Principia Mathematica (Motte Translation, 1848) — part 18 of 45

If there he several bodies consisting of equal particles whose forces are as the distances of the places from each, the force compounded of all the forces by which any corpuscle is attracted loill tend to the comm^on centre of gravity of the attracting bodies ; and ivill be the same as if those attracting bodies, preserving their common centre of gravity, shoidd unite there, and be formed into a globe.

This is demonstrated after the same manner as the foregoing Proposi- tion.

Cor. Therefore the motion of the attracted body will be the same as if the attracting bodies, preserving their common centre of gravity, should unite there, and be formed into a globe. And, therefore, if the common centre of gravity of the attracting bodies be either at rest, or proceed uni- formly in a right line, the attracted body will move in an ellipsis having its centre in the common centre of gravity of the attracting bodies.

SeCo XIII.J of natural philosophy. 237

PROPOSITION XC. PROBLEM XLIV.

If to the several points of any circle there tend equal centripetal forces, increasing or decreasing in any ratio of the distances ; it is required to find the force ivith ivhich a corpuscle is attracted, that is, situate any cohere in a right line ivhich stands at right angles to the plane of the circle at its centre. Suppose a circle to be described about the cen- tre A with any interval AD in a plane to which the right line AP is perpendicular ; and let it be required to find the force with which a corpuscle y

P is attracted towards the same. From any point //^

E of the circle; to the attracted corpuscle P, let yy

there be drawn the right line PE. In the right ^-

line PA take PF equal to PEj and make a per- pendicular FK, erected at F, to be as the force with which the point E attracts the corpuscle P. *■'

And let the curve line IKL be the locus of the point K. Let that curve meet the plane of the circle in L. In PA take PH equal to PD, and erect the perpendicular HI meeting that curve in I ; and the attraction of the corpuscle P towards the circle will be as the area AHIL drawn into the altitude AP. aE.L

For let there be taken in AE a very small line Ee. Join Ve, and in PE, PA take PC, Vf equal to P^. And because the force, with which any point E of the annulus described about the centre A with the interval AE in the aforesaid plane attracts to itself the body P, is supposed to be as FK ; and, therefore, the force with which that point attracts the body P

AP X FK

towards A is as ™ ; and the force with which the whole annulus

AP X FK

attracts the body P towards A is as the annulus and ^^ conjunct- ly ; and that annulus also is as the rectangle under the radius AE and the breadth Ee, and this rectangle (because PE and AE, Ee and CE are pro- portional) is equal to the rectangle PE X CE or PE X F/"; the force with which that annulus attracts the body P towards A will be as PE X

AP X FK Fy* and — -^^^ conjunctly ; that is, as the content under ¥f X FK X

AP, or as the area FK^ drawn into AP. And therefore the sum of the forces with which all the annuli, in the circle described about the centre A with the interval AD, attract the body P towards A, is as the whole area AHIKL drawn into AP. Q.E.D. Cor. 1. Hence if the forces of the points decrease in the duplicate ratio

238 THE MATHEMATICAL PRINCIPLES [BoOK 1.

of the distances, that is, if FK be as p^j, and therefore the area AHIKL

as p-7 — j5tj ; t^6 attraction of the corpuscle P towards the circle will

PA , . AH be as 1 — ptpj ; that is, as p|j

Cor. 2. And universally if the forces of the points at the distances D be

reciprocally as any power D° of the distances ; that is, if FK be as ^,

and therefore the area AHIKL as 5— j — -^^^j^ ^ ; the attraction

1 PA

of the corpuscle P towards the circle will be as 5— — T>jT~n — i*

Cor. 3. And if the diameter of the circle be increased m injinitifni, and the number n be greater than unity ; the attraction of the corpuscle P to- wards the whole infinite plane will be reciprocally as PA" — ^, because the

PA other term pTj^ vanishes.

PROPOSITION XCI. PROBLEM XLV.

To find the attractioii of a corpuscle situate in the axis of a round solid,

to whose several points there tend equal centripetal forces decreasing

in any ratio of the distances whatsoever.

Let the corpuscle P, situate in the axis AB

^,^,JU_^^^ of the solid DECG, be attracted towards that

I i"* solid. Let the solid be cut by any circle as

i I RFS, perpendicular to the axis; and in its

— -j 'b semi-diameter FS, in any plane PALKB pass-

j y-^}- ing through the axis, let there be taken (by

Is: ;>-''' '(^ Prop. XC) the length FK proportional to the

-^^""^^X^-^-^^ force with which the corpuscle P is attracted

towards that circle. Let the locus of the point

K be the curve line LKI, meeting the planes of the outermost circles AL

and BI in L and I ; and the attraction of the corpuscle P towards the

solid will be as the area LABI. Q..E.I.

Cor. 1. Hence if the solid be a cylinder described by the parallelogram

A DEB revolved about the axis AB, and the centripetal forces tending to

the several points be reciprocally as the squares of the distances from the

points ; the attraction of the corpuscle P towards this cylinder will be as

AB — PE + PD. For the ordinate FK (by Cor. 1, Prop. XC) will be

PF as 1 — =r^-. The part 1 of this quantity, drawn into the length AB, de- I R

Sec. XIIL]

OF NATURAL PHILOSOPHY

239

X AB ; and the other part

, drawn into the length PB describes the

scribes the area 1

PF

PR

area 1 into PB — AD (as may be easily shewn from the quadrature of the curve LKI) ; and, in like manner, the same part drawn into the length PA describes the area i into PD — AD, and drawn into AB, the

D

R i:

/

f

A

1 .-•-'""

^/ s

(J

difference of PB and PA, describes 1 into PE — PD, the difference of the areas. Prom the first content 1 X AB take away the last content 1 into PE — PD, and there will remain the area LABI equal to 1 into AB — PE + PD. Therefore the force, being proportional to this area, isas AB — PE + PD.

Cor. 2. Hence also is known the force by which a spheroid AGBC attracts any body P situate externally in its axis AB. Let NKRM be a conic section whose or- dinate ER perpendicular to PE may be always equal to the length of the line PD, continually drawn to the point D in which that ordinate cuts the spheroid. Prom the vertices A, B, of the spheriod, let there be erected to its axis AB the perpendiculars AK, BM, respectively equal to AP, BP, and therefore meeting the conic section in K and M ; and join KM cutting off from it the segment KMRK. Let S be the centre of the spheroid, and SC its greatest semi-diameter ; and the force with which the spheroid attracts the body P will be to the force with which a sphere describ-

ed.vit.Kthedia:neter ABattractsthesa^ebodya/^ >^ ^«^"^^ "^ ^"^^^

AS=^

is to

PS^ -l-CS^— AS^ QiDcjo- And by a calculation founded on the same principles may be

found the forces of the segments of the spheroid.

Cor. 3, If the corpuscle be placed within the spheroid and in its axis, the attraction will be as its distance from the centre. This may be easily collected from the following reasoning, whether the particle be in the axis or in any other given diameter. Let AGOF be an attracting sphe- roid, S its centre, and P the body attracted. // Through the body P let there be drawn the ! semi- diameter SPA, and two right lines DE, \^ FG meeting the spheroid in D and E, P and G ; and let PCM, HLN be the superficies of

240 THE MATHEMATICAL PRINCIPLES [BoOK 1.

two interior spheroids similar and concentrical to the exterior, the first of which passes through the body P, and cuts the right lines DE, FG in B and C ; and the latter cuts the same right lines in H and I, K and L. Let the spheroids have all one common axis, and the parts of the right lines intercepted on both sides DP and BE, FP and CG, DH and IE, FK and LG, will be mutually equal ; because the right lines DE, PB, and HI, are bisected in the same point, as are also the right lines FG, PC, and KL. Conceive now DPF, EPG to represent opposite cones described with the infinitely small vertical angles DPF, EPG, and the lines DH, EI to be infinitely small also. Then the particles of the cones DHKF, GLIE, cut off by the spheroidical superficies, by reason of the equality of the lines DH and EI, will be to one another as the squares of the distances from the body P, and will therefore attract that corpuscle equally. And by a like rea- soning if the spaces DPF, EGCB be divided into particles by the superfi- cies of innumerable similar spheroids concentric to the former and having one common axis, all these particles will equally attract on both sides the body P towards contrary parts. Therefore the forces of the cone DPF, and of the conic segment EGCB, are equal, and by their contrariety de- stroy each other. And the case is the same of the forces of all the matter that lies without the interior spheroid PCBM. Therefore the body P is attracted by the interior spheroid PCBM alone, and therefore (by Cor. 3, Prop. L.XXII) its attraction is to the force with which the body A is at- tracted by the whole spheroid AGOD as the distance PS to the distance AS. U-E.D.

PROPOSITION XCII. PROBLEM XLYL

An attractiiig body being given, it is required to find the ratio of the de- crease of the centripetal forces te7iding to its several points. The body given must be formed into a sphere, a cylinder, or some regu- lar figure, whose law of attraction answering to any ratio of decrease may be found by Prop. LXXX, LXXXI, and XCI. Then, by experiments, the force of the attractions must be found at several distances, and the law of attraction towards the whole, made known by that means, will give the ratio of the decrease of the forces of the several parts ; which was to be found.

PROPOSITION XCIII. THEOREM XLYII.

If a solid be plajie on one side, and infinitely extended on all other sides^ and consist of equal particles equally attractive, whose forces decrease^ in the recess from the solid, in the ratio of any power greater than the square of the distances ; and a corpuscle placed towards either part of tlie plane is attracted by the force of the whole solid ; T say, that tfie attractive force of the whole solid, in the recess from its plane superfi-

Sec. XIII.J

OF NATURAL PHILOSOPHY.

241

L

0 ^

iSU^

^

K

I

H

G '

0

n

m

/

\C

cies, will decrease in the ratio of a power whose side is the distance of

the corpuscle from the plane, and its index less by 3 thaii the index of

the poicer of the distances.

Case 1. Let LGZ be the plane by which the solid is terminated. Let the solid lie on that hand of the plane tliat is to- wards I, and let it be resolved into in- numerable planes mHM, 7?IN, oKO, &c., parallel to GL. And first let the attracted body C be placed without the solid. Let there be dr^^wn CGHI per- pendicular to those innumerable planes^ and let the attractive forces of the points of the solid decrease in the ratio of a power of the distances whose index is the number 7^ not less than 3. Therefore (by Cor. 3, Prop. XC) the force with which any plane mHM. attracts the point C is reciprocally as CH"— ^. In the plane mHM take the length HM reciprocally proportional to CH"— ^, and that force will be as HM. 1\ like manner in the several planes /GL, nIN, oKO, &c., take the lengths GL, IN, KO, &c., reciprocally proportional to CG"— ^, CI"— ^, CK"— 2, &c., and the forces of those planes will be as the lengths so taken, and therefore the sum of the forces as the sum of the lengths, that is, the force of the whole solid as the area GLOK produced infinitely towards OK. But that area (by the known methods of quadratures) is reciprocally as CG"— ^, and therefore the force of the whole solid is reciprocally as CG"-^ aE.D.

Case 2. Let the corpuscle C be now placed on that hand of the plane ZGL that is within the solid, and take the distance CK equal to the distance CG. And the part of the solid LGZoKO termi- nated by the parallel planes /GL, oKO, will at- tract the corpuscle C, situate in the middle, neither one way nor another, the contrary actions of the opposite points destroying one another by reason of their equality. Therefore the corpuscle C is attracted by the force only of the solid situate beyond the plane OK. But this force (by Case 1) is reciprocally as CK"—^, that is, (because CG, CK are equal) reciprocally as CG"- \ aE.D.

Cor. 1. Hence if the solid LGIN be terminated on each side by two in- finite parallel planes LG, IN, its attractive force is known, subducting from the attractive force of the whole infinite solid LGKO the attractive force of the more distant part NIKO infinitely produced towards KO.

Cor. 2. If the more distant part of this solid be rejected, because its at- traction compared with the attraction of the nearer part is inconsiderable,

16

!o N

K I

0

I

242 THE MATHEMATICAL PRINCIPLES [BoOK I.

the attraction of that nearer part will, as the distance increases, decrease nearly in the ratio of the power CG"— ^.

Cor. 3. And hence if any finite body, plane on one side, attract a cor- puscle situate over against the middle of that plane, and the distance between the corpuscle and the plane compared with the dimensions of the attracting body be extremely small ; and the attracting body consist of homogeneous particles, whose attractive forces decrease in the ratio of any power of the distances greater than the quadruplicate ; the attractive force of the whole body will decrease very nearly in the ratio of a power whose side is that very small distance, and the index less by 3 than the index of the former power. This assertion does not hold good, however, of a body consisting of particles whose attractive forces decrease in the ratio of the triplicate power of the distances ; because, in that case, the attraction of the remoter part of the infinite body in the second Corollary is always infinitely greater than the attraction of the nearer part.

SCHOLIUM.

If a body is attracted perpendicularly towards a given plane, and from the law of attraction given, the motion of the body be required ; the Pro- blem will be solved by seeking (by Prop. XXXIX) the motion of the body descending in a right line towards that plane, and (by Cor. 2, of the Laws) compounding that motion with an uniform motion performed in the direc- tion of lines parallel to that plane. And, on the contrary, if there be re- quired the law of the attraction tending towards the plane in perpendicu- lar directions, by which the body may be caused to move in any given curve line, the Problem w^ill be solved by working after the manner of the third Problem.

But the operations may be contracted by resolving the ordinates into converging series. As if to a base A the length B be ordinately ap- plied in any given angle, and that length be a^ any power of the base

An ; and there be sought the force with which a body, either attracted to- wards the base or driven from it in the direction of that ordinate, may be caused to move in the curve line which that ordinate always describes with its superior extremity ; I suppose the base to be increased by a very small

iin ^ ^ . m

part 0, and I resolve the ordinate A + Ou into an infinite series A- +

— OA —TT- ^ ^ 00 A &c., and I suppose the force propor-

n 2nn n ;

tional to the term of this series in which O is of two dimensions, that is, to the term — 5 00 A — „— . Therefore the force sought is as

Sec. Xiy.j OF natural philosophy. 243

mm — m^n m - 2n ^ • ^ ■ 1 ^ . ^^^ — ^^ T-» »" - 2n — A -—;, — . or, which is the same thmo^, as B ^ .

As if the ordinate describe a parabola, m being = 2, and ?i = 1, the force will be as the given quantity 2B% and therefore is given. Therefore with a given force the body will move in a parabola, as Galileo has demon- strated. If the ordinate describe an hyperbola, m being = 0 — 1, and 7i = 1, the force will be as 2 A ^ or 2B =^ ; and therefore a force which is as the cube of the ordinate will cause the body to move in an hyperbola. But leaving this kind of propositions, I shall go on to some others relating to motion which I have not yet touched upon.

SECTION XIY.

Of the motion of veri/ small bodies lohen agitated by centripetal forces tending to the several parts of any very great body.

PROPOSITION XCIV. THEOREM XLYIII.

If tic 0 similar ')nediums be separated from each other by a space termi- nated on both sides by parallel planes, and a body in its passage through that space be attracted or impelled perpendicularly toioards either of those mediums, and not agitated or hindered by any other force ; and the attraction be every xohere the same at equcd distances from, either plane, taken toioards the same hand of the plane ; I say, that the sine of incidence upon either plane will be to the sine of emer- gence from the other plane in a given ratio. Case 1. Let Ka and B6 be two parallel planes, \g and let the body light upon the first plane Aa in the direction of the line GH, and in its whole passage through the intermediate space let it be attracted or impelled ^to wards the medium of in- cidence, and by that action let it be made to de- scribe a curve line HI, and let it emerge in the di- rection af the line IK. Let there be erected IM perpendicular to B/> the plane of emergence, and m:

meeting the line of incidence GH prolonged in M, and the plane of inci- dence Aa in R ; and let the line of emergence KI be produced and meet HM in L. About the centre L, with the interval LI, let a circle be de- scribed cutting both HM in P and Q., and MI produced in N ; and, first, if the attraction or impulse be supposed uniform, the curve HI (by what Galileo has demonstrated) be a parabola, whose property is that of a rec-

^

\1T

R a

p\

V

■•>.^

./•^

,,J^

B

^t

?^=5^

/ 5

\

244 THE MATHEMATICAL PHINCIPLES [BoOK 1*

tangle under its given latus rectum and tlie line IM is equal to the square of HM ; and moreover the line HM will be bisected in L. Whence if to MI there be let fall the perpendicular liO, MO, OR will be equal ; and adding the equal lines ON, 01, the wholes MN, IR will be equal also. Therefore since IR is given, MN is also given, and the rectangle NMI is to the rectangle under the latus rectum and IM, that is, to HM- in a given ratio. But the rectangle NMI is equal to the rectangle PMQ, that is, to the diiFerence of the squares ML^, and PL^ or LI^ ; and HM^ hath a given ratio to its fourth part ML^; therefore the ratio of ML^ — LP to ML^ is given, and by conversion the ratio of LI^ to ML^, and its subduplicate, the ratio of LI to Mli. But in every triangle, as LMI, the sines of the angles are proportional to the opposite sides. Therefore the ratio of the sine of the angle of incidence LMR to the sine of the angle of emergence LIR is given. Q,.E.D.

Case 2. Let now the body pass successively through several spaces ter- minated with parallel planes AabB, BbcC, &c., and let it be acted on by a \ force which is uniform in each of them separ-

\ ately, but diiFerent in the diiferent spaces ; and

g X ^ by what was just demonstrated, the sine of the

c ^s^ jangle of incidence on the first plane Aa is to

\ the sine of emergence from the second plane Bb

in a given ratio ; and this sine of incidence upon the second plane Bb will be to the sine of emergence from the third plane Cc in a given ratio ; and this sine to the sine of emergence from the fourth plane Dd in a given ra- tio ; and so on in infinitum ; and, by equality, the sine of incidence on the first plane to the sine of emergence from the last plane in a given ratio. Let now the intervals of the planes be diminished, and their number be in- finitely increased, so that the action of attraction or impulse, exerted accord- ing to any assigned law, may become continual, and the ratio of the sine of incidence on the first plane to the sine of emergence from the last plane being all along given, will be given then also. Q.E.D.

PROPOSITION XCY. THEOREM XLIX.

The same things being supposed, I say, that the velocity of the body be- fore its incidence is to its velocity after emergence as the sine of emer- gence to the sine of incidence.

Gr Make AH and Jd equal, and erect the perpen-

\t diculars AG, dK meeting the lines of incidence

;\ and emergence GH, IK, in G and K. In GH

A I -■. ^^ a ^^]^g rpjj equal to IK, and to the plane Aa let

c ^ ^ fall a perpendicular Tv. ' And (by Cor. 2 of the

3> |x^^ j Laws of Motion) let the motion of the body be

^^ resolved into two, one perpendicular to the planes

Sec. XIV.] of natural philosophy. 245

A«, Bb, Cc, &Cj and another parallel to them. The force of attraction or impulse, acting in directions perpendicular to those planes, does not at all alter the motion in parallel directions ; and therefore the body proceeding with this motion will in equal times go through those equal parallel inter- vals that lie between the line AG and the point H, and between the point I and the line dK ; that is, they will describe the lines GH, IK in equal times. Therefore the velocity before incidence is to the velocity after emergence as GH to IK or TH, that is, as AH or Id to vH, that is (sup- posing TH or IK radius), as the sine of emergence to the sine of inci- dence. Q.E.D.

PROPOSITION XCYI. , THEOREM L.

The same things being supposed^ and that the ^notion before incidence is swifter than afterioards ; 1 say^ that if the line of incidence be in- clined continnally, the body will be at last reflected, and the angle of reflexion luill be equal to the angle of incidence.

For conceive the body passing between the parallel planes Aa, B^, Cc, &c., to describe parabolic arcs as above; ^ rr

and let those arcs be HP, PQ., Q.R, &c. a^ And let the obliquity of the line of inci- c- dence GH to the first plane Aa be such E" that the sine of incidence may be to the radius of the circle whose sine it is, in the same ratio which the same sine of incidence hath to the sine of emer- gence from the plane T)d into the space D<ieE ; and because the sine of emergence is now become equal to radius, the angle of emergence will be a right one, and therefore the line of emergence will coincide with the plane Jyd. Let the body come to this plane in the point R ; and because the line of emergence coincides with that plane, it is manifest that the body can proceed no farther towards the plane Ee. But neither can it proceed in the line of emergence R«i; because it is perpetually attracted or impelled towards the medium of incidence. It will return, therefore, between the planes Cc, Dd, describing an arc of a parabola Q.R^', whose principal vertex (by what Galileo has demonstrated) is in R, cutting the plane Cc in the same angle at q, that it did before at Q. ; then going on in the parabolic arcs qp, ph, &c., similar and equal to the former arcs Q,P, PH, &c., it will cut the rest of the planes in the same angles at p, /z, &c., as it did before in P, H, (fee, and will emerge at last with the same obliquity at h with which it first impinged on that plane at H» Conceive now the intervals of the planes Aa, B6, Cc, J)d, Ee, (fee, to be infinitely diminished, and the number in- finitely increased, so that the action of attraction or impulse, exerted ac- cording to any assigned law, may become continual; and, the angle of emergence remaining all along equal to the angle of incidence, will be equal to the same also at last. Q,.E.D.

246 THE MATHEMATICAL PRINCIPLES [BoOK I.

SCHOLIUM.

These attractions bear a great resemblance to the reflexions and refrac- tions of light made in a given ratio of the secants, as was discovered by Snellius ; and consequently in a given ratio of the sines, as was exhibited by Des Cartes. For it is now certain from the phenomena of Jupitefs satellites, confirmed by the observations of different astronomers, that light is propagated in succession, and requires about seven or eight minutes to travel from the sun to the ep.rth. Moreover, the rays of light that are in our air (as lately was discovered by Gri?naldus, by the admission of light into a dark room through a small hole, which I have also tried) in their passage near the angles of bodies, whether transparent or opaque (such as the circular and rectangular edges of gold, silver and brass coins, or of knives, or broken pieces of stone or glass), are bent or inflected round those bodies as if they were attracted to them ; and those rays which in their passage come nearest to the bodies are the most inflected, as if they were most attracted ; which thing I myself have also carefully observed. And those which pass at greater distances are less inflected ; and tliose at still greater distances are a little inflected the contrary way, and form three fringes of colours. In the figure s represents the edge of a knife, or any

^/^z^.

ci S

■a

kind of wedge ks^ ; and gowog,fnuvf, emtme, dlsld, are rays inflected to- wards the knife in the arcs owo, nvn, mtm, Isl ; which inflection is greater or less according to their distance from the knife. Now since this inflec- tion of the rays is performed in the air without the knife, it follows that the rays which fall upon the knife are first inflected in the air before they touch the knife. And the case is the same of the rays falling upon glass. The refraction, therefore, is made not in the point of incidence, but gradually, by a continual inflection of the rays ; which is done partly in the air before they touch the glass, partly (if I mistake not) within the glass, after they have entered it ; as is represented in the rays ckzc, biyb, ahxa, falling upon r, q, p, and inflected between k and z, i and 2/, h and x. Therefore because of the analogy there is between the propagation of the rays of light and the motion of bodies, I thought it not amiss to add the following Propositions for optical uses ; not at all considering the nature of the rays of light, or inquiring whether they are bodies or not ; but only determining the tra- jectories of bodies which are extremely like the trajectories of the rays.

Sec. XIV.] of natural philosophy. 247

PROPOSITION XCVIL PROBLEM XL VII.

Supposing the sim of incidence upon any superficies to he in a given ra- tio to the sine of emergence ; and that the inflection of the paths of those bodies near that superficies is performed in a very short space, which may be considered as a poiiit ; it is required to determine such a superficies as may cause all the corpuscles issuing from any one given place to converge to another given place. Let A be the place from whence the cor- -^

puscles diverge ; B the place to which they ^D^

should converge ; ODE the curve line which by its revolution round the axis AB describes A cnm"

the superficies sought ; D, E, any two points of that curve ; and EF, EG, perpendiculars let fall' on the paths of the bodies AD, DB. Let the point D approach to and coalesce with the point E ; and the ultimate ratio of the line DF by which AD is increased, to the line DG by which DB is diminished, will be the same as that of the sine of incidence to the sine of emergence. Therefore the ratio of the increment of the line AD to the decrement of the line DBisgi'ven; and therefore if in the axis AB there be taken any where the point C through which the curve CDE must pass, and CM the increment of AC be taken in that given ratio to CN the decrement of BC, and from the centres A, B, with the intervals AM, BN, there be described two circles cutting each other in D ; that point D will touch the curve sought CDE, and, by touching it any where at pleasure, will determine that curve. GI.E.I.

Cor. 1. By causing the point A or B to go off sometimes in infinitum, and sometimes to move towards other parts of the point C, will be obtain- ed all those figures which Cartesius has exhibited in his Optics and Geom- etry relating to refractions. The invention of which Cartesius having thought fit to conceal, is here laid open in this Proposition.

Cor. 2. If a body lighting on any superfi- cies CD in the direction of a right line AD, -Qt— ^^^^ —

drawn according to any law, should emerge

in the direction of another right line DK ;

and from the point C there be drawn curve ^ q

lines CP, CO,, always perpendicular to AD, DK ; the increments of the

lines PD, QD, and therefore the lines themselves PD, QD, generated by

those increments, will be as. the sines of incidence and emergence to each

other, and /o contra.

PROPOSITION XCVIII. PROBLEM XLVIIL

The sa?ne things supposed ; if round the axis AB any attractive super- ficies be described as CD, regular or irregular, through which the bo- dies issuing from the given place A must pass; it is required to find

248 THE MATHEMATICAL PRINCIPLES. [BoOK I.

a second attractive superficies EF, which may make those bodies con- verge to a give7i place B.

Let a line joining AB cut tlie first superficies in C and the second in E, the point D being taken any how at plea- sure. And supposing the sine of incidence on the first superficies to the sine of emergence from the same, and the sine of emergence from the second super- ficies to the sine of incidence on the same, to be as any given quantity M to another given quantity N ; then produce AB to G, so that BG may be to CE as M — N to N ; and AD to H, so that AH may be equal to AG ; and DF to K, so that DK may be to DH as N to M. Join KB, and about the centre D with the interval DH describe a circle meeting KB produced in L, and draw BF parallel to DL ; and the point F will touch the line EF, which, being turned round the axis AB. will describe the superficies sought. Q,.E.F.

For conceive the lines CP, CQ. to be every where perpendicular to AD, DF, and the lines ER, ES to FB, FD respectively, and therefore QS to be always equal to CE ; and (by Cor. 2, Prop. XC VII) PD will be to QD as M to N, and therefore as DL to DK, or FB to FK ; and by division as DL — FB or PH — PD — FB to FD or Fa — QD ; and by composition as PH — FB to FQ, that is (because PH and CG, QS and CE, are equal), as CE + BG — FR to CE — FS. But (because BG is to CE as M — N to N) it comes to pass also that CE + BG is to CE as M to N ; and therefore, by division, FR is to FS as M to N ; and therefore (by Cor. 2, Prop. XCYII) the superficies EF compels a body, falling upon it in the direction DF, to go on in the line FR to the place B. Q..E.D.

SCHOLIUM.

In the same manner one may go on to three or more superficies. But of all figures the sphjerical is the most proper for optical uses. If the ob- ject glasses of telescopes were made of two glasses of a sphasrical figure, containing water between them, it is not unlikely that the errors of the refractions made in the extreme parts of the superficies of the glasses may be accurately enough corrected by the refractions of the water. Such ob- ject glasses are to be preferred before elliptic and hyperbolic glasses, not only because they may be formed with more ease and accuracy, but because the pencils of rays situate without the axis of the glass would be more accu- rately refracted by them. But the different refrangibility of different rays is the real obstacle that hinders optics from being made perfect by sphseri- cal or any other figures. Unless the errors thence arising can be corrected, all the labour spent in correcting the others is quite thrown away.

BOOK II.

BOOK ir.

OF THE MOTION OF BODIES.

SECTION I.

Of the motion of bodies that are resisted in the ratio of the velocity.

PROPOSITION I. THEOREM I.

If a body is resisted in the ratio of its velocity, the motion lost by re- sistance is as the space gone over in its motio?i.

For since the motion lost in each equal particle of time is as the velocity, that is, as the particle of space gone over, then, by composition, the motion lost in the whole time will be as the whole space gone over. Q,.E.D.

Cor. Therefore if the body, destitute of all gravity, move by its innate force only in free spaces, and there be given both its whole motion at the beginning, and also the motion remaining after some part of the way is gone over, there will be given also the whole space which the body can de- scribe in an infinite time. For that space will be to the space now de- scribed as the whole motion at the beginning is to the part lost of that motion.

LEMMA I.

Quantifies proportional to their differences are continually proportional. Let A be to A — B as B to B — C and C to C — D, (fee, and, by con- version, A will be to B as B to C and C to D, &c. Q.E.D.

PROPOSITION II. THEOREM IL

If a body is resisted in the ratio of its velocity, and m^oves, by its vis in- sita only, through a similar Tnedimn, and the times be taken equal, the velocities in the beginning of each of the times are in a geometri- cal progression, and the spaces described in each of the times are as the velocities.

Case 1. Let the time be divided into equal particles ; and if at the very beginning of each particle we suppose the resistance to act witli one single impulse which is as the velocity, the decrement of the velocity in each of

252 THE MATHEMATICAL PRINCIPLES [BoOK 11.