nbsp;do alfo liiffer the greateftRefraftion in the other.

It will appear alfo by the following Experi- B�f ments, that the Refrangibility and Colour arenbsp;not chang�d by Reflexion.

Experiment 2.] The Rays which, for Example, make the red Part of the oblong colour�d Image, whatever Body they arc reflefted from,nbsp;are always red j that is, all Bodies appear rednbsp;in that Light; In the violet Light they arenbsp;violet i green in the green j and fo on in thenbsp;other Colours.

This will appear by trying it with Vermillion, Orpiment, blue Bice, or Cloth of any Colour,

Experiment Plate quot;KMl. Fig. 3.] Let the Light enter the dark Room thro� two Holes ofnbsp;a Quarter of an Inch Diameter each, in thenbsp;round Plate moveable in the Window } letthefenbsp;Beams be about two Inches afunder, and refle-�ted by the plane Mirror S.

The Plate and Mirror muft be fo difpos�d that the two Beams may fall upon two Prifmsnbsp;A A, which are laid horizontally upon the famenbsp;Frame 5 fo that the oblong Image, produc�dnbsp;by the Refraftion of thofe Prifms, may touchnbsp;one another at their Sides. Let one Prifm benbsp;turn�d a little about its Axis, that the Red ofnbsp;one Image may be juft on the Side of the Violetnbsp;of the other : Let thefe Colours, and none ofnbsp;the reft, be intercepted by a wooden Ruler thatnbsp;has a white Paper pafted upon it, fo that the

I 4 nbsp;nbsp;nbsp;Red

Ito Mathematical Elements Book III.

Red be at R, and the Violet at V, the reft of each Image falling upon the Wall, which muftnbsp;be cover�d with a black Cloth. IfanyPerfonnbsp;ftand at O, and thro� the Prifm B B looks atnbsp;thofe Colours, R and V, (in the Manner de-fcrib�d concerning the Hole, in Experiment 4.nbsp;of the foregoing Chapter ) he will fee the Colours fcparated from one another, the Red atnbsp;r, and the Violet at a; j which therefore, beingnbsp;refle�tcd in going thro� the Prifm B B, undergoes a greater Refra�lion.'

In the firft Experiment of this Chapter we gave a Method, whereby to feparate the heterogeneous Rays better than in other Experi-8(S6 ments j in the following Experiments, Lightsnbsp;of divers Colours become much more homogeneous, which is requir�d in the ftxth Experimentnbsp;of this Chapter.

Experiment Plate XVII. Fig. i.] Let the Sun�s Light enter the dark Room thro� an Holenbsp;of .5. of an Inch 5 the Sun Beam muft be reftedt-ed by the Prifm A A to the convex Lens V,nbsp;which ftands upon a Foot, and is about ? or 4nbsp;Inches wide �, the Length of the incident andnbsp;reflcdled Rays, taken together, is about fevennbsp;Feet ; the Convexity of the Lens is fuch, thatnbsp;the Foci of the Rays coming from the feveralnbsp;Parts of the Hole, at the Diftance of 10 or linbsp;Feet from the Lens, will form the Reprefenta-�7,i tion of the Hole, if a Paper be held there*.nbsp;Juft beyond the Lens place the Prifm B B,nbsp;whereby the Rays (as in the firft Experimentnbsp;of the foregoing Chapter) are difpers�d ; nownbsp;placing the flat Board T, at the Diftance atnbsp;which the Rays, coming from the feveral Points,nbsp;are colledted, there will fall upon it a wellnbsp;tarainated obleng golour�d linage R V, whofe

Book III, of Natural Thilofophy. izr

Length will have a greater Proportion to its Breadth than in other Experiments j and whichnbsp;therefore is made up of Rays more homogeneous and fo much the lefs mix�d, as the Hole * 862nbsp;in the Window is the lefs. By moving forwardnbsp;or backward the Board T, one may find thenbsp;Difiance at which the Image is moft difiindtnbsp;and terminated without the Penumbra.

Now that the Rays, feparated in this Experiment, are homogeneous enough, isprov�d ironi this, that they can be no more difpers�d by anbsp;new Refraftion : And that this is the Propertynbsp;of homogeneous Rays, follows from their equalnbsp;Refrangibility ; which has been already confirm�d by an Experiment and is more fully * 864nbsp;demonftrated by the following one.

Experiment 6Take two Circles of Paper of an Inch Diameter each, and let Light fall upon them in fiich a Manner, chat the one maynbsp;have the homogeneous Rays of any Colournbsp;fitll upon it, and the other may be enlighten�dnbsp;by the Rays of the Sun j if both thefe Papers benbsp;look�d at thro�the Prifm at the Difiance of fomenbsp;Feet, as in the third Experiment of this Chapter, the Circle, enlighten�d by the heterogeneousnbsp;Light, appears oblong, and ting�d with different Colours, as in the fourth Experiment of thenbsp;foregoing Chapter 5 but neither the Colour nornbsp;Figure of the other Circle is chang�d.

Experiment 6-3 Upon a white Paper draw black Lines parallel to one another, and aboutnbsp;the fixteenth Parc of an Inch wide � let thefe benbsp;enlighten�d by throwing upon the Paper the oblong Image, which is deferib�d in the fourthnbsp;Experiment, fo that the Lines may lie long-wifenbsp;in the Image. Befides this you mufl have a convey

I z ^ nbsp;nbsp;nbsp;Mathematical Elements Bo ok III.

vex Lens, about five or fix Inches wide, fu-ftain�d upon a Foot, fuch as are reprefented at V {Plate XIX. 2.) which colledls the red Raysnbsp;that come from a radiant Point, fix Foot difiantnbsp;from the Glafs, at an equal Diltance on thenbsp;other Side. If this I..ens be plac�d at the Di-ftance of fix Feet from the Image above-men-tion�d, the Parts of the black Lines that arenbsp;enlighten�d by the Red falling upon the Paper,nbsp;by means of the Rays which are colledted bynbsp;the Lens, will be exactly reprefented in the rednbsp;Image at the Diltance of fix Feet j but younbsp;mult bring the Paper forward about three Inchesnbsp;and a Half, to make thofe Parts in the Linesnbsp;which are enlighten�d by the Purple, to appearnbsp;dillindt in that Part of the Image which is ofnbsp;the fame Colour : The intermediate Coloursnbsp;give Images at intermediate Diftances3 the Violet is fo weak, that the Threads cannot be rc-prefented in it.

This Experiment does alfo confirm, that the Colour of reflefted Rays is not chang�d by anbsp;new Refradlion thro� the Lens 3 as alfo that thenbsp;molt refrangible Rays are more inflefted thannbsp;the others in palling thro� the Lens.

This laft Experiment does alfo prove, that the ^ different Refrangibility of the Rays is the Caufe thatnbsp;hinders the PerfeSlion of Pelefcopes, For the Focinbsp;of the Points, that are equally dillant, fall atnbsp;different Diftances from the Lens, according tonbsp;their different Colours 5 whence all the Repre-fentationsof the Points quot;are unequally diftantfromnbsp;the Eye-Glafs 3 and therefore they cannot be allnbsp;perfedlly feen thro� it.

868 Concerning the Reflexion of the Rays it is to be obferv�d, .that thofe Rays are more eafily re-fietled totally^ which have the^'eateft Refrangibility 3nbsp;for the greater is the Refradlion of the Rays,

Book III. of Natural Vhilojophy, nbsp;nbsp;nbsp;123

�he lefs Obliqviity is requir�d to make them be totally refleftcd *.nbsp;nbsp;nbsp;nbsp;* 793

Experiment 7. Take a Prifm, plac�d as before in the Experiment where it was obferv�d, *790 that by moving the Prifm about its Axis, thenbsp;I^ays that fitft went thro�, when their Inclination is increas�d, become wholly reflefted gt; butnbsp;if the Prifm be gently mov�d in this Cafe, wenbsp;Iball perceive that the violet Rays are the firfbnbsp;^�hich will be wholly reflefted, then the purplenbsp;Rays, and all the reft in the fame Order, as theynbsp;3tc in the oblong Image of the Sun, fo oftennbsp;tuention�d j which appears, if the reflefted Raysnbsp;tgt;e feparated by the Refradion of the Prifm.

XX.

That the Refrangibllity and Colour ofggp the Rays are not chang�d, by the Mixturenbsp;of Rays of different Refrangibility, has alreadynbsp;been faid*, and now we muff prove it byExpe-timents.

Experiment i. Plate XVI. Fig. 3.] This Experiment muft be made in the fame Manner as the J^hird of the foregoing Chapter 3 here you muffnbsp;tuake the Red R, and the Violet V be confounded and mix�d together, by throwing themnbsp;^ponthe fame Part of the Ruler L L, whichnbsp;thereby will appear of a purple Colour in thatnbsp;�lace. But if a Perfon looks at them thro� anbsp;Prifm, the Colours will appear feparated 3 andnbsp;therefore neither the Colour nor Refrangibilitynbsp;^te chang�d by this Confulion of Colours.

Mathematical Elements Book III.

Experiment z. Plate^Yll. Fig. 2.3 If the oblong colour�d Image of the Sun (ofwhich Mem-tion was made in the firft Experiment of the 18//^ Chap.) falls at RV upon the convex Lensnbsp;L, mention�d in Experiment 4, of the foregoingnbsp;Chap, which muft be placed at fix or fevenFoocnbsp;from the Prifm B B, the divergent Rays willnbsp;converge by the Rcrraftion of the Lens, andnbsp;interfe�l one another at the Diftance of 9 ornbsp;10 Feet at A j if the Board T be placed atnbsp;a greater Dillance, the Rays, which divergenbsp;again after the Interfe6tion, will fill upon it di-fpers�d j and you will again have an oblong colour�d Image j but the Colours by reafon ofnbsp;the Intcrfeftion at A, will appear in a contrarynbsp;Order, and will not be chang�d by having beennbsp;mix�d together at A.

Experiment 3. Plate XVII. Fig. 2.] Every Thing ftill remaining as in the former Experiment, if with a black Paper you intercept fomcnbsp;of the Rays of the Image R V, which changesnbsp;the Mixture (which this Way may be varied innbsp;any Manner) the Colours of the other Rays, thatnbsp;are again feparated, are no Way chang�d.

870 If the Rays of the Sun., as they come to us, are wholly refteSled by any Body, that Body appearsnbsp;White ; but thefc Rays are an Heap, or Parcelnbsp;* 847 of Rays of various Colours * j whence we de-858 ducc, that a Mixture of different Colours makes anbsp;Whitenefs 5 for if the Colours which areobferv�dnbsp;in the oblong Image of the Sun, fo often mention�d, be mix�d and confounded together, innbsp;the fame Proportion as they are in that Image,nbsp;a Whitenefs willbe produced j which alfo proves,nbsp;that in that Refpe�t the Rays are unchangeable.nbsp;The Rays that come from the Sun appear

Book III. of Natural�Philo/b_phy.

white i but, if they are feparated, their Colours are difcover�d ; and, if they be mix�d again, thenbsp;Whitenefs will be reftor�d.

Experiment 4. Plate XVII. Fig. z.J Things being difpos�d as in the two foregoing Experi-inents j let the Board T be plac�d at A, in thenbsp;Very Place where all the Rays of the Image R Vnbsp;are confounded together j there will be aWhite-�iefs at A ; if with a black Paper you interceptnbsp;�he Red of the Image RV, the White will va-niih, and the Colour at A be bluifh j but if younbsp;intercept the violet and blue Rays, the Whitenbsp;becomes reddilh.

Experiment . Plate'KYW.Fig. 3.] Take three triangular Prifms, made of Wood and Plates ofnbsp;Glafs which contain Water, as BB, DD, DDgt;nbsp;the Plates of Glafi in each of them make annbsp;^ngle of about 70 Degrees: The Length of thenbsp;^late is of fix or feven Inches, and their Breadth /nbsp;'f three Inches; thefe Plates are fix�d in anothernbsp;'danner in the Prifms D D, D D, than in the Prifinnbsp;^B, fo as to make their Bafes bigger, that is,nbsp;hefe Prifms are fiiorter and deeper than thenbsp;'ther. Let the Sun�s Rays be refrafted thro� thenbsp;�^rifm B B, as in the 3d Experiment of the i%thnbsp;llhap. And let the oblong Image of the Sun at thenbsp;Pittance of three or four Footfall upon theSur-bice of the Prifm D D, plac�d parallel to thenbsp;Surface of the Prifm BB, out of which theRa)'snbsp;So. In the fecond Prifm the Rays undergonbsp;^ contrary Refraftion than in the firft gt; becaufenbsp;of the Parallelifm abovc-mention�d : And be-oaufe the Edge of the Angle, form�d by theGlafsnbsp;Blates in the Prifm B B, is turn�d upwards, andnbsp;^bat of D D downwards; therefore the firftnbsp;B.efra�tion is deftroy�d by the fecond, and thenbsp;^^ays go out of the Prifm D D parallel to one

1x6 Mathematkal�lerHents Book III.

another, and fall upon R V i for if the Prifms be brought together, fo that the parallel Surfaces may touch one another, the Light will pafsnbsp;thro� a Medium terminated with parallel Planes,nbsp;(which is form�d by the two Prifms join�d together) thro� which Light, of any Degree ofnbsp;Refrangibility, will pafs without Change of Di-* 62S re�lion*. Now the Prifms are feparated, thatnbsp;the heterogeneous Rays may be feparated, before they again become parallel 5 if -thefe colour�d Rays fall upon the third Prifm D D, andnbsp;in palling thro� it undergo a Refraftion like thatnbsp;which they have undergone in palling thro� thenbsp;lirll or fecond Prifm gt; the Rays that go out atnbsp;rv converge, on account of unequal Refra�lionsnbsp;in Rays of diflPerent Colours, and concur at A,nbsp;in which Place alfo Whitenefs will be produc�d,nbsp;as in the foregoing Experiment.

Experiment lt;S.] If an oblong colour�d Image of the Sun be made, after the Manner deferib�dnbsp;in Experiment i. of the 18/^ Ci^ap. and a Per-fon Handing at the Diftance of the Prifm, thatnbsp;refrafts the Light, looks at the Image thro� thenbsp;faid Prifm, as was done in refpe�t of the Holenbsp;in the fourth Experiment of the fame Chap, henbsp;will fee a round and white Image; the fecondnbsp;Refraftion deltroying the firft j for thereby thenbsp;Rays, being again mix�d, enter the Eye, and innbsp;this Cafe make the Objefl: appear white.

Sji 'Ehe Mixture of all the Colours, which are ohfernfd in the oblong Image of the Sun, is not neceffary fornbsp;producing White : The Whitenels of the Sun�snbsp;Rays is a little inclin�d to yellow 5 and thereforenbsp;if Part of the yellow Rays be taken out of thenbsp;Mixture, the White will be the more perfe�t.nbsp;From the Mixture of four or five Colours, in anbsp;Juft Proportion, White will be produc�d,

Primary^ that is, homogeneous, Cvlours^ being 873 Tiiix'd^ produce innumerable Colours^ different fromnbsp;the primary or homogeneous ones. From a Mixturenbsp;of other Colours one may produce a Colournbsp;like that which is homogeneous j but 'when therenbsp;can be no Difference obferv'd by the naked Eye between an homogeneous and a mix'd Colour., one maynbsp;perceive a fenftble Difference thro' a Prifnt.

Experiment 7.^ Thro� a Prifm look at any 874 ftnall Objefts, fuch as Letters upon Paper,nbsp;Plies, and other fuch things; if they be expos�dnbsp;to the common open Light, they will appearnbsp;Confus�d ; but if they be enlighten�d by the ho-tuogeneous Light of the fourth Experiment ofnbsp;this Chapter, they will appear diftindt whennbsp;feen thro� the Prifm.

made an End of confidering ^ ^ what relates to the Rays, whereby Bodies are enlighten�d j before we leave this Sub-jedt, we muft explain a Phasnomenon, whichnbsp;Is too remarkable and common to be pall by innbsp;Silence.

The Iris or Rainho'W is what every Body has '^ften feen ; we rauft explain what is the Caufenbsp;of it, having firft laid down fome Things fornbsp;that Purpofe.

Plate XVIII. Fig. i.] Let there be a denfeMe-diwin encompafs'd 'with a rarer, terminated by the � ' Circle B D F H. Let homogeneous Rays that are pa-t'allei to one another fall upomt,wo,i let AB be onenbsp;of thefe Rays; let the Semidiameter C B benbsp;^'awn and continued to N; it will be perpendicular

t%% Mathematical Elements Book IIL

cular to the Surface feparating the Mediums: ABN therefore is the Angle of Incidence 5nbsp;This Angle is equal to the oppofite vertical Angle CBL, whofeSineisCL, perpendicular to BLnbsp;and going thro� the Center j the Ray is refradtednbsp;*624 towards the Perpendicular�^, and C B M is thenbsp;Angle of Refradlion, whofe Sine is CM drawnnbsp;from C perpendicular to B D: There is the famenbsp;*639 Proportion between fuchLinesas CL and CM*,nbsp;for every Ray, as there is for A B. The Raynbsp;B D does in Part penetrate into the rarer Medium in the Dire�tion D E, and is in Partrefledtednbsp;along D F j and makes the Angle of Reflexionnbsp;f805 CDF equal to the Angle of Incidence *B DCjnbsp;whence B D and D F are equal. The Raynbsp;D F does alfo in Part go out of the denfer Medium along F G, and is in Part reflefted alongnbsp;FHj which in the fime Manner docs in Parcnbsp;go thro� H I, and is in Part refledted : But thisnbsp;Reflexion, and other farther Reflexions andRe-fradtions we fliall not confider gt; they are toonbsp;weak, on account of the feveral Diviflons thatnbsp;the Light has undergone.

Plate XVIII. Fig. i.] The Ray F G, which after one Reflexion goes out of a denfer Medium, makes the Angle GP A with the incidentnbsp;Ray AB, which varies in different incident Rays}nbsp;therefore, tho thefe Rays fhou�d be parallel, theynbsp;will be fcatter'd when they go out., after one Reflexion, as may be feen by the fecond Figure.

The Ray EE, which, being continued, goes thro� the Center C, is not turn�d out of its Waynbsp;*8^06 either by Reflexion or Refradtion *.

As you recede from this Ray, the returning Ray is continually lefs and lefs inclin�d to the incident one. So the Ray D D, which goes outnbsp;of a denfer Medium along d d, and returns a-long the Line, makes a greater Angle with d d,nbsp;2.nbsp;nbsp;nbsp;nbsp;than

Book III. of Natural ^hilojb^hy. nbsp;nbsp;nbsp;ii^

than the intermediate Rays between D D and E E do make with their returning Rays, whichnbsp;go out of the denfer Medium.

Let B B be a Ray, in refpedt of which this 876 Inclination is the leaft of all, that is, whichnbsp;makes the Angle GP A (Fi^. i.) the biggeft ofnbsp;all. Beyond B B the returning Rays are morenbsp;inclin�d to the incident ones j thus A A returnsnbsp;along a a.

From this Difperfion of the returning Rays re- 877 ceding from the denfer Medium, they becomenbsp;Continually weaker and weaker, and their Coternbsp;cannot be perceived throughout the whole Space whichnbsp;they fill^ tho the Colour of the incident Raysnbsp;he vivid, ^he Colour in the returning Rays isnbsp;only fenfible, where the neighbouring Ray s are paral-Icl-i and thofe next to them diverge out very little ; fo that at a great Diftance they are denlenbsp;enough to be perceiv�d. �Fhefe only are faid tobenbsp;efficacious, and will be fuch, when the incidentnbsp;Hays which are near oneanother, being refrafted.nbsp;Concur in the very Point ol Reflexion.

PlateNVlW. Fig.:^l] LetAB, ab, benelgh-houring Rays parallel to one another, falling upon a circular Surface that terminates a denfer Medium} if thefebeingrefraftedalong BD, bd,

^o concur in D the Point of Reflexion, the re-fle�ted ones D F, D/, will make the fame Angles �'vith one another as the incident ones DB,nbsp;therefore the refrafted RaysFG,will be pa- * 627nbsp;tallel * and efficacious*quot;. In this Gale the follow- * ^77nbsp;Ing Method will ferve to determine the Anglenbsp;made by the incident Ray with the returningnbsp;one ; that is, the Angle A P G, which here isnbsp;the greateft of all.

Let us call the Ratio of I to R, that which 378 ts found between the Sines of the Angle of Incidence and Refraftion, jwhen the Light goesnbsp;VoL. II.nbsp;nbsp;nbsp;nbsp;Knbsp;nbsp;nbsp;nbsp;otit

130 Mathematical Elements Book III.

out of a rarer Medium, by which a denfer is encompais�d, into a denfer contain�d in the Circle itfelf. Therefore, having drawn C m perpendicular to ^D, and the Arc �;�with the Center C and Semidiameter Cnbsp;], R :: CL, CM C/, C;� CL�C/=L/,nbsp;CM � C�2 = M�.

Draw B o perpendicular to B L, and alfo B/) perpendicular to B D j and draw hp^ fo that itnbsp;may make a right Angle with Bp j laftly, joinnbsp;together by Lines the Points B, C, and M,nbsp;the Triangles B^o, B CL are fimilar ; for theynbsp;are reftangular, and the Angles o B ^ and CBLnbsp;(the Difference of each of which from a rightnbsp;Angle is the Angle o B C,) are equal.

It may be prov�d in the fame Manner, that the Triangle BMC and B^p are limilar j thenbsp;Triangle Mmn^ which is reftangular at w, isnbsp;alfo fimilar to this, for the Sides M� B p,nbsp;which are perpendicular to the Line B D, arenbsp;parallel j as alfo M m and B becaufe the Linesnbsp;B D, ^ D, are bifefted at M and m into equalnbsp;Parts. Therefore alfo B ^ is the Double of Mnbsp;Wj and Bp the Double oimn.

B Ij, B M : : B 0 = L /, Bp = 2 M � ;: T, 2 R : ; C L, 2 C M, by comparing thefe Proportions with the aforefaid Proportion.

Now fince the Squares of proportional Quantities are themfelves proportional, you will have

BLS CL^ : : BM% 4CM''.

BL�-fCL�gt;:: BC�, BL*�:: BM'�-1-4CM� ^BC�q-3 CM^ BM� = BC�� CMquot; �nbsp;BL X L C��CM�.nbsp;nbsp;nbsp;nbsp;�nbsp;nbsp;nbsp;nbsp;quot;nbsp;nbsp;nbsp;nbsp;By

Book III. of NaturalThilofophy. rji

By fubtrafting the firftand fecondTerm from the third and fourth, (which does not changenbsp;the Proportion) you will have

BCi, BLi, : : ? CMq, LCl�CMa, 3 Ri, Ii � Rij for there is the fame Ratio between C M and L C as between R and I.

If therefore you know the Ratio between R and I, you have the Ratio between the Semidiameter B C, and the Line B Lj which is thenbsp;Sine of the Angle B CL, which Angle thereforenbsp;is given j and by Confequence you have the Arenbsp;B N, and alfo F G, for they are equal.

Having the Sine B L, you have alfo the Sine of the Angle B C M i becaufe (as we have fliewiinbsp;before)

Hence we may eafily find the Arcs NH and Hjp BFj if the former be taken out of the latter,nbsp;and the Remainder be divided into two equalnbsp;Parts, we {hall have, as is well known, the Mea-furc of the Angle A P G.

When the Ratio between I and R varies, the Angle A P G is changed, which therefore becomesnbsp;different, according to the different Refrangibili-ty of the Rays.

If tbe abovefaid Surface be enlightened by hetero~ geneom Rays, as they flow from the Sun, the efficacious Rays of different Colours do not make equalnbsp;�Angles with the incident ones, and fo by the Helpnbsp;ff this Refraction the Colours are feparated.

Plate XVIII. Fig. 4. Experiment i.] Let the Sun�s Light enter the dark Chamber, thro�a Slitnbsp;the moveable round Plate O, and being horizontally reflected from the Looking-glafs S, letnbsp;pafs thro� a Slit in the little Board or Stand T jnbsp;K 2,nbsp;nbsp;nbsp;nbsp;as

Mathematical Elements Book III,

*630 as has been before explain�d.* Take a Phiol exactly cylindric, made of clear Glafs, and fullnbsp;of Water. Let the Beam or Ray at f g fall uponnbsp;the Surface of the Phiol, it will berefra�led in thenbsp;Water towards hi^ and there reflefted, and willnbsp;go out of the Phiol at / m. The Phiol may benbsp;eafily fo placed, as to make thefe Rays efficacious i and, becaufe of the Breadth of the incidentnbsp;Ray or Beam, efficacious Rays of all Coloursnbsp;will go out of the Phiol in the fame Timej fornbsp;they are very little diftant from one another irinbsp;their Incidence. If thefe efficacious Rays fall upon a white Paper, at the Diftance of 4ory Feetnbsp;from the Phiol, they will produce verticalnbsp;or Pillars, of various homogeneous Colours, a-rifing from the efficacious Rays of each Colour jnbsp;if alfo, the Eye be placed any where at the Diftance of fome Feet from the Phiol, as at N, innbsp;thefe efficacious Rays, it will fee in the Phiolnbsp;that Colour whofe Rays enter the Eye, and by anbsp;fucceffive Motion of the Eye, it will perceive all

881 Plate'^Ylll. Fig. f.] But as to thofe Rays, which after a double Refradtion in a denfer Medium emerge, they will be efficacious, if theynbsp;are parallel after the firft Reflexion 3 for thennbsp;FH, � ^, are in the fame Manner inclin�d tonbsp;as BD, bd.f to and therefore fuppofingthenbsp;incident Rays A B, to be parallel, theemer-

In this Cafe is Half of the Difference between the Arcs DE and df, or D B and d b, but their Difference is B ^ �Tid �, if thereforenbsp;this laft be fubtrafted from the other, there willnbsp;remain the Double of the Arc D^, whofe Ti'ipfonbsp;therefore is B If by Lines the Points Djnbsp;and B, be Join�d, the Triangles B E ^ andnbsp;D E will be nmilar, as is known j which there-znbsp;nbsp;nbsp;nbsp;fore

Book III. of Natural Thilofo^hy. 133

fore obtains, tho� the Arcs B D be fo very ftnall, that they may be taken for right Lines.

There is therefore that Ratio between E D and E which obtains between thefe Arcs,nbsp;that is, ED is the third Part of E^, orEBj be-caufe we fuppofe the Arc B ^ to be extremelynbsp;fmali. M D therefore is divided into two equalnbsp;Parts at E; and M E is a third Part of E B.

Now if (as in Fig. 3.) the Triangles B 0 B ^ and M m �, be form�d, M m will be thenbsp;third Part of B and B/gt; the Triple of M � jnbsp;now if, mutatis mutandis., wcapply to this Figurenbsp;what was demonllrated with refpeft to Fig. 3.* * 878nbsp;becaufe in this B � is equal to 3 M �, whofenbsp;Square is 9 M nl, we ihall have

From which Proportion, as was faid of the Arc B N may be difeover�d to which H Gnbsp;is equal j and becaufe in this Cafenbsp;BL, BM : : I, 3 R,

You have alfo the Arc BD, to which (becaufe the Angles of Reflexion are equal to the Angles of Incidence) * D F and F H are equal. *805nbsp;From which Things being found, we maynbsp;cafily deduce the Arcs G F D N, and B H,nbsp;whole half Difference is the Meafure of the Angle H P B, which is form�d by the emergingnbsp;Ray with the incident one j which Angle in thisnbsp;Cafe is the lealt of all that are like it, and is different according to the different Refrangibility ^nbsp;of the Rays. Whence alfo in this Cafe the effi- 00nbsp;cacious Rays of various Colours, fuppofmg the inci- ^nbsp;dent ones paralhl, are feparated after a double Re^nbsp;flexion.

Plate XVIir. Fig. Experiment if] This Experiment is perform�d after the flime Manner as the former, only the Situation of the Phiolmuft

Mathematical Elements Book III.

be a little chang�d, that after two Reflexions in the Phiol, the Rays may come to the Eye ornbsp;Paper towards N.

884 What has been hitherto explain�d, may beap-plied to the Rainbow j for which Phaenomenon it h requir'd^ that Drops of Waterfhould be fufpend-ed in the Air } that the Speblator fhould be placednbsp;with his Back towards the Sun., between it and thenbsp;Drops j and that there fhould be a dark Cloud behindnbsp;the Drops, that the Colours may be more fenfible,nbsp;which are fcarcely perceiv�d, if vivid Light enters the Eve at the fame Time.

If thefe Things being fuppos�d, we conceive each Drop to be cut by planes paffing thro� thenbsp;Sun and the Eye of the Spedtator j what hasnbsp;been demonllrated of a Medium, terminatednbsp;with a circular Surface,* may be applied to eachnbsp;of thefe Sedlions.

Now here we fpcak of Rays that penetrate gg^� out of Air into Water. In red Rays, that is,nbsp;883! thofe which are leafl: of all refrangible, the Ratio between the Sine of the Angle of Incidencenbsp;and the Sine of the Angle of Refraftion, thatnbsp;is, between I and R, is that of 108 to 81 j ornbsp;which is the fame, of 4 to 3 ; with which Numbers, if the Computation be made, the Anglenbsp;*^879 APG {Fig. 3.) will be 42. Deg. z Min. *, andnbsp;the Angle API {Fig. ^.) will be fo Deg. f/nbsp;Min. * If we fpeak of the violet Rays, I andnbsp;R are to one another as i op to 81 j which Numbers give the Angles APG (F/^.3.) of40 Deg.nbsp;i/Min*, and API (Fi^. 4.) of y4 Deg. 7 Min.*nbsp;Plate XIX. Fig. i.] Let the Drops be fuppos�d to be diffus�d in the Air, and enlighten�dnbsp;by the Sun�s Rays, which are parallel to onenbsp;another, and to the Line O F, that pafles thro�nbsp;^^Lye of die Spe�tator. Let the Lines e O,

Book III. of Natural ^hilofo^hy.

let the Quantity of the following Angles be thus, (W2.) eOF of 40 Deg. 17 Min. EOF 4^ Deg.nbsp;z Min. O F fo Deg. f7 Min. EOF 74 Deg.

7 Min. Thefe fame Lines with the incident Rays de, DE, ah^ A B, form Angles that arenbsp;refpeftively equal to the aforemention�d ; therefore, if the Drops be conceiv�d at f, E, B,nbsp;the violet efficacious Rays, after one Reflexionnbsp;in the Drop enter the Eye j and the red efficacious Rays come to the Eye from the Drop Ejnbsp;in like manner after one Reflexion, the other intermediate Colours are obferv�d between e andnbsp;E, in the above-mention�d Order*. After two �nbsp;Reflexions in the Drop, the efficacious red Raysnbsp;come to the Eye from the Drop b-, and the efficacious violet ones from the Drop Bj the intermediate Colours appear between thefe Dropsnbsp;after the fame Manner as between E, e j butnbsp;they are difpos�d in a contrary Order, and bynbsp;realon of their double Reflexion are alfonbsp;Weaker.

Let us conceive a Line at O e, to be revolv�d about a fix�d Line O F, preferving the Anglenbsp;^ O F, and to form a Cone or Part of a conicnbsp;Surface ; In every Situation the Cine eO will,nbsp;with the Rays of the Sun that afq parallel tonbsp;one another, and to the Line OF, form an Angle of 40 Deg. 17 Min. ^therefore the Drops ggj-he diffus'd near Part of the Surface of this Conenbsp;at the fame or different Diftances, the Eye 'willnbsp;fee a violet Arc or Bow : The fame may be faidnbsp;of the other Colours} and therefore Drops beingnbsp;fufpended in the Air, it fees an Arc or Bow ofnbsp;the Breadth �'E, tingd with the homogeneous Colours before-mention�d *, that are difpos�d in the * 35,nbsp;fame Order as in the Experiments with Prifms}nbsp;becaufe the heterogeneous Rays are feparated asnbsp;liiuch in the Drops as in the Prifm*.nbsp;nbsp;nbsp;nbsp;�

13 5 Mathematical Elements Book IIL

88(5 By the fame Way of reafoning it is plain there will be a broader Arc or Bow furrounding thenbsp;fir ft ^ in which the fame Colour swill appear in a contrary Order.) and weaker.

Experiment 3.] Hang up a black Cloth in thd Light of the Sun j and let a Spectator look at itnbsp;Handing between the Sun and the Cloth j thennbsp;if Water be made to fall in fmall Drops betweennbsp;the Spectator and the Cloth, the Spe(51:ator willnbsp;fee a Rainbow, at leail the inner one.

CHAP. XXII.

^ / Of the Colours of thin Elates.

WE pafs on to the Colours of natural Bodies i and before we go any farther, we think it proper to examine thin Plates. Whoever attentively has obferv�d very thin Glafs, ornbsp;Bubbles made of Water thicken�d with Soap,nbsp;muft have perceiv�d feveral Colours in them.nbsp;Rays of Light, by Help of a thin tranfparencnbsp;Plate, are feparated from one another, and ac-o�7 cording to the different ffhicknefs of the Plate, thenbsp;Rays of fome Colours are tranfmitted, and thofe ofnbsp;others are reflebled 3 and the fimevery thin Plate isnbsp;of another Colour, when feenby the tranfmittedRays,nbsp;than when feen by the refleSled ones.

Experiment i. PlatelKVlll. Fig. 5.] Take two Objeft-GlaiTes belonging to long Telefcopes,nbsp;AB and CD 5 and let one of them be laid upon the other 3 then let them be prefs�d together hard, zud in the Middle, where theGlaflesnbsp;touch one another, you will fee a tranfparencnbsp;Spot, which is incompafs�d with colour�d Circles. If the Light, refle�ted by the Air that is

Book lil. of Natural ^hilofophy, 137

between the Glafles, comes to the Eye at O, there will appear a black Spot j and the Colours, which, as you recede from the Center, arenbsp;fo difpofed, that by reafon of the fame Coloursnbsp;coming over again, they may be referr�d to fe-veral Orders, will be as follows j BLACK, Blue,nbsp;White, Yellow, Red: VIOLET, Blue, Greetr,nbsp;Yellow, Red: PURPLE, Blue, Green, Yellow,nbsp;Red: GREEN, Red i which Colours arenbsp;alfo incompafs�d with other Colours v but, as younbsp;recede from the Center, grow continually weaker and weaker.

If the Light pafles thro� the Glafs to the Eye at O, the tranfparent Spot which tranfmitsnbsp;all the Rays is White 5 and by that Series, asnbsp;you recede from the Center, there will appearnbsp;Colours, which are alfo referr�d to fevcralnbsp;Orders contrary to thofe above-mention�d ;nbsp;WHITE, yellowiih Red, Black Violet, Blue:nbsp;WHITE, Yellow, Red, Violet,Blue: GREEN,nbsp;Yellow,Red,bluilhGreen: RED, bluiflr Green,nbsp;which are alfo incompafs�d with weaker Colours.

Experiment z.] Blow up foap�d Water into a Bubble, to form a thin Plate of Water. Letnbsp;this be cover�d with a very clear Glafs, left, bynbsp;the Agitation of the Air, the Colours, tobeob-ferv�d in that Bubble, fhou�d be confounded bynbsp;the Motion of the Water. Such a Bubble, be-caufe the Water continually runs down everynbsp;Way, is very thin at Top, and the Thicknefs innbsp;going down is continually increas�d gt; and fornbsp;the fame Reafon the Thicknefs of the Whole isnbsp;continually diminifti�d. Before the Bubble breaks,nbsp;the Top of it becomes fo thin, as to tranfmicnbsp;the whole Light and appear Black. If in thatnbsp;Cafe the Bubble be obferv�d by reflefted Light,

�38 Mathematical Elements Book III.

when it is enlighten�d by the Reflexion of a whitiiTi Sky, and the extraneous Light is intercepted by placing any black Body behind thenbsp;Bubble } the black Spot above-mention�d willnbsp;appear, and be encompafs�d with the fame colour�d Circles, difpos�d in the fame Order asnbsp;about the black Spot in the former Experiment.nbsp;As the Water defeends, the Rings are continually dilated till the Bubble burtts.

When the extreme Circumference of the Bubble appears red by the reflefted Rays, if it be look�d at fo as to befeenby the tranfmittedRays,nbsp;it will appear blue 5 and generally the Coloursnbsp;produc�d by tranfmitted and reflected Rays, innbsp;thefrme Manner as in the foregoing Experiment,nbsp;are oppofite to one another,nbsp;ggg By comparing thefe Experiments, it follows,nbsp;X.h'Mif u'e increaje theThicknefs of a very thinPlate^nbsp;its Colour will be chang'd^ and there voill he the famenbsp;Changes fucceffvely^andin the fame Order, whethernbsp;the Plate he form'd out of a rarer or a denfer Medium. For in the Plate of Air, between the Glaf-fes, and the watery one in the Bubble, whofenbsp;Thicknefs increafes as it goes farther from thenbsp;S8p Middle, the Colours will be in the fame Order.nbsp;Yet in a denfer Plate a lefsPhicknefsis requir'd thannbsp;in a rarer, to have it ting'd with the fame Colour.

Experiment 3.] Every thing being difpos�d as in Experiment r. if you wet the Edges of thenbsp;Glafles a little on one Side, the Water will bynbsp;degrees infinuate itfelf between the Glafles 5 andnbsp;there will be obferv�d the fame colour�d Circlesnbsp;as in the Air, neither will their Order be changed, but the Circles will be lefs 5 when the Water is got as far as the Center, all the Portionsnbsp;of the Circles in the Water will be feparatednbsp;from the Portions of the Circles in the Air, andnbsp;be all reduc�d into a lefs Space.nbsp;nbsp;nbsp;nbsp;Ehe

Book III. of Natural Thilofiphy, i3(gt;

I�he Colour of a Plate depends upon itsThicknefs* 890 andDenfity. andnotupon the encornpaJJlngMedimn. * 887

Experiment 4.3 If you take a Piece of Ifing-glafs or Talk, fo thin as to have it colour�d, the Colours will not be chang�d by wetting it j thatnbsp;is, if inftead of Air the Plate is incompafs�dwitbnbsp;Water.

Ehe Colour of the fame Plate is fo much the more 8pi tvT/V, asits Denfity differs more from the Denfity ofnbsp;the circumambient Medium. This is prov�d by thenbsp;foregoing Experiinenr, in which the Colours ofnbsp;the Plate, when wet, are more languid thannbsp;thofe of the fame Plate incompafs�d with Air.

In the third Experiment alfo the Colours are lefs vivid than in the fecond 5 in both there is anbsp;Plate of Water j but in the fecond Experimentnbsp;it is inclos�d with Air, and in the other withnbsp;Glafsj but Water and Glals differ lefs in Denfitynbsp;than Air and Water.

If Mediums equally differ in Denfity.^ the Colours 8pz 'iioill be morevivid.^ if the denferbe incompaf dwithnbsp;the rarer : For a very thinGlafs Plate, which isnbsp;Colour�d on account of its Thinnefs, being incompafs�d with Air, the Colours will be morenbsp;vivid than in Experiment i. in which a Plate ofnbsp;Air is incompafs�d with Glafs.

A Plate of the fame Denfity., incompafs'd with the gg� fame Medium., will reflect fo much the more Lightnbsp;as it is thinner. Tet if thefbicknefs be too much di- 854nbsp;'miniflj'd, it does not reflect the Light. All this isnbsp;plain from the foregoing Experiments^ j in thenbsp;three firfl;, the colour�d Circles which are thenbsp;leaf!;, and which are alfo the thinneft, refleftnbsp;Light befl of any; but in the Center, wherenbsp;the Spot is the thinneft of all, there is no fenfi-ble Reflexion ; as this clearly appears in the fe-coud : In the firft there is alfo a very thin Plate

140 Mathematical Elements Book III.

of Air, which does not refle�. Light} for the tranfparent Spot in the Center is much biggernbsp;than thofe Parts of the Surfaces of the GlafTes,nbsp;which immediately touch by the yielding Inwards of the comprefs�d Parts.

8py If there are Plates of the fame Medium^ whofe Tbicknejfes are m an Arithmetical Progrejfon of thenbsp;Natural Numbers i, z, 3, 4, y, 6, 7, ^c. Ifnbsp;the thinnefiof them allrejieSts any homogeneous Rays^nbsp;the fecond will tranfmit the fame., the third againnbsp;will refle�l them, and the Rays will be alternatelynbsp;reflected and tranfmitted j that is, the Plates whofenbsp;Thicknefles in the above-mention�d Progreffionnbsp;anfwer to the odd Numbers i, 3, y, 7, X3c. re-fle�t the Rays which the others tranfmit, wholenbsp;Thicknefles anfwer to the even Numbers z, 4,nbsp;6} 8, ^c.

This Property of the Plates obtains in rcfpe�t of any Sort of homogeneous Rays, with thisnbsp;Difference, that different Thicknelfes are requir�d for different Colours, as has been faid be-*887 fore*gt; the thinneft of all for reflefting Violet,nbsp;and they muff be thickeft for rcflc6ting Red j ifnbsp;the Thicknelfes are intermediate, the Rays ofnbsp;an intermediate Refrangibility are refle�ted, thatnbsp;8p� is, as the Refrangibility of the Ray increafes^ thenbsp;'Thicknefs of the Plate^ that reflects it^ is diminijh'd�

Experiment y.] Let the Experiment be made in a dark Room to produce the oblong Imagenbsp;of the Sun upon a Paper, fuch as is mention�dnbsp;in Experiment i. Chap. 18. Take two Obje�t-Glafles of long Telefcopes (fuch as were mention�d in the firft Experiment of this Chapter}nbsp;let them be prefs�d together, and fo difpos�d,nbsp;that every Angle Colour of the Image above-mention�d may be fuccelfively feen in them as innbsp;a Looking-glafs j that is, that the Glafles may

Ijc

Book III. of Natural Thilofophy. 141

be fucceflively enlighten�d by feveral homogeneous Rays; which may be done, by gently mo-''ing about its Axis the Prifm that feparates the Rays to make the oblong Image. The Rings,,nbsp;tticntion�d in the firft Experiment, appear, butnbsp;*Oore in Number, and only of one Colour j bynbsp;Reafon of the Unchangeablenefs in homogeneousnbsp;Rays:* In the Interftices of thofe Rings the * 863nbsp;Rays are tranfmitted, as plainly appears by holding a Paper behind, upon which the tranfmittednbsp;Rays will come �, the Rings are leafl; of any,nbsp;'vhen they are violet j then are fucceffively dilated, confidcring the following Colours, quitetonbsp;the Red. The Rings being of any Colour, ifnbsp;you meafure exactly the Diameters of the Circles that may be conceiv�d to be in the Middlenbsp;of the Breadth of the Rings, the Squares ofnbsp;their Diameters will be to one another as thenbsp;odd Numbers I, 3, f, �s?c. and meafuring inthenbsp;fame Manner the Diameters of the Circles innbsp;the Middle of every one of the Interftices between the Rings, the Squares of their Diametersnbsp;will be as the even Numbers 2, 4, 6, fj�c. Nownbsp;in ufing fpherical Glafles, the Thickneffes of thenbsp;Rate of Air, in the Circle above-mention�d, arenbsp;as the even and odd Numbers.

jln homogeneous Colour^ in a Plate of any Medi- Spy is faid to be of the firft Order, if the Plate benbsp;the thinneft of all thofe that reflet that Colour, innbsp;a Plate whofe Phicknefs is triple^ it is faid to be ofnbsp;the fecond Order, ^c.

A Colour of the firft Order is the mo ft vivid of^pS any j and fucceffively in the following Orders^ innbsp;the fecond, third, 6cc. it is lefts and left vivid. *

When a Plate of Air is inlighten�d with heterogeneous Rays, as that between the Glafles

14I- Mathematical Elements Book III*

of the Telefcopes, or any Plate like it of any other Subftance, as in Experiment i. and z. feve-ral of the Rings feen in the iaft Experiment arenbsp;confounded together, and that Colour is feennbsp;8pp which is made ot their Mixture j for the famenbsp;Thickn'efs of a Plate is often requir'd^ for refleSlingnbsp;different Colours of virious Orders : So a Plate,nbsp;which refle�ts the Violet of the third Order,nbsp;does alfo refledt the Red of the (econd Order,nbsp;as may be deduc�d from the latt Experiment, ifnbsp;you attentively conhder it : Therefore in thenbsp;fil'd and fecond Experiment, the third violetnbsp;Ring is confounded with the outward Edge ofnbsp;the fecond red Ring, and purple Colour isnbsp;produc�d } yet all the Red of the fecond Ordernbsp;is not abforb�d, becaufe the red Ring is widernbsp;than the violet one.

poo Phe more thePhicknefs of a Plate is increased the more Colours it reflepls^ and different ones^ of different Orders. The violet Plate of the tenth Order falls in with the blue one of the ninth Order,nbsp;and the yellow one of the eighth Order, andlad-ly with the red one of the feventh Order gt; and thenbsp;Colour of the Place is made up of a Mixture ofnbsp;thofe Colours.

Jf in the firft and (econd Experiments a Spe-dlator look obliquely upon the Plates^ fuch as that made of Air, and that which is made of Water,nbsp;the Rings will be dilated as they are feen morenbsp;obliquely, that is, in that Motion of the Eyenbsp;the Colour of the Plate in a determinate Place isnbsp;chanfd gt; yet the Dilatation is greater in the firffnbsp;Experiment 5 which proves, chat the Colour isnbsp;more chanfd by the Obliquity of the Rays^ if thenbsp;Plane be incompafs'd 'with a denfer Medium^ than ifnbsp;it be inclos'd by a rarer Medium.

Plate XVin. Fig. 7.] The Demonftration of which Propofition is eafily deduc�d from the

Laws

Book III: nbsp;nbsp;nbsp;�f Natural Thilofo^hy.nbsp;nbsp;nbsp;nbsp;143

Laws of Refraftion. Let L and / be thin Plates j the laft incompafs�d with a denfer, and the firftnbsp;with a rarer Medium ; let them both be of thenbsp;fame Thicknefs: If the Rays A B, ab^ equallynbsp;inclin�d to the Plates, fall upon them, there willnbsp;be at L a Refraction towards the Perpendicular j * on theccntrary, at I the Rays are refradted * 624nbsp;from the Perpendicular} * and tho BD and b d*nbsp;are equal, ^ f is longer than B C} and thereforenbsp;there is a greater Change in the Motion of thenbsp;Light in the Plate / than in the Plate L. The lt;305nbsp;T^enjity of the Plate L being increas'd^ the Medium,

quot;tvith which it is incompafs�d remaining the fame,^ there 'eoill be a lefs Difference between B C and B D,nbsp;and therefore a lefs Change of Colour } and if the 904nbsp;refrabling Power of the Plate be fo increas'd, thatnbsp;the refradred Rays (whatever be the Obliquitynbsp;of the incident ones) fliall not fenfibly differ,nbsp;there will be no fenfible Difference in the Colour ofnbsp;the Plate, whatever Situation the Eye is plac'd in.

Hence we may eafily deduce, that the Colour pop of fome Plates will vary by changing the Pofition ofnbsp;the Eye, and that the Colour of others is permanent.

CHAP. XVIII.

WHAT relates to the Colours of all Sorts of Bodies may be eafily deduc�d, fromnbsp;What has hitherto been explain�d.

We have fhewn that the Rays of Light have Colours peculiar to themfelves and unchangeable,nbsp;fo as not to be chang�d by Reflexion. *nbsp;nbsp;nbsp;nbsp;*

Therefore the Rays reflected from Bodies, ac-cording as a greater or lefs Rcfra7jgibility is proper to the Colour of the Body itflelf, have a greater ornbsp;lefs Refrangibility.

�44 Mathematical Elemeftts Book III.

Experiment i.] In the Middle of a black Paper fix two fquare Pieces of Ribbon, the one red and the other violet, which muft be join�dnbsp;fo as to touch one another at their Sides in thenbsp;fame Manner as the red and violet Colours innbsp;the third Experiment of Chap. ip. The blacknbsp;Paper muft be fo plac�d, that the Ribbons maynbsp;be well inlighten�d by the Light that comes into the Room thro� the Window : If aSpe�tatornbsp;looks at thefe Ribbons thro� a Prifm, as in thenbsp;Experiment abovc-mention�d, the Colours willnbsp;appear fcparated in the fame Manner as in thatnbsp;Experiment.

Experiment z. Plate XIX. Fig. 2.] Place the two Ribbons mention�d in the former Experiment at R and V j let the laft be violet, the firftnbsp;red ; let them be inlighten�d by the Flame ofnbsp;a Candle } at fix Feet Diftance place the convex Lens V j (of which Mention has been madenbsp;in Exper. 6. Cap. 19.) at the Diftance of aboutnbsp;fix Feet, you will have the Reprefentation ofnbsp;the Ribbon R upon a white Paper at r ; at alefsnbsp;Diftance you will have an exa�; Reprefentationnbsp;of the other at v. You may determine when thenbsp;Reprefentations are exaft, by binding blacknbsp;Threads upon the Surface of the Ribbons gt; fornbsp;thefe Threads appear diftinftly, where the Reprefentation is exaft.

9^J ICh'HZ Bodies have various Colours., hecaufe different Rajs are refleSled from Bodies differently colour'd j and that a Body appears of that Colour.,nbsp;which arifes from the Mixture of the reflePled Rays.,nbsp;may not only be deduc�d from the foregoingnbsp;Experiments, but may be alfo demonftratednbsp;dircftly by others.

B�ok III. of Natural Thilofophy.

Experiment 3.] Take two Bodies of any Kind, the one red, the other blue j let them be enlightened fucceffively, in a dark Place, by the Colours of the coloured Image, which is made bynbsp;the Refradfion of the Prifm ; every one of thenbsp;Colours will be refledted by both j but the rednbsp;Rays are copioufly refledled from the red Body,nbsp;whilft the blue Body refledfs but few of them,nbsp;which plainly appears by comparing both Bodies, when they are enlightened by the red i thenbsp;contrary may be obferved in blue Colours, whichnbsp;are copioufly refledted from the blue Body, whilftnbsp;Only a few are refledfed from the red one.

The Rays, which are not refledled from a Body, penetrate into it, and {here fuf��r innumerable Reflexions and Refradlions, as we havenbsp;explained between 842, and 843 j till atnbsp;length they unite themfelves to the Particles ofnbsp;the Body itfelf.* Therefore a Body grows hot fonbsp;much the fooner, as it refleSls Light iefs copioufly. 908nbsp;For which Reafon a white Body, which refledlsal- 9�9nbsp;moft all the Rays with which it is enlightened, * ��*7^nbsp;heats the floweft, whilfi a black Body, into whichnbsp;almoft all the Rays penetrate, becaufe only fewnbsp;are refledted,* acquires Heat fooner than any other. '*^S45nbsp;To determine that Conftitution of the Surfaces of Bodies, upon which their Colour depends, we muft take Notice of the fmalleft Particles , of which thefe Surfaces are made up:nbsp;Thefe Particles are ttanfparent *, and are fepa-*?^,nbsp;rated by a Medium of different Denfity fromnbsp;the Particles themfelves they are alfo thin, *843nbsp;otherwife the Surface would as it were be covered by a tranfparent Body, * and the Colour *843nbsp;Would depend upon the Particles under thefe.nbsp;Therefore in the Surface of every coloured Bodynbsp;there are innumerable fmall thin Plates j but bynbsp;Vot. II.nbsp;nbsp;nbsp;nbsp;Lnbsp;nbsp;nbsp;nbsp;leffen-

Mathematic at Elements Book III

leflening the Plate, keeping the fame Thicknels, its Properties, as to the Reflexion of Light, arenbsp;not changed j for the leaft Plate, in refpe^ of thenbsp;Rays of Light, is very large : Wherefore, whatnbsp;has been demonftrated in the preceding C^ap.nbsp;may be applied to thefe Plates in the Surfacesnbsp;of Bodies. From whence we deduce the following Conclufions.

Den fty of the Parts of the Body, which are in ^ 9 * * the Surface, between the Pores of the Bodyi^

Ihe Colour is fo much the more vivid and homo-90, geneoHs, as the Parts are thinner.^

912 Cseteris porihus, the aforefaid Parts are of the greateji ^hicknefs when the Body is red-, and of thenbsp;leaf, when violet*

90 . l^his Denfity is lefs in the Tails of Peacocks,and 5i4jnfome Silks, and in general in oW Bodies, whofenbsp;Colour varies according to the different Situation ofnbsp;^�UheEye*

j The Colour of a Body is more ohfcure and darker, when a denfer Medium enters its Pores * for thennbsp;the Parts, upon which its Colour depends, arenbsp;lurrounded by a denfer Medium than before.

We experience this in all Bodies, which are thoroughly penetrated by Water and Oil: Whennbsp;the Bodies are dry, they recover their formernbsp;Colour, unlefs in feme Gafesi, in which foraenbsp;Parts are carried away by the Adfion of the Water, or Oil, or when fome Parts of the Water^nbsp;or Oil, are fo united with the Parts of the Body,nbsp;as to change the Thicknefs of the Plates.

From fuch a Caufe are deduced the Changes in the Colours of fome Liquors, by their beingnbsp;916 mixed with others. Often the faline Barticles, fwint-^nbsp;wing in one Liquid, unite them felves to the faline

Partin

Book III, fgt;f Natural�Philofophy. nbsp;nbsp;nbsp;147

particles fwimming in another ; or the uniteAParticles are feparated by the J�liens of others that are [nferadded on all �which Accounts the Thick-nefs of the particles is ebangedy and together with itnbsp;the Colour of the Liquids*nbsp;nbsp;nbsp;nbsp;�ggg

Sometimes the Colour of a Liquid is different, 517 when feen by reflebied Kays, from what it is, �whennbsp;feen by tranfmitted ones: We have Ihewn beforenbsp;whence this arifes.*nbsp;nbsp;nbsp;nbsp;*^^7

Experiment 4.] An Infufion of Lignum Ne-pbriticum, that is not too much tinged, appears blue by reflefted Rays, but yellow, if the Phial,nbsp;which contains the Infufion, be placed betweennbsp;the Light and the Eye.

Experiment 5,] If you pour Spirit of Vinegar into the Infufion of Lignum Nephriticum, it willnbsp;appear yellow in any Pofition whatever.

In this Cafe the Thicknefs of the Particles is changed, and the Rays, that were tranfmittednbsp;through every one of them, are now intercepted �,nbsp;but though the Liquor is placed between the Eyenbsp;and the Light, it is feen by refledted Rays ^ fornbsp;we may eafily conceive that fuch Rays come tonbsp;the Eye by the various Reflexions which the Lightnbsp;undergoes in the Liquid. But this Colour only isnbsp;fenfible, becaufe the Rays cannot penetrate di-redlly through the Liquid.

From this we may deduce the Reafon, why 918 coloured Liquid in a Glafs, of the Figure of an in-'verted Cone, if it be placed between the Eye andnbsp;the Light, appears of a different Colour, in differentnbsp;Parts of the Veffel; in the lower Part all the Raysnbsp;which are tranfmitted thro�the Particles, are not intercepted i then they are more and more intercepted,nbsp;according as there is a greater Quantity of thenbsp;Liquid between the Eye and the Light j till at

148 nbsp;nbsp;nbsp;Mathematical Elements Book III.

length they come to be all intercepted j and only thofe Rays which are refie��ed by the Particlesnbsp;penetrate the Liquid ; in which Cafe the Colournbsp;coincides with the Colour of the Liquid feennbsp;by the refleded Rays.

919 nbsp;nbsp;nbsp;Clouds often appear very heautifully coloured jnbsp;they conGft of aqueous Particles, between whichnbsp;Air is interfperfed j therefore, according to the va~

920 nbsp;nbsp;nbsp;rioiis '�hicknefsof t�ofe aqueousParticles^xhQCloxxdnbsp;quot;sss will be of a different Colour.*

EXPERIMENTS

E who attentively confiders that *15 Space can be terminated by nonbsp;Bounds, will fcarce be able tonbsp;deny, that the Supreme Almighty Intelligence has every wherenbsp;manifefted the fame Wifdom which He hasnbsp;fliewn to the Inhabitants of the Earth in a imallnbsp;L 3nbsp;nbsp;nbsp;nbsp;Compafs. �

Mathematical Elements Book IV.

Compafs. What I here call a fmall Compafsj imtnenfly exceeds our Comprehenfion j yet it isnbsp;but final], compared with infinite Space.

921 Our Earthy �with fixteen other Bodies (for we have no Knowledge of any more Qmoves in anbsp;determinate Space; neither do thefe Bodies recedenbsp;from, or come nearer to one another, beyondnbsp;their fet Bounds j and their Motions are performed according to unchangeable Laws,

The whole Art of Jftronomy is almoft employed concerning thefe alone j and thefe will be chiefly my Subjeft in this Work : The other Bodies that conftitute the Univerfe, are too fardiftantnbsp;from us, for their Motions (if they are moved)nbsp;to fall under our Obfervations : Of thefe onlynbsp;the lucid Bodies can be perceived by us ; and ofnbsp;thofe only the more remarkable ones, and whichnbsp;are lefs diftant from us than the reft �, even moftnbsp;of fuch as are feen by the Telefcope are invifiblenbsp;to the naked Eye.

They are called fixed, becaufe, as far as can be perceived, they keep the fame Pofition, with re-fpe�t to one another: We muft take Notice ofnbsp;fomething peculiar concerning thefe hereafter.

924 nbsp;nbsp;nbsp;But as to the Planetary Syftem: In thisvieha.venbsp;faid there are feventeen Bodies, which are all fphe~nbsp;ricalOne only pines hy its own Light, the reftnbsp;are opaque, and are vifible only by the Light �whichnbsp;they borrow from that.

525 iThe Sun is that lucid Body, ajtdfar the great eft of �� - alt in the Planetary Syftera; it is fuiefcent in the middle

Definitton' in.

ithefe are divided into two Clajfes ; fix are called 926 the Primary Planets j ten are calk'd the Secondarynbsp;Planets. When we fpeak of the Planets, without any Diftinftion, we always underftand thenbsp;Primary ones.

^he Primary Planets move roundthe Sun^ and are 927 carried at different Diftances from it, in Curvesnbsp;that return into themfelves.

A Secondary Planet revolves round a Primary 928 one^ and accompanies it in its Motion round thenbsp;Sun.

^e Planets in their Motions defcrihe Ellip-tic Lines, not much different from Circles. And all thofe Lines are fixed ; at leaft, there is but anbsp;fmall Change to be obferved in a long Time, innbsp;their Situation.

Plate XXIV. Fig. 3.] An Elliptic Line is 93� formed, if a Thread, whofe Extremities are fixednbsp;in the two Points F and �; is moved, remainingnbsp;ftretched �, as is to befeeninthis Figure, in whichnbsp;the Thread is reprefented zt 7 d f, V E �, F B �.

The Points F, �, are called the Foci ; the Line A a, which paftes through them, and is terminated on each Side by the Circumference of thenbsp;Ellipfe, is called the greater Axis-, and it is thenbsp;greateft Line that can be drawn in the Ellipfis,

The middle Point C of the Axis is the Center of the Ellipfe ; and the lefier Axis D d falls perpendicularly through this Point upon the greater Axis.

Fbe Orbits of all the Primary Planets are in fucb a pofition, that one of their Foci falls innbsp;with the Ceiiter of the Sm -, let the Ellipfe A D

932 nbsp;nbsp;nbsp;^he Biflance between the Center of the Sun andnbsp;the Center of the Qrhit^ is called the Eccentricitynbsp;of the Planet i as F C.

933 nbsp;nbsp;nbsp;In every Kevolution the Planet approaches oncenbsp;to the San, and once recedes from it-, and at itsnbsp;greateft Diftance is at a, the Extremity of thenbsp;greater Axis of the Orbit, and at its leall Diftance,nbsp;in the oppofite Extremity A.

934 nbsp;nbsp;nbsp;That Difiance of the Planet from the Sun, is called the mean Diftance, which is equally differentnbsp;from the greatefi and the leafi. At this Diftancenbsp;the Planet is in the Extremities D, d, of the fmall-er Axis.

935 nbsp;nbsp;nbsp;The Point of the Orbit, in which the Planet is atnbsp;its grearefi Difiance from the Sun, is called thenbsp;Aphelium �, as a.

936 nbsp;nbsp;nbsp;The Point of the Orbit, in which the Planet is atnbsp;its leafi Diftance from the Sun, is caUed the Perihelium.

938 nbsp;nbsp;nbsp;The Line which joins the Apfides, that is, thenbsp;greater Axis of the Orbit, is called, the Line ofnbsp;the Ap�des.

Book IV. of Natural Thilofofhy. i5�3

This Plane is to be continued every Way; and JJironomers confider the Pofition of thenbsp;Planes of the other Orbits, with refped: to this.

The Points.^ in which the other Orbits cut tbe^/^i Plane of the Ecliptic, are called the Nodes.

The Line which joins the Nodes of any Orhit, 542 that is, the common Se�fion of the Plane of thenbsp;Orbit with the Plane of the Ecliptic, is callednbsp;the Line of Nodes.

A planet is not carried with an equal Celerity all the Points of its Orbit-, the le[sit isdiftant from^^nbsp;the Sun, the fwifter is its Motion and the Times,nbsp;in which the feveral Arcs of its Orbit are run thro',nbsp;are to one another, as the Area's formed by Linesnbsp;drawn from the Planet to the Center of the Sun :

The Arcs AB and a E are run through in Times, which are to one another, as the Area�s of thenbsp;mixed Triangles A F B, lt;3: F E.

All the Planets are carried the fame Way, and 94^ their Motion in their Orbits is contrary to thatnbsp;Motion which we obferve daily in all the Ccele-ftial Bodies, by which in one Day they feem tonbsp;be carried round the Earthy of which hereafter.

A Motion, fich as is that of the Planets in their 946 Orbits, is faid to be in Confequentia, and dire�l.

15'4 Mathematical Elements Book IV.

947 nbsp;nbsp;nbsp;cantrary Motion is called, a Motion in Ante-cedentia ^ and fometimes Retrograde.

948 nbsp;nbsp;nbsp;more difiant the Planets are from the Sun^nbsp;the flower they move in their Orhits j fo that thenbsp;periodical Times of the moft diftant ones arenbsp;greater, both becaufe they have a greater Orbitnbsp;�O run through, and a flower Motion.

949 nbsp;nbsp;nbsp;Line which paffes through the Center of thenbsp;Planet, and about which it moves, is called thenbsp;Axis of the Planet.

9J'o quot;�be Planets, at leafl: moft: of them, and the Sun itfelf move round their Axis: There are Two,nbsp;of which, in this refpe�f, Aftronomers have beennbsp;able to make no Obfervations, but which, in allnbsp;Probability, have this Motion.

951 This Motion agrees or confpires with the Motions of the planets in their Orbits, that is, it is in Confequentia-

tion, fo that all the Points of the Axis of a Planet defcribe equal and fimilar Lines.

g54 Plate XX. I.] Aftronomers compare together accurately enough the Diftances of the Planets from the Sun, to give us an Idea of thenbsp;whole Syftem. The Dimenfions of the Orbits arenbsp;reprefented in this Scheme, in which the Pointsnbsp;N N Ihew the Nodes of each Orbit.

955 Ncverthelefs we cannot compare the Dimenfions of this Syftem with any that we know upon thenbsp;Surface of the Earth �, for no Aftronomers willaflert,

tnac

Book IV. of Natural Thilofophy.

But, that the feveral Parts of the Syftem may 956 be compared together, we fuppofe the mean Di-ftance of the Earth from the Sun divided intonbsp;1000 equal Parts, which we makeufeof in mea-furing the other Diftances.

The Sun as was faid before, is agitated by9j;7 a fmall Motion in the middle of the Syftem ^nbsp;it moves round its Axis in the Space of 25nbsp;Days : And its Axis is inclined to the Plane ofnbsp;the Ecliptic, making an Anglp of 87 Degr. 30nbsp;Min.

Mercury 7 is the leaft diftant from the Sun of958 any of the Planets ; its mean Diftance from tlyenbsp;Sun is 387, its Eccentricity 80, the Inclinationnbsp;of its Orbit, that is, the Angle, formed by thenbsp;Plane of its Orbit with the Plane of the Ecliptic,nbsp;is 6 Degr. 52 Min. it performs its Revolutionnbsp;round the Sun in 87 Days, 23 Hours.

The ney^t is Fenus 9, whofe Diftance from the 959 Sun is 723, its Eccentricity 5, the Inclination ofnbsp;its Orbit 3 Deg. 23 Min. It performs its Periodical Motion in 224 Days, 17 Hours ^ andnbsp;its Motion round its Axis is performed in 23nbsp;Hours.

The third Planet in order from the Sun is 960 our Earth � j its mean Diftance from the Sun isnbsp;looo, its Eccentricity 169: It is moved in thenbsp;Plane of the Ecliptic ^ its Periodical Time is 365nbsp;Days, 5 Hours, 51 Min. and the Motion aboutnbsp;its Axis is performed in 23 Hours, 56 Min. 4 Sec.

The mean Diftance of Mars d from the Sun 961 is 1524, its Eccentricity 141, the Inclinationnbsp;of its Orbit i Degr. 52 Min. Its Periodicalnbsp;Time 686 Days, 23 Hours. Its Revolutionnbsp;Inbsp;nbsp;nbsp;nbsp;about

15-6 Mathematical Elements Book IV.

562 Jupiter V, the biggeft of all the Planets, isdi-ftant from the Sun, at a mean Diftance 5201, its Eccentricity z^o. The Inclination of its Orbitnbsp;I Degr. 20 Min. The Periodical Time 43 3 2 Days,

963 nbsp;nbsp;nbsp;The mean Diftance of Saturn b, the mofl:nbsp;diftant Planet from the Sun, is 9538 j its Eccentricity 547 j the Inclination of its Orbit 2nbsp;Degrees 30 Min. The Periodical Time 10759nbsp;Days, 7 Hours. It is encompafled with a Ringnbsp;which does not touch the Planet, but nevernbsp;leaves it. This Ring is not vifible without a Te-lefcope.

The mean Diftance being given, if you add the Eccentricity, you will have the greateft Diftance ; but if you fubftraft the Eccentricitynbsp;from the mean Diftance, you will have the leafl:

964 nbsp;nbsp;nbsp;The three Planets, Mars^ 'Jupiter^ and Saturn,nbsp;which are more diftant from the Sun than thenbsp;Earth, are called the Superior Planets-, Venus andnbsp;Mercury are called the Inferior ones.

Five Planets called Satellites, move about Saturn ; four about Jupiter one about the Earth fvi%.') the Moon.

966 The Satellites, hy Lines drawn to the Center of the Primary Planets, defcribe Area�s about themnbsp;proportional to the T'imesas has been faid of thenbsp;Primary Planets, withrefpe�; to the Center of the

Book IV. of Natural Thilofofhy. i $y

^be Moon moves about the Earth in an Ellipfe^()6y one of whofe Foci is in the Center of the Earth^lt;)6Synbsp;from which the mean Diftance of the Moon is 6o 969nbsp;� Semidiameters of the Earth and a Half; Its Eccentricity is liable to Change j the mean one is ofnbsp;3 Semidiameters and a Third. The Plane of itsnbsp;Orbit forms an Angle with the Plane of the E-cliptic of S Degr. but this Inclination is not con- 9170nbsp;ft ant or always the fame. In the Motion of thenbsp;Moon round the Earth, neither the Line of the Nodes,nbsp;nor the Line of the Jpftdes, is carried in a parallelnbsp;Motion 5 but the Line of the Nodes Weft ward, ornbsp;in Antecedentia-^ the Lines of the Apfides Esamp;viiqxA,nbsp;or in Confequentia -y the firft performs its Revolutionnbsp;in about 9 Years, the fecond in 19. The Periodical Time of the Moon�s Motion about thenbsp;Earth is 27 Days and about 7 Hours ; and it isnbsp;turned about its own Axis exactly in the famenbsp;Time.

Elate XX. Fig. 2.] The firft or inmoft of the 971 Satellites of Jupiter is diftant from fiipiter^nbsp;Center 25 Diameters of Jupiter : It is movednbsp;round Jupiter m one. Day, 18 Hours, 28 Min.

The Diftance of the Second is 4 and an Half Diameters of Jupiter: Its Periodical Time is 13nbsp;Hours, 18 Min.

The Diftance of the Third is 70 Diameters ^ its Periodical Time 7 Days, 4 Hours,

The Fourth is diftant i2j Diameters; It performs its Motion in 16 Days, 18 Hours, 5 Min.

Elate yCN. Fig. 3.] The firft or inmoft Satel-972 lite of Saturn is diftant from Saturn''s Center if ofnbsp;a Diameter of the Ring: Its Periodical Time isnbsp;1 Day, 21 Hours, 18 Min.

The Diftance of the Second is i4: Diameter of the Ring : Its Periodical Time is 2 Days, 17nbsp;Hours, 41 Min.

Mathematical Elements Book IV.

The Diftance of the Third is Diameter of the Ring j its Periodical Motion is 4 Days, 13nbsp;Hours, 47 Min.

The Diftance of the Fourth is 4 Diameters of the Ring j its Periodical Time 15 Days, 22 Hours,nbsp;41 Min.

The Diftance of the Fifth is 12 Diameters of the Ring its Periodical Time is 79 Days, 7nbsp;Hours, 53 Min.

973 Concerningthe Motion of thefe, as alfo of the Satellites of 'Jupiter about their Axes, we cannbsp;hitherto determine nothing certain from Aftro-nomical �bfervations.

If we take Notice of the Diftances and Periodical Times of the Planets, we fhall find that the following Rule holds good in our Syftem, where-ever feveral Bodies are moved round the famenbsp;Point, that is about the Sun, Saturn, and Jupiter;

974(�z^^^� ^he Squares of the Periodical Times are to one another, as the Cubes of the mean Difiancesnbsp;from the Center,

9*15 Plate XX. Fig. 4.] To give a Notion of the Dimenfions of the Bodies themfelves in our Syftem, we have contrived the fourth Figure, innbsp;which all the Primary Planets, amp;ndSaturn*sKlng,nbsp;are defcribed, according to their Dimenfions; thenbsp;Sun, whofe Magnitude exceeds all the reft, is re-prefcnted by the greateft Circle (F/g. i.) that is,nbsp;the Circle which terminates the Figure.

Thefe Dimenfions reprefent the Proportions of the Bodies one to ^nother exaftly enough, if younbsp;except the Earth, \yhich, for the Reafon already

�'955 mentioned,* cannot be fo compared with the o-ther Bodies, as to leave no Room to doubt what Proportion it bears to them.

976 Yet the EartFs Diameter maybe meafured, and contains 3.4oo669Perches, each of which i s equal to

Book IV. of Natural Thilofophy.

other Planets may be compared together, and with the Sun�s Diameter, yet it can�t be determined how many Feet they contain, till afternbsp;Obfervations ftall be made at a proper Timenbsp;hereafter.

Of the Bodies that make up the Planetary 51717 Syftem, the Moon only canbe com^aredto the Earth jnbsp;its Diameter being to the Diameter of the Moon,nbsp;as 40 to 11.

^�he other fecondary Planets are not meafared hy �4lironomers, but it can�t be doubted, but thatnbsp;fome of them are bigger than the Earth.

Befides the Bodies already mentioned, there979 are others in the Planetary Syftem, which are,nbsp;vifible for a Time, as they come near the Sun,nbsp;and then recede from it, and become invifiblejnbsp;they are called Comets. They appear mofi com-^^�nbsp;fnonlywithTails.) andthe '�ail is always turned fromnbsp;the Sun j in their Motion they defcribe Areals, bynbsp;Lines drawn to the Center of the San, proportional *944nbsp;to the T�imes, as has been faid of the Planets.* 966

As to Comets, it is probable that they move in e/- 981 liptic Orbits, that are very eccentric; fo that theynbsp;are invifible, when they are in that Part of thenbsp;Orbit which is moft diftant from the Sun ; whicivnbsp;is deduced from the Periods of fome of them,nbsp;that have been obferved to be pretty regular :

And it is plain from Obfervations, that fome have ^^2. ^efcribed in their Motion Portions of very eccentric Eclipfes, in one of wbofe Foci was the Centernbsp;Pf the Sun.

The Notion, that we have hitherto given of the Planetary Syftem, is founded upon Aftrono-mical Obfervations; and what we have alreadynbsp;faid admits of no Difpute among Aftronomers, ifnbsp;except what relates to the Elliptic Line andnbsp;the Motion of the Earth,

Some

i6o Mathematical Elements Book IV.

Some affirm, that the Orbits of the Planet's are not Elliptic, but that in their Motion theynbsp;delcribe another Oval : Kepler has deduced fromnbsp;^ycho Brab�s Obfervations, that thefe Lines arenbsp;Elliptic i and we lhall fhew in the following Part,nbsp;that no other Curves can be defcribed by thenbsp;Planets.

Thofe that fay the Earth is at reft, have no Aftronomical or Phyfical Argument for a Foundation of their Opinion j that is, don�t reafonnbsp;from Pheemmena : Negletfting the Simplicity ofnbsp;the Syftem, and the Analogy of the Motions,nbsp;they af��rt, that their Opinion is not contrarynbsp;to Obfervations in which they err, as we lhallnbsp;Ihew in the following Part.

CHAP. 11.

Concerning the Apparent Motion.

WHOEVER, after having read the former Chapter, looks upon the Heavens^ will fcarcely believe, that he beholds the Syftemnbsp;which is explained there j and a more exadtnbsp;Confideration of the heavenly Motions will en-983 creafe his Doubt. No Wonder, [ince we' can ob-^nbsp;ferve very little in the Heavens but falfe Appearances.

The common Obferver of the Heavens is a Spedfator, who-x thinking himfelf to be at reft,nbsp;is carried about by various Motions, and beholds Bodies, concerning whofe Diftance andnbsp;Magnitude he makes falfe Judgments. The truenbsp;Syftem of the World was unknown for manynbsp;Ages, even to the molt exaCl Obfervers ofthenbsp;Heavens.

Book IV. of Natural Thilofofhy. nbsp;nbsp;nbsp;i6i

But �w� muft explaili how all Things, which 984 are obferved in the heavenly Bodies, have Placenbsp;in the Syftetu that has been explained, in relpeftnbsp;of a Spectator upon the Earth j that is, we lhallnbsp;deduce the Appearances from the true Motions.nbsp;Which cannot be done, unlefs we firft lay downnbsp;fome general Things concerning the apparentnbsp;Motion in general.

It is certain, that we have no Art, by which we can difcover the true Motion, only the relativenbsp;Motion can be perceived by the Senfcs ; and it isnbsp;that only concerning which we treated in thenbsp;former Chapter. Who can reafonably affirm ornbsp;deny, that all the Bodies, which are known tonbsp;us, are not carried in a common Motion throughnbsp;the immenfe Spaces ?

relative Motim to be dijlingtiijhedfrom the ^35 apparent otie ; for the apparent Motion is the ^nbsp;Change which appears to be in the Situation ofnbsp;the Bodies, and depends upon the Change of thenbsp;Pi�lure in the Bottom of the Eye j for Objedsnbsp;have the fame apparent Relation to one another,nbsp;as their Reprefentations have in the Eye j fortheynbsp;are feen as they are painted in the Eye; *^716nbsp;and the Change in that Pidure, from the Motion of the Bodies, moft commonly differs from thenbsp;Change of the Relation between the Bodiesnbsp;themfelves; as follows from the Formation of thatnbsp;Eidure.

The Heavens are nothing but an immenfe Space,^ pgg yhich cannot he feen, and would appear black, * *845nbsp;ff innumerable Rays of Light, flowing from thenbsp;heavenly Bodies, did not continually penetratenbsp;our Atmofphere. Moft of them come to usnbsp;h'om the Bodies in right Lines, yet a great ma-oy fuffer various Reflexions in the Atmofphere,

^nd enlighten the whole Atmofph�re; which is the Reaton that, in the Day, Bodies are enlight-

Mathematical Elements Book IV.

ened, even without the Reflexion of the Clouds, to which the Rays of the Sun cannot come di-redlly.

Thefe Rays are heterogeneous and white j for there are Bodies enlightened by thefe I^ays whichnbsp;appear white : And thefe Bodies feen throughnbsp;Prifms, at the Extremities, are tinged witli Colours, which does not happen in an homogeneousnbsp;'**74 Colour i * alfo a Circle of white Paper of Half annbsp;Inch Diameter, being put upon black Cloth, ifnbsp;^ it is enlightened by thefe Rays, it will appear ob-long through the Prifm i * and the fame Colours,nbsp;which areobferved in the Rays of the Sun, arenbsp;feen here in the fame Manner : All which Thingsnbsp;would not happen, if the Air, as many think, wasnbsp;a blue Liquid i that is, through which only thenbsp;Sun�s blue Rays, at leaft moftly fuch, pafs.

9817 When we look at the black Sky, the white Kays, beforementioned, enter our Eyes, whence the bluenbsp;Colour of the Sky arifes. Becaufe we are accuftom-ed to fee a Colour where there is a coloured Ob-je�t, we alfo refer the Colour of the Heavens tonbsp;an Objedt j but fincethis is feen equally towards allnbsp;988 Parts, we conceive a concave fpherical Surface, innbsp;whofe Center we are placed j we imagine this Sur~nbsp;face to be opaque, and therefore diftant fromus beyond all vifible Bodies.

When a Body is between a Plane and the Eye, of whofe Diftance we cannot judge, the Body appears to us to be applied to the Plane, whatfo-ever the Diftance is between that and the Plane;nbsp;for there is no Reafon why the Parts of thenbsp;Plane, which are painted at the Sides of the Imagenbsp;of the Body in the Eye, fhould not appear at thenbsp;fame Diftance with the Body.

589 Thence alfo all the celeftial Bodies (of which the leaft diftant from us, (-y/zs.) the Moon) isnbsp;yet fo removed that we can give no Judgment

Book IV. of Natural Thilofophy. 163

of its Diftance are referred to that imaginary *7'^'^ Sphere above-mentioned ^andthey all appear equally diftantif and Jeem to move inthe Surf ace fa concave Sphere. So the Moon appears to be amongftnbsp;the fixed Stars, although its Diftance bears Icarcelynbsp;any fenfible Proportion to the Diftance of Saturn inbsp;and the Diftance even of Saturn itfelf is nothing,nbsp;compared with the immenfe Diftance of the fixednbsp;Stars. It is no Wonder therefore, that the common People know nothing of the Magnitude ofnbsp;the celeftial Bodies, and the Immenfity of thenbsp;Heavens.

We fee from what has been faid, how the Motion of any Body being given, and the Motion of the Earth being known, the apparent Motionnbsp;may be determined.

We have faid that a Sphere is imagined beyond the fixed Stars, in whofe Center is the Spedator : * The Orbit of the Earth is fo fmall,*988nbsp;in refped of the Diameter of this Sphere, thatnbsp;the Center of the Sphere is not fenfibly changednbsp;by the Alteration of the Place of the Spedator,nbsp;whilft he is carried along with the Earth. Wherefore in all the Points of the Earth's Surface, andatg^tsnbsp;any Time, the Inhabitants of the Earth imaginenbsp;the fame Sphere, to which they refer the heavenlynbsp;Bodies , and which hereafter we IhallcalDi'^ Spherenbsp;of the fixed Stars.

Thefe Things being laid down, if we conceive ggx a Line to be drawn through the Earth, and a Body,nbsp;�which, being continued beyond the Body, cuts the a-forefaid Sphere ; we have a Point, to which the a*nbsp;bove-mentioned Body is referred, and which is thenbsp;apparent Place of that Body.

Whilft the Body, or the Earth, or both are Woved, this Line vaovtdi 3\{o, SLX\d the apparentnbsp;Motion is the Line, which is defcribed amongfi thenbsp;fixed Stars by the Extremity of the Line above-

Mathematical Elements Book I V,

mentioned, which goes through the Earth and the Body^ whofe apparent Motion is ohferved.

993 nbsp;nbsp;nbsp;Thereforenbsp;nbsp;nbsp;nbsp;fame Appear anceswiU follow from

the Earth�s 'being moved out of its Place^ as if the Body had heen moved, and the fame alfo may henbsp;deduced from the Motion of both.

994 nbsp;nbsp;nbsp;But if the Body and the Earth he fo moved^ thatnbsp;the Line which paffes through thefe Bodies he carried in a parallel Motion, the Body will feem to henbsp;at reft among ft the fixed i^if^rjjbecaufe in this Cafe,nbsp;the Space, gone through by the End of th�nbsp;Line amonglt the fixed Stars, cannot exceed thenbsp;Space gone through by the Earth; but the Line thatnbsp;is equal to the whole Space which the Earth can gonbsp;through, at fo great a Diftance as the fixed Stars,nbsp;is not fenfible to us.

995 nbsp;nbsp;nbsp;From the Motion of the Earth round its Axis therenbsp;is produced an apparent Motion, which will benbsp;eafily deduced in its proper Place from the Foundation laid down in this Chapter.

That the apparent Motion differs from the relative, and is varied by the Motion of the Spe�la-tor, is what Sailors every Day obferve.

CHAP. III.

0/ the Ehanomena or Appearances of the Sun from the Motion of the Earth in itsnbsp;Orbit.

r J be the Sphere of the fi.xed Stars ; the *991 apparent Place of the Sun will be at s. * Whennbsp;996 the Earth is carried in its Orbit from T to t, thenbsp;^ Sun feins to move in Confequentia, and to runnbsp;�'991 thro the Arcs r,* which meafures the Angle r S snbsp;equal to the Angle T St, fo that the Celerity of

Book IV. of Natural Thilofophy.

the apparent Motion of the Sun depends upon the Celerity of the angular Motion of the Earth,nbsp;with refpe�t to the Center of the Sun 3 whichnbsp;Motion encreafes upon a double Account j onnbsp;account of the Diftance from the Sun being lef-fened, and the Celerity of the Earth being en-creafed : Both which Caufes always concur ; * *9 4nbsp;wherefore the Ineq^iality of the apparent Motion of 997nbsp;the Siin is fenfible. Jn a whole P^-evolution of the 99^3nbsp;Barthf the Sun alfo feenis to run thro' a whole Circle.

^his apparentlVay of the Sun is called theEclip-999 tic Line. It is the Se6Bon of the Sphere of thenbsp;fixed Stars with the Plane of the Ecliptic, fup-pofed to be continued to this Sphere.

This Way is divided into 12 equal Parts, each of which contains 30 Degr. Thefe Parts are callednbsp;the Signs, and are diftinguilhed by thefe Names jnbsp;Aries 7^, Raimisnbsp;nbsp;nbsp;nbsp;Cancer $, Leo

Virgo 'tv, Libra Scorpius ill, Sagittarius Capricorn Aquarius Pifces K. Whence thefe Parts have their Names, we lhall explain when wenbsp;treat of the fixed Stars.

iThe San is longer a going thro'the fix firjl Signs 1000 than the fix laft^ and the Difference is nine Days.

Although a Circle has neither Beginning noriooi End, yet when feveral Points muft be determinednbsp;in it, fome Point mull: be taken as the Beginning j this, in the Ecliptic Line, is the firfi Pointnbsp;of Aries ; we fhall (hew how it may be determined hereafter. It is not fixed to one Place a-mongft the fixed Stars ithtxt�oxetheOrhitsoftheiooznbsp;Planets, which alter fo little, that they may benbsp;looked upon as unchangeable, * don't prejervethe''9'^9nbsp;fame Situation, in refpecl of this Point.

Mathematical Elements Book IV.

1003 nbsp;nbsp;nbsp;Difiatice of the Sun from the firji Point ofnbsp;Aries, meafuredin confequentia, iscalledthQSxai^snbsp;Longitude.

1004 nbsp;nbsp;nbsp;The Longitudes of the other heavenly Bodies arenbsp;nteafured after the fame Manner in the Ecliptic^

100$ they are referred to this Line, hy conceiving a great Circle to pafs through the Body, and cut the Eclipticnbsp;perpendicularly; for the Point, in which the Lclip-tic is cut by this Circle, determines the Longitude of the Body.

1006 nbsp;nbsp;nbsp;The Difiance of a heavenly Body from the Ecliptic is called its Latitude. It is meafured by annbsp;Arc of a great Circle, perpendicular to the E-cliptic, intercepted between the Body and thenbsp;Ecliptic.

1007 nbsp;nbsp;nbsp;Jf'vce imagine a Line to go thro� the Center of thenbsp;Sphere of the fixed Stars, and perpendicular to thenbsp;Ecliptic, the Points, in which this cuts the above-

1008 The Zodiac is a Zone which is imagined in thenbsp;Heavens, which the Ecliptic Line cuts intotwo equalnbsp;Parts, and which, on either Side, is terminated by anbsp;Circle parallel to the Ecliptic Line, and eight Degrees difiant from it. On Account of the fmallnbsp;Inclinations of the Orbits of the Planets, and thenbsp;Moon to the Plane of the Ecliptic, no Bodies ofnbsp;j 009 the Planetary Syftem appear without the Zodiac,

Book IV. of Natural Thilofoj^hy�

^bofe of them that have the fame Longitude are loio faid to he in Conjun�lion.

CHAP. IV.

Of the Thanontena of the Inferior Tlanetsy arifng from the Earth's and their ownnbsp;Motions in their Orbits.

Nate XXI. T E T S be the Sun, A V B � the Fig. 2.] JL/ Orbit of an inferior Planet; letnbsp;T be the Earth in its Orbit, and avh Part ofnbsp;the Sphere of the fixed Stars j the apparent Placenbsp;of the Sun is v.*nbsp;nbsp;nbsp;nbsp;�

If from the Earth there be drawn to the Orbit of the Planet the Tangents T A TB it isnbsp;evident that the Planet, in its apparent Motion,nbsp;is never removed farther from the Sun, than thenbsp;Diftance s a^ v b 1 and that the Planet accompanies it in its apparent Motion round thenbsp;Earth.

^be apparent Diftance of the Nanet from the loiz Sun is called its Elongation v a or V ^ is thenbsp;greateft Elongation: This varies upon two Ac-ioi}nbsp;counts} (viz.) becaufe the Earth and the Planetnbsp;revolve in elliptic Lines.*nbsp;nbsp;nbsp;nbsp;*9,5,

^he Planet performs its Motion fooner than the 1014 Earth j* therefore, in its Motion ^ it paffes ^^-*948nbsp;tween the Earth and the Snn.^ and then moves be~

Mathematical Elements Book IV.

yoiidthe Sun, in refpeSi of the Earth : So that it.is in ConjiiTiSlion �with the Sun in two Manners, butnbsp;never in Oppofition.

That we may have an Idea of the apparent Motion of the planet, we muft conceive the Lines T B h, T Sv, T A a, to move along with thenbsp;Earth j fo that the Points A, V, B, and v, whilftnbsp;the Earth performs its Revolution, may run thro�nbsp;the Orbit of the Planet; but the Planet, whichnbsp;moves fwifter, pafles fdcceffively through thefenbsp;Points over and over.

1015 IThen it is carried in its Orbit from V to D, it feems to move amongft the fixed Stars fromnbsp;V to d: In this Cafe, the apparent Motion is in An-loi�tecedentia, and the Planet is retrograde. In D itnbsp;is faid to be; becaufe it appears, fornbsp;forne S�ime, in the fame Place amongft the fixednbsp;: This obtains, when the Orbit of the Planet, in the Place in which the Planet is, is fo inclined to the Orbit �f the Earth, in the Place innbsp;�which the Earth is, that, if the Line t dhe. drawnnbsp;parallel to the Line T D, and at a fmall Diftancenbsp;from it, T) d be to T t as the Celerity ofthe Planetnbsp;in its Orbit, to the Celerity of the Earth j thefenbsp;*53 Lines are fun through in the fame Time j * andnbsp;the Line, which is drawn through the Earth andnbsp;the Planet, is carried in a parallel Motion ; fornbsp;which Reafon the apparent Place of the Planet isnbsp;^991 not changed. *

Between d and B, the Orbit of the Planet is more inclined to the Orbit of the Earth; wherefore the Extremity of the Line palTing throughnbsp;the Earth and the P�ianet ^although the Planetnbsp;1017 moves fwifter than the Earth) is carried in Confe-qnentia ; towards which Part alfo the apparent Mo-tion ofthe Planet is direSied. * Yet fince the apparentnbsp;Motion of the Sunexceeds the apparent Motion ofnbsp;the Planet, the Elongation is increafed, which

Book IV. of Natural Thilofofhy.

becomes greateft, when the Planet is at Whilft the Planet goes through the Arc B -v, itsnbsp;apparent Motion is alfo Confequentia^ and exceeds the apparent Motion of the Sun to whichnbsp;it is coming, and then goes beyond it,, recedingnbsp;from it, till it comes to A. Between A and �nbsp;the Motion Confequentia is continued ; but thenbsp;Sun, whofe apparent Motion in this Gafe isnbsp;fwifter, as has been explained concerning the Arcnbsp;^ B, comes towards the Planet, and the Elongation is diminiflied. At E, in the fame Manner ioi8nbsp;as at D, the Planet is ftationary, between E andnbsp;V it is again retrograde.

The Orbit of the Planet is inclined to the Plane of the Ecliptic,; * therefore it does not *9^s.nbsp;feem to move in the Ecliptic Line^ fometimesnbsp;lefs, fometimes more diftant from it, and appearsnbsp;to be carried in an irregular Ctirve^ which fometimesnbsp;cuts the Ecliptic.

Plate XXI. Fig, 3.J Let NVN be the Orbit of the Planet, whofe Nodes are N N; let S benbsp;the Sun ; T t the Orbit of the Earth in thenbsp;Plane of the Ecliptic ; the Earth T ; the Planetnbsp;V. If V A be imagined to pafs through the Planet, and to be perpendicular to the Plane of thenbsp;Ecliptick, the Angle VTA, or rather the Arcnbsp;which meafures it, is the Latitude of the Planet : * This is called Geocentric Latitude,, to *1006nbsp;diftinguifh it from the Latitude of the Planetnbsp;feen from the Sun, which is called the Heliocentricnbsp;Latitude, and is in that Cafe the Angle V S A.

Here we fpeak of the Geocentric Latitude, becaufe We examine the Phsenomena as they appear from 'nbsp;the Earth.

When a Planet appears in the Node, it appears 1019 in the Ecliptic Line-, and the Curve, which isde-fcribed by the Planet, by its apparent Motion in thenbsp;Zodiac, cuts the Ecliptic Line ; as the Planet 1120

Mathematical Elements Book IV.

recedes front the Node, its Latitude is encreafed, which is alfo different, accordingto the Situation ofnbsp;the Earth ; fo the Planet remaining at V, thenbsp;Latitude is greater if the Earth be at T, than ifnbsp;it was at t. Now, if the Earth remaining in itsnbsp;Place, we imagine the Planet to be carried fromnbsp;V to V, the Angle z� T B will be lefs than thenbsp;Angle VTA upon a double Account, from thenbsp;Planet coming nearer the Node, and the Spectator being moved farther off.

Now, if weconfider that both the Earth and the Planet are continually moved, we (hall eafily conceive that the Latitude is changed every Momentnbsp;from each Caufe, which fometimes a�t contrari-wife, and fometimes confpire in encreafing andnbsp;diminifhing the Latitude j whence it neceflarilynbsp;follows, that the Apparent Motion is performednbsp;in an irregular Curve, which, as was faid before,nbsp;cuts the Ecliptic as often as the Planet palTesnbsp;the Node, that is, twice in each of its Revolutions : This Curve alfo does not recede from thenbsp;Ecliptic, on either Side, beyond certain Limitsnbsp;in the Zodiac.

We difcover alfo fome remarkable Phacno-mena of the Inferior Planets by means of the Telefcope, which are owing to their Opacity.

Elate XXI. Fig. 4.] Let S be the Sun, T the Earth, A, B, C, v, D, E, F, V, an inferior Planet, ex. gr. Venus in its Orbit. This Planet fliinesnbsp;with Light borrowed from the Sun, and thatnbsp;Hemifphere only which is turned to the Sun isnbsp;enlightned, the other Hemifphere is invifible :nbsp;Therefore that Part only of the enlightened He-mifphere, which is turned to the Earth, can benbsp;feen from it j in V the Planet cannot be feen 5 in vnbsp;it would appear round, if the Sun�s Rays did notnbsp;hinder it from being feen.

Going from \ the Planet continually decreafes at D it has the Figure d i ax. e and � it is drawnnbsp;as it appears at E and F, and continues to de-creafe, till it vanijhes at V, and then again en~nbsp;creafes fucceffively^ changing its Figure^ �till thenbsp;whole enlightened Flemifphere be turned towardsnbsp;Earth.

When the Node is at V, or near it, the Planet lozz appears in the very Disk of the Sun^ and as itnbsp;were applied to it^ and is obferved as a black Spotnbsp;which moves on the Sun's Surface : In this Cafe,nbsp;properly fpeaking, we don�t fee the Planet, butnbsp;we perceive where it intercepts the Sun�s Rays.

The iefs diftant the Planet is from the Earthy the 1023 greater it appears.^ * and the more lucid ; but as it ^751nbsp;comes nearer the Earth, the lucid Part that isnbsp;vifible is lefs �, fo that on one Account the Lightnbsp;encreafes, and on another it is diminifhed; andnbsp;there is a Difiance at which the refle�ied Light isnbsp;greatefi-

CHAP. V.

Concerning the Thcenomena of the Superior Tlanets, arijingfrom their Motions andnbsp;the Motion of the Earth in their reJpeC�nbsp;tive Orbits.

The Apparent Motions of the Superior Planets do in many Things agree withnbsp;what has been explained in refpedl of the Inferiornbsp;Planets, and in many Things difagree. 'Fhey do notiozt^nbsp;always accompany the Sun, but are often cbferv~nbsp;ed in Oppofition ; but in their Oppofition (as hasnbsp;been faid of the Inferior Planets) they do not always feem to he carried in Confequentia, but often lozsnbsp;appear fiationary, and often retrograde.

ijx Mathematical Ele7ne7its BooK IV.

1026 Plate XXII. Fig. I.] Let M he a Superior Planet, for Example, Mars in its Orbit, ATHBnbsp;the Orbit of the Earth. The Periodical Timenbsp;of the Earth is Ihorter than the Periodical Timenbsp;quot;*948 of Mars * j therefore the Earth in its Motion goesnbsp;between it and the Sun ; in which Cafe the Planetnbsp;appears at F, amongft the fixed Stars oppofite tonbsp;the Sun. Through M draw 'the' Lines B M,nbsp;.A M, that touch the Earth�s Orbit, which, beingnbsp;continued, go to G and D in the Sphere of thenbsp;fixed Stars. Let us imagine, that whilft the Planet is carried about in its Orbit, thofe Lines arenbsp;alfo moved; fo that the Points A and B, in whichnbsp;the Lines that go through the Planet touch thenbsp;Orbit of the Earth, perform a Revolution in thenbsp;Periodical Time of the Planet. Now fince thenbsp;Earth revolves fafter, it paffes through the Pointsnbsp;A and B in its Motion. In this Motion the apparent Place of the Planet, feen from the Earth,nbsp;is not removed from the Place of the Planet feennbsp;from the Sun beyond F D and F G. Let T benbsp;fuch a Point in the Orbit of the Earth, thatnbsp;the Line t m, being drawn, may be parallelnbsp;to T M j let T ? be to M m, as the Celeritynbsp;of the Earth to the Celerity of the Planet;nbsp;in which Cafe thefe fmall Lines are gone throughnbsp;*S3inthe fame Time j in the mean time the Planetnbsp;�'994feems to be at reft, * and is faid to be ftationary.nbsp;In the fame Manner it is ftationary when thenbsp;Earth is at H. In the Motion of the Earth between T and H, the Planet appears to move in Jlnte-cedentia �xom E through F, and is faid to be retrograde ; whilft the Earth goes through the reftnbsp;of its Orbit, the Planet is dire�l.

1027. The Phsenomena,-which relate to the Latitude, are like thofe -which have been explained, in re-

'Jupiter and Saturn encompafs the Orbit of the 1028 Earth at a great Diftance j v/herefore almoft theirnbsp;whole Hemifpheres, which are enlightened bynbsp;the Sun, are vifible from the Earth; and therefore thefe planets always appear round.

Eecaufe Mars is lefs diftant, it appears a little gibbous., between the ConjunBion and Oppofition withnbsp;the Sun.

C H A P. VI.

Concerning the Thanomena of the Satellites., from their Motion in their Orbits. Wherenbsp;we Jhall fpeak of the Eclipfes of the Sunnbsp;and Moon.

The Satellites of Jupiter and Saturn io fl/-lo3o ways accompany their Primaries in theirnbsp;Motion, and never appear to recede from, them be~nbsp;yond certain Limits on either Side, which may benbsp;eafily determined from their DHtances from theirnbsp;Primaries j andthey are alternately carried in Ante-cedentia and in Confequentia. Sometimes all ofnbsp;them are on the fame Side of the primary Planet,nbsp;and fometimes the Primary is obferved to be betweennbsp;them j they are all always in the fame Right Line, 1031nbsp;or very little diftant from it. All which Thingsnbsp;may be deduced from this. That the Motion aboutnbsp;the primary Planets is performed in Planets thatnbsp;make fmall Angles with one another, and with thenbsp;Plane of the Ecliptic.

All the Satellites of Saturn or Jupiter are not al- 1032 Ways vifible at the fame Time; fometimes they arenbsp;hid by theirPrimary,oxA often immers'd in its Shadow.

Plate XXII. Eli;. 2.] Such an Immerfion in the 1033 Shadow is called an Eclipfe of the Satellite.

Mathematical Elements Book Iv .

Let S be the Sun, T t the Orbit of the Earth, I quot;fupiter^ Mm the Orbit of a Secondary of �Jupiter. Whilft the Secondary moves from M tonbsp;it undergoes an Eclipfe ; and, not being enlightened by the Sun, becomes invifiblc. If thenbsp;Earth be at T, the Emerfion into the Shadownbsp;is eafily obferved j on the contrary, the Emer-lion is more fenfible, if the Earth be placednbsp;at t.

*034 Amongft the Bodies that accompany Saturn, *963 we have faid that there is a Ring j * concerningnbsp;which it is to be obferved, that an Obferver uponnbsp;the Earth never fees it wider than it is reprefent-ed in the^th Fig. of Plate XX. and that fometimesnbsp;it is invifible j namely, when the Plane of thenbsp;Ring being continued, goes through the Earth �,nbsp;for the Thicknefs of the Ring is not fenfible. Thenbsp;Ring is alfo invifible, when its Plane continuednbsp;paffes between the Earth and the Sun ; for thennbsp;the enlightened Surface of the Ring is turnednbsp;from the Earth: In each Cafe Saturn appearsnbsp;round, yet in the laft Cafe, by reafon of thenbsp;Rays that are intercepted by the Ring, there appears a black Belt upon the Surface of the Planet, like that which is occalioned by the Shadownbsp;of the Ring.

The Phsenomena of the Earth�s Satellite, namely of the Moon, are very remarkable in re-fpe�t to us, and therefore particularly to be explained.

1035 nbsp;nbsp;nbsp;It is very often in Conjun�lion with the Sun,nbsp;and as often in Oppofition to it, but not at everynbsp;Revolution of the Moon in its Orbit ^ for whilftnbsp;the Moon, after one entire Revolution of 27nbsp;Days and 7 Hours, returns again to the Place

1036 nbsp;nbsp;nbsp;amongft the fixed Stars, in which it was in Con-^96ojunftion with the Sun, the Sun is gone from that

Book IV. of Natural Thilofofhy. nbsp;nbsp;nbsp;17 J

therefore theneighlouringConjun�iioiisare fwenty~ nine Days and a Half dijlant from one another.

The Lunar Periodical Month istbel'ime of one *�37 Revolution of the Moon in its Orbit.

The Moon�s Synodical Month, or a Lunation, 1038 is the Hime that the Moon [pends between the twonbsp;next Conjunbiions with the Sun.

^he Moon is invijible in its Con'jnnbiion with the 1039 Sun, becaufe the enlightened Hemifphere is turned from the Earth. Let T (^Plate XXII. Fig. 3.)nbsp;be the Earth, N the Moon between the Sun andnbsp;the Earth, the enlightened Hemifphere will benbsp;W li, which cannot be feen from the Earth.

Whilft the Moon is carried, in its Orbit,/row the 1040 Conjuntiion to the Oppofition, the enlightened Part,nbsp;which is dire�bed towards the Sun, does continually become more and more vifibleto the Inhabitants ofnbsp;the Earth; and in the Points A, B, C, the Moonnbsp;does fucceffively acquire the Figures a, b, c.

At P, in its Oppofition with the Sun, it appears 1041 round �, then going through D, E, F, it decreafes,nbsp;as it is reprefented at d, e,f.

31)0 Oppofition of the Moon with the Sun is called io43 ^he V\i\\-Moon,becaufe the whole Moon appears enlightened.

Mathematical Elements Book IV,

1044 9!he ConjunSiion and Oppojition of a Satellite voith the SuHy are called by the common Name Syzygies.

�043� At A and F, the dark Part of the Moon is a� little enlightened by the Rays that are reflediednbsp;from the Earth ^ and therefore it is feen by a Spectator to whom the Sun is not vifible, that is, innbsp;the firft Cafe, after the Setting of the Sun, and innbsp;the fecond, before its Rife.

1046 nbsp;nbsp;nbsp;When the Light of the Sun is intercepted by thenbsp;Moon, fothat, tn refpeSf of anyObferveruponthenbsp;Earth, the Sun is partly or wholly covered, the Sunnbsp;is faid to undergo an Eclipfe.

Properly fpeaking, this is an Eclipfe of the Earth, on whofe Surface the Shadow or Penumbra of thenbsp;Moon falls.

1047 nbsp;nbsp;nbsp;An Eclipfe of the Moon is the Ohfctirationof thenbsp;Moon by the Shadow of the Earth.

1048 nbsp;nbsp;nbsp;ithe Eclipfe of the Sim is never obferved, exceptnbsp;attheTimeoftheNew-Moon.

loyo Yet the Luminaries are not eclipfed at every one of the Syzygies, becaufe the Moon does not movenbsp;*969 in the Plane of the Ecliptic, * in which the Sunnbsp;and Earth always are ; wherefore, upon Accountnbsp;of the Moon�s Latitude, its Shadow, at the New-Moon, often does not touch the Earth j and it-felf, at the Full-Moon, pafles befide the Shadownbsp;of the Earth.

very little, that is, when it is in the Nod?, or near it, at its Syzygies, an Eclipfe is obferved i in that

Book IV. of Natural Thilofophy. nbsp;nbsp;nbsp;177

Cale the Moon appears to be in the Ecliptic, or very near it i and this it is that has given thenbsp;Name to that Line,

Plate XXI11. Fig. i.] That what relates to the Eclipfe of the Moon may appear the morenbsp;plainly, let OO be the Way of the Moon, R Rnbsp;the Plane of the Ecliptic; the Center of the*904;nbsp;Earth�s Shadow is always in it ; *N is the Node939nbsp;of the Moon�s Orbit.

If the Center of the Earth�s Shadow be at A, the Moon that goes by at F will not be darkened.

' .^the Moon be lefs diftant from the Node at 1052 the Full-Moon, as at G, the Shadow of the Earthnbsp;is at B, and the Moon is darkened in part ^ this isnbsp;called a partial Eclipfe.

If fuppofmg the Shadow at D, the Moon bet�53 Full, the Moon will he �wholly darkned at I, it runsnbsp;into the Shadow at L, and goes out of it at H inbsp;and the Eclipfe is faid to he T)tal.

Fhe Eclipfe is faid to he Central^ when the Cen~tos^ ter of the Moon goes through the Center of the Shadow^ which only happens in the very Node N.

We have hitherto fpoken of the Shadow of the Earth ^ becaufe, when we mention the Earth, wenbsp;underftand its Atmofphere which is joined to it,nbsp;of whichjwe have fpoken elfewhere: * Fhe Shadow *418nbsp;of the Atmofphere is properly confidered in Lunar 105nbsp;Eclipfes i for the Shadow of the Earth itfelf doesnbsp;not reach the Moon.

Plate XXIY. Fig. i.] Let T be the Earth, the Atmofphere about it FD GG DF. The Sun�snbsp;Rays B D, BD, touching the Atmofphere; thefenbsp;go {freight on, and terminate the Shadow of thenbsp;Atmofphere, out of which if the Moon be, it isnbsp;immediately enlightened by the Sun�s Rays, butnbsp;it is not enlightened in the fame Manner all thenbsp;'vhile it is between B D and B D.

VoL. II. nbsp;nbsp;nbsp;Nnbsp;nbsp;nbsp;nbsp;SThe

Mathematical Elements Book IV.

1056 '�he Rays zvhich enter the Atmofpbere obliquely are *617 refra�ted j * and, as they come towards the Earth,nbsp;^ ^ they continually penetrate into a Medium whichnbsp;4^9denfer and denfer, * and therefore are every

E F, penetrate the Atmofphere in the Curves F G, F G, that touch the Earth- All thenbsp;Light between E F, E F, is intercepted by thenbsp;Earth, and the Rays G A, G A, terminate thenbsp;Earth�s Shadow.

The Light between E F and B D, being re-fradfed by the Atmofphere, is fcattered between G A and B D continued, and beyond A, thenbsp;Point of the Earth�s Shadow, the Lights that

1057 nbsp;nbsp;nbsp;come from all Parts are confounded, but are continually weaker and weaker the farther from thenbsp;Earth ; So that the Shadow of the Atmofphere isnbsp;not a perfedl Shadow, but a weak Light, whereby the Moon is vifible in an Eclipfe.

1058 nbsp;nbsp;nbsp;^he- Shadow of the Atmofphere is Conical, be-caufe the Sun�s Diameter is greater than the Diameter of the Atmofphere, which is fcarce biggernbsp;than that of the Earth j and this Cone does notnbsp;reach quite to Mars, as appears from immediatenbsp;Obfervations ; but the Shadow of the Diameter,nbsp;in the Place where it is cut by the Moon�s Orbit, is fcarce one Fourth lefs than the Diameternbsp;of the Earth.

With the fame Reafoning that we have proved, that the Moon comes into the Shadow of the Atmofphere, when the Moon in its full is in thenbsp;Node or near it j it is alfo proved, that the Moon�snbsp;Shadow falls upon the Earth at the New-Moon,nbsp;when the Moon is in the Node or near the Node j

1059 nbsp;nbsp;nbsp;therefore in that Cafe the Sun undergoes an E-clipfe ; concerning which, feveral Things are tonbsp;be obferved.

T nbsp;nbsp;nbsp;Plate

Book IV. of Natural Thilofophy.. 179

Plate XXII. Fig. 4. J Let S be the Sun, T the Moon i let the Shadow of it fall upon any Planenbsp;G H. Ihis Shadow is encompalTed with a Penumbra, for beyond L and E that Plane is enlightened by one entire Hemifphere of the Sun-as you go from L to H, and from E to G, thenbsp;Light is continually diminilhed, and near G andnbsp;H the Rays come to the Plane only from a fmailnbsp;Part of the Sun�s Surface.

Fhis dirninifljed Light.y which eficompajfes the'^^(gt;o ^Shadow G H every Way^ is called the Penumbra.

In the Eclipfe of the Moon, the Shadow of the^^^i Earth is encompalTed with the like Penumbra, butnbsp;this is only fenfible near the Shadow, and therefore has blit a fmall Breadth �, but if an Obfervernbsp;be placed upon a Plane upon which the Shadownbsp;falls, he may obferve the whole Penumbra as is 1062nbsp;the Cafe in the Eclipfe of the Sim. An Obferver Inbsp;or F can only fee the Semidiameter of the Sun, thenbsp;reft of the Diameter is hid by the Moon; and going from L towards H, the Sun is continuallynbsp;more and more hid by the Moon, till it becomesnbsp;wholly invifible in the Shadow' itlelf.

Hence it follows, that there is a Solar Eclipfe, 1063 though the Shadow of the Moon does not touch thenbsp;Earth, provided the Penumbra comes to its Surface.

And alio, that the Eclipfe is not obferved inallthe 1064 Places in which the Sun is vifible-, and that it is 1065nbsp;different, according as the Shadow or a different jnbsp;Part of the Penumbra goes through the Place, innbsp;the Places in which it is obferved.

But the Eclipfe of the Moon is every where the 1066 fame, where the Moon is vifible, during the Eclipfe.

But when the Shadow itfelf of the Moon falls upon 1067 the Earth, the Sun�s Eclipfe is faid to be Fotal-, ifnbsp;�nly the Penumbra reaches the Earth, it is faid to be

Mathematical Elements Book IV.

1068 nbsp;nbsp;nbsp;But, as to particular Places, it is faid-to he tTo-tal in thofe Places where the Shadow pa^es Central^ in thofe where the Center of the Shadow paffes, that is, where the Center of the Moon coversnbsp;the Slinks Center j and laftly, Partial^ where thenbsp;Penumbra only goes by , and this is drawn in Fig. 6.

1069 nbsp;nbsp;nbsp;PtoeXXlI. Fig. 4.1 Fhe wider the Shadow G PInbsp;/�j, the more Places is the Eclipfe of the Sun Fotalnbsp;in, and the longer is the Sun wholly ohfcured. Butnbsp;the Breadth of the Shadow is different according tonbsp;the differenr Diftances of the Moon from thenbsp;Earth, and of the Earth from the Sun.

1070 nbsp;nbsp;nbsp;Jf there be an Eclipfe of the Moon, fuppofng thenbsp;Earth in the Perihelion, and the Moon in thenbsp;Jpogceum, that is, at the greateft Diftancefrom thenbsp;Earth, the Shadow of the Moon does not reachnbsp;the Earth, and the Moon does not cover the wholenbsp;Sun j fucb an one is called anAttniilar Eclipfe, andnbsp;is reprefented in Fig. 5.

CHAP. VII.

Of the Thanomena arifmg from the Motion of the Sun, the Elanets, and the Moon,nbsp;about their Axes,

L by obferving the Spots, which are obferved very often upon the Sun�s Surface. Thefe Spotsnbsp;feem to change their Figure and Situation everynbsp;Day, and fometimes to move fwifter, fometimesnbsp;flower; all which Things may be eafily deducednbsp;from the Motion of a Spherical Surface ; andnbsp;the Sun, which, if it was not moved by fuch anbsp;Motion, would only once in a Year fuccelflvely

turn its whole Surface to the Earth, now fhews it to the Inhabitants of the Earth in lefs than thenbsp;Space of one Alonth.

Such like Phsenomena arife from theKotation oflo*]Z yupiter^ Mars, and Venus, about their Axis, whichnbsp;Motions become fenfible, by obferving the Spots innbsp;the Surface of the Planets.

Whilft the Earth is moved round its Axis, the Obferver,who is carried round,imagines himfelf tonbsp;be at reft, and all the heavenly Bodies to be innbsp;Motion. ^

'fhe Points, in �which the Axis of the Earth, he- *073 ing continued both Ways, touches the Sphere of thenbsp;fixed Stars, are calledthe Poles of the World.

'The apparent Motion, arifingfrom the Motion of the Earth about its Axis, is called The Diurnalnbsp;Motion.

A Plane is conceived to pafs through the Center iQ^e of the Earth, perpendicular to its Axis, and conti-nued every Way, and the Circle, in which it cuts thenbsp;Sphere of the fixed Stars, is called The Coeleftialnbsp;�Equator.

In the Motion of the Earth round the Sun, the io�?6 �Equator is moved but fince the Plane of this

Circles,'whofe Planes go through the Axis of the io7'7 Earth, are called Meridians.

They all pafs through the Poles of the World, and 1078 are perpendicular to the JEqiiator.

i8x nbsp;nbsp;nbsp;Mathematical Elements Book IV.

1079 nbsp;nbsp;nbsp;which is interceptednbsp;between the Equator and a Star, is called the Declination of that Star.

Plate XXIII. Fig. 2 ] Let an Obferver beup-on the Earth 1�, who directs his Sight along Tgt;A; after a little Time, when the Line T A fhall benbsp;be carried by the Motion of the Earth to T a, ifnbsp;the Spedbator dire�s his Sight through the famenbsp;Line, the Body A will appear to have been carried through the Arc a A �, but when the Linenbsp;has returned to its former Situation T A, the Body w ill feem to have performed one whole Revolution. But if he direds his Sight along the Axisnbsp;of the Earth produced, becaufe thatisat relf, thenbsp;Body, which is feenin the Axis, will appear not to

1080 nbsp;nbsp;nbsp;have moved i therefore intbe Polcsof the World thenbsp;'�75 diurnal Fiction is not obferzed. * But that Bodies,

which are near the Poles, are moved round them is plain j and that the Body by its diurnal Motionnbsp;defcribes fo much a greater Circle round the immoveable Pole, as it is farther diftant from it.

1001 Xherefore the whole Sphere of the fixed Stars [eems to revolve about the Axis of the Earth continued,nbsp;in Antecedentia, in that Hme in vchich the Earthnbsp;really turns about its Axis. Therefore the diurnal Motion is common to all the coeleftial Bodies,nbsp;except fo far as it is diilurbed by the Motionsnbsp;above-mentioned.

The jTquator is equally diftant from both Poles, and divides the Eleavens into two Hemifpheres,nbsp;whofe middle Points are the Poles, whichnbsp;therefore are equally diftant frqm the feveral

1082 Points of the JBquator �, therefore the heavenly Bodies which are in the jFquator by their diurnal Motion feem to deferibe the JEquator itfelf, thenbsp;greateft Circle of all that can be deferibed

Book IV. of Natural T*hiloJ�phy.

by the diurnal Motion j the other Bodies defcriheio^'^ Circles parallel to the JEquator.

The Axis of the Earth is inclined to the Plane of the Ecliptic in an Angle of 66 Degr. 31 Min. *

The Poles of the World therefore are diftant from 1084 the Poles of the Ecliptic 23 Degr. 29 Min. and thenbsp;Plane of the Equator makes an Angle with thenbsp;Plane of the Ecliptic of 2^ Degr. 29 Min. Bothnbsp;Planes pafs through the Center of the Earth; butnbsp;fince this may be looked upon as the Center ofnbsp;the fixed Stars, * it follows, that the ^Equator and io8snbsp;the Ecliptic Line are great Circles^ which are inclined *9* snbsp;to each other, and cut one another in two oppofite 993nbsp;Points, in the Beginning of Aries, andthe Beginningnbsp;o/Libra; which Points, in the Way of the Sun,nbsp;are determined by thefe lnterfe6lions.nbsp;nbsp;nbsp;nbsp;^loot

When the Sun is in thofe Points, it fcems to de-tlt;cgt;^(gt; fcribe the Equator hy its diurnal Motionnbsp;nbsp;nbsp;nbsp;when it ^ 087

is carried about in the Ecliptic by its apparentMo- * tion, it continually recedes more and morefromthenbsp;JEquator, and its Declination is increafed, aiid itnbsp;defcribes lefs Circles every^ 'Day; * till it comes ro *ioS;nbsp;its greatefi Difiance from the Equator, which is 23 1088nbsp;Degr. 29 Min. * thenit comes backto the ^Equator*nbsp;again, and goes beyond it alfo 23 Degr. 29 Min.nbsp;advancing towards the oppofite Pole.

Thofe Circles, defcribed by the Sun in its diurnal Motion, which are mofi diftant from the Equator,nbsp;that is, 23 Degr. 19 Min. are called the Tropics.

One touches the Ecliptic Line in the firft Degree of Cancer, and is called the Tropic of Cancer-, the other is called the Tropic of Capricorn, andnbsp;pafles through the firft Point of the Sign Capri~nbsp;corn, and there touches the Ecliptic Line.

Mathematical Elements Book IV.

1090 nbsp;nbsp;nbsp;The Pole of the Worlds which is next to the 'Trope of Cancer, is called the Ar�lic Pole, and alfonbsp;the North Pole Oppofite is called the Antar�tic,nbsp;and alfo the South Pole.

1091 nbsp;nbsp;nbsp;Circles that are delcribed in the diurnal Motion by the Poles of the Ecliptic, that is, by thenbsp;Points which are dift ant from the Poles of the Worldnbsp;23 Degr. 29 yPxn.arectxlled the Polar Circles.

The Ar�tic PolarCircleis that which furrounds the Ardtic Pole ^ the other oppofite one borrowsnbsp;its Name from the Antardtic Pole.

1092 nbsp;nbsp;nbsp;There remains to be explained theMoon^s Motion ahoiit its Jxis, whofe Effepi is, that the famenbsp;Face of the Moon is alwaystiirned towards the Earth.

Plate XXII. Fig. 3.] Let the Moon be at N, the Face which is turned to the Earth is m n i-,nbsp;if the Moon did not turn about its Axis, andnbsp;all its Points were carried through parallel Lines,nbsp;the Line m i would coincide with the Line In innbsp;the Situation of the Moon at B, and theaforefaidnbsp;Hemifphere mni, would heamp;t Imn-, but becaufe,nbsp;whilft the Moon deferibes a fourth Part of itsnbsp;Orbit, it performs likewife of its Revolutionnbsp;round its Axis, the Face, which would be at I mnbsp;is now at m n i, that Is, again turned to the Earth.nbsp;After the fame Manner it is proved, that this famenbsp;Face mni, when the Moon is at P, is feen bynbsp;an Obferver on the Earth, and that it is turnednbsp;towards the Earth at � 3 as alfo jn all other Pointsnbsp;of the Moon�s Orbit.

3093 The Axis of the Moon is not perpendicular to the Plane of its Orhit, but a little inclined to it :nbsp;The Axis keeps its Parallelifm in its Motionnbsp;ropnd the Earth, as has been faid of the Pti-

mary

tnary Planets; * therefore it changes its Situa;ion/95i in refpe�t of an Obferver upon the Earth, tonbsp;whom fonietimes one, fometimes the other Polenbsp;of the Moon is vifible, whence it feems to be agitated by a Sort of libratory Motion. There is an-�o94nbsp;other libratory Motion obferved in the Moon; thenbsp;Motion about the Axis is equable, and it is carriednbsp;in its Orbit with an unequal Celerity ; * therefore *966nbsp;when the Moon is at its Perigcetwt, that is, at itsnbsp;leaft Diftance from the Earth where it is movednbsp;the fwifteft in its Orbit,* that Part of its Surface,*9fi6nbsp;which, on Account of its Motion in the Orbit,nbsp;would be turned towards the Earth, is not whollynbsp;turned from it on Account of its Motion roundnbsp;its Axis; Therefore fome of that Part of the Surface of the Moon, which before was not vifible,nbsp;is feen at the Side; which, when the Moon is atnbsp;its Apogxtm^ is again vifible.

Of the Thanomend which relate to the Surface of the Earth, and its particular Tarts.

WE have explained the Coeleftial Phsenome-na hitherto examined, by confidering the Spectator as carried about by thofe Motionsnbsp;wherewith the Earth is really moved. No\v wenbsp;fhall confider him as placed upon the Surface ofnbsp;the Earth, and carried from one Place to anothernbsp;upon it.

The firft Phtenomena to be here obferved is,^�9J that, by reafon of the Jnterpofitionof the Earth^onenbsp;Half of the Heavens is invifible to the Obfervernbsp;''coho is placed tiponthe Surface of the Earth.nbsp;Definition I.

fthat Circle in the Heavens^vobich feparates the vi-1096 fihle from the invifible Part, wh;n the Rays are

Mathematical Elements Book IV.

not intercepted by the Inequalities of the Earth�s Surface, is called the Horizon.

When the Height, to which the Spe�lator can be railed above the Surface of the Earth, is verynbsp;fmall, compared with the Semidiameter of thenbsp;Earth j the Eye of the Speciator may be lookednbsp;upon as placed in the Surface itfelf.

Plate XXIII. Pig. 4,] Let the Earth be at T, and the Obferver at S, and P E p e the Sphere ofnbsp;the fixed Stars j if you conceive a Plane at H Hnbsp;to touch the Earth and go through S, it will be thenbsp;Plane of the Horizon, whof� Seftion with thenbsp;Sphere of the fixed Stars is the Horizon. A Plane,nbsp;as h h, isconceived to gothrough the Center of thenbsp;Earth, parallel to H H ; theDittance ^ �P is infen-fible, by reafon of the immenfe Dilfance of thenbsp;fixed Stars; therefore the Seftion of that Plane,nbsp;Vv'ith the Sphere above-mentioned, may be takennbsp;�*99 ; for the Horizon.quot;*^

^ �99 If we conceive a Line drawn through the Obferver and the Center of the Earth, which muft neceffarilynbsp;be perpendicular to the Horizon, it will reachnbsp;the Point Z among the fixed Stars, which is callednbsp;ihe Zenith.

11 o i Ehe SePlion which a Plane of the Meridian, that goes through theObferzer, makes with the Horizon,nbsp;is called the Meridian Line �, and is directed fromnbsp;North to South,nbsp;nbsp;nbsp;nbsp;D e f i-

Book IV. of Natural 7hilofo^hy. 187

The Eaftern Part of the Heavens is that, fromf-^oz which we fee the Bodies rife above the Horizon-, andnbsp;the 'Ezik'Sdint is that, inwhichahinedire�isdEaft-'Wards,perpendicular to the Meridian Line, and goingnbsp;throquot;quot; the Obferver, cuts the Sphere of the fixed hars,

The Point, oppofite to this, is called the Weft ii 03 Point i and the Weftern Point of the Heavens isnbsp;oppofite to the Eafiern Part,

contained between the Eaft or IVefl Point, and the Point in which the Star rifesor fets. �'he firftnbsp;is called the Rifmg, and the other the fetcing Amplitude ; And each is either Northern or ijouth-rnbsp;ern Amplitude.

The Height or Altitude of a Star, above the 1105 Horizon, is the Arc of a Circle perpendicular to thenbsp;Horizon, in whofe Center the Spe�lator is, terminated by the Horizon and the Star.

The Difference of the Height of a Star, according 1105 to the different Pofition of the Obferver, as be isnbsp;fuppofed in the Center, or onthe Surface of the Earth,nbsp;is called the Parallax of the Star.

There is only the Parallax of the Moon, which can no�j bedeterminedby Obfervationsgflhe Diftanceof thenbsp;reft of the Bodies in the Planetary Syfteai is toonbsp;great to be compared with the Semidiameter ofnbsp;the Earth ; and the Parallax depends upon thenbsp;Ratio which the Semidiameter of the Earth has

Mathematical Elements Book iVquot;.

to the Diftance of a Planet; therefore even the \io% Parallax of Mars, in Oppofition with the Sun, isnbsp;too [mall for the niceft Okfervations.

1109 nbsp;nbsp;nbsp;Where there is a Parallax, it diminifies as anbsp;Body afeends above the Horizon, and vanifies innbsp;the Zenith.

The apparent Height of the Stars is alfo changed upon another Account, which equally afFeds all the

1110 nbsp;nbsp;nbsp;heavenly Bodies. The Rays are infledted by thenbsp;nil Kefra�iion of the Atmofphere,* and the Stars ap-'*^os(,pear higher than they are yet the higher they

^^fall lefs obliquely on the Surface of theAtmofphere. In the Zenith there is no Refra�lion* even at thenbsp;Diftance of twenty or thirty Degrees from the Zenith, it is not fenfible.

Plate XXIV. Fig. 2.] All thefe Things relate to the Surface of the Earth in general, now wenbsp;muft examine the feveral Parts of it j thefe arede-termined, by referring to the Earth the feveralnbsp;Circles which we have before confidered in thenbsp;lli4Heavens ; fo on the Earth we confider the Asqua-ton, the Meridians, the Fropics, and Polar Circles;nbsp;and thefe Circles divide the Surface of the Earthnbsp;in the fame Alanner as the Sphere of the fixednbsp;Stars is divided by the Circles in the Heavens :nbsp;And therefore the Circles in the Heavens, andnbsp;thofe upon the Earth, do fo mutually correfpondnbsp;with each other, that a Line being drawn fromnbsp;the Center of the Earth to a Circle in the Heavens, it will go through the fame Circle in thenbsp;Earth. If the Poles are P, p, the AEquator willnbsp;be E e, the Tropics T T, 11, and the Polar Circles A A,

^ Fhc Meridian, which goes through a Place, is called the Aleridian of the Place.nbsp;nbsp;nbsp;nbsp;'Ehe

BooklV. of Natural Thilofophy. nbsp;nbsp;nbsp;189

I�he Pi'ane of it is perpendicular to the Horizon; 1116 becaufe it goes through the Center of the Earth,nbsp;and the Obferver.

jd Meridian Line, drawn in any Place, is Part it ip of the M.eridian of the Place.*1101

The Latitude of a Place is its Difiance from 1118 the Mquator ; that is, the Arc intercepted between that Place and the JEquator.

Ey determining the Latitude of the Place, we determine the Circle of Latitude, which goesnbsp;through the Place j now, to determine the Situation of feveral Places, in refpe�l of each other,nbsp;we muft determine Places upon the feveral Circles ; which is done, by fuppofing a Meridian tonbsp;pafs through any remarkable Place, which, by itsnbsp;Sedion, determines a Point upon each Circle ofnbsp;Latitude, from which the Diftances of Places arenbsp;meafured.

^he Meridian above-mentioned, taken at Plea- 1120 fare, is called the Firft Meridian.

The Difiance of a Place from the firfi Meridian,nzi meafured on a Circle of Latitude that goes through anbsp;Place, is called the Longitude of the Place.

Jfironomers refer every thing to the Meridian of uzz the Place in which they make their Obfervations.

In explaining the Phaenomena which relate to the feveral Parts of the Surface of the Earth, wenbsp;ftiall confider the Obferver going from the Pole

Mathematical Elements Book IV.

1123 nbsp;nbsp;nbsp;Plate XXIII. Fig. 3.] When the Spedlator isnbsp;at S, in the very Pole of the Earth the coele-ftial iEquator E e coincides with the Horizon,nbsp;and the Pole of the World P is in the Zenith-, innbsp;that Cafe, becaufe the Circles which are parallelnbsp;to the Horizon, are alfo parallel to the ^Equator jnbsp;all the heavenly Bodies appear to be carried by

�*1083 a Motion parallel to the Horizon, * in Circles which are reprefented by the Lines K a, Bnbsp;Fhe heavenly Bodies in the Hemi/phere EF e never fet, and the others are never vifible. The Ho-

1124 nbsp;nbsp;nbsp;rizon in this Situation is [aid to be parallel^ ornbsp;this Situation is called a parallel Sphere.

1x25 Plate'KXXll. Fig. If an Obferver upon the Earth T recedes from the Pole, and is at S, thenbsp;Horizon is faid to be oblique, or the Sphere isnbsp;oblique 5 then the Axis P �gt; is inclined to the Horizon hh, fo much the more as the Obferver isnbsp;farther from the Pole.

1127 nbsp;nbsp;nbsp;This Height of the Pole is equal to the Latitude.nbsp;The Height of the Pole is the Angle P Tnbsp;whofe Meafure is the Arc P h the Latitude isnbsp;tneafured by an Arc, which upon the Earth cor-

�'iiigrefponds to the Arc ZE in the Heavens: * But it is equal to the Arc Fh for the Complement ofnbsp;each of them, to a Quarter of a Circle, is thenbsp;Arc Z P.

1128 nbsp;nbsp;nbsp;In this Pofition of the Obferver, becaufe thenbsp;.ffiquator is inclined to the Horizon, all the heavenly Bodies are carried by the diurnal Motion innbsp;Circles inclined to the Horizon, reprefented by thenbsp;Lines Ka, Eh.

Book IV. nbsp;nbsp;nbsp;of Natural Thilofophy.nbsp;nbsp;nbsp;nbsp;191

Some of the heavenly Bodies rife and fet at every 1129 Revolution of the Earth; namely, thofe which arenbsp;between the Parallels to the Equator B h andnbsp;hi-, becaufe all the Parallels between thofe Twonbsp;are cut by the Horizon.

The Planes of the Equator and the Horizon go through the Center of the Earth j therefore thefenbsp;Circles cut one another mutually into two equalnbsp;Parts, and Half of the jTquator is above the Horizon j therefore the heavenly Bodies, which are in 1130nbsp;the Equator, are above the Horizon during half anbsp;Revolution of the Earth about its Axis-, * and, on ^loSinbsp;Account of the Equability of the Motion aboutnbsp;the Axis, are invifible during an equal Time.

Thefe alfo rife due Eafi, and fet due Weft fthat n is, in the very Points of the Eaft and Weft;)fornbsp;the Se�tion of the Planes of the iTquator andnbsp;the Horizon, is perpendicular to a Plane perpendicular to both thefe Planes ; and this laft Planenbsp;is the Plane of the Meridan of the Place.* Where-fore the above-mentioned Sedtion is perpendicularnbsp;to the Meridian Line,* and confequently goesnbsp;through the Eaft and Weft Points.*nbsp;nbsp;nbsp;nbsp;*noa

Bodies between the Equator and a Parallel 'Bh, 1105 which touchesthe Horizon, as in the Circle A con- 1132nbsp;tinne longer above than below the Horizon ; andnbsp;this Difference is fo much the greater, the more thenbsp;Circle A a approaches that Pole, which is above thenbsp;Horizon. On the contrary, as the Body goes towardsnbsp;the oppofite Pole, its Time of Continuance above thenbsp;Horizon is the longer.

This Inequality of the Time that a Body is above ^ t34 and below the Horizon encreafes as the Height of thenbsp;Pole does, becaufe of the Diminution of the Angle made with the Horizon by the JEquator andnbsp;its Parallels.

Bodies, whofe Diftance from the Pole is eaiial to 1135 the Height ofitj never fet-, for fuch is the Diftance

Mathematical Elements Book IV,

of the Circle B which touches the Horizon, but has no Part of it below the Horizon.

*^36 It appears by the fame Reafoning, that Bodies, whofe Dijiance from the oppofite Pole does not exceed the Height of the Pole, never rife above thenbsp;Horizon, and are always invifible.

^^37 Bodies, whofe Diftance E Z from the JEquator is equal to the Height of the Pole, go through thenbsp;Zenith Z j for E Z is equal to the Latitude of thenbsp;Place to which the Height of the Pole isnbsp;*iii6equal. *

1138 Plate XXIII. Fig. 5.] When aSpediator S has receded as far as he can from the Pole, he comesnbsp;to the ^Equator, whofe Points are equally diftantnbsp;* 107 5 from each Pole ; * then the Axis P p is in the Ho-�'Hrizon, with which the JEquator makes a rightnbsp;*1075 Angle,* for which Reafon the Ho�zo� �/o

The Horizon cuts into two equal Parts all the Circles, that are parallel to the JEquator, whichnbsp;are reprefented by the Lines A a, B ^ ; thereforenbsp;w^cjall the heavenly Bodies, at every Revolutiojt of thenbsp;Earth, rife, andfet, and are vifible and invifiblenbsp;during equal Fimes.

1140 Fhe .^Equator itfelf goes through the Zenith, and therefore all the Bodies that are in it pafs throughnbsp;it alfo.

If what we have explained concerning the Diurnal Motion be applied to the Bodies of whofe other apparent Motions we have fpoken before,nbsp;the PhsDnoraena will be eafily determined fromnbsp;the Motions joined together : Thofe that relatenbsp;to the Sun are more remarkable than the reft�nbsp;and therefore more particularly to be explained.

Book IV. of Natural Thtlofophy.

We call a Natural Day the '�ime elapfed between * ^4^ the Kecefs of the Sun fromthe Meridian of a Place,nbsp;and its next Keturn to the fame Meridian.

This Day differs from the Time of the Kevolution 1142 of the Earth about its Jxis, which Times wouldnbsp;be equal, if the Sun appeared immoveable amongftnbsp;the fixed Stars j but whilft by the Diurnal Motion, in the Time of one Revolution of the Earthnbsp;about its Axis, the Sun is carried round fromnbsp;Eaft to Weft, that is, in Antecedentiaf it is carried by a contrary Motion in the Ecliptic,* whereby it comes later to the Meridian.

But as the Sun does not every Day go through an equal Space in the Ecliptic,*' all the Naturalnbsp;Days do not equally exceed the Revolution of the 1143nbsp;Earth about its Axis j therefore thefe Days arenbsp;unequal to one another.

Natural Days are unequal alfo upon another Account, namely, by reafon of the Inclination ofnbsp;the Ecliptic in refpeft to the jEquator j whencenbsp;it follows, that the Annual Courfe of the Sun isnbsp;Unequally inclined to the ^Equator in differentnbsp;Points ; and, though the Sun fhould equally gonbsp;forward every Day in the Ecliptic, the naturalnbsp;Days would not equally exceed the Time of thenbsp;Revolution about the Axis j for if the Motion ofnbsp;the Sun be refolved into two Motions,* of whichnbsp;the one is parallel to the Equator, and the othernbsp;perpendicular to it, the firft: is only to be con-fidered in determining the Excefs above-mentioned, and that it is unequal, is plain from the different Inclination above-mentioned.

Every Natural Day is divided into twenty-four 1144 equal Parts which are called Hours. Each Hour is

VoL. II, nbsp;nbsp;nbsp;Onbsp;nbsp;nbsp;nbsp;divided

194 Mathematk�l Elements Book IV.

divided into fixty Minutes ; and� each Minute into fixty fecond Minutes^ or Seconds �, and fo on. Thatnbsp;thefe Parts of 1'ime vary in different Days.^ appearsnbsp;plainly from what has been * faid j they are bynbsp;Aftronomers reduced to Equality, by confideringnbsp;the Number of Hours in the whole Revolution ofnbsp;the Sun in the Ecliptic, and dividing the wholenbsp;Time into as many equal Parts as there are Hours,nbsp;twenty-four of which are taken for one Day.

1145 nbsp;nbsp;nbsp;The Time, whofe Parts are by this Method reducednbsp;to Equality, is called Mean Time, and that Reduction is called the .Equation of Time.

1146 nbsp;nbsp;nbsp;always make ufe of the Days and Hoars ofnbsp;the Mean Time in determining the Periods of thenbsp;heavenly Motions.

1148 nbsp;nbsp;nbsp;in Oppofition to Night. In determining the Lengthnbsp;of Artificial Days, we fioall not attend to the Equation of Time.

and follows its Setting-, this is that dim Light which we commonly call Morning and Eveningnbsp;Twilight.

1151 nbsp;nbsp;nbsp;The Twilight is produced by the Atmo/phere,nbsp;which is enlightened by the Sun's Rays, and whofenbsp;Particles refieSithe Light every Way ^ from whencenbsp;fome Rays come to us, though the Sun be depreffednbsp;eighteen Degrees below the Horizon.

1153 nbsp;nbsp;nbsp;In the oblique Sphere the Days are longer orjhorter,nbsp;according to the different Defiance of the Sun ffom

Book IV. of Natural Thilofophy.

Degr. 29 Min.*

The Sun is in the JBquator about the iitb of

quot;h M ... I. nbsp;nbsp;nbsp;-T ^ 1/1 f\-f Cnbsp;nbsp;nbsp;nbsp;1154

recedes from the JEquator towards the Poles 23 r�----- nbsp;nbsp;nbsp;^yI;� *nbsp;nbsp;nbsp;nbsp;'�^7

'0. Ue �un �S tn ive jntjuuiui uvum, ive iita oj March, and. the xzth of September, and then thenbsp;Bay is equal to the Nighty* which happens allnbsp;o ver the Earthy except juft at the Poles.

^hofe Poifits of the Ecliptic, in which it is cut 115$ hy the JBquator,* are called the .ffiquino�tialnbsp;Points-^ becaufe the Sun is in thofe Points, whennbsp;the above-mentioned Equality of Day and Nightnbsp;happens.

Thefe Points of the Ecliptic, in which the 'Tropics touch that Circle,^ are called the Solfticial Points i becaufe for a few Days, when the Sunnbsp;Comes to thofe Points, and goes beyond them, itnbsp;does not fenlibly change its Declination, and thenbsp;Length of the Days not fenfibly vary.

Under the Poles, if there be any Inhabitants iijr� there, they can only once in a Year fee the rifingnbsp;and the fetting Sun, and only one Bay with onenbsp;Night make up their whole Tear. The Sun continues above the Horizon all the while it goesnbsp;through one Half of the Ecliptici* the reft of thenbsp;Time It is hid under the Horizon. But ytt their jnbsp;Bay is lengthened upon account of the Kefrahiion fnbsp;and the Twilights lafi very long, for they laft asnbsp;long as the Declination of the Sun towards thenbsp;hidden Pole does not exceed 18 Degr.*nbsp;nbsp;nbsp;nbsp;*11^1

At the Arfiic Pole, in the firft fix Signs, from 1158 Aries to Libra, the Sun is above the Horizon �,nbsp;therefore at that Pole the Bay exceeds the Nightnbsp;nine Natural Bays,* hefidestbe Diminution of the.^.^^Q^nbsp;Night on Account of the iLefraSHon.*nbsp;nbsp;nbsp;nbsp;-quot;m?

Mathematical Elements Book IV

Thefe general Things which relate to the different Pofitions of the Horizon being explained� fome more particular Things are to be examined.

Zones. The firft is contained between the twoitro-ii6opics TT, tt^ (Plate XXIV. Fig. 2.) and called the Torrid Zone j there are two Temperate Zones,nbsp;and two Frigid Zones. The Northern Temperate Zone is terminated hy the Tropic of Cancernbsp;TT, and the Ardic Polar Circle A A. Thenbsp;1161 Southern Temperate Zone is contained betweennbsp;tt^ the Tropic of Capricorn, and the Polar Circlenbsp;1162^^. q'he Frigid Zones are circumfcrihed by thenbsp;Polar Circles, and the Poles are in the Centers ofnbsp;them.

1163 nbsp;nbsp;nbsp;In the horrid Zone, twice a Tear, the Sun goesnbsp;through the Zenith at Noon.* For the Elevation

*1160 Qf pQjg jg lefsthan 23 Degr. 29 Min.* and the 1117 Dillance of the Sun from the J�quator towardsnbsp;the Pole, which is above the Horizon, is twice innbsp;*1087 a Year equal to the Height of the Pole.* For

1164 nbsp;nbsp;nbsp;namely under the Tropics, the Sun comes to thenbsp;.'*�087 Zenith only once in a whole Tear.*

1165 nbsp;nbsp;nbsp;Height of the Pole exceeds the greateft Diftancenbsp;*1087 of the Sun from the JEquator;* and therefore to

their Inhabitants the Sun never gees through the 1162 Zenith.* Yettbe fame Day theSunrifesto a greater

1166 nbsp;nbsp;nbsp;Height, the lefs the Height of the Pole is ; becaufenbsp;^1137 thereby the Inclination of the Circles of the

1167 nbsp;nbsp;nbsp;In the Torrid Zone, and in the Temperate Zones,nbsp;^^'^0 every Natural Day the Sim rifes and fets j for the

1168 nbsp;nbsp;nbsp;the Height of the Pole.* Yet everywhere but undernbsp;^loijthe JEquator* the Artificial Days are unequal to

11611151*113� nbsp;nbsp;nbsp;greater.

But in the Polar Circles^ juft where the Tempe- ^^^9 rate Zones are feparated from the Frigid ones, thenbsp;Height of the Pole is equal to the Diftance ofnbsp;the Sun from the Pole, when it is in the neighbouring Tropic ; * and therefore in that Cafe, *1089nbsp;that is, once a Tear, the Sun in its diurnal Motion '�9inbsp;performs one entire devolution'without goiji^ downnbsp;under the Horizon.

But every where in a Frozen Zone the Height of 1170 the Pole is greater than the leaft Diftance of thenbsp;Sun from the Pole �* therefore, durhig fome Revo- * '0*9nbsp;lutions of the Barth, the Sun is at a Diftance from 'nbsp;the Pole which is lefs than the Pole�s Height, andnbsp;during all that Time, it does not fet, nor fo muchnbsp;as touch the Horizon.* But where the Diftance *1135nbsp;from the Pole, as the Sun recedes from it, doesnbsp;exceed the Height of the Pole or Latitude ofnbsp;the Place, * the Sun rifes or fets every Natural *1117nbsp;Day;* then in its Motion towards the oppofite *gt;119nbsp;Pole, it flays in the fame Manner below the Horizon, 1171nbsp;as was faid of the Motion above the Horizon.^ 156

Thefe Times, in which the Sun makes entire Revolutions above the Horizon, and below it,nbsp;in its diurnal Motion, are fo much the greater,nbsp;that is, the longefl Day and Night lafl the iongefl,nbsp;the lefs the Place in the Frigid Zone is diflantnbsp;from the Pole, till at laft, at the Pole itfelf, theynbsp;take up the Time of the whole Year.

Form the fame Caufes, namely, the Obliquity of the Ecliptic in refpe�f of the JBquator, by which are occafioned all the Things whichnbsp;relate to the Inequality of Days, which is different in different Places; we alfo deduce thenbsp;Difference of Seafons, which fucceed one another every Year �, 1 ftiall fpeak of them firft in

19S nbsp;nbsp;nbsp;Mathematical Elements Book IV.

refpe�t to the Frigid and Temperate Zones, and then in refpeft to the Torrid Zone.

The Rays of the Sun communicate Heat to the Air, not only when they come diredtly from:nbsp;the Sun, but when they are reflefted irregularlynbsp;�'sSifrom Bodies or the Surface of the Earth.*

7 his Effe�t is fo much greater as the Rays ftrike the lefs obliquely^ againft the Surface of thenbsp;Earth i and that upon a double Account, i. Ifnbsp;^ you refolve the Motion of the Light into twonbsp;^^^Motions, *one pf which is parallel, and the othernbsp;perpendicular to the Surface of the Earth, thenbsp;Light a�fs upon Bodies only by this lall Motion,nbsp;which diminifties as the Obliquity encreafes.nbsp;2. There are more Rays adingatone Time uponnbsp;the fame Part of the Surface of the Earth, thenbsp;more diredly they come upon it.

Hence we deduce, that the Caufes of Heat en-creafe �when the Days encreafe^ by the Sun coming towards the Pole, which is above the Horizon j becaufe the Sun does daily afeend to a greaternbsp;Height; fo that to the diminished Obliquity isnbsp;added the longer Continuance of the Sun abovenbsp;the Horizon, both which concur to the encrea-fmg of the Heat ; the Nights alfo are diminilhednbsp;as the Days encreafe, and the Height that is produced by Day has lefs Time to decreafe in.

In the Northern Zones, as follows from this, the Caufe of the Heat is the greateft of all whennbsp;*^�5�the Sun comes to the Tropic of Cancer* Tet thenbsp;^^T^Heat is not always the greateft where the Caufenbsp;of Heat is the greateft-, for the Heat encreafesnbsp;as long as that which is acquired by Day isnbsp;not wholly deftroyed by Night j for though thenbsp;daily Agumentations be diminilhed, as long asnbsp;there is an Augmentation the Heat encreafes.nbsp;1175 The moft intenfe Cold is not upon the Ihorteft

Pay,

Day, in which the Obliquity of the Sun�s Rays is the greateft, and the Abfence of the Sun thenbsp;longeft ; but the Gold encreafes, as long as thenbsp;Diminution of Heat does laft concerning whichnbsp;one may reafon in the fame manner as concerningnbsp;the Encreafe of Heat.

^he Tear is divided into four Sea fans; the hottefi nrjS is called the Summer-, the coldeft the Winterthenbsp;temperate Seafon that follows the Winter, Spring jnbsp;andthe Autumn comes in between Summer and^ Winter.

In the Northern Regions, in the Beginning of Spring, the Sun appears to be in the Beginning ofnbsp;Aries. Jn the Beginning of Simmer the Sun comesnbsp;to the Tf-opic of Cancer. When the Sun enters Libra, the Autumn begins. Jn the Beginning of Winternbsp;the Sun performs its diurnal Motion in the Iropic ofnbsp;Capricorn ; all which may be eafily deduced from 1174nbsp;what has been explained.*nbsp;nbsp;nbsp;nbsp;*117?

In the Southern Regions, the Summer happens in the Time of the Winter above-mentioned, andtbeynbsp;have their Spring whilft the former have their Autumn-, andfo of the other Seafons.

The general Caufes, upon which the Divifion above-mentioned depends, are often difturbed bynbsp;Caufesrelating to particular Places -, efpeciallynbsp;the Torrid Zone, of which we faid we muft treat ofnbsp;feparately. In moft Places of this Zone there arenbsp;only two Seafons obferved, (viz.) Summer and Winter, which are chiefly dijiinguijhed by dry and wetnbsp;Weather.

When the Sun comes to the Zenith of any Place, there are almoft continual Rains j upon which Account the Heat is diminiflied, which Time is referred to, or called Winter. As the Sun recedes, thenbsp;Rains diminilh, the Heat is encreafed, and thatti^inbsp;Time is referred to Summer.

LOO nbsp;nbsp;nbsp;Mathematical Elements Book IV.

182 In the Middle of the horrid Zone there are two Summers^ and as many Winters �, becaiife the Sunnbsp;'1163 comes tip tivice to the Zenith. *

Towards the Sides of the Zone, tho� the Sun comes twice to the Zenith, yet fince there is butnbsp;a fmall Time between its coming to it the firftnbsp;and fecond Time, both the Winters are confounded into one j wherefore only two Seafons in anbsp;Year are obferved there.

CHAP. IX.

Concerning the Thanomena arijingfrom the Motion of the Axis of the Earth.

WE have faid that the Axis of the Earth is carried by a parallel Motion * i wenbsp;have not confidered a fmall Motion, whereby it is really moved, of which we fhall nownbsp;fpeak.

The Axis of the Earthy keeping the Inclination 1183 of 66 Degr. 31 Min. to the Plane of the Ecliptic^nbsp;revolves in Antecedentia, that is, is fucceffivelynbsp;1184 carried towards all Parts; and its Extremities,nbsp;(viz.') the Poles of the World defcribe Circles roundnbsp;the Poles of the Ecliptic, from Eaft to Weft. Andnbsp;this Revolution is performed in the Time of aboutnbsp;25000 Tears � which Period is called the Greatnbsp;Tear.

Becaufe the Earth is looked upon as immoveable by its Inhabitants, this Motion is referred to the heavenly Bodies, as has been faid of the othernbsp;Motions. Therefore whilft the Poles of thenbsp;World are moved about the Poles of the Eclipticnbsp;in Antecedentia, and pafs fucceffively thro� all thenbsp;Points that are 23 Degr. 29 Min. diftant fromnbsp;thefe Poles, thefe Points themfelves, or rather

the

Eook IV. of Natural Thilofophy.

the fixed Stars that are in them, come towards the Poles of the World fucceffively, and feem tonbsp;be carried in Confeqnentia, and to defcribe Circlesnbsp;which are really defcribed by the Poles of thenbsp;World about the Poles of the Ecliptic, which,nbsp;being placed in Centers, alone are at reft. Fornbsp;together with the Stars above-mentioned thenbsp;reft of the Stars (becaufe they keep the fame Situation in refped to one another) * do alfo ap- *915nbsp;pear to be moved.

Therefore the whole Sphere of the fixed Stars 11%^ feems to tnove^ in Confequentia, about an Axisnbsp;paffing through the Poles of the Ecliptic; and eachnbsp;Star apparently defcribes a Circle parallel to thenbsp;Ecliptic ; by which Motion the Latitude of thenbsp;Stars is not changed.

The Plane of the JEquator makes a right Angle with the Axis of the Earth j therefore, by the aforefaid Motion of its Axis, the Sedion of thenbsp;Plane of the Alquator, with the Plane of the 1186nbsp;Ecliptic, is moved round j wherefore the firftnbsp;Points of Aries and Libra *, which are always op-polite, move through the whole Ecliptic Line in thenbsp;Space of about 25000 in Antecedentia: Yetnbsp;they they are looked upon as immoveable by thenbsp;Inhabitants of the Earth, who imagine that thenbsp;fixed Stars themfelves are moved in Confequen'

CHAP. X.

Concerning the fixed Stars.

WE have faid that the fixed Stars are lucid Bodies, removed fo far off, that theirnbsp;Biftjnces can be compared with no Diftances in

ao^ Mathematical Elements- Book IVl

1187 nbsp;nbsp;nbsp;the Planetary Syftem. For Aftrommers have notnbsp;heen able, by their nicefi Obfervations, to obfervenbsp;the Poles of the World carried out of their Place innbsp;the annual Motion of the Earth, although they de- �nbsp;fcribe Circles in the Heavens which are equal to

Thatthe fixed Stars are at animmenfe Diftance, is alfo proved by Obfervations with the Help of

llSpTelefcopes. If any fixed Star, even the lucid and confpicuous, be beheld with a 1�elefcofe, throughnbsp;which the Diameter of the Sun would appear equalnbsp;to the Diameter of the Earth�s annual Orbit, itnbsp;will appear to he a lucid Point, without any fenfi-hle Magnitude ; for all the fixed Stars appear lefsnbsp;when they are feen through Telefcopes than theynbsp;do to the naked Eyej for it is only their Twinkling which makes them appear to have any fenfi-ble Magnitude.

1190 nbsp;nbsp;nbsp;��hatthe Stars may be diftinguijhed, they are referred to various Figures, which are imagined in thenbsp;Heavens, and are called Conftellations.

1191 nbsp;nbsp;nbsp;Twelve Conftellations are imagined to be in thenbsp;Zodiac, which are called the Sines of the Zodiac;nbsp;they receive their Names from the Animals or

1192 nbsp;nbsp;nbsp;Things which they reprefent : Aries, Taurus, Ge-^nbsp;mini. Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius, Capricornus, Aquarius, Pifces. Thefe Signsnbsp;have given their Names to twelve Parts of the E-

In the Time of Hipparchus, the Sedlions of the Ecliptic and Alquator. were between the Conftellations of Pifces and Aries, and Virgo and Libra,nbsp;and the Conftellations gave their Names to thofe

Book IV. of Natural Thilofofhy, nbsp;nbsp;nbsp;^03

Parts of the Ecliptic, which palTed through each Conftellation. And the Parts of the Ecliptic, ftippo-it^inbsp;fin^ the Beginningof Anes and Libra, in the Inter-fe flions of the J.quator and the Ecliptic, have keptnbsp;the Names �which they had at that Time, thoquot; thefenbsp;Interfe�iions be carried from their old Places-,nbsp;whence the Sun is faid to be in Taurus, when itnbsp;moves amongft the Stars of the Conftellation Aries.

In the Northern Region are the follo-wingConfieU 1194. lations ; the lefferBear, the greater Bear, the Bra~nbsp;gon, Cepheus, the Hounds, Bootes, the Northernnbsp;Crown, Hercules, the Harp, the S'wan, the Lizard,nbsp;CaJJlopeia, Camelopardus, Perfeus, Andromeda, thenbsp;Triangle, the leffer Triangle, the Fly, Auriga, Pe-gafus, or the Flying-Horje, Eqtinleus, the Dolphin,nbsp;the Fox, the Goofe, the Arrow, the Eagle, Antinous,nbsp;Sobiesky^s Shield, Serpentarius, the Serpent, Mountnbsp;Mcenalus, Berenice�s Hair, the lejfer Lion, the Linx.

In the Southern Region of the Heavens, are the followingConfiellations, many of which are invifible 1195nbsp;to us : * namely the Whale, the River Eridamis, thenbsp;Hare, Orion, the great Dog, Rhinoceros, the leffernbsp;Dog, the Ship Argo, Hydra, the Sextant of Urania,nbsp;the Clip, the Crow, the Centaur, the Wolf, the Altar, the Southern Crown, the Southern Fijh, thenbsp;Phosnix, the Crane, the Indian, the Peacock, thenbsp;Bird of paradife, the Southern Triangle, the Crofs,nbsp;the Fly, the Chamceleon, King Charle�j Oak, thenbsp;flying FiJh, the Toucan, or American Goofe, Hydrus,nbsp;or Water-Serpent, Kiphias,or the Sword-Filh.

The Stars are not equally lucid, and they areiigq *'eferred hy A^xonomcxstofixClaffes, the moll lucid

Mathematical Elements Book. IV,

cid are called Stars of the firft Magnitude j others are faid to be Stars of the fecond Magnitude, others of the third, 6�e. to the fixthnbsp;Magnitude.

1198 nbsp;nbsp;nbsp;Some are not referred even to this lafi Clafs, andnbsp;are called Nebulous Stars.

1199 nbsp;nbsp;nbsp;There is alfo a certain Zone or Belt obferved innbsp;the Heavens^ tw^/V^isnotevery where of the famenbsp;Breadth, and goes round the whole Heavens, andnbsp;in fome Places is feparated, fo as to become dou-

1200 nbsp;nbsp;nbsp;ble. From its Colour it is called the milky Way.nbsp;It is plain from Obfervations, by the Help of thenbsp;Telefcope, that this Way is an Affemblage of innumerable fixed Stars., which cajinot be feen by thenbsp;naked Eye, either becaufe they are lefs than thenbsp;other Stars, or more diftant.

1201 nbsp;nbsp;nbsp;Towards the Antarctic Vole there are two Nubecu-^nbsp;Ice, of the fame Colour as the Milky Way, which arenbsp;alfo Heaps of fmall Stars, and cannot be feen without a Telefcope. Befides the Stars, which arenbsp;obferved in thefe Nubeculac, and in the Milky

1202 nbsp;nbsp;nbsp;Way, to whatfoever Part of the Heavens you pointnbsp;the Telefcope, you may difcover fmall Stars in anbsp;great Number, which are not vifible to the nakednbsp;Eye. Very often an Heap of Stars appears to thenbsp;naked Eye to be but one Star.

1203 nbsp;nbsp;nbsp;Amongd the Stars, fome are vifible and invifiblenbsp;by Fits, and obferve regular Periods; others arenbsp;fuccellively fometimes more lucid, fometimes of anbsp;duller Light, and to be feen only by the Help ofnbsp;a Telefcope, and that at certain Times.

J204 Yet they are not equally bright at every Period. Sojiietimes Stars have appeared faddenly, exceedingnbsp;the brighteji in Light, which afterwards, fuccej-'nbsp;finely decreafing, have vanifioed in a Jhort Time,nbsp;and fill remain invifible.

12oj: Belides the Stars, we obferve in the Heavens fe-vptal whitijk Spot Sywhich are in fome Meafure lucid,

and tnvifible to the naked Eye-, their Light is referred to the Stars which are in them, or they are looked upon as nebulous Stars.

What thefe Spots are, cannot be determined ; perhaps they are a Congeries of Stars, whichnbsp;have the fame Relation to the Telefcopic Stars,nbsp;as thofe which form the Milky Way, have tonbsp;thofe which are feen to the naked Eye.

BOOK IV.

Part II.

The Fhyfical Caufes of the Cele-ftial Motions.

Aving explained the Motions of the heavenly Bodies, and the Phsenome-na arifing from them, we muft nownbsp;examine by what Laws thefe Motionsnbsp;are performed.

We have before laid down the Laws according to which the Motions of Bodies are direfted.

If we add one to thefe, we fliall fee the whme 126 Contrivance by which that vaft Machine, thenbsp;Planetary Syftem, is governed.nbsp;nbsp;nbsp;nbsp;^nbsp;nbsp;nbsp;nbsp;,

Book IV. of Natural Thilofifhy. nbsp;nbsp;nbsp;zoj

the Square of the Diftance : That is, all Bodies mutually attrad: or tend towards each other, withnbsp;the Force which belongs to each Particle ofnbsp;Matter ading upon each Particle �, and the Force,nbsp;with which a Body ads upon others, is compounded of the joint Forces of all the Particles of whichnbsp;the Body confifts j fo this Force encreafes in thenbsp;feme Proportion as the Quantity of Matter, andnbsp;is unchangeable in every Particle ; it is always thenbsp;feme at the fame Diftance; but the Diftance en-creafing, the Force decreafes as the Square of thenbsp;Diftance encreafes.

IVe call this Force Gravity^ when we confider a Body which of itfelf tends towards another �, becaufenbsp;this Force is called by this Name near the Earth�snbsp;Surface.*nbsp;nbsp;nbsp;nbsp;*7j

Btit when we confider a Body ^towards which an- taio othertends-, we call this Force AttraBion. We meannbsp;the feme Effed by thefe Names, and nothing butnbsp;the Effed; for finceall Gravity is reciprocal, * it�^wsnbsp;is the fame to fay, all Bodies gravitate mutuallynbsp;towards one another, as that Bodies mutuallynbsp;altrad one another, or mutually tend towards eachnbsp;other.

We look upon this Effed as a Law of Nature*, *4 becaufe it is conftant, and its Caufe is unknownnbsp;to us, and cannot be deduced from Laws thatnbsp;are known, as we fhall Ihew by and by. Now,nbsp;that there is fuch a Gravity, is to be proved fromnbsp;Phaenomena.

All the Primary Planets are kept in their Orbits by Forces, which tend towards the Center of the Sun quot;; therefore there is a Force by which�quot;mnbsp;the Planets are carried towards the Sun, andnbsp;whereby the Sun tends reciprocally towards eachnbsp;of them*: ^hat is, the Sun and Planets gravi- 1211nbsp;tate mutually towards each other.nbsp;nbsp;nbsp;nbsp;*126

xo8 Mathematical Elements Book IV-

1212 nbsp;nbsp;nbsp;After the fame Manner it is plain, that the Satellites of'Jupiter andSaturngravitate towards each

*9^6 other j as alfo that the Satellites of Saturn gravi-tate towards their primary Planet, and that towards � them*

1213 nbsp;nbsp;nbsp;Moon and the Earth zKo gravitate tovoardsnbsp;*966each other*

All the Secondary Planets gravitate towards ^^6the Sun. For they are all carried by a regularnbsp;*214 Motion about their Primary Planets, as if the Primary Planets were at reft ; whence it is plain, thatnbsp;they are carried about by the common Motionnbsp;with the Primary Planets ; that is, that the famenbsp;Force, by which the Primary Planets are everynbsp;Moment carried towards the Sun, alt;fts upon thenbsp;Secondary ones, and that they are carried towardsnbsp;the Sun with the fame Celerity as the Primary Planets. Even the Irregularities of the Secondarynbsp;Planets, which are fo fmall, as only to be fenfi-ble with refpe�t to the Moon, confirm this Gravity of the Secondary Planets towards the Sun ;nbsp;for we fhall (hew hereafter, that all the Irregularities are caufed by the Change of the Moon�snbsp;Gravity towards the Sun, at a different Diftance jnbsp;and becaufe the Lines, in which the Earth andnbsp;Moon tend towards the Sun, are not altogethernbsp;parallel.

1216 Ffom the Gravity of the Secondary Planets towards the Sun, it follows, that the Sun gravitates '^x�f.towards them.*

12117 refpe�t of the Gravitation of the Primary Planets towards one another, Aftronomers havenbsp;obferved, that Saturn changes its Way when it isnbsp;neareft to 'Jupiter, which is far the greateft of allnbsp;the Planets; fo that it is plain from immediatenbsp;Obfervations, that Jupiter and Saturn gravitatenbsp;towards each other,

Book IV. of Natural �Philofophy. 2,09

Jupiter alfo in this Cafe, as Flanifieed has ob-i2i8 ferved, difturbs the Motion of the Satellites of Saturn, attradling them a little to itfelf j whichnbsp;proves that theie Satellites gravitate towards Jupiter, and Jupiter towards them. From all whichnbsp;Confiderations * compared together, it follows,*�^''nbsp;that the feventeen Bodies, of which the Planetarynbsp;Syftem is made up, mutually gravitate towards 1114�nbsp;each other, although no immediate Obfervationsnbsp;can be made concerning the Gravitation of eachnbsp;particular one towards the reft.*

The fecond Part of the Law is,* that Gravity *1107 is proportional to the Quantity of Matter j thatnbsp;is, that all the Particles of Matter gravitatenbsp;towards each other; and therefore that thenbsp;Law of Gravity is univerfal, and that every Bodynbsp;a�ls upon all other Bodies ^ which is deducednbsp;from Phaenomena.

The Forces of Gravity are as the Qualities of 1219 Motion which they generate, * and thefe Quan-tities, in unequal Bodies that are equally fwift, arenbsp;to one another as the Quantities of Matter ^nbsp;therefore fince unequal Bodies, at the fame Di-ftance from the attracting Body, move equallynbsp;fwift by Gravity,* it is evident, that the15nbsp;Forces of Gravity are proportional to the Quantity of Matter. We find the fame in all Bodies 1^20nbsp;near the Earth's Surface, which have a Gravitation towards the Earth, proportional to theirnbsp;Quantity of Matter.* But the mutual Gravitynbsp;of all thefe Bodies towards one another is not fen-fible-, becaufe it is exceeding fmall in refpeCf ofnbsp;their Gravity towards the Earth; and thereforenbsp;cannot difturb their Motion arifmg from theirnbsp;Gravity towards the Earth ; * at leaft, fo as tonbsp;make any fenfible Change in the Direction ofnbsp;their Motions.

aio Mathematical Elements Book IV.

We fliall prefently Ihew, by another Method, that this univerfal Gravity of all the Particles ofnbsp;Matter, whereby they a�t upon one another,nbsp;�'iziirnay be proved from Phsenomena.*

The third Part of the Law, which we examine, is, that Gravity decreafes when the Diffance en-creafes, and is inverfly as the Square of the Di-ftances, which alfo follows from Phaenomena.

Bodies, upon which Gravity a�ls according to their Quantity of Matter, as in our Syftem, arenbsp;moved with an equal Celerity in the fame Circum-ftances, as we faid before; fo that it is no Matter,nbsp;whether the Bodies are greater or lefsj and theynbsp;are moved as if they were equal. But in thisnbsp;Cafe, if the Force towards a Point decreafes innbsp;an inverfe Ratio of the Square of the Diftancenbsp;from that point, and the Bodies move at variousnbsp;Diftances from it, and are kept in Circles bynbsp;that Force j the Squares of the Periodical Timesnbsp;will be to one another, as the Cubes of the Di-�^139ftances.* Which is demonftrated by Geometers to obtain (in refpeft of the mean Diftances)nbsp;in Elliptic Lines, whofe Forces are direftednbsp;to their Foci. But this is the Cafe in Bodiesnbsp;which revolve about the Sun^ Saturn and 'Ju-'*97^piter-, * whence it follows, that the Force ofnbsp;Gravity, receding from the Centers of thefenbsp;Bodies, decreafes in an inverfe Ratio of thenbsp;Squares of the Diftances.

1222 By this Reafoning, fuppofing Gravity proportional to the Quantity of Matter, we demon-ftrate, that it decreafes in an inverfe Ratio of the Square of the Diftance. And by the famenbsp;Reafoning, fuppofing the Diminution of Gravitynbsp;to be in this Proportion, it follows, that Gravitynbsp;is proportional to the Quantity of Matter, as isnbsp;very evident.

Book IV. of Natural T^hilofophy. 211

But we prove by another Argument, that the Diminution of Gravity, fo often mentioned, is innbsp;an inverfe Ratio of the Square of the Diftance inbsp;fo that there can remain no Doubt concerningnbsp;the two Laws of Gravity, which we now treatnbsp;of.

The Planets are moved in Orbits at and are kept in them by Forces, which are di- *9^9nbsp;redbed to an Excentric Point; * but it is plain,nbsp;that this would not obtain, if the central Forcenbsp;did not encreafe in an Inverfe Ratio of the Squarenbsp;of the Diftance.*nbsp;nbsp;nbsp;nbsp;^ 3

It follows from the fame Reafoning, that re- 1224 ceding from the Center of the Earth, Gravity de-creafes according to the fame Law. For thenbsp;Moon is retained in its Orbit by a Force whichnbsp;tends towards the Center of the Earth, that is,nbsp;to an Excentric Point * And though the Line of *967nbsp;the Apfides is not carried by a parallel Motion, 5Cgt;6nbsp;its Agitation is fo fmall, if we confider everynbsp;Revolution, that it may be looked upon here asnbsp;quiefcent ; For if we compute the Force whichnbsp;keeps the Moon in its Orbit fo agitated, we fliallnbsp;find the Diminution of the Force of Gravity, innbsp;refpedb of the Moon, to differ very little from annbsp;Inverfe Ratio of the Square of the Diftance ;nbsp;and we ftialll fhew hereafter, that this Difterencenbsp;depends upon the Adtion of the Sun.

And no Doubt will remain concerning this 1225 Diminution, if we confider, that the Moon is keptnbsp;in its Orhit by that very Force^ wherewith Bodiesnbsp;are carried towards the Earth, near the EartFs Snr~nbsp;face-, which is diminiftied, according to the Law ofnbsp;Diminution fo often mentioned. The Mean Diftance of the Moon is 601 Semidiameters of thenbsp;Earth : We have before fhewn, that a Diameternbsp;of the Earth contains 3400669 Khynland Perches whence knowing the Periodical Time, *976

V OL. II. nbsp;nbsp;nbsp;Panbsp;nbsp;nbsp;nbsp;we

112 nbsp;nbsp;nbsp;Mathematical Elements Book IV.

we eafily difcover that the Moon, in one Minute of Time, goes through 16425 i Kbynland Perches of her Orbit. This Arc is not the hundredth Part of one Degree, and may be lookednbsp;upon as its Subtenfe ; therefore the Diameter ofnbsp;its Orbit is to this Arc, as the Arc itfelf is to itsnbsp;Verfed Sine^ which is difcovered to be of 15,nbsp;736 Khynland Feetj and it is the Space whichnbsp;the Moon and Earth would go through in one Minute, coming to one another by their mutualnbsp;Attraflion. The Celerity with which a Bodynbsp;comes to another by the Force of Gravity, depends upon the Force with which it is attra�lednbsp;by that other, all whofe Particles of Matter at-trad: it �, therefore the Celerities of the Moonnbsp;and Earth, as they come towards each other,nbsp;are inverfly as the Quantities of Matter in them jnbsp;which is alfo deduced from the equal Quantity ofnbsp;*i,.6Motion that is in each Body.* Therefore bynbsp;65 this Proportion we difcover how much of thenbsp;aforefaid Space (15, 736 Feet) is gone throughnbsp;by the Moon : As the Quantity of Matter, in bothnbsp;Bodies, is to the Quantity of Matter in the Earthynbsp;fo is the Space gone through by both Bodies, innbsp;their mutual Accefs towards each other, to thenbsp;Way gone through by the Moon only. Thenbsp;Quantities of Matter in the Moon and in thenbsp;Earth, as we lhall (hew hereafter, are to onenbsp;another as i, to 39, 37; and 40, 37, is to 39,nbsp;37, as 15, 736 to 15, 344, the Space gone thro�nbsp;by the Moon j which therefore would be gonenbsp;through in one Minute by any Body, which at thenbsp;Moon�s Diftance Ihould be impelled by Gravitynbsp;towards the Earth. This Force increafing in annbsp;Inverfe Ratio of the Square of the Diftancenbsp;from the Center, the Space gone through in thenbsp;fame Time at the Diftance of a Semidiameternbsp;of the Earth, that is, on its Surface, will be

Book IV. of Natural ^h�lofiphy, nbsp;nbsp;nbsp;xi3

in every Motion equably accelerated, as here (for we confider the Force as removed from thenbsp;Earth�s Center to the Diftance of its Surface)nbsp;the Squares of the Times are as the Spaces gonenbsp;through in the Fall;* by dividing the Number by *131nbsp;60 X 60, that is 3600; we have the Space gonenbsp;through by a Body in one Second, near thenbsp;Earth�s Surface, by the Force with which thenbsp;Moon is kept in its Orbit, which is difcoveredtonbsp;be 15, 6. Khynland Feef.

Now if we examine the Gravity which we daily find in all Bodies near the Earth�s Surface,* *7^nbsp;it is plain, from what has been faid concerningnbsp;the Motion of Pendulums,* and from Experi-*M7nbsp;ments made upon Pendulums, that a Body innbsp;falling goes through 15, 6. Khynland Feet innbsp;one Second of Time, and therefore falls withnbsp;the Force by which the Moon is kept in its Orbit.

In this Computation we have negledled to confider the Adtion of the Sun, becaufe it isnbsp;fmall, and fometimes encreafes, fometimes di-minifhes the Gravity of the Moon towards thenbsp;Earth.

We have confidered the Centers of the Bodies in examining the Law of the Diminution of Gravity,nbsp;although Gravity belongs to all the Particles of Bodies; becaufe it is plain by Mathematical Demon-ftration, that the AElion of a fpberical Bodynbsp;wbicbjiuevery Part, tbe Particles, that are equallynbsp;diftant from tbeCenter,are bomogeneous, andwbicbnbsp;is made up of Particles, towards which there is anbsp;Gravity that decreafes, receding from each of themnbsp;in an Inverfe Ratio of the Square of the Diftance')isnbsp;direlfed towards the Center of the Body, and recedingnbsp;from it, is diminifhed in the fame Inverfe Ratio ofnbsp;the Square of the Difiance ; So that fuch a Body

Mathematical Elements Book IV.

a�ls, as if all the Matter, of which it confifts, was collected in its Center. Whence we dechice thenbsp;following Conclufions.

1227 nbsp;nbsp;nbsp;That on the Surfaces of Bodies, in which thenbsp;Matter is homogeneous at equal Diftances from thenbsp;Center, the Gravity is diredily as the ^lantity of

'*07 Matter, and inverjly as the Square of the Diameter �* J�08 for in thefe Bodies the Diftances from the Centernbsp;are as the Diameters,

1228 nbsp;nbsp;nbsp;That on the Surfaces of Bodies that are fpherical,nbsp;homogeneous, and equal, the Gravities are as thenbsp;Denfitiesof the Bodies-, for the Diftances from thenbsp;Center are equal; in which Cafe the Forces of

-1229 rical, unequal, homogeneous, and equally deitfe, the Gravities are inverfly as the Squares of the Diame-ters becaufethe Diftances from the Center arenbsp;��^�7 in the Ratio of thofe Diameters,* the Gravitiesnbsp;alfo are direftly as the Cubes of their Diametersnbsp;for the Quantities of Matter in Spheres are in thatnbsp;Ratio : And the Ratio compounded of that di-redf Ratio of the Cubes of the Diameters, andnbsp;the inverfe Ratio of their Squares, in the dire�lnbsp;Ratio of the Diameters tbemjelves.

�230 Therefore if both the Denfties and the Diame~ ters differ, the Gravities on the Surface will he in a

*1228 Ratio compounded of the Denfities* and the Diame-t�!^lt;jters.^ Therefore dividing the Gravity on the Surface by the Diameter, you will have/i'e

1231 nbsp;nbsp;nbsp;fity which confequently is in a direhl Ratio ofnbsp;the Gravity on the Surjace, and an Inverfe Rationbsp;of the Diameter.

1232 nbsp;nbsp;nbsp;If a Body be placed in a Sphere that is homogeneous, hollow, and every where of the fame Thick-nefs, wherefoever it be placed, it has no Gravity,nbsp;the oppofite Gravities mutually deftroying one

another precifely: Whence it follows, that z� lt;�;; 1233 homogeneous Sphere^ a Body, coming towards thenbsp;Center, gravitates towards the Center only fromnbsp;the Adion of the Sphere, whofe Semidiameternbsp;is the Diftance of the Body from the Center,nbsp;which Gravity decreafes in coming towards thenbsp;Center, in the Katio of the Diftance from the Cen- *1219nbsp;ter j for all Matter, which is at a greater Diftancenbsp;from the Center, forms an hollow Sphere, innbsp;which the feveral Actions on a Body deftroy eachnbsp;other.*

We have faid that the Gravity, which we have hitherto explained, is to be taken for a Law ofnbsp;Nature, becaufe we don�t know the Caufe ofnbsp;it j and becaufe it depends upon no Caufe thatnbsp;is known to us, which will evidently appear, ifnbsp;we attend to what follows.

(Viz.') That Gravity requires the Prefence of the attraBing Body ; fo the Satellites o� Jupiter,nbsp;ex. gr. gravitate towards Jupiter, wherefoever it

That the Diftance remaining the fame,the Cele- 1235 rity with which Bodies are carried hy Gravity, depends upon the ^lantity of Matter in the attra�l-ing BodyAnd that the Celerity is not change d,lct 1236nbsp;the Mafs of the gravitating Body be what it will* *1207

Befides that, if Gravity depend upon any known 1237 Law of Motion, it ought to be referred to a Strckenbsp;from an extraneous Body j and becaufe Gravity isnbsp;continual, a continual Stroke would be required.

If there be fuch a Sort of Matter continually ftriking againft Bodies, it muft of Neceffity benbsp;fluid, and very fubtile, fo as to penetrate all Bodies ; for Bodies, that are any how fhut up innbsp;others, are heavy.

Now let a Mathematician confider, whether a Fluid fo fubtile as freely to penetrate the Poresnbsp;of all Bodies, and fo rare, as not fenfibly to hin-P 4nbsp;nbsp;nbsp;nbsp;der

Mathematical Elements Book IV.

der the Motion of Bodies (for in a Place void of Air, the Motion of a Pendulum will be continued very long) can impel vaft Bodies towardsnbsp;one another with fo much Force ? Let him explain how this Force increafes in a Ratio of thenbsp;Mafs of the Body towards which another is car-�*1235 ried.*

Lajily, Let him fliew, what feems moft difficult to me, how all Bodies, in any Situation whatfoever (if the Diftance, and the Body to-wards which the Gravitation is, remain the fame)nbsp;are carried with the fame Velocity j* that is,nbsp;how a Liquid which can only aft on the Surfaces,nbsp;either of the Bodies themfelves, or their internalnbsp;Particles, to which it is not hindered fromnbsp;coming by the Interpofition of other Particles,nbsp;can communicate fuch a Quantity of Motion tonbsp;Bodies, which in all Bodies exaftly follows thenbsp;Proportion of the Quantity of Matter in them ;nbsp;aftd which in this Chapter we have proved to obtain every where in Gravity, and which we havenbsp;*77 demonftrated by a direfl Experiment, in refpeftnbsp;1238 of the Gravity near the Earth�s Surface.*

Yet we don�t fay, that Gravity does not depend upon any Stroke, but that it does not follow fromnbsp;that Stroke, according to any Laws known to ns,nbsp;and we confefs that we are entirely ignorant of thenbsp;Caufe of Gravity.

CHAP. XII.

Of the Celefial Matter ; �where a Vacuum is proved.

Having explained the Laws whereby the whole Planetary Syftem is governed,nbsp;feveral Things muft be firft laid down, beforenbsp;we proceed to the Pbyfical Explication of the

Book IV. of Natural Thilofophy. nbsp;nbsp;nbsp;r\y

Syftetn. We mull begin, by faying foniethingof the Celeftial Matter, that is, of the Mediumnbsp;in w hich the Bodies that make up the Syftem arenbsp;moved, which would be done infew Words, if allnbsp;Philofophers agreed that there is a Vacuum.

We have before proved that a Vacuum is pof-fible now we are to demonftrate that there is *13 really one. From only confidering Motion we can 12'^^nbsp;deduce a Vacuum and this is a very common andnbsp;ufual Way of proving it: To fee the Force ofnbsp;which Argument, we muft confider, that indeednbsp;all Motions are not impoffible without a Vacuum^nbsp;but moft of thofe which are daily obferved inbsp;which might be fully evinced by a longer Difcuf-fion : But it feemstome to be fo evident fromnbsp;the following Confideration, that it would be ufe-lefs to add much more.

The Figure of the leafl Particles is unchangeable; for the Particle, whofe Figure may be changed, confifts of fmaller Particles, which arenbsp;moved in refpe�t to each other j and therefore ifnbsp;it has a changeable Figure, it is not one of thenbsp;leaft Parts.

But if the Figure of thefe Particles be unchangeable, and a Body can move between them, without fuch a Separation of the Particles as tonbsp;leave a void Space, this will depend on the Figurenbsp;of the Particles, and the Relation which theynbsp;have to one another, which a Mathematiciannbsp;will not deny; Therefore, if keeping Things innbsp;this State (as to their Figure and Relation) thenbsp;Particles are encreafed, even in that Cafe Bodiesnbsp;may be moved without a Vacuum.

Now fuppofmg the fmalleft Particles encreafed to the Bignefs of a Cubic Foot, whatever benbsp;their Figure and Relation with the other Parts,nbsp;which we fuppofe encreafed in the fame Proportion as the firfti let any one confider, whether

ii8 Mathematical Elements Book IV.

Bodies of any Bignefs can be carried between thofe Parts in Right Lines, and in all forts of Curves,nbsp;and yet never feparate the Particles fo as to leavenbsp;Vacuities between them.

We cannot conceive how the ihoft fubtile Parts are made, and therefore often attribute to themnbsp;fuch Properties as do not follow from their Figure jnbsp;but thefe Errors will be corredied, by imaginingnbsp;the Particles encreafed.

We have faid that Matter is ina�livej* fome difpute about the Word, but no Man denies thenbsp;1 hing ; whence it follows, that a Body cannotnbsp;move through a Fluid, without undergoing a Re-*3'9 fiftance,* and confequently a Retardation.* Thenbsp;^330 Refinance arifing from the Inertia of the Matternbsp;fwhich Refinance alone is here confidered) depends upon the (Quantity of Matter to be removednbsp;out of its Place, which is the fame, whether thenbsp;parts of a Fluid be greater or iefs, if the Celeritynbsp;of the Body remains the fame: Whence it follows, that in determining what relates to thenbsp;Refinance, we mun have no Regard to the Subtility of the Fluid, as long as it cannot go throughnbsp;the Pores of Bodies for if we come to fuchnbsp;a Finenefs of Parts, that a Fluid fhall partly penetrate a Body, it will lefs refiR the Body.

Now let us fuppofe any Ball or fpheric Body to be moved along in a Medium of the famenbsp;Denfity as icfelf, and fo clofe, that the Parts ofnbsp;the Medium cannot pafs through the Pores ofnbsp;the Bodyj this Body will be retarded every Moment, fo that its Velocity at laR will be reducednbsp;to half (as may be proved by a Mathematicalnbsp;Demonttration) before the Body has gone throughnbsp;twice the Length of its Diameter.

Book IV. nbsp;nbsp;nbsp;of Natut al Thilofojgt;hy,

In order to apply this Propofition to a Motion in a very fubtile Fluid, which freely penetratesnbsp;the Pores of all Bodies, and fills all Places,nbsp;we muft conceive a fpherical Body without anynbsp;Pores at all j and that fuch a Body may be fup-pofed by imagining all the Particles of Matternbsp;clofely joined, no body will deny.

That the Refiftance of fuch a Body in any Fluid does not depend upon the Bignefs of thenbsp;Parts of the Fluid, and is the fame, whether thenbsp;Parts of the Fluid be equal, or any how unequal,nbsp;is evident.

If every thing be full of Matter, this Body can only mov.e through a Fluid of the fame Den-fity as itfelf j for it mull run againft all the Matter which is in thofe Places through which it paffes, and in them the Matter is without Interfti-ces as it is in the Body j therefore it will lofenbsp;half its Velocity, before it has run through thenbsp;Length of twice its Diameter.

firft Suppofition, when we conceived it to be without Pores; for the Body being encreafednbsp;the Surface has not been changed, but only dilated by the Interpofition of Pores : So that innbsp;both Cafes the Body will undergo the fame Refiftance

Now let us fuppofe the Body to be encreafed, the Quantity of Matter remaining the fame,nbsp;and the Body continuing homogeneous j that is,nbsp;let there be Pores in the Body, through which thenbsp;moft fubtile Particles of Matter may pafs verynbsp;freely ^ and let thefe Pores be equally difperfednbsp;all over the Body. If the Body, thus changed,nbsp;be moved, the very fubtile Fluid, of which wenbsp;fpeak, will not run againft the whole Surface, butnbsp;only thofe Parts of the Surface which are between the Pores, which Parts being taken together, becaufe we fuppofe the Body homogene-are equal to the Surface of the Body in the

220 nbsp;nbsp;nbsp;Mathematical Elements Book IV.

fiftance-^ from the Impulfe upon the Surface j and the Refiftance on the dilated Body is greater fromnbsp;the Fluid running againft the internal Parts ofnbsp;the Body. Wherefore this Body will fooner lofenbsp;half its Motion in the fecond, than in the firftnbsp;Cafe i that is, before it has run through the Lengthnbsp;of two Diameters of the firft fuppofed Bignefs inbsp;and therefore it lofes a greater Quantity of Motion, in going through two Diameters of thenbsp;Bignefs fuppofed in the fecond Cafe.

But this is contrary to Experience ; for a homogeneous Ball of Gold, or Lead, amp;c. lofes a much lefs Quantity of Motion than what we havenbsp;mentioned, in Water or Air; whence it follows,nbsp;that the Suppofition that all Things are full ofnbsp;Matter, is falie. Therefore there is a Vacuum.

Pbccnomena relating to Gravity ; by which it follows that it is proportional to the Quantity of Matter. If all was full of Matter, Gravity wouldnbsp;ad equally every Way, and the Forces which arenbsp;direded towards oppofite Farts would deftroy onenbsp;another; and therefore no fenfible Gravitynbsp;would be obferved, which is contrary to Experience.

1142 The Motions of the heavenly Bodies do not depend upon the Motion of the Celeftial Mathis* ter, if there be fuch a Matter whereby is overthrown. theOpnion of thofe which fay that the heavenly Bodies are carried along by the common Motion of the Matter which fills our Syftem. This Opinion is alfo overthrown by the Motion of the Comets : If there was a iVledium in the Syftemnbsp;which carried about the Planets in its Motion,nbsp;and alfo the Comets, it would at leaft fenfiblynbsp;difturb thefe laft in their Motions, whilft they

Book IV. of Natural Thilofofby. 2^l

come almoft diredly towards the Sun, or directly recede from it, or are carried in Antecedentia; that is, in a Motion contrary to the Motion ofnbsp;that Matter. Now as this Motion is not difturb-ed, but follows the Way which depends uponnbsp;Gravity, as it is obferved, it is plain, that ifnbsp;there be any Celeftial Matter, and that it is innbsp;Motion, it does not exert a fenfible ACtion onnbsp;the Bodies of the Planetary Syftem, which is al-fo deduced from the fmall Refiftance of fuch anbsp;Medium j for by comparing the moft ancient Ob-fervations with the modern, it does not appearnbsp;that the Planets are fenfibly retarded in theirnbsp;Motions. Yet in Air the Refiftance is fenfible;nbsp;wherefore the Denfity of the Medium, in whichnbsp;the Planets fliould move, muft be almoft im-menfly lefs ; therefore the Planetary Syjiem isnotnbsp;filled^ unlefs it be hy fuch a fubtile Medium.

But we may from the Divifibility of Matter deduce, that a Quantity of Matter, how fmall fo-ever it be, may be difperfed all over the Planetary Syftem, leaving but very fmall Inter-ftices. *nbsp;nbsp;nbsp;nbsp;*29

CHAP. XIII.

Concerning the Motion of the Earth.

Besides the Queftion that has been handled in the foregoing Chapter, there isnbsp;alfo another to be examined, before we proceednbsp;to the Explication of the whole Syftem.

That no Doubt may be made concerning the Syftem, which has been explained in the firftnbsp;Chapter of this Book, we muft here prove thenbsp;Motion of the Earth, concerning which it is nonbsp;Wonder that many have doubted ; for the Celeftial

ail nbsp;nbsp;nbsp;Mathematical Elements Book IV.

leftial Motions cannot be determined by us, but by Obfervationsmade by Obfervers on the Earthy,nbsp;and the fame Phsenomena appear, whether thenbsp;Bodies themfelves be moved, or the Spedatornbsp;*993 be moved fo that it is not to be proved bynbsp;immediate Obfervations, whether the Motion ofnbsp;the Earth is to be referred to the heavenly Bodies or not.

the Earth is carried about the Sun^ is de~ duced from the Analogy of the Motions, and from annbsp;Examen of the Laws of Nature.

As to what relates to the Analogy of the Motions, it is to be obferved, that Satellites revolve about yupiter and Saturn, which are lefs thannbsp;the central Body; that the Moon revolves aboutnbsp;the Earth, than which it is lefs. Lafily, Thatnbsp;the Sun has revolving about it lefs Bodies thannbsp;itfelf, as Mercury, Venus, Mars, fupiter, andnbsp;Saturn. Now if the Earth revolves with the

1245 nbsp;nbsp;nbsp;reft, then every where in our Syftem the leffer Bodies tnove abontthe greater: Now there would be annbsp;Exception in this Rule, in refpebi of the Sun, ifnbsp;thatva^ Body was to goromid fofmallaBody asthe

About the Sun, Jupiter Saturn, about which feveral Bodies revolve, thofe move the floweft which

1246 nbsp;nbsp;nbsp;moft difiant from the central Body, and according to this Rule, that the Squares of the Periodical Times follow the Ratio of the Cubes of the

''974Diftancesj* which Rule taamp;y be applied to the Earth, if it be carried about the Sun with the reftnbsp;of the Planets, as appears, if its Periodical Timenbsp;(namely the Time in which the Sun appears to perform an entire Revolution) and its Diftance fromnbsp;the Sun be compared with the Diftances and Periodical Times of the reft of the Planets.

Now this Rule has only one Exception, if the Sun be moved about, the Earth is at reft. In this

Book. IV. of Natural Thilofophy.

Cafe Mercury, Venus, Mars, Jupiter, and Saturn, are fubje�t to this Rule in their Motions, as alfonbsp;the five Satellites of Saturn, and the four Planetsnbsp;that accompany Jupiter only the Moon and thenbsp;Sun vi'ould move above the Earth in a Proportionnbsp;quite different �, and then the Celerity of thenbsp;Sun would not only be greater than is requirednbsp;by this Law, but its Velocity would at lealt benbsp;fix and twenty Times greater than that of thenbsp;Moon, though it be removed to a vaft Diftancenbsp;from the Earth, in refped: to the Moon�s Diftance :

And therefore in this refpedb, the Analogy of the Celeftial Alotions would be difturbed.

To thefe Arguments 1 fhall add others, whereby it will clearly appear, that the Motion of the Earth is a neceffary Confequence of the Lawsnbsp;of Nature, which are deduced from Phenomena.

therefore the Body moved is retained in a Curve by a Force, which is diredfed towards the Centernbsp;of the other. * Now as Re-alt;ftion is alwaysnbsp;equal to Adion,* unlefs the Laws of Nature,nbsp;which obtain conftantly every where, be whollynbsp;overturned, thefe two Bodies tend towards onenbsp;another with equal Motions ; that is, with Celerities that are inverfly as their Maftes;* which *65nbsp;is alfo immediately deduced from the Law ofnbsp;Gravity.*nbsp;nbsp;nbsp;nbsp;�'1135

The Quantity of Matter in the Earth is next to nothing, in comparifon to the Quantity of

Matter

All Bodies gravitate towards one another, * 124? therefore the Sun and Earth do; but the Mo-^nbsp;tion, whereby thefe Bodies tend towards onenbsp;another, is deduced from diredt Obfervations.nbsp;Whichfoever of thefe Bodies moves about thenbsp;other, deferibes Area�s by Lines drawn to thenbsp;Center of it, proportional to the Times, whichnbsp;evident from Aftronomical Obfervations ;

Mathematical Elements Book IV.

Alatter in the Sun, as we fhall fhew in the following Chapter wherefore the Sim miift move very (lowly whilft the Earth conies towards it verynbsp;fwiftly.

Whence it follows, that the Earth is carried round the Sun, left it Ihould fall upon the Sunnbsp;by that very violent Motion whereby it is retained in its Orbit.

Two Bodies, that are carried towards one another by any Force, will at laft concur, or continually recede from one another, unlefs each of them be fo moved, as to have a centrifugal Forcenbsp;equal to the Force whereby it is carried towardsnbsp;the other Body �, but as the Bodies, whichnbsp;gravitate towards one another, tend towardsnbsp;*i26each other with equal Forces,* or what is thenbsp;*6 5 fame,* with Celerities that are inverfly as thenbsp;�*gt;13^Quantities of Matter j* thefe Bodies cannot per-i248fevere in their Motions about one another, unlefsnbsp;both of them be fo moved, as to have equal centrifugal Forces, which does not happen, unlefs theynbsp;both revolve in equal Times about their commonnbsp;Center of Gravity ; That is, if this Propofitionnbsp;be applied to the Sun and Earth, unlefs theynbsp;both move about a Point, whofe Diftance fromnbsp;the Center of the Sun is to its Diftance fromnbsp;the Center of the Earth, as the Quantity ofnbsp;^ Matter in the Earth is to the Quantity of Mat-^��^ter in the Sun, they cannot perfevere in theirnbsp;Motions about one another:* This Pointernbsp;Center of Gravity mull: of Confequence be very near to the Sun�s Center. Now fincenbsp;whichfoever of thefe Bodies moves, it perfe-veres in its Motion about the other, it follows,nbsp;that both of them are affeded by the Motionsnbsp;above-mentioned, and that the Sun is moved

Book IV. of Natural Thilofophy. nbsp;nbsp;nbsp;'i.'iS

but a little, whilft the Earth defcribes a very great Orbit. Whence it follows, that the Motion ofnbsp;the Earth cannot be denied by any one, whonbsp;reafons from the Laws of Motion that are deducednbsp;from Phsenomena.

Having proved annual Motion of the Earth 1249 and brought back the Earth amongft the Planets,nbsp;there remains but little Difficulty in relation tonbsp;the Motion of it about its Axis-, for no body,nbsp;that believes the annual Motion, doubts of this;nbsp;a great many, which allow of the Motion aboutnbsp;the Axis, deny the annual Motion; therefore itnbsp;will be enough to obferve by the By, that allnbsp;the Planets, concerning which any Obfervationnbsp;could be made in refpe�t of this Motion, donbsp;move about their Axes and that the Earth hasnbsp;fuch a Motion, the uniform diurnal Motion innbsp;Bodies, at any Diftances, does plainly enough Jhew.

To which we muft add, that the Celerity of the fixed Stars, going through one whole Revolutionnbsp;lefs than 24 Hours, can hardly be more probablenbsp;than it is conceivabl�;

This Motion alfo is difagreeable to the Nature of all the heavenly Bodies for, if they are car- 1250nbsp;ried round, they muft every Day, with an equable Motion, de�cribe Circles that have the Earthnbsp;for their Center; that is, they muft, by Linesnbsp;drawn to the Center of the Earth, fweep throughnbsp;Area�s proportionable to the Times, and be retained in their Orbits by Forces which are diredednbsp;towards the Center of the Earth,* and bynbsp;which (by reafon that Aftion and Re-acftion * *126nbsp;are equal) the Earth muft alfo be continually at-tra�led towards thofe Bodies j fo that it muft ne-cejfarily he agitated by a very violent Motion 5 whencenbsp;it appears that the diurnal Motion muft notnbsp;be referred to the heavenly Bodies, but to thenbsp;Rotation of the Earth about its Axis.

Mathematical Elements Book IV.

1251 Thofe that obftinately affirm that the Earth is at Reft, objed; that Bodies, upon the Surfacenbsp;of the Earth, muft (on account of their centrifugal Force) recede from the Earth, along anbsp;^'7 Tangent to a Circle parallel to the JEquator.*nbsp;We anfwer, that the Bodies in the Places wherenbsp;they are, are carried round with the fame Motionnbsp;as the Surface of the Earth and therefore that,nbsp;in refpe�t of the Points of the Surface, they en-^ deavour to recede in Lines perpendicular to thenbsp;Axis but alfo that Bodies by Gravity tend to thenbsp;Center of the Earth and therefore by a Motionnbsp;compounded of both thefe, the Body is conti nu-190 ally moved, or endeavours to move ;* but becaufenbsp;19S the firft Motion is extremely fmall in refpedl ofnbsp;the other, a heavy Body is turned but very little out of its Diredion towards the Center, andnbsp;the Gravity is a little diminilhed, fo much thenbsp;more as the Place is more diftant from thenbsp;Pole; which agrees with Experience. We (hallnbsp;hereafter Ihew, when we come to fpeak of thenbsp;Figure of the Earth, that the above-mentionednbsp;Diredfion of heavy Bodies is every where diredl-ed perpendicularly to the Surface of the Earth.nbsp;A Body, which is thrown upwards, is adlednbsp;upon, not only by the Motion wherewith it isnbsp;thrown upj but it is alfo carried by the Motion that is imprefled to the Perfon or Machine that impels the Body ; that is, it is carriednbsp;by the Motion which is common with the Surface of the Earth j and therefore the Body movesnbsp;in the fame Line (the Line being carried onnbsp;with the Surface of the Earth) as it would do ifnbsp;the Earth was at Reft.

CHAP.

CHAP. XIV.

BEFO RE we proceed to the PhyficalExplanation of the Syftem, we muft determine the Quantities of Matter in fome Bodies, and their Diverfities j which being known, thenbsp;EfFe�fs of the Laws, by which thefe Bodies arenbsp;governed, will more eafily appear.

The Quantities of Matter, in different Bodies, are to one another, as the Gravities at the famenbsp;Diftance from thefe Bodies which Gravitiesnbsp;are to one another inverlly as the Squares of thenbsp;Periodical Times of the Bodies revolving aboutnbsp;thofe different Bodies at the fame Diftance.�*^

By multiplying the Quantities which are in this Ratio by the fame Quantity, (wz.) by the Cubenbsp;of this Diftance, the Ratio of thefe Quantities willnbsp;not be changed j which are therefore to one another as the Quotients of the Divilions of the a-bove-mentioned Cube, by the Squares of the Periodical Times aforefaid : But the Quotient of fuchnbsp;a Divifion is found for any Body, by dividingnbsp;the Cube of the other Diftance j let it be whatnbsp;it will, by the Square of the Periodical Time ofnbsp;the Body revolving at that Diftance; for fuchnbsp;Quotients are equal to one another, for all Bodies that revolve about the fame Body at anynbsp;Diftancesj as follows from the Equality of thenbsp;Ratio between the Cubes of the Diftances, andnbsp;the Squares of the Periodical Times at thofenbsp;Diftances.* From which we deduce, that thenbsp;^lantities of Matter^ in any Bodies in our Syftem^ 125anbsp;are to one another dire�ily as the Qt�es of thenbsp;Lift antes at which other Bodies revolve about thefe,

Mathematical Elements Book IV.

Thefe Things are demonftrated by fetting a-fide the Agitation of the Central Body, whofe Quantity of Matter is enquired after.

By reafon of the Sun�s Magnitude, in refpe�fc of V^enus^ ex. gr. which alone we confider of thenbsp;Planets, the Sun is fcarce moved by the A�tionnbsp;^�3 5 of that Planet.* And Venus may be conliderednbsp;as moving about a quiefcent Body.

The Satellites of Jupiter and Saturn are indeed carried by the common Motion along with the Primary Planets, but by reafon of the Magnitude of the Primary Planets, they are carriednbsp;about them as about Bodies that are at reft.

But the Moon afts fenfibly enough upon the Earth and moves it; wherefore before we can compute the Motion of the Moon by the Help of thenbsp;*1251 aforefaid Rule,* in order to compare the Quanti-1253 ty of Matter in the Earth with the Quanti*nbsp;ties of Matter in the Sun, Jupiter, and Saturn,nbsp;we muft determine the Dijiance at which thenbsp;Moon would move about the Earth, if it was atnbsp;Reft ^'that is, not carried about by the Aliion ofnbsp;the Mooti) in the fame Periodical '�ime in whichnbsp;it now performs its Kevolution. Here alfo we don�tnbsp;take Notice of the Motion that is common to thenbsp;Earth and Moon, by which they are both carried about the Sun.

The Moon perfeveres in its Motion about the Earth; therefore the Earth and Moon are movednbsp;about a common Center of Gravity ; as followsnbsp;from what has been demonftrated concerning thenbsp;*1248 Earth and the Sun,* and the Moon (with thatnbsp;Force with which it tends towards the Earth)nbsp;revolves in an Orbit whofe Sepai-Diameter is thenbsp;Diftance of the Moon from the aforefaid common

Book IV. of Natural Thilofiphy, nbsp;nbsp;nbsp;zx9

Let L be this Diftance of the Moon from the common Center of Gravity 3 T the Diftance ofnbsp;the Earth from the fame Center 3 L T therefore is the Diftance of the Moon from the Earth,nbsp;and is 60 { Semi-Diameters of the Earth ; for herenbsp;we confider the mean Diftance. Let D be thenbsp;Diftance which we would have, at which thenbsp;Moon, by its Gravity towards the Earth, wouldnbsp;move about the Earth, if it was at Reft, in thenbsp;fame Time in which it is now moved aboutnbsp;the common Center of Gravity at the Diftancenbsp;L.

By reafon of this Equality of the Periodical Times, the Force, whereby the Moon would benbsp;kept in its Orbit at the Diftance D, is to thenbsp;Force whereby it is kept in its Orbit at thenbsp;Diftance L, as D to L.*nbsp;nbsp;nbsp;nbsp;*131

But the Force whereby the Moon would tend to the Earth, and be kept in its Orbit at thenbsp;Diftance D, is to the Force whereby it is nownbsp;kept in its Orbit at the Diftance L -f- T, asnbsp;IT Tquot;! to D^*. Thereforenbsp;nbsp;nbsp;nbsp;��

Confequently D�' = L x L T^, and O'quot; x L -f- T = Lx L T�= : Whence we deduce the followingnbsp;Proportion :

ThereforJ^M^fD � :\VT,^is to thefirft of two mean Proportionals between L -f- T and L.

L -f- T is to L, as the Quantity of the Matter in the Earth and Moon taken together, to thenbsp;Quantities of Matter in the Earth alone 3**134nbsp;which Quantities of Matter, as we fliall ftiew ^3 5nbsp;hereafter, are to one another, as 40, 37. tonbsp;Q 3nbsp;nbsp;nbsp;nbsp;39� 37-

Mathematical Elements Book IV

39, 37. and the firft of two mean Proportionals to thefe Numbers 1340,035;; therefore 40, 37. isnbsp;to 40,035. as 604 to the Diltance required,nbsp;which is found to be 60 Semi-Diameters of thenbsp;Earth.

Concerning this Operation it is to be noted, that the Diftance D cannot be difcovered, unlefsnbsp;the Ratio between the Mafs of the Moon and thenbsp;Earth be known, which cannot be determinednbsp;unlefs the Ratio between the: Denfity of the Sunnbsp;and the Earth be found : To difcover which, itnbsp;is neceflary that the Diftance D be known.nbsp;Wherefore D is difcovered at firft by Trials, andnbsp;is exa�lly determined by Approximation. Butnbsp;it is certain, that this is 60 Semi-Diameters ofnbsp;the Earth ; becaufe, this being fuppofed, it isnbsp;found that the Ratio between the Quantitiesnbsp;of Matter of the Moon and the Earth is asnbsp;I to 39, 37. as we fhall fee hereafter; by making ufe of which Proportion, this Diftance isnbsp;difcovered to be 60 Semi-Diameters, as we havenbsp;Ihewn.

The Diftance of Fenus from the Center of the Sun is 723, and its Periodical Time is 5393nbsp;quot;^959 Hours.

The Fourth Satellite of Jupiter is diftant from �.he Center of Jupiter 12, 507. fuch Parts ofnbsp;which Venus is diftant from the Sun 723. Thenbsp;Periodical Time of this Satellite is 402 Hours,nbsp;�^9713; Minutes.*

The Fourth Satellite of Saturn is diftant from the Center of Saturn 9, 292. of the fame Parts;nbsp;*97iand its Periodical Time is 382 Hours 41 Min.*

Laftly, the Diftance of the Moon is 60 Semi-Diameters of the Earth from its Center, and

Book IV. of Natural Tbilofofhy, 2,31

If you divide the Cubes of thefe Diftances re- 1254 fpe�tively by the Squares of the Periodical Times,nbsp;you will have, in the Quotients, Numbersnbsp;which are to one another as the Quantities ofnbsp;Matter in the aforefaid Central Bodies ;nbsp;which Quotients are to one another as the following Numbers;

We have alfothe Proportion of the Diameters 1256 of thefe Bodies from Aftronomical Obfervations,nbsp;as follows.

Diame- ? Of the Sun.Of'jupiter. Of Saturn. OfheEarth ters J loooo. 1077.nbsp;nbsp;nbsp;nbsp;889.nbsp;nbsp;nbsp;nbsp;104.

If the Quantities of Matter abovefaid be di-ia57 vided by the Squares of the Diameters, the Quotients will be to one another as the Weights onnbsp;the Surfaces of the aforefaid Bodies and thefe*�^^7nbsp;Quotients are as the following Numbers.

If you divide thefe Numbers by the Diame-1259 ters, you will have the Proportion of the Den-fities of thofe Bodies.*nbsp;nbsp;nbsp;nbsp;*1131

Den-} Of the Sun. Of Jupiter. OfSaturn.OftheEarth 1260 fitiesS 10000.nbsp;nbsp;nbsp;nbsp;7404.nbsp;nbsp;nbsp;nbsp;60U.nbsp;nbsp;nbsp;nbsp;39214-

Q 4 nbsp;nbsp;nbsp;We

2-3 2- nbsp;nbsp;nbsp;Mathematical Elements Book IV.

It is not probable that the aforefaid Bodies are homogeneous. We fhall ftiew in relation to thenbsp;Earth, in the 17th Chapter, that it is denfer towards the Center than towards the Surface ^ fromnbsp;whence it follows, that the Denfities cannot benbsp;exa�tly determined: Wherefore we only determinenbsp;the mean Denjities, that is, which the Bodies wouldnbsp;x^fiihave, if the fame Bodies, keeping the fame ^lan-tity of Matter and Bulk which they now have, Jhouldnbsp;become homogeneous.

1262 �Ehe above-mentioned Proportion quot; between the gt;2.(gt;oDenfities, in refpebi of all the Bodies, and the reftnbsp;of the Computations in refpebi of the Sun, Jupiter,nbsp;and Saturn, are free from any fenfible Error : Whennbsp;they are compared with the Earth there maybe famenbsp;Error, which muft becorre�fed by Obfervationstonbsp;be made hereafter ; for we fuppofe the Diftance ofnbsp;the Moon (which is 60 Semi-Diameters of thenbsp;Earth) to be 2, 909 fuch Parts, of which Venusnbsp;^ � is diftant from the Sun 723, that is of which thenbsp;Earth is diftant from the Sun 1000 ; * which Diftance of the Moon is difcovered by fuppofmg thenbsp;Horizontal Parallax of the Sun 10quot;, which can^nbsp;not be looked upon as abfolutely true, althoughnbsp;it be deduced from the moft exadl Obfervationsnbsp;that have hitherto been made, of the Parallax ofnbsp;Mars, when it is neareft of all to the Earth, whichnbsp;is too fmall to leave us without Sufpicion ofnbsp;*iio?fome Miftake.*

But the Error, in not determining truly the Proportion between the Semi-Diameter of the Earth and the Diftance from the Sun, does not changenbsp;the determined Denfity of the Earth, as is deduced from Computations made about it.

For it follows from thefe, that the Denfities of Bodies are to one another in a Ratio compounded

EooklV. of Natural Thilofophy. nbsp;nbsp;nbsp;^33

pounded of the diredt Ratio of the Cubes of the Diftances of the Bodies carried about, andnbsp;the inverfe Ratio of the Squares of the Periodicalnbsp;Times of thefe revolving Bodies �* as alfo ofnbsp;the inverfe Ratio of the Cubes of the Diametersnbsp;of the Central Bodies whofe Denfities are required i* the Ratio, compounded of thefe, is com-**^57nbsp;pounded of the direft Ratio of a Fradfion whofenbsp;Numerator is the Cube of the Diftance of thenbsp;revolving Body, and whofe Denominator is thenbsp;Cube of the Diameter of the Central Body, andnbsp;the inverfe Ratio of the Square of the Periodicalnbsp;Time of the Body carried about. But you havenbsp;fuch a Fradlion, if you know the Ratio betweennbsp;the Diameter of the Central Body and the Diftance of the revolving Body from that Center,nbsp;although this Diftance can be compared with nonbsp;other j but this Ratio is given in refped� of thenbsp;Earth and Moon, as well as in refpedl of thenbsp;other Bodies j wherefore alfo the Ratio of thenbsp;Denfity of the Earth to the Denfities of thenbsp;other Bodies is exadtly difeovered.

The Thyfical Explanation of the whole �Planetary Syjiem.

Jz6,

1206,

1207, 1208

1262

*1206

IN the firft Part of this Book we have fhewn, what are the Motions of the Bodies in thenbsp;Planetary Syftemi now we muft explain hownbsp;thefe Motions follow from the Laws of Nature �*nbsp;that is, how thefe Bodies, being once put in Motion, perfevere in thofe Motions which weobferve'nbsp;Let us conceive the Sun and Mercury to be leftnbsp;to themfelves, and they will come together �*nbsp;but if they be projedled, they may revolve about

Mathematical Elements Book IV.

�*iJ4^�and defcribe immoveable Elliptic Lines,* and jjioSperfevere in that Motion ^ for it is plain, by Ma-54' thematical Demonftration in that Cafe, that thenbsp;Bodies will defcribe Ellipfes about the commonnbsp;Center of Gravity, fimilar to that which the onenbsp;of them could defcribe with the fame Forcesnbsp;about the other, if it was at Reft : This Center,nbsp;*235on account of the Sun�s Magnitude,* is verynbsp;little diftant from the Center of the Sun it-felf.

Let us conceive befides, Venus to be projedled at a greater Diftance from the Sun, it will a little difturb the Motion of Mercury, which alfo,nbsp;by its Adlion upon Venus, will turn it a little out of the Way, and both will draw the Sun,nbsp;fometimes the fame Way and fometimes differentnbsp;Waysj but we find ail thefe Irregularities are in-fenfible, if we cpnfider the Magnitude of thenbsp;Sun j and therefore that thefe three Bodies tendnbsp;towards a Point that is between them near the Sun,nbsp;which therefore is very little diftant from thenbsp;common Center of Gravity of them all.

If the Earth, Mars, and the other Planets be fucceflively proje�ted at different Diftances from

1264 nbsp;nbsp;nbsp;the Sun, the fame Reafoning will hold good.nbsp;Whence it follows, thsX all the Planets are revolved about the common Center of Gravity of all thenbsp;Bodies �which compofe the Syfiem, which is but littlenbsp;diftant from the Sun, and that the Planets do not

1265 nbsp;nbsp;nbsp;fenfibly difturb one another in theirMotions; wherefore they all defcribe the fame Line fingly, whichnbsp;they would defcribe about the Sun, if every one ofnbsp;them was alone with the Sun in the P lanetary Syfiem, that is, immoveable Ellipfes ; For it isnbsp;plain, that thefe will be defcribed by the Force

^1108 of Gravity;* and it is proved by Mathematical 541 Demonftration, that no other immoveable ex-1nbsp;nbsp;nbsp;nbsp;centric

Book IV. of Natural Thilofofhy. 235�

centric Lines can be defcribed by a Central Force afling equally at equal Diftances.

It will alfo more plainly appear, that all the Planets tend to a Point near the Sun, if we con-lider that the Quantity of Matter in the Sun isnbsp;a thoufand times, and more, greater than thenbsp;Quantity of Matter in Jupiter^ which is far thenbsp;greateft of all the Planets.*nbsp;nbsp;nbsp;nbsp;*'^5?

When all the Planets move, though they move 1226 the Sun but little, yet they do move it, and drawnbsp;it differently according to their different Situation in refpedt of one another j whence there arifesnbsp;a fmall Motion in the Sun, which always dependsnbsp;upon the Motion already acquired,and the Changenbsp;which happens in it from the Adlion above-mentioned, which varies every Moment,

It is owingto this Agitation of the Sun, that the 1267 Planets difturb one another lefs in their Ellipticnbsp;Motions round the Sun, than if the Sun was at Rednbsp;in the Middle of the Syftem. If Jupiter, ex. gr.nbsp;was equally diftant from Mercury and the Sun, itnbsp;would attrad: both thofe Bodies to itfelf with annbsp;equal Celerity;* whence the Situation, in refpe�tnbsp;of the Sun, is lefs changed than if the Sun wasnbsp;not agitated by this Motion, and Mercury onlynbsp;was attrafled by Jupiter : According to the various Diftances of Mercury and the Sun fromnbsp;Jupiter, the one or the other is more attrafted,nbsp;and there is always a lefs Change in their refpec-tive Situation when both are carried the famenbsp;Way, than if (the Sun being at Reft) Mercurynbsp;only fhould be carried towards Jupiter.

This Reafoning may be applied to all the A�tions of the Planets, that are more diftantnbsp;from the Sun, upon thofe that are lefs diftant.

As to what relates to the Adlion of thofe that are nearer, upon thofe that are farther from thenbsp;Sun, according to the different Situation they

Mathematical Elements Book IV

draw a Planet to the Sun, or drive it from the Sun, and in confidering one whole refpe�live Revolution, that is, the Motion from one Conjunction to another, the Difturbance is lefs thannbsp;if the Sun was immoveable.

1268 nbsp;nbsp;nbsp;The Magnitude of the Sun, compared withnbsp;the reft of the Bodies of our Syftem, is the Rea-fon (as appears by what has been already demon-ftrated) that the Planets ditturb one another butnbsp;little i but fince the Magnitude is not infinite,nbsp;thefe mutual Aftions muft not be wholly overlooked.

We have faid, that it appears by Aftronomi-cal Obfervations, that Jufiter alters the Way of Saturn \n'htn it is neareft toitj* why this Difturbance is more fenlible than the reft, is deducednbsp;from the Law of Gravity.

1269 nbsp;nbsp;nbsp;The Aftions of J-upiter upon Saturn when itnbsp;is neareft to it, and of the Sun upon the famenbsp;Planet, by which it is kept in its Orbit, are tonbsp;one another dire��y as the Quantities of Matter innbsp;'Jupiter and the Sun,* (viz.') asp, 248. to 10000,*

*1255 and inverfly as the Squares of the Diftances of Jupiter and the Sun from Sat urn-, that is, directlynbsp;as 81 to 16 i for the Diftances of Saturn and

1270 nbsp;nbsp;nbsp;Jupiter (rom the Sun are almoft as 9 to 5 j wherefore,nbsp;nbsp;nbsp;nbsp;Saturn, Diftances

of Saturn from Jupiter and the Sun are as 4 to 9. The Ratio compounded of the two afore-faid Ratio�s is as 749 to 160000, or as i to 214.nbsp;This of Jupiternbsp;nbsp;nbsp;nbsp;with theGravityof

We don�t here confider the Force by which Jupiter attra�is the Sun, for the Orbit of Saturnnbsp;is not changqd by it, and what we had to explain was, why Aftronomers obferve Saturn to

Book IVk of Natural�Philofofbj. 23 7

be turned out of th� Way j yet by the A�lion of Jupiter upon the Sun, the Sun is brought nighernbsp;to Saturn, and the refpedive Situation of thefenbsp;Bodies is more difturbed, than is difcovered bynbsp;Aftronomical Obfervations. The Force withnbsp;which Jupiter in the aforefaid Pofition attradfsnbsp;the Sun, and with which therefore the Sun is at-tradted towards Satmquot;n,\stotht Force with whichnbsp;Jupiter attradls Saturn, as 16 to 255* thatis,nbsp;as 479 to 749, which Number exprefles thenbsp;Force with which Saturn tends towards Jupiter^nbsp;when the Gravity of Saturn towards the Sun isnbsp;exprefled by 160000. If we colledt into onenbsp;Sum the Forces of Jupiter, by which it attradlsnbsp;Saturn und. the Sun; the Force, by which, fromnbsp;the Interpofition of Jupiter, thefe Bodies tendnbsp;towards each other, will be to the Gravity ofnbsp;towards the Sun, as 1228 to T 60000 ; butnbsp;this Gravity is to the Gravity of the Sun towardsnbsp;Saturn, as 160000 to 67, 5.* wherefore the^izojnbsp;mutual Accefs, or Approach of the Sun and Soitmnnbsp;is to the Encreafeof this Approach by the Abiionnbsp;Jupiterinterpofed, as i6oo67to 1228, or, as 130nbsp;to I.

ThisDifturbance is remarkable, and far the great-eft of any that happens in the Motion of any of the primary Planets; This alfo obtains only in thisnbsp;one Cafe of the Conjundfion ; for when Jupiternbsp;recedes from Saturn, the Difturbance of the Motionnbsp;of Saturn, in a ftiort Time, becomes infenfible.

In the fame Pofition of Jupiter, when it is neareft to Saturn, the Force of Saturn, althoughnbsp;it be the greateft of all in this Cafe, does not fonbsp;fenfibly alter the Way of Jupiter about the Sun.

The Adlion of Saturn, attradting Jupiter, is to its Adtion by which it attradls the Sun, as 81 tonbsp;Id;-quot;' therefore it mrnamp;.s Jupiter with greaternbsp;Celerity; and fince they are both attradled the

Mathematical Elements Book IV

fame Way, the Difference of thefe Forces is the Force with which (from the Affion of Saturn')nbsp;*175 Jupiter and the Sun are feparated from each other* inbsp;which is therefore to the Gravity of the Sun towards Saturn,, as 65 to 16; but this Gravitynbsp;of the Sun towards Saturn is to the Gravity of 'Jupiter towards the Sun, as 4, 223. tonbsp;10000*, and as 25 to 81* j that is, as 106 tonbsp;810000, or as 16 to 122756 3 therefore the di-fturbing Force of Saturn is to the Gravity ofnbsp;Jupiter towards the Sun, as 65 to 122756, ornbsp;1272 as I to 1888 i therefore by the greateft A�iion ofnbsp;Saturn, the Gravity of Jupiter towards the Sun isnbsp;diminijhed only by ts�j� fart, which Difturbance isnbsp;infenfible.

The other mutual Difturbances of the Planets are much lefs, as will appear by determiningnbsp;that, which is the greateft of them all, (wz.)nbsp;that of Mars by 'fupiter, which is difcovered bynbsp;the fame Sort of Computation as the foregoing.

The Diftance of Jupiter from Mars and the Sun, when Mars is between the Sun and Jupiternbsp;*961 in the fame Line, are as 7 to 10*, whereforenbsp;961 the Forces with which Jupiter attralt;fts thefenbsp;*i28ogodies, are as 100 to 49*, the Difference ofnbsp;which Forces is to the Gravity of the Sun towards Jupiter, as 51 to 49. This Gravity ofnbsp;the Sun towards 'Jupiter, is to the Gravity ofnbsp;�*1207 Mars towards the Sun, as 9, 248 to 1000*, andnbsp;12.55 as 9 to 100* j that is, as 83 to 1000000 j or asnbsp;�*1208 4p to 590443 j and the difturbing Force of 'Jupiter is to the Gravity of Mars towards the Sun,nbsp;as 51 to 590443 ; or as I to 11577 � whereforenbsp;the Gravity �� Mars towards the Sun, is diminijhednbsp;onlyrrhTf Part by the A�iion of Jupiter when near-eji to it.

BooklV. of Natural Thilofophy. nbsp;nbsp;nbsp;239

Although thefe Difturbances, arifing from the 1274 ASiions of the Planets ttpou each other^ be verynbsp;fmall,and although thofe which happen in a diflfer-rent Pofition of the Planets do in fome Meafurenbsp;compenfate each other, yet the Proportion in whichnbsp;the Force, which keeps the Planets in their Orbits decreafes, is a little changed by thefe A�lions,nbsp;fo that it does not decreafe exa�lly in an inverfenbsp;Ratio of the Square of the Diftance : Thereforenbsp;although the Orbits are at Reft as to Senfe, afternbsp;a great many Kevolutionsy a fmall Change is oh- *245nbsp;ferved in their Situation.*nbsp;nbsp;nbsp;nbsp;929

From all this it follows, that if we fuppofe the*^?^ Planets at firft once projefted at theDiftances fromnbsp;the Sun at which they are moved, they will, bynbsp;the Laws already explained, perfevere in thofenbsp;Motions ; and the Excentricity of the Orbits depends upon the Celerity and Diredtion of the firftnbsp;Projedlion. But thefe Motions may be preferv-ed very long, by reafon of the fmall Reliftancenbsp;of the Celeftial Matter.

It is alfo plain, why, by Lines drawn to the Center of the Sun, they defcribe Areas proportionable to the Times ; namely, becaufe all othernbsp;Gravities, in the Syftem, are very fmall in refpedbnbsp;to the Gravity towards the Sun*; therefore by thisnbsp;Gravity alone, it is that they are retained in theirnbsp;Orbits j whence follows this Proportion of thenbsp;Areas*. And alfo the Motion in Elliptic Lines,'nbsp;which are carried on very flowly, follows from thenbsp;Law of Gravity 5 and thefe Lines would alfo benbsp;immoveable, if the Planets gravitated only towards the Sun * but this flow Motion of the�*241nbsp;Orbits is deduced from the Adfion of the Pla- 12o�nbsp;nets upon one another * Now in refpeft to the'*1214nbsp;Proportion which is obferved between the Cubesnbsp;of the Diftances, and the Squares of the Periodical Times, it is alfo deduced from the Law of

Mathematical Elements Book IV^

IGravity �* fo that if we add to thefe what we *,j^�jhave faid of the Defle�tion of Saturn.,* nothingnbsp;ii7�will remain to be explained in refpe�tto the primary Planets.

1276 nbsp;nbsp;nbsp;That the Motion of Comets depends upon the Lawnbsp;of Gravity, is alfo deduced from Obfervations jnbsp;and in refpe�t of them, as has been faidconcerning the Planets, the Sun�s Gravity prevails, andnbsp;by that Gravity they defledi: from a re�tilinear

'*5*oCourfei* but that the Curvature of their Way depends upon the fame Gravity, follows fromnbsp;this � That a Body, by that Gravity, will de-fcribe an Ellipfe, or a Parabola, or an Hyper-*�^*bola ;* which Lines it appears that thofe Cometsnbsp;*^�*have defcribed, whofe Trajedories have be�nnbsp;determined.

1277 nbsp;nbsp;nbsp;Hhe Satellites of Jupiter and Saturn are movednbsp;*944 hy the fame Laws about their Primaries, as the

^00 'Primaries are moved abotit the Sun-,* wherefore the 974 Explication of thofe Motions * may be alfo refer-�*1^75 red to them j for in thefe three Cafes, fmallernbsp;Bodies are revolving at different Diftances a-bout a much greater Body : Namely, Satellitesnbsp;about yupiter and Saturn and Primary Planetsnbsp;about the Sun.

1278 nbsp;nbsp;nbsp;Whilft fecondary Planets are moved about aPri^nbsp;mary one, it is evident that they may all be movednbsp;with one common Motion, whereby the refpeftivenbsp;Motions with which they are moved, in refpeftofnbsp;each other, will not be difturbed, becaufe a Bodynbsp;may at the fame Time be moved by different Im-

*iz5preflions: * The Motion that a Primary Planet has, in common with its Satellites, is the Motionnbsp;of a Primary Planet about the Sun.

ed fometimes fafter, and fometimes flower, according to the different Pofition of the Primary j and

they alfo often concur towards the Sun�s Center in different Dire(5lions. Thefe Irregularities,nbsp;which are very fmall, cannot be obferved in thenbsp;Satellites of fupiter and Saturn, though they benbsp;really like thofe which are obferved in the Motionnbsp;of the Moon : The leaft Deviation of this laft isnbsp;very fenfible to us. But that the Irregularities ofnbsp;the Moon exaftly follow from the Theory ofnbsp;Gravity, will appear in the next Chapter.

The Thyfaal Exflication of the Mootis Motion.

IT is certain that the Moon and Earth having 1280 once a proje�bile Motion given them, theynbsp;can perceive in their Motion about their commonnbsp;Center of Gravity if they be carried any Way *ii6jnbsp;by a common Impreffion dire�ted in parallel Lines,nbsp;as was faid of the Satellites Jupter and Saturn,*nbsp;this Motion will not difturb the Motion aboutnbsp;the common Center of Gravity, which will follow that Dire�lion only, becaufe in refpeft ofnbsp;the two Bodies it is at Reft. But the Bodiesnbsp;are carried by a Motion compounded of that Im-preflion, and of the Motion about the commonnbsp;Center of Gravity * that is, they are whirlednbsp;about that Center as it is carried along, as before its Motion when it was at Reft. If everynbsp;Moment new Impreflions common to both thenbsp;Bodies ad: upon them, the Way of the Centernbsp;of Gravity may be changed every Moment,nbsp;which Change will be like that which the Bodiesnbsp;themfelves would undergo, if they had no refpe-dive Motion.

Mathematical Elements Book IV.

Hence we deduce, that if whilft the Moon and Earth are whirled round their common Center ofnbsp;Gravity, they be both projected, the Way of thenbsp;Center of Gravity, by the Aftion of the Sunnbsp;ading upon both Bodies, is the fame as a Body,nbsp;projeded in the fame Manner, would defcribenbsp;about the Sun.

1281 nbsp;nbsp;nbsp;Whence it follows, that the Moon difiiirhs thenbsp;Motion of the Earthy and that the common Centernbsp;of Gravity of thofe Bodies defcribes that Orbitnbsp;about the Sun, which we have hitherto faid, thatnbsp;the Earth defcribed-, becaufe we looked the Adlion

1282 nbsp;nbsp;nbsp;of the Moon �, but the Earth defcribes an irregularnbsp;Curve.

1283 nbsp;nbsp;nbsp;Fig.^. Elate XXIV.] Let the Sun beat Sj andnbsp;the common Center of Gravity of the Moon Q,nbsp;and the Earth at M, at the Time of the full Moon,nbsp;be at F: After one whole Lunation, that is, thenbsp;next full Moon, let that Center be at A and Jetnbsp;F D A be the Orbit which we call that of thenbsp;Earth j but in which it is the Center of Gravitynbsp;above-mentioned that does really move.

If this Lunation be divided into four equal Parts, after the firll, the Center of Gravity willnbsp;be at E, the Moon at P, and the Earth at L jnbsp;after the fecond Part of the Time, at New-Moon, the Center of Gravity will be at D, thenbsp;Moon at R, and the Earth at I ^ in the follow-ing Quadrature, the Center of Gravity will benbsp;at B, the Moon at O, and the Earth at H. Laftly,nbsp;at Full-Moon the Center of Gravity being at A,nbsp;the Moon will be at N, the Earth at G: Allnbsp;which follows from the Revolution of the Earthnbsp;and Moon about their common Center of Gravity, whilft it is carried in the Orbit about thenbsp;Sun.

Therefore we fee the Earth moves in the Curve M L IH G, which is twice inflefted in

each

Book IV. of Natural ^hilofophy. nbsp;nbsp;nbsp;2,43

each Lunation ; which Curve alfo does not return into itfelf, becaufe the Infledlions in the feveraJnbsp;Revolutions about the Sun do not coincide; fornbsp;12 Lunations, and a third Part of another, arenbsp;performed every Year.

tthis Irregularity of the Motion of the Earth which is deduced from the Laws of Nature, is toonbsp;fmall to become fenfible inAflronomicalObfervations �nbsp;wherefore we may without any Error fay, thatnbsp;the Center of the Earth itfelf defcribes the Orbit F D A i for M F, or D I, the greateft Di-ftanceof the Earth from that Orbit, is about thenbsp;40th Part of the Diftance I R, which Diflancenbsp;itfelf is not the 300th Part of the Diftancenbsp;FS.

In explaining what relates to the Moon, we alfo 1285 n-egleU the Confideration of the Motion of the Earthnbsp;about the common Cettter of Gravity above-mentioned,nbsp;but we fuppofeit to revolve at the Diftance of �odie-midiameters from the Center of the Earthbecaufe,nbsp;as we have before demonftrated,* fuch is the Di-fiance at which, in its periodical Time, itcould revolve about the Earth at Reft, or be carried alongnbsp;in an Orbit in which it (hould notbe difturbedbynbsp;the Moon�s Adtion. By this Method, the Moon�snbsp;Irregularities will be.much more eafilydifcovered jnbsp;for they are the fame ; as is evident, whether thenbsp;Moon moves about the common Center of Gravity of the Moon and Earth, or about the Center of the Earth itfelf.

Plate XXV. Eig. i.] Let S be the Sun; Tthe *286 Earth; and the Orbit of the Moon A L B / j andnbsp;laft of all, let the Moon be at A in the Quadrature inbsp;it tends towards the Sun in the Direftion A S,nbsp;in the fame Manner, and with the fame Forcenbsp;that the Earth is carried �towards S along T S,nbsp;becaufe the Diftances A S and T S are equal:

This Force may be reprefented by T S or A 5,

R a nbsp;nbsp;nbsp;whereby

Mathematical Elements Book IV.

whereby the Moon endeavours to defcend along A Sj and is refolved into two Forces, by drawingnbsp;the Parallelogram A D S T j fo that the Moonnbsp;will endeavour to move in the Diredions ADnbsp;and A T, by Forces reprefented by thofenbsp;�*i9z Lines.

By the Force which a�ts along AD, the Moon is carried with the fame Celerity and the famenbsp;Way of the Earth j by reafon of the equal andnbsp;parallel Lines T S and A D : Wherefore by thisnbsp;Motion, the Relation between the Moon andnbsp;Earth is not changed; but the Force along AT

1287 nbsp;nbsp;nbsp;confpires with the Gravity of the Moon towards thenbsp;Earth, and this Gravity is encreafed hytbe ASlionnbsp;of the Sun, whentLe Moon is in the^ladratures �,nbsp;and the Augmentation or Addition is to the Gravitynbsp;of the Earth towards the Sun, as A^ the Moon'snbsp;Difiance from the Earth is to T S, the Earth's Difiance from the Sun.

1288 nbsp;nbsp;nbsp;'VS the Earth's Difiance from the Sun remainingnbsp;the fame, the above-mentioned Addition of Gravity encreafes, and diminifJees in the Katioofthe Linenbsp;A T, the Difiance of the Moon from the Earth.

But this Diftance of the Moon from the Earth A T, if it remains the fame, and T S be encreaf-ed, then AT will be lefs in refpe�t of AS :nbsp;Therefore though there fhould be no Change in thenbsp;Force, whereby the Earth and Moon fall towardsnbsp;the Sun, the Addition will be lefs, and fo muchnbsp;lefs as T S is greater; that is, it will be inverfly asnbsp;T S; but the Force of Gravity does not remainnbsp;the fame, when T S is encreafed, but is diminilh-ed; wherefore alfo in that refpedl, the above-mentioned Addition is diminilhed, and in the famenbsp;Ratio with that Force of Gravity; thereforenbsp;it is the inverfe Ratio of the Square of thenbsp;*1108 Diftance TS;* if this Diminution be added tonbsp;. that above-mentioned, we fee that the Addition oi

which

Book IV. of Natural ^hilofophy, 2,45'

which we fpeak, follows the inverfe Ratio of the 1289 Cube of the Difiance of the Earth from the Sun.

^be Difiance of the Earth from the Sun remain^ IZ90 ing the fame^ the Gravity of the Moon towards thenbsp;Earth decreafes more flowlyin the ^adrattires^ thannbsp;according to the inverfe Ratio of the Square of thenbsp;Difiance from the Center of the Earth; for if thenbsp;Addition in that Cafe Ihould follow the inverfenbsp;Ratio of the Square pf the Diftance, as thenbsp;Gravity from the A�fion of the Earth does,* this*gt;io*nbsp;Ratio would not be difturbed j but the Additionnbsp;encreafes when the Gravity itfelf is diminilhed;nbsp;wherefore the Addition, when the Diftance isnbsp;encreafed, is always greater than is required, andnbsp;confequently the Diminution the lefs.

This Addition is determined in the mean Difian-\2gi ces of the Moon from the Earth., and of the Earthnbsp;from the Sun: Let A T and T S be thefe meannbsp;Diftances; the Addition required is to the Gravitynbsp;of the Earth towards the Sun, as A T to TS^s�^uS?nbsp;the Gravity of the Earth towards the Sun, is tonbsp;the Gravity of the Moon towards the Earth (htcamp;w�Qnbsp;thefe Bodies are retained by thefe Gravities innbsp;their Orbits) dire�tly as T S to T A, and inverjlynbsp;as the Squares of the Periodical fines of the Earthnbsp;about the Sun, and of the Moon about the Earth:nbsp;Therefore the Addition required is to the Gravity 1136nbsp;of the Moon towards the Earth, in a Ratio compounded of thefe Ratio�s that is, the above-mentioned inverfe Ratio of the periodical Timesnbsp;of the Earth and Moon, the other Ratio�s deftroy-ing one another. Thefe Times are given, andnbsp;their Squares are inverfly as i to 178,73.

Fig. I ] in which Situation the Sun attradls the Moon and Earth in the fame Line, but not equally jnbsp;the Moon it draws with a greater Force, be-caufe it is lefs diftant from it: The Difference ofnbsp;R 3nbsp;nbsp;nbsp;nbsp;thofe

Mathematical Elements Book IV.

The Forces whereby the Moon at L, and the Earth at T, tend towards the Sun, are to onenbsp;another as the Squares of the LinesS T and SLnbsp;and the Difference of the Forces, that is, the di-fturbing Force, is to the Force by which thenbsp;Earth defcends toward the Sun, as the Differencenbsp;of thofe Squares to the Square of the Line L S;nbsp;that is, nearly as double I. T to L S or T S ; fornbsp;I2p3 thefe Lines do but very little differ from one another ; and the Diffeyence of the Squares, whofenbsp;Koots differ but little, is keeping the Proportionnbsp;double that which is beHveen the Roots.

If therefore T S, as before, reprefents the Force whereby the Earth defcends towards the Sun, L /nbsp;will reprefent the difturbing Force and diminifh-ing� Gravity, when A T reprefents the difturb-?iiS�ing Force in the Quadratures.* Let the Moon benbsp;1294 at / j it is again (together with the Earth) attracted by the Sun in the fame Line ; but the Earth,nbsp;becaufe lefs diftant, moves more fwiftly to-*iioS wards the Sun j * fo that there is a Forcenbsp;which feparates the Earth from the Moon,nbsp;namely, the Difference of the Forces which attract the Moon and the Earth j which Forcenbsp;always aCts contrary to the Gravity of the Moonnbsp;towards the Earth, and diminifhes it, in thenbsp;fame Manner as has been demonftrated from thenbsp;greater Gravity of the Moon towards the Sun,nbsp;fuppofing it at L. At / alfo the feparatingnbsp;Force fcarce differs from the feparating Forcenbsp;at L; for this Force, as we have fhewn, is proportional to the Difference of the Squares ofnbsp;the Lines T S and L S ; and that (as appears bynbsp;fuch another Demonffration) proportional tonbsp;Inbsp;nbsp;nbsp;nbsp;the

Book IV. of Natural Thihfofhy. nbsp;nbsp;nbsp;X47

the Difference of the Squares of the Lines I S and T S ; which Difference, by reafon that L /nbsp;is very fmall in refped: of TS, fcarcediffer amongnbsp;themfelvesj fo that the Force which diminifhesnbsp;the Moon�s Gravity at /, is alfo reprefented bynbsp;L/.

Yet the Difiurhing Force is fomething p^eater at 1295 the Conjun�iion L, than at the Oppojition I � fornbsp;fuppofing the Differences between the Rootsnbsp;to be equal, the Squares, keeping the Proportion,nbsp;will differ fo much the more, the lefs they are -and fo keeping the Proportion, the Forces differnbsp;more at L and T, than at T and I which alfo arenbsp;lefs*

From this we conclude, that the Force which diminijhes the Gravity of the Moon in the Sy%ygies^nbsp;is double that which encreafesit inthe ^ladratures �nbsp;namely�, asl^ltoK^. Wherefore in the Syzy-gies, the Gravity of the Moon from the Adionnbsp;of the Sun is diminifhed by a Fart, which is tonbsp;the whole Gravity, as i to 89, 36 ; for in thenbsp;Quadratures, the Addition of Gravity is to thenbsp;Gravity, as i to 178,73*

In the Syzygiesy the difturhing Force follows the ^ 29 7 fame Proportion with half added to it j that is, ^nbsp;with the difturbing Force in the Quadratures * jnbsp;it is thtrt�oxQ direSlly as the Diftance of the Moonnbsp;from the Earth* and inverfly as the Cube of thenbsp;Difiance of the Earth from the Sun*

At the Syzygies the Gravity of the Moon towards 1298 the Earth, receding from its Center, is more dimi-nifioed, than according to the inverfe Ratio of thenbsp;Square of the Difiance from that Center-, for itnbsp;would be diminifhed in that Ratio, if the Forcenbsp;to be taken away followed the fame Ratio ; butnbsp;on the contrary, as it encreafes when the Diftancenbsp;becomes greater,* the Diminution is, always*1,57nbsp;greater than in that Ratio.

^48 nbsp;nbsp;nbsp;Mathematical Elements Book IV.

1299 Plate'KXN. Fig. i.] Laftly, let the Moon be at F, in any intermediate Place between the Sy-zygy and Quadrature, it is drawn towards thenbsp;Sun along F S j by which fince it is lefs diftantnbsp;than the Earth T, it is attrafted with more Forcenbsp;than the Earth ; Let the Force, with which thenbsp;Moon tends to the Sun, be to the Force, withnbsp;which the Earth is carried towards it, as F M tonbsp;T S, which alfo before has been made ufe of tonbsp;exprefs the fame Gravity of the Earth. Drawnbsp;the Parallelogram F FI MI, whofe Diagonal isnbsp;F M, and whofe Side F H is parallel and equalnbsp;to the Line T S. The Motion of the Moon towards the Sun is refolved into two Motions, onenbsp;along FFI, the other along F I, and thefe Linesnbsp;denote the Forces whereby the Moon endeavoursnbsp;�*192 to move along them.* The Motion along F H isnbsp;common to the Moon and the Earth, which withnbsp;an equal Force, and in a Line parallel to it, doesnbsp;alfo tend to the Sun : So that by this Motionnbsp;of the Moon, the Situation of it, in refpe�l ofnbsp;the Earth, will not be changed, and the difturb-ing Force will be only the Motion alongnbsp;F I.

By reafon of the immenfe Diftance of the Sun, the Part MS of the Line MF is fmall, in refpeftnbsp;of the Whole, and the Angle FS T, where it isnbsp;the greateft, as AST, is hardly more than thenbsp;6th Part of a Degree: Whence it follows, thatnbsp;the Lines M I and S N are very near one another, and that the Points I and N are fcarce fen-fibly diftant, and may without any fenfible Error be confounded together; which Error, not-withftanding how little foever it need be regarded, in Confideration of onewholeRevolution,nbsp;is compenfated by a contrary Error when thenbsp;Moon is at E. Therefore the difturbing Forcenbsp;is exprefled by F N.

Book IV. of Natural Thilofophy. 149

It is to be obferved, when only the Part E F 1300 of the Line ES is confideredj that it is to be looked upon as parallel to the Line L /, becaufe ofnbsp;the fmall Angle which thefe Lines make. Fromnbsp;the Point N, draw N Q, perpendicular to the 1301nbsp;Line F T, continued if Need be, in which thenbsp;Moon gravitates towards the Earth ; and letnbsp;the Re�tangular Parallelogram E P N Q benbsp;drawn: Let us conceive the Force along F N re-folved into two others, afting in the Dire�fionsnbsp;F Q and FP, and reprefented by thefe Lines *19�nbsp;By the Force along F Q^, the Force of Gravity isnbsp;diminifhed, in the Cafe reprefented by this Figure;nbsp;but it is encreafed when the Point falls between F and T; but by the Force along F P,nbsp;the Moon in its Orbit is drawn towards the nextnbsp;Syzygy L, and the Motion of the Moon is accelerated or retarded, according as this Force con-fpires with, or ads contrary to the Moon�s Motion.

Near a Syzygy, the Gravity of the Moon is diminifhed, and the Line F Q, which follows thenbsp;Proportion of this Diminution, grows lefs, receding from the Syzygy till it vanilhes, at the Di-ftance of about 54 Degr. 44 Min. from it : At anbsp;greater Diftance of the Moon from the Syzygy,nbsp;falls in between F and T ; and the Gravitynbsp;of the Moon towards the Earth is encreafednbsp;by the Sun�s Adion. The Force along FPnbsp;vanifhes in the Syzygy L; receding from it, itnbsp;encreafes quire to the Odant, which is thenbsp;middle Point between the Syzygy and the Quadrature ; and then it diminifhes again till it vanilhes quite at E.

Between B and /, or I and A, the difturbing 1302 Motions are determined in the fame Manner, asnbsp;in A L B, the oppofite inferior Part of the Orbit ; at � and F, the Diminution of Gravity is

Mathematical Elements Book IV.

equal, and in that Pofition it is drawn in the Orbit with an equal Force towards the Syzygy /,nbsp;with that with which at F it is impelled towardsnbsp;the Syzygy L.

1303 Hence it follows, that in the Motion of the Moon fromthe Syzygy to the ^ladrature^ between L andnbsp;B, as alfo between / and A j the Gravity of thenbsp;Moon towards the Earth is continually encreafedynbsp;lio^and the Moonis continually retarded in its Motion.nbsp;But in the Motion from the S^iairature to the Sy-zygy-) between B and/, as alfo between A and L,nbsp;every Moment the Moon's Gravity is diminifhedy andnbsp;its Motion in its Orbit is accelerated.

You may determine the Forces upon which thefe Effects depend, by comparing them with thenbsp;known Force, whereby Gravity is encreafed in thenbsp;�*1291 Quadratures,* and which is reprefented by thenbsp;Moon�s Diftance from the Center of the Earth.nbsp;1305 The Lines M I, HF, ST, are equal by Con-ftru�lion 3 therefore when the Points I and N arenbsp;confounded, M N is equal to S T, and M S isnbsp;equal to NT. The Lines MF andS T reprefentnbsp;the Forces, whereby the Moon at F and thenbsp;Earth at T are carried towards the Sun S ; therefore they are as the Square of the Line T Snbsp;�*^1108 to the Square of the Line F Sj* wherefore asnbsp;F G is the Difference of thofe Lines, F M andnbsp;T S differ from one another double the Linenbsp;�2^93 Q Pj* and adding G F to the Line FM, the Difference between G M and T S, that is, S willnbsp;be triple the Line FG; and therefore this isnbsp;alfo the Quantity of the Line N T : Now as F Enbsp;*1300 is double F G therefore N T will be to F E asnbsp;Three to Two.

Let F T be continued, if Need be, and from E draw EV perpendicular to it 3 the Trianglesnbsp;E V F and N Q T, W�hich are reflangular, willnbsp;be fimilar, by reafon of the alternate Angles

Book IV. of Natural Thilofophy. nbsp;nbsp;nbsp;151

V F E and (^TN :* Therefore NT is to F that is, Three is to Two, as NQ, equal to F P,nbsp;is to E V 5 which therefore is proportional to twonbsp;third Farts of the Force, which is exprefled bynbsp;F P 5 but E V is the Sine of the Angle E T Vnbsp;at the Center, which is double the Angle E F Vnbsp;at the Circumference, equal to the Angle F T L,nbsp;which is the Diftance of the Moon from thenbsp;Syzygy. Therefore, as the Radius T A, orTE, 1306nbsp;is to a Sim and an half of double the Diftance ofnbsp;the Moon from the Syzygy, namely, F P, fo thenbsp;Addition of Gravity in the ^ladratnres (which isnbsp;exprefled by the Radius T A) is to the Force whichnbsp;accelerates or retards the Moon in its Orbit.

The Computation of this Diminution of Gravity, and of its Encreafe at a lefs Diftance from the Quadratures, is deduced from the fame Principles.

This Diminution is reprefented by the Line F Q, which is equal to T, minus the Radius jnbsp;but from the Confideration of the Triangles a-bove-mentioned, V F taken once and an half isnbsp;equal to Q T ; therefore V T and an half, withnbsp;half the Radius added to it, exprefles the requirednbsp;Diminution of Gravity ; and the Radius is to the 1307nbsp;Sum or Difference of once and a half the Co-fine ofnbsp;double the Diftance of the Moon from the Syzygynbsp;and half the Radius : As the Addition of Gravitynbsp;in the ^ladratures, to the Diminution or Encreafenbsp;of Gravity in that Situation of the Moon, cornerji-ing which the Commutation is made.

We make ufe of the Difference of the Co-fine from half the Radius, when the Angle, whofe the Co-fine is, is greater than a right Angle j becaufe in that Cafe we make ufe of thenbsp;Co-fine of the Complement of the Angle tonbsp;two right Angles ; When in this fame Cafe thenbsp;Co'fine and a half, which we make ufe of, is

Mathematical Elements Book IV.

greater than half the Radius, the Quantity found is to be added that is, encreafes the Gravity,nbsp;which obtains every where between the Quadrature, and 35 Degr. i6 Min. from it.

1308 ^hefe Forces^ whatever is the Figure of the Moon^s Orbit, are exactly determined j for theynbsp;are compared with the Addition of Gravity in thenbsp;Quadratures, fuppofing the Moon in the Quadrature to be at the fame Dilhance from the Earth, atnbsp;which it really is in the Place which is confider-ed; but this Addition is difcovered in every

Though it be foreign to the Purpofe of this 'Work, to give a Computation of the Moon�snbsp;Motion, I thought it neceflary to explain in anbsp;few Words, what is the Method whereby tonbsp;difcover the Forces that govern the Moon j be-caufe the more exactly we know the Forces, thenbsp;more eafily we fhall conceive their general Ef-fed.

Now to examine the Moon�s Motion, w�C muft fingly confidcr its feveral Irregularities ;nbsp;which to do without Confufion, we muft removenbsp;feveral Irregularities, and conceive the Moonnbsp;as moving in a Circle about the Earth, innbsp;which Curve it is plain, that it can be retain-�*1-1�, ed by Gravity.* This Motion is ditiurbed bynbsp;the A�lion of the Sun, and the Orbit is more con~nbsp;1-^c^vex in the ^ladratures than in the Syzygies. Thenbsp;Convexity of a Curve, which a Body defcribesnbsp;by a central Force, is fo much the greater, asnbsp;the central Force does more ftrongly every Moment turn the Body out of the Way j it is.ilfothenbsp;greater, the more llowly the Body moves be-caufe the central Force, ading the longer, has anbsp;greater Effed in infleding the Way of the Body. From contrary Caufes the Convexity of thenbsp;Curve is diminilhed. Both concur in encreafing

the Convexity of the Orbit in the Quadratures, * and diminiftiing it in the Syzgics.*

From this it follows, that the circular Figure 1310 of the Moon�s Orbit is changed into an Oval,nbsp;whofe greater Axis goes through the Quadratures jnbsp;fo that the more convex Parts are in the Quadratures. WhereforeMoon is lefs diftantfromnbsp;the Earth at the Syzygies^ and more at the ^ladra-tures^ and it is no Wonder that the Moon comesnbsp;towards the Earth, when its Gravity is dimi-nifhed i becaufe the Accefs is not the immediatenbsp;EffetP of this Diminution, but of the Inflexion ofnbsp;the Orbit towards the Quadratures.

The Motion of the Moon, taking away the Aftion of the Sun, is not in a Circle, but in annbsp;Ellipfe, one of whofe Foci coincides/with thenbsp;Center of the Earth j * for the Orbit of thenbsp;Moon is Excentric, and it is retained in it by thenbsp;Force of Gravity.

Therefore what has been demonflrated cannot be exadly applied to the Moon�s Motion j for asnbsp;the Forces, which generate the Deviations explained, do really ad upon the Moon, the Ellipfe, which the Moon would defcribe if the Sunnbsp;was taken away, is changed, and c^eteris paribus, 1311nbsp;the Proportions of N�i309, 1310, maybe appliednbsp;to the Moon's Motion.

In the ^ladratures and Syzygies, the difturbing 1312 Force ads in the fame Line as the Force of Gravity towards the Earthnbsp;nbsp;nbsp;nbsp;therefore the Force which

continually ads upon the Moon, and retains it in its Orbit, is direded towards the Center ofnbsp;the Earth, and the Moon defcribes Area's bynbsp;Lines drawn to the Center, of the Earth proper-tional to thel�imes.*

Plate XXV. Fig. i.] In other Points of the Or- 1313 hit, as F, befides the Force which ads in thenbsp;Line F T, there is alfo another, whofe Dire-

^^5'4 Mathematical Elements Book IV. \

'301 jj perpendicular to FT, * which is here re- ^ prefented by F P : The Diredion of the Forcenbsp;compounded of both, is direfted fometimes fide-wife to the Line F T, and does not tend to thenbsp;quot;**9� Center of the Earth wherefore the Jre^^s, hynbsp;^ Lines drawn to the Center of the Earthy are notex-a�lly proportional to the linies* In the Odfants,

F P is the greateft of all �, and the Force, which is reprefented by that Line, is to the Gravity of thenbsp;Moon towards the Earth, in that Point, in thenbsp;^ Mean Diftances of the Sun and Moon, as i to

119,49.* Wherefore the Diredionof the Force, ' ' compounded of the Adtions of the Sun and Earthnbsp;upon the Moon, makes an Angle of above half anbsp;Degree with the Line F T.

The Motion of the Moon is fulnedi to feveral other Irregularities ; fo that it defoibes a Curvenbsp;wholly irregular; which Afironomers, in ordernbsp;to fubjedt it to the moft exadt Computationsnbsp;^3 ^4 that can be made, do reduce to an Ellipfe whichnbsp;they conceive to he agitated by various Motions^ aitinbsp;alfo to he changeable^ left the Moon jhould go outnbsp;of it.

In refpedl to central Forces we have obferved, that a Body does not defcribe Ellipfe, if thenbsp;central Force, by which it is retained in its Orbit, decreafes in any other Ratio than the Inverfenbsp;Ratio of the Square of the Diftanccj but that thenbsp;Curve may be often reduced to a moveable El-j lipfe :* Concerning which it is to be obferv-^ ^ ed, that the Ellipfe^ in that Motion, turns aboutnbsp;1316 one of its Foci, and the Motion of the Ellipfe isnbsp;directed the fame iVay as the Motion of the Body innbsp;it, when the central Force decreafes fafter than in thenbsp;Inverfe Katio of the Square of the Diftance. But ifnbsp;the central Force decreafes flower as you recede fromnbsp;the Center, the Ellipfe is carried the contrary iVay;nbsp;as thefe Things may be demonftrated Mathematically.nbsp;nbsp;nbsp;nbsp;Hence

Book IV. of Natural ^hilofophy. 25:5

Hence it follows, the Orbit of the Moon cannot be referred to an Elliptic Orbit, unlefsnbsp;you fuppofe it agitated by four Motions everynbsp;Revolution ; that is, unlefs the Line of theAp-lldes, which goes through the Center of the Earth,nbsp;goes forwards twice, and backwards twice.

*ithe Apfides of the Moon go forward wbe?: the Moon is in the Syzygies^* or rather whilft the'quot;�5r?nbsp;Moon moves between the Points, which are 54 129*nbsp;Degr. 44 Min. diitant from them. * Intbe^ta-*'^ilt;gt;7nbsp;dratures, and between the Points diltant fromnbsp;them 35 Degr. 16 Min. The Apfides gohackwards^t^i^nbsp;that is, move in Antecedentia*nbsp;nbsp;nbsp;nbsp;*1316

The Forces, upon which the Progrefs and Re-gre� of the Apfides depend^ are the Forces which difturb the Motion of the Moon, which have 1319nbsp;been before explained ; therefore, fince the difturb-ing Force in the Syzygies, is double the difturb-ing Force in the Quadratures the Progrefs, *11^6nbsp;confidering one entire Revolution of the Moon^ ex~nbsp;ceeds the Regrefs, cceteris paribus.

In a Circle whofe Center is in the Center of the Forces, the Diminution of the Force, in receding from the Center, produces no Ef��ft ;nbsp;becaufe in fuch a Line the Body does not recede from the Center ; therefore the Effect of thisnbsp;Diminution is fo much the greater, as the Curvenbsp;defcribed by the Body differs more from fuch anbsp;Circle.

In an Elliptic Orbit, one of whofe Foci co-1320 incides with the Center of the Forces, the Curvature 1n the Apfides differs moft of all from fuchnbsp;a Circle, and the Effefi of the Diminution of thenbsp;Force, in receding from the Center of the Porces,nbsp;is the greatefi of all. If this Orbit is but a little 1321nbsp;Excentric at the Ends of the leffer Axis^ the El-lipfe differs very little from the Circle above-mentioned, zndthe Effe�l of the Diminution is tbeleafinbsp;*gt;f all.nbsp;nbsp;nbsp;nbsp;The

7.^6 nbsp;nbsp;nbsp;Mathematical Elements Book IV.

1322 nbsp;nbsp;nbsp;The Progrefs and Kegrefs of the Apfides dependnbsp;upon the Proportion, according to which thenbsp;Force of Gravity decreafes, receding from the

*'315 Center of the Earth;* it is therefore the EffeSi of Diminution of the Central Force.

1323 nbsp;nbsp;nbsp;Plate XXV. Fig. 2.] This Motion of thenbsp;Apfides, which we have explained, undergoesnbsp;feveral Changes; the Apfides go forward fafiejl ofnbsp;all in a Kemlution of the Moon^ fuppofing the Linenbsp;of the Apfides in the ^adratures �,* and in that

1324 nbsp;nbsp;nbsp;Siippofing the Line of the Apfides to he in the ^ita-Hfattires, the Apfides are carried in Confequentia,

* 1318 fwifteft in the ^adratures; * and in this Cafe, one entire Revolution of the Moon, the Kegrefsnbsp;exceeds the P rogrefs.

1325 nbsp;nbsp;nbsp;Whilftthe Earth is carried along in its Orbit,nbsp;the Line of the Apfides does fucceflively go throughnbsp;all Situations in refpeft of the Sun ; wherefore,nbsp;confidering a great many Revolutions of the Moon

*ni9taken together, the Apfides go forwards �* and it is plain from Observations, that in the Space of aboutnbsp;eight Years, the Line of the Apfides performsnbsp;one entire Revolution.

1326 nbsp;nbsp;nbsp;Sthe Excentricity of a Body is encreafied, if thenbsp;central Force, the Diminution being continued,nbsp;decreafes fafter than before, whilftthe Body is carried from the lower to the upper Apfis ; fornbsp;then it is every Moment lefs attrafted, than ifnbsp;the Force did not decreafe; and therefore it recedes the more. The Excentricity of the Orbitnbsp;is alfo thereby encreafed, in the fame Cafe, in

Book IV. of Natural Thilofophy. nbsp;nbsp;nbsp;^5'7

the Motion from the upper to the lower Apfis, becaufe in this Cafe, coming towards the Center,nbsp;the Force encreafes fo much the fafter, as thenbsp;Body defcends more towards the Center j fonbsp;that in each Cafe, the Dift�erence between thenbsp;greateft and leaf; Diftance from the Center ofnbsp;the Forces, may become greater, and the Ex-centricity be thereby encreafed. By the fame 1327nbsp;Reafoning it appears, that the Excentricity is di-minijhed^ when the Central Force is encreafed, innbsp;the Motion of the Body from the lower to the upper Apfisi and likewife when that Force is dimi-niftied in the Motion from the upper Apfis to thenbsp;lower i that is, when it decreajes more (lowly thannbsp;before, in the receding from the Center.

Applying this to the Moon�s Motion, it ap-1328 pears, that the Excentricity of the Orbit, every Ke-�voliition, undergoes various Changes. Fhat it is 1329nbsp;the greateft of all, when the Line of the Jpftdes isnbsp;in the Sy%ygies-, becaufe the Forces in the Apfides,nbsp;being compared, do decreafe fafter than in an in-verfe Ratio of the Square of the Diftance,* *quot;'198nbsp;whence this Addition follows,* which prevails�'i 320nbsp;in this Pofition;* 'E��t the Orbit is the leaft Ex-*ii^onbsp;centric of all, when the Line of the Apftdes is innbsp;the ^tadratures., the Diminution of the Excentricity prevailing.*nbsp;nbsp;nbsp;nbsp;*1290

We have faid that the Moon moves in a Plane *527 inclined to the Plane of the Ecliptic , that the 1321nbsp;Line of the Nodes is carried round in Anteceden-tia-f and that the Inclination of the Orbit isnbsp;not conftantj* thefe EiFedts are alfo deduced fromnbsp;the Adbion of the Sun upon the Moon.

Byreafonof the fmall Inclination of the Moon�s Orbit, the Forces which we have hitherto con-ftdered adling in the Plane of the Ecliptic, not regarding the Inclination of the Orbit, may (without any fenfible Error) be referred to the Plane

2^8 nbsp;nbsp;nbsp;Mathematical Elements Book IV.

1330 nbsp;nbsp;nbsp;Motions before explained : But there is a Forcenbsp;�which removes the Moon from the Plane of the Or-

*13'4 hit * fo that we muft conceive that Plane to be agitated, otherwife the Moon would go out ofnbsp;the Orbit.

1331 nbsp;nbsp;nbsp;Plate XXV. Fig. i.j Let the Moon be at F jnbsp;if we attend to what has been faid above con-

��1299cerning the Actions of the Sun,* it isplain, that the Plane of the Parallelogram F H M I goesnbsp;through the Line T S, which joins the Centers ofnbsp;the Earth and Sun ; and therefore it is in thenbsp;Plane of the Ecliptic ; fo that the Point N, tonbsp;which is diredted the Force F N, difturbing onnbsp;account of the Adfion of the Sun, is in thatnbsp;Plane.

1332 nbsp;nbsp;nbsp;Plate'yiX.Y. Fig.Let the fame Force benbsp;reprefented by F 1; at F let F R be raifed perpendicular to the Plane of the Orbit and imagine the Parallelogram FRI/quot;, whofe SideF? isnbsp;in the Plane of the Orbit, and whofe Diagonalnbsp;is F I i the difturbing Force along F 1 is refolvednbsp;into two, in the Diredlions F R and F i which

*i92thefe Lines reprefent,* and of which this laft adfs in the Plane of the Orbit: So that we mull:nbsp;refer to this what relates to the difturbing Force,nbsp;of which we have treated in N� 1299 ; for thenbsp;Lines F i and F1 fcarce differ, and the Plane ofnbsp;the Parallelogram F RI i is perpendicular to thenbsp;Plane of the Moon�s Orbit.

1333 nbsp;nbsp;nbsp;The Line F R muft be determined, which re-prefents the Force that adfs perpendicular to thenbsp;plane of the Orbit, and removes the Moonnbsp;from that Plane : Now the Relation of the Linenbsp;FR or Ir, to the Radius ET, is the Ratio ofnbsp;the difturbing Force, which is fpoken of here,

In the Cafe of this Figure, in which the Line 1334 of Nodes N n is in the Quadratures, we find outnbsp;FR ; becaufe I T (which is N T of Fig. i.) isnbsp;given,* and becaufe IT is to �?, or to F R, as *132^5nbsp;the Radius to the Sine of the IncJination of thenbsp;Orbit.

But in every Cafe the Force muft be deterniin-1335 ed which drives the Moon out of the Plane- letnbsp;us therefore fuppofe the Line of Nodes carried tonbsp;the Situation M whereby, every thing elfenbsp;remaining as before, 1 i is changed. To M 7�nbsp;continued, if Need be, let z X and IX be drawnnbsp;perpendicular, which make an Angle equal to thenbsp;Inclination of the Plane of the Orbit.

The Ratio between ET and Iz ; that is, tbeiSS^ Ratio between the Addition of Gravity in the ^la-drattires^ and tbeForce^ which we feek, which removes the Moon out of its Orbit, is compounded ofnbsp;the Ratio�s of the Line ET to Tl, of the Linenbsp;TI to IX, and laftly of the Line 1X to 1 z. Thenbsp;firft is the Ratio between the Radius and threenbsp;Times the Sine of the Diftance of the Moon fromnbsp;the Quadrature the fecond is the Ratio of the�'isosnbsp;Radius to the Sine of the Angle I TX, that is,nbsp;of the Diftance of the Node from the Syzygy.

Laftly, the third is the Ratio of the Radius to the Sine of the Inclination of the Orbit: Andnbsp;the Ratio, compounded of thefe, is the Ratio ofnbsp;the Cube of the Radius to three times the Prcdutlnbsp;of the Sines of the Diftances of the Moon fromnbsp;the ^tadrature, and of the Node from the Syzygy,nbsp;as aifo of the Inclination of the Plane. To thisnbsp;Force is alfo to be referred Nquot; 1308.

SFhis Force vanijhes in the ^mdratures, becaufe 1227 the Point I coincides with the Point T, which *nbsp;is the Center of the Earth, and the Line Iz va-nilhes ^ the Lines FI and Ft concurring in thenbsp;Plane of the Orbit, which alfo follows fromnbsp;the Computation above-mentioned i* the Sine of *11,6

Mathematical Elements Book IV.

the Diftance of the Moon from the Quadrature vanifhing, and confequently, the whole Produ�lnbsp;which is multiplied by that Sine.

133* That fame Produft vaniflies alfo, and with it the Force which it reprefents, when the Sine ofnbsp;the Diftance of the Node from the Syzygy va-nijhes, that is, fiippofing the Line of the Nodes innbsp;the Syzygies : It is alfo deduced from this, thatnbsp;the Line of Nodes N n {Plate XXV. 5.) continued goes through the Sun, wherefore the Sunnbsp;is in the Plane of the Orbit itfelf, and thereforenbsp;cannot draw the Moon but in that Plane.

*339 itbe Force which �we examine^ is increafed as the Moon advances towards the Syzygy^ and asnbsp;^^'^the Node recedes from it*

1340 Plate XXV. Fig. 6.] Let P p he the Plane of the Ecliptic, P A the Orbit of the Moon j whennbsp;the Moon is come to A, that is, is receded anbsp;little from the Node, it is removed out of thenbsp;Plane of the Orbit, and in the fccond Momentnbsp;it is not carried along AB (the Continuation ofnbsp;the Orbit P A) but along A b becaufe it comesnbsp;towards the Plane of the Ecliptic along B ^ jnbsp;therefore it is moved as if it came from a more di-I34iftant Node/). Whence it appears, that

gobackward^whilji the Moon moves in its Orbit ^ as long as it recedes from the Node ; The Nodes alfo go back, whilft the Moon is going to the op-pofiteNodej becaufe as the Moon is continuallynbsp;driven out of its Orbit towards the Plane of thenbsp;Ecliptic, it is continually diredled to a Point lefsnbsp;diftant, and comes fooner to the Node, than if,nbsp;not being agitated by fuch a Motion, it had continued in Motion with the fame Celerity.

1342 Co7ifidering one entire Kevolntion of the Moon caeteris paribus, the Nodes tnove in Antecedentianbsp;^l^^ofatifteft of all, when the Moon is in the Syzygies,*nbsp;then flower and flower, till they are at Re/l, whennbsp;. '337 the Moon is in the ^.adratv.res.nbsp;nbsp;nbsp;nbsp;Whilft

Book IV. of Natural Thilofophy. nbsp;nbsp;nbsp;x6r

Whilft the Earth is carried round the Sun (even when we do not attend to the above-mentioned Motion of the Nodes) the Liiie of Nodes 1343nbsp;does fucceflively acquire all polTlble Situations innbsp;refpe�l of the Sun ; and every Tear goes twicenbsp;throughthe iSy%ygies^andtwicetbro�� tbe^iadratures.

If now we confider feveral Revolutions of the AIogu, the Nodes in one whole Revolution go backnbsp;very faji, the Nodes being in the ^ladratiires *1339nbsp;then flower^ till they come to reft, when the Linenbsp;of Nodes is in the Syzygies*nbsp;nbsp;nbsp;nbsp;'*133?

By the fame Force with which the Nodes are 1345 moved, the Inclination of the is alfo changed ^nbsp;it is increafed as the Moon recedes from the Node-,nbsp;and diminiftsed as it comes to the Node,

Plate X-XN. Fig. 6.] Fot the Angle bpL 131346 kfs than the Angle A PL, and for the famenbsp;Caufe it is continually diminiflied, and the Inclination becomes greater ^ but when the Moon isnbsp;come to the greateft Diftance from the Planenbsp;of the Ecliptic, and is going towards the oppofitenbsp;Node, the Direction of the Moon is continuallynbsp;inflefted towards the Plane of the Ecliptic, andnbsp;lefs inclined to it than if it continued in Motionnbsp;in its Orbit : Let N^tz be the plane of the E-cliptici the Curve N � the Orbit of the Moon ;nbsp;by the Force whereby the Moon is continually removed out of it, the Way of the Moon is changed, and it goes in the Curve N p, which isnbsp;more inclined to Npn at N than at pj fo thatnbsp;we muft conceive the Inclination of the Plane ofnbsp;the Orbit to be twice changed f whilft the Moonnbsp;moves from one Node to the other, therefore thisnbsp;happens four times in each Revolution of the Moon, 1347nbsp;it is twice dminifljed, and twice again encreaCed.

Plate XXV. Fig. 4.] Suppoftng the Nodes N n, 1348 to be in the ^ladratiires, the Forces which in onenbsp;Revolution encreafe the Inclination, and diminifh it,nbsp;are equal to one another ^ for by reafon of the

2.6z Mathematical Elements Book IV.

equal Diftance of each Node from the Syzygies, the Forces that change the Inclination at N Dnbsp;and 71 E are equal to the Forces in the correfpon-'issedent Points in the Arcs Jjn and EN ;* By thenbsp;former the Inclination is encreaied, by the latternbsp;quot;� 345 it is diminifhed the Diminution of the Anglenbsp;of Inclination, on account of the firft, is reftorednbsp;by the A�tion of the fecond, and here it is notnbsp;In the Motion above-mentioned* of thenbsp;Line of Nodes in refpedl to the Sun, which depends upon the parallel Situation of this Line,nbsp;the Node N is carried to the Syzygy �. Whennbsp;(/or Ex.') the Line of Nodes is come to the Situation M OT, the Moon in its Recefs from thenbsp;Nodes goes through the Quadratures N�, innbsp;which the Force which changes the Inclinationnbsp;�'� 3 37 vanilhes,* and near which it is the lealt of all ;*nbsp;*jj36but in coming towards the Nodes, the Moon isnbsp;every where diftant from the Quadratures, andnbsp;�*i33lt;gt;a greater Force a�ts upon it;* therefore coti-j'i\'^)fideTing one entire Revolution, theEncreafe of thenbsp;Angie of Inclination exceeds its Diminution j*nbsp;that is, that Angle is encreafed, or which is thenbsp;fame, the Inclination is diminijloed �, which obtainsnbsp;every where in the Motion of the Nodes from thenbsp;^ladratures to the Syzygies.

1351 nbsp;nbsp;nbsp;clination of the Plane of the Orbit is the leaf of all-,nbsp;for if/ the Motion of the Nodes from the Syzygies tonbsp;the ^ladratures, the Plane of the Orbit is continually more and more inclined for in that Cafenbsp;as the Moon goes to the Node, it paffes throughnbsp;the Quadratures ; in its Recefs from them thenbsp;Ivloon is diftant from the Quadratures, and innbsp;one �whole Revolution of the Moon., the Force which

'*�3 37 encreares the Inclination * exceeds that which '*� 345 diminifnes it 3* therefore Inclination is en-J ^yzereafed; and it is the greatef of all when the Nodes

Book IV. of Natural Thilofophy. 2.63

the Angle made by the Plane of the Orbit with ^ the Plane of the Ecliptic is terminated *

AUthe Errors in the Mooli's Motiojij that we have 1353 explained, are fometbinggreater in the Conjunctionnbsp;than in the Oppofition.*nbsp;nbsp;nbsp;nbsp;*1194

All the difturbing Forces are determined by dif- i354 covering their Relation with the Addition ofnbsp;Gravity in the Quadratures for which Reafon *'^�^

they all undergo'the fame Changes as that Ad-dition does j that is, they are inverfly^ as the '' Cube of the Difiance of the Sun from the Earth f *�*9nbsp;�which when it remains the fame, they are as thenbsp;Difiance of the Moon from the Earth* Confider-ing all the difitirhing Forces together, the Ditninu-1355nbsp;tion of Gravity prevails j* which follows imtnedi-*-^'-^*^nbsp;ately from the Progrefs of the Apfides ; for it ap- quot;'97�nbsp;pears from thence, that confidering feveral Revolu- �nbsp;tions together, the Effed of the Diminution of�,^,^nbsp;Gravity exceeds the Effed of the Encreafe of it *nbsp;nbsp;nbsp;nbsp;131 s

Therefore Motion of the Moon being confider-1356 ed in general, the Gravity of the Moon towards thenbsp;Earth is diminifioed, coming near the Sun i* there- *�35?nbsp;fore when it is lefs attraded by the Earth, it re-cedes more from it than it would recede, if there wasnbsp;no fuch Diminution of Gravity: Therefore innbsp;that Cafe the Moon�s Diftance is encreafed, 3813517nbsp;alfo the Periodical Fime f and that Time is the �'22 nbsp;greatefi, as aljo the Difiance of the Moon^exteris pari-busj thegreatefi,when the Earth is in thePerihelion,* '*gt;3 54nbsp;becaufe then it is leall diftant from the Sun.

CHAP. XVII.

Concerning the Figures of the Tlanets.

IF we confider the Figures of the Planets, we fliall find that they have fuch Figures,nbsp;which follow from thefe very Laws by which

Mathematical Elements Book TV.

the Syftem is governed ; which is very agreeable to that admirable Order which we obferve everynbsp;where, that no Forces a�b upon the Planets tonbsp;135S deftroy them j that is, that the Figure of a Planet,nbsp;whether it be a Primary or Secondary Planet, is jucbnbsp;as it zvoitld acquire, if it wholly conffied of fluidnbsp;Matter ; which agrees with the Phsenomena.

Whence it follows, that all the Primary and Secondary Planets are Spherical j for they confiltnbsp;of a Matter whofe Particles gravitate towards onenbsp;�'ii'^anocher;^ from which mutual Attraction a Sphe-rical Figure is generated in the fame Manner, as anbsp;Drop becomes round from another Sort of At-''34tra�lion of the Parts.*

1360 nbsp;nbsp;nbsp;Fbis Spherical Figure of the Planets is not changednbsp;from their Motion round the Sun, or from the Motio7inbsp;of the Secondary Planets about their Primary ones ;nbsp;becaufe all the Particles are carried by the famenbsp;Motion : But this Figure undergoes fome Changenbsp;by the Motion round the Axis, and fo much thenbsp;greater as this Alotion is fwifter.

1361 nbsp;nbsp;nbsp;Plate XXV. Fig. 7.] Let PP be the Axisnbsp;of a Planet Ee the Diameter of the JEquatornbsp;perpendicular to the Axis ; let there be a Canalnbsp;P C E filled with a Liquid j this Fluid will de-feend by its Gravity in both Legs towards C,nbsp;and will not be at reft, till the PreflTure in bothnbsp;Legs be equal. If the Planet be at reft, the

*i3 59Flcight of the Fluid in both Legs will be equal:* but if the Planet be moved about its Axis Pp,nbsp;all the Liquid in the Leg CE wdll endeavour tonbsp;recede from the Center by its centrifugal Force,*nbsp;�223 which Force aCls contrary to Gravity,* andnbsp;therefore diminiOies ^the Gravity j fo that therenbsp;is no .Equilibrium till C E exceeds C P. Nownbsp;if the Canal be taken away, the lateral Preffurenbsp;pf the Fluid, of which the Planet confifts, doesnbsp;pot change the Gravity towards Cj nor the Difference

Book IV. of Natural Thilofophy. nbsp;nbsp;nbsp;^65'

C P j* therefore the Planet is every where higher in the JEquator than in the Poles, and acquires,nbsp;hy its Motion round its Jxis, the Figure of a 1^62nbsp;Spheroid deprejfed in its Poles j for the Elevation,nbsp;is continually diminifhed as you go towards thenbsp;Pole, becaufe the centrifugal Force is diminifhednbsp;by reafon of the Diminution of the Diftancenbsp;from the Axis.*nbsp;nbsp;nbsp;nbsp;*23,

If what has been demonftrated be compared with the Phsenomena, it will appear why all thenbsp;Bodies in our Syftem are ipherical^* but that *924nbsp;this Figure is not exa�l, but a little changed bynbsp;their Motion round their Axis,* though this can- *'361nbsp;not be obferved in moft of them, may be deducednbsp;from Obfervations made upon J-upiter and thenbsp;Earth. Jfironomers have ohferved that the Jxis oftl6%nbsp;Jupiter is Jborter than its Equatorial Diameter ;nbsp;although this Planet be the greateft of all the Planets, it is moved the fwifteft about its Axis i*nbsp;and therefore this Difference may be obferved.

Fbe Elevation of the Earth at the ^.quator ZJ1364 � determined by ns, although perhaps to the Inhabitants of the other Planets, if there are any, itnbsp;may not be more fenfible than the Elevations onnbsp;Marsmdi Venus are to us, which are fo fmall thatnbsp;we cannot perceive them.

Suppofe the Earth to be fluid, it will acquire 1365 the aforefaid fpheroidical Figure;* If the Partsnbsp;cohere towards the Center, the Pofition of thenbsp;other Parts will not be changed thereby, nor willnbsp;it be changed, if in fome Places the Parts coherenbsp;togetherquite to the Surface j fo that the Surfacenbsp;of the Sea muft neceflfarily acquire a fpheroidical Figure deprefled at the Foies. But fincenbsp;the Shores are every where but a little elevatednbsp;above the Surface of the Sea, it is certain thatnbsp;the Continent acquires the fame Figure.

Mathematical Elements Book IV.

Now to meafure this Elevation, that is, how much the Diameter of the ^Equator of the Earthnbsp;is longer than the Axis, we muft confider itsnbsp;Motion round its Axis in the Space of 23 Hours,nbsp;�*56056 Min. 4 Sec.* and fuppofing the Earth homogeneous, the Computation will be made in thenbsp;following Manner,

1366 The Periphery of the Earth is 128,202,185 Khynland Feet j therefore in one Second of Time,nbsp;a Point cf the ,ffiquator goes through 1488 Feet;nbsp;the verfed Sine of which Arc is o, 054, a Spacenbsp;which could be gone through by a Body in fuchnbsp;a Time by the centrifugal Force.

By a Gravity, a Body, in one Second, as we have (hewn before, falls through 15,607 Khynlandnbsp;Feet; but thefe Experiments were made at thenbsp;Diftance of 48 Degr. from the iEquator � e (Platenbsp;XXV. Fig. 7.) at the Point A, the centrifugalnbsp;Force at E is to the centrifugal Force at A, asnbsp;C E to C A, for thefe Lines are very little dif-*z32ferent at AB:* Let this centrifugal Force benbsp;A h ; having drawn the Perpendicular b a x.onbsp;C A continued, let the Force through A ^ be re-folved into two other Forces diredfed along A anbsp;*192 and a b the Gravity is ditniniflied only by thenbsp;former, and A ^ is to the Force diminifhing it,nbsp;as G A to A B, by reafon of the fimilar reftan-gular Triangles Aba and ABC, which havenbsp;their oppofite vertical Angies equal at A; therefore the centrifugal Force at the ^Equator, withnbsp;which a Body in one Second goes through 0,054;nbsp;is to the Force which diminifhes the Gravity atnbsp;A, in a duplicate Proportion of the Radius A Gnbsp;to A B, which is the Co-fine of the Latitudenbsp;A E of 48 Degr. So that from this diminilhingnbsp;Force the Body in one Second goes through 0,0243;nbsp;wherefore, if the Earth was at reft, in falling itnbsp;would not go through 15, 607 Feet, but 15,632;

with

Book IV. of Natural Thilofophy.

with which Gravity a Body falls under the Poles, becaufe thefe Points are not moved. At thenbsp;jEquator, by the centrifugal Force, a Body goesnbsp;through o, 054. and falls as much in the fame Timenbsp;from the Height of ly, 57^ Feet i whence it appears that the Gravity under the Poles is, tonbsp;the Gravity under the JEquator, as 289 tonbsp;288.

If Fig. 7. reprefents the Figure of the Earth, the Weight of a Column of Liquid C E will benbsp;to the Weight of a Column of Liquid CA, thenbsp;Earth being at reft, as 289 to 288 j for otherwife,nbsp;the Earth moving, there will not be an JEquilibri-mn; becaufe jh of the Column C E is fuftained bynbsp;the centrifugal Force ; for the centrifugal Forcenbsp;decreafes as you come towards the Center, in thenbsp;Ratio of the Diftance, * in which Ratio alfo thenbsp;Gravity decreafes ;* fo that in all the Points of *1133nbsp;the Column, the fame Part of the Weight is fuftained, as towards the Surface.

Whence we deduce, that tbe Height CP at the 1367 Pole is, to the Height EC at the Equator, as 229nbsp;to 230 ; for fuppofing this Ratio between thenbsp;Axis and the Alquatorial Diameter, if a Computation be made of the Gravities in the Placesnbsp;P and E, the Earth being at reft, they are foundnbsp;to be to one another, asii2i, 71.10 1120, 71 jnbsp;which Ratio obtains every where in correfpon-dent Points, that is, which are diftant from thenbsp;Center as C P to PE; becaufe in both I^egsnbsp;the Gravity decreafes in Proportion to the Diftance from the Center.* You have the Weight *1133nbsp;by multiplying the Quantity of Matter by thenbsp;Gravity ; for the Weight encreafes in a Rationbsp;of both ; By multiplying 1121, 71. by 229, andnbsp;1120, 71, by 230. the Produds are to one ano-\ther as 288 to 289 ; which is the Ratio of thenbsp;Weights before difeovered. The mean Diameter

Mathematical Elements Book IV.

*97�of the Earth is 3,400,669 Perches,* therefore the Axis PP is 3,393,261, and the ^Equatorial Diameter E is 3,408,078 Perches, which exceeds

1368 nbsp;nbsp;nbsp;the Axis by 14,817 Perches (wz.) tjo, and thenbsp;Equator is more elevated by 7408,5.

1369 nbsp;nbsp;nbsp;In this Computation, as we have faid, wenbsp;have looked upon the Earth as homogeneous;nbsp;but if it be more denfe towards the Center, thenbsp;Matter which is added to it may be looked uponnbsp;as a feparate Eody, from whofe Center the Pointsnbsp;P and E are unequally diftant, and towardsnbsp;which therefore the Bodies P and E ha^e a dif-

�*i226 ferent Gravity^* and the Difference is fo much the greater as thefe Differences are greater; andnbsp;it will be alfo fo much the greater in refpeft ofnbsp;the whole Gravity, as the Quantity of Matternbsp;w'hich is added, or which is the fame, as thenbsp;Denfity is greater towards the Center.

It is plain that the Forces of Gravity, at the poles and the A^quator, differ from one anothernbsp;more than Part, by comparing together Experiments made at feveral Diftances from the iEqua-tor, by the Help of Pendulums, by which thenbsp;Forces of Gravity may be compared together,nbsp;*1643$ we have fhewn^* and which Difference is trulynbsp;nearly double that which is found by Computa-

1370 nbsp;nbsp;nbsp;tion ; whence it follows, that the Elevation of thenbsp;Equator is nearly double that which we have de^

1371 nbsp;nbsp;nbsp;of the Earth, vie^2i\\^e:ex.ha.t heavy Bodies do notnbsp;tend direblly to the Earth's Center^ unlefs at thenbsp;Poles and the JEquator, hut every where perpendicularly to the Surface of the Spheroid; for anbsp;Liquid will not be at reft, unlefs its upper Surfacenbsp;forms a right Angle with the Direction of heavy

Book IV. of Natural ^hilofophy. ^6^

alfo deduce this Diredion of heavy Bodies from 1372 the centrifugal Force (flate X.X.Y.Fig. 7.) Thenbsp;Body A, by its Gravity, tends towards C, andnbsp;is carried by its centrifugal Force along A ^ :

This Force at the Point A is to the Gravity along AC, as I to 430, 8. having formed a Parallelogram with the Sides A c and Ab, fuppofingnbsp;thefe to one another, as 430, 8. to i, the Diagonal will fhew the Diredion of heavy Bodies,*,50nbsp;forming a fmall Angle with the Line AC. Thenbsp;Force along A h encreafes as you go towardsnbsp;the jBquator, whereby this Angle is encreafed,nbsp;but is diminilhed by the Encreafe of the Angienbsp;C Ah-, fo that in the JEquator, where the centrifugal Force is greateft, the Direftion of heavynbsp;Bodies coincides with E C i at the Pole it coincides with PC; becaufe there is no centrifugalnbsp;Force there.

In this fpheroidical Figure, the Latitude of tbefi^Ji Place is determined by an Angle^ as A C E, whichnbsp;is made with the Equator, by a Line drawn fromnbsp;the Place of the Center. Dividing this whole Arcnbsp;PAE, by this Method, into 90 Parts, that !$,nbsp;into Degrees, it will eafily appear, that going\-^yj^nbsp;towards the Poles., the Degrees are encreafed on thenbsp;Surface; but this Difference is fo very fmall,nbsp;that in meafuring Degrees that are not very di-ftant, it cannot be difcovcred ; becaufe the Error, arifing from the Make and Ufe of the In-ftruments, exceeds this Difference. Whence Degrees meafured at the South and North of France^nbsp;as alfo in England, differ little from one another,nbsp;and the middle one is the leaft of all ; whereforenbsp;nothing can be concluded concerning the Earth�snbsp;Figure from thefe Meafttres.

CHAP.

CHAP. XVIII.

The Thyfical Explanation of the Motion of the j^xts of the Earth.

That the Nodes of the Moon go back, that is, are moved in Antecedentiaf andnbsp;that the Inclination of its Orbit is liable tonbsp;�**347 Change,* we have already demonftrated ; letnbsp;us conceive feveral Moons, to be at the famenbsp;Diltance, revolving in equal Times about thenbsp;Earth, in a Plane inclined to the Plane of thenbsp;Ecliptics it is plain they will all be agitated bynbsp;the fame Motions : Let us conceive the Number of the Moons to be encreafed, fo as to touchnbsp;one another, and form a Ring, whofe Parts cohere s whilft one Part of the Ring is attra�fed,nbsp;to encreafe the Inclination, the other Part isnbsp;1376 agitated by a contrary Motion, to diminifh itsnbsp;quot;*'345 Inclination s* the greater Force in this Cafe pre-1377 vails, that is, in the Motion of the Line of Nodes,nbsp;from the ^ladrattires towardstheSyzygies, the Inclination of the King is diminijhed in each of itsnbsp;*1349 Ke-volutions �* and it is the leaf of all, -when thenbsp;'^iis�Line of Nodes is intheSyzygies* On the contrary,nbsp;its Inclination is encreafed, when the Line of Nodesnbsp;is carried from the Syzygies towards the ^ladra-�* 15 31 tures and it is the greateft of all, when the Line ofnbsp;^a^the Nodes is in thefe lajip The Line of thenbsp;Nodes is continually carried in Antecedentia, iin-f^lfefs when it is at reft in the Syzygies.*

nijhed, its Motions will not be changeds becaufc they depend upon Gravity, which a�ls equally up-�'1017 on every Particle of Matter.*

Book IV. of Natural �Philofophy. 2,71

If the Diameter of the Kinghe diminijhed^ thefe 1380 Motions are diminifhed in a Ratio of this Diminution,�*^ but none of them wholly vanifli; and *�35 nbsp;it is agitated by the fame Motions.

Let us conceive the Earth to be fpherical �, and *38i in the Plane of the Equator, which makes annbsp;Angle of 23 Degr. 29 Min. with the Plane ofnbsp;the Ecliptic, let there be a Ring, revolving innbsp;the fame Time as the Earth ; let it be diminifhednbsp;fo as to touch the Earth, and cohere with itjnbsp;by this the aforefaid Motion of the Ring willnbsp;not be dellroyed j for fince the Earth is kept innbsp;a determinate Situation by no Force, it yields tonbsp;the Impreflions of the Ring, whofe Agitationsnbsp;are yet diminifhed, the Matter to be moved being encreafed, and the moving Power remainingnbsp;the fame.

And this is truly the Cafe, for the Figure of the Earth is fpherical, encompafled with a Ringnbsp;at the JEquator, whereby the Earth is more elevated towards the .^quator,�*^ the Line of Nodes *1370nbsp;of which Ring is the Section of the Planes ofnbsp;the iEquator and Ecliptic. Whence we deducenbsp;the following Conclufions,

In the JEqtiinoxeSy the Inclination of the JE-i^8z quator is leaf: of allj�*^ and therefore the Incli-*iij';nbsp;nation of the Axis is the greateft ; for it makes anbsp;right Angle with the Plane of the JEquator.* *1075nbsp;The Inclination of the iEquator is encreafed, thatnbsp;is, the Inclination of the Axis is diminifoed^ tillti^inbsp;the Sun comes to the Solftices, where this Inclinationnbsp;of the Axis is leaf of alf and that of the iEquatornbsp;the greateft.* Therefore twice in a Tear the ln-*imnbsp;clination of the Axis of the Earth is diminifhed .^andnbsp;twice encreafed. Andthe Seliionof the Plane of thenbsp;JEquator with the Plane of the Ecliptic^ which is 1385nbsp;at reft in the Mqninoxes.^ the reft of the Time isnbsp;moved in Antecedentia.�^

The

2-72- Mathematical Elements Book IV.

1386 The Plane of the JPquator is enclined to the ^ Plane of the Moon�s Orbit; for it makes a fmallnbsp;*969 Angle with the Plane of the Ecliptic Andnbsp;therefore the Moon afts in the fame Manner upon the Ring as the Sun ; and although the Moonnbsp;be lefs, yet, becaufe it is much lefs diftant thannbsp;the Sun, it exerts a greater Aftion upon the Ring.nbsp;^387 Wherefore alfo the Inclination of the Axis of thenbsp;Earth to the Plane of the Moon�s Orbit* (andnbsp;confeqiiently to the plane of the Ecliptic') is twicenbsp;changed in every Revolution, and twice refiored bynbsp;the ASiion of the Moon : And the SeSiion of thenbsp;Plane of the JEqiiator, with the Plane of the Moon�snbsp;*1385 Orbit,* is carried in Antecedentia ; From whichnbsp;Motion neceffarily follows, that the Seftion ofnbsp;the Plane of the JEquator, with the Plane of thenbsp;Ecliptic, changes its Place.

1388 The Changes of the Inclination of the Axis are too fmall to be obferved; but the change of Place of thenbsp;Line of the Equinoxes, and the Motion of the Axisnbsp;which follows from it, being always carried thenbsp;fame Way, at laft become fe72fible �, and fromthefenbsp;'^5follow the Phsenomena before explained.*

Concerning the Tides.

That we may explain the Tides from the Principles already laid down, we mull con-lider, that the Earth, as alfo all Bodies near it,nbsp;'1206gravitate towards the Moon;* therefore thenbsp;Particles of Water in the Earth�s Surface whichnbsp;tend towards the Center of the Earth (for herenbsp;we negle�l the Confideration of N� 1371.) arenbsp;carried with it towards the Moon. Since alfonbsp;the folid Mafs of the Earth is carried towardsnbsp;the Moon, according to the Laws, w hich would

book IV, of Natural T^bilojbphy. nbsp;nbsp;nbsp;2,73

obtain, if all the Matter of which it confifts was colledled in its Center what has been demon-*nbsp;firated in Chap. 16. of the ASiion of the Sun uponnbsp;the Moon falling towards the Earth, whilft with ijponbsp;the Earth it goes towards the Sun, may be appliednbsp;to the Adiion of the Moon upon the Particles ofnbsp;Water in the Earth's Surface^ which do not cohere with the Mafs of the Earth, but tend towards its Center, and continually with its Mafsnbsp;fall towards the Moon; by which Force, as wenbsp;ha ve fhev/n,* the Earth is keptinits Orbit, about *iigonbsp;the common Center of Gravity of the Earth andnbsp;the Moon.

Plate XXV. Fig. i ] Let S be the Moon ^ 1391 ALBL/the Surface of the Earth, whofe Mafsnbsp;tends towards the Moon, as if it was all colledt-ed at T j by the Afffion of the Moon, the Particles of Water, A and B, acquire a greaternbsp;Gravity towards T on the contrary, the Parti-*11*7nbsp;cles at L,/, lofe of their Gravity j* whence we**^9�nbsp;deduce, that if the whole Earth was covered withnbsp;Water, there would not be an JEquilibrium^ un-lefs this Water was more elevated in the Pointsnbsp;L and /, than in a whole Circle 90 Degreesnbsp;diftant from thefe Points j and therefore paf-ling through the Points A and B. Therefore by 1392nbsp;the Allion of the Moon, the Water acquires anbsp;fpheroidical Figure, formed by the Revolution ofnbsp;an Oval about its greater Axis, which, being continued, goes through the Moon.

Let us fuppofe the Moon in the JEquator ; all the Setftions of the Earth which are parallel tonbsp;the JEquator, as they are alfo parallel to the Axisnbsp;of the Spheroid* are Oval, whofe greater Axes �1391nbsp;pafs through the Meridian of the Moon whencenbsp;it follows, that the Earth being at Refi, in anynbsp;Circle of Latitude, the Water is more elevatednbsp;V o L. II.nbsp;nbsp;nbsp;nbsp;Tnbsp;nbsp;nbsp;nbsp;in

a74 nbsp;nbsp;nbsp;Mathematical Elements Book IV.

in the Meridian in which the Moon is^ and in th^ oppojite Meridian^ than in the intermediate Places.

1394 The Lunar Day is the 1�ime /pent between the Moony�s going from the Meridiaii and coming to itnbsp;again. This Day is divided into 24 Lunarnbsp;Hours : It is 50 Minutes longer than the Natural Day.

From the Motion of the Earth round its Axis, every Lunar Day, every Place pafles throughnbsp;the Meridian of the Moon and the oppolite Meridian, that is, twice palles through that Place,nbsp;ijpj;where the Water is raifed by the Abiion of thenbsp;Moon.^ and twice through that Place, where thenbsp;*1393 Water is difperfed by the fame A�tion* and fonbsp;in a Lunar Day the Sea is twice elevated^ and twicenbsp;depreffed^ in any afjigned Place.

1396 By the Motion of the Earth round its Axis, the elevated Water continually recedes from thenbsp;Meridian of the Moon 3 yet by the Action ofnbsp;the Moon, the Axis of the Spheroid paflesnbsp;*13 91 through the Moonj* therefore the Water is continually agitated, that the Elevation, which (onnbsp;Account of the Motion of the Earth) is removed, may be brought under the Moon. Therefore the Water continually flows from A and Bnbsp;{Plate XXV. Fig. i.) towards L and /, whilft,nbsp;by the Motion of the Earth, the Elevation isnbsp;carried from L towards B, and from / towards A �, that is, between L and B, as alfonbsp;between / and A, there are two contrary Motions, by which the Water is accumulated 3 fonbsp;that the greateft Elevations are betw'een thefenbsp;Points (i^ia,) 'not diretdly under the Moon, butnbsp;on one Side of that Point, and likewife afide ofnbsp;I39i7the oppofite Point. That is, in any Place, th^nbsp;Water mofi elevated^ two or three Hours after the

Book IV. of Natural ^hilofophy* ^75'

The Elevation tovoards the Moon a little exceeds 1398 the oppofite one.* The Afcent of the Water isdimi- *1390nbsp;nified, as you go toward the Poles, hecaiife therenbsp;is no Agitation of the Water there.

What has been demonftrated, in relation 101399 the Moon, may be applied to the Sun; therefore 1400nbsp;from the Action of the Sun, every natural Day, the 1401nbsp;Sea is twice elevated, and twice deprejfed*. This �'139snbsp;Agitation is much lefs, on account of the imnienfenbsp;Diftance of the Sun, than that which depends upon the Moon-, yn it is fnbje�i tothe fame Laws. I402

The Motions, which depend upon the ASiions 0/1403 the Moon and Sun, are not diftinguifhed, but confounded; and from the Adlrion of the Sun, thenbsp;Lunar Tide is only changed which Change variesnbsp;every Day, by reafon of the Inequality between 1404nbsp;the Natural and Lunar Day*.nbsp;nbsp;nbsp;nbsp;�*^1394

IntbeSyzygies, the Elevation, from the A�lions 1405 of both Luminaries, concur, and the Sea is mo-renbsp;elevated; the Sea afcends lefs in the ^ladratures;nbsp;for where the Water is elevated by the Adtionnbsp;of the Moon, it is difperfed by the Adfion of thenbsp;Sun, and fo on the contrary. Therefore, whilftiC[.o6nbsp;the Moon pajfes from the Syzygy to the ^ladratiire,nbsp;the daily Elevations are continually diminifoed:

On the contrary, they are enc-reafed when the Moon moves the ^ladrature to the Syzygy. At a New 1407nbsp;Moon alfo, cxteris paribus, the Elevations.arenbsp;greater, and tbofe that follow one another the fame 1:,nbsp;Day, are more different than at a Full-Mooji.* 1401nbsp;The greatefi and leaf Elevations are not ohferved, 1408nbsp;till the fecond or third Day after the New or Fullnbsp;Moon ; becaufe the Motion acquired is not prefent-Jy deitroyed from the Attrition, and othernbsp;Caufes, by which acquired Motion the Afcent ofnbsp;the Water is encreafed. Although the Adionbynbsp;VoL. II.nbsp;nbsp;nbsp;nbsp;Tznbsp;nbsp;nbsp;nbsp;which

which the Sea is raifed be diminiihed j fomewhat like to what we have demonftrated elfewhere con-^�74 cerning Heat*.

*4�9 If now we confider the Luminaries receding from the Plane of the jEquator, we fhall perceive that the Agitation is diminilhed, znAbecomesnbsp;lefsy according as the Declination of the Luminaries becomes greater. Which plainly appears, ifnbsp;we conceive them to be in the Poles ; for thennbsp;the Axis of the fpheroidical Figure coincidesnbsp;with the Aftion of the Earth ; and all the Sections that are parallel to the .Equator, are perpendicular to the Axis of the Spheroid i andnbsp;therefore circular. So that the Water, in everynbsp;Circle of Latitude, will have every where thenbsp;fame Elevation ; and fo in the Motion of thenbsp;Earth, the Height of the Sea is not changed innbsp;particular Places. If the Luminaries recedenbsp;from the Poles, it is eafy to find, that the Agitation will be more and more encreafed, till it benbsp;the greateft of all, the Spheroid revolving aboutnbsp;a Line perpendicular to the Axis, the Axis ofnbsp;the Spheroid being fuppofed in the Plane of thenbsp;jEquator.

1410 nbsp;nbsp;nbsp;Hence it is plain, why in the Syzygies, near thenbsp;.^qtiinoxes.^ the 'ifides are obferved to be the great-eft^hoth. Luminaries being in or near the JEquator.

1411 nbsp;nbsp;nbsp;fhe ASiions of the Moon and Sun are greater, thenbsp;* n-i'^lefs thofe Bodies are difiant from the Earth* j butnbsp;*'39� vvhen the Difrance of the Sun is lefs, and it is

in the South Signs, often both the greateft ASqui-noCtial Tides are obferved in that Situation of the Sun j that is, before the Vernal, and afternbsp;the Autumnal J�quinox, which yet does notnbsp;happen every Yearj becaufe fome Variation maynbsp;arife from the Situation of the Moon�s Orbit,nbsp;and the Diftance of the Syzygy from the AEqui-liox.

Book IV. of Natural ^hilofofloy. nbsp;nbsp;nbsp;^77

Plate XXV. Fig. 8.] In Places difiant from th� ^Equator.) as the Luminaries recede from the JEqna.-tor, the Elevations, that happen the fame Day, arenbsp;unequal. Let PP be the Axis of the Earth, EEnbsp;theAiquator, L / a Circle of Latitude; A B thenbsp;Axis of the fpheroidical Figure which the Waternbsp;forms: When a Place in the Circle LI is givennbsp;at L or I, it is given in the fame Meridian withnbsp;the Axis of the Spheroid, and the Water is moltnbsp;elevated, in both Cafes j yet at L it is morenbsp;elevated than at I; for Cl exceeds Cl, whichnbsp;Lines meafure the Heights of the Waters, thatnbsp;is, the Diftances from the Center : Thefe Linesnbsp;would be equal if A L and Bl (which are thenbsp;Diftances from the Axis of the Spheroid) werenbsp;equal. But Cl is lefs, becaufe Bl exceeds AL,nbsp;which arifes from the Inclination of the Axis ofnbsp;the Spheroid to the ^Equator.

Js long as the Moon is on the fame Side of the 1413 Equator in any Place, that is, towards the Linenbsp;C A continued, the Elevation of the M'ater is oh- 1414nbsp;ferved to be greatefi every Day, after the Moon hasnbsp;faffed the Meridian of the Place �, for there is thenbsp;greateft Elevation when the.Place is come to L jnbsp;but if the .Equator feparates, or is hetvceen thenbsp;Moon and the Place, of which we fpeak, that is, ifnbsp;the Moon be towards the Line CB continued,nbsp;the Water again at L will come to the greateftnbsp;Height, and, every Day, the greateft Elevation ofnbsp;the Sea will he, after the Moon has pajfed throughnbsp;the oppoftte Meridian.

All Things, which have been hitherto explained, would exadly obtain, if the whole Surface of the Earth was covered with Sea �, but fince the Sea is not every where, fome Changesnbsp;arife from thence, not indeed in the open Sea jnbsp;becaufe the Ocean is extended enough to benbsp;fubje�i: to the Motions we have fpoken of. But

Mathematical Elements Book IV.

i^lSthe Situation of th� Shores^ the Streights^ and many other Ehings defending upon the particular Situationnbsp;�ftbe Places^ difiurbthefe general Rules. Yet it isnbsp;plain from themoft general Obfervations, that thenbsp;Tide follows the Laws which we have explained.nbsp;What remains is, to determine the Forces withnbsp;which the Sun and Moon adf upon the Sea, thatnbsp;it may appear, that they are able to produce thenbsp;Effeds which we have mentioned i and that thenbsp;Adions of thofe Bodies, upon Pendulums andnbsp;other Bodies, are infenfible.

1416 The Encreafe or Addition to the Gravity of the Moon in the Quadratures, from the Adionnbsp;of the Sun, is to the Gravity of the Moon to-*1291 wards the Earth, as i to 178, 73*. In whichnbsp;Computation we have fuppofed the mean Di-ftance of the Moon from the Center of thenbsp;*1285 garth to be 60 Semidiameters of the Earth* :nbsp;Therefore the Gravity of the Moon is to thenbsp;Gravity of the Earth�s Surface, as i to 60 ^ 60nbsp;�''1208 =3600*. Therefore the above-mentioned Encreafe is to the Gravity on the Earth�s Surface, asnbsp;I to 643428, in which Computation there is annbsp;Error to be correded.

This Computation would be exad, if the Encreafe, of which we fpeak, was to the Force, with which the Earth defeends towards the Sun,nbsp;as the Diftance of the Moon (which is 60 Semidiameters of the Earth) to the Diftance ofnbsp;�*1287 the Earth from the Sun* j but it is as the truenbsp;mean Diftance of the Moon, which is 6oj Semidiameters of the Earth to the Diftance of thenbsp;Earth from the Sun. Wherefore the Encreafe,nbsp;that we have juft determined, ought to be T33nbsp;Part encreafed, and will be to the Force of Gravity on the Earth�s Surface, as i rh to 643428,nbsp;or as I to 638110,4.

This Encreafe of the Gravity of the Moon, in the Quadratures from the Adion of the Sun,

Book IV. of Natural Wilofofhy. 2-79

is to the Encreafe of the Gravity of the Water on the Earth�s Surface, in Places which are 90nbsp;Degr. diftant from the Sun (from the famenbsp;Adion of the Sun) as 603 to i* Therefore thisquot;^'!^*nbsp;Encreafe of the Gravity is to the Gravity of thenbsp;Water, as i tonbsp;nbsp;nbsp;nbsp;The Diminution of

the Gravity, under the Sun, and in the oppolite^ Place, is double this Encreafe * j therefore it is tonbsp;the Gravity as i to 1930 839, and the whole*4 7nbsp;Change in the Gravity^ arifing from the A6iion ofnbsp;the Sun, is to the Gravity itfelf, as i to 12868560.

In order to compare the A�fion of the Moon 1418 with the Aftion of the Sun, we muft make Experiments in Places, in which, by reafon of thenbsp;Narrownefs, the Sea is fenfibly raifed. Nearnbsp;Briftol at the Autumn and Spring, at whichnbsp;Times the Agitation of the Sea is greateft*, the *1410nbsp;Water afcends in the Syzygies about 45 Feet,nbsp;more or lefs, in the Quadratures about 25 Feet,nbsp;more or lefs. Which Numbers are to one another as 9 to 4.

The Determination of the Forces, which we would find, if the greaceft and leaft Elevationsnbsp;were exa�tly at the Time of the Syzygies, wouldnbsp;be very eafy, which we have Ihewn before notnbsp;to happen fo*.nbsp;nbsp;nbsp;nbsp;quot;^14�*

The Diftance of the Moon from the Syzygy, or the Quadrature, is not always the fame in thenbsp;greateft or leaft Elevation j for this Diftance varies,nbsp;becaufe the Moon is fometimes more, and fome-times lefs diftant from the Meridian, when itnbsp;goes through the Syzygy or Quadrature. Thenbsp;mean Diftance of the Moon from the Syzygy,nbsp;or Quadrature, to which the aforefaid Ob-fervations ought to be referred, is about 18nbsp;Degr. 30 Min. fo that the whole Adfion of thenbsp;Sun neither confpires with the Aftion of thenbsp;Moon in the Syzygies, nor afts contrary to itnbsp;in the Quadratures. Alfo in fuch a Cafe, if atnbsp;T 4nbsp;nbsp;nbsp;nbsp;the

Mathematical Elements Book IV.

the Syzygy, both the Luminaries be in the 2L-quator, in the faid Diftance from the Quadrature, the Declination of the Moon is 22Degr. 13 Min.nbsp;more or left �, whereby the Force of the Moon tonbsp;�*HC9move the Seaisdiminifhed*. BeCdes, cxteris pa~nbsp;ribus^ the Diftance of the Moon from the Earthnbsp;*i3gt;oat the Syzygies is lefs than at the Quadratures �*nbsp;*5quot; whence alfo the A�lion of the Moon isdiminifti-*1411 ed at the Quadratures;* By attending to ail whichnbsp;Things we may difeover, that the mean Force ofnbsp;1/^l^the Sun.) to move theSea^ is to themean Forceoftbenbsp;Moon to move the fame, as 1 to 4,4813'. But thenbsp;Force of the Sun is to the Force of Gravity, asnbsp;� to 12868560;* wherefore the Force of the Moonnbsp;is to the fame Force of Gravity as l to 2871485.

1420 nbsp;nbsp;nbsp;From whence it follows, that thefe Forces of thenbsp;Moon and Sun are too fmall to be fenfible innbsp;Pendulums and other Experiments; but it is eafilynbsp;proved, that they are capable of agitating thenbsp;Sea.

By diralnilhing the Gravity 20? Part, the Sea *1368 is raifed to the Height of 88902 Khynland, Feet*.nbsp;^*'7 For each Perch contains 12 Feet; whence, by the

1422 nbsp;nbsp;nbsp;the Sun changes the Height of the Sea two Feet,nbsp;*1^19 and that the ABion of the Moon changes it 8, 95;*

and that from the joined Action of both, the mean Agitation is of about eleven Feet,

1423 nbsp;nbsp;nbsp;ty well with �bfervations ; for in the open Ocean,nbsp;as the Sea is more or lefs open, the Water isnbsp;raifed to the Height of Six, Nine, Twelve, ornbsp;Fifteen Feet ; in which Elevations alfo therenbsp;is a Difference arifing from the Depth of thenbsp;Waters. But thofe Elevations, which far exceednbsp;thefe, happen, where the Sea violently enters into

x/^ia^Streights orGulphs; where the Force is not broken^ till the Water rifes higher,

CHAP.

CHAP. XX. �

TH E Forces of the Sun and Moon, for 142J giving Motion to the Sea, are to one another, in a Ratio compounded of the Ratio of thenbsp;Quantities of Matter in thefe Bodies* (for all the *1107nbsp;Particles of Matter ad:) and the inverfe Ratio ofnbsp;the Cubes of the Diftances of the Sun and Moonnbsp;from the Earth*nbsp;nbsp;nbsp;nbsp;*1390

The Quantities of Matter are in a Ratio com-pounded of the Ratio of the Bulks, that is, of the Cubes of the Diameters, and the Ratio ofnbsp;the Denfitiesi* wherefore the Forces above-mentioned are diredly as the Denfities and the 79nbsp;Cubes of the Diameters, and inverfly as thenbsp;Cubes of the Diftances.

The apparent Diameters of Bodies, that is, the Angles under which they are feen, encreafenbsp;as the Diameters themfelves, and diminilh as thenbsp;Diftances j that is, they are diredly as the Diameters, and inverfly as the Diftances j thereforenbsp;the Ratio compounded of the Ratio�s of thenbsp;Cubes of the apparent Diameters of the Sun andnbsp;Moon, and of the Ratio of the Denfities, will benbsp;the Ratio of the Forces, whereby thofe Bodies ad;nbsp;upon the Sea. Therefore the Denfities of thofe 1426nbsp;Bodies are direSily as the Forces^ whereby theynbsp;move the Sea, and inverfly as the Cubes of theirnbsp;apparent Diameters ; and dividing the Forces bynbsp;the Cubes of thofe Diameters, you have the Ratio of the Denfities.

The

Mathematical Elements Book IV.

The Force of the Sun is to the Force of the ^i4'9Moon, as a to 4, 48154- the mean apparentnbsp;Diameter of the Sun is 32 Min. 12 Sec. and thenbsp;mean apparent Diameter of the Moon is 31 Min.

161 Sec. That is, they are to one another, as 14273864 to 3753. Therefore t�e Denfity of the Sunnbsp;is to the Moonquot;s Denfity^ as loooo 48911: whichnbsp;Denfity of the Moon may be compared with thenbsp;�*iilt;)oDenfities o?fupiter^ Saturn, and the Earth*, andnbsp;the Moon is denfer than the Earth.

The Quantities of Matter in two Bodies are to one another, in a Ratio compounded of thenbsp;*79 Denfities and Bulks;* that is, if the Body be anbsp;Sphere, in a Ratio compounded of the Denfitiesnbsp;and the Cubes of the Diameters,

1428 Vhe Denfites of the Moon and Earth are to one another, as 48911 to 39214^; the Diametersnbsp;1260 as II to 40, 2. therefore the ^lantities of Matter,nbsp;inthofe Bodies, areas i to 39, 13. Though thenbsp;Denfities be difeovered, if you fuppofe the Bodiesnbsp;to be homogeneous; yet the Quantities of Matternbsp;will be rightly defined, though the Bodies are notnbsp;homogeneous; for we determine the Denfity whichnbsp;that Body would have, if the Matter, of whichnbsp;the Body really confifts, was equally diffufed allnbsp;over it.

142P Vhe Gravities, on the Surfaces of the Earth and Moon, are determined, by multiplying the Denfi-*1230 ties by the Diameters*, that is, they are to onenbsp;another, as 2, 93 to i, or as 407,8 to 139, 2.nbsp;Which Number alfo does exprefs the Relation ofnbsp;Gravity on the Surface of the Moon, with thenbsp;Gravity on the Surfaces of the Sun, Jupiter andnbsp;Saturn.*

1430 S^he common Center of Gravity of the Moon and Earth, about which both Bodies are moved, isnbsp;determined; for its Diftance from the Center of

Book IV. of Natural Thilofofhy. nbsp;nbsp;nbsp;aSs

the Earth, is to the Difiduce between th� Centers of both Bodies, as the Quantity of Matter in thenbsp;Moon to the Quantity of Matter in both Bodies j* therefore 40, 13. is to i: as the Diftancenbsp;of the Moon from the Earth, is to the requirednbsp;Diftance of the Center of Gravity, from the Centernbsp;of the Earthy which is found to be of 5126950nbsp;Perches^ as is deduced from the known Diameter of the Earth*, and the Diftance of thenbsp;Moon.

Gravity upon the Moon^s Surface, as 10, 7. to 1432 980028, or as I to 91524, the Gravity beingnbsp;changed on the Earth�s Surface, by iK7l48T Partnbsp;the Water is raifed 8, 95 Feetj* and therefore if *141�nbsp;Gravity was to be changed Part, the Eleva-tionedwould he of28o, 7 Feet, as it is found by thenbsp;Rule of Three. If, keeping this Diminution ofnbsp;Gravity, we confider a lefs Body, this Height

muft

To determine the Figure of the Moon, we 1431 muft examine what Figure it would have, if knbsp;was fluid*. If we confider the Moon alone at *1358nbsp;reft, it would be fpherical:* If we confider the *13^9nbsp;Aiftion of the Earth upon the Moon, the Moonnbsp;would acquire the Figure of a Spheroid, whofenbsp;Axis would go through the Earth*. The Force *'5^1nbsp;of the Earth, for changing the Figure of thenbsp;Moon, is to the Force of the Moon upon the ,nbsp;Earth, as 39, 13. to i.* and as the Diameter ofnbsp;the Moon to the Earth�s Diameter,* which are *1390nbsp;to one another as ii to 40, 2. and it is a Ratio 1354nbsp;compounded of thefe 10 7. to i. This Forcenbsp;of the Moon is to the Gravity upon the Earth�snbsp;Surface, as i to 2871485 which Gravity, on'^Hionbsp;the Earth�s Surface, is to the Gravity on thenbsp;Surface of the Moon, as 407, 8, to 139, 2.*nbsp;as 2871485, to 980028 : Wherefore the ASlion ofnbsp;the Earthy for changing the Moon^s Figure, is to the

^84 Mathematical Elements Book IV.

muftbe diminifbed, in Proportion to the Diameter: Therefore, from theA6i:ion of the Earth, the Elevation of the Moon is of 76, 8 Feet j and if thenbsp;*43 3 Moon be homogeneous, there will not he an Equilibrium^ unlejs the Axis of the Spheroid exceeds thenbsp;Diameter^ which is perpendicular to it^ 153, 60nbsp;Feet.

J434 ��'he Elevation of the Moon^ from the Acfrion of the Earth, may be difcovered by one Angle Proportion, by knowing the Elevation of the Seanbsp;from the Moon�s A��on j for thefe Elevations arenbsp;in a duplicate inverfe Katio of the Gravities on thenbsp;Surfaces of thofe Bodies.

I43jr If, fuppofing this to be the Figure of the Moon, we conceive the Parts to cohere, there will not be annbsp;Mqiiilibriim between the Parts of the Moon, un-lefs the Axis of the Spheroid be directed towardsnbsp;the Earth ^ whence we fee the Reafon, why thenbsp;Moon always turns the fame Face towards thenbsp;Earth ^ by which continual Agitation, the Moonnbsp;1436 has at leaft acquired the Motion about its Axis ofnbsp;�*9/0 which we have before fpoken which Motionnbsp;�c9iw?{/? neceffarily be performed in the fame Time, asnbsp;the Moon performs one Kevolution �, for from thenbsp;Adlion above-mentioned, itmuft neceffarily adaptnbsp;itfelf to fuch a Celerity ; for if the Celerity wasnbsp;greater, it would be continually retarded by thenbsp;Force, whereby the fame Face is always direftednbsp;towards the Earth j and, if this Celerity wasnbsp;Jefs, it would be continually accelerated. Yetnbsp;this Force is not great enough, fenfibly to di-fturb the .Squibility of the Motion acquired a-143 *7 bout the Axis, every Revolution ; Thereforenbsp;the Motion about the Axis is equable, though the Moonnbsp;quot;*9^6 he moved in its Orbit by an unequal Motionquot;^. Thenbsp;Pofition alfo of the Moon's Axis cannot be fonbsp;changed by the Force above-mentioned, as to

Book IV. of Natural Thilofophy. a8f

become perpendicular to the Plane of the Orbit, when its Inclination is changedi* there-fore the Axis of the Moon is fometimes inelined^^i^nbsp;to the Plane of the Orbity as we have beforenbsp;{hewn*

The End of the Fourth Book.

D E X.

�the Niimhers (i.) and (2.) refer to the firft and fe-cond Volume, p. ftands for Page^ and n. for the Number in the Margin.

Acceleration of heavy Bodies (i.)p- 52. n. 129. and the following Acceleration of Bodies rolling downnbsp;upon an inclined Plane.) p. 58. n. 144. and fol. 135. andnbsp;fol.

^Equation of time. (2.) p. 194, 1145-Alquator (2.) p. 181. n. 1075. p. 188. �. 1114. JEquinox (2.) p. 15. n. 1155.

rhel EX.

Animals cannot live without Air (i.) p, 232. 0,463.

^he Line of Apfides, p, 152. n. 938.

� nbsp;nbsp;nbsp;even in a Vacuum, ibid. n. 36.

between two Glafs Planes, p. 14. n. 37, 38. Afterifms, p. 202. n. 1190. and fobnbsp;Atmofphere (1.) p. 209. n. 418.

---�- the Laws of it, ibid.

Fbe Space of AttraSiion (jzl) p. 31. n. 632.

Axioms concerning Motion (i.) p. 21, 0,58. andfol. Axes of a Planet (2.) p. 154. n 949.

Fhe Motion of the Axis of the Earth: See the Preceffion of the JLquinoxes.

___the PerfeBion of one, p. 24. n. 102.

Black Bodies grow warm focner than Bodies of other Colours (2. p 145. n. 909.

Burning

The I NTgt; E X

��-compared together, p. 112. n. 231, and fol. Centrifugal i'bw (i.) p. 104. n. 218.

S'bis laji Colour is always unchangeable, p. Ii7.n. 863; and fol. p. 123. n. 869.

^be Order of Colours, according to their different Re-frangibility, p. 117. n. 861.

Why Colours vary in fame Bodies according to the Situation of the Lye, ibid. 914.

The Change of Colours in Liquids mixed, p. 146. n. 916. A Liquid appearing of a different Colour, according as itnbsp;is feen by refledied or tranfmitted Rays, p. 147. n. 917.nbsp;The Colours of the~Glpuds, ibid. 148. n. 920.

The 1 NT) E X.

21?^ Conjun�lion af the heavenly Bodies (2.) p. 167. n- loio.

Degrees of Latitude encreafe, as you conic neare? the Boles (2.) p. 269. n. 1374.

and fol./gt;� 1S2. nbsp;nbsp;nbsp;1081. and fol. p. 200.nbsp;nbsp;nbsp;nbsp;1183.

The r NT) E X.

The Eh�icity of Plates of Metal, p. 129. n. 262, andfol. The Elafticity of a Ball, p. 131. n. 266.

Conjehhires ahot.t the Caufe of it, p. 7. n. 563. and fol. The Elongation of Planets (2.)p. 167. n.lol2.

Fire contained in Bodies (2.) p. 3. n. 552. and fol. The Action of Air tipon Fire (2.) p. 12. n. 577- $� ^7-n. 593. p. 18. n. 594.

The

The 1 NTgt; E X.

n. 459-

Gravity (i.) p. 25. �. 73. (2.)/). 2^6.n. 1206. andfol. The Pheenomena of Gravity (i.) /gt;. 25. n. 72. p. 25.nbsp;�� 75) 77

Propofitions relating to Gravity,/). 213. n. 1226. andfol. Gravity not to be afcribed to any Imprefiion known to us,nbsp;p. 316. n. 1238.

Gravity upon the Surfaces of Planets, p. 231. n. 1258.

U 2 nbsp;nbsp;nbsp;Hard

The INDEX.

Heat is not in Proportion to the Fire, p. 15. n.588. Heat makes Bodies emit Light or JJoitie, p. i6. n. 591,

Heterogeneous Body (^id) p. 146. n.286. Heterogeneous Rays (2.) p. 109. n. 850.nbsp;Hornogeneous Body (i.) p. 146. n. 285.

Fhe Inertia of a Body (i.) p. $. n. 12. iriscr Rainbow, when prodiiced{zd) p. 134 n. 884,885,nbsp;T�wo frequently appear at a Fime, p. 136. n. 886.nbsp;Explanation of the Motion of Light in forming a Rainbow, p. 127. n 875. and fol.

The I NT) E X.

PerfeBion of a magic Lantern C^-) �� 102. n. S39. Latitude of an heavenly Body (2.) p. 166. n, iqo6.

The Celerity lt;)� Light in different Mediums, p. 37. n. 640. Inflexion of Light, p. 24.nbsp;nbsp;nbsp;nbsp;611. and fol.

Compound Machines, p. 47. n. 122. and fol. ji Machine by which the Properties of the ll'edge arenbsp;demonftrated (i.) p. 44-

A Machine to demonfirate the Properties of an inclined Plane (i.) p. 57. n. 143.

��to make Experiments eoncerning Fluids fiponting out, p. I7jr. n. 356. p. 182. n. 370.

Various Machines, the EffeB of which depends upon the Aciioncf the Air (i.) p. 223. n.467. and fol.

�to jloew the RefraBion of Light, p. 29. n. 630. p. 38 in the Beginning, p 49. p. $$ n 702. p. 58.nbsp;n, 708.

Mars

The INDEX.

The SluantitiesofMattexin Playlets (2..') p 23i.n. 1255. The Medium of Light (i .) p. 7. n. 15.

Mercury jloines in a Vacuum (2.) p. ii. n, 571. and fol. The Planet Mercury (2.') p. 155. n. 958.

The Phcemmena of the Moon (2.) p. 174. n. 1035. and fol.

Why it appears bigger near the Horizon, p. 67. n. 735-Jt turns round its Axis, p. I57.n. 970. 184. n. 1092.

Jt always turns the fame Face towards the Earth, p. 184-n. 1092.

Eclipfe of the Moou (2.) p. 176. n. lofi'j.

--when it happens, ibid. n. 1049.1051.

-Partial p. i77-

------Total, ibid. � 1053

�-Central, ibid. ??. 1054.

The Gravity of the Moon towards the Earth, p. 211.

n. 1225.

The

The 1 N Tgt; E X.

The Gravity of the Moon in its Surface, p. ibid.n. 1429. The iVeigbt of the Moon, p. ibid. n. 1429.

The Difiance of the Moon, if in its periodical Time it �was carried round the Earth, and the Earth floodftill,nbsp;p. 228. n. 1253.

The Determination of the Forces that difliirh the Moon, �). 245' n. 1291. p. 247. n. 1296, p. 251. n- 1306,nbsp;1307. p. 259-nbsp;nbsp;nbsp;nbsp;1336.

The comparing of Motions (i.) p 22. �. 62, and fol. Accelerated Motion Ct f p. 51.nbsp;nbsp;nbsp;nbsp;127.

Motioa

The I N Tgt; E X.

-------in Confequentia, p. 153. n. 946. nbsp;nbsp;nbsp;'nbsp;nbsp;nbsp;nbsp;�*'

Opacity depends upon the Pores f2.') p. lof n Sia Ocftave (i.) p. 2SS 527.nbsp;nbsp;nbsp;nbsp;'

Motions �/Pendulums, p. 61. n. 1^4 and fol. Penumbra (2.) p. 179. n. 1060,nbsp;percuffionnbsp;nbsp;nbsp;nbsp;C^Op 67. n. 166.

The I N Tgt; E X.

��'they revolve about the Center of Gravity of the whole Syfiem, p. 234. n. 1264.

Motions andthe Difiances of the SecondaryPhr\ets,p. 156. n. 965. and fol, p, 191. �. 1130, and fol.

The Aliions of Eoviitrs compared, p. 24. n, 67. and fol. The ASiionsof oblique Powers determined, p, 91. n. 196.nbsp;and fol.

The Prffceffion of the Equinoxes. See JBqainox.

The J N D E X.

Pulley Cl.) P 27 � 82, 83. fgt;. 40. �. and fol. Common Pumps (i.) p.zi�.n

Black Bodies refledt no Light, p. 107. n. 845. Refradbion of Light (2 )9. 26. n. 616.

��its Laws, p. 28. n. 624. and fol A conflant Proportion between the Sine of Incidence andnbsp;the Sine cf Refradiion, p. 37. n. 639.

The Laws of Refradbion in Mediums divided by the plane Surface, p. 39. n. 643, in Mediums divided bynbsp;a fpherical Surface, p. 44. n. 660.

Refradbion of the Stars, p. i88. n. 1110. and fol. Refrangibility different in different Rays (2.) p. 108.nbsp;n. 847.

ed, p. iz%. n. 868. nbsp;nbsp;nbsp;Repul�

The I NT) E X.

Reiilbnce of Fluids (i.) p. 162. n. 319. and fol. Retardation of Bodies afcending �vertically (i.) p. 55.nbsp;n. tss.

��of a Body rifing upwards, p. 172. n. 351. and fol. Retardations compared witbone another, p. 165. n.331.nbsp;and fol.

Ring of Sattirn (2.) p. 156. n 963. p. 174. n. 1034. Pi��ngofthe Stars (^2.') p, 186, n. 1097.

The I NTgt; E X.

The

The I N T) � X.

an Explanation of it according to the Principles of Natural Philofophy (^2.) p. 235. n. 1275.

p. 77. n. 776.

The Forces of Influences determined^ P-273' n. 1416. andf. Time (i.) p. 20. n. 51, 52,

Tubes fliled with Gun-powder (2.) p.22. n. 609. Tranfparent Body (2,) p. 104. n. 840.

The Parts of a� Bodies tranfparent, ihid. n. 841. Tranfparent Bodies become opaque uponthe Separation ofnbsp;their PartSy p 106. n 844.

��its real Exifience in Nature (2 ) p.217. n. 1239. and fol.nbsp;nbsp;nbsp;nbsp;Vcio-

The I E X.

Velocity (i.) nbsp;nbsp;nbsp;20.nbsp;nbsp;nbsp;nbsp;53.

�.-728.

Zcquot;

��of the Temperate ones, p. 196. n. 1165. and fol. p. 199. �.II 76. and fol.

f='V

S, »• ,

ï' 1

.r

.-'X'' —

i�

ELEMENTS

Natural Philofophy,

Confirm�d by Experiments;

INTRODUCTION

Sir I s A A c N E w T 0 K�s Philofophy.

L 0 ND 0 N:

THOM AS

Lord PARKER,

Baron of ^Macclesfield^

Lord High Chancellor of Great Britain^ See,

My Lord,

iv DEDICAT 10 N.

DEDICJTIO N. V

vj D EDICJTION.

fliip

DEDICATION, vij

PREFACE.

X nbsp;nbsp;nbsp;The PREFACE.

The PREFACE,

THE

CONTENTS

Second Volume.

BOOK III.

BOOK III.

The CONTENTS.

BOOK IV.

BOOK IV.

Part II. The Phyfieal Caufes of the Celeftiai

Motions.

The C o N T E N T S.

Mathe-

Natural Philofophy,

EXPERIMENTS.

%, nbsp;nbsp;nbsp;Mathematical Elements Book III,

CHAP. II.

Book III. of Natural Thilofophy. . 5

4 nbsp;nbsp;nbsp;Mathematical Elements Book III.

Book Iir of Natural �Philofofhy. nbsp;nbsp;nbsp;^

5 Mathematical Elements Book III.

Book in. of Natural ^hilojbphy. nbsp;nbsp;nbsp;f

Mathematical Elements Book III.

Book III. of Natural�Philofophy. nbsp;nbsp;nbsp;9

Mathematical �l�ments Book lit

Book III. of Natural Thilofofhy. Il

1 i Mathematical Elements Book IIL

Book III. of Natural Thilofophy.

CHAP. III.

Mathematical Elements Book kf�.

Book III. of Natural TbHofo^hy. !gt;

Mathematical Elements Book III.

Thus

Book III. of Natural Thzlofo_phy. 17

i8 Mathematical Elements Book III,

CHAP. IV.

Of the Dilatation occajion�d hy Beat.

Book III. of Natural Thtlofo^hj.

to Mathematical Elements Book III.

ï/^r:

•- a-

\ ..

Book III. of Natural ^hilofophy. ai

%% Mathematical Element:^ Book III,

BOOK

Part II.

Concerning the Inflexion, Refra�lion, and Reflexion of Light.

Concerning the Inflexion of the Rays of Light.

C4

Mathematical Elements Book III.

Book III. of Natural Vhilofophy.

2-6 Mathematical Elements Book III.

CHAP. VI.

Definition IT.

Book III. of Natural Thilofophy. 17

Mathematical Elements Book III.