■^^m^ 
 
 ^^fy'^:ffW 
 
 ^^, .s<^ 
 
 1 <'.«•- 
 
 J' 
 
 m^^^^^^^^^M 
 
 
 
 
 ■SB 
 
 
 
 s^^^ S^m^ 
 
.^^^ ^^^3^ 
 
 ^^^ 
 
 irf9i 
 
 
 iti^^i^ 
 
ELEMENTS 
 
 OF 
 
 §^^^, 
 
 ECHANICAIL FMIILOSOPHY,. 
 
 BEING THE SUBSTANCE OF 
 
 A COURSE OF LECTURES 
 
 ON THAT 
 
 SCIENCE. 
 
 By JOHN ROBISON, LL.D. 
 
 PROFESSOR OF NATURAL PHILOSOPHY IN THE 
 
 UNIVERSITY OF EDINBURGH, 
 
 FELLOW OF THE ROYAL SOCIETY OF EDINBURGH, 
 
 OF THE IMPERIAL ACADEMY OF SCIENCES AT ST. PETERSBURGJ1> 
 
 AND OF THE PHILOSOPHICAL SOCIETIES OF MANCHESTER 
 
 AND NEW YORK, &C. &C. 
 
 VOLUME FIRIST, 
 
 INCLUDING 
 
 DTNAMICS AND ASTRONOMT, 
 
 EDINBURGH: 
 
 Printed for 
 
 ARCHIBALD CONSTABLE & CO. EDINBURGH i 
 
 T. CADELL & W. DAVIES, AND 
 
 LONGMAN HURST REES 5c ORME, , 
 
 LONDON. 
 
 1804. 
 
D. WlUIson, Printer, ^^j ^; JA 
 
 Craig's Close, Edinburgh. 
 

 ADVERTISEMENT. 
 
 Jl he foil-owing pages contain the fubftance of 
 a Courfe of Lecliires, which have been read by 
 me during the annual feflions of the Colleges, 
 ever fince the year 1774. Any perfon, well ac- 
 quainted with Natural Philofophy mud be fen- 
 fible that, in the ihort fpace of a fix months fef- 
 "fion, juflice cannot be done to the various branches 
 of this extenfive fcience. I found that I mufl 
 either treat in a loofe manner fubjedls which re- 
 <juire and admit of ftrid reafoning, or mufl omit 
 fome articles ufually taught in this clafs ; and I 
 was induced to prefer the latter method, becaufe I 
 was of opinion that a loofer manner of proceed- 
 ing is neither fuitable to the Inftitution in this 
 Univerfity, nor calculated to convey ufeful know- 
 ledge. In one feiTion I omitted the confidera- 
 tion of Magnetilin and Eledricity, and in the 
 next feflion thefe were treated of, and Optics was 
 jomitted. 
 
 ^ But 
 
 336 
 
ly ADVERTISEMENT. 
 
 But this plan was not always acceptable. T 
 was therefore mduced to print thefe Elements, 
 in the hopes of being able to fhortcn the ledure, 
 and thus to include all the articles of the courfe. 
 I fliall now think myfelf at liberty to leclure in a 
 more popular manner, as the.ftudent, by confult- 
 ing the text-book, will find the demonftratioiLof 
 what was only fketched in the lecture of the day. 
 
 Such being the intention in this publication, 
 the reader will fee in what refpeds, and for what 
 reafons, it may differ a httle from a formal fyf- 
 tem of Natural Philofophy. It is intended that 
 it ihall contain a fyftem. But all the articles will 
 not be treated with the fame minutenefs. The 
 experience of thirty years has enabled me to judge 
 what articles are more abflrufe or intricate, and 
 require a more detailed difcuffion. 
 
 The general doctrines of Dynamics are the 
 bafis of Mechanical Philofophy, diftinguifliing it 
 from every other "department of fcience. They 
 are nearly abflrad truths, containing the laws of 
 human judgement concerning all thofe phenomena 
 which we call mechanical. We fliall fmd thefe 
 la.\vs nearly as fjmple and precife as the propor- 
 tions in geometry, and that they carry with them 
 
At)VERTISEMENT. V 
 
 a fimilar accuracy, wherever they can be properly 
 apphed. We fnall have the pleafure of feeing the 
 complete fuccefs of this application, to veiy exten- 
 five and important articles of the fcience. 
 
 Thefe doctrines being fo important, and fo 
 fufceptible of accurate treatment, nothing is omit- 
 ted here that is neceffary for their full eftablifii- 
 ment ; and hence this occafions the firfl part of 
 the courfe to be very minute and particular. But, 
 afterwards, a more familiar mode of difcuffion 
 may be admitted. If the ftudent make himfelf 
 familiarly acquainted w^ith the principles of Dyna- 
 mics, it is hoped that he will find httle difficulty 
 afterwards,' in the apphcation of thefe abflrad 
 do&ines to the inveftigation of the laws of me- 
 chanical nature, or to the explanation of fubor- 
 dinate phenomena. For this reafon, it is not in- 
 tended to annex the mathematical demonftration 
 to every propofition in the fubfequent parts of the 
 courfe. This will not be omitted, however, when 
 either the difficulty or importance of the fubiect 
 feems to require it. 
 
 The ftudent mull be mindful that this book 
 will not fuperfede the neceffity of carefully at- 
 tending to the lecture. Many things, illuftrative 
 and intereftirig, will be heard in the clafs, which 
 ^ 2 have 
 
Vt ADVERTISEMENT. 
 
 have no place here. It will alfo contribute to 
 liis improvement, if he accuflom himfelf to take 
 notes in the clafs ; and he is advifed to take par- 
 ticular notice of fuch formulae, or other fymbols 
 of mathematical reafoning, as occur in the lec- 
 ture. Thefe will frequently give a compendious 
 expreflion of a procefs of reafoning which he 
 may otherwife find very difficult to remember with 
 diilinclnefs. 
 
 In applying the abftract dodrine of Dynamics 
 to the mechanical hiflory of nature, fome arrange- 
 ment mud be adopted which may facilitate the 
 talk. It is propofed, in this courfe of ledures, to 
 arrange the mechanical appearances as much as 
 poffible in the order of their generality or extent. 
 It will be found that this is, in fa(5l, arranging, 
 them by the great diflinguiihing powers of natu- 
 ral fubftances, by which this generality of event 
 is efFeded. 
 
 All the mechanical phenomena that we obferve 
 are effected, 
 
 1 . By gravity. 
 
 2. By cohefiom 
 
 3. By magnetifm. 
 
 4. By electricity. 
 
 j;. By the affections of light. 
 
 Hence 
 
ADVERTISEMENT. VU 
 
 Hence is faggefted the following arrangement 
 of the articles which will be treated of in this 
 Gourfe of lectures. 
 
 L Gravity. 
 
 1. As it is feen in the celeflial motions- 
 its law of aclion difcovered by Sir Ifaac 
 Newton — applied by him, with great fuc- 
 cefs, to the explanation of all the pheno^ 
 mena — univerfal gravitation. 
 
 2, As it is obferved on this globe — motion 
 
 of falling bodies — of projectiles — theory 
 of gunnery. 
 
 11. Cohesion. 
 
 Corpufcuiar forces — Theory of Bofcovich. 
 
 Mechanical qualities of tangible matter — bo- 
 dies are folid — or fluid — and thefe differ 
 exceedingly in their mechanifm. 
 
 Mechanism of Solid Bodies, 
 
 Laws of the excitement of corpufcuiar forces. 
 i. Motion in free fpace — impulfion — diredt 
 
 —oblique — preceilion of the equinoxes — 
 
 force of moving bodies. 
 2» Motion in conftrained paths. 
 
VHl ADVERTISEMENT. 
 
 3. Rotation — centrifugal force. 
 
 4. Solidity combined with gravity — liability 
 — theory of arches and domes. 
 
 5. Motion on inclined planes. 
 
 6. Motion of pendulums — meafure of gravity 
 — meafure of time. 
 
 7. Theory of machines — or Mechanics com- 
 
 monly fo called — mechanic powers — com- 
 pound machines — maxims of conflruction. 
 
 Of friction. 
 
 Of the action of fprings. 
 
 Mechanism of Fluid Bodies* 
 
 1. Coherent fluids — Hydrostatic s^ treating 
 of the preffure and equilibrium of fluids — 
 Hydraulics, treating of the motion, im- 
 pulfe, and refiftance of fluids. 
 
 Hydraulic machines. 
 
 Conflruction and working of fhips. 
 
 2. Expanfive fluids — Pneumatics, treating 
 of the preflTure of the air — its elaflicity — its 
 motion, impulfe, and refiflance — Pneumatic 
 machines — found — theory of mufic — adion 
 of gunpowder — theoiy of artillery, and of 
 mines — account of the fleam engine. 
 
 III. 
 
ADVERTISEMENT. IX 
 
 III. Magnetism. 
 
 General laws of the phenomena — theory of 
 iEpinus — Gilbert's terreflrial magnetifm — 
 mariner's compafs — variation — dip of the 
 needle — artificial magnetifm. 
 
 iV. Electricity. 
 
 General laws. 
 Theory of iEpinus. 
 Thunder — aurora borealis, &G. 
 Galvanic phenomena. 
 
 V. Optics. 
 
 Mathematical laws — catoptrics — dioptrics. 
 
 Vifion — optical inflruments. 
 
 Newtonian difcoveries concerning colours. 
 
 Phyfical optics — further difcoveries of New- 
 ton — mechanical nature of light — mutual 
 adlion of bodies and light. 
 
 Province, and hiftory of natural philofophy. 
 
 Edinburgh, QEioler 31. 1804. 
 
THE READER IS RE(^UESTED TO COTIRECT THE FOLLOWING 
 ERRORS OF THE PRESS. 
 
 .^-7;:, 
 
 Line 
 
 . 
 
 
 Pa^e 
 
 Line. 
 
 
 6 
 
 12 
 
 c 
 
 c 
 
 231 
 
 1 
 
 ( ) 
 
 (345) 
 
 
 13 
 
 c 
 
 c 
 
 2J9 
 
 19 
 
 c?.ruI.->t:on 
 
 calculation 
 
 9 
 
 8 
 
 AY^aT^b 
 
 AFB, an 
 
 .26 
 
 10 
 
 fin^' 
 
 p 1) n 
 
 3S 
 
 17 
 
 LKIH 
 
 LIH 
 
 33^' 
 
 17 
 
 to 
 
 of 
 
 
 18 
 
 KICB, KHDB 
 
 LICB,LHDB 
 
 3«4 
 
 9 
 
 confidering 
 
 ccnfider 
 
 
 i3 
 
 KG 
 
 IG 
 
 385 
 
 7.7 
 
 relre; ts 
 
 rerreat 
 
 47 
 
 II 
 
 ^Z 
 
 '.^ 
 
 394 
 
 26 
 
 poUente 
 
 pellente 
 
 
 la 
 
 v 
 
 if 
 
 395 
 
 9 
 
 art: 
 
 is 
 
 
 20 
 
 A P 
 
 AC 
 
 398 
 
 28 
 
 MP 
 
 MR 
 
 53 
 
 14 
 
 to tn 
 
 from E to ffz 
 
 418 
 
 4 
 
 clotted 
 
 dotted 
 
 
 15 
 
 
 
 tn 
 
 449 
 
 12 
 
 PI 
 
 GI 
 
 65 
 
 14 
 
 AVjV 
 
 ADO 
 
 
 13 
 
 PF— PG 
 
 PH— PI 
 
 64 
 
 17 
 
 AK 
 
 BK 
 
 
 16 
 
 PF'—PG' 
 
 PH'— PI' 
 
 73 
 
 23 
 
 ultimate 
 
 alternate 
 
 45* 
 
 26 
 
 Fij 64. 
 
 Fig 64. N^i. 
 
 75 
 
 a 
 
 defleciiny fo rces 
 
 deflexions 
 
 458 
 
 28 
 
 after P 
 
 (Fig.64.N^a.) 
 
 79 
 
 i» 
 
 para! Id 
 
 particle 
 
 471 
 
 18 
 
 L.'^gr 
 
 E.Qy 
 
 Sz 
 
 la 
 
 divcfting 
 
 dividing 
 
 476 
 
 19 
 
 BDF 
 
 PDF 
 
 549 
 
 4 
 
 S 
 
 C 
 
 Note. 
 
 Boiruet 
 
 BoOiit 
 
 ^S3 
 
 a8 
 
 equal, 
 
 equally 
 
 591 
 
 26 
 
 S 
 
 5 
 
 188 
 
 11 
 
 AC! 
 
 AE[ 
 
 59* 
 
 3 
 
 Cof.* 2 X 
 
 Cof ^x 
 
 189 
 
 20 
 
 A EM 
 
 ACM 
 
 647 
 
 19 
 
 corrofponds 
 
 correfpo||i 
 
 190 
 
 la 
 
 MI 
 
 Mi 
 
 
 
 
 
 
 
 CORRECTIONS FC 
 
 R THE FIGURES. 
 
 
 Figure. 
 
 
 
 
 
 
 
 9 
 
 Draw EQ. 
 
 
 
 
 
 
 37 
 
 44 
 46 
 
 64 
 ^5 
 
 71 
 
 73 
 76 
 
 Draw ED. The line ES was, drawn (295.) perpendicular to IC. Tn {e€t. 29S, 
 it is fuppofcd to be perpendicular to i C. The two perpendiculars would 
 not he diftinguilhable. 
 
 e (hould be d 
 
 Produce BS to M 
 
 Draw SM^ perpendicular to PN" 
 
 D fhould be in the crofling of I i and e q 
 
 Draw Ap 
 
 The upper S fhould be s 
 
 Infert G at the crofling of EQ and N^S 
 
 Write / to the left of F, on the outfide of all. 
 
 THE BOOKBINDER IS DESIRED TO PLACE THE PLATES 
 AS FOLLOWS : 
 
 Plate I. to face page 48. 
 
 a. to face page 72. 
 
 3. to face page 80. 
 
 4. to face page T44. 
 
 5. to face page 160, 
 
 6. to face page 16. 
 
 Plate 7. to face page i8a. 
 
 8. to face page 224. 
 
 9. and fucceeding ones to be placed 
 
 agreeably to the engraved re- 
 fcri^nce at th. L<j;-> of each. 
 
EXPLANATION OF SYMBOLS 
 
 usJed in the following pages. 
 
 {a) A. HE fymbol ^.: ^ expTeffes the ratio or proper- 
 t»n of a magnitude a to another magnitude i> of the fame 
 kind, fuch as two lines, two furfaces, two weights, velo- 
 cities, times, &c. 
 
 (h) a'.b '=z c :d. The ratio of ^ to ^ is equal to, or 
 is the fame with, that of c to d, — This is ufually read, 
 a is toh as Q to d. 
 
 [c) ab \d> the producl of two numbers, or the rect- 
 angle of two lines, a and b, 
 
 {d) a :sib IS 2. fymbol made up of the fymbol : of 
 proportion, and the fymbol = of equality. It m^ans that 
 a ifjcrea/es or decrenfes at the fame rate with by fo that if b 
 become double or triple, &c. of its primitive value, the 
 contemporaneous a is alfo double, triple, Sec. of its firfl 
 value. 
 
 A a , this 
 
J^ ifXPLANATlON OF SYMBOLS. 
 
 This Is a fhort way of writing A : a zz B : hj in which 
 A and a are fuccefiive values Of one changeable magni- 
 tude, and B, b, the correfponding or fimultaneous values 
 of the other. In this fymbol, a and b may be magnitudes 
 of different kinds, which cannot hold with refpec^ to the 
 fymbol a : b, becaufe there is no proportion between 
 magnitudes of different kinds, as between a yard and a 
 pound, an hour and a force, &c. This may be called 
 the fymbol of a proportional equation. 
 
 (e) ab'.cd expreffes the ratio compounded of the 
 ratio of ^ to r and that of b to d. It therefore expreffes 
 the ratio of the product of the numbers a and b to that 
 of the numbers c and d. In like manner, it represents 
 the proportion of tVv^o reftangles, a and b being the fides 
 of the fir ft J and c and d the fides of the fecond. In the 
 fame manner a b c : d e f \s tho: ratio compounded of thofe 
 of a to d, of b to e, and of c to /-, and fo on, of any 
 number of ratios compounded together. (See Euclid, 
 VI. 23.) 
 
 ( / ) a :b =: -:- means that a is to b In the ijiverfe 
 ^^ c a 
 
 proportion of c to r/, or, that a : b ■= d : c. It is plain 
 
 that if c be doubled or trebled, the fraction - is reduc- 
 
 c 
 
 ed to one half or one third, &c. fo that - or -7 are in- 
 
 c a 
 
 creafed in the fiime proportion that c ox d arc diminifli- 
 ed. 
 
EXPLANATION OF SYMBOLS. f 
 
 (g) a :h = -:- means that the ratio of ^ to ^ is the 
 
 c d 
 
 fame with that of the fra<£lion - to the fraction - , or 
 
 that the ratio of « to ^ is compounded of the direct ratio 
 oi c to d and the inverfe or reciprocal ratio of e to f. It 
 is the fame with a\h ■=. cf-.d e, 
 
 (h) X = - means that x increafes at the fame ratio 
 ./ 
 
 that y diminiflies, and is equivalent to X : :v = ^ ; - , 
 or ecjuivalent to X : ;v = j? : Yv 
 
 (i) X =^ ~ means that .v varies in the ratio com.- 
 pounded of the direct ratio oi y and the inverfe ratio of z. 
 
 {k) x' \y' expreffes the proportion between the differ- 
 ence of tM^o fucceflive values of x and the dfference of the 
 two correfponding values of y» It is equivalent to the 
 fatio of X — X to Y — ;;. 
 
 (/) Suppofe that, in the continual variation of x 
 and J', thefe fimultaneous and correfponding differences 
 are always in the fame ratio \ then x' :_y' is a conftant ra- 
 tio. Thus, Let AD and AF (fig. A) be two right lines 
 diverging from A, and let B C, Br, B D, be fucceiTive 
 values of Xy and the parallel ordinates C E, c e^ D F be 
 correfponding values of y. Draw E G and eg parallel 
 to A D, and confequently equal to C D and c Z), then 
 C D and G F are correfponding differences of tlie fuc- 
 ceiTive 
 
O EXPLANATION dF SYMBOLS. 
 
 cefTive values of a: and )'. So are c D and g F. Now it 
 is plain that CD:GF = (rD:^F, and x' : y' is a con- 
 ftant ratio. 
 
 (m) But it more frequently happens that the ratio 
 x' : y is not conftant. ThuS;, if the line E ^ F (fig. B) be 
 an arch of a curve, fuch as a hyperbola, of w^hich A is 
 the centre, we know that C D has not the fame ratio to 
 G F that c D has to g F, and that the ratio of .v' to / 
 continually increafes as the point C or r approaches ^o 
 D. We know that while C is above D, the ratio of 
 C D to OF, or ^ D to ^ F is lefs than that of the fub- 
 tangent T D to the ordinate D F. But when c' gets be- 
 low D, the ratio of E'G', or r'D, to G'F is greater 
 than that of T D to D F ; and the difference of thefe 
 ratios increafes, as c feparates from D on either fide. 
 The ratio of a;' to y', therefore, approximates to that of 
 T D to D F as r approaches to D from either fide. For 
 this reafon, the ratio of T D to D F has been called the 
 ultimate ratio of the evanefcent magnitudes .v' and y^ as 
 the magnitudes x' and^))' are continually diminifhed, till 
 both van'ipj together, when c coalefces with D. If, again, 
 we conceive the point* C to fet out, either upM^ard or 
 dow^nward, from D, the ratio T D : D F is called the 
 prime ratio of the nafcent magnitudes x' andjy'. 
 
 We know alfo that the ratio of the fubtangent tc 
 to the ordinate r ^ is lefs than that of T D to D F, and 
 that the ratio of the fubtangent to the ordinate increafes 
 continually, as D is taken further from the vertex V of 
 
 the. 
 
EXPLANATION OF SYMBOLS. J 
 
 tJie liypcrbola. But we know alfo that it never is fo 
 great as the ratio of A D to D/ (the ordinate produced 
 to the asymptote) but approaches nearer to it than by 
 any difFcrence that can be affigned. For this reafon, 
 A D : T)f has been called the ultimate ratio of the fub- 
 tangent and ordinate — in the fame manner, the ultimate 
 ratio of D F to Df has been faid to be the ratio of e- 
 4^uality. 
 
 (w) But, in thefe two cafes, the employment of the 
 term ultimate is rather improper, becaufe this ratio is never 
 attained. Perhaps the term limiting ratio, alfo given it by 
 Sir Ifaac Newton, is more proper in both thefe cafes. 
 TD : D F is the limiting ratio of sk' \y'y or the limit, to 
 which the variable ratio of the nafcent, or, evanefccnt 
 magnitudes x' and y' continually approaches. 
 
 [o) Sir Ifaac Newton, the author of this way of con- 
 fidering the variations of magnitude, has exprefTed by a 
 particular fymbol this limiting ratio of the variations x 
 and y\ He expreflcs it by x : y. It is not the ratio of 
 any x' to any y', however fmall, but the limit to which 
 their ratio continually approaches. When we chance to 
 employ the terms ultimate or prime, Vv^e defire to be un- 
 derftcod always to mean this limiting ratio. The fo- 
 reign m.athematicians employ the fymbol dx:dy, in 
 which d means the infinitely or incomparably finail diiFer,. 
 ence between tv/o fucceeding values of x ox y. 
 
 We, 
 
S EXPLANATION OF SYMBOLS^ 
 
 We have been thus particular in defcribing this view 
 of the variations of quantity, becaufe without a know- 
 ledge of fome of thofe limiting ratios, it is fcarcely pof- 
 iible to advance in mechanical philofophy. 
 
 (/)) The cafe already mentioned, namely TD:DF 
 = x' :y'y occurs very frequently in our inveftigations. 
 
 And, in like manner, if the arch B F be reprefented 
 by the fymbol 2, we have » : z = T D : T F, and ;,' : z 
 = DF:TF. 
 
 Alfo, if E g be drawn parallel to the tangent / ^, we 
 have E ^ to E g ultimately in the ratio of equality. For, 
 tecaufe the triangles tee and E d g are fimilar, we have 
 Ed:Ei=::tc:te, that is, = x : c , that is, = C r : E f, 
 or E ^ : E f, and therefore, ultimately, E e = E ^. 
 
 [q) Such limiting ratios may alfo be obtained in 
 turves that are referred to a pole or focus, inftead of 
 an abfciffa. Thus, let B F G (fig. C) be an ellipfe, whofe 
 centre is C, and focus D. Let F ^ be a very fmall arch 
 of the curve. Draw D F and D e, and about the pole D, 
 with the diftance D^, defcribe the circular arch Eeg, 
 cutting F D in g. Draw the tangent F T, and D T per- 
 pendicular to DF. Now, reprefenting FD by x, FB 
 by z, and the circular arch ^ E hy y, it is plain that 
 ; : ;;;=FD:FT, and « :; = FD:DT. All this is very 
 evident, being demonftrated by the fame reafoning as in 
 the cafe of the hyperbola referred to its axis or abfcifla 
 
EXPLANATION OF SYMBOLS. «> 
 
 (r) Another limiting ratio, of very frequent occur- 
 rence, is the following. Suppofe two curves AB. and 
 ah (fig. D) round the fame pole F, from which are drawn 
 two ri^ht.liiies FA, F B, cutting both lines in A, a, B, 
 and^. Let-FB, by revolving round F, continually ap-». 
 ptoach to FA. Let it come, for example, into the.fima* 
 tion Y cQ very near to F A a. Let S and s repreient the 
 mixtii-ineai fpaces AFBwF^. Then S' and/ may ex- 
 ^refs the fpaces AFC and aFr. It is plain that the li^ 
 mitiflg ntiQoi A F C to aV c is that of F A^ to F a", and 
 we may fay tkat' S • x =i F A* : F a-. 
 
 (j) The laft example v/hich fl:iall be mentioned is 
 of almofl continual occurrence in our inveftigations.-rr- 
 Let FHIC andy/j^ (fig..E) be two curves, having the 
 abfciflk A E and a e. Let thefe abfcilTs be divided into 
 an equal number of fm.aii equal parts, fuch as A B, B C, 
 DE and ah, be, de\ and let ordinates be drawn through 
 the points of divifion. And on thefe ordinates, as bafes, 
 let parallelograms, fuch as A B L F, B C N G, &c. and 
 ah If, hcng, &c. be infcribed, and others, fuch as 
 ABGM, ACHO, and ahgm, acho, &c. be circum- 
 fcribed.— It is ainrmed, i/?, that if the filbdivifion be 
 carried on without end, the mixtiiineal areas A E K F 
 and aekf are, ultimately, in the ratiO of equality to the 
 fum of all the infcribed, or of all the circum.fcribed pa- 
 rallelograms j and, zdly, that the ratio of the fpace 
 A E K F to the fpace a e kf is the limiting ratio cf the 
 
 B fum 
 
1* EXPLANATION OF StMBOtJT. 
 
 fum of all the parallelograms (infcribed or circumfcribed) 
 m A E K F to tlie fum of thofe in a e hf.- 
 
 I/?, Mske D S and d s equal to A F and af, and 
 drav/ S R, s r, parallel to A E, ^ e. It is evident that 
 the parallelogram S R K Q is equal to the excefs of aH 
 the circumfcribed over all the infcribed parallelograms. 
 Therefore, by continuing the fubdivifion of AE with-^ 
 out end, this parallelogram may be made fmaller than 
 any fpace that can be affigned. Therefore the infcribed 
 and circumfcribed parallelograms . are ultimately in the 
 ratio of equality — or equality is their limiting ratio. The 
 fpace A E K F is greater than all the infcribed, and lefs 
 than all the circumfcribed parallelogranls, and is nearly 
 the half fum of both. Therefore, much more accurately 
 is equality the limiting or ultimate ratio of A E K F to 
 either fum. The fame muft be true of the other figure. 
 
 idly, Since each mixtilineal figure is ultimately equal 
 to its parallelograms, it is plain that both have the fame 
 ratio vrith the fums of the parallelograms* 
 
 {t) Cor. If the ordinates which are dravv-n through 
 fimilarly fituated points of the two abfcilTx, be in a con- 
 ftant ratio, the areas are in the ratio compounded of the 
 ratio of A E to ae, and that of A F to afy or are as 
 AExAF to a e X afi This is evident. For, by the 
 fuppofition, C N : r /? = A F : ^y. And, fince the number 
 of parallelorgrams is the fame in both figures, B C and 
 he are fimilar parts of AE and ae\ that is, BC :^r=: 
 A E : ^ c. Therefore BCNG:^<r^^ = AExAF:«^ 
 
IKPLANATION OF SYMBOLS, l,X 
 
 "X af. Since this is true of every correfponding pair of 
 parallelograms, it is true of their fums, and of the mix- 
 tilineal fpaces A E K F and a s kf, which are ultimately 
 equal to thofe fums. 
 
 (//) It may be thought that In thefe cafes wher€ th©^ 
 limiting ratio is not an ultimate ratio adually attained, 
 there remains fome fmall error. The foreign mathemati- 
 cians feem to acquiefce in this, and content themfeives 
 with afluming dx or d y as infinitely fmall ; inferring 
 from thence that the remaining error is infinitely fmall, 
 fo that it will not amount to a fenfible quantity, thougl^ 
 multiplied by any number, however great. But this con- 
 ceflion leads them neceffarily into the fuppofition of quan- 
 tities infinitely fmaller than quantities already afTumed as 
 infinitely fmall ; a fuppofition plainly abfurd or unintel- 
 ligible. But no error whatever lurks in this method of 
 limiting ratios. For it is all founded on the following 
 unqueftionable axiom, 
 
 {v) If the ratio of ^7 to ^ be greater than any ratio 
 whatever that is lefs than the ratio of c to d, but lefs 
 than any ratio luhatever which is greater than that of 
 c to dy then « is to ^ as r is to d. 
 
 For if a be not to ^ as tr to d^ let ^ be to ^ as m to ?i. 
 Then if in : n be greater than c.dy a:!; is lefs than m : n. 
 If m\n be lefs than c : d^ then ^ : ^ is greater than m : tiy 
 both which confequences are contrary to the conditions 
 aiTumed. Therefore a : b muj} be c : <^ 
 
 ^ % The 
 
IZ EXPLANATION OF SYMBOLS. 
 
 The propofidon (j-) may be demonftrated in tliisvva^. 
 The fpace AEKF is to a e if in a greater r^tio' than 
 that of the parallelograms iiifcribed in the firft to thofe 
 circumfcribed on the fecond, but in a lefs ratio than that 
 of the parallelograms circumfcribed on the firft to thofe 
 Hifcribed in the fecond. We perceive, by continuing the 
 fubdivifion of the two abfciiile, that this holds true with 
 regard to every ratio that is either greater or lefs than that 
 gf A E K F to « ^ kf. Thus, the prcpofition is demon-, 
 ilrated without the fmalleffc room for error. 
 
 (w) This doctrine of limiting ratios is of die greateit 
 fervice in the phyfico-maxhernatical fcienoej. Nature 
 prefents magnitudes in a continual change:. The velo- 
 city of a falling body, and the line of its fall, nre in- 
 creafing together. — As a piece of iron approaches a mag- 
 net, its diftanee, its velocity, and the force by which it 
 is urged, all vary together, and. there is an indiflbluble 
 relation between their refpeclive fimultaneous variations. 
 Thefe variations alfo are the immediate meafures of their 
 rates of variation. Hence it is plain that, by knowing 
 thefe rates, we can learn the whole change, and by ob- 
 ferving the whole change we* can infer the rate of va- 
 riation •, juft as the navigator learns his dby's progrefs by 
 heaving the log every hour, in order to difcover thi^' ifcip'g 
 rate cf failing, and converfely. • i- V '♦: !' 
 
 (x) The letters F, V, T, &c., will be ufed to'.exprfifo 
 
 Force, Velocity, Time, and oth^r magnitudes. .Thus, 
 
 . , F,A 
 
EXPLANATION OF SYMBOLS, 1 3 
 
 F, A exprefles the force ading In the point A. F, A B 
 is the force acting along the line A B. 
 
 (y) A proper notation, and arrangement of the fym-r 
 bols, greatly affift our conceptions in mathematical rea- 
 foning. When ratios are compounded (a thing perpe- 
 tually occurring in our difquifitions) it is extremely con- 
 venient to recolle£i: th^t the ratio, which is compounded 
 of many numerical ratios, is the fame with that of the 
 produft of all the antecedenta to the produd of all the 
 Gonfequents. 
 
 Thus, if a : b = c : d 
 
 and e :f= g : /^ 
 
 and 2 : k = / : m 
 
 and n : 0— p \ q 
 
 then a e i n \ bfk q-=. c g I p \ d h m q. 
 If we ufe lines, we can go no farther without fubfti- 
 lutions than three fuch compofitions, becaufe fpace has 
 but three dimenfions. All our pra£lical ufes of the, 
 doctrines mufl be profecuted by means of arithmetical 
 calculations, although fome linear ratios, fuch as that of 
 the diameter of a circle to its circumference, or that of 
 the diagonal of a fquare to its fide, cannot be accurately 
 exprefled by numbers. But, as we know perfectly what 
 fubftitutions may be made in every cafe where more than 
 three ratios are compounded, fo as to obtain accurate 
 ratios, no mathematician obje6ls to this method of mere-, 
 ly exprefling the comp9rition. 
 
MECHANICAL PHILOSOPHY. 
 
 INTRODUCTION 
 
 i. iVlAN is Induced by an inftindllve principle, im« 
 planted in his mind by the Author of Nature, to confider 
 every change obferved in the condition of things as an. 
 Effect, indicating the agency, chara6teriring the kind, 
 and meafuring the degree of its Cause. 
 
 2. The kind and degree of the caufe are, therefore, 
 inferred from the obferved kind and degree of the change 
 which we confider as its effe£t. 
 
 3. The appearances in the material world, exhibited 
 in the changes of motion which we obferve, are called 
 Mechanical Appearances, or Phenonema, and the 
 caufes, to the agency of which we afcribe them, are call- 
 ed Mechanical Causes, 
 
 4. Mechanical Philosophy is t\\Q lludy of the 
 ftiechanical phenomena of the univt-rfe, in order to dif- 
 Cover their caufes, and by their means to explain fubor- 
 
 dinate 
 
i6 Oi' MOTION. 
 
 dinate phenomena, and to improve arts, and thus increafe 
 man's power over nature. 
 
 This definition of the ftudy points out Motion, with 
 all its affe6bions and variettesj as the objects oF our firft 
 attention, a knowledge of thefe being indifpenfably ne^ 
 ceflary for perceiving and. appreciating its changes, from 
 which alone we are to derive all our knowledge of their 
 caufes, the mechanical powers of nature. 
 
 OF MOTION. 
 
 5. In mptlon we qbferve the fuaefftve appearance of 
 the thing moved in different -^'SLXtb.oi fpace. Therefore, in 
 pur idea of Motion are involved the ideas or conceptions 
 of Space and of Time. 
 
 6. Space is cbnceived by us as a quantity <, that is. It 
 may be conceived as great or little. It is one of that 
 fmall clafs of quantities of which we have the cleared 
 and mod diftin£i; conceptions. We conceive them as 
 magnitudes made up of their own diftinguifhable parts, 
 and meafurable by one of thefe as a unit. We cannot 
 conceive fo clearly of heat, or preflure, or many other 
 things \yhich are magnitudes, capable of increafe and di- 
 minution, but not diilinguilhable into feparate parts. 
 
 7. In our fimplefl conception of fpace, it is mere ex=- 
 tenfion-, we think of nothing but a diftance between 
 two places. This is the mod Aifual conception of it in 
 
 mechanical 
 
% 
 
 ^ 
 
OF MOTION* i^ 
 
 mechanical dlfqulfitions — the path along which a thing 
 moves ; and we fay, figuratively, that the thing defcribes 
 this path. 
 
 But the geometer confiders fpace as having not on- 
 ly lengtli, but alfo breadth, and he then calls it a fur^ 
 face; and, in order to have a complete notion of the 
 capacioufnefs of a portion of fpace, he confiders not on- 
 ly its length and breadth, but alfo its thicknefs— and 
 fuch fpace he calls 2^ folid fpace. But, by folid, he means 
 nothing but the fufceptibility of meafure in three ways. 
 He calls it extenfion of three dimenfions. 
 
 But, in pure mechanics, we feldom have occafion to 
 oonfider more than one dimenfion of fpace. — In our 
 invefliigations, however, we make ufe of geometrical rea- 
 fonings, which include both furfaces and follds — but our 
 reafoning always terminates in a mechanical theorem, of 
 which diftance alone is the fubje^l. 
 
 8. The adjoining parts or portions of fpace are dif- 
 tinguifiied or feparated from one another by their miu- 
 tual boundaries. Contiguous portions of a line are fe- 
 parated by points — contiguous portions of a furface are 
 feparated by lines — and contiguous portions of a folid 
 are feparated by furfaces. 
 
 9. Thefe boundaries are not parts of the contiguous 
 portions of fpace, but are common to both. They are 
 the places where the one portion of fpace ends, and the 
 •ther begins. It is of importance to have very clear 
 
 C notions 
 
OF MOTION* ff 
 
 mechanical difqulfitlons — the path along which a thing 
 moves J and we fay, figuratively, that the thing defcribes 
 this path. 
 
 But the geometer confiders fpace as having not on- 
 ly lengtli, but alfo breadth, and he then calls it a fur^ 
 face: and, in order to have a complete notion of the 
 capacioufnefs of a portion of fpace, he confiders not on- 
 ly its length and breadth, but alfo its thicknefs— and 
 fuch fpace he calls a foUd fpace. But, by foiid, he means 
 nothing but the fufceptibility of meafure in three ways. 
 He calls it extenfion of three dimenfions. 
 
 But, in pure mechanics, we feldom have occafion \.o 
 oonfider more than one dimenfion of fpace. — In our 
 inveftigations, however, we make ufe of geometrical rea- 
 fonings, which include both iurfaces and follds— but our 
 reafoning always terminates in a mechanical theorem, of 
 which diflance alone is the fubje^l. 
 
 8. The adjoining parts or portions of fpace are dif- 
 tinguifhed or feparated from one another by their mu- 
 tual boundaries. Contiguous portions of a line are fe- 
 parated by points — contiguous portions of a furface are 
 feparated by lines — and contiguous portions of a folid 
 are feparated by furfaces. 
 
 9. Thefe boundaries are not parts of the contiguous 
 portions of fpace, but are common to both. They are 
 the places where the one portion of fpace ends, and the 
 •ther begins. It is of importance to have very clear 
 
 C notions 
 
i^ or MOTION. 
 
 notions of this diftiridtion, for great miftakes have arife» 
 in mechanical difcuffion-s by not attending to it. 
 
 10. We cannot conceive fpace as having any bounds^ 
 and it is therefore faid to be infinite, or unbounded. 
 
 11. A portion of fpace may be eonfrdered in relation 
 to its fituation among other portions. This may be call- 
 ed the RELATIVE PLACE of the Body which occupies 
 this portion of fpace. It may alfo be called its Situation. 
 
 Or it may be confidered as a determinate portion af 
 infinite fpace, the individuality or identity of w^hich con- 
 frfts entirely in its being there. This is called the abso- 
 lute PLACE of the body vt^hich occupies this portion of 
 infinite fpace. — It is plain that in this fenfe, fpace is im- 
 moveable — that is, we cannot conceive this identical por- 
 tion of fpace as removed from where it is, to another 
 place — for whatever be taken from thence, fpace remains. 
 Yet we always proceed on the contrary fuppofition in 
 our a61:ual meafurements. If we find that three appli- 
 cations of a foot rule to one line completely exhauft it, 
 and that fix applications are required for another line, 
 we affirm that the laft is double of the firft. But this 
 really proceeds on another fuppofition, viz. that the ru/e, 
 though it do not aliuays occupy the fame fpace, yet, in every 
 fituation, it occupies an equal fpace. Granting this, the 
 conclufion is juil. It will afterwards appear that this 
 remark on the immobiHty of fpace is of importance in 
 mechanical difcufiion^. 
 
 ^.2> 
 
OF MOTION. ^^ 
 
 12. We do not perceive the abfolute place of any ob- 
 jeft. — A perfon in the cabni of a flilp does not confider 
 the table as changing its place while it remains faftened 
 to the fame plank of the deck. Few perfons think that 
 a mountain changes its place while it is obferved to re- 
 tain the fame fituation among other objecSls. On the 
 other hand, moll men think that the ftars are continually 
 changing their places, although we have no proof of it, 
 and the contrary is almoft certain. 
 
 13. We acquire our notions of time by our faculty 
 of memory, in obferving the fucceffions of events. 
 
 14. Time IS conceived by us as unbounded, conti- 
 Tiuous, homogeneous, unchangeable in the order of its 
 parts, and divifible without end. 
 
 15. The boundaries between fucceffive portions of 
 time may. be called instants, and minute portions of it 
 may be called momefits. 
 
 1 6. Time is conceived as a proper quantity, made up 
 cf, and meafured by, its own parts. In our a£lual mea- 
 furements, we employ fome event, Vv^hich we imagine 
 always to require an equal time for its accomplifliment ; 
 and this time is emiployed as a unit of time or duration, 
 
 in the fame manner as we employ a foot rule as a unit 
 
 * • • • 
 
 of extenfion. As often as this event is accomplifhed du- 
 ring fome obferved operation, fo often do we imagine 
 
 C 2 that 
 
4^' OF MOTION. 
 
 that the time of the operation contains this unit. It is 
 thus that we affirm that the time of a heavy body falling 
 1 44 feet, is thrice as great as the time of falling 1 6 feet j 
 becaufe a pendulum 39-I- inches long makes three vibra- 
 tions in the firft cafe, and one in the laft. 
 
 17. There is an analogy betvi^een the affefticns of 
 fpace and time fo obvious, that, in mofi languages, the 
 fame words are ufed to exprefs the affe6i:ions of both.— 
 Hence it is that time may be reprefented by lines, and 
 meafured by motion ; for uniform motion is the fimplell 
 fucceflion of events that can be conceived. 
 
 18. All things are placed in fpace, in the order of 
 fituation. — All events happen in time, in the order of 
 fueceffion. 
 
 19. No motion can be conceived as inilantaneous. 
 For, fince a moveable, in paffing from the beginning to 
 the end of its path, pafles through the intermediate 
 points ; to fuppofe the motion along the moll minute por- 
 tion of the path inftantaneous, is to fuppofe the move- 
 able in every intervening point at the fame inftant. — 
 This is inconceivable, or abfurd. 
 
 20. Absolute Motion is the change of abfolute 
 pilace. Relative Motion is the change of fituation 
 among other objeds. Thefe may be differept, and even 
 contrary 
 
 21^ 
 
OF MOTION. 5f 
 
 ?,i. The relative motions of things Ire the dljfferences 
 of their abfolute motions, and cannot, of themfelves, tell 
 us what the abfolute motions are. The detection and de- 
 termination of the abfolute motions, by means of obfer- 
 vations of the relative motions, are often talks of great 
 difficulty. 
 
 22. Mathematical knowledge is indifpenfably requi- 
 fite for the fuccefsful fludy of mechanical philofophy. 
 On the other hand, the confideration of motion, in all its 
 varieties of fpace, direction, and time, is purely mathe- 
 «iatical, and carries with it, into all fubjeds, the moll 
 incontrovertible evidence. 
 
 23. Motion is fufceptible of varieties in refpe6t of 
 quantity and of direBion, 
 
 24. That aife£lion of motion which determines its 
 quantity, is called velocity. Its moft proper meafure i« 
 the length of the line uniformly defcribed during fome 
 given unit of time. Thus, the velocity of a fhip is 
 afcertained, when we fay that fhe fails at the rate of fix 
 miles per hour. 
 
 25. The DIRECTION of a motion is the pofitlon of 
 the ftraight line along which it is performed. A motion 
 is faid to be in the dire^lion AB (fig. i.) when the tiling 
 moved palTes along that Ymefrom A towards B. In com- 
 mon difcourfe we frequently exprefs the diredion others 
 
 wife. 
 
^ OF MOTION. 
 
 wife. Thus we fay a wefterly wind, although it moves 
 caftward. 
 
 26. In rectilineal motion, the direiJ^lion remains the 
 fame, during the whole time of the motion. 
 
 27. But if the motion be performed along two con- 
 tiguous ftraight lines AB, BC (fig. 2.) in fucceflion, 
 the direction is changed in the point B. From B Cy the 
 prolongation of A B, it is changed to B C. 
 
 This change may be called deflection -, and this 
 deflection may be meafured, either by the angle cBC, or 
 by a line c C drawn from the point r, to which the move- 
 able would have arrived, had its motion remained un- 
 changed, .to the point C, at which it aClually arrives in 
 tlie fame time. 
 
 When a moveable defcribes the fides of a polygon, 
 there are repeated deflections, with undefleCted motions 
 intervening. 
 
 28. But if the motion be performed along a curve 
 line, fuch as ADBEC (fig. 3.) the direction is co^iti- 
 mially changing. The direction in the point B is that 
 of the tangent B T, that direction alone lying between 
 any pair of polygonal directions, fuch as B C and B r, 
 or B D and B E, however near we take the points A 
 and C, or D and E, to the point B. 
 
 29. A curviiineal motion fuppofes the deviation and 
 
 deflection 
 
eF MOTIOV'. ^3 
 
 deflection to be continual, and a continual deflection 
 conftitutes a curvilineal motion. 
 
 1. Of Uniform Motmts. 
 
 30. In our general conceptions of motion, in which 
 we do not attend to its alterations, the motion is fuppofed 
 to be equable and rectilineal ; and it is only by the de- 
 viations from fuch motion that we are to obtain the marks 
 and meafures of all changes, and therefore of all chang- 
 ing caufes, that is, of the mechanical powers of nature. 
 Let us therefore fix the characters of uniform or un- 
 changed motion. 
 
 31. In uniform motions , the velocities are in the pro* 
 portions of the /paces defcribed in the fame, or in equal 
 times. 
 
 For thefe fpaces are the meafures of the velocities, 
 and things are in the proportion of their meafures. 
 
 Let S and s reprefent the fpaces defcribed in the time 
 T, and let V and v reprefent the velocities. We have 
 Ae analogy V : •y = S : x. This may be exprefled by the 
 proportional equation v == s, 
 
 32. /// uniform motions 'luith equal velocities, the times 
 Mre in the proportion of the fpaces defcribed. during their 
 currency. 
 
 For, in uniform motions, equal fpaces are defcribed 
 in equal times. Therefore the fucceffive portions of time 
 
 are 
 
24 0¥ MOTION*. 
 
 are equal, in which equal fpaces are fucceilively defcrib- 
 ed, and the fums of the equal times mail have the fame 
 proportion as the correfponding fums of equal fpaces. 
 Therefore, in all cafes that can be reprefentcd by num- 
 bers, the propofition is evident. This may be extended 
 to all other cafes, in the fame w^ay that Euclid demon- 
 flrates that triangles of equal altitude are in the propor- 
 tion of their bafes. 
 
 33. Thefe proportions are often exprefled thus : 
 ^* ^l)e velocities are proportional to the fpaces defcribed in 
 " equal times.— ^The times are proportional to the fpaces de- 
 ** fcribed ivith equal velocities. " Proportion fubiifts only 
 betvi^een quantities of the fame kind. — But nothing more 
 is meant by thefe inaccurate expreflions, than that the 
 proportions of the velocities and times are the fame vi^ith 
 the proportions of the fpaces. 
 
 34. It is on this authority that uniform motion is 
 univerfal-ly employed as a meafure of time. — But it i^ 
 not eafy to difcover whether a motion which may be pro- 
 pofed for the meafure is really uniform — fandglafs — 
 clepfydra — ^fundial — clock — revolution of the ftarry hea- 
 vens. 
 
 35. //; uniform motions y the fpaces defcribed are in the 
 ratio compounded of the ratio of the velocities and the raiU 
 »f the times. 
 
 Let the fpace S be defcribed with the velocity V, in 
 the time T^ and Jet the fpace s be defcribed with tlie ve- 
 locity 
 
OF MOTIOM. I5 
 
 ioclty v, "m the time t, Let another fpace Z be defcribed 
 
 in the time T with the velocity v. 
 
 Then, by art. 3 1 , we have S : Z = V : i> 
 And, by art. 32, Z : j =: T : /• 
 
 'Therefore, by compofuion of ratios (or by VI. 23. Eucl.-^ 
 
 we have = V x T : -u X / = S X Z : / X Z j that is, 
 
 = S:j-. 
 
 36. This is frequently exprefled thus : " The /paces 
 •** defcribed iv'ith a iimform motion are proportional to the 
 ** produEls of the times and the velocities. " — Or thus : 
 
 37. '* 27>^ f paces defcribed ivith a iwifonn motion atf 
 *' proportiofiat to the reElangles of the tunes and the veloci' 
 «* ties.'' 
 
 Thefe are all equivalent expreffions, <lemonfl:rated by 
 the fame compofition of ratios. By products or re6l- 
 angles of the times and velocities, is meant the produces 
 ,of numbers, which are as the times, multiplied by num- 
 bers, which are as the velocities ; or the re61:angle, whofe 
 bafes are as the times, and whofe heights are as the ve- 
 locities. — 'There are feveral other jnodes of exprefling 
 thefe propofitions^ 
 
 58. Cor^ I. If the fpaces defcribed in ttuo uniform tm- 
 tions be equaly the velocities are in the reciprocal proportion 
 of the times. 
 
 For, in this cafe, the produ£l:s V T and v t are equal, 
 ?.i>d therefore V ; i) ;=; / ; T, or V : v =: p^ : -. Or, be- 
 
Z6 OF MOTION. 
 
 caufe the rectangles AC, DF (fig^ 4.) are in this cafe 
 equal, we have (by Eucl. VI. 14.) AB:BF = BD:BC, 
 that is Y :v = t:'T, 
 
 39. In uniform viQtionSy the times are as the fpuces^ 
 iireEllyy and as the velocities, inverfely. 
 
 For, by art. 35, S : / = V T : -y / ^ 
 therefore S -y / = j- V T 
 
 and T'.t-^v:sY 
 
 and t =- 
 
 ' <v 
 
 40. In uniform motiofis, the velocities are as the fpaces^ 
 ilreBlyy and as the times , itiverfely. 
 
 For, as before, Svt z=z sY'V 
 therefore V : i; = S / : j- T 
 
 TT S* J" 
 
 or V : i; =: — : - 
 
 and 
 
 V t 
 
 s 
 t 
 
 41. It is evident that the abfolute magnitudes of the 
 ipace and time do not change the values of the refults of 
 thefe proportions, provided both are changed in the fame 
 ratio. The value of ^^i, or of ^J^', is the fame 
 •with ^ of a foot per fecond. Therefore, if s' be taken 
 to exprefs an extremely minute portion of fpace de- 
 fcribed with this velocity in the minute portion of time 
 
OF MOTION. 27 
 
 f\ we ftill have tlie velocity v accurately expreffed by 
 
 J"' s' 
 
 ~ . Alio - is the accurate expreffion of the time t\ 
 
 t' 
 
 ■ 2. Of Variable Motions. 
 
 42. It tarely happens that the phenomena of nature 
 prefent to our obfervation motions perfectly uniforms 
 Yet we diilinftly conceive them, with all their proper- 
 ties J and the deviations from thefe are the only marks 
 and meafures of the variations, and, therefore, of the 
 changing caufes. Therefore it is plain, that it is of the 
 firft importance that all thefe deviations be thoroughly un- 
 derftood. 
 
 43. If a body continue to move uniformly in the 
 fame direction, its motion, or condition in refpedl: to 
 motion, is unchanged. Its condition, therefore, mufh 
 be allowed to be the fame in any two portions of its path^ 
 however diftant they may be. The ditFerence of place 
 does not imply any change, becaufe a change of place Is 
 involved in the very conception of motion. If, therefore, 
 two bodies be moving with the fame velocity in this 
 path, or in two lines parallel to it, their condition in re- 
 fpe£l: of motion mud be allowed to be the fame. They 
 have the fame direction, and move at the fame rate; 
 No circumftance, therefore, feems to enter into our con- 
 ception of the ftate of a body, in refpe£l of motion, 
 except its velocity and its direiSlion. Changes in one 
 
 Da* <»r 
 
5f 6^ ACCELERATED 
 
 or botli of tlicfe circumflances conilituts all the cKa'n'resr 
 of which this condition is fufccptiblc. We fliall firlt 
 eonfider changes of velocity. 
 
 Of Accelerafed and Retarded Motions, 
 
 44. Every one is fenfiblo that a falling ftone Is car-' 
 ried downward with greater rapidity in every fuccefTive 
 inoment of its fall. During the firft feCond of its fall, 
 we know that it falls 16 feet \ during the next, it falls- 
 48 ; during the third, it falls 80 ; during the fourth, 
 112; and fo on: falling, during every fecond, 32 feet 
 more than during the preceding. 
 
 Such a* motion is, with propriety, called an ACCE-- 
 LERATED MOTION. On the contrary, an arrow Pnot per- 
 pendicularly upward is obferved to rife with a motiort 
 continually retarded. Thefe bodies are therefore con- 
 ceived to be in differejit ftates of motion in every fuc- 
 ceeding inftant. The velocity of the falling body is con- 
 ceived to be greater in a certain inftant than in any pre- 
 ceding inftant. Mechanicians fay that when It has fallen 
 144 feet, .its velocity is thrice as great as w^hen it has 
 fallen only 16 feet. But It is plain that this inference 
 cannot be made directly, from a comparifon of the fpaces 
 defcribed in the following moments; for in- thefe, it 
 falls 1 1 2 and 48 feet : nor from tlie fpaces defcribed m 
 the moments immediately preceding ; for in thefe, the 
 body fell 80 and 16 feet. The alTertion however fup- 
 jpofes that this variable condition, called Velocity, is fuf- 
 
 ceptible 
 
AND' RETAllDED MOTIONS. ft^ 
 
 #cptlole of an accurate meafure In every inftant, although 
 in no moment, however fliort, does the body defcrihe 
 uniformly a fpace v/hich may be taken as the meafure of 
 its' velocity at the beginning of that moment. The fpace 
 defcribed in any moment is too great for mcafuring the 
 velocity at the beginning of the mom.enf, and too fmall 
 for the meafure of the velocity at the end of it. Yet 
 its mechanical condition is not known till we obtain fuch 
 a meafure. 
 
 In a motion, like this, couUnually accelerated, there 
 €an be no fuch meafure. In an inftant, no fpace is de-^ 
 fcribed, for this recjuires time. But the body has, ii7 
 «hat inftant, what may be called a potential velocity, 
 a certain determination, however imperfe6lly con- 
 ceived by ^us, which, if not changed, would caufe it to 
 defcribe, and v/ould be indicated by its actually defcrib- 
 jng, a certain fpace uniformly, during a certain afllgn- 
 able portion of time. At another inftant, it has another 
 determination, by which, if not changed, another fpace 
 would be uniformly defcribed in the fame, or an equal 
 portion of time. It is in the difference of thofe two 
 determinations that its difference of mechanical ftate con- 
 fifts. The fpaces which would thus Ix? uniformly de- 
 fcribed, are th^ marks and meafures of thofe determi- 
 nations, and mufh therefore be fought for with the mod 
 fcrupulous care, as the meafures of thofe velocities ; and 
 the proportions of thole fpaces mufh be taken as the pro- 
 portions of the velocities. Tins refcarcli i^ effeded L y 
 th6 following propofition. 
 
 4i 
 
^6 OF ACCELERATED 
 
 45. Let the ilraight line ABD (fig. 5.) be deicribed 
 witli a motion any how co7n'mually varied, and let it be re- 
 quired to determine the proportion of the velocity in the 
 point A to the velocity in any other point C. 
 
 Let the right line ah d reprefent the time of this mo- 
 tion along the path AD, fo that the points «, h^ c, d, 
 may mark the inftants of the moveable's being in A, B, 
 C, D, and the portions ab^b Cy c d, may exprefs the times 
 of defcribing A B, B C, C D, that is, may be in the pro- 
 portion of thoie times. Moreover, let a e, perpendicular 
 to a d, exprefs the velocity of the moveable at the inflant 
 SI, or in the point A. 
 
 Let e g /j be a line, fo related to the axis a d, that the 
 areas a hfe, h c gfy c dh g, comprehended between the 
 ordinates ae^ b fy eg, d h, all perpendicular to « r/, may 
 be proportional to the fpaces A B, B C, CD, defcribed 
 in the times a b, be, c d, and let this relation obtain in 
 every part of the figure. 
 
 It is then affirmed that the velocity in A is to the 
 velocity in B, or C, or D, as <3 ^ to bf, or eg, or d h, 
 &c. In other words. 
 
 If the abfeijfa 2l A of a curve e g h be proportmml to 
 the time of ajiy motion, and the areas interrupted by parallel 
 ordinates be proportional to the fpaces defcribed, the velocities 
 are proportional to thofe ordinates. 
 
 Make b c and c d equal, fo as to reprefent very fmall 
 and equal moments of time, and make p a equal to one 
 of them, and complete the redlancle p a e q. This will 
 reprefent the fpace uniformly defcribed in the moment 
 
 /^3 
 
AND RETARDED MOTIONS. 3I 
 
 p a, with the velocity a e (35.) Let P A be the portion 
 of the fpace thus uniformly defcribed in the moment p 6i, 
 Let tlie Hues i m, k n, parallel to a d, make the re61:angles 
 h c m i and c d n k^ refpedlively equal to the areas b c gf 
 ^nd c d h g. 
 
 If the motions along the fpaces P A and B C had 
 been uniform, their velocities vuould have been propor- 
 tional to the fpaces defcribed (3i.)j becaufe the times p a 
 and b c are equal. That is, the velocity in A vi^ould be 
 to the velocity in C, as the reftangle p a e q to the area 
 h c gfy that is, -a^ p a e q to be m /, that is, as the bafe 
 a e to the bafe c w, becaufe the altitudes p a and b c 
 are equal. 
 
 But the motion along B C is not reprefented here as 
 uniform. For the line f g h diverges from the axis b d^ 
 the ordinate c g being greater than b f. Therefore the 
 fpaces, which are meafured by thofe areas, increafe fafter 
 than the times, and the figure reprefents an accelerated mo- 
 tion. Therefore the velocity with which B C would be 
 uniformly defcribed during the moment b c, is lefs than 
 the velocity at the end of that moment, that is, at the in- 
 ftant c, or in the point C of the path. It muft therefore 
 be reprefented and meafured by a line greater than c m. 
 
 We prove, in the fame manner that c k reprefents 
 and meafures the velocity with which C D would be uni- 
 formly defcribed during the moment c d. Therefore, 
 fince the motion along C D is alfo accelerated, the velo- 
 city at the beginning of that moment is lefs than the ve- 
 locity witl]^ which it would be uniformly defcribed in the 
 
 fame 
 
^2 OF ACCELERATED 
 
 fame time, and mufl be reprefented by a line lefs tlian 
 ck. 
 
 Therefore the velocity in A is to that in C in a lefiij 
 ratio than that of ^ ^ to c w, but in a greater ratio 
 than tliat oi a e to c k. But, in this example, as long 
 as the inftant b is prior and d pofterior, to the inftant c, 
 c m is lefs, and r ^^ is greater, than c g. Therefore the 
 velocity in A is to that in C in a ratio that is greater 
 than any ratio lefs than that of a e to c gy but lefs than 
 any ratio greater than that of ^ ^ to eg. And, confe-^ 
 quently, the velocity in A is to that in C d.s a e to c g. 
 (Symb. (v) 
 
 Since this can be proved in the fame manner with 
 refpe£l to the velocity in any other point D, the propofi- 
 tion is demonftrated. 
 
 It is plain that the reafoning would have been pre- 
 cifeiy the fame, had the motion along BCD been re^ 
 tarded. 
 
 45. Cor, I . The veloclftes m different points of the 
 path A D are in the ultimate ratio of the /paces defcribed 
 in equal Jmall moments of time. For, drawing g parallel 
 to a dy the velocity in the inilant a is to that in the in^ 
 ftant <: as a e to c g^ that is, as the rectangle p e to the 
 rectangle c Oy that is^ 2iS p a e q to c d h g very nearly. 
 As the moments are diminiflied, the difference g h be- 
 tween the rectangle c go d and c gh dy diminifties, nearr 
 ly in the duplicate ratio of the moment ; fo that if the 
 fnomeot be taken f , -f , or ^ of ^ d^ the error goh is re- 
 
 dufeu 
 
anC retarded motions. 33 
 
 duccd to I, or -i-, or V3-. The ultimate ratio oi c g d 
 to c g h d is plainly the ratio of equality, and the corol- 
 lary is manifeft. That is, the velocity in A is to that 
 in C in the ultimate ratio of P A to B C defcribed in 
 equal fmall moments. 
 
 47. It often happens that we cannot afcertain this 
 ultimate ratio, although we can meafure the fpaces de- 
 fcribed in A^ery fmall moments. We are then obliged to 
 take thefe as meafures of the velocity. The error is re- 
 duced almoft to nothing, if we take the half fum of the 
 fpaces B C and C D for the meafure of the velocity in 
 the point C ; or, which is the fame thing, if we take 
 B C for the meafure of the velocity in the middle of the 
 moment b c. For the fpaces B C and C P are meafured 
 by the areas b f g c and c g h d, which is very nearly 
 equal to the rectangle b i d. Now b c gt^ or c d gy 
 is the half of it; and it is evident by this propofition^ 
 that the velocity in A is to that in C, as the rectangle 
 p a e q to the rectangle b c g t, or c d g. 
 
 48. Cor. 2. The momentary increments of the fpaces 
 defcribed are in the ratio compounded of the ratis of ihs 
 velocities and the ultimate ratio of the moments. 
 
 For the increments P A, C D, are as the re£tangles 
 p e and c ultimately (35.) j 2nd thefe are in the ratio 
 compounded of the ratio of the bafe « ^ to the bafe d 0, 
 and the ultimate ratio of the altitude p a to the altitude 
 c d. This may be expreffed by the proportional equa- 
 tion i =f v i. 
 
 E 4^- 
 
34 OF ACCELERATED 
 
 49. Confcquently -y =^ 4- , and i'^ - , The equa«^ 
 
 tion s =^ V i, V =^ ~ , and i = - feem to be the fame 
 with thofe in art. 41. But, in art. 4 1 , the fmall fpaee s' 
 was defcribed uniformly, and the equations were abfo-^ 
 lute. In the articles 4S. and 49. j does not reprefent a 
 fpace uniformly defcribed. But s : J exprefies the ulti- 
 mate ratio of S' to s\ when they are diminifhed conti«- 
 nually, and vanifh together. Th-e meaning of the equa-- 
 tion s =^ V t therefore is, that the ultimate ratio of S' to 
 }' is the fame with that of VT' to vt\' 
 
 50. The converfe of this propofition rrtay be thus ex- 
 prefTed : 
 
 If the ahfcijfa 2. A cf the line e f h reprefent the tinte 
 ef a moticn along the line A B D, and if the ordinates a e, 
 b f , c g, Sec. be as the velocities in the poittts A, B, C, 
 &c. theti the areas are as the fpaces defcribed. This i* 
 mod expeditioully demonftrated, indiredlly, thus : 
 
 If the fpaces A B, A D be not pioportional to the 
 areas a bfe^ a dh e^ they mud be proportional to fome 
 other areas ahf^e, adh' e, of another Ime ^y'/^V pafl- 
 ing through e. But, if fo, then, by art. 45, the velocity 
 in A is to that in B as a e to bf. But the velocity ii> 
 A was ftated to that in B as ^^ to bf. Therefore a e% 
 hf=. a e : bf'y which is abfurd. Therefore, &c. 
 
 f 51. The only immediate obfervation that we caii 
 m^ke on thefe variable motions' is the relation between 
 
AND ©.ETARDED MOTIONS, 25 
 
 flie ipace defcrlbed and the time which elapfcs. The 
 preceding propofitions teach us how to infer from this 
 relation the mechanical condition of the body, to which 
 condition we have given the name Velocity, which, how- 
 ever, more properly denominates the efFecl and meafuri? 
 of this condition or determinatioru 
 
 The fame inference may be made in another way. 
 Inftead of taking the uniform motion along a line to rc- 
 f refent the uniform lapfe of time, Sir Ifaac Newton often 
 Teprefents it by the uniform increafe of an area during 
 the motion along the line taken for the abfciiTa. The 
 .velocities, or determinations to miction in the different 
 points of this line, will be found inverfely proportional 
 to the ordinates of the curve which bounds this area. 
 
 Thus, let a point move along the ftraight Hne A D 
 (fig. 6.) with a motion any how continually changed, 
 and let the curve LKIH be fo related to AD that the 
 area^ I C B is to the area Jt H D B a^ the time of mov- 
 ing along B C ;:o that of moving along B D -, and let this 
 i>e true in every point of the line A D. Let C c, D d 
 be two very fmail fpaces defcribed in equal times, draM^ 
 the. ordinates i<-, h d, and draw ii, h I perpendicular to 
 K C, H D. 
 
 It is evident that the areas IC.^i and HD J/7 are 
 .equal, becaufe they reprefent equal moments of time. It 
 is alfo plain that as the fpaces C c and D d are continually 
 diminilhed, the ratio of ICci and HTi d h to the rec- 
 |:angles kCci and IVi dh continually approaches to that 
 of pi^uaUty, and that the ratio of equality is the limiting or 
 Jl 1 ultmi^^^ 
 
2^6 OV ACCELERATED 
 
 ultimate ratio. Therefore, fince the areas iCc t and 
 HD ^^ are equal, the rectangles kC c i and ITi dh are 
 ultimately in the ratio of equality. Therefore their bafes 
 i c and h d are inverfely as their altitudes C c and D dy 
 that is, I c \h d=-T) d\Q,c. But C c and D d being de- 
 fcribed in equal times, are ultimately as the velocities in 
 e and d (46). Therefore /' c and h d are inverfely as the 
 velocities in c and d. Becaufe this may be fimilarly de- 
 monftrated in refpecl of every point of the abfciiTa, the 
 propofition is demonftrated. 
 
 52. It now appears that in all cafes in u'hich we can 
 difcover by obfervation the relation between the ipaces 
 defcribed and the times elapfed during the defcription, 
 we difcover the velocities and the mechanical condition 
 of the moveable. To make any practical application of 
 our conclufions, we muft always have recourfe to arith- 
 metical calculations. Thefe are indicated by the alge- 
 braic fymbols of our geometrical reafonings. We repre- 
 fent any ordinate c g oi fig. 5. by v, and the portion cd 
 of the abfciffa by 't , and the area c dh g^ or rather, its e- 
 qual, the rectangle c d g, by -y / . And iince this redl- 
 angle is as the correfponding portion C D of the line of 
 motion, and C D is reprefented by j , v/e have die equa- 
 tion ]•=: V ], 
 
 We may now afTume as true, all the mathematical 
 confequences of thefe reprefentations. Therefore / =-, 
 as in art. 41. For the algebraic fymbols are the repre- 
 fentations of arithmetical operations, and they reprefent 
 
AND RCTARDUD MOTIONS. 57 
 
 the Operations of geometry more remotely, and only be- 
 caufe the area of a re(£langle Is analogous to the produ£l 
 of numbers which are proportioned to its fides. If we 
 life the fymbol j v t to reprefent the fum of all thefe 
 rectangles, it will exprefs the whole area adhe, and 
 will alfo exprefs the whole line of motion A D, and we 
 
 may ftate the equation s = y ^ t . In like manner J ~ 
 will be equivalent to f t, that is, to t, and will ex- 
 prefs the whole time ad. It is alfo eafy to fee that - 
 reprefents the ordinate D H of the line L K I H of fig. 6, 
 becaufe any portion D ^ of its abfcilTa is properly repre- 
 fented by j, and the ordinates are reciprocally propor- 
 tional to the velocities, that is, are proportional to the 
 quotients of fome conilant number divided by the veloci- 
 ties, and therefore, to -. Now i being reprefented by 
 
 the redtangie kQci, which is alfo reprefented by j x -, 
 
 v/e have / = — , and t = /— , as before. 
 
 Such fymbollcal reprefentations will frequently be 
 employed in our future difcuffions, and will enable us 
 greatly to fhorten our manner of proceeding. 
 
 53. There is one cafe of varied motion, which has 
 very particular and ufeful charaders, namely, when the 
 line efg h of fig. 5. is a ftraight line. Let fig. 7. re- 
 prefent this cafe of motion along the line A D, and let 
 ^a^ h c^ c d reprefent equal moments of time, in which 
 
3^ OP ACCELERATED 
 
 the moveable defcribes PA, B C, CD; draw /?;/, gn,es 
 parallel to the abfcifs a d. 
 
 It is evident that m g and n h are equal, or that equal 
 increments of velocity are acquired in equal times. AU 
 fo eq^er^e s are proportional to qf^ r g^ s hy and there- 
 fore the increments qf, rgy s h, of velocity, are propor- 
 tional to the times ab, a c, a d, in v/hich they are ac^ 
 quired. 
 
 This motion, may vi^ith great propriety be called UNi- 
 PORMLY ACCELERATED, in which the velocity increafe^ 
 at the fame rate wdth the times, and equal increments are 
 gained in equal times. 
 
 If the hne e h cut the abfciffa in fome point ^», it vv^ill 
 reprefent a motion uniformly accelerated from reft, dur^ 
 ing the time v d, and will give us the relations between 
 the fpaces, velpcliies and times in fuch motions. 
 
 Frpm this manner of expreffing thefe relations, It folr 
 lows that, z';; motioJis uniformly accelerated from a fate of 
 r^Jly the acquired vekc't'es are proportional to the times 
 from the begirifiing of the motion* For a <?, bf c gy d h, 
 ^eprefent the velocities acquired during the times v a, 
 V by V Cy V dy and are in the fame proportion v/ith thofs 
 lines. 
 
 54. Alfo, the momentary increments of 'uelocity are as 
 the moments in ivhich they are acquired ; or the increment f 
 of velocity are as the increments of time, 
 
 55. Alfo, the fpaces defcribed from the beginning of the 
 mqtion are as the fquares of the times. For the fpaces ale" 
 
 reprefent€4 
 
AND RETARDED MOTIONS. 3^ 
 
 r'eprefented by the triangles v a ey vbf, v c g, &c. and 
 tf a e -.v bf zz V a^ ',v b' &c. 
 
 Remark. 
 ITiis gives us the oftenfible character of an uniform- 
 ly accelerated motion. For all that we can immediately 
 obferve in a motion, is a fpace defcribed, and a time e- 
 lapfed. Velocity is not an obfervation, but the name of 
 an obferved relation between the increafe of the fpace 
 and that of the time. The fpace defcribed in the time 
 
 V b h obferved to be to that in the time t; ^, as v b^ to 
 
 V d\ We can reprefent the proportion of v h^ and v dr 
 by the triangles -u ^/ and vdhy which have the fams 
 proportion. We then fee that the points v,fy h are in 
 a ftraight line, and therefore bf and dh 2XQ 2,% vb and 
 
 V dy that is, when we obferve a motion fuch that the 
 fpaces defcribed are proportional to the fquares of the 
 times, we are certain that the velocities are as the times 
 from the beginning of the motion, and that the incre- 
 ments of velocity are as the increments of the times, and 
 therefore the motion is uniformly accelerated. 
 
 56. Alfo, the increments of the fpaces are as the incre- 
 merits of the fquares of the times (counted from the be- 
 ginning of the motion), that is, v bf—v a e :v d h — v c g 
 = V b^ — -v a^ : v J^— ^; r*. 
 
 57. Alfo, the fpaces defcribed from the beginning of the 
 motion are as the fquares of the acquired velocities. For 
 
 V a e \v bfzz a #' ; bf, 
 
 58- 
 
4d OF ACCELERATED. 
 
 58. Alfo, the momentary increments of the. /paces are ns 
 the momentary iticreme?its oj the. fq-uares of .the velocities. 
 Yoxbcgf'.cdhg^c g'-'—hf : d h'^c g' &c. This lail 
 is a corollary of frequent ufe, as it often happens that we 
 can only obfcrve momentary changes. 
 
 59. Alfo, the fpcice defcrihed during any portion of time, 
 hy a motion imiformly accelerated from rej}^ is one half of 
 the /pace uniformly defcribed in the fame time luith the final 
 velocity of the accelerated motion^ For the triangle v dh 
 meiafures the fpace defcribed in the time t* ^ by the ac- 
 celerated motion, and the recSlangle v d hH meafures the 
 fpace uniformly defcribed in the time v d with the velo- 
 city d h. 
 
 Here it is to be remarked, that eg h d is only half of 
 the difference between the rectangles v d hH and vcg G. 
 If we make dh ■=. v dy then v d h}rl and v c g G will be 
 the fquares of the velocities dh and eg. In this cafe, 
 /; hy the increment of velocity, is alfo equal to g «, and 
 dn X nh \sz=: c g X f^h. Employing v and v to exprefs 
 velocity and its momentary increment, vv will be the ex- 
 preffion of the re£langle eg X n h. Now 2 t^ -J is the 
 ufual expreffion of the increment of the fquare of velo- 
 city. As halves are proportional to their wholes, v v is 
 always proportional to 2 vv, and is generally ufed to ex- 
 prefs the variation of v^. But we muft keep in mind that 
 it is only the half of it. 
 
 60. And the fpace defcribed during any portion of the 
 iinie of the accelerated motion^ is equal to that ivhich would 
 
 h 
 
A"ND RETARDED MOTIONS- 4I 
 
 ,k' (lefcribed in the fame time ivhh the mean between the 
 lielociiies at the beginning and end of this portioji of time. 
 J-QXhdhf- bd X eg, 
 
 Thefe properties of uniformly accelerated motion will 
 be found of very great fervice in the inveftigation of all 
 other varied motions, particularly in cafes where an ap- 
 proximation is all that can be effected without very te- 
 dious and complicated procefies. 
 
 61. Acceleration may be confidered as a meafureabie 
 q-uantity. A ftone falling in the vertical line, much fooner 
 acquires a great velocity, than u-hen rolling down a flope, 
 and all are fendble that the acceleration is lefs as the de- 
 clivity is more gentle. 
 
 If we fuppofe the acceleration to be always llie fame, 
 the conception that we have of this conftancy is, furely, 
 that in equal times equal increments of velocity are ac- 
 quired j and, confequentiy, that the augmentations of ve- 
 locity are proportional to the times of acquiring them*r 
 This being fuppofcd, that acceleration muft furely be ac- 
 counted double or triple, &c. in which a double or triple 
 velocity is acquired ; and, in general, the augmentation 
 of velocity uniformly acquired in a given time, mull be 
 taken for the meafure of the acceleration, 
 
 62. Cor. Therefore accelerations are proportional to the 
 fpaces defcribcd in equal times with motions unifortnly ac' 
 celerated from a fuite of rejl^ (in v/hich the velocities gra- 
 dually increafe from nothing). For, in this cafe the fpaces 
 gre the halves of what would be uniformly defcribed in. 
 
 F XU 
 
^2 OF ACCELERATED 
 
 the fame time with the acquired iinal velocities, and are 
 therefore proportional to thefe velocities (31), that is, to 
 the accelerations, feeing that thefe velocities werr, uni- 
 formly acquired in equal times. 
 
 On the other hand;, tliat acceleration muft be reckon- 
 ed double or triple of another, in wlilch a given aug- 
 mentation of velocity is uniformly acquired in one half 
 or one third of the time. For, if a given augmentation 
 of velocity be acquired in half of the time, then, if the 
 fame acceleration be continued during the remaining half 
 of the given time, another equal augmentation v/ili be 
 acquired, the acceleration being conftant. The whole 
 augmentation acquired in the fame time will be double, 
 and therefore the acceleration is double. The fame thing 
 muft be granted for any other proportion. 
 
 63. Therefore, we muft fay that acceleraiio?is are pro^ 
 poriional to the increments of velocity uniformly acquired^ 
 dlreSllyf and to the times in luhich they are acquired ^ in^ 
 vcr/ely. 
 
 ■ Y V 
 A : fl rr — : - . 
 i t 
 
 Or, v/e may exprefs it by tlie proportional equation 
 
 . "J' 
 a -- ~ . 
 ' t 
 
 It is to be remarked here, that this relation betvv'een 
 
 the Acceleration, Velocity, and Time, is not confined to the 
 
 cafe of a motion pafhng through all degrees of velocity- 
 
 from nothing to tlie final magnitude v^ but is equally 
 
 true (in uniformly accelerated motio-ns) with rcfpect to. 
 
AND RETARDED MOTIONS. 43 
 
 ^uy momentary cliange of velocity. For, fince the ve- 
 locity iiicreaies at the lame rate vi^lth th^ time, we have 
 V '.d' — t \V {y and /' exprelFmg the fimultaneoiis incre- 
 
 xiients of velocity and time). Therefore the fymbols - 
 
 and -T- have the fame value, and therefore a ^ — , 
 t^ ' t' 
 
 ^4. On the othei* hand, fince the augmentation of 
 velocity is the meaf^re of the acceleration, and is there- 
 fore proportional to it, and nnce in uniformly accelerated 
 motions, the velocity increafes at the fame rate vi^ith the 
 times, it foliovv^s that the augmentations of velocity are 
 as the accelerations and as the times, jointly. This gives 
 the proportional equation v =^ a iy 
 and ''o' === a t'* 
 
 65. Since ail that we can obferve in a motion is a 
 fpace defcribed, and a time elapfed during the defcrip- 
 tion, it is defireable to have a meafure of acceleration 
 expreffed in thefe terms only. 
 
 This is eafily obtained. We have feen in art. 62. 
 that, when the velocity has uniformly increafed from 
 nothing, the fpaces defcribed in equal times are very 
 proper meafures of acceleration. And, in uniformly ac- 
 celerated motions, the fpaces are as the fquares of the 
 times (56). Therefore, v/hen the acceleration remains 
 
 the fame, the fraction — muil remain ef the fame v»lue, 
 
 ^nd a is proportional to — .. 
 
 F Q^ Therefore^ 
 
44 ^^ AC6EtTllLAT*Kti 
 
 Therefore, acceleratiofis are proportional to the Jpatcs 
 dcfcrihed ivith a motioti uniformly accelerated from reji^ 
 direcllji and to the fquares of the tunes y invcrfely. 
 
 66. Farther^ fince « = - (6a) we have a= — : but 
 ' ^ ' vt 
 
 ^j / == J-, therefore « = — , This gives us another mea- 
 
 fure of acceleration, viz. Accelerations are directly as the 
 fquares of the velocities^ and inverfely as the fpaces along 
 which the velocities are uniformly augmented. 
 
 67. On the other hand, fince, vi^hen the fpaces are 
 equal, we have a = v'^; and, in uniformly accelerated 
 motions, that is, when a remains conftant, if the fpace 
 is increafed in any proportion, v^ increafes in the fame 
 proportion ; it follows that v'' increafes in the propor- 
 tion, both of the acceleration and of the fpace. There- 
 fore we have, in general, ^'^ ^ a s. 
 
 Again (as in art. 64, 6^) we fliall have t>- = /? S, 
 and V^ — v^ == aS -— a s, or =^ S — j, which we may 
 exprcfs in this manner vv' =^ as'. That is, the morne?!- 
 iary change of the fquare of the velocity^ in a motion unU 
 formly accelerated^ is proportional to the acceleration and to 
 the fpace y jointly. This will be found a moft important 
 theorem. 
 
 Thus we fee that the acceleration conti-nued during ^ 
 given time /, or /', produces a certain augmentation of 
 the fimple velocity ; but the acceleration continued along' 
 a given fpace x, or 'S, produces a certain augmentation- 
 
AND RETARDED MOTIONS. 4^ 
 
 •f die fquare of the velocity. This obfcrvation will be 
 found of very great importance in niechanical philofophy, 
 
 68. Hitherto the acceleration has been confidered as 
 conllant — that is, we have been confidering only fuch mo- 
 tions as are uniformly accelerated j but thefe are very rare in 
 tlie phenomena of nature. Accelerations are as variable 
 as velocities, fo that it is equally difficult to find an ac- 
 tual meafure of them. 
 
 Yet it is only by changes of velocity that we get any 
 information of the changing caufe, or the mechanical 
 power of nature. It is only from the continual acce- 
 leration of a falling body, that we learn that the power 
 which makes it prefs on our hand, alfo prelles the body 
 downward, while it is falling through the air ; and it is 
 from our obferving that it acquires equal increments of 
 velocity in equal times, that we learn that the dov/nward 
 prefflire of gravity on it is the fame, whatever be \^:\q ra- 
 pidity of its defcent. No rapidity withdraws it in 'Cn^i 
 fmalleft degree from the action of its gravity or weight. 
 This is valuable information ; for it is very unlike all our 
 more familiar notions of preliures. We feel that all 
 fuch preffures as we em.plovj have their acceleratlnn- 
 power diminiflied as the body yields to them. A flream 
 of water or of wind becomes lefs and lefs efFeclive 
 as the impelled bodies move more rapidly away, and, 
 although they are Hill in the ilream, there is a limit- 
 ing velocity which they cannot pafs, nor ever fully at- 
 tain. It is of tl:ie greated confeq^uence therefore to ob- 
 
 triia 
 
'4^ O? ACCELERATED 
 
 tain acciirat:e mcafures cf acceleration^ even wlien con* 
 tinually varying. 
 
 We may obtain this in the very fame way that we 
 get meafures oi a velocity which varies continually. We 
 can conceive a line to increafe along with our velocity, 
 and to hicreafc precifely at the fame rate. It is evident 
 tliat this rate of increase of the velocity is the very thing 
 that we call Acceleration, juft as the rate at v/hich the 
 line now mentioned increafes is the very thing that we 
 call Velocity. We have only therefore to confider the 
 areas of fig. 5. or the line A D of that figure, as repre- 
 fenting a velocity ; then it is plain that the ordinates to 
 tlie line e g h, which we demonftrated to be proportional 
 to the rate of variation of this area, will reprefent, or be 
 proportional to the variation of this velocity, that is, to 
 the acceleration. Hence the following propofition. 
 
 69. If the ahfdjfa 7i A of a curve line e g h reprefent 
 the time of a motion^ mid if tho areas a b f e, a c g e, 
 a d h e, &c. are prcportioned to the velocities at the in- 
 Jlanis b, c, d, &c. then the ordinates a e, b f , eg, d h, 
 &c. are proportional to the accelerations at the infants 
 a, b, c, d, &c. 
 
 This is demonftrated precifely in the famiC manner as 
 m art. 45. and we need not repeat the procefs. We 
 have only to fubflitute the word acceleration for the word 
 velocity. 
 
 From this propofition, we may deduce fome corol- 
 laries which are of continual ufe m every mechanical 
 difcuiTion. 
 
 70. 
 
AND RETARDED MOTIONS. 47 
 
 70. The momentary 'vKrements of vehcity arc as the 
 (iccelcratlor.Sy and as the moment s^ jointly. 
 
 For, the inerement of velocity in the moment c d (for 
 example) is accurately reprefented by the area c dh ^y or 
 by the reclanclle c d nk ^ and c d accurately reprefents 
 the n^.oment. Alfo, the ultimate ratio of c k to fuch an- 
 other ordinate b /, is tlie ratio of eg to bf (45) ; that is, 
 the ratio of the acceleration in the indant c to the accele- 
 ration in the inftant b. Therefore the increment of velo- 
 city daring the moment pa is to that during the moment 
 c d ?iS p n X a e to c d X d g, — We may exprefs this by 
 the proportional equation v =^ a t. 
 
 71. Converfcly. The acceleration o is proportional 
 
 to - , agreeably to what v/as fhown v/hen the motion h 
 
 uniformly accelerated (63). 
 
 When, from the circum fiances of the cafe, we can 
 
 meafure the area of this figure, as it is analogous to the 
 
 fum of all the infcribed redangles, we may exprefs it by 
 
 J aty and thus we obtain .the whole velocity acquired 
 
 during the time A P, and we fay v ^ ja t. 
 
 It frequently happens that we know the Intenfities 
 (or at lead tlieir proportions) of the accelerating powers 
 cf nature in the diiFerent points of the path, and we want 
 to learn the velocities in thofe points. This is obtained 
 by means of the following propcfition : 
 
 72. If the abfnjfa AE cf a line ace (fig. 8.) he the 
 /pace along ivhich a body is moving ivilh a motion coniinuallf 
 
 varied^ 
 
4.8 OV ACCELERATED 
 
 n^/iried, and if the ordi?iates An, B ^, Cr, S:c. he propor- 
 tional to the accchrat'ions in the points A, ]>, C, &c. then, 
 the areas AV>ba, AD day A¥.ea, &c. are proportional 
 to the auginefiiatiom of the fqiiare of the velocity in A at the 
 points B, D, E, &c. 
 
 Let B C, CD, be two very fmall portions of the line 
 AE, and draw bf eg, panillel to AE. Then, if we 
 fuppofe that the acceleration B b continues through the 
 fpace BC, the rectangle BbfC will exprefs the aug.- 
 mentation made on the fquare of the velocity in B (67 ). In 
 like manner, C ^ ^ D will exprefs the increment of the 
 fquare of the velocity in C ; and, in like manner, the 
 rectangles infcribed in the remainder of the figure will 
 feverally exprefs the increments of the fquares of the 
 velocity acquired in ipoving over the correfponding por- 
 tions of the abfcifTa. The v/hole augmentation thercr 
 fore of the fquare of the velocity in A (if there be any 
 ■velocity in that point) during the pafiage from A to B, 
 is the aggregate of thefe partial augmentations. Tlie 
 fame muil be affirmed of the motion from B to E. 
 Nov/, vviien the fubdivifion of A E is carried on without 
 end, it is evident that the ultimate ratio of the area 
 J^JLe a to the aggregate of infcribed recStangles, is that 
 of equality ; that is, when the acceleration varies, not 
 by ftarts, but continually, the area AB b a will exprefs 
 the augmentation made on the fquare of the initial ve- 
 locity in A, during th.e motion along A B. The fame 
 muil be affirmed of the motion along B E. — Therefore 
 ffie intercepted areas A B b a, B,D d b, BEe d, are pro^ 
 
 portional 
 
AND RETARDED MOTIONS. 4f 
 
 portional to the changes made on the fquarcs of the ve- 
 locity in A, B, and D. 
 
 73. Cor. I. If the moveable had no velocity in A, 
 the areas AB^^, AD da, &c. are proportional to the 
 fquares of velocity acquired in B, D, &c. 
 
 74. Cor. 2. The momentary change on the fquare 
 of the velocity is as the acceleration and increment of 
 the fpace jointly, or, vi^e have vv=: as ', and thus we find 
 that virhat vi^e demonflrated ftridly in uniformly accele- 
 rated motions (67) is equally true when the acceleration 
 ■continually changes. 
 
 75. Cor. 3. Since v/e found vv equal to half th? 
 increment of the fquare of the velocity {^g), it follows 
 
 Tchat the area AlLea, or the fluent fas is only equal to 
 
 V*, v^ . . . • 
 
 , fuppofing V and V to be the velocities in A 
 
 and R. 
 
 76. All that has been faid of tlie acceleration of 
 motion is equally applicable to motions that are retarded, 
 whether uniformly or unequably ; the momentary varia- 
 tions being decrements of velocity inftead of increments. 
 A moveable, uniformly retarded till it is brought to reft, 
 will continue in motion during a time proportional to the 
 initial velocity ; and it v/ill defcribe a fpace proportional' 
 ip the fquare of this velocity ; and the fpace fo defcribed 
 
 G is 
 
lilD RETARDED MOTIONS. 4f 
 
 portional to the changes made on the fquarcs of the ve- 
 locity in A, B, and D. 
 
 73. Cor. r. If the moveable had no velocity in A, 
 the areas A B ^ ^, AD da^ &c. are proportional to the 
 iquares of velocity acquired in B, D, &c. 
 
 74. Cor, 2. The momentary change on the fquare 
 of the velocity is as the acceleration and increment of 
 the fpace jointly, or, we have vv=: as '^ and thus we find 
 that what we demonilrated ftridly in uniformly accele- 
 rated motions (67) is equally true when the acceleration 
 continually changes. 
 
 75. Cor, 3. Since we found vv equal to half th^ 
 increment of the fquare of the velocity (59), it follows 
 
 that the area A^ea, or the fluent fas is only equal to 
 
 V* ; v^ . . . • 
 
 , fuppofing V and V to be the velocities in A 
 
 and E. 
 
 76. All that has been faid of the acceleration of 
 motion is equally applicable to motions that are retarded, 
 whether uniformly or unequably ; the momentary varia- 
 tions being decrements of velocity infl^ad of increments. 
 A moveable, uniformly retarded till it is brought to reft, 
 will continue in motion during a time proportional to the 
 initial velocity ; and it v/ill defcribe a fpace proportional' 
 ift the fquare of this velocity *, and the fpace fo defcribed 
 
 G l^ 
 
$0 Of compound motions, 
 
 i? one half of what It would have defcribed in the fame 
 time with the initial velocity undiminiflied, Sec, Sec. &c. 
 Having nov/ obtained proper marks and meafures of 
 all variations of velocity, it remains to obtain the fame 
 for all changes of diredion. Thus we fhall obtain a 
 knowledge of the greateft part of thofe motions y^hich the 
 fpontaneous phenomena of nature exhibit to our view. 
 Jt is very doubtful whether we have ever feen a motiou 
 ftri£lly re£lilineal. 
 
 3. Of Compound Motions. 
 
 79. In our endeavours to obtain a general mark ov 
 <:hara6leriftic of any change of motion, it is evident that 
 when the change is fuppofed to be the fame in any two 
 or more inftances, the oftenfible marks muft be the fame, 
 whatever have been fehe previous conditions of the two. 
 moveables. There muft be obferved, in all the cafes of 
 change, fome circumftance in the difference between the 
 former motions and the new motions, which is precifely 
 the fame, both In refpe^l: of kind and of quantity, that is, 
 in refpeft of direction and of velocity. We may there- 
 fore fuppofe one of the bodies to have been previoufly 
 at reft. In this cafe, the whole change produced on it, 
 is unqueftionably the very motion which we fee it ac- 
 <juire, or the determination to this motion. 
 
 Therefore, In the firft place, a change cf motion is, 
 itfelf, a motion, or determination to motion. In the cafe 
 nc^^ mentioned, it is the new motion, and that only. 
 
 But 
 
OF COMPOUND MOTIONS. 5 J 
 
 But It is by no means the new motion in every other 
 cafe. For, if the previous condition of the body has 
 been different from that of a body at reft, and if thefame 
 change has been produced on it, the 72ew condition muft 
 alfo be different from the new condition of the other, 
 and therefore the tiew condition cannot be the change^ be- 
 caufe this is fuppofed to be the fame in both cafes. But, 
 farther, when the fame change is made in any previous 
 motion, we muft fee, in the difference between the for- 
 mer motion and the new motion, fomething that is equi- 
 valent to, or the fame with, this motion produced in the 
 body that was previoufly at reft, and which has received 
 the fame change. And alfo, the difference between the 
 new motions of thefe two bodies muft be fuch as fhall in- 
 dicate the difference between thefe previous conditions of 
 each. 
 
 AfFuming therefore as a principle, that the change of 
 motion is itfelf a motion, let us endeavour to find out a 
 motion, which alone fhall produce that difference from 
 the former motion which is really obferved in the new 
 motion, in all cafes whatever. This, undoubtedly, is the 
 proper mark and meafure of the change. 
 
 Something very analogous to theie indifpenfable con- 
 ditions may be obferved in the following motions. Sup- 
 pofe the ftraight line EI (fig. 9-) lying eaft and weft, 
 crofTed by the line E K from north to fouth. Let the 
 line EK (which we fuppofe to be material, fuch as a 
 rod or wire) be carried along the line EI in a minute, 
 keeping always parallel to its firft pofition, that is, al- 
 
 G 1 ways 
 
|i' OF coMPOiWD Morio:^?. 
 
 ways lying north and fouth. At the end of 20" It wiS 
 have the pofition Ggy, its end E having moved uniform- 
 Ij along y of E I ^ at the end of 40" it will have the po- 
 fition Mh xy ^ having defcribed y of E I ^ and at the 
 end of the" minute it will have the pofition I i. 
 
 In the mean time, let the line E I (alfo fuppofed ma- 
 terial) move uniformly from north to fouth, keeping al- 
 \vays parallel to its firft pofition EI. At the end of 
 20" it will have the pofition mgn, its end E having 
 moved along y of E K. At the eiid of 40'^ it has the 
 pofition o/jp, and E is y of E K. And at the end of 
 the minute, it has the pofition Ky;^?. 
 
 It is plain that the common interfe£lion of thefe twa 
 lines will always be found in the diagonal E i of the pa- 
 rallelogram E K n ; for Emg G is a parallelogram fi- 
 milar to E K i I, becaufe E G : E I = E /« : E K. In like 
 manner EohH Is a parallelogram fimilar to E K i I. Thefe 
 parallelograms are therefore about a common diameter 
 EL 
 
 Further, the motion of the point of interfecllon of 
 thefe lines is uniform j becaufe E G : E I == E^ : E /, and 
 EH;EI = ]E/6:Ei, &c. j and therefore the fpaces E^, 
 E y6, E i are proportional to the times. 
 
 And thus it appears that the Interfecllon of two lines, 
 each of which moves uniformly In the dlredlion of the 
 other, moves uniformly In the dIre£lion of the diagonal 
 of the parallelogram formed by the lines In their firft or 
 bft pofition, and that the velocity of the interfecllon is 
 to tlie velocity of each of the motions of the lines as tlie 
 
 diagpi^ 
 
OF COMPOUND MOTIONS, 5'^ 
 
 iiiagonal' Is to the fide in whofe diredlioii the motions are- 
 performed. 
 
 This motion of the interfedion may, with great pro- 
 priety of language, be faid to be conltituted by, or com- 
 pounded of, the two motions in the diredion of the fides. 
 For the point g of the Hne G y is, at the inflant, mov- 
 ing eaftward, and the fame point g of the hne m gn is 
 moving fouthward. Therefore, if the point g be confi- 
 dered as a point of both Hnes (as if it were a ring em- 
 bracing both) it partakes, in every inflant, of both mo- 
 tions. 
 
 It is alfo evident that the point g feparates from G 
 in the fame direction, and with the fame velocity, as if 
 E K had remained at reft, and the ring had moved to ;/; . 
 Alfo it feparates from the point o at the fame rate, and 
 in the fam.e direction as if it had moved from E to G. 
 The motion along E i therefore contains both of the mo- 
 tions along EI and along EK, and is really identical 
 with a motion comipounded of thofe motions, plainly in- 
 dicating both, or the determination to both. According- 
 ly, we fay that in every fituation of the point of inter- 
 fe<Sl:ion, its velocity is compounded of the velocity E I 
 and the velocity E K. If thei-efore E I has been a pre- 
 vious motion, that is, if a body v/as moving fo that, had 
 its motion continued unchanged, it would have defcribed 
 EI uniformly in a minute, but we obferve that after 
 coming to E, it turns afide, and defcrlbes E i uniformly 
 in a minute, we fliculd fay that the change which ic 
 faftains in the point E, is a motion E K. For, if the 
 
 body 
 
^4 OP COMPOUND MOTIONS. 
 
 body had been previoufly at reft in E, and we obferve it: 
 delcribe E K in a minute, then the motion E K is, un- 
 qiieftionably, the. change which it has fuftained. The 
 motion E i is not the change -, for had E L been the 
 primitive motion, the fame motion E i would have re- 
 fulted from compounding the motion E M with it. 
 Now, fmce E L is different from E I, it is impoffible that 
 the fame change can make the new conditions the fame. 
 
 Moreover, there is no other motion, which, by com- 
 pounding it with E I, will produce the motion E /. 
 
 And laftly, the motion E K is the only circumftance 
 of famenefs between changing the motion E I into the 
 motion E i, and giving the motion E K to a body pre- 
 vioufly at reft. 
 
 .After a mature confideration of all thefe conditions, 
 we may fay, that 
 
 J[ change of motion is that motion whichy by compofi^ 
 iion with the former ftate of motion^ produces the neiv 
 motion, 
 
 80. This compofition of motion is ufually prefenteA 
 to the mind in a way fomewhat different. A body is 
 fuppofed to move uniformly in the direction E I, while 
 the fpace in which this motion is performed is carried 
 uniformly in the direflion E K. But we cannot conceive 
 a portion of fpace to be moved out of its place. We 
 can conceive the compofition very diftindly by fuppof- 
 mri a man walking along a line E I drawn on a field of 
 Ice, while the ice is floating in the diredion E K. This 
 will produce the very motion E /, and affords the cleareft 
 
 notion 
 
OP COMPOUND MOTIONS. 55"" 
 
 DOtion of the compofition. If one man ftands ftlll, and 
 another walks in the direction and with the velocity E I, 
 and a third in the direction and with the velocity E L, 
 while the ice floats in the direction and with the vcio- 
 city E K, then the new condition of the firft man will 
 be the motion E K, that of the fecond will be E z, and 
 that of the third will be E O. There can be no doubt 
 of thefe three men having fuftained the very fame ch.ange 
 of motion. Now, the only circumftance of famenefs in 
 thefe three new conditions is the compofition of their 
 former condition with a motion E K. 
 
 The refleding reader will perceive, however, that 
 this way of illuftrating the fubjecfl:, by the motion oa 
 moving ice, is not precifely a compofition of two deter^ 
 rninations to motion. This is completed in the firft in- 
 ftant. As foon as the m^otion in the direction and vvath 
 the velocity E i begins, there is no need of further ex- 
 ertion ; the motion will continue, and E / will be de- 
 fcribed. But it ferves very well to exhibit to the mind 
 the mathematical compofition of tivo motio?iSy which is all 
 we want at.prefent. We have fhewn that, in the refult 
 of this combination, ail the charaderiftics of the two 
 determinations are tp be found, bccaufe the point of in, 
 terfe£lion, whether wc confider it as a material exift- 
 ence, or as a mere mathematical conception, partakes o£ 
 both motions. There is a phyfical queftion which will- 
 come under confideration afterwards, that is very differ^ 
 ent from the prefent, namely, Whether two natural 
 powers, which are known to be produdive, feparately, 
 
 of 
 
56" or COMPOUND MOTIONS. 
 
 of two determinations of a body to two dlfliin^l motions, 
 will, by their joint action, produce a determination to 
 that motion which is compounded of thofe which they 
 would produce feparatcly ? — This is a queftion of very 
 difficult folution ♦, but we trufl that the notions already 
 acquired will enable us to give an anfwer with confidence. 
 
 8 1 . Thus then have we obtained a general mark or 
 charafteriftic of a change of motion, perfectly confonant 
 with our mark and meafure of every moving caufe, 
 jiamely, the very motion which we conceive it to pro- 
 duce. Nay, perhaps what we have juft now eftablifhed 
 is the foundation even of our former meafures. For 
 every acceleration, or retardation, or deflection, may be 
 confidered as a new motion, compounded with the for- 
 mer. This is not a mere fubftitution, to aid the imagi- 
 nation ; for it is, almoft always, the very fa6l. For what 
 we take for the beginning of motion, in all our actions 
 on bodies, and all our obfervations of the bodies which 
 furround us, is in fa£\: only a change induced on a mo- 
 tion already exifting, and exceedingly rapid. This re- 
 fults from the motion of rotation, by which we are car- 
 ried round the axis of the earth, and even this is com- 
 pounded with the motion of revolution round the fun, 
 What we confider as changes of motion, and therefore 
 as the proper meafures and marks of the changing caufes, 
 the powers of mechanical nature, are indeed changes, 
 and the very changes that we imagine. But they are 
 fey no means changes of the motions that we imagine. 
 
 We 
 
OF COMPOUND MOTIONS. ^^ 
 
 Vfe fliall foon learn, tliat if \vc meafure or eftlmate the 
 changes of motion in the way now propofed, all our de- 
 dudlions will be perfectly conformable to the appear* 
 ances of nature, and the inferences of their caufes per- 
 fectly confiftent and legitimate, giving us accurate know- 
 ledge of thofe caufes. And we {hall find that m other 
 way of ejlimating and meafurhig the chmiges of motion will 
 have thefe qualities. Thus we demonftrate the juflnefs 
 of our principle, and that it gives a fufficient ground for 
 mechanical fcience. 
 
 82* Since the actual compofition of motion is fo 
 general in the phenomena of the univerfe, that it obtains 
 in every motion and change of motion that we can pro- 
 duce or obferve, and fince the chara£l:eriftic which we 
 have afmmed of a change of motion is the fame, what-^ 
 ever the previous motion may have been, and therefore 
 is equally appHcable to motions Vv^hich are really fimple, 
 and fuch as we obferve them, it is plain that a know- 
 ledge of the general refults of this compofition of mo- 
 tion muft greatly promote our knowledge of mechanical 
 nature. We ihall therefore conlider them* in order. 
 
 83. The general theorem., to which all others may 
 be reduced, is the following. 
 
 Tiuo uniform motions^ having the direciions and velo- 
 cities reprejented by the fides EI, E K, of a parallelograniy 
 compofe a imiform motion in tie diagonal. This is already 
 iJeinonilrated, For the motion of the poinf of interfec- 
 
 H tio^ 
 
-g OF COMPOUND MOTIONS- 
 
 tion of tliefe two lines, while each moves uniformly in 
 all its points, in the direftion of the other, is, in every 
 inftant, compofed of thefe two motions, and is the fame 
 as if a point defcribed E I uniformly, while E I is uni- 
 formly carried in the dire£lion EK. And this motion ^ 
 is along the diagonal E z, and is uniform, as has been al- 
 ready fhewn. Alfo, becaufa E I and E i are defcribed 
 in the fame time, the velocities of the motions along 
 E I, E K, and E /, are proportional to thofe lines. 
 
 84. Cor. I. The compound motion Ez is in the 
 fame plane with the two constituent or stmfle mo- 
 tions E I and E K. For a parallelogram lies all in one 
 plane. 
 
 85. Cor. 2. The motion E / may arife from the 
 com.pofition of any two uniform motions, which have the 
 diredtion and velocities reprefented by the fides of any 
 parallelogram E L i M, or EI/ K, of which E / is the 
 diagonal. 
 
 Cafes frequently occur, where we know the direc- 
 tions of the two fimple motions which compofe an ob- 
 ferved motion, but do not know the proportion of their 
 velocities. The velocity is afcertained by this prcpofi- 
 tion, bccaufe the direction of the three motions, viz. the 
 two fimple and the compound motions, determines the 
 fpecies of parallelogram, and the ratio of the fides. 
 
 Sometimes we have tlie direction and the velocity of 
 one of the fimpb motions, and therefore its proportion 
 
OF COMPOUND MOTIONS. 1^^ 
 
 to that of the obierved compound motion. The direc- 
 tion and velocity of the other is alfo found by this pro- 
 pofition, becaufe thefe data alfo determine the paral- 
 lelogram. 
 
 The motion in the diagonal is evidently equivalent to 
 the motions in the fides combined. Thus, if the move- 
 able firft defcribe E I, and then I i (or E K), it will be 
 in the fame point i as if it had defcribed E /. There- 
 fore E / is frequently called the EquivALENT motion, thef 
 RESULTING motion. 
 
 It frequently gives great affiftance in our inveftiga- 
 tions, if we fubftitute for an obferved motion fuch mo- 
 tions as will produce it by compofition. This is called 
 the RESOLUTION OF MOTIONS. It is in this way that 
 the navigator generally computes the fhip's change of 
 fituation at the end of a day, in which Ihe has perhaps 
 failed in many different courfes. He confiders how 
 much he has gone to the eaflv/ard, or weftward, and 
 how much to the northward or fouthward, on each 
 courfe •, and he then adds together all his eaftings, and 
 all his fouthings, and then fuppofes that the fliip has 
 failed for the whole day on that unvaried courfe which 
 would be produced by the fame eafting and fouthing 
 combined. 
 
 In like manner, it is very uieful for the mechanician 
 to confider how much his obferved motion has advanced 
 the body in fome particular dired:ion, E F, for examples 
 (fig. lo). To do this, he confiders the motion AB as 
 compcfed of a motion A C parallel to the given line E F, 
 
 H 3 and 
 
00 01* COMPOUND MOTIONS. 
 
 and another motion A D perpendicular to E F, A B form- 
 ing the diagonal of a parallelogram A C B D, of which 
 one fide A C is parallel, and the other A D is perpendi- 
 cular to E F. It is plain that the motion A D neither 
 promotes nor obftrufts the progrefs in the dire6lion E F, 
 and that the body has advanced in the direction of E F, 
 juft as much as if it had moved from a to b, inftead of 
 moving from A to B. 
 
 This proceeding is called estimating a motion in a. 
 given direftion, or reducing it to that dire6lion. 
 
 In like manner, the mechanician h faid to eftimate 
 a motion A B (fig. ii.) in a given plane E F G H, when 
 he confiders it as compofed of a motion A D perpendi- 
 cular to that plane, and A C parallel to it. The lines 
 D A, B C being drawn perpendicular to the plane, cut 
 it in two points a and <^, and A C is parallel to a b. 
 
 86. Any number of motions A B, AC, AD, A E 
 (fig. 12.) may be thus compounded, forming a motion 
 A F. The method for afcertaining the motion refulting 
 from this compofition is as follows. A B, compounded 
 with A C, produces the motion A G. This, compound- 
 ed with A D, produces A H j and this, compounded 
 with A E, produces A F. 
 
 The fame final fituation F will be found by fuppofing 
 all the motions A B, AC, AD, A E, to be performed 
 In fucceffion. Thus the moveable defcribes A B ; then 
 B G, equal and parallel to A C ; then G H, equal and 
 parallel to AD; and then, HF, equal and parallel to AE. 
 
 Note. 
 
OF COMPOUND MOTIONS. 6t 
 
 A 
 
 Note. — It Is not neceflary that all thcfe motions b(? 
 In one plane. 
 
 87. Three motions A B, A C, A D (fig. 13.) which 
 have the direclion and proportions of the fides of a pa- 
 ralleloplped, compofe a motion in the diagonal A F of 
 that parallelopiped ; for A B and A C compofe A E, and 
 A E and A D compofe A F. 
 
 The mine-furveyor proceeds in this way. Like the 
 navigator, he fets down any gallery of the mine, not di- 
 rectly by its real pofition, but enters his table with its 
 calling or wefting, and with its northing or fouthing. 
 But he alfo keeps an account of its rife or dip. He re- 
 fers all his meafures to three lines, one running eafl ami 
 weft, one running north and fouth, and one running per- 
 pendicularly up and down. Thefe three lines are evi- 
 dently like the three angular boundaries A B, A C and 
 A D of a redlangular box. 
 
 This is now the conftant procedure of the mecha- 
 nician, in his more elaborate inveftigations. It was firft 
 praClifed (we think) by M*Laurin, in the excellent phy- 
 fico-mathematlcal fpeculations which are to be found In 
 his Treatlfe on Fluxions. The mechanician refers all mo- 
 tions to three co-ordinate lines A B, AC, AD, which 
 are perpendicular to each other, and his ultimate refult 
 is the diagonal A F of fome parallelopiped. 
 
 88. Hitherto we have confidered the compofitlon of 
 uniform ^notions only. But any motions may be com- 
 pounded. 
 
6Z OF COMPOUND MOTIONS. 
 
 pounded, as we may eafily conceive, by luppofiiig a man 
 to walk on a field of ice along any crooked path, while 
 the ice floats down a crooked ftream. 
 
 Thus, a uniform motion in the direclion AB (fig. 14.) 
 may be compounded with a uniformly accelerated mo- 
 tion in the diredion AC. Such a motion is obferved 
 when we fee a ftone fall from the mail-head of a fhip 
 failing fteadily forward in the direftion A B ; for this 
 ftone will be obferved to fall down parallel to a plummet 
 hung from the maft-head. The real motion of the ftone 
 will therefore be a parabolic arch A l^fgy which A B 
 touches in A ; for while the maft-head defcribes the e- 
 qual lines A B, B F, F G, the ftone has fallen to /S and (f> 
 and y, and the line A C A' has got into the pofitions 
 BB', FF', GG', fo that A (p is four times A/3; and 
 A y is nine times A /S. Therefore A /3, A ^, A y, are 
 as the fquares of /3 b, cpfj and ygy and the line A bfg is 
 a parabola. 
 
 It is in this way that a nail in the fole of a cart- 
 wheel defcribes a cycloid, while the cart moves along a 
 fmooth plane. This is the compofition of a pros^relTive 
 motion with an equal circular motion. The geometrical 
 lectures of Dr Barrow contain many beautiful examples 
 of fuch compofitions of motion ; and it was by intro- 
 ducing this procefs into mathematical reafoning, that this 
 celebrated geometer gave a new department to the fcience, 
 which quickly extended it far beyond the pale of the 
 ancient geometry of the Greeks, and fuggefted to Sir 
 Ifaac Newton his dcdrine of Fluxions. 
 
 89. 
 
OF COMPOUND MOTlONfS. ^53 
 
 89. When two motions, however variahle, are com- 
 pounded, we dlfcover the direclion and velocity of the 
 compound motions in any inftant, if we know the direc- 
 tion and velocities of each of the fimple motions at that 
 hi Q ant. For v.^e may fuppofe, that, at that inflant, each 
 motion proceeds unchanged. Then we conftrudl a pa- 
 rallelogram, the fides of which have the directions and 
 proportions of the velocities of the fimple motions. The 
 diagonal of this parallelogram will exprefs the direftion 
 and velocity of the compound motion, 
 
 90. On the other hand, knowing the direction and 
 velocity of the compound motion, and the dire^ions of 
 each of the fimple motions, we difcover their velocities. 
 
 91. When a curvllineal motion AD V (fig. 15.) re- 
 fults from the com.pofition of tv/o motions, whofe direc- 
 tions we know to be AC and A F, we learn the veloci- 
 ties of the three motions in any point D, by drawing 
 the tangent D I, and the ordinate D h parallel to one of 
 the fimple motions, and from any point L in that ordi- 
 nate, drawing LI parallel to the other motion, cutting 
 the tangent in I. The three velocities are in the propor- 
 tion of the three lines IL, LD, and ID. This is of 
 very frequent ufe. 
 
 Since the phenomena are our only marks and mea- 
 fares of their fuppofed caufes, it is plain that every mif- 
 take with refpe6l to a change of motion, is accompanied 
 by a miflake in pur iniference of its caufe. Such mif- 
 
 takes 
 
6'4 OF COMPOUND MOTIONS. 
 
 takes are avoided with great difficulty, becaufe the mo- 
 tions "which we obferve are, at all times, extremely dif- 
 ferent from what we take them to be. A book lying on 
 the table feems to be at reft ; but it is really moving 
 with a prodigious fpeed, and is defcribing a figure very 
 like the figure defcribed by a nail in the nave of a coach- 
 wheel while the carriage is going over the fummit of a 
 gentle rifing. We imagine that we are at reft, and we 
 judge of the motion of another body merely by its change 
 of diftance and diredtion from ourfelves. 
 
 Thus, if a (liip is becalmed at B (fig. i6.) in a part 
 of the ocean where there is an unknown current in the 
 direction B D ; and if the light of another fhip is feen 
 at A, and if A really fails to C while B floats to D, 
 A will not appear to have failed along A C, but along 
 A K ", for when B is at D, and A at C, A appears at C, 
 having the bearing and diftance D C. Therefore, if A K 
 be made equal and parallel to D C, it will have the fame 
 bearing by the compafs, and the fame diftance from B 
 that C has from D ; and therefore the fpeclator in B, 
 not knowing that he has moved from B to D, but be- 
 lieving himfelf ftill at B, muft form this opinion of tlie 
 motion of A. — In the fiime manner it muft follow, that 
 our notions of the planetary motions muft be extremely 
 diftcrent from the motions themfelves, if it be true that 
 this earth is moving to the eaftward at the rate of nearly 
 twenty miles in every fecond. It would feem a defpe- 
 rate attempt therefore for us to fpeculate concerning the 
 powers of nature by which thefe motions are regulated. 
 
 And, 
 
OF COMPOUND MOT^ON^* <^ 
 
 ^nd, accordingly, nothing can be conceived more fan- 
 taftical and incongruous than the opinions formerly en- 
 tertained on this fubje£l. But Mathematics affords ^t. 
 clue by which we are conducted through this labyrinth. 
 
 92. l^he motion of a hcdy A -relative to^ or as fee/i. 
 frotUy another body B, which is alfo in motion, is compound^ 
 ed of the real motion of A, and the oppoftte to the real mo- 
 tion of "&. (Fig. 16.) 
 
 Join A B, and draw A E equal and parallel to S P, 
 and complete the parallelogram ACFE, and join ED 
 and D C. Alfo produce E A till A L is equal to A E 
 or B D, and complete the parallelogram LACK, and 
 draw A K and B K. Had A moved along A E while B 
 moves along BD, they wouW have been at E and D nt 
 the fame time, and would have the fame bearing and 
 diftance as before. If the fpedlator in B is infenfible of 
 his own motion, A will appear not to have changed its 
 place. It is well known that tv/o {hips, becalmed in an 
 unknown current, appear to the crews to remain at reft. 
 It is plain, therefore, that the real pofition and diftance 
 D C are the fame with B K, and that if the fpe£l:ator iu 
 B imagines himfelf at reft, the line A K will be confi- 
 dered as the motion of A. This is evidently compofed 
 of the motion A C, which is the real motion of A, and 
 the motion A L, which is equal and oppofite to the mo- 
 tion B D. 
 
 ^3 . In like manner, if B H be drawn equal and o^" 
 piofite to A C, and the parallelogram B H G D be com* 
 
 I pleted. 
 
6"(5 Otf COMPOUKD MOTIONS. 
 
 pleted, and BG and A G be drawn, the diagonal B G 
 will be the motion of B relative to A. (92.) Now, It is 
 plain that K A G B is a parallelogram. The relative po- 
 fition and diftances of A and B at the end of the motion 
 are the fame as in the former cafe. B appears to have 
 moved along B G, which is equal and oppofite to A K. 
 Therefore, the apparent or relative motions of two bodies 
 are equal and oppofite, whatever the real motions of both 
 way bey and therefore give no immediate information 
 concerning the real motions, 
 
 94. It needs no farther difcufiion to prove the fame 
 propofitions concerning every change of motion, viz. that 
 the relative change of motion in A is compofed of the 
 real change in A, and of the oppofite to the motion, or- 
 change of motion in B. 
 
 Suppofe the motion B D to be changed Into*-- B ^. 
 This has arifen from a compofition of the motion B D 
 with another D ^ •, draw C » equal and oppofite to D ^, 
 and complete the parallelogram E C » «. The diagonal 
 E X is the apparent or relative change of motion. For 
 the bearing and diftance ^ C is evidently the fame with 
 D X, becaufe the lines ^ C and D « which join equal an4 
 parallel lines are equal and parallel. 
 
 ^^, Therefore, if no change happen to A, but if 
 
 the motion of B be changed, the iTxOtion of A will appear 
 
 to be equally changed in the oppofite diredlion. 
 
 • Hence we draw a very fortunate conclufion, that the 
 
 pbferved or relative changes of motion are equal to the 
 
 real 
 
(5F COMPOUND MOTIONS* 6y 
 
 veal changes. But we remain ignorant of its diredlion, 
 becaufe we may not know in which body the change 
 has happened. E s is the apparent cJiange of motion of 
 the body A, becaufe E C was the apparent motion be- 
 fore the change into E k. Complete the parallelogram 
 AC K ec. The diagonal A k would have been the mo- 
 tion of A, had its motion A C fuftained the compofition 
 or change A tx,. It is plain that either the motion D ^, 
 compounded with B D, or the motion A oe. compounded 
 with AC, will produce the fame apparent or relative change 
 of motion. Still, however, it is important to learn that 
 the apparent and real changes are the fame in magni- 
 tude ; becaufe they give the fame indication of the mag- 
 nitude of the changing caufe. 
 
 96. It is evident that if we knov/ the real motion of 
 B, we can difcover the real motion of A, by confidering 
 its apparent motion E C as the diagonal of a parallelogram 
 of which one fide E A is equal and oppofite to the known 
 motion B D. It mufh tlierefore be A C. 
 
 97. In like manner, if any other circumftances have 
 afiured the fpedtator in B, that A C is the true motion 
 of A, which had appeared to him to move along A K, 
 he muft confider A K as the diagonal of a parallelogram 
 A L K C, and then he learns that B has moved over a 
 line BD, equal and oppofite to AL. It was in this 
 manner that Kepler, by obfervations on the planet Mars, 
 difcovered the true form of the earth's orbit round the 
 Sun. 
 
0i 
 
 OF CURVILINEAL MOTIOxNb^^ 
 
 98. If equal and parallel motions be compounded 
 with all and each of the motions of any number of bo- 
 dies, moving in any manner of v/ay, their relative mo- 
 tions are not changed by this fupenndu6lion. For, by 
 Compounding it with the motion of any one of the bo- 
 dies, which we may call A, the real motion of A is in- 
 deed changed. But its motion relative to another body 
 B, or its apparent motion as (een from B, is compound- 
 ed of the real change (94. )> and of the oppofite to the 
 real change in B, that is, oppofite to the real change in 
 A, and therefore deftroys that change, and the relative 
 motion of A remains the fame as before. — In this man- 
 ner, the motions and evolutions of a fleet of (hips in a 
 current which equally afFefts them all, are not changed, 
 or are the fame as if made in ftill water. The motions 
 m the cabin of a fhip are not afFe6led by the (hip's pro- 
 greffive motion 5 nor are the relative motions on the fur- 
 face of this globe fenfibly afFe£\ed by Its revolution 
 round the fun. We (hould remain for ever igix>rant of 
 all fuch common motions, if we did not fee other bo- 
 dies which are not aiFe£led by them. To thefe we re- 
 fer, as to fo many fixed points, 
 
 4. Of Motions continually DeJieBed. 
 
 99. A curvilineal motion is a cafe of continual de- 
 fle£lion. It is fufceptible of infinite varieties, and its 
 modifications and chief properties are of diflicult inVef* 
 tigatiou. 
 
 The 
 
OF CURVILINEAL MOTIOKS. <5g 
 
 The fimpleft cafe of curvilineal motion is that of 
 Uniform, motion in a circular arch. Here, the deflediona 
 in equal times from re£lilineal motion are equal. But, 
 fhould the velocity be augmented, it is plain that the 
 momentary deflection is alfo augmented, becaufc a greater 
 arch will be defcribed, and the end of this greater arch 
 deviates farther from the tangent •, but it is not eafy to 
 afcertain in what proportion it is increafed. When one 
 uniform rectilineal motion AB (fig. 17.) is defleCled in- 
 to another BC, we afcertain the linear defleCtion by 
 drawing a line from the point c, at which the body would 
 have arrived without deflexion, to the point C, to which 
 it really does arrive. And it is the fame thing whether 
 we draw dD, or c C, in this manner, becaufe thefe lines, 
 being proportional to B J, B c, will always give the fame 
 meafure of the velocities (41.), and the lines of deflec- 
 tion are all parallel, and therefore aflure us of the di- 
 redlion of the defleCtion in the point B. But it Is other- 
 wife in any curvilineal motion. We never have cIJ^icC 
 =r B ^ : B r ; moreover, it Is very rarely that ^ D, c C, 
 &c. are parallel. We know not therefore which of thefe 
 lines to feleCt for an Indication of the dIreClion of the 
 defleClIon at B_, or for a meafure of its magnitude. 
 
 Not only does a greater velocity In the fame curve 
 caufe a greater deflection, but alfo, If the path be more 
 incurvated, an arch of the fame length defcribed with 
 the fame velocity, deviates farther from the tangent. 
 Therefore, if a body move uniformly in a curve of va- 
 riable curvature, the defledlion will be greater wnere 
 ihe curvature is greatier. 
 
70 OF CURVILINEAL MOTIONS. 
 
 We may learn from thefe general remarks, that iha 
 dire(5lions and the meafures of the clefle6l:ions by which 
 a body deviates continually into a curvilineal path, can 
 be afcertained, only by inveftigating the ultimate pofi- 
 tions and ratios of the lines which join the points of 
 the curve with the fmiultaneous points of the tan- 
 gent, as the points ^ and C are taken nearer and nearer 
 to B. Some rare, but important cafes occur, in which 
 the lines joining the fimultaneous points c and C, d and ^, 
 &c. are parallel. In fuch cafes, the deflexion in B 
 is certainly parallel to them, and they are cafes of the 
 compofition of a motion in the direction of the tan- 
 gent with a motion in the direction of the lines c C, d ^, 
 &c. But, in moil cafes, we mull difcover the direction 
 of the deflection in B, by obferving what dire£lion the 
 lines d ^, c Cy &c. taken on both fides of B, continually 
 approximate to. The following general propofition, dif- 
 covered by the illufhrious Newton, will greatly facilitate 
 this refearch. 
 
 100. If a body defcribe a curve line ABCDEF 
 (fig. 18.) ivhich is all in one plane y and if there he a point 
 S /// this plafiCy Jo fituatedy that the lines S A, SB, S C, 
 &c. drawn to the curve, cut off areas A S B, A SC, A SD, 
 &c. proportional to the times of defer ibing the arches A B, 
 AC, AD, &c. then are the defleBions always direEled 
 to this point S. 
 
 Let us firft fuppofe that the body defcribes the poly- 
 gon ABCDEF, formed of the chords of this curve, 
 and tliat it defcribes each cliord uniformly, and is de- 
 
 fleaecl 
 
OF CURVILINEAL MOTIONS. ^t 
 
 fleacd only in the angles B, C, D, &c. Let us alfo (for 
 the greater fimplicitv of argument) fuppofe that the fides 
 of this polygon are dcfcribed in equal times, fo that (by 
 the hypothecs) the triangles A SB, BSC, CSD, &c. 
 are all equal. 
 
 Continue the chords A B, B C, &c. beyond the 
 arches, making B c equal to A B, and C d equal to B C, 
 and fo on. Join r C, ^ D, &c. and draw <; S, ^ S, &c. 5 
 alfo draw C b parallel to <: B or B A, cutting B S in ^, 
 and join h A, and draw C A, cutting B ^ in c^. Laftly, 
 make a fimilar conftruclion at E. 
 
 Then, becaufe r B is equal to B A, the triangles 
 A S B and B S ^, are equal, and therefore B S ^ is equal 
 to B S C ; but they are on the fame bafe S B. There- 
 fore they are between the fame parallels •, that is, r C is 
 parallel to B S, and B C is the diagonal of a parallelo- 
 gram B ^ C r. The motion B C therefore is compounded 
 of the motions B c and B hy and B ^ is the deflection, by 
 which the motion B r is changed into the motion B C ; 
 therefore the deflection in B is dire<£l:ed to S. — By fimi- 
 lar reafoning / F, or E i, is the defledtion at E, and is 
 likewife dire£led to S *, and the fame may be proved con- 
 cerning every angle of the polygon. 
 
 Let tlie fides of this polygon be diminifhed, and their 
 number increafed without end. The demonftration re- 
 mains the fame, and continues, when the polygon ex- 
 haufts or coalefces with the curvihneal area, and its fides 
 with the curvilineal arch. 
 
 Now, when the whole areas are proportional to the 
 tiiiies, equal areas are defcribed in equal times ; and 
 
 thereforej 
 
^t OF CURVILIJ^EAL MOTIONS, 
 
 therefore, in fuch motion, the defleclions are always dU 
 rented to S. 
 
 This point S may be called the centre of dejleSlion. 
 
 I o I . If the dejleEllon by luhich a curve line A D F is 
 defcribed, be continually direEled to a fixed pointy the figure 
 njuill be in one plans ^ and areas will be defcribed round that 
 point proportional to th times. For B C is the diagonal 
 of a parallelogram, and is in the plane of S B and B c 
 (84.) j and r C is parallel to B S, and the triangles SBC, 
 S B o and S B A, are equal. Equal areas are defipribed 
 in equal times ; and therefore areas are defcribed pra^ 
 portional to the times, ^g. &c. 
 
 102. Cor, I. The velocities in different points of the 
 eurve are inverfely proportional to the perpendiculars S r 
 and S t (fig. 19.) drawn from S on the tangents A r, E t 
 in thofe points of the curve. For, becaufe the elementary 
 triangles A S B, E S F, are eqiial, their bafes A B, E F, 
 are inverfely as their altitudes $ r^ St. Thefe bafes, 
 being defcribed in equal times, are as the velocities, ^nd 
 they ultimately coincide vi^ith the tangents at A and E. 
 Therefore the velocity in A is to that in E as S ^ to S r* 
 
 T03. Cor. 2, 7'he angular velocities round S are in- 
 'perfely as the fquares of the difiances. For, if we defcribe 
 round the centre S the fmall arches B cf, F ^, they may 
 be confidered as perpendiculars on S A and S E ; alfo 
 with the diflance S F defcribe the arch g h^ It is evident 
 
 that 
 
^^' 
 

OF CURVILINEAL MOTIONS. 73 
 
 that g /: IS to Y d as the angle A S B to the angle E S F. 
 
 Now, fmce the areas A S B, E S F, are equal, we have 
 
 B ^, : F ^ =r S E : S A. 
 
 But g /j :B cc = S E : S A 
 
 therefore ^ y6 : F ^ = S E^ : S A* 
 
 and ASB:ESF_- SE^:S A* 
 
 104. AVe now proceed to determine the magnitude 
 of the defle£tion, or, at leaf!:, to compare its magnitude 
 in B, for example, with its magnitude in E. .In the po- 
 lygonal motion (fig. 18.) the defleftion in B is to that 
 in E as the line B <!? to the line E z ; for B ^ and E i are 
 the motions, which, by compofition with the motions 
 B c and 'Efy make the body defcribe B C and E F. 
 Therefore, when the fides of the polygon are diminiflied 
 without end, the ultimate ratio of B ^ to E f is the ratio 
 of the deflection at B to the defledion at E. 
 
 In order to obtain a convenient expreihon of this ul- 
 timate ratio, let A B C X Z Y be a circle pafling through 
 the points A, B, C, and draw B S Z through the point S, 
 and draw C Z, A Z. 
 
 The triangles B C ^ and A Z C are fimilar ; for C ^ 
 was drawn parallel to r B or B A. Therefore the angle 
 C ^ B is equal to the ultimate angle ^ B A or Z B A, 
 which is equal to the angle Z C A, being fubtended by 
 the fame chord Z A ; alfo C B ^, or C B Z, is equal to 
 C A Z, (landing on the fame chord C Z. Therefore, the 
 remaining angles ^ C B and C Z A are equal, and the 
 triangles are fimilar j therefore B^:CA=:BC:AZ. 
 
 K Now, 
 
OF CURVILINEAL MOTIONS. 73 
 
 that ^ Z? Is to F ^ as the angle A S B to the angle E S F. 
 
 Now, fmce the areas A S B, E S F, are equal, we have 
 
 B «: : F ^ = S E : S A. 
 
 But o^ /; : B « = S E : S A 
 
 therefore ^ y?^ : F ^ = S E^ : S A* 
 
 and ASB:ESFr= SE^:S A* 
 
 104. We now proceed to determine the magnitude 
 of the defledtion, or, at leaf!:, to compare its magnitude 
 in B, for example, with its magnitude in E. In the po- 
 lygonal motion (fig. i8.) the deflexion in B is to that 
 in E as the line B <^ to the line E i ; for B b and E i are 
 the motions, which, by compofition with the motions 
 B c and Ey^ make the body defcribe B C and E F. 
 Therefore, when the fides of the polygon are diminiflied 
 without end, the ultimate ratio of B Z* to E i is the ratio 
 of the deflection at B to the defleclion at E. 
 
 In order to obtain a convenient expreflion of this ul- 
 timate ratio, let A B C X Z Y be a circle pafling through 
 the points A, B, C, and draw B S Z through the point S, 
 and draw C Z, A Z. 
 
 The triangles B C ^ and A Z C are fimilar •, for C b 
 was drawn parallel to r B or B A. Therefore the angle 
 C ^ B is equal to the ultimate angle ^ B A or Z B A, 
 which is equal to the angle Z C A, being fubtended by 
 the fame chord Z A ; alfo C B />, or C B Z, is equal to 
 C A Z, {landing on the fame chord C Z. Therefore, the 
 remaining angles ^ C B and C Z A are equal, and the 
 triangles a.re fimilar ; therefore B^:CA:=BC:AZ. 
 
 K Now, 
 
f4 0^ CURVILINEAL MOTIONS. 
 
 Now, fince, by continually diminifhing the fides of 
 the polygon, the points A and C continually approach to 
 B, and C A continually approaches to ^ A or to 2 r B, 
 or 2 C B, and is ultimately equal to it ; alfo A Z is ulti- 
 mately equal to B Z. Therefore, ultimately, B 5 : 2 B C 
 
 = B C : B Z, and B ^ X B Z = 2 B C% and B Z' = -^-~- 
 
 In like manner, at the point E, v/e fliall have E / 
 
 2 E F ' 
 ultimately equal to —^ — , Ez being that chord of the 
 
 circle through D, E, and F, which pafies through /'. 
 
 Iherefore B ^ : il ; zz _, ^ : — ^=; — . 
 n Zj E 2 
 
 The ultimate circle, when the three points A, B, C, 
 coalefce, is called the circle of equal curvature, or 
 the EQUicuRVE CIRCLE, coalefcing with the curve in B 
 in the mod clofe manner. The chord B Z of this circle, 
 which has the direction of the deflecSlion in B, may be 
 called its deflective chord. 
 
 Since B C and E F are defcribed in equal times, they 
 are proportional to the velocities in B and E. There- 
 fore, we may exprefs this propofition In the following 
 words : 
 
 In curvilweal motions y the dejleBlons in different points 
 ef the curve are proportional to the fqiiare of the velocities 
 in thofe points^ direclly, and to the dejleEiive chords of the 
 equicurve circles in thofe points, iiiverfely. 
 
 It mufl be here remarked, that this theorem is not 
 limited to curvilineal motions, in which the deflections 
 sre always directed to one fixed point, but extends to 
 
 all 
 
OF CURTILINEAL MOTIONS. 75 
 
 all curviilneal motions whatever. For it may evidently 
 be exprelTed in this manner ; The defleBing forces are ul- 
 timately proporUo7ial to the fquares of the arches defcribed 
 in equal times, direSfly, afid to the dejleclive chords of the 
 equiciirve circle, inverfely. 
 
 The equable defcription of areas only enabled us to 
 fee that the lines B C and E F were defcribed in equal 
 times, and therefore are as the velocities. 
 
 It will be convenient to have a fymbolical expreflion 
 of this theorem. Therefore, let the defleftive chord of 
 the equicurve circle be reprefented by c, and the deflec- 
 tion by d, the theorem may be exprelTed by 
 
 , . "y ^ , 2 arch "* 
 d =^ — 5 or rf = 
 
 c c 
 
 105. Remark. — The line B i^ is the linear deflec- 
 tion, by which the uniform motion in the chord A B is 
 changed into a uniform motion in the chord B C, or it is 
 the deviation c C from the point where the moveable 
 would have arrived, had it not been deflected at B. But, 
 in the prefent cafe of curvilineal motion, the lines B h 
 and Br exprefs the meafures of the velocities of thefe 
 motions, or the meafures of the determinations to them. 
 B r is to B ^ as the velocity of tlie progreflive motion is 
 to the velocity of the deflection, generated during the 
 defcription of the arch B C. But, becaufe the defle<51:ion 
 'in the arch has been continual, and becaufe it is to be 
 meafured, like acceleration, by the velocity which is ge- 
 nerated uniformly during a given moment of time, it 
 
 K 2 jiiay 
 
']6 OF CURVILINEAL MOTIONS. 
 
 may be meafured by the velocity generated during the 
 defcrlption of tne arch B C Its meafure therefore will 
 be double of the fpace through which the body is ac- 
 tually defle£i:ed in that time from the tangent in B. The 
 fpace defcrlbed will be only one half of B b, or it will 
 be BT). Now, this is really the cafe ; for the tangent is 
 ultimately parallel to O C, and bife^ls r C ; fo that al- 
 though the deflection from the tangent to the carve is 
 only half of the defleftion from the produced chord to 
 the curve ; yet the velocity gradually generated is that 
 which will produce the defleclion from the produced 
 chord, or is that which conftitutes the polygonal motion 
 in the chords. 
 
 It is perfecJy legitimate, therefore, to reafon from 
 the fubfultory deflections of a polygonal motion to the 
 continual deflections in a curvilineai motion ; for the de- 
 fieftions in the angles of the polygon have the fame ra- 
 tio to one another with the deflections in the fame points 
 of the curve. But we muft be careful not to confound 
 the defledions from the tangent with thofe from rhe 
 chords. This has been done by eminent mathematicians. 
 Tor the employment of algebraical expreflions of the in- 
 crements of the abfciflis and ordinates of curves, ahvay>^ 
 gives the true cxpreflion of the deflections in a polygonal 
 motion. But, when we turn our thoughts to the figures, 
 vmd to the curvilineai motions them fe Ives, we naturally 
 think of the deflcCtions (fuch as we fee them) from tlie 
 tangent to the curve. We then make geometrical in- 
 ferences, which are true only when aflirmed fA the cur- 
 vilineai 
 
OF CURVILINEAL MOTIONS. '"^y 
 
 vilineal motions. We are apt to mix and confound thefe 
 inferences with the refiilts of the fliixionary calculus, 
 which always refer to the polygon. By thus mixing 
 quantities that are incongruous, fome celebrated mathe- 
 maticians have committed very grofs miftakes. 
 
 It is, in general, moft convenient, and furely mofl 
 natural, to ufe the ultimate ratio of the actual deflec- 
 
 BC" 
 
 tions from the tangent, or T.-y- ; ^nd this even gives us 
 
 its meafure in feet or inches, when we know the dimen- 
 fions of the figure defcribed. Thus we know that, in 
 one minute, the Moon, when at her mean diftance, de- 
 flefts 193 inches from the tangent to her orbit round 
 the Earth, and that the earth deviates 424 inches in the 
 fame time from the tangent of her orbit round the Sun. 
 
 1 06. The velocity in any point of a curvilineal mo- 
 tion is that which would be generated by the delie6lioii 
 in that point, if continued through ~ of the deflective 
 chord of the equicurve circle. Let x be the fpace along 
 which the body muft be accelerated in order to acquire 
 the velocity B C. 
 
 We have Bb\ or 4 B O^ : B C^ = B O : :v (57) and 
 
 , , BC^xBO BC^ J Be 
 
 tnereicre^ = — ^^-^, = -g-^, and 4. =-gg-, or 
 
 B O : B C = B C : 4 .V. But B O : B C = B C : B Z. 
 
 Therefore a: = ^ B Z, 
 
 ' ReCAPITU" 
 
yI br curvilineal tviotions. 
 
 Recapitulation. 
 Thus have we obtained marks and mcafures of all the 
 principal affe£lions of motion. 
 
 The acceleration « is - (71) or -r— (72) or — (65) 
 t s f- 
 
 The momentary variation of velocity v= at (71) 
 
 The momentary variation of the fquare of velocity 
 
 2vv^2a s (72) 
 
 arc ^ 
 The momentary defle£l:ion d = , ' (105) 
 
 The deflective velocity = ^— (104) 
 
 But, in order to apply the dodlrines already eflablifhed 
 w^ith the accuracy of vi'hich phyfico-mathematical fubjefts 
 are fufceptible, it is necelTary to feleft fome point in any 
 body of fenfible magnitude, or in any fyftem of bodies, 
 by the pofition or motion of v^^hich we may form a jufh 
 notion of the pofition and motion of the body or fyftem. 
 It is evident that the condition which afcertains the pro- 
 priety of our choice, is, that the pofition 3 dijlance, or mo- 
 t'lon of this point fiall he a medium or average of the poft- 
 iions, dijiancesy aud motions of every particle of ?natter in the 
 fjfetnblage, 
 
 107. This will be the cafe, if the point be fo fituated 
 that, if a plane be made to pafs through it in any direc- 
 tion ivhatever, and if perpendiculars be drawn to this 
 plane from every particle of matter in this aflemblage, 
 the fum of all the perpendiculars on one fide of this 
 
 plant? 
 
OF THE CENTRE OF POSITION. 79 
 
 plane Is equal to the fum of all the perpendiculars on the 
 other. 
 
 That there may be found in every body fuch a point, 
 is demonftrated (after Bofcovich) in the Eticycl. JBrlLi/i, 
 Art. Portion ( Centre of). 
 
 Let P (fig. 20.) be a point fo fituated, and let OR 
 be a plane (or rather the feftion of a plane, perpendicu- 
 lar to the plane of the paper) at any diftance from the 
 body. The diftance V p of P from this plane, is the 
 average of all the diftances of each particle. For, let the 
 plane A P B pafs through this point, parallel to the plane 
 O R. The diftance C S of a parallel C from this plane 
 is DS — DC, orP/> — DCj and the diftance GT of 
 a particle G is H T -f G H, or P/ + G H. Let ;; be 
 the number of particles betvi^een O R and A P 5 and let 
 # be the number on the other fide of A P ; and let m be 
 the number of particles in the w\\o\q body, that Is, let 
 in:=zn -\- 0. It Is evident that the fum of all the diftances, 
 fuch as C S Is // X ^ p minus the fum of all the diftances, 
 fuch as C D. Alfo o X P/), plus the fum of the diftances 
 GH, is the fum of all the diftances GT. Nov/, the 
 fum of the lines GDIs equal to that of all the lines G H, 
 
 and therefore ?/ + o X P/, or m X P/, is equal to the 
 fum of all the lines C S and G T, and V p is tlie /;/•* part 
 «f this fum, or the average diftance. 
 
 Now, fuppofe the body to have approached to the plane 
 Q R (fig. 2 1 ), and that P is now at tt. It is plain that the 
 diftance 7rp is again the average diftance, and in x 7rp 
 is the fum of all the new diftances. The difference from 
 
 the 
 
to OF THE CENTRE OF POSITION. 
 
 the former fum Is m X V tt, and confequently }?i X P ^J" is 
 the fum of the approaches of every particle j and P tt is 
 the fn^^ part of this fum, or is the average of them all. 
 The diftance, pofition, and motion of this point is there- 
 fore the average pofition, diftance, and motion of the 
 whole body. The fame demonftration will apply to any 
 fyftem of bodies. The point P is therefore properly 
 chofen. 
 
 1 08. Since the point P is the fame, in vi^hatever 
 direflion the plane APB is made to pafs through it, it 
 follows tliat the laft propofition is true, although the body 
 may have turned round fome centre or axis, or though 
 the bodies of u'hich the fyftem confifts may have changed 
 tlieir mutual pofitions. 
 
 109. The point P, thus felecled, may, with great 
 propriety, be called the centre of position of the body 
 or fyftem. 
 
 no. If A and B (fig. 22.) be the centres of pofi- 
 tion of two bodies A and B, and if a and b exprefs the 
 numbers of equal particles in A and B, or their quanti- 
 ties of matter, the common centre C of this fyftem of 
 two bodies lies in the ftraight line A B joining their re- 
 fpe6live centres, and A C : C B = ^ : ^. This is evident. 
 
 III. If a third body D, whofe quantity of matter 
 is r/, be added, the common centre of pofition of thi$ 
 
 fvftem 
 
OF THE CENTRE OF POSITION* -81 
 
 fyllem of tliefe three bodies lies in the ftraight line D C, 
 joining D with the centre of the other two, and D E : E C 
 :=a + hid. 
 
 In like manner, if a fourth body be added, the common 
 centre of pofition is in the line joining it with the centre of 
 the other three, and the diilance of the fourth from this 
 common centre, is to the diilance of that from the com- 
 mon centre of the tbr^e, as the matter of all the three 
 to the matter of the fourth— And the fame thing is true 
 for every addition. 
 
 112' If the particles or bodies of any fyflem be 
 inoving uniformly in firiiight lines, with any velocities 
 and directions whatever, the centre of the fyftem is ei- 
 ther at reft, or it moves uniformly in a ftraight line. 
 
 For, let one of the bodies D move uniformly from 
 D to F. Join F with the centre C of the remaining bo- 
 dies, and make Cf to Ff as the matter in F is to that 
 in the remaining bodies. It is plain that E y is parallel 
 to D F, and that DF:E/=:A + B:D. In like man- 
 ner, may the motion of the centre be found that is pro- 
 duced by that of each of the other bodies. 
 
 Biit tliefe motions of the centre F are all uniform 
 and r<^diJlneal. Therefore, the motion compounded of 
 tiiem all is uniform and ro6lilineal. 
 
 It m.ay happen that the motion refulting from this 
 corapofition may be nothing, by reafon of the coatrariety 
 of fome individual motions. In this cafe, the centre 
 will remain in the fame point. 
 
 This obtains alfo, if the centres of any number ofc 
 U l)odie$ 
 
OF THE CENTRE OF POSITION* Zl 
 
 fyilem of tliefe three bodies lies in the ftraight line D C, 
 joining D with the centre of the other two, and D E : E C 
 
 In like manner, if a fourth body be added, the common 
 centre of pofition is in the line joining it with the centre of 
 the other three, and the diftance of the fourth from this 
 common centre, is to the diitance of that from the com- 
 mon centre of the thr^e, as the matter of all the three 
 to the nicitter of the fourth— And the fame thing is true 
 for every addition. 
 
 112' If tlie particles or bodies of any fyflem be 
 moving uniformly iu ilraight lines, with any velocities 
 and dircftions whatever, the centre of the fyftem is ei- 
 ther at reft, or it moves uniformly in a ftraight line. 
 
 For, let one of the bodies D move uniformly from 
 D to F. Join F with die centre C of the remaining bo- 
 dies, and make Cf to ¥f as the matter in F is to that 
 in the remaining bodies. It is plain that E ^ is parallel 
 to D F, and that D F : E/ = A + B : D. In like man- 
 ner, may the motion of the centre be found that is pro- 
 duced by that of each of the other bodies. 
 
 But tliefe motions of the centre F are all uniform 
 and r^dillneal. Therefore, the motion compounded o£ 
 tliem all is uniform and rcclilineal. 
 
 It m.ay happen that the motion refulting from this 
 compofition may be nothing, by reafon of the contrariety 
 o£ jfome individual motions. In this cafe, the centre 
 will remain in the fame point. 
 
 This obtains alfo, if the centres of any number ofc 
 X* J)odie» 
 
'8f2 OF THE CENTRE OF POSITION. 
 
 bodies move uniformly in right lines, whatever may have 
 been the motion of each body, by rotation or otherwife. 
 The motion of the common centre w^ill ftill be uniform 
 and re£lilineal. 
 
 113. Cor. I. The quantity of motion of fuch a fyf- 
 tem, is the fum of the quantities of motion of each body 
 reduced (85.) to the dire6lion of the centre's motion, 
 and it is had by multiplying the quantity of matter in the 
 whole fyftem by the velocity of the centre. 
 
 114. Cor. 2. This velocity of the centre is had by 
 reducing the motion of each particle to the direction of 
 the centre motion, and divefting the fum of the reduced 
 motions by the quantity of matter in the fyftem. 
 
 115. If equal and oppofite quantities of motion be 
 any how imprefled on any two bodies of fuch an af- 
 femblage, the motion of the centre of the whole is not 
 afFeifbed by it. For the motion of the centre, arifmg 
 from the motion of one of the bodies, being compound- 
 ed vv^tli the equal and oppofite motion of the other, the 
 diagonal of the parallelogram becomes a point, or thefe 
 motions defbroy one another, and no change is induced 
 thereby, in the motion of the centre. The fame thing 
 muft be faid of equal and oppofite quantities of motion 
 being imprefled on any other pair of the bodies, and, in 
 •ihort, on every pair that can be formed in the aflem- 
 blage. Therefore the propofition is ftill true. 
 
 MECHA- 
 
MECHANICAL PIIILOSOP^IY. 
 
 PART I. Section I. 
 
 OF MATTER. 
 
 1 1 6. JL HE term matter exprefles that fubflance 
 of which all things which we perceive by means of our 
 fenfes are conceived to confift. It is almofl fynonymous, 
 in our language, with body. Material and corpo- 
 real feem alfo fynonymous epithets. 
 
 117. Senfihle bodies are ufually conceived as con- 
 fifting of a number of equal particles or atoms of this 
 fubftance. Thefe atoms may alfo be fuppofed fimilar in 
 ^11 their qualities, each poffeffing fuch qualities as diftin- 
 ^uifli them from every thing not material. 
 
 118. But we are eiitirely ignorant of the e/Tential 
 qualities of matter, and cannot ailirm any thing concern- 
 ing it, except what we have learned froi7i obfervatioii. 
 To us, matter is a mere phenomenon. Cut we mufl: af- 
 
 L 2 certaiu 
 
84 OF MATTER. 
 
 certain with precifion the properties which we fele£l as 
 diftin£live of matter from all other tilings.. 
 
 1 19. All men feem agreed in calling that alone mat- 
 ter, which excludes all other fubftances of the fame kind, 
 or prevents them from occupying the fame place, and 
 which requires the exertion of what we call force to re- 
 move it from its place, or anyhow change its motion. 
 Thefe two properties have been generally called soli- 
 dity or IMPENETRABILITY, and INERTIA Or MOBILITY, 
 
 Mere mxobility, however, is not, perhaps, peculiar to mat- 
 ter I for the mind accompanies the body in all its changes 
 of fituation. "When mobility is afcribed to matter, as a 
 diftinguifhing quality, we always conceive force to be re- 
 quired. We are ccnfcious of exerting force in moving 
 even our own limbs. In like manner, extenfion, and fi- 
 gure, and divifibility, although primary quahties of mat^ 
 ter, are common to it with empty fpace. 
 
 120. Mobility in confequence of the exertion of 
 force may be ufed as a charafteriftic of matter, or of 
 an atom of matter. All polTefs it — and probably all pofTefs 
 it alike, their fenfible ditFerences being the confequence 
 of a difference in the combinaticns of atoms to form a 
 particle. 
 
 121. A particle of matter under the influence of a 
 moving force, is the objccr of purely mechanical con- 
 templation, and trie coniideration of thc'ckiuges of mo- 
 tion 
 
OF MATTER. ^f 
 
 tioil which refuk from its condition as thus defcribed 
 may be called the mechanism of the phenomenon. 
 
 122. Perhaps all changes of material nature are 
 cafes of local motion (though unperceived by us) by th© 
 influence of moving forces. Perliaps they cannot be faid 
 to be completely underflood, till it can be fhewn how tlia 
 atoms of matter have changed their fituations. Perhaps 
 the folution of a bit of filver in aqua fortis is not complete^ 
 ly explained, till we (liew, as the mechanician can (hew 
 with refpe£^ to the fatellites of Jupiter, how an individual 
 atom of filver is made to quit its connexion with the 
 reft, and by what path, and with what velocity in every 
 inftant of its motion, it gets to its final ftate of reft, in 
 a diftant part of the veflel. But thefe motions are not 
 eonfidered by the judicious chemift. He confiders the 
 phenomenon as fully explained, when he has difcovered 
 «11 the cafes in which the folution takes place, and has 
 defcribed, with accurate fidelity, all the circumftances of 
 the operation. 
 
 123. We have derived our notions of solidity or 
 IMPENETRABILITY chicfly from our fenfe of touch. The 
 fenfations got in this way feem to have induced all men 
 to afcribe this property of tangible matter to the mutual 
 contaft of the particles — and to fuppofe that no diftance 
 is interpofed between them. 
 
 124. But the compreflibility and elafticity of alj 
 Jcnown bodies, their contradion by cold, and many ex- 
 
 ^mplea- 
 
3(5, OF MAtTEK. 
 
 amples of chemical union, in which the ingredients oc- 
 cupy lefs room when mixed, than one of them did before 
 mixture, feem incompatible with this conflitution of tan- 
 gible matter. Did air confift of particles, elaftic in the 
 fame manner that blown bladders are, it would not be 
 fluid when comprefled into half of its ufual bulk, be- 
 caufe, in this cafe, each fpherule would be comprelTed 
 into a cube, touching the adjoining fix particles in the 
 whole of its furfaces. No liquid, in a ftate of fenfible 
 compreflion, could be fluid ; yet the water at the bottom 
 of the deepeft fea is as fluid as at the furface. Some op- 
 tical phenomena alfo fhew incontrovertibly that very 
 flrong prefl"ure may be exerted by two bodies in phyfical 
 or fenfible conta6l, although a meafurable diftance is ftill 
 interpofed between them. On the whole, it feems more 
 probable that the ultimate atoms of tangible matter are 
 not in mathematical contact. 
 
 125. Bodies are penetrated by other matter in con- 
 fequence of their porofity. Therefore the fame bulk 
 may contain different quantities of matter. 
 
 126. Density is a term, which, in flri^l language, 
 cxprefles vicinity of particles. But, v/hen ufed by the 
 mechanician as a term of comparifon, it expreflcs the 
 proportion of the number of equal particles, or the quan- 
 tity of matter, in one body, to the number of equal par- 
 tiv'les in the fame bulk of another body. 
 
 127. 
 
• F MATTER. 87 
 
 127. Therefore the quantity of matter (frequently 
 galled the mass) is properly exprefled by the product of 
 numbers exprefling the bulk B and the denfity D. If M 
 be the quantity of matter, 
 
 then M = B D 
 
 MECHA- 
 
MECHANICAL PHILOSOPHY. 
 
 PART L Section IL 
 
 DYNAMICS. 
 
 128. JJynamics is that department of phyfico- 
 mathematical fcience which contains the ahftracl doc- 
 trines of moving forces ; that is, the neceffary refults of 
 the relations of our thoughts concerning motion and the 
 caufes of its produftion and changes. 
 
 129. Changes of motion are the only indications of 
 the agency, the only marks of the kind, and the only 
 meafures of the intenfity of thofe caufes. 
 
 130. "We cannot think of motion, i7i ahJlraElo^ as -st 
 thing, properly fo called, that can fubfifl feparately, but 
 as a quality, or rather as a condition, of fome other thing. 
 Therefore we confider this condition as permanent, like 
 the fituation, figure or colour of the thing, unlefs fome 
 caufe of change exert its influence on it, 
 
 13'' 
 
a&YNlMICSu S^ 
 
 13 f. Looking round us, we cannot fail of ohferving 
 \\\\t the changes m the ftate or condition of a body in 
 refpe«£t of motion, have a diftin61: and conftant relation to 
 the fituation and diftance of fome other bodies. Thus, 
 fche motions of the Moon, or of a ftone projected through 
 the air, have an evident and invariable relation to the 
 Earth. A magnet has the fame to iron« — an electri- 
 fied body to any body near it — a billiard ball to ano- 
 ther billiard ball, &c. &c. Such feeming dependences^ 
 may be called the mechanical relations of bodies. They 
 are, unqueftionably, indications of properties, that is, of 
 diftinguifliing qualities. Thefe accompany the bodies 
 wherever they are, and are commonly conceived as m- 
 herent in them 5 and they certainly afcertain and deter- 
 mine what we call their mechanical nature. The me- 
 chanician will defcribe a magnet, by faying that it at- 
 traiHis iron. The chemift will defcribe it, by faying that 
 it contains the martial oxyd in a particular proportion of 
 metal and oxygen* 
 
 132. Philofophers are not uniform, liowever, in their* 
 reference of the qualities indicated by thole obferved re- 
 lations. Magnetiiin is a term exprelfing a certain clafs 
 of phenomena, which are relations fubfifting between 
 magnets and iron ; but many reckon it a property of the 
 magnet, by which it attracts iron ; others imagine it a 
 property of the iron, by which it tends to the magnet. 
 This difference generally arifes from the intereil we take 
 in the phenom.enon ; both bodies are probably affecSted a- 
 like^ and the property is diftinclive of both : For, in all 
 
 M €%fs3 
 
9© MECHATSilCAL FORCES 
 
 cafes that have yet been obferved, we find that the in- 
 dicating phenomenon is obferved in both bodies ; — the 
 jnagnet approaches the iron, and the electrified body ap- 
 proaches tlie other. The property therefore is equally 
 inherent in both, or perhaps in neither •, for there are 
 fome philofophers, who maintain that there are no fuch 
 mutual tendencies, and that the obferved approaches, or, 
 in many cafes, mutual feparations, are effe6ted by the 
 extraneous impulfion of an aethereal fluids or of certain 
 miniftering fpirits, intrinfie or extrinfic. 
 
 133. Tliefe mechanical afFe£lions of matter have 
 been very generally called powers or forces •, and the 
 body conceived to poiTefs them is faid to act on the re- 
 lated body. This is figurative or metaphorical language. 
 Power, and force, and a61:ion, cannot be predicated, in 
 their original ftrift fenfe, of any thing but the exertions 
 of animated beings •, nay, it is perhaps only the exerted 
 influence of the mind on the body which we ought to 
 call action. But language began among fimple men ; 
 they gave thefe denominations to their own exertions 
 with the utmoll propriety. To move a body, tliey found 
 tliemfelves obliged to exert their Jlrength^ or force^ or 
 pDiuery and to aci. When fpeculative men afterwards 
 attended to the changes of motion obferved in the meet- 
 ings or vicinity of bodies, and remarked that the phe- 
 nomena very much refembled the refults of exerting 
 their own (trength or force ; and when they would ex- 
 prefs this occurrence of nature, it was eafier to make ufe 
 
 . of 
 
IN GENERAL. pt 
 
 t)f an old term, than to make a new one for things 
 ■which fo much refembled ; becaufe there are always fuch 
 differences in other circumstances of the cafe, that there 
 is little danger of confounding them. We are not to 
 imagine that they thought that inanimate bodies exerted 
 ftrength, as they themfelves did. This was referved for 
 much later times of refinement. — In the progrefs of thij 
 refinement, the word power or force was employed to 
 ^xprefs any efficiency ivhatever ; and we now fay^ the power 
 of aqua fortis to dilTolve filver- — the force of argument — 
 the action of motives, &c. &c. 
 
 To this notion of conveniency we mufl :afa:ibe, not 
 only the employment of the words power and forcey to 
 exprefs efficiency in general, but alfo of the terms attract 
 i'lon^ repitlfton^ impulfion^ preffiure, &c. all of which are 
 metaphorical, unlefs when applied to the adiions of ani- 
 mals. But they are ufed as terms of diftin£tion, on ac- 
 count of the refemblance between the phenomena and 
 thofe which we obferve when we pull a thing toward 
 us, pufh it from us, kick it away, or forcibly comprefs 
 it. 
 
 134. IMuch confufion has arifen from the unguard- 
 ed ufe of this figurative language. Very flight analogies 
 have made fome animate all matter with a fort of mind, 
 a oc-TTs^ "^vyc'^i while other refemblances have made other 
 fpeculatiils materialize intelledl itfelf. 
 
 The very names which we give to thofe powers 
 
 •v/hich we fancy to be inherent in bodies, iliew that we 
 
 M 2 know 
 
92 FORCE, IMPULSE, 
 
 know nothing about them. Thefe names either, Wlit 
 magnetifm, exprefs a relation to the particular fubllances 
 which we imagine poiTefs the power, or they exprefi 
 fomething of the effect which fuggefted their exiftence. 
 Of this laft kind are cohefion,, gravity ^ &c. They are 
 almoft all verbal derivatives, and fhould be coniidered 
 by us merely as abbreviated defcriptions or hints of the 
 phenomena, or as abbreviated references to certain bo- 
 dies, but by no means as any explanation of their na- 
 ture. The terms are the worffe by having fome meaning. 
 For this has frequently mified us into faife notions of the 
 manner of aiding. Perhaps tlie only ftrict application 
 of the term action is to the effect produced by our 
 exertions in moving our own limbs. But we think that 
 ^e move other bodies, becaufe our own body, which is 
 the immediate inftrument of the mind, is overlooked, 
 like the plane in the hand of the carpenter, attending to 
 the plank which he drefles. 
 
 135. Forces have been divided into impulsions andi 
 ?RESSURES. Impulnons are thofe which produce the 
 changes of motion by the collifion of moving bodies. 
 Preflure is a very familiar idea, and perhaps enters into 
 every clear conception that we can form of a moving 
 force, when we endeavour to fix our attention on it. 
 We know that preffure is a moving force ; for, by prelT- 
 ing round the handle of a kitchen jack, we can urge the 
 fly into any rapidity of motion. Even when one ball 
 puts another in motion by hitting it, we thiTik tliat fome- 
 
PRESStTRfi. 95 
 
 thing preciiely like our own prefTure Is the hnmediate 
 producer of the motion ; for if the ball is compreflible, 
 we fee it dimpled by the blow. Gravity, or elafticity, 
 and the like, are called preffing powers *, becaufe a ball, 
 Jying on a mafs of foft clay, makes a pit in it, and, if 
 lying on our hand, it excites the fame feeling that an- 
 other man would do by prefhng on our hand. There 
 are fome Indeed who call fuch powers, as gravity, mag- 
 netifm, and ele61:riclty, solicitations to motion. We 
 fhall foon fee that this claffification of forces is of no 
 ufe. 
 
 136. Preflure and impulfion are thought to be ef- 
 fentially diftingulfhed by this clrcumfcance, that, in or- 
 der to produce a finite velocity In a body by preflure, it 
 muft be continued for fome time — as when we urge the 
 fly of a jack into fwlft motion by prefTnig on the handle ; 
 whereas Impulfion produces it In an inftant. — ^They are 
 alfo difllngulflied by another circumftance. The impel- 
 ling body iofes as much motion as the impelled body has 
 gained j fo that there feems fometliing like the transfu- 
 fion of motion from the one to the other. Accordingly, 
 It is called the communication of motion. But we 
 fliall find that neither the Inftantaneous production of mo^ 
 tion by impulfe, nor the transfufioa of It into the body, 
 are true. 
 
 137. Some again think that Impulfion is the only 
 ^ufe of motion, faying, * Nihil movetur nifi a contigua. 
 
 ' et 
 
fj^ MECUANICAL FORCE 
 
 ' et moto ; ' and they have fuppofed ftreams of aether, 
 which urge heavy bodied dovi^nward — which impel the 
 iron and magnet toward each other, &c. 
 
 138. But a third feci of mechanicians fay that 
 forces, a£ling at a diftance, as we fee in the phenomena 
 of gravitation, magnetifm, and ele(^ricity, are the fole 
 Caufes of motion ; 9nd they affert that fuch forces are 
 exhibited, even in the phenomena of fenfible contact, 
 preiTure,.- and im.pulfion. 
 
 139. The only fafe procedm-e is to confider all the 
 forces which we obfeive in aclioh as mere phenomena. 
 The cpnftitution of our mind makes us infer the agency 
 of a caufe, whenever we obferve a change. But, whe- 
 ther the exertion of force fhall p^^oduce motion or heat, 
 we know not, except by experience, that is, by obferva- 
 tion of the phenomena. Nor vrill fpeculations about the 
 intimate nature of thefe forces, and their manner of a61:- 
 iiig, contribute much to our nfefui knowledge of mecha- 
 nical nature. We gain all that is poffible concerning the 
 nature of thofe faculties which accompany matter, or are 
 fuppofed to be its inherent properties, by noticing the 
 LAWS according to which their exertions proceed. With- 
 out a knowledge of thefe laws, the other knowledge is 
 of no value. 
 
 140. It is alfo from the change of motion alone that 
 we learn the direBion of any force. . Thus, by obferving 
 
 that 
 
IN GENERAL. 95* 
 
 tliat an arrow is retarded during its afcent tKrough the 
 air, but accelerated during its fall, we infer, or learn, 
 that the force of gravity a6ts downwards. 
 
 141. When a force is known to be In a£lIon, and 
 yet its chara<Sleriftic motion does not follow, we fuppofe 
 that it is oppofed by a force acting in the oppofite direc- 
 tion. Thus the agency of that other force is dete(S^ed, 
 and its intenfity may be meafured. Thus, the force with 
 which the parts of a ftring cohere, or with which a 
 fpringey body unbends, are detected by their fupporfing 
 a weight — and the magnitude of the weight Is the mea- 
 fure of the cohefion of the ftring, or of the elafticity 
 of the fpring. 
 
 142. But the body in which this oppoling force is 
 thus dete6i:ed, is alfo faid to refijl the force to which it 
 is oppofed. This is figurative language, and, as ufed in 
 mechanical philofophy, it is generally improper. In 
 wreftling, when my antagonift exerts his ftrength, to 
 prevent his being thrown down, I am fenfible of his ex- 
 ertion, and I thus learn that he refifts. But fhould I 
 feel no more exertion neceffary than if he were a mafs 
 of lifelefs matter, I fhould not think that he refifted. 
 In the mechanical operations of nature, a force of any 
 kind always produces its full efteft, agreeably to the cir- 
 cumftances of the cafe, and can do no more. The force 
 is indeed expended in producing its effect, becaufe mat- 
 ter is not moved without force. The weight lying on a 
 
 fpring, 
 
p6 MECHAKICAL FORCE 
 
 fpring, and keeping it in a flate of tenficn, is as com- 
 pletely meafured by the degree of tenfion which fupports 
 it, as this tenfion is meafured by the fupported weight ; — 
 neither can, with propriety, be faid to refill. Silver is 
 faid to refill the dilTolving pow^r of aqua regia, but not 
 that of aqua fortis ; yet the dilTolving power of aqua 
 fortis is expended, and that of aqua regia is not. All 
 this is very inaccurate employment of words, and this> 
 inaccuracy has done much harm in natural philofophy. 
 The word inertia, which had been employed by Kepler 
 and Newton, to exprefs the indifference of matter as to 
 motion or reft, or its tendency to retain its prefent ftate, 
 has got other notions annexed to it by fubfequent writers, 
 and has been called a force, vis inertia. Mr Rutherfurth, 
 in his Syftem of Natural Philofophy, lectures which he 
 read in the Univerfity of Cambridge with great applaufe, 
 is at pains to fliew that matter is not merely indifferent, 
 but RESISTS every change of motion, by exerting what 
 he calls the force of ina£livityy by which it preferves its 
 condition unchanged. But, furely, this is as incongru- 
 ous as to fpeak of a fquare circle. Yet is inertia con- 
 fidered as a real exiftence, and is faid to be proportional 
 to the quantity of matter in a body. When we find that 
 we muft employ twice as much force to move A with a 
 certain velocity as to move B, we fay that A contains 
 twice as much matter, becaufe we fee that it has twice 
 as much inertia. Is it not enough to fay that we judge 
 A to have twice as much matter, becaufe all matter re- 
 quires force to move it ? — this is its chara<Sleriftic. Should 
 
IN GENERAL. 97 
 
 we find that we can move a thing by a very wifh, or a 
 •command, we fliould not think it matter. Inertia, taken 
 in this fenfe, as expreffing the neceihty of what we call 
 iorce, in order to change the motion of matter, is jiift 
 one of thofe general phenomena by which it is known to 
 us. Whether this force be, in every cafe, external to 
 the material atom, or whether fome of the obferved 
 powers of body may not be inherent in it, is a queftion 
 of Metapiiyiics, and is probably beyond the reach of our 
 faculties. But naturaliils have generally fuppofed that' 
 the atom is purely palhve and indiiferent, and that all its 
 powers are fuperadded to the mere material atom. 
 
 Thefe doubts and difficulties in the ftudy have all 
 arifen from the introduQion of the notion of rcjlflancey 
 or force exerted by matter, in order to remain as it is. 
 It would have been iiifinitely better to have employed the 
 term reaction, becaufe this is the expreffion of the very 
 facl 5 for, in all the phenomena of changed motion, there 
 "is obferved an equal change in oppofite directions in the 
 two afting bodies. Iron approaches to the magnet — 
 the magnet to the iron. In tlie coilifion of bodies, the 
 impelling and impelled are obferved to fuflain equal and 
 oppofite changes. But in mofl, and probably in all, we 
 difcover that thofe changes are brought about by forces 
 familiarly known to us in other ways ; and no method 
 has been difcovered, by v/hich we may learn whether the 
 nvhole of the change is owing to thofe mutual forces, or 
 whether fome part is to be afcribcd to inertia. 
 
 ^ T43. 
 
98 MECHANICAL FORCE. 
 
 143. When the body B is always obfervecl to ap- 
 proach to A, and no intermediat-e caufe can be afTigned, 
 A is faid to attra6i: B. Thus a magnet is faid to attra<£^ 
 a piece of common iron. But if B is always obferved to 
 fnun A, or to feparate from it, A is faid to repel B, 
 Tiius one eledlrified body repels another. 
 
 144. Mechanical forces are confidered as meafurable 
 magnitudes. But, fmce they are not objects of our per- 
 ception, but only inferences from the phenomena, it is 
 plain that we can neither meafure nor compare their 
 magnitudes diredlly. Having no knowledge of their 
 Bgency, nor any mark of their kind^ except the change of 
 motion which w^e confider as their effe^l, it is only in 
 this change of motion that we muft look for any meafure 
 of their magnitude or intenfity j — this is alfo the only 
 mean of comparifon. Now, change of motion, involv- 
 ing no ideas but of fpace and time, affords the mxoft 
 perfecl: meafurement. We cannot find a better mea- 
 fure ; nay, it is improper to employ any other ; and the 
 moft eminent philofophers, by employing other mea- 
 fures, founded on their fancied knowledge of the intimate 
 nature of mechanical force, have advanced mofl incon- 
 gruous opinions, which have fpoiled the beauty of the 
 fcience. We fhall therefore adhere ftrictly to the mea- 
 fure fuggefted by this rcafoning, and fliall call that a 
 double or triple force, which, by its fimilar action, dur- 
 ing the fame tim.e, produces a double or triple change of 
 motiorij whether it accelerates, or retards, or defle£fs a 
 
 motion 
 
LAV/S OF MOTION. 99 
 
 ttiotioti already going on. We exprefs tins notion In the 
 mofl; fimple manner by (laying, tirat we confider force 
 merely as fomething that is proportional to the change of 
 velocity. 
 
 Of the Laws of Motion, 
 
 145. Such being our notions of motion, and of the 
 Caufes of its production and changes, there are certain 
 refults, which, by the conftitution of our minds, necef^ 
 farily arife from the relations of thefe ideas. Thefe are 
 laws of human judgement, independent of all experience 
 of external nature, juil as it refults from the laws of 
 judgement that the three angles of a right lined triangle 
 are equal to two right angles, although there fhould not 
 be a triangle in the univerfe. 
 
 Some of thefe laws may be intuitive, and may be 
 called axioms ; others, etjualiy neceflary truths, may not 
 be fo obvious, and may require fteps of argument. 
 
 There are three fuch laws, firit propofed in precife 
 terms by Sir Ifaac Newton, which feem to give a fuffi- 
 cient foundation for all the do£lrines of Dynamics, and to 
 which, as to firft principles, we may appeal for the ex- 
 planation of every mechanical phenomenon of nature. 
 
 Ftrf Law of Motion. 
 
 I4<5. Every body co7itinues at refl^ or in uniform rec* 
 tilineal motion^ unlefs affeSled by fome mechanical force. 
 
 N 2 If 
 
loo FIRST LAW 
 
 If we adhere to our inference of tlie agency of force 
 only from an obferved change of motion, and to this in- 
 ference from every fuch change, and if M-e grant that we 
 have no notion of a force independent of a change of mo- 
 tion, this law feem.s little more than a tautological pro- 
 pofition. For, unlefs we fuppofe the agency of a me- 
 chanical force, we do not fuppofe a change of motion, 
 that is, the ab fence of mechanical agency is the abfence 
 of a change of motion, and the body continues in its 
 former ftate of red or motion. But philofophers have 
 attempted to demonfcrate this law in various ways. 
 
 147. Some confider it as a neceffary truth, in the 
 . nature of the thing. A body, they fay, can neither ac- 
 celerate, nor retard, nor defle<£l, becaufe the event is but 
 one, and there is no caufe of determination whether it 
 fhall accelerate, or retard, or defied, nor whether to 
 
 . the right or to the left, or which determines any one de- 
 gree of any of thofe changes. This fort of proof is ob- 
 fcure and unfatisfadlory. 
 
 148. Others choofe to conllder it as a phyfical law, as 
 an univerfal faft, for which, perhaps, we can give no rea- 
 fon. They offer numerous proofs by induftiou. Thus, 
 a coach being fuddenly accelerated, or checked in its 
 progrefs, or turned out of its courfe, the fTtters are thrown 
 towards the back, or the front of the coach, or to one 
 fide, fhewing, in all cafes, a tendency to continue in 
 their former condition in refpedl of motion. Number- 
 
 . , lefs 
 
OF MOTION. lOl 
 
 lels examples may be given of the fame marks of this 
 tendency to continue in the former ftate. 
 
 149. But it may be obje6ted, that it is very far 
 from being a matter of univerHil experience. Whoever 
 grants the truth of the Copernican defcription of tlio 
 planetary motions, will alfo grant that we perhaps never 
 faw one inilance, either of reil or of uniform re£lili- 
 neal motion. Our moft familiar obfervations fhew an 
 evident tendency to reft, a fort of fluggiflmefs in all 
 matter. For it is a fa6i:, that all motions gradually di- 
 minifh, and, in a fliort time, terminate in reft. No 
 force feems neceflary for maintaining a ftate of reft. 
 But motion, they fay, is a violent ftate, the continual 
 production of an efFe£l, and therefore requiring a con- 
 tinuation of the caufe. Motion therefore requires the 
 continual exertion of the caufe. They fay that a body 
 in motion continues in it, only by the continual agency 
 of a force infufed into it in giving it the motion, and 
 inherent in it while in motion. They call it the inherent 
 force — VIS injita corpori moto, 
 
 150. But this is contrary to our clcareft experience, 
 and to any diftinO: notions that we can form of motion 
 as an effeCl of force. We are not confcious of any ex- 
 ertion, in order to continue our motion in Hiding or 
 fkating on, fmooth ice; and when any obftru61:ion comes 
 in our way, we feel dift'mBly our natural tendency to 
 continue our*fpeed undiminifhed — ^we feel that we muft 
 
 refift 
 
i02 FlUST LAW 
 
 refifl a tendency to fall forwards — we feel all obftrtic- 
 tions as checks on our fpeed, and think timt if the ice were 
 perfeBly fmooth, we fhould go on for ever. It is equally 
 contrary to our notions of a moving force. By its in- 
 ftantaneous action, it produces motion, that is, a fuc- 
 cefTive change of place, otherwife it produces nothing. 
 Or if, in any inftant of its action, it .do not produce a 
 continuing motion, it cannot produce it by continuing 
 to a£i:. Continuation of motion is implied in our very 
 idea of motion. In any inftant, the body does not 
 move over any fpace *, but it is in a certain condition 
 (however imperfectly underftood by us) or has a certain 
 determination, which we call velocity, by which, if not 
 hindered, a certain length of path is pafled over in a fe- 
 cond. This muft be eiFe£l:ed by the inftantaneous action 
 of the moving caufe, otherwife it is not a caufe of mo- 
 tion. In Ihort, motion is a Jlate or condition, into which 
 a body may be put, by various caufes, but by no means 
 a thing which can be infufed into a body, or taken out 
 of it. 
 
 Should it be faid that we have full evidence of a 
 force refiding in a moving body, by obferv^ing its impui- 
 five power, which is not to be found in the fame body 
 at reft, we may anfwer, that there are forces refiding in 
 moving bodies, but that they are equally inherent in 
 them when at reft, but that motion is necellary, in or- 
 der that thefe forces may be able to exert their adtion 
 on the other body long enough to produce a ienfible ef- 
 fed. Motion in the impelling body is not the caufe of 
 
 ' that 
 
OF MOTION. 103 
 
 tliaf of* the body impelled by it, but only an occafwn or 
 cpportunlty for the forces to act eifeclually, and without 
 which tjie other body would withdraw itfelf from the 
 acT^icn. The bow-firing mull continue prefling the arrow 
 forwards — the Piammer mufl follow the nail, that it may 
 drive it to the head by one blow. This will be clearly 
 fhewn as we proceed. 
 
 The gradual diminution and final ceflation of all mo- 
 tions mentioned above is granted, but is eafily explained 
 by flating the obftruclions. The diminution is obferved 
 to be precifely what fliould arife from thofe obftru£lions, 
 on the fuppofition that if there were no obflrucStion, 
 there would be no diminution. For example, where we 
 can fhew that the obllruclion is only half, the diminu- 
 tion of motion is only one half. This would not be, if 
 there were any diminution where there is no obflruc- 
 tion. A pendulum is foon brought to reft when vibrat- 
 ing in water \ it vibrates much longer in air ; and flill 
 longer in the exhaufted receiver of an air-pump. The 
 planets have continued for many ages v/ithout ihe fmallefl 
 perceptible diminution of their motions. 
 
 151. Another fe6l of philofophers deny this law al- 
 together, and affirm that matter is elTentially prone to 
 motion. Every body, when at liberty, begins to move, 
 and continually accelerates this fpontaneous motion. Bo- 
 dies are fo far from being iluggifh, that they are perpe- 
 |:ually a6:ive, 
 
 152. 
 
104 IIRST LAW 
 
 152. All thefe differences of opinion may be com- 
 pletely fettled, by adhering to the principle, that * every 
 * change is an effeB.^ It is a matter of facl, that the 
 human mind always confiders it as fuch. Therefore, the 
 law is ftri(Slly deduced from our ideas of motion and its 
 caufes •, for, even if it were effential to matter gradually 
 to diminiili its motion, and, at lafl, come to reft, this 
 would not invalidate the law, becaufe our underftanding 
 would confider this diminution as the indication of an 
 eflential, or, at leaft, a univerfal "property " of matter. 
 We fliould afcribe it to a natural retarding force, in the 
 fame way that we give tliis name to the weight of an 
 arrow difcharged ftraight upwards. The nature of exift- 
 ing matter would be confidered as the caufe, and we 
 fnould eftimate the law of its action as we have done in 
 the cafe of gravity ; and, as in that cafe, we fnould ftill 
 fuppofe that were it not for tliis particular property, the 
 material atom would continue its motion for ever un- 
 diminifhed. 
 
 This is quite fufTicient for all the purpofes of mecha- 
 nical philofophy. Nay, if wc aflumed any thing elfe in 
 this cafe, we fliould be led into continual blunders. 
 Should we fay that a body maintains its motion undimi- 
 nifhed folely by the aftion of an inherent force, we fliould 
 be obliged to adopt the opinion, that when one body in 
 motion impels another, part of this force is transfufed from 
 the impelling into the impelled body, and all the abfurdi- 
 tics which are neceHarily attaclied to this opinion. 
 
 Therefore, to conclude on this fubjecl:, let us confi- 
 der motion merely as a ftate or condition, into which 
 
 matter 
 
OF MOTION. It)5 
 
 matter m?.y be brought by various caufcs, and which, 
 like Its whitenefs or roundnefs, will remain, till fome ef- 
 ficient caufe fhall change it. This we have called the 
 mechanical condition of the hody^ and have fettled the 
 meaning of the term with fuflicient precifton. It con- 
 fifts In its velocity and direftion, and in no other cir- 
 cumftance. 
 
 In the next place, let us confider the change which 
 may be induced on it as confiiling folely in a change in 
 thefe circumftances, and this change as the only indica- 
 tion, the only mark, and the only proper meafure of the 
 changing caufe, that is, of tlie force (for we are confi- 
 dering mechanical caufes only). It is evident that^ as 
 far as this procedure will carry us, we acquire certain 
 knowledge, fufceptible of mathematical treatment. In 
 order to make our talk ufeful, we muft endeavour to learii 
 whether the deviations from uniform motion follow re- 
 gular laws — what the laws are — and to what bodies they 
 refer. 
 
 154. The deviations from uniform motion are dif- 
 coverable only by a comparifon with uniform motions. 
 But we cannot tell whether a propofed motion be uni- 
 form, unlefs we have an accurate meafure of time. For 
 it is to be learned only by obferving the proportions of 
 the fpaces, and thofe of the times, and by obferving that 
 thofe proportions are the fame. To obtain a meafure of 
 time, various contrivances have been employed. They 
 are all to this purpofc — An event is feleded, in which 
 
 O we 
 
ro6 SECOND LAW 
 
 we have no reafon to think tliat any variation occurs irj 
 the operation of thofe caufes which effe£luate its ac- 
 compUlhment. It is then prefumed that it will always 
 be accomplifhed in equal times. The rotation of the 
 heavens, in twenty-three hours and fifty-fix minutes and 
 four feeonds, has been agreed on as the ftandard to 
 which all other contrivances are referred or compared, 
 and their accuracy is eftimated by their agreement with- 
 this ftandard. 
 
 Second Law of Motion, 
 
 155. Every change of mottm is proportional to the 
 force imprejpdy and is made in the direBion of that force. 
 
 This alfo is little more than a tautological propofi- 
 tion. If a force is to be meafured only by the change 
 which it makes in the motion of a body, the propofitiont 
 is only a repetition of this meafure in different terms ; 
 for, furely, quantities are proportional to their accurate 
 meafures. Indeed, this would have been a fufficient de- 
 monftration, had not philofophers attempted it in another 
 way, which has given rife to a great fchifm in the efti- 
 mation of forces. They have attem.pted to demonftrate 
 it as an application of the undoubted maxim, that effecls 
 are proportional to their caufes. But it is eafy to fee that 
 this application cannot be made •, for it prefuppofes that 
 we know the proportion of the forces, and that of their 
 caufes, and that we perceive thofe proportions to be the 
 fame.-^Now, in moft cafes, this is impoflible ; for the 
 
 forces 
 
OT MOTION. 107 
 
 iorces are not obje£ls of our obfervatlon. We know 
 nothing of their proportions. When Newton fays that 
 gravity at the furface of tlie earth is 3600 times greater 
 than at the moonj he proves it by fhewing that the de- 
 flection caufed by it in a fecond, at the earth's furface, 
 is 3600 times greater than that of the moon. But this 
 is begging the queftion, or afluming this propofition as 
 true, unlefs this law of motion be admitted as an axiom. 
 There are very few cafes indeed, where we can fhew 
 that forces are proportional to the changes of motion 
 produced by them *, yet fuch cafes are not altogether 
 wanting. Thtis, a fpring flilyard can be "made, the rod 
 of which is divided by hanging on, in fucceflion, a num- 
 ber of perfeftly equal weights. The elaflicity of the 
 fpring, in its different flates of tenfion, is proportional 
 to the prefTures of gravity which it balances. — Should 
 we find that, at Quito in Peru^ a lump of lead draws 
 out the rod to the mark 3 1 2, and that, at Spitzbergen, 
 it draws it to 313, we feem entitled to fay that the pref- 
 fure of gravity at Quito is to its prelTure at Spitzbergen 
 as 312 to 313, on the authority of etFe6i:s being propor- 
 ;jional to their caufes. 
 
 But fuch cafes are extremely rare, becaufe it is fel- 
 dom that a natural power, accurately meafured in fome 
 other way, is ivhclly employed in producing the obferved 
 motion. Part of it is generally expended in fome other 
 M'ay, and therefore we frequently fee that the inotions 
 are not in the fame proportion with the fuppofed forces, 
 ^ut even though this could be ftrid;ly done, this would 
 O 2 pnlj 
 
I08 SECOND LAW 
 
 only be the proof of a general law or fact, whereas ths 
 pretenfions of the philofophers aim at a proof of it a 
 priori^ of an abftradl truth, 
 
 156. Sir Ifaac Newton feems to confider it only as 
 a phyfical law. In this fenfe, we are not without very 
 good arguments. 
 
 I. A ball moving with a double, triple, or qiiadruple 
 velocity, genierates in another, by impulfe, a double, or 
 triple, or quadruple velocity, or the fame velocity in a 
 double, &c. quantity of matter, and the ball lofes the 
 fame proportions of its own velocity. 
 
 II. Two bodies, mxceting wath equal quantities of 
 motion, mutually flop each other. 
 
 III. Two forces, which, by ailing fimilarly during 
 equal times, would produce equal velocities in fome third 
 body, will, by acting together during the fame timey pro- 
 duce a double velocity, 
 
 IV. If any preffure, acling for a fecond, produce a 
 certain velocity, a double preiTure, afting during a fe- 
 cond, will produce a double velocity in the fame body. 
 
 V. A force, which we know to a£l: equably, pro^ 
 duces equal increments of velocity in equal times, whatr 
 ever thefe velocities may be. 
 
 In all thefe examples, we fee the forces in the fume 
 proportion with the change of m.otion fimilarly produced 
 by them. 
 
 157. Bujt, about the middle of the 17th century, 
 pr Robert Hooke, Fellow of the Royal Society of Lon- 
 
 don^ 
 
OF MOTION. 109 
 
 don, difcoveved a vaft colledllon of fa(Sls, in which the 
 forces feemed to be in a very diflerent proportion. 
 
 1 . In the production of motion. Four fprings, e- 
 qual in ftrength, and bent to the fame degree, generated 
 only a double velocity in the ball which they impelled ; 
 nine fprings produced only ^ triple velocity, &c. 
 
 2. In the extin6bion of motion. A ball moving 
 with a double velocity will penetrate four times as deep 
 into a uniformly refifting mafs ; a triple velocity will 
 make it penetrate nine times as far, &c. 
 
 Thefe are but two inftances of an immenfe colieftion 
 of fa£l:s to the fame purpofe, and they are clofely con- 
 nected with the moft important applications of dynami- 
 cal fcience. 
 
 158. Mr Leibnitz eagerly availed himfelf of thefe faCiis, 
 as authority for declaring himfelf the difcoverer of the 
 real nature and meafure of mechanical aClion and force, 
 which he faid had hitherto been totally miftaken by 
 philofophers ; and he affirmed that the inherent force of 
 a body in motion was in the proportion, not of the ve- 
 locity, but of the fquare of the velocity. John Ber- 
 noulli, his zealous champion, warmly fupported him in 
 this argument, adducing a variety of the moft fimple 
 facts, all confirming this relation between the inherent 
 force of a body in motion and its velocity. They far- 
 ther fupported it by many m.etaphyfical confiderations, 
 relating to the procedure of nature in generating this 
 force and velocity, and the way in which it may be ex- 
 tinguifhed. The moil cogent argument offered by Leib- 
 nitz 
 
r I© SECOND LAW 
 
 nitz is, that the force inherent in a moving hody is to he 
 eftimated by all tliat it is able to do before its motion is 
 completely extinguiflied. When, therefore, it penetrates 
 four times as far, it fhould be confidered as having pro- 
 duced a quadruple effeft. The mechanicians of Europe 
 were divided in their opinions ; the Germans adhering 
 to that of Leibnitz, and the Britifh and French to that 
 of Des Cartes, who firft afhrmed the relation which we 
 have adopted as a fecond law of motion. We fhall fee 
 prefently, that, in the Leibijitzian meafure, many things 
 are gratuitoufly aflumed, many contradiftions are incur^ 
 red, and, finally, that it is cn/y becaufe forces are ajfumed 
 as proportional to the velocities nvhich they generate^ that 
 the facfs ohjerved by Hookcy and employed by Leibnitz, come 
 to be proportional to the fqiiares of the fame velocities. It 
 ihall only be noticed at prefent, that when Leibnitz af- 
 fumes the quadruple penetration as the proof of the qua- 
 druple force of a body having t\yice the velocity, he does 
 not confider that a double time is employed in this pene- 
 tration. Now, a double force adting equably during a 
 double time, fhould produce a quadruple efFedl. This 
 circumHiance is neglected in one and all of the fafts ad- 
 duced by Mr Leibnitz. It may be added, that his fol- 
 lowers, as well as him.felf, agree with us in every confe-? 
 quence which we draw from the meafure adopted by us. 
 They grant that a force which produces a uniformly ac-r 
 celerated motion is a conftant force, and they agree with 
 the Carteiians in all the valuations of accelerating and 
 defle£liiig forces, and have been amon^ the moft affi^u-: 
 
OF MOTIOJ^. 1 1 i 
 
 Ous and fuccefsful cultivators of the Newtonian philofo- 
 phy, which proceeds entirely on the meafure of moving 
 forces by the velocity which they generate. 
 
 159. We muft here obferve that wc are confidering 
 nothing but moving forces. When a ball has had a cer- 
 tain velocity given it, whether impelled by the air in a 
 pop-gun, or by a fpring, or ftruck off by a blow, or urged 
 forward by a flream of wind or water, or has acquired 
 it by falling, we conceive that in all thefe cafes it has 
 fuftained the fame adliion of moving force. Perhaps 
 preflure is the only difiincl notion we can form of force ; 
 but it is experience only that has informed us that pref- 
 fure produces motion, but does not produce heat or 
 fweetnefs. Produftion of motion is a circumftance in 
 which all mechanical forces may agree, while they may 
 differ in many others. By, or in, this circumftance of 
 refemblance, they may be compared, and get a name ex- 
 preffing this comparifon ; namely, moving force. There- 
 fore this particular faculty of preffure, elaflicity, &c. may 
 be meafured by the change of motion which prefigure pro- 
 duces. And whatever may be the proportions of pref- 
 fure on the quiefcent body, v/e may take it for granted 
 that the prefiure aBually exerted in the production of mo- 
 tion m.ay be meafured by the magnitude of the change of 
 motion. This is really the only change of mechanical 
 condition efFefted by the prefTure in the body moved by 
 it •, therefore it m.ay be meafured by the velocity. Ac- 
 cordingly, we find that v/hen the lame change of velo- 
 
 citv 
 
1 12 SECOND LAW 
 
 city is produced by preflure on a foft clay ball, the fame 
 preffure has really been exerted, whether the velocity has 
 been augmented from 99 to 100, or diminifhed from 4 
 to 3. For the fame dimple will be obferved in both 
 cafes. Nay, all our actions on the furface of this globe 
 are proofs of this. A ball fuflains the fame dimple whe- 
 ther we impel it, at noon-day, to the weftward or to the 
 eaftward, north or fouth, or though this fhould be done 
 at midnight ; yet the real velocities at noon and midnight 
 differ by nearly twice the velocity of a cannon ball bat- 
 tering in breach. This could not be, if the changes of 
 motion were not proportional to the exerted preflures. 
 
 160. The fame conclufion may be deduced from 
 our notions of a conftant or invariable force. It is furely 
 a force which produces equal eiFe61:s, or changes of mo- 
 tion, in equal times. Now, equal augmentations of mo- 
 tion are furely equal augmentations of velocity. We find 
 this notion of an invariable accelerating force confirmed 
 by what we obferve in the cafe of a falling body. This 
 receives equal additions of velocity in equal times ; and 
 we have no reafon to think that this force is variable. 
 We fhould therefore infer, that whatever force it imparts 
 in one fecond, it will impart four times as much in four 
 feconds.^ So it does, if we allow a quadruple velocity 
 to indicate a quadruple force 5 but in no otlier eflimation 
 of force. 
 
 To all this may be added, that although four fprings, 
 applied to an ounce ball, impel it only twice as faft as 
 
 one 
 
6^ MOTION. 11^ 
 
 one fpring will do, yet they will give the fame velocity 
 to a four ounce ball which one fpring gives to an ounce 
 ball. And we can demonftrate, to the fatisfa£lion of 
 Mr Leibnitz, that, in this lall cafe, the four fprings adl 
 during the fame time with the fingle fpring. 
 
 1(5 1. Therefore, finally, a change of motion^ in all its 
 drcumjlances of velocity and direBion, is the proper meafure 
 of a changing force. 
 
 But it is alfo the proper meafure of a moving force. 
 For bodies in different ftates of motion may fuftain one 
 and the fame change of motion. Now, fuppofe one of 
 thefe bodies to be previoufly at reft, the change which it 
 fuftains is the fame thing with the motion which it ac- 
 quires. Therefore the force which produces any change 
 of motion in a body already moving, is the fame with the 
 force which produces a motion equivalent to this change, 
 in a body previoully at reft, in which cafe it is, (imply, 
 a moving force. 
 
 It feemed necelTary to be thus particular in the ac- 
 count of this conteft about the meafure of forces, becaufe 
 Mr Leibnitz's opinion has influenced the fentiments of 
 many writers of reputation -, and feme of them, particu- 
 larly Gravefande and Mufchenbroek, have mixed it a 
 good deal with their pra£li^cal deductions. There could 
 not have been any difpute, had not philofophers allowed 
 themfelves to confider force as fomething exifting in 
 body, whereas the term is never ufed to exprefs any rea- 
 lity except the phenomenon v/hich they conceived to bs 
 
XI4 SECOND LAW 
 
 its full effe£i and adequate meafure. It is quite aHaW" 
 able to meafure afcefjJJofia/, or penetrating force, by the 
 afcenfion and the penetration, and to remark that thcfe are 
 as the fquare of the velocity. But this muft not be con- 
 fidered as the general, or the beft, meafure of force, and 
 particularly of moving force. This inuf be meafured by 
 the fimple change of motion which is produced by it. 
 And this meafure has the advantage of being equally ap- 
 plicable to the phenomena of afcenfion and penetration, 
 as we fhall fee very foon. We may now enounce it in a 
 different form, adapted to the chara£leriftic and meafure 
 of a change of motion, which was fliown in art. 79. to 
 be the moft proper. 
 
 Law of the Changes of Motion. 
 
 162. In every change of a motion from AB (fig. 23.} 
 to A D, the new motion A D is compounded of the former 
 motion A B, and of the ?}7otion A C, luhir.h the changing 
 force produces in a body at reft. 
 
 For it was fliewn in art. 79. that the change in any 
 motion is that motion which, when compounded with 
 the former motion, produces the new motion ; and, in 
 art. 81, that the new motion is that compounded of the 
 former motion and the changing motion. Nov/, fince 
 the change of motion is the characleriftic and the mea- 
 fure of the changing force (161.), determining both its 
 direclion and its intenfity, or the velocity produced by it, 
 the propofition follows of courfe. 
 
 163. 
 
0"F MOTION. 115 
 
 163. It was remarked in art. 80, tiiat the compofi- 
 lion of motions, and the limihir compofition of forces, 
 are two very different things. The firft is a truth, purely 
 mathematical, and as certain as any theorem in geome- 
 try. The fecond is a phyfical queftion entirely, depend- 
 ing on the nature of the m.echanical forces which exift in 
 tlie uniyerfe. We do not clearly fee that two forces, 
 each of which will feparately produce motions havino- 
 the dire6lions and velocities expreffcd by the fides of a 
 parallelogram, will, by their joint a6lion, produce a mo- 
 tion in the diagonal. The demonflirations given of this 
 propofition by aimofl all the writers of Elements are alto- 
 gether inconclufive, being ali fimilar to the cafe of a man 
 walking on a field of ice, while the ice floats down a 
 Itream. This is only the compofition of motions. Other 
 writers, endeavouring to accommodate their reafonings 
 to phyfical principles, have afilimed poftuiates that ap- 
 pear gratuitous. The firft legitimate demonftration was 
 given by Dan. Bernoulli, in the Comment. Petropol. Vol. L 
 But it employs a feries of many propofitions, fome of 
 which are very abftrufe. Mr D'Alembert gre?.dy fimpH- 
 fied and improved this demonftration, in a Memoire of 
 the Acad, des Sciences 1769. But this alfo requires 
 many propofitions. Fonfenex and Pviccati, in vol. III. 
 of the Mcmoires of the Academy of Turin, have given 
 another very ingenious one. lyAIembert has alfo im- 
 proved this demonftration, and has given another, in the 
 fume Memoires, and one in his Dyiamique. The firft is 
 y^ry refined and obfcure, and the fecond does not feem 
 P 2 very 
 
llS OF THE COMPOSITION 
 
 very conclufive. An attempt is made in the Encyclcp. 
 Britan, SuppL § Dynamics, to combine Bernoulli's, 
 D'Alembert's, and one by F. Frifi, which is more expe- 
 ditious than either of the two firft, and appears legiti- 
 mate» The demonftration given in this place is undoubt- 
 edly com.plete, if the reafoning be complete that is em- 
 ployed in art. 79, to prove that the motion which, when 
 compounded with the former miction, produces the new 
 motion, is the true change of motion. We apprehend it 
 to be fo. 
 
 164. We have moft abundant proof of this law of 
 motion, if we confider it merely as a phyfical law, or 
 univerfal fa£l. 
 
 I. Nothing is more famihar than the joint aftion of 
 different forces. Thus, we frequently fee a lighter drag- 
 ged in different dire6lions by two track-ropes, on differ- 
 ent fides of the canal, and the lighter moves in an inter- 
 mediate direction, in the fame manner as if it were drag- 
 ged by one rope in that dire£lion. 
 
 In like manner, we may obferve that if a ball, mov- 
 ing in a particular direction, receive a ftroke athwart this 
 direction, it takes a direclion which lies between that of 
 the primitive motion and that of the tranfverfc ftroke. 
 
 165. 2. If a point or particle of matter A (fig. 23.) 
 be urged at once by two preffures, in the directions A B 
 and A C, and if A B and A C are proportional to the 
 mtenfities of thofe prefRires, the joint action of tliefe 
 
 two 
 
01 MOVING FORCES. 1 1? 
 
 two prefTures is equivalent to the action of a" third jpref- 
 fure, in the dire<£lion of the diagonal AD, having its 
 intenfity in the proportion of A IX This is completely 
 proved, by obferving that the point A MnW be withheld 
 from moving, by a preffiire A E, equal and oppofite to 
 A D. Now, we know that prellures are moving forces, 
 and produce velocities (when acting fimilarly during e- 
 qual times) proportional to their intenfities. Therefore, 
 the proportion is true with refpe^b to prefTures confider- 
 cd merely as preflures, and alfo with refpe£l to the mo- 
 tions produceable by their compofition, 
 
 1 6(5. 3. A ball fufpended by a thread, and drav/n 
 afide from its quiefcent pofition, is urged downwards by 
 its weight, and is fupported obliquely by the thread. 
 We can fay precifely what are the directions and intenfities 
 of the forces which incite it to motion in any pofition, 
 and what velocities will refult from them, upon the fup- 
 pofition of the truth of this propofition. And we can 
 tell what number of ofcillations it will make in a day. 
 It is a fa£l, that, when every thing is executed with 
 care, the num.ber of vibrations will not differ from our 
 computation by one unit in a hundred thoufand. 
 
 4. Laftly, the planetary motions, computed on the 
 fame principles of the compofition of forces, exhibit no 
 fenfible deviation from our calculations, after thoufands 
 of years. 
 
 There is nothing therefore that we can reft on with 
 greater confidence, than the perfedl agreement between 
 
1 l8 OF THE COMPOSITION 
 
 the compofition of motions and the compofition of the 
 forces which would, feparately, produce thofe motions, 
 and are meafured by the velocities which they generate. 
 
 It particularly deferves remark, that if we meafure 
 moving forces by the fqu^^res of the velocities which 
 they generate, the com.pofition is impofiible •, that is, two 
 forces reprefented by the fides of a parallelogram made 
 proportional to the fquares of the velocities, will not 
 compofe a force which can be reprefented by the dia- 
 gonal. Yet nature fhev.s the exaft compofition of forces, 
 on the fuppofition that they are as the velocities. 
 
 Therefore, finally, whether we confider this propo- 
 fition as an abflraft truth, or as a phyfical law, it may 
 be confidered as fully eftabliflied. Its converfe is th.e 
 following. 
 
 167. ^he force 'which changes the motion AB inte 
 A D, is that which 'would produce in a quiejcent body the 
 motion A C, whichy when compounded with A B, produces 
 the motion obferved A D. 
 
 168. A force which will produce in a quiff cent body 
 a motion having the direclion and velocity reprefented by 
 AO, if applied to a body moving with the velocity and in 
 the direHion A B, will change its motion into the motion 
 A D, the diagonal of the parallelogram A B D C. For the 
 new motion muft be that compounded of A B and A C 
 (30.), that i&, muft be AD (83.) 
 
 From 
 
OF MOVING FORCES. I Ip 
 
 From thefe two propofitions combined arifcs a third, 
 which is the moft general ; viz. 
 
 169. If a body A be urged at once by two forces^ 
 which would, feparate/y, caufe it to defcribe A B and A C, 
 
 Jides of a parallelogram A B D C, the body will, by their 
 joint aBion, defcribe the diagonal A D in the fame time. 
 For, had the body been already moving with the velocity 
 and in the direction A B, and had it been afted on in A 
 by the force AC, it would defcribe AD in the fame 
 time (i68.)« Now, it is immaterial at what time it got 
 the determination by which it would defcribe A B. Let 
 it therefore be at the inftant that the force A C is appHed 
 to it. It muft defcribe A D, becaufe its mechanical con- 
 dition in A, having the determination to the motion AB, 
 is the fame as in any other point of that line. 
 
 170. Cor, Two forces, acting on a body in the 
 fame, or in oppofite directions, will caufe it to move 
 with a velocity equal to the fum, or to the difference, of 
 the velocities which it would have received from the 
 forces feparately. For, if A C approach continually to 
 A B, by diminifhing the angle B A C, the points C and 
 D will at laft fall on c and d, and then A D is equal to 
 the fum of AB and AC. But if the angle B AC in- 
 creafe continually, the points C and D will, at laft, fall 
 on X and ^, and then A ^ becomes equal to the difference 
 of A B and A C. 
 
 In the laft cafe, it is evident that if A C be equal to 
 A B, the point D or ^ will coincide with A, and there 
 
 wii; 
 
I20 OP THE COMPOSITION 
 
 will be no motion, the two forces being equal, and act- 
 ing in oppofite (lire6lions. 
 
 171, In fucli a cafe, the equal and oppofite forces 
 AC and AB are f^id to balance each other, and, in 
 general, thofe forces which, by their joint operation, pro- 
 duce no change of motion, are faid, in like manner, to 
 balance each other •, and they are accounted equal and 
 oppofite, becaufe each produces on the body a change 
 of motion equal to what it would produce on a body at 
 reft, and at the fame time equal to the motion produced 
 by the other force on a body at reft. Thefe two mo- 
 tions are therefore equal and oppofite, and therefore the 
 forces are fo. 
 
 172. We may now apply to the motions produced 
 by the combined action of forces all that v/as demonftrat- 
 ed concerning the affeftions of compound motions, in 
 the articles 83, 84, 85, 86, 87, 88, 89, & 90. 
 
 But, in making this transference, we muft carefully 
 attend to the effential difference between the compofition 
 of motions and the compofition of forces. In this laft, 
 the compofition is complete, as foon as the body has 
 gotten the determination to move in the diagonal with 
 the proper velocity, and after this there is no more com- 
 pofition. The body then moves uniformly, till fome force 
 change its condition. But, in the compofition of two or 
 more motions, the two conftituent motions are fuppofed 
 to continue, and hy their contmuance onh^ does the com- 
 
 poun«l 
 
OF FORCES. I2t 
 
 poiiiul motion exift. If any force can generate a finite 
 velocity by its inllantaneous a6lion (which does not ap- 
 pear poflible), two fucli forces generate the determina- 
 tion in the diagonal in an inftant. But if the aftion mufl 
 continue for feme time, in order to generate the veloci- 
 ties A B or AC, the joint aclion muft continue during 
 the fame time, in order to produce the velocity AD, 
 Alfo, it is neceffary that, during the whole time of their 
 joint aftion, the moving powers of the two forces mufh 
 retain the fame proportion to each other, although they 
 may perhaps vary in their intenfity during that time. 
 From not attending to this circumftance, many experi- 
 ments, which have been made in order to compare this 
 doftrine with the phenomena, have exhibited refults 
 which deviiite greatly from it. The experiments made 
 "by the combination of preflures, fuch as weights pulling a 
 body by means of threads, agree with this do61:rine with 
 the utmoft precifion, it being always found that two 
 weights pulling in the direftions A B, AC, and propor- 
 tional to thofe lines, are exa<Slly balanced by a third 
 weight in the proportion of A D, and pulling in the di- 
 reftion AE. By thefe, the compofition of prejfures is 
 moft unexceptionably proved ; and, feeing that we have 
 fcarcely any other clear conception of a moving force, 
 thefe experiments may be confidered as fufficient. But 
 we need not ftop here ; for we have the moft diftin6t 
 proof, by experiment, that prefHires produce motions in 
 proportion to their intenfities by their fimilar a61:Ion dur- 
 ing equal times. The planetary motions, in w^hlch the 
 
 Q diredlions 
 
J 22 COMPOSITION 
 
 dire£^ions and intenfities of the compounded forces arr 
 accurately known as moving forces, complete the proof 
 of the phyfical law, by their exquifite agreement with 
 the calculations proceedmg on the principles of this doc- 
 trine. This perfe£t agreement muft be received as a full 
 proof of the propriety of the meafure of a moving force 
 which we have affumed. Any other meafure would give 
 refults widely different from the phenomena. 
 
 173. The force which fmgly produces the motion 
 in the diagonal, may be faid to be equivalent to the 
 forces which produce the motions in the fides of the pa- 
 rallelogram. It may alfo be called the compound force, 
 and the resulting force ; and the forces which act in 
 the direftion of the fides, may be called the simple 
 FORCES, or the constituent forces. 
 
 174. ^he two conjlituefit forces and their refulting 
 force aEl in one plane ; and they are proportional to the three 
 fides of a triangle haying their dire5lionSy or of any fimilar 
 
 triangle (84). 
 
 175. Each force is proportional to thefne of the angle 
 contained by the direclions of the ether two. For the fides 
 of any triangle are as the fiiics of tlie oppofite angles. 
 
 176. A force a£ling in the dire£i:ion parallel to any 
 line B D does not affe£l the approach toward that line, 
 or its recefs from it, occafioned by the adion of another 
 
 force 
 
OF FORCES. 123 
 
 Force AC. For, becaiife the motion AD is uniform, 
 the points 5 and e, to which the body would have gone 
 by the force A B, are at the fame diftance from B D 
 with the points d and e to^which it reilly goes in the 
 fame time, by the joint adion of the forces AB and 
 AC 
 
 177. A body under the influence of any number of 
 forces AB, AC, AD, AE, (fig. 12.) will defcribe the 
 line A F, determined as in article 86. 5 and A F will ex- 
 prefs the equivalent or refulting force, both in refpecl of 
 ^iredlion and intenfity. 
 
 178. Any force AB may be conceived as refulting 
 from the joint a£l;ion of two or more forces having any 
 directions whatever, and their intenfities may be com- 
 pared as in art. 85. 
 
 1 79. Forces may be ejhmaUd in the direction of a 
 given line or plane, or may be reduced to that direction, 
 as in art. 35. 
 
 1 80. Any number of forces, acting on a particle of 
 matter, will be balanced by a force equal and oppofite to 
 their refulting or equivalent force. 
 
 181. If any number of forces are in equilibrio, and 
 are eftimated in, or reduced to, any one diredlion, or in 
 one plane, the reduced forces are in equilibrio. 
 
 Q 2 Te 
 
J 24 THIRD LAW 
 
 To thefo two laws of moLion, which we have iiU. 
 tempted to fhew to be neceflary confequences of the re-, 
 lations of thofe conceptions which we form of motion 
 and of mechanical force, arid alfo to be unlverfal facts or 
 phyfical laws, Sir Ifaac Newton has added another, or 
 
 Third Law of Motioih 
 
 182. The aclions of bodies on one another are ahvays 
 ■inutuali equali and in contrary dircclions. It is ufually 
 exprefled thus — Readion is always equal and contrary t'j^ 
 
 aciion. 
 
 This is indeed a fa£V, obferved without exceptioD, 
 in all tlie cafes which we can examine with accuracy. 
 Sir Ifaac Newton, in the general fcholium or remark on 
 the laws of motion, feems to confider this equahty of 
 action and reaction as an axiom deduced from the rela- 
 tions of ideas. But this feems doubtful. Becaufe a 
 magnet caufes the iron to approach towards it, it does 
 not appear that we neceflarily fuppofe that iron alfo at- 
 tracts the magnet. The faft is, that although many ob- 
 fervations are to be found in the writings of the ancients 
 concerning the attractive power of the magnet, not one 
 of them has mentioned the attra(Stive power of the iron. 
 It is a modern difcovery, and Dr Gilbert is, I think, the 
 earlieft writer, in whofe works we meet with it. He 
 aiErms that this mutual attraction is obferved betv^een 
 the magnet and iron, and betv/een ail electrical fub- 
 ftances and the light bodies attracted by them. Kepler 
 
 noticed 
 
OF MOTION. 125 
 
 iiGticct-l tlils mutu.il influence between the Earth and the 
 Moon. Wailis, Wren, and Huyghens, firft diftin£tly 
 ailirmed the nuitunl, equal, and contrary a6]:ion of folid bo- 
 dies in tkelr collifions -, and it l?as been confirmed by innu- 
 merable obfervations. Nay, fuice that time, Sir Ifaac New- 
 ton himfelf on\y prefumed that, becaufe the Sun attraded 
 the planets, thefe alfo attracted the Sun j and he is at much 
 pains to point out phenomena to afcronomers, by which 
 this mny be proved, when the art of obfervation fhali 
 be fufliciently improved. Thefe mufh be put on the 
 fame footing with the phenomena by which the mutual 
 a(!:l:ions of the planets are proved. Now, this laft a£Hon 
 was altogether a prefumption, although the proof was by 
 far the mofl; eafy. The difcovery and complete demon- 
 ilration of this, as a phyfical law, is certainly the moll 
 illullrious fpecimen of Newton's genius and nice judge- 
 ment. 
 
 We mufl receive it therefore as a lav/ of motion, 
 with refpecl to all bodies on which we can make ex- 
 periment, or obfervation fit for deciding the queftion. 
 
 183. As it is an univerfal law, we cannot rid our- 
 felves of the perfuafion that it depends on fome general 
 principle, which influences all the matter in the univerfe. 
 It powerfully induces us to believe that the ultimate a- 
 toms of matter are all perfectly alike — that a certain col- 
 leflion of properties belong, in the fame degree, to every 
 atom — and that all the fenfible ditFerences of fubftance 
 -yv'hich we obferve arife from a diiFerent combination of 
 
 priipary* 
 
126 g:eneral observations 
 
 primary atoms in the fon nation of a particle of thofe 
 fubftances. A very flight confideration may fliew us 
 that this is perfectly poflible. Now, if fuch be the con- 
 ftitution of every primary atom, there can be no adlion 
 of any kind of particle, or collection of particles, on an- 
 other, u^hich will not be accompanied by an equal re- 
 a£lion in the oppofite diredlion. Nothing can be clearer 
 than this. This therefore is, in all probability, the ori- 
 gin of this Third Law of Motion. 
 
 1 84. The aim of the Newtonian philofophy, which 
 we profefs to follow, is to invefligate the laws obferved 
 in the produ£lion of natural efl'eO:s, and to comprehend 
 any propofed phenomenon in one or other of thofe laws. 
 We then account it as explained. 
 
 Thefe general, but ftill fubordinate, laws are to be 
 cftabliilied only by obfervation and experiment ; but 
 when fo eftablifhed as far as obfervation extends, it b 
 only by means of fome obferved analogy that we can 
 ufe them as explanations of many other phenomena. 
 With this we muft reil fatisfied, becaufe it feems im- 
 pofTible for our faculties to difcover the efficient caufes 
 of thofe general laws, fo as to be able to demonftrate 
 that t y wuj} be fuch as we obferve. But in the eftab- 
 liihm^^^t of them as mere matters of fa61:, we may ob- 
 ferve them to be of various extent, and that fome are 
 iubdivifions of others. In this fubordination, v/e can dif- 
 
 cern 
 
ON THE LAWS OF MOTION. I72 
 
 C€rn much order, harmony and beauty, and our minds 
 are left deeply imprefled with admiration of the wifdom 
 and fkill of the contrivance, by which this magnificent 
 fabric is fitted for the accomplifhment of a great and 
 beneficent purpofe. 
 
 185. The three axioms, and, indeed, the two firft, 
 feem to include the whole firft principles of Dynamics, 
 and enable us, without other help, to accomplifh every 
 purpofe of the fcience. Some authors of eminence have 
 thought that there were other principles, which influ- 
 enced every natural operation, and that thefe operations 
 could not be fully underflood, nor an explanation pro- 
 perly deduced, without employing thofe principles. Of 
 this kind is the principle of oeconomy of action, or 
 SMALLEST ACTION, affirmed by Mr Maupertuis to be 
 purfued in all the operations of nature. This philofopher 
 fays, that the perfe£l; wifdom of Deity muft caufe him 
 to accomplifh every change by the fmalleft poffible ex- 
 penditure of power of every kind ; and he gives a theo- 
 rem which he fays exprefTes this oeconomy in all cafes of 
 mechanical action. He then afTerts that, in order to 
 fhew in what manner fuch and fuch bodies, fo and fo 
 fituated, fhall change each other's condition, we muft 
 find what change in each will agree with th*s value of 
 the fmalleft a6lion. He applies this to the folution of 
 many problems, fome of which are intricate, and p-ives 
 folutions perfectly agreeable to the phenomena. 
 
 But the fact is, that the theorem was fuggefted by 
 the phenomena, and is only an induction of particulars. 
 
 It 
 
128 GENERAL OBSERVATIONS 
 
 It is a law, of a certain extent, but by no means a firft 
 principle ; for the law is comprehended in, and is fub- 
 ordinate, by many degrees, to the three laws of motion 
 now ellabliflied. It is no juft exprefiion of a minimum 
 of a£^ion ; and he has obtained folutions, by its means, 
 of problems, in which its elements are akogetlier fuppo- 
 fititious, which is proof fufTicient of its nullity and im- 
 propriety. 
 
 1 86. Mr D' Alembert and Mr De la Grange have alfo 
 given general theorems, which they call firft principles, 
 and which they tliink highly neceffary in dynamical dif- 
 quifitions. Thefe, too, are nothing but general, but very 
 fubordinate laws, mod ingenioufly employed by their au- 
 thors in the folution of intricate problems, where they 
 are really of immenfe fervice. But flill they are not 
 principles ; and a perfon may under ft and the mechanique^ 
 analyiique of De la Grange, by ftudying it with care, 
 and yet be very ignorant of the real natural principles of 
 mechanifm. All thefe theorems are only ingenious com- 
 binations of the fecond and third Newtonian Laws of 
 Motion. 
 
 187. The application or employment of thefe laws 
 is to a twofold purpofe. 
 
 I . To difcover thofe mechanical powers of natural 
 fubftances which fit them for being parts of a permanent 
 univerfe. We accomplifh this by obferving what changes 
 of motion among the neighbouring bodies always accom- 
 pany 
 
OF ACCELERATING FORCES. np 
 
 pnny thofe fubflances, wherever they are. Thefe changes 
 are the only charaderlflics of the powers. It is thus 
 that we difcDver and defcVibe the power of magnetlfm, 
 gravity, &-c. 
 
 2. Having obtained the mechanical chara6Ver of any 
 fubftance, vfe afccrtain what will be the refult of its be- 
 ing in the vicinity of the bodies mechanically allied to it, 
 or we afcertain what change will be induced on the con- 
 dition of the neighbouring bodies. 
 
 To fave us a great labour, which mud be repeated 
 for every queflion, if we make immediate application of 
 the laws of motion to the phenomenon, it will be ex- 
 tremely convenient to have in readinefs a few general 
 rules, accommodated to the more frequent cafes of na- 
 tural operations. The mechanical powers of bodies oc- 
 cafionally accelerate, retard, and defie6l the motions of 
 other bodies. Therefore it is proper to premife the prin- 
 cipal tlieorems relating to the a6tion of accelerating, 
 retarding, or deflecting forces. They have got thefe 
 names, becaufe we know nothing of their nature, or of 
 the m.anner in which they are effective, and therefore 
 name them, as we meafure them, by the phenomena 
 which we confider as their eiFefts, 
 
 Of udccekrat'mg and Retarding Forces. 
 
 1 88. Since we have adopted the changes of motion 
 tis the marks and meafures of the forces, it is evident 
 that every thing already faid of accelerations and retarda- 
 
 R tions 
 
i^O ACCELERATING rORCE^. 
 
 tions is equally defcriptive of the effeds of accelerating 
 and retarding forces. Therefore, 
 
 If the abfdjfa ad (Jig. 5.) reprejent the time of any vio- 
 tioHy and if the areas abfe, acge, &c. are as the velo- 
 cities at the injiants b, c, &c. the orciinates a e, b f , eg, &c. 
 are as the accelerating forces at thofe infants (69). 
 
 189. Cor. I. The momentary change of velocity 
 
 is as the force f and the time / jointly, which may be 
 
 thus expreffed (71.) 
 
 Vy or — ^', =ft. 
 
 Alfo, the accelerating or retarding force is proportional 
 
 to the momentary variation of the velocity, direftly, and 
 
 to the moment of time in which it is generated, in- 
 
 verfely (71.) 
 
 ^ . "y . — iy 
 
 ;~-ry or ~ —r. 
 
 t t 
 
 Indeed, all that v/e know of force is that it is fome- 
 thing which is alvv'ays proportional to -. 
 
 190. Cor. 2. Uniformly accelerated or retarded mo- 
 tion is the indication of a confant or invariable accelerating 
 force. For, in this cafe, the areas ahfe^ ^^t-'g^s &c. 
 increafe at the fame rate with the times ab, ac, &c. 
 and therefore the ordinates a Cy bf c g^ &c. muft all be 
 equal ; therefore the forces reprefented by them are the 
 fame, or the accelerating force does not change its in- 
 tenfity, or, it is conftant. If, therefore, the circum- 
 
 ftances 
 
ACCELERATING FORCES. 13! 
 
 fiances mentioned in articles 54, 55, 56, 57, 58, 59, 60, 
 61, are obferved in any motion, the force is conftant. 
 And if the force is known to be conftant, thofe propofi- 
 tions are true refpe£ling the motions. 
 
 191. Ccr. 3. No finite change if velocity is generated 
 in an inftant by any accelerating or retarding force. For 
 the increment or decrement of velocity is always ex- 
 prefled by an area, or by a product y/, one fide or fa61:or 
 of which is a portion of time» As no finite fpace can 
 be defcribed in an inftant, and the moveable muft pafs 
 in fucceflion through every point of the path, fo it muft 
 acquire all the intermediate degrees of velocity. It muft 
 be continually accelerated or retarded. 
 
 192. Cor. 4. The change of velocity produced io 
 a body in any time, by a force varying in any manner, is 
 the proper meafure of the accumulated or whole action 
 of the force during this time. For, fince the mom.en- 
 tary change of velocity is exprefled by ft, the aggregate 
 of all thefe momentary changes, that is, the whole change 
 of velocity, muft be exprefled by the fum of all the 
 quantities ft. This is equivalent to the area of the 
 figure employed in art. 188, and may be exprefled by 
 
 > 
 
 193. If the ahfcijfa A E (fig. 8.) of the line zee bt 
 ihe path along ivhich a body is urged by the aBion of a fore e^ 
 
 R % %'arying 
 
132 ACCELERATING FORCES. 
 
 varying in a7iy manner y and if the ordinates A <7, B b, C c^ 
 &c. be proportional to the intejifities of the force in the dif- 
 ferent points of the pathy the intercepted areas ivill be pro- 
 portional to the changes made on the fquare of the velocity 
 during the motion along the correfpondijig portions of the 
 path. 
 
 For, by art. 72, tlie areas are in this proportion when 
 the ordinates are as the accelerations. But tlie accelera- 
 tions are the mea hires of, and are therefore proportional 
 to, the accelerating forces. Therefore the propofition is 
 manifeft. 
 
 194. Cor. I. The 2riomentary change on the fquare 
 of the velocity is as the force, and as the fmall portion 
 of fpace along which it a^fls, jointly; 
 V V =^ f s 
 
 and /=f-^ 
 
 V V 
 
 s 
 
 I 195. It deferves remark here, that as the moment- 
 ary change of the fniiple velocity by rny force f depends 
 only on the time of its afticn, it being r=y if (^89.), fo 
 the change on the fquare of the velocity depends en the 
 fpace, it being — f s. It is the fame, whatever is the 
 velocity thus changed, or even though the body be at 
 reft when the force begins to act on it. Thus, in every 
 fecond of the falling of a heavy body, the velocity is aug- 
 mented 32 feet per fecond, and in every foot of the fall, 
 the fquare of the velocity increafcs by 64. 
 
 50$. 
 
ACCELERATING FORCES. I3-3 
 
 196. The whole area AY. e a, exprefled ^f J f h 
 expreffes the whole change made on the fquare of the 
 velocity wliich the body had in A, whatever this velocity 
 may have been. We miay therefore fuppofe the body to 
 have been at red in A. The area then meafures the 
 fquare of the velocity which the body has acquired in 
 the point E of its path. It is plain that the change on 
 v^ is quite independent on the tim.e of action, and there- 
 fore a body, in paffing through the fpace A E with any 
 initial velocity whatever, fuftains the fame change of the 
 fquare of that velocity, if under the influence of the 
 fame force. 
 
 197. This propofition is the fame with the 39th of 
 the Firfl Book of Newton's Principia, and is perhaps the 
 mod generally ufeful of all the theoremiS in Dynamics, 
 in the folution of pra6lical queltions. It is to be found, 
 without denionflration, in his earlieil writings, the Op^ 
 tical Leftures, which he delivered in 1669 and foUow- 
 ing years. 
 
 198. One important ufe may be niade of it at pre- 
 fent. It gives a complete folution of all the fac^s which 
 were obferved by Dr Hooke, and adduced by Leibnitz 
 with fuch pertinacity in fupport of his meafure of the 
 force of moving bodies. All of them are of precifely 
 the fame nature with the one mentioned in art. 157, or 
 with the faft, " that a ball projected direflly upwards 
 ^* with a double velocity, will rife to a quadruple height, 
 
 " and 
 
134 ACCELERATING FORCES. 
 
 *' and that a body, moving twice as faft, will penetrate 
 ** four times a$ far into a uniformly tenacious mafs. " 
 The uniform force of gravity, or the uniform tenacity 
 of the penetrated body, makes a uniform oppofition to 
 the motion, and may therefore be confidered as a uni- 
 form retarding force. It will therefore be reprefented, 
 in fig. 8, by an ordinate always of the fame length, and 
 the areas which meafure the fquare of the velocity loft 
 will be portions of a rectangle AEia. If therefore A E 
 be the penetration necelTary for extinguifhing the velo- 
 city 2, the fpace AB, necelTary for extinguifliing the 
 velocity I, muft be ;^ of AE, becaufe the fquare of i 
 is 4 of the fquare of 2. 
 
 199. What particularly defer ves remark here, is, 
 that this propofition is true, on/y on the fiippofition that 
 forces are proportional to the velocities generated by them in 
 equal times. For the demonftration of this propofition 
 proceeds entirely on the previoufly eftablifhed meafure 
 of acceleration. We had nj =ft ; therefore v v ^ft v. 
 But / «u == J- ; therefore v v ==' f s, which is precifely 
 this propofition. 
 
 200. Thofe may be c^WqA ft7nilar points of fpace, 
 and fimilar inilants of time, which divide given portions 
 of fpace or time in the fame ratio. Thus, the beginning 
 of the 5th inch, and of the 2d foot, are fimilar points 
 of a foot, and of a yard. The beginning of the 21st 
 
 minute, 
 
ACCELERATING FORCES. I35 
 
 minute, and of the 9th hour, are fimilar inftants of an 
 liour, and of a day. 
 
 Forces may be faid to a£l; ftmilarly when, in fimilar 
 inltants of time, or fimilar points of the path, their in- 
 tenfities are in a conftant ratio. 
 
 201. Lemma, If two bodies be fimilarly acceler- 
 ated during given times a c and h k (fig. 24.), they are 
 alfo fimilarly accelerated along their refpeftive paths A C 
 and H K. 
 
 Let a, by c be inftants of the time a c, fimilar to the 
 inftants />, /, k of the time h k. Then, by the fimilar 
 accelerations, we have the force a e \h 1-=. b f : i m. This 
 being the cafe throughout, the area af is to the area h m 
 as the area a g x.o the area h n (Symbols (/)). Thefe 
 areas are as the velocities in the two motions (71.) 
 Therefore the velocities in fimilar inftants are in a con- 
 ftant ratio, that is, the velocity in the inftant b is to that 
 in the inftant z, as the velocity in the inftant c to that in 
 the inftant l. 
 
 The figures may now be taken to reprefent the times 
 •f the motion by their abfciflse, and the velocities by 
 their ordinates, as in art. 45. The fpaces defcribed are 
 now reprefented by the areas. Thefe being in a con- 
 ftant ratio, as already fliewn, we have A, B, C, and 
 H, I, K, fimilar points of the paths. And therefore, in 
 fimilar inftants of time, the bodies are in fimilar points 
 ef the paths. But in thefe inftants, they are fimilarly 
 accelerated, that is, the acceleratioyis and the forces are 
 
 in 
 
J3(J ACCELERATING FORCE?. 
 
 in a conftant ratio. They are therefore in a cOnftanS 
 ratio in fimilar points of the paths, and t\\Q, bodies are 
 fimilarly accelerated along their refpective paths (200.). 
 
 202. If two particles of matter are fimilarly urged 
 h\ accelerating or retarding forces during given iimes^ the 
 whole changes of velocity are as the forces and times jointly ; 
 or \ ==i t. 
 
 For the abfciflk a c and /; k will reprefent the times, 
 and the ordinates a e and h I will reprefent the forces, 
 and then the areas will reprefent the changes of velocity, 
 by art. 70. And thefe areas are as ^ r "%. a e tohl X h ly 
 (by Symbols (/. Cor.) 
 
 Hence ^ = 7^, and f = - . 
 
 203. If two particles of matter are fimilarly impel* 
 led or oppofed through given fpaces, the changes in the 
 fquares of velocity are as the forces and fpaces jointly ; or 
 
 4fs. 
 
 This follows, by fimilar reafoning, from art. 72. 
 
 It is evident that this propofition applies dire£i:ly to 
 the argument fo confidently urged for the propriety of 
 the Leibnitzian meafure of forces, namely, that four 
 fprings of equal ftrength, and bent to the fame degree, 
 generate, or extinguiih, only a double velocity. 
 
 204. If two particles of matter are fimilarly impelled 
 through given fpaces ^ the fpaces are as the forces and the 
 fquares of the times jointly^ 
 
 For 
 
OF MOTIVE FORCES. !37 
 
 For the moveables are fimilarly urged during the times 
 of their motion (converfe of 2oi.) Therefore v =f ftf 
 and v^ ==/* /* j but (203.) v^ =rfs. Therefore fs =f 
 \pt\ and/4/^\ 
 
 J" X 
 
 Cor. /*=T> and /=-. That is, the fquares of 
 
 the times are as the fpaces, directly, and as the forces, 
 inverfely ; and the forces are as the fpaces, dire£lly, and 
 as the fquares of the times, inverfely. 
 
 205. The quantity of motion in a body is the fum 
 of the motions of all its particles. Therefore, if all are 
 moving in one dire6tion, and with one velocity v, and if 
 m be the number of particles, or quantify of matter, m v 
 will exprefs the quantity of motion q, or 5^ == in v, 
 
 206. In like manner, we may conceive the ac- 
 celerating forces y, which have produced this velocity v 
 in each particle, as added into one fum, or as combined 
 on one particle, by article 170. They will thus compofe 
 a force, which, for diftindlion's fake, it is convenient to 
 mark by a particular name. We fliall call it the motive 
 FORCE, and exprefs it by the fymbol p. It will then be 
 confidered as the aggregate of the number m of equal ac- 
 celerating forces /, each of which produces the velocity 
 V on one particle. It will produce the velocity tn v, and 
 the fame quantity of motion q. 
 
 207. Let there be another body, confif?-ing of n 
 particles, moving with on^ velocity u. Let the moving 
 
 S force 
 
IjS' OF MOTIVE FORCES'. 
 
 force.be reprefented by tt. It is meafured in like maimer 
 
 by n lu Therefore we have, piTizzmv.n u^ and v : u =: 
 
 P '^ -i • 
 /- : - ; that is. 
 
 m n 
 
 The velocities luhich may be produced by the Jtmilar ac- 
 tion of different motive forces^ in the fame time, are di- 
 reEily as thofe forces ^ and inverfely as the quantities of matter 
 to which they are applied. 
 
 In general. 
 
 V == — 
 m 
 
 And f being = — > /=^ — . - 
 
 t ' ' mt 
 
 Remark. 
 
 208. In the appHcation of the theorems concerning 
 accelerating or retarding forces, it is necefTccry to attend 
 carefully to the diftin61:ion between an accelerative and a 
 motive force. The caution neceflary here has been ge- 
 nerally overlooked by the writers of Elements, and this 
 has given occafion to very inadequate and erroneous no- 
 tions of the a£lion of accelerating powers. Thus, if a 
 leaden ball hangs by a thread, which pafles ever a pulley, 
 and is attached to an equal ball, moveable along a hori- 
 zontal plane, without the fmallef obftru£l:ion, it is known 
 that, in one fecond, it will defcend 8 feet, dragging the 
 other 8 feet along the plane, with a uniformly accelerated 
 motion, and will generate in it the velocity 16 feet per 
 fecond. Let the thread be attached to three fuch balls. 
 We know that it will defcend 4 feet in a fecond, and 
 
 Ejenerate 
 
or DEFLECTING FORCES. I39 
 
 generate the velocity 8 feet per fccond. Mod readers 
 arc difpofed to think that it fliould generate no greater 
 velocity than 54- feet per fecond, or y of 16, becaufe it 
 is applied to three times as much matter (207.) The er- 
 ror lies in confidering the motive force as the fame in 
 both cafes, and in not attending to the quantity of mat- 
 ter to which it is applied. Neither of thefe conje£lures 
 is right. The motive force changes as the motion acce- 
 lerates, and in the firft cafe, it moves two balls, and in 
 the fecond it moves four, Tlie motive force decreafes 
 fimilarly in both motions. "When thefe things are confi- 
 dered, we learn by articles 202 and 207, that the mo- 
 tions will be precifely what v/e obferve. 
 
 Of Defle&lng Forces^ in gaiei^aL 
 
 209. It was obferved, in art. 99, that a curvilineal 
 motion is a cafe of cotitinual defle£l:ion. Therefore, when 
 fuch motions are obferved, we know that tke body is un- 
 der the continiml influence of fome natural force, a6ling 
 in a direction v.'hich crofles that of the motion in every 
 point. We muft infer the magnitude and direction of 
 this deflecting force by the magnitude aiid dire6lion of 
 the obferved deflection. Therefore, all that is afl^irmed 
 -concerning deflections in the 99th and fubfequent articles 
 of the Introduction, may be afHrmed concerning defleCt- 
 4ng forces. It follows, from what has been eftablifhed 
 concerning the action of accelerating. forces, that no force 
 -can produce a finite change of velocity in an inftant, 
 
 ^ 2 JS'ow, 
 
140 OF DEFLECTING FORCES 
 
 Now, a defle<5lion is a compofition of n motion already 
 exifting with a motion accelerated from reft by infenfible 
 degrees. Suppofing the defle6^ing force of invariable 
 direftion and intenfity, the defleclion is the compofition 
 of a motion having a finite velocity with a motion uni- 
 formly accelerated from reft. Therefore the linear de- 
 flection from the rcftilineal motion muft increafe by in- 
 fenfible degrees. The curviiineal path, therefore, muft 
 have the line of undefleCled motion for its tangent. To 
 fuppofe any finite angle contained between them would 
 be to fuppofe a polygonal motion, and a fubfultory de- 
 fledion. 
 
 Therefore no finite change of d'lreclion can he produced 
 by a defieEiing force in an tnftant. 
 
 210. The moft general and ufeful propofition on 
 this fubje6l is the following, founded on art. 104. 
 
 The forces by luhtch bodies are defieBed from the tangents 
 in the different points of their curviiineal paths are propor- 
 tional to the fquares of the velocities in thofe points y direEily^ 
 and inverfely to the defieBive chords of the equicurve circles in 
 the fame points. "We may ftill exprefs the propofition by 
 the fame fymbol 
 
 where f means the intenfity of the defledliing force. 
 
 211. We may alfo retain the meaning of the propofi- 
 tion exprefled in article 105, where it is fhewn that the ac- 
 tual 
 
IN GENERAL. 14! 
 
 tual linear ileflc6lion from the tangent is the third pro- 
 portional to the dcflc6live chord and the arch defcribed in 
 a very fmall moment. For it was demonftrated in that 
 article (fee fig. 1 8.) that BZ:BC = BC:BO. 
 
 We fee alfo that B h, the double of B O, is the mea- 
 fure of the velocity, generated by the uniform action of 
 the defle61:ing force, during the motion in the arch B C 
 of the curve. 
 
 212. The art. 106. alfo furnifhes a propofition of 
 frequent and important ufe, viz. 
 
 The velociiy in any point of a curvilinear fnotion is that 
 which the deflecling force in that point nvoidd generate i?i 
 the body by uniformly impelling it along the fourth part of 
 the defective chord of the equicurve circle. 
 
 Remark. 
 
 213. The propofitions now given proceed on the 
 fuppofition that, when the points A and C of fig. 18, 
 after continually approaching to B, at laft coalefce with 
 it, the laft circle which is defcribed through thefe three 
 points has the fame curvature which the path has in B. 
 It is proper to render this mode of folving thefe queftions 
 more plain and palpable. 
 
 If A BCD (fig. 25.) be a material curve or mouldy 
 and a thread be made faft to it at D, this thread may be 
 lapped on the convexity of this curve, till its extremity 
 meets it In A. Let the thread be now unlapped or 
 EVOLVED from the curve; keeping it always tight. It i& 
 
 plaija 
 
142 O? DEFLECTING 
 
 plain that its extremity A will defcribe another curve 
 line Khc. All curves, in which the curvature is neither 
 infinitely great nor infinitely fmall, may be thus defcribed 
 by a thread evolved from a proper curve. The proper- 
 ties of the curve hb c being known, Mr Huyghens (the 
 author of this way of generating curve lines) has (hewn 
 how to conftruO: the evolved curve ABC which will 
 produce it. 
 
 From this genefis of curves we may infer, \sty that 
 the detached portion of the thread is always a tangent to 
 the curve ABC; idly^ that when this is in any fituation 
 B 3, it is perpendicular to the tangent of the cuiTe Kh e 
 in the point b^ and that it is, at tlie fame time, defcrib- 
 ing an element of that curve, and an element of a circle 
 * 3 X, whofe momentary centre is B, and which has B h 
 for its radius. 3^/y, That the part b A oi the curve, be- 
 ing defcribed with radii growing continually fhorter, is 
 more incurvated than the circle b «, which has B b for its 
 conftant radius. For fimilar reafons the arch be oi the 
 curve A ^ <: is Itfs incurvated than the circle « b ». 4/^/^', 
 That th^ circle ub x, has the fame curvature that the curve 
 has in by or is an equicurve circle. B^ is the radius, 
 and B the centre of curvature in the point b. 
 
 ABC is the curva evoluta or the evolute. 
 Abe IS fometimes called the involute of ABC, and 
 fometimes its evolutrix. 
 
 214. By this way of defcribing curve lines, we fee 
 clearly that a body, when pafiing through the point b of 
 
 tl>e 
 
CENTRAL FORC&ff. I^^ 
 
 the curve Kbc may be confidercd as in the fame flate, in 
 that inftant, as in pafling through the fame point b of 
 the circle ocBki, and the ultimate ratio of the defle6tions 
 in both is that of equality, and they may be ufed indif- 
 criminately. 
 
 The chief difficulty in the application of the preced- 
 ing theorems to the curvilineal motions which are ob- 
 ferved in the fpontaneous phenomena of nature, is in af- 
 certaining the direction of the defle(fl;ion in every point 
 of a curvilineal motion. Fortunately, however, the moft 
 important cafes, namely thofe motions, where the defle6l^- 
 ing forces are always directed to a fixed point, afford a 
 very accurate method. Such forces are called by the ge- 
 neral name of 
 
 Central Forces. 
 
 215. If bodies defer ibe circles with a uniform motion 
 the defleEling forces are always direBed to the centres of the 
 sirclesy and are proportional to the fquare of the velocities 
 direBlyy and to their diflances from the centre^ inverfely. 
 
 For, fmce their motion in the circumference is uni- 
 form, the areas formed by lines drawn from the centre 
 are as the times, and therefore (100) the deflediions 
 and the deflecting forces (209) are directed to the centre. 
 Therefore, the defledive chord is, in this cafe, the dia- 
 meter of the circle, or twice the diftance of the body 
 from.the centre. Therefore, if we call the diilance from 
 the centre J, we have/ 4 -j, 
 
 2^4 
 
144 OF CENTRAL FORCES. 
 
 2 1 6. Thefe forces are alfo as the dtftanceSy direBly, 
 and as the fquare of the time of a revolution^ inverfely. 
 
 For the time of a revolution (which mny be called 
 the PEPviODic time) is as the circumference, and there- 
 fore as the diftance, directly, and as the velocity, in- 
 
 verfely. Therefore /=-, and-y^^-, and v"^ ~ ~ . 
 
 ,V' . d 
 
 and-^--. 
 
 217. Thefe forces are alfo as the difiances^ and the 
 fquare of the angular velocity^ jointly. 
 
 For, in every uniform circular motion, the angular 
 Telocity is inverfely as the periodic time. Therefore, 
 
 calling the angular vefocity <7, « ^ == - , and - == d a^, 
 
 and therefore f==da'', 
 
 218. The periodic time is to the time of falling along 
 half the radius by the uniform aElion of the centripetal force 
 in the circumference^ as the circumference of a circle is to 
 the radius* 
 
 For, in the time of falling through half the radius, 
 the body would defcribe an arch equal to the radius (59), 
 becaufe the velocity acquired by this fall is equal to the 
 velocity in the circumference (212.) The periodic time 
 is to the time of defcribing that arch as the circumference 
 to the arch, that is, as the circumference is to the radius. 
 
 219. When a body defcribes a curve ivhlch is all in 
 $ne plane ^ and a point isfofituated in that plane, that a line 
 
 drawn 
 
TjjtJ^.J. 
 
 Fift.-'4. 
 
CENTRAL FORCES. I45 
 
 dinwrJ from it io the body ilefcrihes round that pobit areas 
 proportional to the times y the dejle cling force is ahuays dire^' 
 ed to that point (100.) 
 
 220. Coiiverfely. If a body is defeBed by a force al- 
 iVi^ys direBed to a fixed pointy it will defcribe a curve line 
 lying in one plane ivhich paffes through that pointy and the 
 line joining it ivith the centre of forces will defcribe areas 
 proportional to the times (101.) 
 
 The line joining the body with the centre is called 
 the RADIUS VECTOR. The defle£i:ing force is called 
 CENTRIPETAL, or ATTRACTIVE, if its direction be al- 
 ways toward that centre. It is called repulsive, or 
 CENTRIFUGAL, if it be directed outwardsyr^;« the centre. 
 In the firft cafe, the curve v/ill have its concavity toward 
 the centre, but, in the fecond cafe, it will be convex to- 
 ward the centre. The force which urges a piece of iron 
 towards a magnet is centripetal, and that which caufes 
 two ele(Slrical bodies to feparate is centrifugal. 
 
 221. The force by which a body may be ?nade to 
 defcribe circles round the centre of forces ^ with the jaiigU' 
 lar velocities which it has in the different points of its 
 €urvili?ieal path, are inverfely as the cubes of its diflances 
 from the centre of forces. For the centripetal force in 
 circular motions is proportional to da'^ (217.) But when 
 the deflediions (and confequently the forces) are diredled 
 
 to a centre, we have ^ =f 7T (103-) ^^^ ^^ =f 7:;? there- 
 
 fore da' 4^^ X -j^y 4 73> therefore/=f 77. 
 
 d'^ ^ ' • d' 
 
 ,/3' therefore/ 4^ 
 
 This 
 
Cr.NTRAL FORCES. I45 
 
 dinivfi from it to the body defcribes round that point areas 
 proportional to the t'uneSy the defecting force is always dire^- 
 ed to that point ( 1 00.) 
 
 220. Converfely. If a body is defcFted by a force aU 
 ivays direEied to a fixed pointy it ivill defcribe a cur'De line 
 lying in one plafie ivhich paffes through that pointy and the 
 line joining it ivith the centre of forces ivill defcribe areas 
 proportional to the times (101.) 
 
 The line joining the body with the centre is called 
 the RADIUS VECTOR. The deflefting force is called 
 CENTRIPETAL, 01 ATTRACTIVE, if its direction be al- 
 ways toward that centre. It is called repulsive, or 
 CENTRIFUGAL, if it be directed outwardsyr^;^ the centre. 
 In the firfl; cafe, the curve v/ill have its concavity toward 
 the centre, but, in the fecond cafe, it will be convex to- 
 ward the centre. The force which urges a piece of iron 
 towards a magnet is centripetal, and that which caufes 
 two eledrical bodies to feparate is centrifugal. 
 
 221. The force by which a body may be made to 
 defcribe circles round the centre of forces^ with the migu- 
 lar velocities which it has in the different points of its 
 €'arvili?ieal path, are inverfely as the cubes of its diflances 
 from the centre of forces. For the centripetal force in 
 circular motions is proportional to da'^ (217.) But when 
 the deiledions (and confequently the forces) are direiled 
 
 to a centre, we have a ^= ~ (103.) and a^ =f -rj, there- 
 fore da' ^d X j^y "^ J5' therefore/4 j]- 
 
 T Thift 
 
J4(> CENTRAL I^ORCES-. 
 
 This force is often called centrifugal, ihe centrifugal 
 force of circular motion, and it is conceived as always act- 
 ing in every cafe of curvilineal motiorr, and to act in op- 
 pofition to the centripetal f6rce which produces that mo- 
 tion. But this is inaccurate. We fuppofe this force, 
 merely becaufe we muft employ a centripetal force, jult 
 as we fuppofe a refifing vis inerti?^, becaufe we mull em- 
 f)loy force to move a body. 
 
 222. If a body defcrihe a curve line ABC ^_y means 
 of a centripetal (fig. 26.) force directed to S, and varying 
 according to fome proportion of the difances front it, and if 
 another body be impelled toward S in the ftraight line a b S 
 hy the fame force y and if the two bodies have the fame velo- 
 eity in any points A and a which are equidiftafit from S, 
 they nvill have equal velocities in any other tvjo points C ajul 
 c, which are alfo equidiftant from S. 
 
 Defcribe round S, with the diftance S A, the circu- 
 lar arch Afl!, which will pafs through the equidiftant 
 point a. Defcribe another arch B b, cutting off a fmall 
 arc A B of the curve, and alfo cutting A S in D. Draw 
 D E perpendicular to the curve. 
 
 The diflances A S and a S being equal, the centri- 
 petal forces are alfo equal, and may be reprefented by 
 the equal lines A D and a b. The velocities at A and a 
 being equal, the times of dcfcribing A B and a b will be 
 as the fpaces (31). The force ab h Vv-holly employed 
 in accelerating the rectilineal motion along a S. But 
 the force A D, being tranfverfe or oblique to the motion 
 
 along 
 
CENTRAL FORCES. 147 
 
 along A B, is not wholly employed in thus accelerating 
 the motion. It is equivalent (173) to the two forces AE 
 and ED, of which ED, being perpendicular to AB, 
 neitlier promotes nor oppofes it, but incurvates the mo- 
 tion. The accelerating force in A therefore is A E. It 
 was Ihewn, in art. 71, that the change of velocity is as 
 the force and as the time jointly, and therefore it is 
 as A E X A B. For the fame reafon, the change of the 
 velocity at a is as ab X ab, or ab^. But, as the angle 
 A D B is a right angle, as alfo A E D, we have A E : A D 
 = AD:AB, and AExAB = ADS = a b\ There- 
 fore, the increments of velocity acquired along A B and 
 ab are equal. But the velocities at A and a were equal. 
 Therefore the velocities at B and b are alfo equal. The 
 fame thing may be faid of every fubfequent increafe of 
 velocity, while moving along B C and b c •, and therefore 
 the velocities at C and c are equal. 
 
 The fame thing holds, when the deflefting force is 
 directed in lines parallel to a S, as if to a point S' infi- 
 nitely diftant, the one body defcribing the curve line 
 V A' B', v.diile the other defcribes the ftraight line V S. 
 
 223. The propofitions in art 102. and 103. are alfo 
 true in curvilineal motions by means of central forces. 
 
 When the path of the motion is a line returning into 
 itfelf, like a circle or oval, it is called an orbit ; other- 
 wife it is called a trajectory. 
 
 The time of a complete revolution round an orbit is 
 called the periodic time. 
 
 T 2 224. 
 
148 CENTRAL rORC£S. 
 
 224. The formula / =i — ferves for difcoveriiig the 
 
 law of variation of the central force by which a body 
 defcribes the different portions of its curvilineal path , 
 
 and the formula y"=f - ferves for comparing the forces 
 
 by which different bodies defcribe their refpective orbits. 
 
 225. It muff always be rcmcm.bered, in conformity 
 to art. 105, that/=r— or f — — ~ exprcffes the li- 
 near deileftion from the tangent, which may be taken 
 
 2 v^ 
 
 for a mcafure of the deflecting force, and that f~ 
 
 c 
 
 2 arc ^ 
 or y = — — expreffes the velocity generated by this 
 
 force, during the defcription of the arc, or the velocity 
 which may be compared dire£i:ly with the velocity of the^ 
 motion in the arc. The laff is the mod accurate, becaufe 
 the velocity generated is the real change of condition. 
 
 226. A body may ^lefcrihey by the acticn of a cen- 
 tripetal forcCy the direBion of which pajfs through C 
 (fig. 27.) a figure VPS, which figure revolves {in its 
 own plane) round the centre of forces C, /;/ the fame man^ 
 tier as it dfcribes the quiefcent figure^ provided that the 
 angular mrjion of the body in the orbit be to that of the or- 
 bit itfelf in any confiant ratio^ fuch as that of m to n. 
 
 For, if the direction of the orbit's motion be the fame 
 with that of the body moving in it, the angular motioii 
 
eXNTRAL FORCES. 14^ 
 
 €f the body in every point of its motion is increafed in 
 the ratio of m to // + ^^h ^i^d it will be in the fame ratio 
 in the different parts of the orbit as before, that is, k 
 will be inverfely as the fquare of the diftance from S ( 103). 
 Moreover, as the diflances from the ceiltre in the fimal- 
 tancous pofitions of the body, in tlie quiefcent and in 
 the revolving orbit, are the fame, the momentary incre- 
 ments of the area are as the momentary increments of the 
 angle at the centre ; and therefore, in both motions, the 
 areas increafe in the conftant ratio of in to m -f ;/ (^03). 
 Therefore the areas of the abfolute path, produced by the 
 compofition of the two motions, will fiiili be proportional 
 to the times; and therefore (101) the defle61:ing force 
 mnft be directed to the centre S ; or, a force fo directed 
 will produce this compound motion. 
 
 227. The differ e?jcei hetiveen ihe forces by ivhich a 
 kcdy may be made to move in the quiefcent and in the move- 
 able orbit are in ihe inverfe triplicate ratio of the diflances 
 from the centre of forces. 
 
 Let VKSBV (hg. 27.) be the fixed orbit, and 
 up h b u the fame orbit moved into another pofition ; and 
 let N p « N N / Q V be the orbit defcribed by the body in 
 abfolute fpace by the compofition of its motion in the 
 orbit with the motion of the orbit itfelf. If the body be 
 fuppofed to defcribe the arch V P of the fixed orbit while 
 the axis V C moves into the fituation u C, and if the arch 
 :.'/» be mnde equal to VP, then p will be the place c£ 
 
 the 
 
1^0 <;entral forces. 
 
 die body in the moveable orbit, and in the compound 
 path Y p. If the angular motion in the fixed orbit be to 
 the motion of the moving orbit as m to ;/, it is plain that 
 the angle VCP is to VC/> as m to m -\- «. Let PK 
 and pk he t-wo equal and very fmall arches of the fixed 
 and moving orbits. P C and p c are equal, as are alfo 
 K C and k C, and a circle defcribed round C with the 
 radius C K will pafs through h. If we now make V C K 
 to V C « as m to m -\- n \ the point n of the circle Yikn 
 will be the point of the compound path, at which the 
 body in the moving orbit arrives when the body in the fixed 
 orbit arrives at K, and p n is the arch of the abfolute 
 path defcribed while P K is defcribed in the fixed path. 
 
 In order to judge of the difference between the force 
 which produces the motion P K in the fixed orbit and 
 that which produces p n in the abfolute path, it muft be 
 pbferved that, in both cafes, the body is made to ap- 
 proach the centre by the difference between C P and C K. 
 This happens, becaufe the centripetal forces, in both 
 eafes, are greater than what would enable the body to 
 defcribe circles round C, at the diilance C P, and with the 
 fame angular velocities that obtain in the two paths, viz, 
 the fixed orbit and the abfolute path. We fliall call the 
 one pair of forces the circular forces^ and the other the 
 srbital. I^et C and c reprefent the forces which would 
 produce circles, with the angular velocities which obtain 
 in the fixed and moving orbits, and let O and o be the 
 forces which produce the orbital motions in thefe two 
 paths. 
 
 Thefe 
 
CENTRAL FORCES. I5I 
 
 Thefe tilings being premifetl, it is plain that — ^ is 
 equal to O — C, bccaiife the bodies are equally brought 
 toward the centre by the difFerence between O and C 
 and by that between and c. Therefore — O is e- 
 qual to c — C. * The difFerence, therefore, of the 
 forces which produce the motions in the fixed and 
 moving orbits is always equal to the difFerence of the 
 forces which would produce a circular motion at the fame 
 diflances, and with the fame angular velocity. But the 
 forces which produce circular motions, with the angular 
 motion that obtains in an orbit at different diftances from 
 the centre of forces, are as the cubes of the diftances 
 inverfely (221). And the two angular motions at the 
 fame difiance are in the conftant ratio of m to m -f- n. 
 Therefore the forces are in a confcant ratio to each other, 
 and their differences are in a conftant ratio to either of 
 the forces. But the circular force at different diftances 
 is inverfely as the cube of the diftance (221). Therefore 
 the difference of them in the fixed and moveable orbits is 
 in the fame proportion. But the difference of the orbital 
 
 forces 
 
 CO CO 
 
 * For let A 0, A O, A r, A C reprefent the four forces 
 /?, O, c, and C. By what has been faid, we find that c z=: 
 O C. To each of thefe add O c, and then it is plain that 
 c/ O = ^ C, that is, that the difFerence of the circular forces c 
 and C is equal to that of the orbital forces and O. 
 
1^2 CENTRAL FORCES. 
 
 forces is equal to that of the circular. Thei'efore, finaliy, 
 the chfFerence of the centripetal forces by whic'i a body 
 may be retained in a nxed orbit, and in the hniic orbit 
 moving as determined in article 226, is always in the in- 
 verfc triplicate ratio of the diftances from the centre of 
 forces. 
 
 In this example, the motion of the body in the orbit 
 is in the fame direction with that of the orbit, and the 
 force to be joined with that in the fixed orbit is always 
 additive. Had the orbit moved in the oppofite dire61:ion, 
 the force to be joined would have been fubtraftive, un- 
 lefs the retrograde motion of the orbit exceeded tv/icc 
 the angular motion of the body. But in all cafes, the 
 reafoning is fimiiar. 
 
 228. Thus we have confulered the motions of bo- 
 dies influenced by forces directed to a fixed point. But 
 we cannot conceive a mere mathematical point of fpace 
 as the caufe or occafion of any fuch exertion of forces. 
 Such relations are obferved only between exifling bodies 
 or mafles of matter. The propofitions which liave been 
 demonftrated may be true in relation to bodies placed in 
 thofe fixed points. That continual tendenxy towards a 
 centre, which produces an equable defcription cf areas 
 round it, becom.es intelligible, if we fuppofe fome body 
 placed in the centre of forces, attracting the revolving 
 body. Accordingly, we fee very remarkable example* 
 of fuch tendencies towards a central body in the motions 
 of the planets /•■und tjie Sun, :md of the fatellites round 
 the priraiary planet. 
 
 But, 
 
CENTRAL FORCES. f53 
 
 But, fince it is a univerHil hO: that all the relations 
 l>ctwcen bodies are mutual, we are obliged to fuppofe 
 that whatever force inclines the revolving body towards 
 the body placed in the centre of forces, an equal force 
 .(from whatever fource it is derived) inclines the central 
 hody toward the revolving body, and therefore it cannot 
 remain at reft, but rnufl move towards it. The notion 
 of a fixed centre of forces is thus taken away again, and 
 we feem to have demonftrated propofitions inapplicable 
 to any thing in nature. But more attentive confidera- 
 tion will fliew us that our propofitions are moft ftri^lly 
 applicable to the plienomena of nature. 
 
 229. For, in the firfl; place, the motion of the com- 
 mon centre of pofition of two, or of any number of bo- 
 dies, is not affeded by their mutual adions. Thefe, be- 
 ing equal and oppofite, produce equal and oppofite mo- 
 tions, or changes of motion. In this cafe, it follows from 
 art. 115. that the ilate of the common centre is not af- 
 fected by them. 
 
 230. Now, fuppofe two bodies S and P, fituated at 
 the extremities of the line SP (fig. 28.) Their centre 
 of pofition is in a point C, dividing their difiiance in fuch 
 a manner that S C is to C P as the number of material 
 atoms in P to the number in S ( 1 10.) or S C : P C = P : S. 
 Suppofe the mutual forces to be centripetal. Then, be- 
 ing equal, exerted between every atom of the one, and 
 fvery particle of the other, the vis motrix may be ex- 
 
 U prcfied 
 
1-4 CENTRAL FORCES. 
 
 prefled by P X S. This mufc produce equal quantities 
 of motion in each of the bodies, and therefore muft pro- 
 duce velocities inverfely as the quantities of matter (127). 
 In any given portion of time, therefore, the bodies will 
 move towards each other, to / and />, and S j- will be to 
 Vp as P to S, that is as S C to P C. Therefore we fliall 
 ftill have /C:/C = SC:PC. Their dilVances from 
 C will always be in the fame proportion. Alio we fhall 
 have S C : S P = P : S + P, and j- C :/) C = P : S + P 5 
 and therefore S C : S P = j C : j P. Confequently, in 
 whatever manner the mutual forces vary by a variation 
 cf diftance from each other, they will vary in the fame 
 manner by the fame variation of diftance from C. And, 
 converfely, in whatever maimer the forces vary by a 
 change of diftance from C, they vary in the fame man- 
 ner by the fame change of diftance from each othe.\ 
 
 Let us now fuppofe that when the bodies are at S 
 and P, equal moving forces are applied to each in the 
 oppofite diredions S A and P B. Did they not attra£l 
 each other at all, they would, at the qw^i of fome fmall 
 portion of time, be found in the points A and B of a 
 ftraight line drawn through C, becaufe they will move 
 with equal quantities of motion, or with velocities S A 
 and P B inverfely as their quantities of matter. There- 
 fore S A : P B = S C : P C, and A, C, and B are in a 
 ftraight line. But let them now attract:, when impelled 
 from S and P. Being equally attra<fted tov/ard each other, 
 they will defcribe curve lines S a and P by fo that their 
 deflexions A a and B b arc as S C and P C ; and we ftiall 
 
 have 
 
CENTRAL FORCES. I^^ 
 
 liave flC:^C = SC:PC. As this Is true of every part 
 of the curve, it follovi^s that they defcribe fimilar curves 
 round C, which remains in its original place. 
 
 Laflly^ If the motion of P be conlidered by an ob- 
 ferver placed in S, unconfcious of its motion, fmce he 
 judges of the motion of P only by its change of direction 
 and of diftance, we may make a figure which will per- 
 fectly reprefent this motion. Draw the line E F equal 
 and parallel to PS, and EG equal and parallel to ab. 
 Do this for every point of the curve S a and Vb, We fliall 
 then form a curve F G fimilar to the curves S a and P bj 
 having the homologous lines equal to the fum of the ho- 
 mologous lines of thefe two cur\^es. Thus the bodies will 
 defcribe round each other curve lines which are fimilar and 
 equal (lineally) to the lines which they defcribe round 
 their common centre by the fame forces. They may ap- 
 pear to defcribe areas proportional to the times round 
 each other j and they really defcribe areas proportional 
 to the times round their common centre of pofition, and 
 the forces, which really relate to the body which is fiip" 
 tofed to be central, have the fame mathematical relation 
 to their common centre. 
 
 Thus it appears that the mechanical inferences, drawn 
 from a fuppofed relation to a mere point of fpace, are 
 true in the real relations to tlie fuppofed central body, 
 although It Is not fixed In one place. 
 
 231. The time of defcrlbing any arch FG of the 
 
 curve defcrlbed round the other body at reft in a centre 
 
 U 2 of 
 
1^5 CENTRAL FORCES. 
 
 of forces (where we may fuppofc It forcibly withhelc{ fronf 
 moving) is to the time of defcribing the fimiiar arch P 3 
 round the common centre of pofition in the fubdupHcat(? 
 ratio of S 4- P to S, that is, in the ratio of 'Z S + P' to 
 V~S. For the forces being the fame in both motions^ 
 the fpaces dcfcribed by their fmiilar aftions, that is, their 
 deflections from the tangent are as the fquares of the 
 times T and t (204). That is, H G : B ^ =r T* : /% and 
 
 Hence it follows that the two bodies S and P are 
 moved in the fame way as if they did not atl on each 
 other, but were both acted upon by a third body, placed 
 in their common centre C, and ailing with the fame 
 forces on each ; and the Law of variation of the forces by 
 a change of diftance from each other, and from this third 
 body, is the fame. 
 
 232. If a body P (fig. 29-.) revolve around another 
 body S, by the adlion of a central force, while S moves 
 in any path A S B, P will continue to defcribe areas pro- 
 portional to the times round S, if every particle m P be 
 affe<Sled by the fame accelerating force that a6ls, in that 
 mftant, on every particle in S. For, fuch a61;ion will 
 compound the hme motions Vp and S j- with the mo- 
 tions of S and P, whatever they are ; and it was fhown 
 in art. (98.) that fuch compofition does not afFed thei* 
 relative motions. This is another way of making a body 
 defcribe the limie orbit ia motion which it defgribes whife 
 the orbit is fixed (226), 
 
MECHANICAL PHILOSOPHY. 
 
 P A R T IL 
 
 THE MECHANICAL HISTORY OF NATURE- 
 
 INTRODUCTION. 
 
 I33. VV E have now confidered in fufficient de- 
 tail thofe general Confequences which refult from the re- 
 lations of the Ideas that we have of Matter and Motion, 
 and of the Caufes of its changes. Thefe confequences are 
 the metaphyfical or abftracEl dodrines of Mechanical 
 Philofophy. They are, in reality, defcriptions, not of 
 external nature, but of the proceedings of the human 
 mind in contemplating or ftudying it. Being Independ- 
 ent of all experience of any thing beyond our own 
 thoughts, they form a body of demonftrative truths. If 
 this has been made fufficiently complete, that is, if all 
 the poflible mechanical changes are comprehended in the 
 three propofitions which we called the Laws of Motion,- 
 we ftiould now be in a condition to conlider every change 
 of motion, and every changing caufe, which nature pre- 
 Unu tQ our yiew| whether in order to inveftigate and 
 
 difcover 
 
1^8 MECHANICAL HISTORY 
 
 ilifcover natural Forces hitherto unknown, and to give 
 an account of the Laws by which their ad ion is regu- 
 lated, or to explain complicated phenomena, by referring 
 them to the operation of fome known forces. 
 
 234. Both of thefe purpofes are to be attained by a 
 careful obfervation of the phenomena, AH circumftances 
 of coincidence or refemblance among them are to be 
 taken notice of, and confidered as indications of a fimi- 
 larity in their Caufes. The more extenfive the obferved 
 coincidence of appearances is, the more general muft the 
 affection of matter be which is the caufe of this refem- 
 blance. If any fimilarity is univerfally obferved, it muft 
 be confidered as the indication of a mechanical quality 
 that is competent to all matter. 
 
 235. This confideration points out to us a principle 
 for arranging the mechanical phenomena of the univerfe. 
 Thofe fhould be firft confidered that are mofl: general. 
 Thus are we made acquainted with the molt general 
 mechanical properties of Bodies, which extend their in- 
 fluence to phenomena in all the fubordlnate clailes, and 
 modify even that circumflance which forms the parti- 
 cular clafs. Our previous acquaintance with thofe ge- 
 neral properties will enable us to free the more particu- 
 lar phenomena from part of that complication which 
 makes the ftudy of them more difficult ; and then to 
 confider apart thofe circumftances of the phenomena 
 which are indications of qualities lefs general. 
 
 23d. 
 
IN GENERAL. I59 
 
 236. The moft general phenomenon that we ob- 
 ferve is the cunllmeal motion of bodies in free fpace. 
 The Globe which we inhabit, the Sun, and all his at- 
 tending Planets and Comets, are continually moving in 
 curve-lined paths. And thefe curvilineal motions are 
 compounded with all the other motions that are per- 
 formed on the furface of this Globe. When a cannon 
 bullet is difcharged in a foutherly direction with the ve- 
 locity of 1500 feet in a fecond, it is at the fame time 
 carried ealtward, nearly at the fame rate, by the rota- 
 tion of the Earth ; and by its revolution in a year round 
 tlie Sun, it is moving eaflward, more than fixty times 
 as fail. Such being the condition of the vifible univerfe, 
 it appears that the defleciiing forces, by which all thefe 
 bodies are kept in their curvilineal paths, muft be ac- 
 knowledged to have the moil extennve influence. The 
 phenomena which are the indications of thefe forces, 
 claim the firft place in the Mechanical hiftory of Nature. 
 Thefe are obferved in the celeflial motions, and Aftro- 
 nomy is tlierefore the firft department of that hiftory to 
 which we fhall turn our attention, 
 
 237. This order of ftudy has other advantages be- 
 fides this fcientific propriety. It is that part of the ftudy 
 of material nature in which the underftanding of man 
 has been moft fuccefsful. It is perhaps owing to the 
 unexceptionable proofs, which Aftronomy alone affords 
 of the perfect conformity of our abftradt do<Sl:nnes with 
 the real ftate of the world, that thofe doctrines have 
 
 been 
 
l6o meCHANICAL HISTORT 
 
 been admitted as a jufl expofition of the elements of 
 Univerfal Mechanics, and thus have given us a ground- 
 work, on which we can proceed with confidence in ex- 
 plaining the mechanical phenomena of this fublunary 
 world. 
 
 Aflronomy Is alfo the department of natural fcience 
 that Is the moft eafily comprehended with the dlftincl- 
 nefs and accuracy that deferve the name of fcience. 
 Here we have a clear and adequate idea of the fubjedt, 
 and a diftin(Sl feeling of the validity of the evidence by 
 v/hich any propofition is fupported. In the fimplelt pro- 
 pofition of common Mechanics, or Hydraulics, the fub- 
 jeO: under confideration has a degree of complication 
 not to be found in the moft abftrufe propofition in Aftro- 
 nomy. Accordingly, the knowledge which we can ac- 
 quire in Aflronomy approaches near to the certainty of 
 firft principles ; while in thofe other departments it is on- 
 ly a fuperficial knowledge of fome very general property 
 that we are able to acquire. 
 
 Aftronomy is therefore recommended to our firft no*- 
 tice, by the univerfality of the powers of nature that are 
 indicated by the planetary motions, — by the fuccefsful- 
 nefs of the inveftigation^ — and by the eafy accefs wliich 
 it gives us to the elementary principles of all Mechanical 
 ftience, 
 
 ifECHi** 
 
MECHANICAL HISTORY 
 
 NATURE. 
 
 Sect. I. ASTRONOMY. 
 
 238. /\sTRONOMY was firft ftuclied as an art, fub- 
 fervient to the purpofes of focial life. Some knowledge 
 of the celeftial motions was neceffary, in every ftate of 
 fociety, that we might mark the progrefs of the feafons^ 
 which regulate the labours of the cultivator, and the 
 migrations of the fhepherd. It is necelTary for the re- 
 cord of pad events, and for the appointment of national 
 meetings. 
 
 While the motions of the heavenly bodies afford us 
 the means of attaining thefe ufeful ends, tliey alfo pre- 
 fent to the curious philofopher a feries of magnificent 
 phenomena, the operation of the greateft powers of ma- 
 terial nature *, and thus they powerfully excite his curio- 
 fity with refpe£i: to their caufes. This circumflance a- 
 lone makes the celeftial motions the proper obje6ts of 
 attention to a ftudent of Mechanical Philofophy, and he 
 has lefs concern in the b'^;autiful regularity and fubordi- 
 nation which have made theni fo fubfcrvient to the pur- 
 pofes of Navigation, of Chronology, and the occupations 
 of rural life. 
 
 X But 
 
MECHANICAL HISTORY 
 
 NATURE. 
 
 Sect. I. ASTRONOMY. 
 
 238. x\sTRONOMY was firft ftudled as an art, fub- 
 fervient to the puipofes of focial life. Some knowledge 
 of the celeftlal motions was neceffary, in every (late of 
 fociety, that we might mark the progrefs of the feafons^ 
 which regulate the labours of the cultivator, and the 
 migrations of the fhcpherd. It is neceffary for the re- 
 cord of pad events, and for the appointment of national 
 meetings. 
 
 While the motions of the heavenly bodies afford us 
 the means of attaining thefe ufeful ends, tliey alfo pre- 
 fent to the curious philofopher a feries of magnificent 
 phenomena, the operation of the greateft powers of ma- 
 terial nature *, and thus they powerfully excite his curio^ 
 flty with refpe£l to their caufes. This circumftance a- 
 lone makes the celeftial motions the proper objects of 
 attention to a ftudent of Mechanical Philofophy, and he 
 has lefs concern in the b'^autiful regularity and fuboi'di- 
 nation which have made theni fo fubfervient to the pur- 
 pofes of Navigation, of Chronology, and the occupations 
 of rural life, 
 
 X But 
 
1 62 ASTRONOMY. 
 
 But the purpofes of the Mechanical philofopher 
 cannot be attained without attending to that beauty, 
 regularity, and fubordination. Thefe features are ex- 
 hibited in every circumftance of the celeflial motions 
 that renders them fufceptible of fcientific arrangement 
 and inveftigation ; and a philofophical view cannot 
 be taken, without the fame accurate knowledge of the 
 motions that is wanted for the arts of life. It muft 
 be added, that fociety never would have derived the 
 benefits which it has received from Aftronomy, without 
 the labours of the philofopher : For, had not Newton, 
 or fome fuch exalted genius as Nevv^ton, fpeculated a- 
 bout the defledVing forces which regulate the motions of 
 the Solar fyflem, we never iliould have acquired that 
 exquifite knowledge of the mere phenomena that is ab- 
 folutely neceffary for fome of the mod important ap- 
 plications of them to the arts. It v/as thefe fpecula- 
 tions alone that have enabled our navigators to proceed 
 with boldnefs through untried feas, and in a few years 
 have almoft completed the furvey of this globe. And 
 thus do w^e experience the moll beneficial alliance of 
 Philofophy and Art. 
 
 Since the motions of bodies are the only indica- 
 tions, charafteriftics, and meafures of moving forces, 
 it is plain that the celeftial motions muft be accurately 
 afcertained, that we may obtain the data wanted for the 
 purpofe of philofophical inference. To afcertain thefe 
 is a tafk of great difficulty ; and it has required the con- 
 tinual effqrts of many ages to acquire juft notions of the 
 
 motions 
 
ASTRONOMY. 163 
 
 motions exhibited to our view in the heavens. For the 
 fame general appearances may be exhibited, and the fame 
 perceptions obtained, and the fame opinions will be 
 formed, by means of motions very different ; and it is 
 frequently very difficult to fele£l thofe motions which 
 alone can exhibit every obferved appearance. If a per- 
 fon who is in motion, imagines that he is at reft, and 
 aflumes this principle in his reafonings about the effects 
 of the motions which he perceives, he miftakes the con- 
 clufions which he draws for real perceptions ; and c.ills 
 that a deception of fenfe, which is really an error in 
 judgement. Errors in our opinions concerning the mo- 
 tions of the heavenly bodies, are necelTarily accompanied 
 by falfe judgements concerning their caufes. Therefore, 
 an accurate examination of the motions which really 
 obtain in the heavens, muft precede every attempt to in*- 
 veftigate their caufes. 
 
 The moft probable plan for acquiring a juft and fa- 
 tisfatiory knowledge of thefe particulars, is to follow the 
 fteps of our predeceflbrs in this ftudy, and firft to con- 
 fider the more general and obvious phenomena. From 
 thefe we muft deduce the opinions which moft obvioufly 
 fuggeft themfelves, to be corrected afterwards, by com- 
 paring them with other phenomena, which may happen 
 to be irreconcileable with them. 
 
 X 2 Aflronomical 
 
164 ASTRONpMICAL PHENOMENA, 
 
 JJlronomical Phenomena. 
 
 239. To an obferver, whofe view on all fides i^ 
 bounded onjy by the fea, the heavens appear a concave 
 fphere, of which the eye is the centre, ftudded with a 
 great number of luminous bodies, of which the Sun and 
 Moon are the moft remarkable. This fphere is called 
 
 the SPHERE OF THE STARRY HEAVENS. 
 
 The only diftances in the heavens which are the 
 immediate obje61:s of our obfervation, are arches of 
 great circles paffing through the different points of the 
 ilarry heavens. Therefore, all aftronomical computations 
 and meafurements are performed by the rules of fpheri- 
 cal trigonometry. 
 
 240. We fee only the half of the heavens at a time, 
 the other half being hid by the earth, on which we are 
 placed. The great circle H B O D (fig. 30.), which fe- 
 parates the vifible hemifphere H Z O from the invifible 
 hemifphere H N O, is called the horizon. This is mark- 
 ed out on the flarry heavens by the fartheft edge of the 
 fea. The point Z immediately over the head of the 
 obferver is called the zenith ; and the point N, diame- 
 trically oppofite X.0 it, is called the nadir. 
 
 241. The zenith and nadir are poles of the horizon. 
 
 242. 
 
CIRCLES OF THE SPHERE, .&C. 1^5 
 
 242. If an obferver looks at the lieaveiis, while a 
 plummet is fufpended before his eye, the plumb line 
 will mark out on the heavens a quadrant of a circle, 
 whofe plane is perpendicular to the horizon, and which 
 therefore pafles through the zenith and nadir, o.nd 
 through two oppofite points of the horizon. ZONH 
 and ZBND are fuch circles. They are called verti- 
 cal CIRCLES and azimuth circles. 
 
 243. The ALTITUDE of any celeftial phenomenon 
 fuch as a ftar A, is the angle A C B, formed in the plane 
 of the vertical circle Z A N, by the horizontal line C B 
 and the line C A. This name is alfo given to the arch 
 A B of the vertical circle which meafures this an^le. 
 
 o 
 
 The arch Z A is called zenith distance of the phe- 
 nomenon. 
 
 244. The azimuth of the phenomenon is the angle 
 O C B, or O Z B, formed between the plane of the ver*- 
 tical circle Z A B paffing through the phenomenon, and 
 the plane of fome other noted vertical Z O N. The arch 
 C B of the horizon, which meafures this angle, is aifo 
 frequently called the Azimuth. 
 
 245. The ftarry heavens appear to turn round the 
 earth, which feems pendulous in the centre of the Iphere ; 
 and by this motion, the heavenly bodies come into view 
 in the eaft, or rise ; they attain the greateft altitude, or 
 culminate, and difappear in the weft, or set. This is 
 Called the first motion. 
 
 246. 
 
l66 ASTRONOMICAL PHENOMENA. 
 
 246. This motion is performed round an axis N S 
 (fig. 31.), pafling through two points N, S, called the 
 poles of the world. In confequence of this motion, a 
 celeftial objeci A defcribes a circle A D B F, through 
 the centre C of which the axis N S pailes, perpendicu- 
 larly to its plane. This motion msy be very diftinctly 
 perceived as follows. Let a point, or fight, be fixed in 
 the infide of a fky-light fronting the north, and inclined 
 fouthwards from the perpendicular at an angle equal to 
 the latitude of the place. An eye placed at this point 
 will fee the ftars through the glafs of the v/indow. Let 
 the points of the glafs, through wliich a ftar appears from 
 time to time be marked. The marks will be found to 
 lie in the circumference of a circle, the centre of which 
 will mark the place of the pole in the heavens. 
 
 247. Thofe ftars which are fartheft from the poles 
 will defcribe the greateft circles ; and thofe will defcribe 
 the largeft poflible circles which are in the circumference 
 of the circle ^ W Q E, which is equidiftant from both 
 poles. This circle is called the equator, and, being a 
 great circle, it cuts the horizon in two points, E, W, 
 diametrically oppofite to each other. They are the eaft 
 and weft points of the horizon. 
 
 248. If a great circle A N Q S j5^ pafies through 
 the poles perpendicularly to the horizon H W O E, it 
 will cut it in the north and fouth points ; and any ftar A 
 will acquire its greateft elevatioft when it comes to the 
 
 femicircle 
 
DIURNAL REVOLUTION. l6j 
 
 femicircle NA S, and its greateft deprelTion when it 
 comes to the femicircle N B S ; and the arch D A F of 
 its apparition will be bife6ted in A. 
 
 249. If the circle A D B F of revolution be be- 
 tween the equator and that pole N which is above the 
 horizon, the greatell portion of it will be vifible ; but 
 if it be on the other fide of the equator, the fmalleft por- 
 tion will be vifible. One half of the equator is vifible. 
 Some circles of revolution are wholly above the horizon, 
 »nd fome are wholly below it. A liar in one of the 
 firfl is always feen, and one in the iaft is never feen. 
 
 250. The diftance A^ of any point A from the e- 
 quator is called its declination, and the circle AD B F, 
 being parallel to the equator, is called a parallel of 
 
 DECLINATION. 
 
 25 1. The angle ^ C H, contained by the planes of 
 the equator and horizon, is the complement of the angle 
 N C O, which is the elevation of the pole. 
 
 252. The revolution of the ftarry heavens is perform- 
 ed in 23^ 56' 4". It is called the diurnal revolution. 
 No appearance of inequality has been obferved in it ; 
 and it is therefore alllimcd as the mod perfe6l meafure 
 of time. 
 
 253. The time of the diurnal apparition or difpari- 
 •tion of a point of the ftarry heavens is bifecied in the 
 
 inftant 
 
l6S ASTRONOMICAL PHENOlVIENil. 
 
 inflant of its culmination or greatefl deprefTion. TfiC 
 fun, therefore, is in the circle N A S Q at noon. For 
 this reafcn the circle N A S Q is called the meridian. 
 
 254. A phenomenon whofe circle of diurnal revo- 
 lution A D B F is on the fame fide of the equator with 
 the elevated pole, is longer vifible than it is invifible. 
 The contrary obtains if it be on the other fide of the 
 equator. 
 
 255. Any great circle NA^S, or NBLS (fig. 32.), 
 pafTmg through the poles of the world, is called an hour 
 
 CIRCLE. 
 
 256. The angle ^CL, or jENL, contained be- 
 tween the plane of the hour-circle NBLS, paffing 
 through any phenomenon B, and the plane of the hour 
 circle N jE S, pafling through a certain noted point ^ 
 of the equator, is called the right ascension of the 
 phenomenon. The intercepted arch JEh oi the equa- 
 tor, which meafures this angle, is called by the fame 
 name. 
 
 257. In afligning the place of any celeftial phenome- 
 non, we cannot ufe any points of the earth as points of 
 reference. The ftarry heavens afford a very convenient 
 means for this purpofe* Mod of the ftars retain their re- 
 lative fituations, and may therefore be ufed as fo many 
 points of reference.: T^e application of this to our purpofe 
 
 requires. 
 
CELESTIAL GLOBES AND MAPS. I^p* 
 
 Tequires a knowledge of the pofitions of the ftars. This 
 may be acquired. The difference between the meridional 
 altitude of a liar B, and of the equator, gives the arch 
 A J?', intercepted between the equator and the parallel 
 of declination, or circle of diurnal revolution A B D, de- 
 fcribed by the (tar. And the time which elapfes between 
 the palTage of this ftar over the meridian, and the paf- 
 fage of that point M of the equator from which the right 
 afcenfions are computed, gives the arch ^ L of the equa- 
 tor v/hich has paiTed during this interval. Therefore, an 
 hour circle N L S being drawn through the point L of 
 the equator, and a circle of revolution A B D being 
 drawn at the obferved diftance A iE from the equator, 
 the place of the liar will be found in their Lnterfediion 
 B. 
 
 258. Globes and maps have been made, on which 
 the reprefentations of the ftars have been placed, in pofi- 
 tions fimilar to their real pofitions ; and catalogues of the 
 ftars have been compofed^ in which every ftar is fet down 
 with its declination and right afcenfion, this being the 
 moft convenient arrangement for the practical aftrono- 
 mer. Their longitucles and latitudes (to be explained af- 
 terwards) are alfo fet down, in feparate columns. The 
 moft noted of all thefe is the britannic catalogue, 
 conftru6:ed by Dr Flamftead, from his own obfervations 
 in the Royal Obfervatory at Greenwich. ' This catalogue 
 contains the places of 3030 ftars. It is accompanied by 
 2 coUetlion of maps, known to all aftronomers by the 
 
 y litis 
 
I/* ASTRONOMICAL PHENOMENA, 
 
 title of ATLAS cELESTis. An ufeful abridgement of 
 both has been publiflied by Bode in Berlin^ and by Fortin 
 at Paris, in fmall quarto. Two planifpheres have alfo 
 been publifhed by Senex, in London, conilruclied from 
 the fame obfervations, and executed with uncommon ele- 
 gance ; as aifo a particular map of that zone of the 
 heavens to which all the planetary motions are limited. 
 This is alfo executed with fuperior elegance and accu- 
 racy. The place of any phenomenon may be afcertained 
 in it within 5' of the truth, by mere infpe61:ion, without 
 calculation, fcale, or compaiTes, No aitronomer fliould 
 be unprovided with it. 
 
 259. All thefe reprefcntations and defciiptions of 
 the ftarry heavens become obfolete, in fome meafure, in 
 confequence of a gradual change in the declination and 
 right afcenfion of the ftars. But, as this may be accu- 
 rately computed, the maps and catalogues retain their 
 original value, requiring only a little trouble in accomo- 
 dating them to the prefent (late of the heavens. The 
 Britannic Catalogue and Atlas are adjufted to the ftate of 
 the heavens in 1690 ; and the planifpheres, &c. by Senex 
 are the fame. The editions of Paris and Berlin are for 
 
 1750- 
 
 260. In tliefe maps and catalogues, it has been 
 found convenient to diftribute the ftars into groups, call- 
 ed CONSTELLATIONS ; and figures are drawn, which com- 
 prehend all the ftars of a group, and give them a fort of 
 
 (:onnexion 
 
CATALOGUES OF THE STARS. I7I 
 
 connexion and a name. Each flar is diftinguifhed by its 
 number in the conftellation, and alfo by a letter of the 
 alphabet. Thus, the moll: brilliant ftar in the heavens, 
 the Dog ftar, or Sirius, is known to all aftronomers as 
 N° 9., or as <«, canis tnajoris. The numbers always refer 
 to the Britannic catalogue, it being confidcred as claflical. 
 
 261. Since the publication of that work, however, 
 great additions have been made to our knowldge of the 
 ftarry heavens, and feveral Catalogues and Atlafes have 
 been publifhed in different parts of Europe. Of the ca- 
 talogues, the moft efteemed are, i. a fmall catalogue of 
 389 ftars, the places of which have been determined with 
 the utmoft care by Dr Bradley, at the Greenwich Obfer- 
 vatory ; 2. a catalogue of the fouthern ftars by Abbe 
 de la Caille ; 3. a catalogue of the zodiacal ftars by To- 
 bias Mayer at Gottingen 5 and, lajlly, a new atlas celeftis, 
 confifting of a catalogue and maps of the whole heavens, 
 And. contahiing above 15,000 ftars, by Mr Bode of Berlin. 
 The Rev. Mr Fr. Wollafton publiftied, in 1780, a fpe- 
 cimen of a general aftronomical catalogue of the fixed 
 ftars, arranged according to their declinations, folio, Lon- 
 don, 1780. This is a moft valuable work, containing 
 the places of many thoufand ftars, according to the cata- 
 logues of Flamftead, La Caille, Bradley, and Mayer. 
 Thefe being arranged in parallel columns, we fee the dif- 
 ferences between the determinations of thofe aftrono- 
 mers, and are advertifed of any changes which have oc- 
 curred in the heavens. The catalogue is accompanied 
 
 Y 2 by 
 
fjZ ASTRONOMICAL PHENOMENA. 
 
 by dire£tions for profecutlng this method of obtaining; a 
 minute furvey of the whole ftarry heavens. 
 
 In the valuable aftronomlcal tables publifhed in 177^ 
 by the academy of Berlin, Mr Bode has given a fimilar 
 fynopfis of the catalogues of Flamftead, La Caille, Brad- 
 ley and Mayer, not indeed fo extenfive, nor fo minute, 
 as WoUafton's, but of great ufe. 
 
 262. Having thus obtained maps of the heavens, 
 the place of a cclcftial phenomenon is afcertained in a 
 variety of ways. i. By its obferved diftance from two 
 known ftars. 2. By its altitude and azimuth. 3. Moil 
 jtccurately, by its right afcenfion and declination. 
 
 263. This lad being the mod accurate method of 
 afcertaining the place of any celeftial phenomenon, ob- 
 fervations of meridional altitude, and of transits over 
 the meridian, are the mod important. For an account of 
 the manner of conducing thefe obfervations, and a de- 
 fcription of the inftruments, we may confult Smith's 
 Optic-,, Vol. IL ; Mr Vince's Treatife of Praaical Aftro- 
 nomy ; La Lande's Aftronomy, &c. The mural quad- 
 rant, transit instrument, and clock, are there- 
 fore the capital furniture of an obfervatory -, to which, 
 however, fliould be added an equatoreal instrument 
 for ofcferving phenomena out of the meridian. Other in- 
 ftruments, fuch as the equal altitude instrument^ 
 the rhomboid.^ L reticula, the zenith sector, and 
 one or two moie, are fitted for aftronomers on a voyage. 
 
 264* 
 
DETERMINATION OF MERIDIAN, ScC. 1 73 
 
 264. The pofition of the meridian, and the latitude 
 ©f the obfervatory, muft be accurately determined. Va- 
 rious methods of determining the meridian. The moil 
 accurate is to view a circumpolar ftar through a telefcope 
 which has an accurate motion in a vertical plane, and to 
 change the pofition of the telefcope till the times which 
 elapfe between the fucce-Tive upper and lower tranfits of 
 the flar are precifely equal. The inftrument is then in 
 the plane of the meridian (fig. 33.) 
 
 265. In order to find the declination of a pheno- 
 menon more readily, it is convenient to know the incli- 
 nation of the axis of diurnal revolution NS (fig. 31.) to 
 the horizon, or the elevation of the pole N. The befl 
 method for this purpofe is to obferve the greateft eleva- 
 tion I O, and the leaft elevation K O, of fome circum- 
 polar ftar. The elevation of the pole N is half the fum 
 of thofe elevations. 
 
 266. The elevation of the pole is diiTerent in dif- 
 ferent places. An obferver, fituated 6^\ ftatute miles 
 due north of another, will find the pole elevated about a 
 degree more above his horizon. From obfervations of 
 this kind, the bulk and fiiape of the earth are deter- 
 mined. For it is plain that 360 times 69^ miles muft be 
 the circumference of the globe. It is found to be nearly 
 an elliptical fpheroid, of which the axis is 7904 miles, 
 and the greateft diameter 7940y miles. This deviation 
 from perfect fphericity has been difcovered by meafuring, 
 
 in 
 
174 ASTRONOMICAL PHENOMENA. 
 
 in the way now mentioned, a degree of the meridian m 
 different latitudes. One was meafured in Lapland, in 
 latitude 66'^ 20', and it meafured 122,457 yards, exceed- 
 ing dpi miles by 137 yards. Another was meafured at 
 Peru, crofnng the very equator. It contained 121,027 
 yards, falling Ihort of 6^\ miles by 1293 yards, and 
 wanting 1430 yards, or almoft a mile, of the other. 
 Other degrees have been meafured in intermediate la- 
 titudes ; and it is clearly eftablifhed, that the degrees 
 gradually increafe, as we go from the equator towards ei- 
 ther pole. 
 
 267. The length of a degree is the diftance between 
 two places where the tangents to the furface are inclined 
 to one another one degree, or where two plumb lines, 
 which are perpendicular to the furface of ftanding water, 
 will, when produced downwards, meet one another, in- 
 tercepting an angle of one degree. The furface of the 
 ftill ocean is therefore lefs incurvated as we approach the 
 poles, or it requires a longer arch to have the fame cur- 
 vature. It is a degree of a larger circle, and has a 
 longer radius. Perfons who do not confider the thing at- 
 tentively, are apt to imagine, from this, that the earth is 
 fliaped like an egg ; becaufe, if we draw from its centre 
 lines CN (fig. 33. N" 2.) CO, CP, C Q, equally in- 
 clined to one another, the arches NO, OP, P Q, will 
 gradually increafe from N towards Q. If thefe lines 
 make angles of one degree with one another, they will 
 meet the furface in points that are farther and farther 
 
 afunder^^ 
 
FIGURE OP THE EARTH. I75 
 
 afunder, and the degree will appear to increafe as we 
 approach the points E and Q, which we fuppofe, at pre- 
 fent, to be the poles. But let fuch perfons refleft, that 
 if thefe lines from the centre are produced beyond the 
 furface, they cannot be plumb lines, perpendicular to the 
 furface of {landing water. But if an ellipfe N E S Q 
 (fig. 33. N° 2.) be made to turn round its fhorter axis 
 N S, it will generate a figure flatter round N and S than 
 at E or Q. If we draw two lines a D and b B perpen- 
 dicular to the curve in a and h^ and exceedingly near one 
 another, they will be tangents to a curve A B D F, by 
 the evolution of which the elliptic quadrant E c N is de- 
 fcribed. A E ie the radius of curvature of the equate- 
 real degree of the meridian E « N. N F is the radius of 
 the polar degree, and « D is the radius of curvature at 
 the intermediate latitude of a^ &c. All thefe radii are 
 plumb lines, perpendicular to the elliptical curve of the 
 ocean. 
 
 Thefe plumb lines therefore do not meet in the centre 
 of the earth, as is^ commonly imagined, but meet, in 
 fuccelTion, in the circumference of the evolute A B D F. 
 The earth is not a prolate fpheroid like an tggy but an 
 ^hlate fpheroid, like a turnip or bias bowl. 
 
 268. Since the axis of diurnal revolution pafle$ 
 through the centre of the earth, it marks on its furface 
 two points, which are the poles of the earth. Thefe 
 are in the extremities of the axis of the terreftrial fphe- 
 roid. In like manner, the plane of the celeftiai equatof 
 
 pafling 
 
176 ASTRONOMICAL PHENOMENA. 
 
 paffing through the centre of the earth, divides it into 
 two hemifpheres, the northern and fouthern, feparated 
 by the terrejlrial equator. Alfo the hour circles, paffing 
 through the earth's centre, mark on its furface the ter- 
 reftria] meridians. 
 
 269. The pofition of a place on the furface of the 
 earth is determined by its latitude, or diftance from 
 the terreftrial equator, and its longitude, or the angu- 
 lar diftance of its meridian, from fome noted meridian. 
 
 270. Aftronomical obfervations are made from a 
 point on the furface of the earth, but, for the purpofes 
 of computation, are fuppofed to be made from the cen- 
 tre. The angular diftance between the obferved place 
 A (fig. 34.) of a phenomenon S in the heavens, as feen 
 from a place D on the Earth's furface, and its place B, 
 as viewed from the centre, is called the parallax of 
 the phenomenon. 
 
 ' 271. Befides the motion of diurnal revolution, com- 
 mon to all the heavenly bodies, there are other motions, 
 which are peculiar to fome of them, and are obferved by 
 us by means of their change of place in the ftarry hea- 
 vens. Thus, while the ftarry heavens turn round the 
 Earth from eaft to weft in 23** 56' 4", the Sun turns 
 round it in 24". He mtijl^ therefore, change his place 
 to the eaftward In the ftarry heavens. The Moon has 
 •an evident motion eaftward among the ftars, moving her 
 
 Qwn 
 
FIXED STARS — PLANETS. ^ I77 
 
 own breadth In about an hour. There are five flars 
 which are obferved to change their places remarkably in 
 the heavens, and are therefore called planets, or wan- 
 derers ; while thofe which do not change their relative 
 places are called fixed stars. The planets are Mer- 
 cury, Venus, Mars, Jupiter, and Saturn. To thefe 
 we mull now add the planet difcovered in 1781 by Dr 
 Herfchel, which he called the Georgian Planet, in ho- 
 nour of his Sovereign, the diflinguifhed patron of Aftro- 
 nomy. Aftronomers on the continent have riot adopted 
 this denomination, and feem generally agreed to call it 
 by the name of the difcoverer. M. Piazzi, at Palermo, 
 has difcovered another, and M. Olbers, at Bremen, a 
 third, which they have named Ceres and Pallas. None 
 of the three are vifible to the naked eye. 
 
 272. Planets are dlftinguifhable from the fixed ftars 
 by the fteadinefs of their light, while all the fixed ftars 
 are obferved to twinkle. The following fymbols are fre- 
 quently ufed : 
 
 For the Sun - - - - O 
 the Moon «---}) 
 Mercury - - - ^ 
 
 Venus - - - - ^ 
 
 the Earth - - - - j 
 
 Mars - - - - ^ 
 
 Jupiter - - - - -4 
 
 Saturn - - - - i) 
 
 Herfchel - - - ?L 
 
 z Th^ 
 
fj^ ASTRaNOMICAL PHENOMENA. 
 
 The motions of thefe bodies have become iiiterefting 
 on various accounts. Iti order to acquire a knowledge 
 ©f their motions more eafily, it is convenient to abllraft 
 Gur attention from the diurnal motion, common to all, 
 and attend only to their proper motions among the fixed 
 ftars* 
 
 Of the proper Motmis of the Sim. 
 
 273. We cannot obferve the motion of the Sun a- 
 mong the fixed flars immediately, on account of his 
 great fplendour, which hinders us from perceiving the 
 ft?.rs in his: . neighbourhood. Bat we can obferve the 
 iuftant of his coming to the meridian, and. liisL meridional 
 altitude (257.) The Sun mufli be in that point of the 
 heavens which paffss the meridian at that inflant, and 
 with that altitude. Or we can obferve the point of the 
 heavens which comes to the meridian at midnight, witli 
 a declination as far on one fide of the equator as the 
 Sun's obferved declination is on the other fide of it. 
 The Sun muft be in the point of the heavens which is 
 diametrically oppofite to this point. By taking either of 
 thefe methods, but particularly the firft, v/e can afcer- 
 tain a feries of points of the heavens through which the 
 Sun pafies. Thefe are found to be in the circumference 
 of a great circle of the fphere AS V W (fig. 35.), which 
 cuts .the celeftial equator in two oppofite points A, V, 
 and is inclined to it at an angle of 23° 28' 10" nearly. 
 This circle, or Sun's path, is called the ecliptic. 
 
 274. 
 
ANNUAL MOTION OF THE SUN. 1 79 
 
 274. In confequence of the obliquity of the ediptic, 
 the Sun's motion in it is accompanied by a change in 
 the Sun's declination and right afcenfion, by a change 
 in the length of the natural day, and by a change of the 
 feafons. Therefore, the revolution of the Sun in the 
 ecliptic is performed in a year. 
 
 275. The points V, A, are called equinoctial 
 POINTS ; bec^ufe, when the fun is in thefe points, his 
 circle of diurnal revolution is the celeftial equator, and 
 therefore the day and night are equal. The point V, 
 through which he palies in the month of March, is call- 
 ed the VERNAL EQUINOX, and the point A is called the 
 AUTUMNAL EQUINOX. The points S and W, where he 
 is fartheft from the equator, are called the solstitial 
 POINTS, S being the fummer, and W the winter folftice. 
 The parallels of declination pafhng through the folfli- 
 tial points are called tropics. 
 
 276. Right afcenfion is always computed eaftward 
 on the equator, from the vernal equinox. 
 
 277. The ecliptic paiTes through the conftcllations 
 Aries, diftinguiflied by the fymbol ^ 
 Taurus ----- \/ 
 Gemini - - jp - - U 
 Cancer ----- 05 
 
 Leo p^ 
 
 Virgo ----- irj^ 
 
 Z 2 Libra, 
 
l8o ASTRONOMICAL rHENOMENA. 
 
 Libra, difllnguiihed by the fymbol £if 
 
 Scorpio ----- trt 
 
 Sagittarius - _ - - ^ 
 
 Capricornus r - - - l^y 
 
 Aquarius ----- ^ 
 
 Pifces - - - - . X 
 
 Thefe conftellations are called the signs of the 
 
 ZODIAC ; and a motion from weft to eaft is faid to be 
 
 DIRECT, or IN CONSEQUENTIA siGNORUM, while a con-f 
 
 trary motion is called retrograde, in antecedentia 
 
 GIGNORUM. 
 
 278. The changes of the feafons were attributed by 
 the ancients to the influence of the ftars which were 
 feen in the different feafons of the year. 
 
 279. The pofition of the ecliptic is invariable, and 
 a complete revolution is performed in 365 days, 6 hourSj 
 9 minutes, and 1 1 feconds. 
 
 280. If fucceflive obfervations be made of the Sun's 
 crofTing the equator, it will be found that the equinoftial 
 points are not fixed, but move to the weftward about 
 50" in a year, fo that they would make a complete re- 
 volution in about 25,972 years. This is called the pre- 
 cession of the equinoxes. 
 
 281. Sir Ifaac Newton made a very ingenious and 
 important inference from this aftronomlcal fa6t. If we 
 
 know 
 
SYDEREAL AND TROPICAL YEAR5. l8l 
 
 know tlie fltimtion of the cquinoclial points at the time 
 of any hlftorical event, tlic date of the event may 
 be difcovered. He thinks that this pofitlon at the 
 time of the Argonautic expedition may be inferred 
 from the defcriptlon given by Aratus of the ftarry hea- 
 vens. The poet defcribes a celeitial fphcre by vrhich 
 Chiron, one of the heroes, directed their motions ; and 
 from this he deduces data for a chronology of the heroic 
 or fabulous ages. But, fmce the equino£liai points fliift 
 only at the rate of a degree in 72 years, and the 
 Greeks were fo ignorant, for ages after that epoch, that 
 they did not know that the pofitions of the (tars were 
 changeable, it does not appear tliat much reliance can 
 be had on this datum. We cannot, from the defcrip- 
 tlon by Aratus, be certain of the portion of the vernal 
 equinox within five or fix degrees. This makes a differ^ 
 cnce of 400 years in the epochs. 
 
 282. The axis of diurnal revolution is not always 
 the fame, and the poles of the heavens defcribe (in 25,972 
 years) a circle round the pole of the ecliptic, diilant 
 from it 23° 28' 10" nearly, 
 
 283. On account of the wefiierly motion of the e- 
 quinodlial points, the return of the feafons mud be ac- 
 comphfiied in lefs time than that of the Sun's revolution 
 round the heavens. The feafons return after an intcrv?.! 
 ^^ 3^5^ 5" 43' 45". This is called a tropical year, to 
 diftinguifh it from the interval 365^^ 6" 9 ii", called a 
 5YDEREAL year. 
 
 284. 
 
lS2 ASTRONOM-ICAL PHENOMENA. 
 
 284. Aftronomcrs have chofen to refer the places 
 of the heavenly bodies to the ecJiptic, on account of its 
 inability, rather than to the equator. For this purpofe, 
 great circles, fuch as PV^, PA/), (fig. 36.) are drawn 
 through the poles P, />, of the ecliptic* Thefe are called 
 ECLIPTIC MERIDIANS. The arch AB of one of thefc 
 circles, intercepted between a phenomenon A and the 
 ecliptic, is called the latitude of the phenomenon ; 
 and the arch V B, intercepted betvv^een the point V of 
 the vernal equinox and the point B, is called the longi- 
 tude of the phenomenon. This is fometimes exprefTed 
 in degrees and minutes, and fometimes in figns, (each 
 
 = 30°.) 
 
 285. The motion of the Sun in the ecliptic is not 
 uniform. On the firfl of January his daily motion is 
 nearly 1° i' 13". But on the firft of July, his daily mo- 
 tion is 57' 13". The mean daily motion is 59' c8". 
 The Sun*3 place in the ecliptic, calculated on the fup- 
 pofition of a daily motion of ^g' 08", will be behind his 
 obferved place, from the beginning of January to the be- 
 ginning of July, and will be before it, from the begin- 
 ning of July to the beginning of January. The greateft 
 difference is about 1° $^' 32", v/hich is obferved about 
 the beginning of April and Oftober ; at which times, 
 the obferved daily motion is 59' 08". 
 
 286. This unequable motion of the Sun appeared 
 to the ancient aftronomers to require feme explanation. 
 
 It 
 
UNEQUABLE MOTION OF THE SUN. iS 
 
 It had been received as a firft principle, tliat the celeftial 
 motions were of the mof!: perfect kind — and this per- 
 fection was thought to require invariable famenefs. 
 Therefore the Sun muft be carried uniformly in the cir- 
 cumference of a figure perfedly uniform in every part. 
 He muil therefore move uniformly in the circumference 
 of a circle. The ailroncmers therefore fuppofcd that 
 the Earth is not in the centre of this circle. Let AbV d 
 (fig. 37.) reprefent the Sun's orbit, having the Earth in 
 E, at fome diflance from the centre C. It is plain that 
 if the Suii's motion be uniform in the circumference, 
 defcribing every dny 59' 08", his angular motion, as {^tn 
 from the Earth, muft be flower when he is at A, his great- 
 eft diftance, than when neareft to the Earth, at P. It is 
 alfo evident that the point E may be fo chofen, that an 
 arch of 59' 08" at A fhall fubtend an angle ut E tliat is 
 only 57' 13", and that an arch of 59' 08'' at P fhall 
 fubtend an angle of 61' 13". This will be accompliflied, 
 if we m.ake EP to E A as 57' 13" to 61' 13'', or nearly 
 as 14 to 15. This v/as accordingly done ; and this me- 
 thod of folving the appearances was called the eccentric 
 hypothefis. E C is the eccentricity, and PE is to 
 P G nearly as 28 to 29. 
 
 287. But although this hypothefis agreed very v/eli 
 with obfervation in thofe points of x\\q orbit where the 
 Sun is mod remote from xki^ Earth, or neareft to it, 
 it was found to differ greatly ir* other parts of tlie orbit, 
 and particularly about half way between A and P. A- 
 
 ilronomcrs. 
 
UNEC^ABl.E MOTION OF THE SUN. 1^3 
 
 It had been received as a firft principle, that the celeftial 
 nictions were of the moft perfect kind — and this per- 
 fection was thought to require invariable famenefs. 
 Therefore tlie Sun muft be carried uniformly in the cir- 
 cumference of a figure perfe£lly uniform in every part. 
 He muil therefore move uniformly in the circumference 
 of a circle. The aflroncmers therefore fuppofcd that 
 the Earth is not in the centre of this circle. Let AbV d 
 (fig. 37.) reprefent the Sun's orbit, having the Earth in 
 E, at fome diftance from the centre C. It is plain that 
 if the Sun's motion be uniform in the circumference, 
 defcribing every day 59' 08", his angular motion, as ittn 
 from the Earth, muil be flower when he is at A, his great- 
 eft diftance, than when neareft to the Earth, at P. It is 
 alfo evident that the point E may be fo chofen, that an 
 arch of 59' 08" at A fhali fubtend an angle ut E tliat is 
 only 57' 13", and that an arch of 59' 08" at P fhall 
 fubtend an angle of 61' 13". This will be accornpliflied, 
 if we m.ake EP to E A as 57' 13" to 61' 13", or nearly 
 as 14 to 15. This v/as accordingly done ; and this me- 
 thod of folving the appearances was called the eccentric 
 kypothefts. E C is the eccentricity, and PE is to 
 P G nearly as 28 to 29. 
 
 287. But although this hypothefis agreed very well 
 with obfervation in thofe points of the orbit where the 
 Sun is moft remote from the Earth, or neareft to it, 
 it was found to differ greatly in other parts of the orbit, 
 arid particularly about half way between A and P. A. 
 
 ftronomcrs. 
 
lg4 ASTROKOMICAL PHENOMENA. 
 
 ftronomers, after trying various other hypothefes, were' 
 obliged to content themfelves with reducing the eccen- 
 tricity confiderab^y, and alfo to fuppoie that the angular 
 motion of 59' 08" per day was performed round a point 
 c on the other fide of the centre, at the fame diftance 
 with E. This, however, v/as giving up the principle of 
 perfect motion, if its perfe(Slion confifted in uniformity j.^ 
 for, in this cafe, the Sun cannot have an uniform mo- 
 tion in the circumference, and alfo an uniform angular 
 motion round e. Eefides, even this amendment of the 
 eccentric hypothecs by no means agreed with the ob- 
 fervations in the months of April and October: but 
 they could not make it any better. 
 
 288. Allronomical computations are made on the 
 fuppofition of uniform angular motion. The angle pro- 
 portional to the time is called the mean motion, and 
 the place thus computed is called the mean place. 
 The differences between the mean places and the ob- 
 ferved, or true places, are called equations. They 
 are always greateft when the mean and true motions arc 
 equal, and they are nothing when the mean and true 
 motions ditfer nioft. For, while the true daily angular 
 motion is lefs than the mean daily motion, the obferved 
 place falls more and more behind the calculated place 
 every day, and although, by gradually quickening, it lofes 
 lefs every day, it flill lofes, and falls ftill more behind ; 
 and when the true daily motion has at laft become equal 
 to the mean, it lofes no more indeed, but it is now the 
 
 farthed 
 
MEAN TIME — EQUATION OF. 1 85 
 
 fartheft behind that can he. Next day it gains a little 
 ti the loft ground, but is ftill behind. Gaining more and 
 more every day, by its increafe of angular motion, it at 
 laft comes up with the calculated place ; but row, its 
 angular motion is the greateft poffible, and differs moll 
 from the equable mean motion, 
 
 289. Thefe computations are begun from that point 
 of the orbit where the motion is lloweft, and the mean 
 angular diftance from this point is called the mean ano- 
 maly. A table is made of the equations correfponding 
 to each degree of the mean anomaly. The true anomaly 
 is found by adding to, or fubtra^iing from the computed 
 mean anomaly, the equations correfponding to it. 
 
 In this manner may the fun's longitude, or place in 
 the ecliptic, be found for any time* 
 
 290. In confequence of the obliquity of the ecliptic, 
 and the fun's unequal motion in it, the natural days, or 
 the interval betv/een two fucceflive palTages of the fun 
 over the meridian, are unequal ; and if a clock, which 
 meafures 365** 5^ 48' 45" in a tropical year, be compared 
 from day to day with an exa£t fun dial, they will be 
 found to differ, and will agree only four times in the 
 year. This difference is called the equation of time, 
 and fometimes amounts to 16 minutes. The time fhewn 
 by the clock is called mean solar time, and that fliewn 
 by the dial is called true time and apparent time. 
 
 A a 291. 
 
r86 ASTRONOMICAL PHENOM'ENA^ 
 
 291. The change in the fun's motion is accompa- 
 nied by a change in his apparent diameter, v/hich, at the 
 beginning of January, is about 32' 39", and at the begin- 
 ning of July is about 31' 34", y^ lefs. This mu^ be 
 afcribed to a change of diftance, which muft always be 
 fuppofed inverfely proportional to the apparent diameter. 
 
 292. By combining the obfervations of the fun's 
 place in the ecliptic with thofe of his diftance, inferred 
 from the apparent diam.eter, and by other more decifive, 
 but lefs obvious obfen^ations, Kepler, a German aftro- 
 nomer, found that his apparent path round the earth is 
 an ellipfe, having the earth in one focus, and having the 
 longer axis to the fhorter axis as 200,000 to 199,972. 
 
 The extremities A and P of the longer axis of the 
 fun's orbit ABPD (fig. 37.) are called the apsides. 
 The point A, where the fun is fartheft from the earth 
 (placed in E), is called the higher apfis, or apogee. 
 P is the lower apfis, or perigee. The diftance E C 
 between the focus and centre is called the eccentri- 
 city, and is 1680 parts of a fcale, of wliich the mean 
 diftance ED is 100,000. 
 
 293. Kepler obfervedy that the fun's angular motion 
 in this orbit was inverfely proportional to the fquare of 
 his diftance from the earth •, for he obferved the fun's 
 daily change of place to be as the fquare of his apparent 
 diameter. Kence, be inferred that the radius vedor E B 
 defcribed areas proportional to the times (103.) 
 
 294- 
 
CALCULATION OF THE SUN*S PLACE. I 87 
 
 294. From this he deduced a method of calculating 
 the fun's place for any given time. Draw a line E F 
 from the focus of the ellipfe, which fliall cut off a fe£lor 
 A E F, having the fame proportion to the whole furface 
 of the ellipfe, which the interval of time between the 
 fun's laft paffage through his apogee, and the time for 
 which the computation is made, has to a fydereal year j 
 F will be the fun's true place for that time. This is 
 called KEPLER^s problem. 
 
 This problem, the mod interefting to aftronomers, 
 has not yet been folved otherwife than by approximation, 
 or by geometrical conftrudtions which do not admit of 
 accurate computation. 
 
 295. Let ABPD (fig. 37.) be the elliptical orbit, 
 having the earth in the focus E. A and P, the extremi- 
 ties of the tranfverfe axis, are the apogee and perigee of 
 the. revolving body, B D is the conjugate axis, and C 
 the centre. It is required to draw a line E T which 
 fhall cut off a feclor AET, which has to the whole el- 
 lipfe the proportion oi m to n -, m being taken to n in 
 the proportion of the time elapfed lince the body was in 
 A to the time of a complete revolution. 
 
 Kepler, who was an excellent geometer, faw that this 
 would be effected, if he could draw a line E I, which 
 fhould cut off from the circumfcribed circle AbV d the 
 area A E I, which is to the whole circle in the fame pro- 
 portion of m to n. For, then, drawing the perpendi- 
 cular ordinate I R, cutting the ellipfe in T, he knew 
 that the area AET has the fame proportion to the el- 
 A a 2 lipfe 
 
1 88 ASTRONOMICAL PHENOMENA. 
 
 lipfe that A E I has to the circle. The proof of this 
 is eafy, and it feems greatly to fimplify the problem. 
 Draw I C through the centre, and make E S perpendi- 
 cular to I C S. The area A E I confifts of the circular 
 feftor A C I, and the triangle C I E. The fe6tor is equal 
 to half the rectangle of the radius C I and the arch A I, 
 
 G A X I A 
 
 that is, to '■ . The triangle C I E is equal to 
 
 eixES CAxES ^, , . . ., ^ 
 , or . Inereiore it is evident 
 
 that, if we make the arch I M equal to the ftraight line 
 E S, the feftor ACM will be equal to the circular area 
 A C I, and the angle ACM will be to 360 degrees, as 
 in to «. 
 
 295. Hence we fee that k will be eafy to find the 
 time when the revolving body is in any point T. To 
 find this, dravv^ the ordinate R T I ; draw I C S and E S, 
 and make IM = ES. Then, 360° is to the arch AM 
 as the time of a revolution to the time in which the body 
 moves over A T. This is (in the aftronomical language) 
 finding the mean anomaly vi^hen the true anomaly is 
 given. The angle A C M, proportional to the time, is 
 called the mean anomaly, and the angle A E T is the 
 TRUE ANOMALY. The angle A C I is called the ano- 
 maly OF THE ECCENTRIC, Or the ECCENTRIC ANO- 
 MALY. 
 
 297. But the aftronomer wants tae, true anomaly 
 ^<:orrcfponding to a given mean anomaly. The procef> 
 Iiere given cannot be reverfcd. We cannot tell how 
 
 mucin 
 
KEPLeVs PROBLEM. I 89 
 
 «nucK to cut cfF from tlie given mean anom?Jy A M, fo 
 as to leave A I of a proper magnitude, becaufe tlie indif- 
 penfable meafure of M I, namely E S, cannot be had till 
 I C S be drawn. Kepler faw this, and fild that his 
 problem could not be folved geometrically. Since the 
 invention of fluxions, however, and of converging fe- 
 riefes, very accurate folutions have been obtained. That 
 given by Frifitts in his Cofmographia is the uune in prin- 
 ciple with all the mod approved methods, and the iovro, 
 in which it is prefented is peculiarly fimple and neat. 
 But, except for the conftru^lion of original tables, tl-efe 
 methods are rarely employed, on account of the labo- 
 rious calculation which they require. Of all the dire6i: 
 approximate folutions, that given by Dr IMatthew Stew- 
 art at the end of his Traclsj Phyftcal and Mathematical^ 
 publifhed in 1761, feems the moft accurate and elegant \ 
 and the calculations founded on it are even iliorter than 
 the indiredl methods generally employed. His conftruc- 
 iion is as follows. 
 
 298. Let the angle A E M be the mean anomaly, 
 join EM, and draw Cz parallel to it, and MO perpendicu- 
 lar to C /. If the orbit is not more eccentric than that 
 of Mars, make the arch i I equal to the excefs of the 
 arch M i above its fine I^.'I O. Then A I is the eccent- 
 ric anomaly correfponding to the mean anomaly A M, 
 and the ordinate I R will cut the ellipfe in T, io that 
 AET will be the true anomaly required. The error 
 will not amount to two feconds in any part of fuch or- 
 bits. 
 
ipo ASTRONOMICAL PHENOMENA. 
 
 bits. But, for orbits of greater eccentricity, another ftep 
 is neceflary. Join E/, and draw C Q parallel to E /, meet- 
 ing the tangent i O in Q. Let D reprefent the excefs of 
 the arch M i above its fine MO*, and inftitute the 
 following analogy, fin. M C i : tan. i C Q = D : / 1, tak- 
 ing i I from / towards M. The point, I, will be fo fi- 
 tuated that the fector A E I is very nearly equal to the 
 feftor ACM, or A I is the eccentric anomaly corref- 
 ponding to the mean anomaly A M. The error will not 
 amount to one fecond, even in the orbit of Mercury. 
 
 The demonftration of this conftrudilon is by no means 
 abftrufe or difficult. Draw I S, and M I. The triangles 
 z C E and i C M are evidently equal, being on one bafe 
 i C, and between the parallels i C and M E. For fimilar 
 reafons, the triangles z S I and /EI are equal. There- 
 fore the triangle / C E, together with the fegment in- 
 cluded between the arch Mb i and the chord M /, will 
 be equal to the circular feftor / C M. 
 
 Now it is plain, from the conftruci:ion, that S i :C i 
 = SE:/Q, =MO:iQ, =Sr^'-^^rO:iL There- 
 fore S/X>I = C/ X Ml^i — Ci X MO. But Ci X 
 
 Mbi 
 
 * This excefs muft be expreffed in degrees, minutes, or 
 fecondo. The radius of a circle is equal to an arch of 206,265 
 feconds. The logarithm of this number is 5.3 14425 1. There- 
 fore we fhall obtain E S, or the feconds in E S, by adding 
 this logarithm to the logarithms of E C (AC being unity), 
 and the logarithm of the fine of A C I. The fum is the lo- 
 garitlim of tlie feconds in E S. 
 
KEPLER'S PROBLEM. I9I 
 
 M /'/ is equal to twice the fe£lor M C /, and C t X M O 
 is equal to twice the triangle M C /. Therefore S i X il 
 is equal to twice the fegnient contained between M b i 
 and the chord M /. Therefore this fegment is equal to 
 the triangle i S I, or to the triangle / E I. Therefore the 
 fpace C / 1 E C is equal to the feclor / C iM, and the 
 feaor A E I to the fedor A C M. 
 
 The calculation founded on this con{lru6tion is ex- 
 tremely fimple. In the triangle M C E, the fides M C 
 and C E are given, with the included angle MCE; and 
 the angles OE M, C M E are fought. Moreover, A E is 
 the fum of the given fides, and P E is their difference, 
 and A C M is the fum of the angles M and E. There- 
 
 ^ E -f M E — INI 
 
 fore A E : E P = tan. — ■ : tan. : and thus E 
 
 2 2 
 
 and M, or their equals, A C i and M C /', are obtained. 
 
 In the next place, in the triangle i C E, the fides i C, 
 
 C E, z-zA the included angle / C E, are given, and the 
 
 angle E / C is fought. We have, in the fame manner as 
 
 before, AC/ equal to the fum of the angles E and /, 
 
 E j_ / E i 
 
 and therefore A E : E P = tan. — -^ : tan. . Thus 
 
 2 2 
 
 the angle E / C, or its equal, i C Q, is obtained, and then, 
 
 a 1 • T T\ ^ Q TA tan. / C O 
 
 the arch z 1 = D x -^^irh j = D X -^ — '^ittt^- • 
 
 MO' ~- ^ im. MC 
 
 In the very eccentric orbits of tl-e comets, this brings 
 us vaftly nearer to the truth than any of the indireft me- 
 thods we know does by the firil ftep. So near indeed, that 
 the common method, by the rule offalfe pofition^ may now 
 be fafely employed. If the point, I, has been accurately 
 
 found, 
 
iC)2 ASTRONOMICAL PHENOMENA. 
 
 found, it is plain that if to the arch A I we add E S, 
 that is, E C X fin. A C I, we obtain the arch A M with 
 which we began. But if I has not been accurately de-' 
 termined, A M will differ from the primitive A M.- 
 Therefore, make fome fmall change on A I, and again 
 compute A M. This will probably be again erroneous. 
 Then apply the rule of falfe pofition as ufual. The er- 
 ror remaining after the firfl ftep of Dr Stewart^s procefs 
 is always fo moderate that the variations of A M are very 
 nearly proportional to the variations of A I ; fo that two 
 fteps of the rule will generally bring the calculation 
 within two or tliree feconds of the truth. The aflrono- 
 mical ftudent will find many beautiful and important 
 propofitions in thefe mathematical trails. The propofi- 
 tion juft now employed is in page 398, Sec. 
 
 299. Aftronomers have difcovered, that the line AP 
 moves flowly round E to the eaftward, changing its place 
 about 25' 56" in a century. This makes the time of a 
 complete revolution in the orbit to be 365(1 6^ 15' 20". 
 This time is called the anomalistic year. 
 
 Of the proper Motions of the Moon, 
 
 300. Of all the celeftial motions, the moft obvious- 
 are thofe of the Moon. We fee her fliift her fituation 
 among the ftars about her own breadth to the eaftward 
 in an hour, and in fomewhat lefs than a month fhe makes 
 a complete tour of the heavens. The gentle beauty of 
 
 her 
 
THE MOTIONS OF THt MOON. i'/J, 
 
 hex appearance during the quiet hours of a ferr^ne night, 
 has attracted the notice, and we may fay the affcc'tions 
 of all mankind ; and (lie is juftly ftyled the O '.een of 
 Heaven. The remarkable and diftin^^ changes of her 
 appearance have afforded to all ^mple nations a moft 
 convenient index and meafure of time, bot)^ for record- 
 ing pad events, and for making any future appointments 
 for bufinefs. Accordingly, we find, in the firft hiftr-nes 
 of all nations, that the lunar motions were the firft ftu- 
 died, and, in fome degree, underftood. It feems to have 
 been in fubferviency to this ftudy alone that the other 
 appearances of the ftarry heavens were attended to ; and 
 the relative poiitions of the ftars feem to have interefted 
 us, merely as the means of afcertaining the motions of 
 the Moon. For we find all the zodiacs of the ancient 
 oriental nations divided, not into 12 equal portions, cor- 
 refponding to the Sun's progrefs during the period of fea- 
 fons, but into 27 parts, correfponding to the Moon's daily 
 progrefs, and thefe are exprefsly called th^ houses or man- 
 sions of the Moon. This is tha diftribution of the zodiac 
 of the ancient Hindoos, the Perfians, the Chinefe, and even 
 the Chaldeans. Some have no divlfion into 12, and thofe 
 who have, do not give names to 1 2 groups of ftars, but 
 to 27. They firft defcribe the fituation of a planet in 
 one of thefe manfions by name, noting its diftance from 
 fome ftars in that group, and thence infer in what part 
 of which twelfth of the circumference it is placed. The 
 divifion into 12 parts is merely mathematical, for the 
 purpofe of calculation. In all probability, therefore, this 
 
 B b was 
 
194 ASTRONOMICAL PHENOMENA. 
 
 was an after-thought, the contrivance of a more culti- 
 vated age, well acquainted with the heavens as an object 
 of figiit, and beginning to extend the attention to fpecu- 
 lations beyond the firfl: conveniences of life. 
 
 301. When the Moon*s path through this feries of 
 manfions is carefully obferved, it is found to be (very 
 nearly) a great circle of the heavens, and therefore in a 
 plane pafling through th^ centre of the earth. 
 
 302. She makes a complete revolution of the heaven§ 
 in 27^ 7'^ 43' 12", but with fome variations. Her mean 
 daily motion is therefore 13° 10' 25", and her horary mo- 
 tion is 32' 5(5''. 
 
 303. Her orbit is inclined to the plane of the ecliptic 
 in an angle of 5° 8' 45", nearly, cutting it in two points 
 called her nodes, diametrically oppofite to each other j 
 and that node through which fhe pafTes in coming from 
 the fouth to the north fide of the ecliptic, is called thj? 
 
 ASCENDING NODE. 
 
 304. The nodes have a motion which is generally 
 weftward, but with confiderable irregularities, making d 
 complete revolution in about 6803'^ 2"^ 55' 18", nearly 
 18} years. 
 
 305. If we mark on a celeftial globe a feries of 
 points where the Moon was obferved during three oj 
 
 four 
 
LUNAR EQUATIONS PARALLAX. I95 
 
 four revolutions, and then lap a tape round the globe, 
 covering thofe points, we flrall fee that the tape crofles 
 the ecliptic more wefterly every turn, and then crofles 
 the lafl round very obliquely -, and we fee that by conti- 
 nuing this operation, we fhall completely cover with the 
 tape a zone of the heavens, about ten or eleven degrees 
 broad, having the ecliptic running along its middle. 
 
 306. The Moon moves unequally in this orbit, her 
 hourly motion increafing from 29' 34" to 36' 48", and 
 the equation of the orbit fometimes amounts to 6° 18' 32" ; 
 fo that if, fetting out from the point where her horary 
 motion is floweft, we calculate her place, for the eighth 
 day thereafter, at the rate of 32' 56" per hour, we fliall 
 find her obferved place fhort of our calculation more 
 than half a day's motion. And we fhould have found 
 her as much before it, had we begun our calculation from 
 the oppofite point of her orbit. 
 
 307. Her apparent diameter changes from 29' 26" 
 to 33' 47", and therefore her diftance from the Earth 
 changes. This diftance may be difcovered in miles by 
 means of her parallax. 
 
 She was obferved, in her paflage over the meridian, 
 by two aftronomers, one of whom was at Berlin, and the 
 other at the Cape of Good Hope. Thefe two places are 
 diftant from one another above 5000 miles j fo that the 
 obferver at Berlin faw the Moon every day confiderably 
 more to the fouth than the perfon at the Cape. This 
 B b 2 diiFerence 
 
loO ASTROXOMICAL PHENOMENA. 
 
 difForc;ice of apparent declination is the meafure o£ the 
 angle DSC (fig. 34.) fubtended at the Moon by the 
 line rD of 5443 miles, between the obfeiTers. The 
 angles ST) c imd S rD are given by means of the Moon's 
 obfeiTcd altitudes. Therefore any of the fides S D or 
 S c m-riy be computed. It is found to be nearly 60 fe- 
 midiameters of the earth. 
 
 3c8» By combining the obfcrvations of the Moon's place 
 in the heavens with thofe of her apparent diameter, we dif- 
 cover that her orbit h nearly an ellipfe, having the Earth 
 in one focus, and having the longer axis to the fhorter 
 axis nearly as 91 to 89. The greatef! and lead diftanccs. 
 are nearly in the proportion of 21 to 19. 
 
 309. Her motion in this ellipfe is fuch, that the line 
 joining the Earth and Moon defcribes areas which are 
 nearly proportional to the times. For her angular hourly 
 motion is obferved to be as the fquare of her apparent 
 diameter. 
 
 3 1 o. The line of the apfides has a flow motion eaft- 
 ward, completing a revolution in about 3232^^ ii'^ 14' 30'', 
 nearly 9 years. 
 
 311. While the Moon is thus making a revolution 
 round the heavens, her appearance undergoes great 
 changes. She is fometimes on our meridian at midnight, 
 and, therefore, in the part of the heavens which is op- 
 
 pofite 
 
LUNAR PHASES — ECLIPSES. 197 
 
 pofite to the Sun. In this fituation, fiie is a complete lu- 
 minous circle, and is faid to be full. As fhe moves- 
 eaftward, (lie becomes deficient on the weft fide, and, 
 after about 7y days, comes to the meridian about fix in 
 the morning, having the appearance of a femicircle, 
 with the convex fide next the Sun. In this (late, her 
 appearance is called half j.ioon. Moving fi:iil eafi:vi?-ardy 
 file becomes more deficient on the weft fide, and has now 
 the form of a crefcent, with the convex fide turned to- 
 wards the Sun. This crefcent becomes continually more 
 fiender, till, about 14 days after being full, ftie is fo 
 near the Sun that fhe cannot be feen, on account of his 
 great fplendour. About four days after this difappear-^ 
 ance in the morning before funrife, fhe is feen in the 
 evening, a little to the eaftward of the Sun, in the form 
 of a fine crefcent, with the convex fide turned towafcl 
 the Sun. Moving ftill to the eaftward, the crefcent be- 
 comes more full, and when the Moon comes to the me-, 
 ridian about fix in the evening, fire has again the appear- 
 ance of a bright femicircle. Advancing ftill to the eaft- 
 ward, flie becomes fuller on the eaft fide, and, at laft, 
 after about 29^ days^ (lie is again oppofite to the Sun, 
 and again full. 
 
 312. It frequently happens that the Moon is eclip- 
 sed when full ; and that the Sun is eclipfed fome time 
 between the difappearance of the Moon in the morning 
 on the weft' fide of the Sun, and her reappearance in the 
 evening on the eaft fide of the Sun. This eclinfe of the 
 
 Sun 
 
198 ASTRONOMICAL PHENOMENA." 
 
 Sun happens at the very time that the Moon, in the coun^ 
 of her revolution, paffes that part of the heaVens where 
 the Sun is. 
 
 313. From thefe obfervations, we conchide, i. That 
 the Moon is an opaque body, vifible only by means of 
 the Sun's light illuminating her furface ; 2. That her or- 
 bit round the Earth is nearer than the Sun's. 
 
 314. From thefe principles all her phases, or ap- 
 pearances, may be explained (fig. 39.) 
 
 315. When the Moon comes to the meridian at 
 mid-day, f!ie is faid to be new, and to be in conjunc- 
 tion with the Sun. When fhe comes to the meridian 
 at midnight, flie is faid to be in opposition. The line 
 joining thefe two fituations is called the line of the 
 SYZiGiEs. The points where fhe is half illuminated are 
 called the quADRATUREs ; and that is called the firft 
 <juadrature which happens after new moon. 
 
 316. When the Moon is half illuminated, the line 
 .EM (iig. 39.) joining the Earth andMoon, is perpendicular 
 
 to the line MS, joining the Moon and Sun. By obferv- 
 ing the angle S E M, the proportion of the diftance of 
 the Sun to the diftance of the Moon may be afcertained. 
 This method of afcertaining the Sun's diftance was 
 propofed by Ariilarchus of Samos, about 264 years be- 
 fore the Chriilian sera. The thought was extremely in- 
 genious, 
 
SYNODICAL REVOLUTION — LUNATION. I99 
 
 i^enious, and flrictly juft j and tliis was the firil oLfervar 
 tion that gave the aftronomcrs any confident gucfs at the 
 very great diftance of the Sun. But it is impofifible to 
 judge of the half illumination of the Moon's diilc v^^itli 
 fufficient accuracy for obtaining any tolerable meafure. 
 Even now, when afTifted by telefcopes, we cannot tell 
 to a few minutes when the boundary between light and 
 darknefs in the Moon is exaftly a llraight line. When 
 this really happens, the elongation S E M wants but 9' 
 of a right angle, and when it is altogether a right angl^ 
 there is no fenfible change in the appearance of the 
 Moon. All that the ancient aftronomers could infer 
 from their bed eftimation of the bifeclion of the Moon 
 was, that the Sun was, for certain, at a much greater 
 diftance than any perfon had fuppofed before- that time. 
 Ariftarchus faid, that the angle S E M was not Icfs than 
 87 degrees, and therefore the Sun was at leaft twenty 
 times farther off than the Moon. But aftronomers of the 
 Alexandrian fchool faid, that the angle S E M exceeded 
 89°, and the Sun was fixty times more remote than the 
 Moon. Modern obfervations fliew him to be near four 
 hundred times more remote. 
 
 317. This fucceflion of phafes is completed in a 
 period of 29** 1 2^* 44' 3", called a syxodical month 
 and a lunation. 
 
 It may be afkcd here, how the period of a lunation 
 comes to differ from that of the Moon's revolution round 
 ihe Earth, which is accompliftied in 27*^ 7'' 43' 12" ? This 
 
 is 
 
2G0 ASTRONOMICAL rHE^:O^IE^'A. 
 
 is owing to tlie Sun's change of place during a revolu- 
 tion of the Moon. Suppofe it new Moon, and there- 
 fore the Sun and Moon appearing in the fame place of 
 the heavens. At the end of the lunar period, the Moon 
 is again in that point of the heavens. But the Sun, in 
 the mean time, has advanced above 27 degrees; and 
 fomewhat more than two days mufl: elapfe before the 
 Moon can overtake the Sun, fo as to be feen by us as 
 new moon. 
 
 318. The period of this fuccefTion of phafes may 
 be found within a few hours of the truth in a very fhort 
 time. V^^e can tell, within four or five hours, the time 
 of the Moon being half illuminated. Suppofe this ob- 
 ferved in the morning of her laft quarter. We fliall fee 
 this twice repeated in ^g days, which gives ipd 12* for 
 a lunation, w^anting about three fourths of an hour of the 
 truth. About 433 years before the Chriflian rera, Me- 
 ton, a Greek aftronomer, reported to the dates aflembled 
 at the Olympic games, that in nineteen years there hap- 
 pened exaftly 235 lunations. 
 
 319. The lunar motions are fubjeft to feveral irre- 
 gularities, of which the following are the cliief : 
 
 320. I. The periodic month is greater when the Sun 
 js in perigee thr.n when in apogee, the greatefl: dilFer- 
 ence being about 24 minutes. Tycho Brahe firll re- 
 m?a*ked this anomaly of the lunar mot!on3, and called 
 
 the 
 
LUNAR INEQUALITIES. 20l 
 
 tlie correction, (depending on the Sun's place in his or- 
 bit), the ANNUAL EQUATION of the Moon. 
 
 321. 2. The mean period is lefs than it was in an- 
 cient times. 
 
 322. 3. The otbit is larger when the Sun is in pe- 
 rigee than when he is in apogee^ 
 
 323. 4. The orbit is more eccentric when the Sun 
 is in the Hne of the lunar apfides ; and the equation of 
 the orbit is then increafed nearly 1° 20' 34". This change 
 IS called the evection. It was difcovered by Ptolemy. 
 
 324. ^. The incHnation of the orbit changes. 
 
 325. 6. The moon's motion is retarded in the firft 
 and third quarters, an(J accelerated in the fecond and 
 laft. This anomaly was difcovered by Tycho Brah6, 
 who c^lls it the variation. 
 
 32<5. 7. The motion of the nodes is very unequal. 
 
 Of the Calendar. 
 
 327. Aftrdnomy, like all other fciences, was firft 
 
 pra£tifed as an art. The chief object of this art was to 
 
 know the feafons, which, as v/e have feen, depend either 
 
 C c immediately. 
 
i02 ^ASTRONOMICAL PHENOMENA. 
 
 immediately, or more remotely,, on the Sun's motion in 
 tlie ecliptic. A ready method for knowing the feafon 
 feems, in all ages, to have been the chief incitement to 
 the (ludy of aftronomy. This muft direct the labours of 
 the field, the migrations of the fhepherd, and the journies 
 of the traveller. It is equally ncceflary for appointing all 
 public meetings, and for recording events. 
 
 Were the ftars vifible in the day time, it would be 
 eafy to mark all the portions of the year by the Sun's 
 place among them. When he is on the foot of Caftor, 
 it is midfummer ; and midwinter, when he is on the 
 bow of Sagittarius. But this cannot be done, becaufe 
 his fplendour eclipfes them alL 
 
 328. The bell approximation which a rude people 
 can m.ake to this, is to mark the days in which the ftars 
 of the zodiac come firft in fight in the morning, in the 
 eaftern horizon, Im.medlately before the Sun rife. As he 
 gradually travels eaflward along the ecliptic, the brighter 
 ftars which rife about three quarters of an hour before 
 the Sun, may be feen in fucceffion. The hufbandman 
 and the fiiepherd were thus warned of the fucceeding 
 talks by the appearance of certain ftars before the Sun. 
 Thus, in Egypt, the day was proclaimed in which the 
 Dogftar was firft feen by thofe fet to watch. The in- 
 habitants immediately began to gather home their wan-' 
 dering flocks and herds, and prepare themfelves for the 
 inundation of the Nile in twelve or fourteen days. Hence 
 that ftar was called the IVatch-dog^ Thoth, the Guar- 
 dian of Egypt. 
 
 This 
 
THE KALENDAR — HELIACAL RISING. 20^ 
 
 /rhis was therefore a natural commencement of the 
 period of feafons in Egypt ; and the interval between the 
 fucceffive apparitions of Thoth, has been called the na- 
 tural year of that country, to diitinguitli it from the 
 civil or artificial year, by -vi hich all records overe kept, 
 but which had little or no alliance with the feafons. It 
 has alfo been called the Ca?iicular year. It evidently 
 depends on the Sun's (ituation and diftance from the 
 Dog-ftar, and muft therefore have the fame period with 
 the Sun's revolution from a ftar to the fame ftar again. 
 This requires 365(16^9' li", and differs from our pe- 
 riod of feafons. Hence we muft conclude that the 
 rifing of the Dog-ftar is not an infallible prefage of the 
 inundation, but will be found faulty after a long courfe 
 of ages. At prefent it happens about the 12th or nth 
 of July. 
 
 This obfervation of a ftar's firft appearance in the 
 year, by getting out of the dazzling blaze of the Sun, is 
 called the heliacal r'lfing of the ftar. The ancient alma- 
 nacks for dire£ling the rural labours were obliged to give 
 the detail of thefe in fucceffion, and. of the correfponding 
 labours. Hefiod, the oldeft poet of the Greeks, has 
 given a very minute detail of thofe heliacal rifings, or- 
 namented by a pleafing defcription of the fucceflive oc- 
 cupations of rural life. This evidently required a very 
 confiderable knowledge of the ftarry heavens, and of the 
 chief circumftances of diurnal motion, and particularly 
 the number of days intervening between the firft appear- 
 ance of the different conftellations. 
 
 C c 2 Such 
 
204 ASTRONOMICAL PHENOMENA. 
 
 Such an almanack, however, cannot be expecicd, ex-^ 
 cept among a fomewhat cuhivated people, as it requires 
 a long continued obfervation of the revolution of the 
 heavens in order to form it j and it muft, even among 
 fuch people, be uncertain. Cloudy, or even hazy wea- 
 ther, may prevent us for a fortnight from feeing the ita^s 
 we want. 
 
 329. The Moon comes mofl opportunely to the aid 
 of fimple nations, for giving the inhabitants an eafy divi- 
 fion and meafure (jf time. The changes in her appear- 
 ance are fo remarkable, and fo diftinft, that they cannot 
 be confounded. AccorcUngly, we find that all nations 
 have made ufe of the lunar phafes to reckon by, and for 
 appointing all public m.eetings. The feftivals and facred 
 ceremonies of fimple nations were not all di61:atcd by fu- 
 perftition •, but they ferved to fix thofe divifions of time 
 in the memory, and thus gave a comprehenfive notion of 
 the year. All thefe feftivals were celebrated at particu- 
 lar phafes of the INIoon — generally at new and full Moon. 
 Men were appointed to watch her firft appearance in the 
 evening, after paving been feen in the morning, rifmg 
 a few minutes before the Sun. This was done in confe- 
 crated groves, and in high places j and her appearance 
 was proclaimed. Fourteen days after, the feftival was ge- 
 nerally held during full Moon. Hence it is that the firfl 
 day of a Roman month was named kalendje, the day 
 to be proclaimed. They faid pridicy tertioy quarto, &c. 
 ^nte calfndas yiecmenias Marfias ; the third, fourth, Sic. 
 
 before 
 
THE ANCIENT KALENDAHS. 205 
 
 before proclaiming the new Moon of March. And the 
 affemblage of months, with the arrangement of all the 
 feftlvals and facrifices, was called a kalendarium. 
 
 As fuperftitlon overran all rude nations, no meeting 
 was held without facrifices and other religious ceremo- 
 nies — the watching and proclaiming was naturally com- 
 mitted to the prieds — the kalendar became a facred thing, 
 connejSled with the worfhip of the gods — and, long be- 
 fore any moderate knowledge of the celeftial motions had 
 been acquired, every day of every Moon had its particu- 
 lar fantlity, and its appropriated ceremonies, which could 
 not be transferred to any other, 
 
 330. But as yet there feemed no prccife dlflindion 
 of months, nor of what number of months fliould be af^ 
 fembled into one group. Moil nations fccm to have ob- 
 ferved that, after 1 2 Moons were completed, the feafon 
 was pretty much the fame as at the beginning. This 
 was probably thought exa6l enough. Accordingly, in 
 mod ancient nations, we find a year of 354 days. But a 
 few returns of the winter's cold, when tliey expe£led 
 heat, would fiiew that this conje^lure was far from being 
 corre^ ; and now began the embarraflment. There was 
 no difficulty in determining the period of the fcafons ex- 
 aclly enough, by means of very obvious obfervations. 
 Almofl any cottager has obferved that, on the approacli 
 of winter, the Sun rifes more to the right hand, and fets 
 more to the left every day, the places of his rifing and 
 fetting coming continually nearer to each other 5 and 
 
 that. 
 
2C6 ASTRONOMICAL PHENOMENA. 
 
 that, after rifing for two or three days from behind the 
 fame object, the places of rifing and fetting again gra- 
 dually feparate from each other. By fuch familiar obfer- 
 vations, the experience of ah ordinary life is fufficient 
 for determining the period of the feafons with abundant 
 accuracy. The difficulty was to accomplifh the recon- 
 ciliation of this period with the facred cycle of months, 
 each day of which was confecrated to a particular deity, 
 jealous of his honours. Thus the Hierophantic fcience, 
 and the whole art of kalendar-making, were neceflarily 
 entrufted to the priefts. We fee this in the hiftory of 
 all nations, Jews, Pagans, and Chriftians. 
 
 331. Various have been the contrivances of difFer- 
 GT.t nations. The Egyptians, and fome of the neighbour- 
 ing Orientals, feem early to have known that the period 
 of feafons confiderably exceeded 12 months, and con- 
 tained 365 days. They made the civil year confill of 
 12 months of 30 days, and added 5 complementary days 
 without ceremonies ; and when more experience con- 
 vinced them that the year contained a fraction of a day 
 more, they made no change, but made the people believe 
 that it was an improvement on their kalendar that their 
 great day, the firft of Thoth, by falling back one day in 
 four periods of feafons, would thus occupy in fucceffion 
 every day of the year, and thus fan<Slify the whole in 
 1461 years, as they imagined, but really in 1425 of their 
 civil years. We have but a very imperfe£l knowledge of 
 the arrangement of their feftivals. Indeed they were to- 
 tally diuerent in almoft every city. 
 
 It 
 
KALENDAR OF THE GREEKS. 207 
 
 It is important to the aftronomer to know tliis me- 
 thod of reckoning ; becaufe all the obfervatlons of Hip- 
 parchus and Ptolemy, and all thofe which they have 
 quoted from tl:e Chaldeans, Perfians, &c. are recorded 
 by it. In An. Dom. 940, the firft day of Thoth fell on 
 the firft of January, and another Egyptian year com- 
 menced on the 31(1 o^ December of that year. From 
 this datum it is eafy to reckon back by years of 365 days, 
 and to fay on what day of what month of any of our 
 years the ift day of Thoth falls, and this wandering 
 year commences. 
 
 332". The Greeks have been much more puzzled 
 v/ith the formation of a lunifolar year than the Egyp- 
 tians. Solon got an oracle to dire(St his Athenians (594 
 years before our fera), Qvciv y.xmc r^ix, kutx 'ha<6v, kxtsc Zg- 
 A^v»3v, Kxt Kxrx vifAi^xq. The meaning of which feems to 
 be, to regulate their year by the Sun, or feafons, their 
 months by the Moon, and their feftivals by the days. 
 Obferving that 59 days made two months, he made thefe 
 alternately of 30 and of 29 days, 7rA«d«<, and xo<a«<, full, 
 and deficient j and the 30th day of a month, the r^mKi?^ 
 was called m KXi nx, no^nytx, as it belonged to both 
 months. 
 
 But this was not fufliciently accurate j and the Olym- 
 pic games, celebrated on every fourth year, during the 
 full Moon neareft to midfummer day, had gone into 
 great confufion. The Hierophants, wliofe proclamation 
 to all the ftates aflembled the chiefs together, had not 
 
 IkiU 
 
208 ASTRONOMICAL PHENOMEXl. 
 
 ikill enough to keep tliem from gradually falling into the 
 autumn months. Injudicious corrections were made 
 from time to time, by rules for inferting months to hring 
 things to rights again. It deferves to be remarked here, 
 that this is the way in which the ancient aftronomy im- 
 proved, before the eftablifliment of the Alexandrian 
 fchool. It was not by a more accurate obfervation of the 
 motions, as in modern times, but by difcovering the 
 errors, when they amounted to an unit of the fcale on 
 which they were meafured. The aftronomers then im- 
 proved their future computations by repeatedly cutting 
 off this unit of accumulated error. 
 
 333. All thefe contrivances were publicly propofed 
 at the meeting of the States for the Olympic Games. 
 This was an occafion peculiarly proper, and here the 
 fcheme of Meton was received with jult applaufe. For 
 Meton not only gave his countrymen a very exa6t deter- 
 mination of the lunar month, but accompanied it with a 
 fcheme of intercalation, by which all their feftivals, re- 
 ligious and civil, were arranged fo as to have very fmall 
 dlflocations from the days of new and full Moon. As 
 this had hitherto been a matter of infuperable difficulty, 
 Meton was declared viftor in the firft department, a fta- 
 tue was decreed him, and his arrangement of the fefti- 
 vals was infcribed on a pillar of marble, in letters of gold. 
 This has occafioned the number expreffing the current 
 year of the cycle of 19 years (called the Metonic cycle) 
 ro be called the Golden Number. This fcheme of Me- 
 
 ton's 
 
^ICTONIC CYCLE ROMAN YEAR. 209 
 
 ton's was indeed very judicious, though intricate, becaufe 
 he arranged the interpolation of a month fo as never to 
 remove the firft day of the month tvi^o days from the 
 time of new Moon, whereas it had often been a week. 
 
 The Metonic cycle commenced on i6. July, 493 years 
 before the beginning of the Chriftian lera, at 43 m.niutes 
 pafl 7 in the morning, that being the time of new Moon. 
 The firft year of each cycle is that in which the full 
 Moon of its firft month is the neareft to the fummer 
 folftice. 
 
 334. The Roman kalendar was in a much worfe 
 condition than the rudeft of the Greeks. The fuperfti- 
 tious veneration for their ceremonies, or their pafTion for 
 public fports, had diverted the attention of the Romans 
 (who never were cultivators or graziers) from the feafons 
 altogether. They were contented with a year of ten 
 months for feveral centuries, and liad the moft abfurd 
 contrivances for producing fome conformity with the fea- 
 fons. At laft, that accomplifhed general, Julius Crefar, 
 having attained the height of his vaft ambition, refolved 
 to reform the Roman kalendar. He was profoundly 
 Jkilled in aftronomy, and had written fome diftertations 
 on different branches of the fcience, which had great re- 
 putation, but are now loft. He had no fuperftitious or 
 religious qualms to difturb him, and was detennined co 
 make every thing yield to the great purpofe of a kalen- 
 dar, its ufe in direcSling the occupations of the people, 
 and for recording the events of hiftory. He took the 
 
 Dd help 
 
210 ASTRONOMICAL PHENOMENA. 
 
 help of Sofigenes, an aftronomer of the Alexandrian 
 fchoolj a man perfectly acquainted with all the difcover- 
 ies of Hipparchus and others of tliat celebrated academy. 
 
 Thefe eminent fcholars, knowing that the period of 
 feafons occupied 365 days and a quarter very nearly, 
 made a fliort cycle of 4 years, containing three years of 
 365, and one of 366 days ; thus cutting off, in the Gfccian 
 manner, the error, when it amounted to a whole day, 
 Cxfar refolved alfo to change the beginning of the year 
 from March, where Romulus had placed it in honour 
 of his patron Mars, to the winter folftice. This is cer- 
 tainly the moft natural way of eftimating the commence- 
 ment of the year of feafons. What we are moft anxious 
 to afcertain is the precife day when the Sun, after hav- 
 ing withdrawn his qheering beams, and expofed us to 
 the uncomfortable cold and ftorms of winter, begins to 
 turn toward us, and to bring back the pleafures of fpring, 
 and by his genial warmth to give us the hopes of ano- 
 ther feafon of ' productive fertility. * Ca^far therefore 
 
 chofe 
 
 * In almoft all nations this feafon is diftingiiilhed by fefti- 
 vities of various kinds. Many of thcfe were incorporated with 
 the religious ceremonies of the Chriftian Church by our ecclc- 
 fiaflics, becaiife they faw that the people were too much wed- 
 ded to them, to rehnquifh them with good hu.mour. Among 
 ourfelves, there are pretty evident traces of druidical fuperfti- 
 tion. We know that, in ancient times, the chief dniid, at- 
 tended by crowds of the people, went into the wcods in the 
 night of the winter fclftice, and with a golden fickle cut a 
 
 jjranch 
 
JULIAN KALENDAK. ill 
 
 ehofe for the beginning of liis kalcndar, a year in which 
 there was a new Moon following clofe upon the v/inter 
 folftice. This opportunity was afforded him in the fe- 
 cond year of his di£i:atorfhip, and the 707th year from 
 the foundation of Rome. He found that tliere would be 
 a new Moon 6 days after the winter folfliice. He made 
 this new Moon the i ft of January of his firft year. But, 
 to do this, he was obliged to keep the preceding year 
 dragging on 90 days longer than ufual, containing 444 
 
 D d 2 days, 
 
 branch of the mifelto of the oak, called Ghiah in Celtic, and 
 carried it in triumph to the frxred grove. The people cut for 
 themfelves, and earned home their prize, confecrated by the 
 druid. At prefent, the pews of our churches, and even the 
 chambers of our cottages, are ornamented with this plant at 
 Chriilmas. In France, till wichin thefe 150 years, there were 
 ftili more perceptible traces. A man perfonating a prince 
 {^Rgi follet) fet out from the village into the woods, bawling 
 out, Au Gin menc% — le Roi le ve>it. The monks followed in 
 the rear with their begging-boxes called f'lre-Iir'u They rattled 
 them, crying tire-Piri ; and the people put money into them., 
 under the fiction that it was for a lady in laboun People in 
 difguife (Guifards) forced into the houfes, playing antic 
 tricks, and bullied the indwellers for money, and for choice 
 victuals, crying tire-Uii — tire-Ilrl — maim du hlanc, et point Ju 
 Lis, They made fuch riots, that the Bi(hop of SoiiTons re- 
 prefented the enormities to Louis XIV., and the practice was 
 forbidden. May not the guifcarts of Edinburgh, with their cry 
 of " Hog menay, troll loUay ; gie's your white bread, none 
 of your gray, '* be derived from this ? 
 
2X1 ASTRONOMICAL PHENOMENA. 
 
 days, indead of the old number 354. As all thefe day* 
 were unprovided with folemnities, the year preceding 
 Ciefiir's kalendar was called the year of confufion, Cicfar 
 alio, for a particular reafon, chofe to make his firft year 
 Gonfift of 366 days, and he inferted the intercalary day 
 between the 23d and 24th of February, choofing that 
 particular day, as a feparation of the luftrations and other 
 piaculurns to the infernal deities, which ended with the 
 23d, from the worfhip of tlie celeftial deities,, which took 
 place on the 24th of February. The 24th was \kiQfcxtus 
 ante kalendas neomenias Martias. His inferted day, an- 
 fwering civil purpofes alone, had no ceremonies, nor 
 any name appropriated to it, and was to be confidered 
 merely as a fupernumerary fexttis ante kalendas. Hence 
 the year which had this intercalation was ftyled an an- 
 nus bijfext'disf a bifiextile year. With refpecl to the 
 reft of the year, C^efar being alfo Pontifex Maximus 
 (an office of vaft political importance), or rather, hav- 
 ing all the pov/er of the ftate in his own perfon, or- 
 dered that attention ftiould be given to the days of 
 the month only, and that the religious feftivals alone 
 fliould be regulated by the facred college. He affigned 
 to each month the number of days which has been con- 
 tinued in them ever fnice. 
 
 335. Such is the fimple kalendar of Julius Csefar. 
 Simple however as it was, his inftru£l:ions were mifun- 
 derftood, or not attended to, during the horrors of the 
 civil wars. Inftead of intercalating every fourth year, 
 the intercalation was thrice made on every fucceeding 
 
 third 
 
JULIAN KALEXDAR — CHRISTIAN ^RA. 213 
 
 third year. The mlftake was dlfcovcred by Auguflus, 
 aiul corre6led in the bell manner poflible, by omitting 
 three intercalations during the next twelve years. Since 
 that time, the kalendar has been continued without in- 
 terruption over all Europe till 15S2. The years, confift- 
 ing of 365-4: days, were called Julia?i years ; and it was 
 ordered, by an edicl of Auguftus, that this kalendar 
 fliall be ufcd through the whole empire, and that the 
 years Ihall be reckoned by the reigns of the different em- 
 perors. This edi6l was but imperfedly executed in the 
 diftant provinces, Vv-here the native princes were allowed 
 to hold a vaffal fovereignty. In Egypt particularly, al- 
 though the court obeyed the edicl, the people followed 
 their former kalendars and epochs. Ptolemy the aftro- 
 nomer retains the reckoning of Hipparchus, by Egyptian 
 years, reckoned from the death of Aljxander t\\Q Great. 
 We muft underfland al^ thefe modes of computation, in 
 order to make ufe of the ancient afS:ronomical obferva- 
 tions. A comparifon of the different epochs will be 
 given as we finifli the fubje6l:. 
 
 336. The lera adopted by the Roman Empire when 
 Chriftianity became the religion of the ftate, was not 
 finally fettled till a good while after Conllantine. Dio- 
 jiyfius Exiguus, a French monk, after confulting all pro- 
 per documents, confiders the 25th of December of the 
 forty-fifth year of JuUus Casfar as the day of our Sa- 
 viour's nativity. The ift of January of the forty-fixth 
 year of Ciefar is therefore the beginning of the jcra now 
 
 ufed 
 
214 ASTRONOMICAL PHENOM£nA* 
 
 ufcd by the Chrlftian world. Any event happening in 
 this year is dated a?mo Domini primo. As Cxfar had 
 made his firfl year a bifTextile, the year of the nativity 
 was alfo biiTextiie ; and the firft year of our xra begins 
 tlie fliort cycle of four years, fo that the fourth year of 
 our cera is bilTextile. 
 
 That we may connect this sera with all the others 
 employed by aftronomers or hiftorians, it will be enough 
 to know that this firll year of the Chriftian sera is the 
 4714th of the Julian period. 
 
 It coincides with the fourth year of the 194th Olym- 
 piad till midfummer. 
 
 It coincides with the 753d ab iirhe cojidita^ till 
 April 2 1 St. 
 
 It coincides with the 748th of Nabonaflar till Au- 
 gufh 23d. 
 
 It coincides with the 324th civil year of Egypt, reck- 
 oned from the death of Alexander the Great. 
 
 In the arrangement of epochs in the aftronomical 
 tables, the years before the Chriflian sera are counted 
 backwards, calling the year of the nativity o, the pre- 
 ceding year i, &c. But chronologifts more frequently 
 reckon the year of the nativity the fird before Chrift. 
 Thus, 
 Yearsof C2erar...4i, 42, 43, 44, 45, 46, 47, 48, 49 
 
 Aftronomers 4, 3, 2, i, o, i, 2, 3, 4 
 
 Chronologifts 5, 4, 3, 2, i, i, 2, 3, 4 
 
 This kalendar of Julius Caefar has manifeft advan^ 
 tages in refpe6l of fimplicity, and in a, fbort time fup- 
 
 pjanted 
 
ERRORS OF THE JULIAN KALENDAR. 21^ 
 
 planted all others among the wcftern nations. Many 
 other nations had perceived that the year of feafons 
 contained more than 365 days, but had not fallen on 
 eafy methods of making the correclion. It is a very re- 
 markable fact, that the Mexicans, when dlfcovered by the 
 Spaniards, employed a cycle which fuppofed that the year 
 contained 365^ days. For, at the end of fifty-two years, 
 they add thirteen days, which is equivalent to adding one 
 every fourth year. In their hieroglyphical annals, their 
 years are grouped into parcels of four, each of which 
 has a particular mark, 
 
 337. But although the Julian conftru£lion of the 
 civil year greatly excelled all that had gone before, it was 
 not perfect, becaufe it contained 11' 14^" more than the 
 period of feafons. This, in 128 years, amounts exa6lly 
 to a day. In 1582, it amounted to 12^* 7*". The equi- 
 noxes and folftices no longer happened on thoie days of 
 the month that were intended for them. The celebra- 
 tion of the church fcftivals was altogether deranged. 
 For it mud now be remarked, that there occurred the 
 fame embarraiTment on account of the lunar months, as 
 formerly in the Pagan world. 
 
 The Council of Nice had decreed that the great fef- 
 tival, Eafter, fliould be celebrated in conformity with the 
 Jewifh paflbver, which was regulated by the new moon 
 following the vernal equinox. All the principal fcftivals 
 are regulated by Eafter Sunday. But by the devialiou 
 gi the Julian kalendar from the feafons, and the words 
 
 of 
 
2l6 ASTRONOMICAL PHENOMENA. 
 
 of the decree of the Nicene Council, the celebration of 
 Eafter loft all connexion with the Paflbver. For the de- 
 cree did not fay, * The firft Sunday after the full moon 
 following the vernal equinox, but the firft Sunday after 
 the full moon following the 2 1 ft of March. * It fre- 
 quently happened that Eafter and the Paftbver were fix 
 weeks apart. This M^as corrected by Pope Gregory the 
 XIII. in 1582, by bringing the 21ft of March to the 
 equinox again. He firft cut off the ten days which had 
 accumulated fince the Council of Nice ; and, to prevent 
 this accumulation, he directed the intercalation of a bif- 
 fextile to be omitted on every centurial year. But the 
 error of a Julian century containing 36525 days. Is not 
 a whole day, but 1 8" 40'. Therefore the correction in- 
 troduces an error of 5*" 20'. To prevent this from ac- 
 cumulating, the omiflion of the centurial intercalation i» 
 limited to the centuries not divifible by four. Therefore 
 1600, 2000, 2400, &c. are ftill biflextile years; but 
 1700, 1800, 1900, 2100, 2200, &c. are common years. 
 There ftill remains an error, amounting to a day in 144 
 centuries. 
 
 The kalendar is now fufficiently accurate for all pur- 
 pofes of hiftory and record — and even for aftronomy, 
 becaufe the tropical year of feafons is fubje<Sl to a perio- 
 dical inequality. 
 
 338. A correftion, much more accurate than the 
 Gregorian, occurred to Omar, a Perfian aftronomer at 
 the court of Prince Gelala Eddin Melek Schah. Omar 
 
 propofed 
 
PERSIAN KALENDAR. 217 
 
 ^ropofed always to delay to the thirty-tliird year the in- 
 tercalation which ihould have been made in the thirty- 
 fecond. This is equivalent to omitting the Julian inter- 
 calation altogether on the 128th year. This method is 
 extremely fimple, and fcrupuloufly accurate. For the 
 error of ii' 15" of the Julian year amounts precifely 
 to a day in 128 years. It differs from the truth only 
 one minute in 120 years. This correftion took place in 
 A° D' 1079, at the fame time that the Arab Alhazen 
 was reforming the fcience of aftronomy in Spain. 
 
 The Gregorian kalendar, however, has lefs chance of 
 being forgotten or miftakeni Centurial years are re- 
 markable, and call the attention, even by the unufual 
 found of the vt^ords. The thirty-fecond year has nothing 
 remarkable, and may be overlooked. 
 
 339. It now appears that certain attentions are ne- 
 ceffary for avoiding miflakes, when we would appeal ta 
 very diftant obfervations. We muil know the accurate 
 interval, however large. Although one hundred Julian 
 years contain 36525 days, we m.uft keep in mind that 
 between 1500 and 1600 ten days are wanting j and that 
 each of the centuries 170Q and 1800 al fo want a day. 
 The interval from the beginning of our sera and A. D. 
 1582 needs no attention; but that between 1-505 and 
 1805 wants twelve days of three Julian centuries. 
 
 340. "We mild alfo be careful, in ufing the ancient 
 obfetvations, to conneft the years of out Lord with the 
 
 E ^ > years 
 
-2lg ASTRONOMICAL PHENOMENA, 
 
 yeavs before Chrift in a proper manner. An eclipi^' 
 mentioned by an aftronomer as having happened on the 
 ift of February anno 3tio A. C. muft be confidered as 
 happening in the forty-fecond year of JuHus Ga^far. But 
 if tlie fame thing is mentioned by a hiftorian or chrono- 
 logiil, it is much more probable that it was in the forty- 
 third year of C?efar. It was chiefly to prevent all am- 
 biguities of this kind that Seaiiger contrived what he 
 called the Julia7i period. This is a number made by 
 multiplying together the numbers called the Lunar or 
 Metonic c\cley the folar cycle^ and the indi^'ion. The 
 ijnar cycle is 19, and the firfl: year of our Lord was 
 the fecond of this cycle. The folar cycle is 28, being 
 the number of years in which the days of the month 
 return to the fame days of the week. As the year con- 
 tains fifty-two weeks and one day, the firil day of the 
 year (or any day of any month) falls back in the week 
 one day every year, till interrupted by the intercalation 
 in a bilTextile year. This makes it fall back two days in 
 that year j and therefore it will not return to the fame 
 day till after four times feven, or twenty-eight years. 
 The firft year of our Lord was the tenth of this cycle. 
 The iNDiCTiON is a cycle of fifteen years, at the begin- 
 ning of which a tax was levied over the Roman Empire. 
 It took place A. D. 312; and if reckoned backward, it 
 would have begun three years before the Chriflian aera. 
 The year of this cycle for any year of the Chriflian sera, 
 will tlaerefore be had by adding three to the year, and 
 dividing by fifteen. The produdl of thefe three num- 
 bers 
 
JULIAN PERIOD — .T.RAS. 2 Tf> 
 
 bers is 7980 ; and it is plain that this number of years 
 muft elapfe before a year can have the fame place in all 
 the three cycles. If therefore we know the place of 
 thefe cycles belonging to any year, we can tell what year 
 it is of the Julian period. 
 
 The firft year of our lera was the fecond of the lunar 
 cycle, the tenth of the folar, and the fourth of induc- 
 tion, and the 4714th of the Julian period. By this we 
 may arrange all the remarkable seras as follows. 
 
 j-p- 
 
 -^ra of the Olympiads . . . 3938 
 Foundation of Rome . . . 3961 
 
 NabonalTar 39^7 
 
 Death of Alexander , . . 4390 
 Firft of Julius Caefar . . . 4669 
 A. Dom. 1 4714 
 
 341. Did theMetonic cycle of the Moon correfpond 
 exa6i:ly with our year, it w^ould mark for any year the 
 number of years which have elapfed fince it was new 
 moon on the ift of January. But its want of perfe£b 
 accuracy, the vicinity of an intercalation, and the lunar 
 equations, fometimes caufe an error of tw^o days. It is 
 much ufed, however, for ordinary calculations for the 
 Church holidays. To find the golden number, add one 
 to the year of our Lord, divide the fum by 19, the re- 
 mainder is the golden number. If there be no remain- 
 der, the golden number is 19. 
 
 E e 2 343. 
 
 
 
 (L 
 
 I. 
 
 A. C. 
 
 18 
 
 5 
 
 8 
 
 IIS^II^ 
 
 13 
 
 9 
 
 I 
 
 1S^^7S3 
 
 19 
 
 15 
 
 7 
 
 74^>747 
 323^324 
 
 21 
 
 H 
 
 4 
 
 44> 45 
 
 10 
 
 2 
 
 4 
 
 
220 ASTRONOMICAL PHENOMENA. 
 
 342. Another number, called EpaO:, is alfo ufed 
 for facilitating the calculation of new and full moon in 
 a grofs way. The epa6l is nearly the moon's age on the 
 id of January. To find it, multiply the golden number 
 by II, add 19 to the product, and divide by 30. The 
 remainder is the epa£fe. 
 
 Knowing, by the epaL>, the Moon's age on the ift 
 of January, and the day of the year correfponding to any 
 day of a month, it is eafy to find the Moon's age on that 
 day, by dividing the double of the fum of this number 
 and the epa£l by 59. The half remainder is nearly the 
 Moon's age. 
 
 Although thefe rude computations dp not correfpond 
 with the motions of the two luminaries, they defer ve 
 notice, being the methods employed by the rules of the 
 Church for fettling the moveable Church feftivals. 
 
 Of the proper Motions of the Planets, 
 
 343. The planets are obferved to change their fitua^ 
 tions in the ftarry heavens, and move among the figns 
 pf the zodiac, never receding far from the ecliptic. 
 
 Their motions are exceedingly irregular, as may be 
 feen by fig. 65. A, which reprefents the motion of the 
 planet Jupiter, from the beginning of 1708 to the be- 
 ginning of 1 7 r 6. E K reprefents the ecliptic, and the 
 initial letters of the months are put to thofe points of 
 the apparent path where the planet was fcen on the firfb 
 .day of each month. 
 
 It 
 
INEQUALITIES OF THE PLANETS. 221 
 
 it appears that, on the ift of January 1708, the pla- 
 net was moving llowly eaftward, and became ftatlonary 
 about the middle of the month, in the fecond degree of 
 Libra. It then turned weftwaid, gradually increafmg its 
 wefterly motion, till about the middle of March, when 
 it was in oppofition to the Sun, at R, all the while de- 
 viating farther from the ecliptic toward the north. It 
 now flackened its wefterly motion every day, and was a- 
 gain ftationary about the 20th of May, in the twenty-fecond 
 degree of Virgo, and had come nearer to the ecliptic. 
 Jupiter now moved eaftward, nearly parallel to the eclip- 
 -tic, gradually accelerating in his m^otion, till the begin- 
 ning of 06tober, when he was in conjundlion with the 
 Sun at D, about the eleventh degree of Libra. He now 
 flackened his progreflivc motion every day, till he was a- 
 gain ftationary, in the fecond degree of Scorpio, on the 
 1 2th or 1 3 th of February 1 709. He then moved weft ward, 
 was again in oppofition, in the twenty-feventh degree of 
 Libra, about the middle of April. He became ftation^ 
 ary, about the end of June, in the twenty-firft degree 
 of Libra ; and from this place he again proceeded eaftr 
 ward -, was in conjunction about the beginning of Novem- 
 ber, very near the ftar in the fouthern fcale of Libra ; 
 and, on the ift of January 17 10, he was in the twenty- 
 fourth degree of Scorpio. 
 
 This figure will very nearly correfpond with the ap- 
 parent motions of the planet in the fame months of 1803 
 and 1804. Jupiter will go on in this manner, forming 
 g loop in his path in every thirteenth month ; and he in 
 
 in 
 
222 ASTRONOMICAL PlIEKOMENA. 
 
 in oppofitioii to the Sun, wlicn in the mkldle of each 
 ioop. His regrefs in each loop is about lo degrees, and 
 his progreflive motion is continued about 40°. He gra- 
 dually approaches the ecliptic, crofTes it, deviates to the 
 fouthward, then returns tou ards it ; crofTes it, about fix 
 years after his former crolhng, and in about twelve years 
 comes to vi'here he was at the beginning of thefe ob- 
 fervations. 
 
 344. The other planets, and particularly Venus and 
 and Mercury, are ftill more irregular in their apparent 
 motions, and have but few circumftances of general re- 
 femblance. 
 
 The firft general remark which can be made on thefe 
 intricate motions is, that a planet always appears largeft 
 when in the points R, R, R, v/hich are in the middle 
 of its retrograde motions. Its diameter gradually di- 
 minifhes, and becomes th^ leaft of all wlien in the points 
 D', D', D', which are in the middle of its diredl mo- 
 tions. Hence we infer that the planet is neareft to the 
 Earth when in the middle of its retrograde motion, and 
 fartheft from it when in the middle of its dire£l motion. 
 
 It may alfo be remarked, that a planet is always in 
 conjunaion M'ith the Sun, or comes to our meridian at 
 noon, when in the middle of its direct motions. The 
 planets Venus and Mercury are alfo in conjunction with 
 the Sun Mhen in the middle of their retrograde motions. 
 But the planets Mars, Jupiter, and Saturn, are always 
 in oppofition to the Sun, or come to our meridian i^t 
 
 midnight. 
 
AKCrEXT PLANETARY THEORY. 223 
 
 niidnight, when in the miJcUc of tlielr retrograde mo- 
 tions. Their fituations aUb, when (tationary, are always 
 fimibr, relative to the Sun. Thefe appearances in all 
 the planetary motions have therefore an evident rela- 
 tion to the Sun's place. 
 
 345. The ancient aftronomers were of opinion tliat 
 the perfection of nature required ail motions to be uni- 
 form, as far as tlie purpofe in view would permit. The 
 planetary motions muit therefore be uniform, in a figure 
 that is uniform ; and the allronomers maintained that 
 tlie obferved irregularities were only apparent. Their 
 method for reconciling thefe with their principle of per- 
 fection is very obvioully fuggefted by the reprefentatioii 
 here given of tlie motion of Jupiter. They taught that the 
 planet moves uniformly in the circumference of a circle 
 q r s (fig. 40.) in a year, while tlie centre Q of this circle 
 is carried uniformly round the Earth T, in the circum- 
 ference of another circle Q A L. The circle O A L is 
 called the deferent circle, and qrs is called the 
 EPICYCLE. They explained the deviation from the eclip- 
 tic, by faying that the deferent and the epicycle were in 
 planes different from that of the elliptic. By various 
 trials of different proportions of the deferent and the 
 epicycle, they hit on fuch dimenfions as produced the 
 quantity of retrograde motion tliat was obferved to be 
 combined with the general progrefs in the order of the 
 figns of the zodiac. — But another inequality was obferv- 
 ed. The arch of the heavens intercepted between two 
 
 fucceffiv'^ 
 
224 ASTRONOMICAL PHENOMfiNit." 
 
 fucceflive oppofitions of Jupiter, (for example), was ob- 
 lerved to be variable, being always lefs in a certain part 
 of the zodiac, and gradually increafing to a maximum 
 ftate in the oppofite part of the zodiac. 
 
 In order to correfpond with this second inequality, 
 as it was called, and yet not to imply any inequality of 
 the motion of the epicycle in the circumference of the 
 deferent circle, the aflronomers placed the Earth not in, 
 but at a certain diftance from, the centre of the defer- 
 ent ; fo that an equal arch between two fucceeding op- 
 pofitions fliould fubtend a fmaller angle, when it is on 
 the other fide of that centre. Thus, the unequal mo- 
 tion of the epicycle was explained in the fame way as 
 the Sun's unequal motion in his annual orbit. The 
 line drawn through the Earth and the centre of the de- 
 ferent is called the line of the planet's apsides, and its 
 extremities are called the apogee and perigee of the de- 
 ferent as in the cafe of the Sun's orbit (292.) In this 
 manner, they at laft compofed a fet of motions which 
 agreed tolerably well with obfervation. 
 
 The celebrated geometer Apollonius gave very judi- 
 cious direclions how to proportion the epicycle to the 
 deferent circle. But they feem not to have been attend- 
 ed to, even by Ptolemy ; and the aflronomers remained 
 very ignorant of any method of conflru£tion which a- 
 greed fufEciently with the phenomena, till about the 
 thirteenth century, when the doctrine of epicycles was 
 cultivated with more care and fkill. 
 
 A very full and diftinft account is given of all the 
 ingenious contrivances of the ancient aflronomers for 
 
 explaining 
 
-/■// 
 
 aT 
 
 <?c; 
 
 or 
 
 Ji 
 
 
 ^ 
 
 ■^ J 
 
 CI 
 
MOTIONS AND PHASES OF VfeNtJS. 225 
 
 explaining the irregularities of the celeftial motions, in 
 the firft part of Dr Small's Hiftory of the Difcoveries of 
 Kepler, piibliflied in 1803. 
 
 Of the Motions of Venus and Mercury, 
 
 346. Venus has been fometimes feen moving acrofs 
 the Sun*s difk from eail to weft, in the form of a round 
 black fpot, with an apparent diameter of about 59'^ A 
 few days after this has been obferved, Venus is feen in 
 the morning, rlfmg a little before the Sun, In the form 
 of a fine crefcent, with the convexity turned toward the 
 Sun. She moves gradually weftward, feparating from 
 the Sun, with a retarded motion, and the crefcent be- 
 comes more full. In about ten weeks, flie has moved 
 46° weft of the Sun, and is nov/ a femicircle, and her 
 diameter is 26". She now feparates no farther from the 
 Sun, but moves eaftward, with a motion gradually ac- 
 celerated, and fhe gradually diminifties in apparent dia- 
 meter. She overtakes the Sun, about pi months after 
 }iaving been i^Q^^s. on his difk. Some time after, VenuS 
 is feen in the evening, eaft of the Sun, round, but very 
 imall. She moves eaftwardj and increafes in apparent 
 diameter, but lofes of her roundnefs, till (he gets about 
 46° eaft of the Sun, when (he is again a femicircle, hav- 
 ing the convexity toward the Sun. She now moves weft- 
 ward, increafing in diameter, but becoming a crefcent, 
 like the waneing Moon ^ and, at laft, after a period of 
 nearly 584 days, comes again into conjunction v/itli the 
 Sun, with an apparent diameter of 59", 
 
 F i 347. 
 
220 ASTRONOMICAL PHENOMENil, 
 
 347. From tliefe phenomena Me cojiclude that thef 
 Sun is included within the orbit of Venus, and is not 
 far from its centre, while the Earth is without tliis orbit. 
 Therefore, while the Sun revolves round the Earth, 
 Venus revolves round the Sun. 
 
 The time of the revolution of Venus round the Sun 
 may be deduced from the interval which elapfes between 
 two or more conjunctions, by help of the following 
 theorem : 
 
 348. Let two bodies A and B revolve uniformly in 
 the fame direction, and let a and b be their refpeclive 
 periods, of which b is the leaft, and / the interval be- 
 tween two fucceflive conjunctions or oppofitions. 
 
 Then h = — - — , and a = 
 
 For the angular motions are inverfely proportional to 
 the periodic times. Therefore the angular motions of 
 
 A and B are as — and -r . And, fmce they move in the 
 a 
 
 fame direction, the fynodical or relative motion is the 
 diiference of their angular motio^ns. Therefore the fun- 
 damental equation is -r =:: - . Flence 7 = - + ~ > 
 
 ^ bat b t ' a' 
 
 = — — , and h — —— . Alio - = -r — 7^ = —nry 
 at a -^ t a b t to 
 
 and a-=z . 
 
 t—^h 
 
 We may alio calculate the fynodical period /, when 
 we know the real periods of each. For — = -r — - = 
 
 a — b , a b 
 
 — r- , and / = y 
 
 a b a — b 
 
 This 
 
TEUIOD AND ANOMALIES OF VENUS. 22^ 
 
 This gives for the periodic lime of Venus round the 
 Sun 224^ 16'' 49' 13". 
 
 349. But it is evident that if this angular motion is 
 not uniform, the interval between two fuccefi'ive conjunc- 
 tions may chance to give a falfe meafure of the period. 
 But, by obferving many conjunctions, in various parts of 
 the heavens, and by dividing the intei*val between the 
 firfl and Jail by the number of intervals between each 
 (taking care that the firft and laft fliall be nearly in the 
 fifme part of the heavens), it is evident that the inequa- 
 lities being diilributed among them ail, the quotient may 
 be taken as nearly an exzGt medium. Hence arifes the 
 great vakie of ancient obfervations. In eight years we 
 liave five conjunctions of Venus, and flie is only i° 32' 
 lliort of the place of the firft conjunction. The period de- 
 duced from the conjunCl ions in 1 761 and 1769, fcarcely 
 differs from that deduced from the conjunctions in 1639 
 and 1 76 1. But the other planets require more diftant 
 obfervations. 
 
 350. Venus does not move uniformly in her orbit. 
 For, if the place of Venus in the heavens be obferved in 
 a great number of fucceflive conjunctions with the Sun 
 tat which time her place in the ecliptic, as feen from tlie 
 Sun, is either the Sun's place, as feen from the Earth, 
 or the oppofite to it), we find that her changes of pbcc 
 are not proportional to the elapfed times. By obferva- 
 tions of tJiis kind, we learn the inequality of the angular 
 
 F f 2 motion 
 
aa3 ASTRONOMICAL PHENOMEKA- 
 
 inotion of Veniis roun4 the Sun, and hence can find the 
 equations for every point of the orbit of Venus, and can 
 thence deduce the pofition of Venus, as feen from the 
 Sun, for any given inftant. 
 
 This however requires more obfervatlons of this kind 
 than we are yet poUelTed of, becaufe her conjunflions 
 happen fo nearly in the fame points of her orbit, that 
 great part of it is left without obfervations of this kind. 
 But we have other obfervations of almoft equal value, 
 pamely, thofe of her greatefl: elongations from the Sun. 
 There is none of the planets, therefore, of which the 
 equations (which indeed are yery fmall) are more ac- 
 curajtely deterniined, 
 
 351. We can now determine the form and pofition 
 of the orbit. For we can ohferve the place of the Sun, 
 or the pofition of the line ES (fig. 41.), joining the 
 Earth and Sun. We knoyv the length of this line (291.) 
 We can ohferve the geocentric place of Venus, or the 
 pofition of the line E D joining the Earth and Venus. 
 And we can compute (350.) the heliocentric place of 
 Venus, or the pofition of the line S C joining Venus 
 and the Sun. Venus muft be in V, the interfe6tion of 
 thefe two lines 5 and therefore that point of her orbit is 
 determined. 
 
 352. By fuch obfervations Kepler difcovered that the 
 orbit of Venus is an ellipfe, having the Sun in one focus, 
 the femitranfverfe axis being 72333, and the eccentri- 
 city 
 
PHENOMENA Or VENUS AND MERCURY. 229 
 
 city 510, mcafuretl on a fcale of which the Sun's mean 
 diftance from the Earth is loccoo. 
 
 353. The upper apfis of the orbit is called the 
 APHELION, and the lower apfis is called the perihelion 
 of Venus. 
 
 354. The line of the apfides has a flow motion eafl- 
 ward, at the rate of 2° 44' 46" in a century. 
 
 355. The orbit of Venus Is inclined to the ecliptic 
 at an angle of 3° 20', and the nodes move weftward about 
 3 1" in a year. 
 
 356. Venus moves in this orbit fo as to defcribe 
 round the Sun areas proportional to the times. 
 
 357. The planet Mercury refembles Venus in all 
 the circumftances of her apparent motion ; and we make 
 fmiilar inferences v/ith refpecfl to the real motions. His 
 orbit Is difcovered to be an elllpfe, having the Sun in one 
 focus. The femitranfverfe axis is 38710, and the ec- 
 centricity 7960. The apfides move eaftward i° 57' 20" 
 in a century. The orbit is inclined to the ecliptic 7*^. 
 The nodes move weftward 45" in a year. The periodic 
 time is 87^ 23*^ 15' 37" j and areas are defcribed propor- 
 J;ional to the times. 
 
 0/ 
 
230 ASTRONOMICAL PHENOMENA. 
 
 Of the proper Motions of the Superior Planets, 
 
 358. Mars, Jupiter, and Saturn, exhibit phenomena 
 confiderably different from thofe exhibited by Mercury 
 and Venus. 
 
 1 . They come to our meridian both at noon and at 
 midnight. When they come t*o our meridian at noon, 
 and are in the ecUptic, they are never feen crofling the 
 Sun's difk. Hence we infer, that their orbits include 
 both the Sun and the Earth. 
 
 2. They are always retrograde when in oppofition, 
 and dire(Sl when in conjunftlon. 
 
 The planet Jupiter may ferve as an example of the 
 way in which their real motions may be invefligated. 
 
 359. Jupiter is an opaque body, vifible by means 
 of the reflecfted light of the Sun. For the fliadows of 
 fome of the heavenly bodies are fometimes obferved on 
 his dilk, and his (Iradow frequently falls on them. 
 
 360. His apparent diameter, v/hcn in oppoHtlon, is a- 
 bout 46", and, when in conjuniflion, it is about 3 1 ", and his 
 dilk is ahvavs round. Hence we infer, tliat he is ncarell 
 when in oppofition, and that his lead and grenteft diftance 
 are nearly as two to three. The Earth is, therefore, far 
 removed f) om the centre of his motion j anc, if we en- 
 deavour to explain his motion by means of a deferent 
 
 circle 
 
EPICYCLICAL THEORY OF JUPITER. 23! 
 
 circle and an epicycle ( ), the radius of the deferent muft 
 be about five times the radius of the epicycle. 
 
 361. Since Jupiter is always retrograde when in op- 
 pofition, and direct when in conjun£i:ion, his pofition, 
 with refpecl to the centre of his epicycle, mufl be fmii- 
 iar to the pofition of the Sun with refpe<Sl to the Earth. 
 His motion, therefore, in the epicycle, has a dependence 
 on the motion of the Sun *, and his motion, as feen from 
 the Sun, muft be fimpler than as feen from the Earth, 
 
 Hi* pofition, as feen from the Sun, may be accurately 
 cbferved in every oppofition and conjunclion. 
 
 It was very natural for the ancient aftronomers of 
 Greece to infer, from what has been faid juft now, that 
 the pofition of Jupiter, in refpect of the centre of his 
 epicycle, was the fame as that of tlie Sun in refpe<^ of 
 the Earth, not only in oppofition and conjunclion, but 
 in every other fituation. For, in twelve years, we fee it 
 to be fo in the oppofitions obferved in 1 2 parts of the 
 heavens, and in 83 years we fee it in 76 parts. It is very 
 iinprobable, therefore, that it fliould be otherwife in the 
 intervals. 
 
 The motion of a fuperior planet may be explained 
 upon thefe principles iu the following manner : 
 
 Let T (fig. 40.) be the Earth, and x/Bk^ i^ y ^x be 
 the Sun's orbit. Alfo, let A, B, C, D, E, F, G, H, I, 
 be the places of the centre of the epicycle in the circum- 
 ference of the deferent when the fun is in «, /3, *, ^, g, (p, 
 7, Xy «5 make A a parallel to T «^ and B b parallel to 
 
232 ASTRONOMICAL PHENOMENA, 
 
 T/3, and Cc parallel to T x, &c., and make thefe linej 
 of a length that is duly proportioned (by the Apollonian 
 rule) to the radius T A of the deferent circle. 
 
 When the Sun is in ««, /s, k, &c. the centre of the 
 epicycle is in A, B, C, &c. and the planet is in a, b, c, 
 &c. -, and the dotted curve abed efg h a k is its path 
 in abfolute fpace between two fucceeding oppofitions to 
 the Sun, viz. in a, and in k. 
 
 26 2' If we make the radius of Jupiter's deferent 
 circle to that of the epicycle, as 52 to 10, the eplLyclical 
 motion arifmg from this conftruftion will very nearly 
 agree with the obfervation. Only we may obferv^e tliat 
 the oppofitions which fucceed each other near the con- 
 ftellation Virgo, are lefs diftant from one another than 
 thofe obferved in the oppofite part of the heavens ; fo 
 that the centre of the epicycle feems to move flower in 
 the firft cafe than in the laft. To reconcile this with the 
 perfe£l uniformity of the motion of that centre in the 
 circumference of the deferent circle, the ancient aflrono- 
 mers faid that the earth was not exactly in the centre of 
 the deferent, but fo placed that the equable motion of 
 the centre of the epicycle appeared flower, becaufe it is 
 then more remote ; and after various trials, they fixed on 
 a degree of eccentricity for the deferent, which accorded 
 better than any other with the obfervations, and really 
 differed very little from them. Copernicus fliews that 
 their hypothefis for Jupiter never deviates more than half 
 a degree from obfervation, if it be properly employed. 
 They found that the epicycle moved round the de- 
 ferent 
 
THEORIES or THE SUPERIOR PLANETS. 23 1 
 
 fercnt in 4332^ days, with an equation gradually in- 
 creafing to near 6 degrees ; fo that if the place of the 
 epicycle be calculated for a quarter of a revolution from 
 the apogee, at the mean rate of 5' per day, it will be 
 found too far advanced by near ten weeks motion. 
 
 363. But the ancient aftronomers had no fuch data 
 for determining the abfolute magnitude of the deferent 
 circles and epicycles for the fuperior planets, as Mercury 
 and Venus afforded them. The rules given them by 
 Apollonius only taught them what proportion the epicycle 
 of each planet mud have to its deferent circle, but gave 
 no information as to the abfolute magnitude of either, 
 or the proportion between the deferent circles of any two 
 fuperior planets. Accordingly, no two ancient aftrono- 
 mers agree in their meafures, farther than in faying that 
 Saturn is farther off than Jupiter, and Jupiter than 
 Mars. This they inferred from their longer periods* 
 All that they had to take care of was to make their fizes 
 fufficiently different, fo that the epicycles of two neigh- 
 bouring planets (hould not crofs and juftle each other. 
 Yet they might eafily have come very near the truth, by 
 a fmall and v^ry allowable addition to their hypothefis of 
 epicyclical motion, namely, by fuppofmg that the epicycle 
 of each planet is equal to the Sun's orbit. This was 
 ijuite allowable. 
 
 364. If we do this, we ihall deduce confequences 
 that are very remarkable, and which would have put tlie 
 
 G g ancient 
 
\ 
 
 134 ASTRONOMICAL PHENOMENA. 
 
 ancient aftronomy on a footing very near to perfection. 
 For, if Cr (fig. 40.) be not only parallel to Tx, but 
 alfo equal to it, then CT*;^ is a parallelogram, and kc 
 is equal and parallel to T C. The bearing (to exprefs it 
 as a mariner) and diftance of Jupiter from the Sun, is at 
 all times the fame with the bearing and diftance of the 
 centre of his epicycle from the Earth \ and Jupiter is al- 
 ways found in an orbit round the Sun, equal and fimilar 
 to the deferent orbit round the Earth. Thus, u a is e- 
 qual to TA; /33 to TBj ac to T C, &c. with refpefi: 
 to all the points of the looped curve. If the Earth be in 
 the centre of the deferent, the diftance of Jupiter from 
 the Sun is always the fame, and he may be faid to de» 
 fcribe a circle round the Sun, while thfe Sun moves 
 round the Earth. Nay, it refults from the equality of 
 Ka to T 6t, of B ^ to T /S, &c., that whatever eccen- 
 tricity, or whatever form it has been thought neceffary 
 to aflign to the deferent, the diftances eta, ^b, x c, &c. 
 will ftill be refpeaively equal to TA, TB, TC, &c. 
 The circle which the aftronomers called the deferent, 
 becaufe it is fuppofed to carry Jupiter's epicycle round 
 the Earth, may be fiippofed to accompany the Sun, be- 
 ing carried round by him in a year, the line of its ap- 
 fides (362.) keeping parallel to itfelf, that is, in our 
 figure, to T A. And thus, the motion of Jupiter round 
 the Sun will be iftcomparably more fimple than the 
 looped curve round the Earth ; for it will be precifely 
 the motion which was given by the aftronomers to the 
 centre of Jupiter's epicycle. The motion of Jupiter in 
 
 abfolute 
 
CALCULATION OF JUPlTER^S PLACE. ^^35^ 
 
 abiolute fpace is indeed the fame looped curve in both 
 cafes; but the way of conceiving it is much more fimple. 
 
 365. This fuppofition of the equ-'lity of Jupiter's 
 epicycle to the Sun's orbit, and the parallelifm of C ^ 
 to T « in every pofition of Jupiter, are fully verified by 
 the modern difcoveries of his fatellites. Thefe little pla- 
 nets revolve round him with perfect regularity, and their 
 fhadows frequently fall on his dilk, and they are often 
 obfcured by his fhadow. This fhews the pofition of Ju« 
 piter's fhadow at all times, and, confequently^ Jupiter's 
 pofition in refpecl of the Sun. This we fimd at all times 
 to be parallel to the fuppofed pofition of the centre of 
 his epicycle. Thus kc is found parallel to T C. 
 
 366. We now can tell the precife point in which 
 Jupiter is found in any moment of time. Having made 
 the radius T x to the radius T A in the due proportion 
 of 10 to 52, and having placed the Earth at the proper 
 diftance from the centre of the deferent Q A L, we can 
 calculate (298.) the pofition and length of the line T« 
 joining the Earth with the Sun. We can draw the line 
 T C to the fuppofed centre of Jupiter's epicycle, having 
 learned the law or equation of the fuppofed motion of 
 that centre by our obfervation of his oppofitions in all 
 quarters of the ecliptic (362.), and we then draw «V 
 parallel to it. This muft pafs through Jupiter, 0¥ Jupi- 
 ter mud be fomewhere in this line. We obferve Jupi- 
 ier^ however, in the direclion T Z. Jupiter muft there- 
 
 G p- ^ ;fore 
 
23<^ ASTRONOMICAL PHENOMENA. 
 
 fore be In the interfeCbion c of the lines « V and T Z. 
 And then we can meafure r«, Jupiter's diftance from, 
 the Sun. 
 
 367. Kepler, by taking this method with a ferles 
 of obfervations made by Tycho Brahe, difcovered that 
 Jupiter was always found In the circumference of an el- 
 Hpfe, having the Sun in its focus. Its femitranfverfe 
 axis is 520098, the mean diftance of the Earth from the 
 Sun being fuppofed 1 00000. Its eccentricity is 25277. 
 Its inclination to the ecliptic is 1° 20', and the nodes 
 move eaftward about i' in a year. 
 
 368. The revolut'ion in this orbit is completed iii 
 4332y days, and areas are defcribed proportional to the 
 times. 
 
 369. Proceeding in the fame manner, we difcove? 
 that the planets Mars, Saturn, and the one difcovered 
 by Dr Herfchel in 1781, are always found in the circum- 
 ference of ellipfes, with the Sun in one focus, and de- 
 fcribe round him areas proportional to the times. 
 
 The chief circumftances of their motions are ftated 
 as follows : 
 
 Mean Dijiance. 
 
 Georgian planet 1908584 
 Saturn - - - - 953941 
 Mars ----- 152369 
 
 370. 
 
 EcceniricHy, 
 
 Period in Dayt, 
 
 90738 
 
 30456,07 
 
 53210 
 
 lo'JS9y^7 
 
 I4218 
 
 686,98 
 
IMPROVE/MENT OF THE ANCIENT THEORIES. 237 
 
 370. Two Other bodies have been lately dete<£led in 
 the planetary regions, revolving round the Sun in orbits 
 which do not leem very eccentric, and feem placed be- 
 tween thofe of Mars and Jupiter. The firft: was obferved 
 in 1 80 1 by Mr Piazzi of Palermo, and by him named 
 Ceres. The other was difcovered in 1802 by Mr OI- 
 berg of Bremen, who has called it Pallas. They are ex- 
 ceedingly fmall, and we have feen too little of their mo- 
 tions as yet to enable us to ftate their elements with any 
 precifion. 
 
 371. Thus it has been difcovered, that, while the 
 Sun revolves round the Earth, the fix planets now men- 
 tioned are always found in the circumferences of ellipfes, 
 having the Sun in one focus ; and that they defcribe 
 round the Sun areas proportional to the times. 
 
 372. But now, inftead of fuppofing that the centre 
 ©f a fmall epicycle is carried round the circumference of 
 a greater deferent circle, different for each planet, we 
 may rather confider the Sun's orbit round the Earth as the 
 only deferent circle, and fuppofe that the planets defcribe 
 their great elliptical epicycles round him with diiTerent 
 periods, while he moves round the Earth in a year. The 
 real motions of the planets are ftill the fame looped curves 
 in both cafes. For, in either cafe, the motion of a pla- 
 net is compounded of the fame motions. But the latter 
 fuppofition is much more probable. We can fcarcely 
 conceive the motion of Jupiter in the epicycle q r s zs 
 
 having 
 
IMPROVE/MENT OF THE ANCIENT THEORIES. 237 
 
 370. Two Other bodies have been lately dete<£^ed in 
 the planetary regions, revolving round the Sun in orbits 
 which do not feem very eccentric, and feem placed be- 
 tween thofe of Mars and Jupiter. The liril was obferved 
 in 1 801 by Mr Piazzi of Palermo, and by him named 
 Ceres. The other was difcovered in 1802 by Mr OI- 
 berg of Bremen, who has called it Pallas. They are ex- 
 ceedingly fmall, and we have feen too little of their mo- 
 tions as yet to enable us to flate their elements with any 
 precifion. 
 
 371. Thus it has been difcovered, that, while the 
 Sun revolves round the Earth, the fix planets now men- 
 tioned are always found In the circumferences of ellipfes, 
 having the Sun in one focus ; and that they defcribe 
 round the Sun areas proportional to the times. 
 
 372. But now, inflead of fuppofing that the centre 
 ©f a fmall epicycle is carried round the circumference of 
 a greater deferent circle, different for each planet, we 
 may rather confider the Sun's orbit round the Earth as the 
 only deferent circle, and fuppofe that the planets defcribe 
 their great elliptical epicycles round him with diiTerent 
 periods, while he moves round the Earth in a year. The 
 real motions of the planets are ftill the fame looped curves 
 in both cafes. For, in either cafe, the motion of a pla- 
 net is compounded of the fame motions. But the latter 
 fuppofition is much more probable. We can fcarcely 
 conceive the motion of Jupiter in the epicycle q r s zs 
 
 having 
 
23S ASTRONOMICAL PHENOMENi. 
 
 having any pKyfical reLitlon to its centre, a mere matfie- 
 matical point of fpace. We cannot confider this point aS 
 having any phyfical properties that fliall influence the 
 motions of the planet. This point alfo is fuppofed to be 
 in motion J carrying with it the influence by which the 
 •planet is retained in the circumference of the epicycle. 
 This is another inconceivable circumftance. This com- 
 bination of circles, therefore, cannot be confidered as 
 any thing but a mere mathematical hypothefis, to furnifh 
 fome means of calculation, or for the delineation of the 
 looped path of the planet. Accordingly, the firfl pro- 
 pofers of thefe epicycles, fenfible of the mere nothing- 
 nefs of their centre, and the impoffibiiity of a nothing 
 moving in the circumference of a circle, and drawing a 
 planet along with it, farther fuppofed that the epicycles 
 were vaft folid tranfparent globes, and that the planet 
 was a luminous point or (lar, flicking in the furface of 
 this globe. And, to complete the hypothefis, they fup- 
 pofed that the globe turned round its centre, carrying 
 the planet round with it, and thus produced the dire£l 
 and retrograde motions that we obferve. Ariftotle taught 
 that this motion was effefted by the genius of the planet 
 refiding in the globe, and directing it, as the mind of 
 man dire6ls his motions. But, further, to account for 
 the motion of this globe in the circumference of the de- 
 ferent, the ancient philofophers fuppofed that the defe- 
 rent was alfo a vaft cryflalline, or, ^ leafl, tranfparent 
 material fpherical fliell, turning round the earth, and that 
 this ill ell was of fufficient thicknefs to. receive the epi~ 
 
 cvclit 
 
EPICYCLICAL AlACHINERY OF CALLIPPUS. ^239 
 
 cyclic globe within its folid fubftance, not adhering, but 
 at liberty to turn round its own centre. This hypnthefis, 
 though more like the dream of a feverifli man than the 
 thoughts of one in his fenfes, was received as unqueflion- 
 able, from the time of Ariitotle till that of Copernicus. 
 it is fcarcely credible that thinking men (hould admit its 
 truth for a minute, even in its moil admiffible form. 
 But, as the art of obferving improved, it was found ne- 
 ceflary to add another epicycle to the one already admit- 
 ted, in order to account for an annu.,1 inequshty in the 
 epicycplical motion. This w^as a fmall tranfparent globe, 
 placed where Ariftotle placed the planet, ziv} the planet 
 vas ftuck on its furface. Even this v/as found infuffi- 
 cient, and another fet of epicycles were added, till, in 
 fhort, the heavens were filled with folid matter. It is 
 needlefs to fay any more of this epicyclical doctrine and 
 machinery. 
 
 373. But the other mode of conceiving the planetary 
 motions, while it equally furniflies the means of cacula- 
 tion or graphical operation, has much more the appear- 
 ance of reality. The Sun's motion is round the Earth, 
 which we are naturally difpofed to think the centre of 
 the world 5 and the planets revolve, not round a mathe- 
 matical point, a nothing, but round the Sun, a real, and 
 very remarkable fubflance. 
 
 374. Kepler, to whom we are indebted for this dil- 
 co-yery of the elliptical motions, and the equable defcrip- 
 
 tion 
 
C40 ASTRONOMICAL PHENOMENA. 
 
 tion of areas, alfo obferved that the fquafes of the pe^ 
 riodic times in thefe cllipfes are proportional to the cube's 
 of the mean diftances from the Sun. He alfo obferved 
 the fame analogy M'ith refpecft to the Sun's period and 
 diflance from the Earth. 
 
 375. The diftances here alluded to, arc all taken 
 from a fcale of equal parts, of which the Sun's mean 
 diflance from the Earth, contains 1 00000. But aftro- 
 nomers wifn to know the abfolute quantity of thofe dif- 
 tances in fome known meafures. This may be learned 
 by means of the parallax of any one of the planets. 
 Thus, let Mars be in M (fig. 42.), and let his diftance 
 from fome fixed ftar C be obferved by two perfons on 
 the furface of the Earth at A and B. The difference 
 G D of the obferved diftances C G, CD, will give the 
 angle D M G, or its equal A M B. The angles M A B 
 and MBA are given by obfervation, and the line A B 
 is given ; and therefore A M, and confequently E M, 
 may be computed in miles. 
 
 The tranfit of Venus acrofs the Sun's diflc aftbrds 
 much better obfervations for this purpofe. For, at the 
 time, Venus is much nearer to the Earth than Mars is 
 when in oppofition, their diftances from us being nearly 
 as 28 to 52. Therefore the diftance between the ob- 
 fervers will fubtend a larger angle at Venus. This may 
 be meafured by the diftance between the apparent tracks 
 of Venus acrofs the Sun's difk. A fpe^lator in Lap- 
 land, for example, fees Venus move in the line CD 
 
 (fig. 
 
DIMENSIONS OF THE SOLAR SYSTEM. 24I 
 
 (fig. 43.)) while one at the Cape of Good Hope fees her 
 move in the line A B. Alfo, as C D is a fhorter chord 
 than A B, the tranfit will occupy lefs time. This differ- 
 ence in time, amounting, in fome fortunate cafes, to 
 many minutes, will give a very exa6l meafure of the in- 
 terval betvv'een thofe two chords; 
 
 376. The tranfits in 1761 and 1769 were employed 
 for this purpofe, at the earneft recommendation of Dr 
 Edmund Halley. From thofe cbfervations, combined 
 with the proportions deduced from Kepler's third law, 
 we may aiTume the following diflances from the Sun in 
 Englifh ftatute miles, as pretty near the truth. 
 
 The Earth 03,726,900 
 
 Mercury 36,281,700 
 
 Venus <^7>795>5oo 
 
 Mars 142,818,000 
 
 lupiter 487,472,000 
 
 Saturn 894,162,000 
 
 Georgian Planet 1,789,982,000 
 
 Of the Secondary Pla?tets, 
 
 377. Jupiter is obferved to be alwa)rs accompanied 
 by four fmall planets called satellites, which revolve 
 rouml him> while he revolves round the Sun. 
 
 Their diflances from Jupiter are mefifured by means 
 of their greatefl elongations, and their periods are dif- 
 covered by their eclipfes, Vv'hen they come into his fha- 
 
 H h dow. 
 
242 ASTRONOMICAL PHENOMENl. 
 
 dow, and by other methods. They are obfer\'ed to d"e- 
 icribe ellipfes, having Jupiter in one focus ; and they de- 
 icribe areas round Jupiter, which are proportional to the 
 times. Alfo the fquares of their periods are in the 
 proportion of the cubes of their mean diftances from 
 Jupiter. 
 
 378. It has been difcovered by means of the eclipfes 
 of Jupiter's futclHtes, that light is propagated in time, 
 "and employs about 8' 11" in moving along a line equal 
 to the mean diilance of the Earth from the Sun. 
 
 The times of the revolutions of thefe little bodies 
 had been fludied with tlie greateft care, on account of the 
 eafy and accurate means which their frequent eclipfes gave 
 us for afcertaining the longitudes of places. But it was 
 found that, after having calculated the time of an eclipfe 
 in conformity to the periods, which had been moft ac- 
 curately determined, the eclipfe happened later than tlie 
 calculation, in proportion as Jupiter was farther from 
 the Earth. If an eclipfe, when Jupiter is in oppofition, 
 be obferved to happen precifely at the time calculated j 
 an eclipfe three months before, or after, when Jupiter 
 is in quadrature, will be obferved to happen about eight 
 minutes later than the calculated time. An eclipfe hap- 
 pening about fix weeks before or after oppofition, will 
 be about four minutes later than the calculation, when 
 thofe about the time of Jupiter's oppofition happen at the 
 exadt time. In general, this retardation of the eclipfes 
 is obferved to be exactly proportional to the increafe of 
 
 Jupiter's 
 
PROGRESSIVE MOTION OF LIGHT. S^.-? 
 
 Itiprter's dlftance from the E.irtli. It is tlie fame with 
 refpe£t to all the fatellites. This error greatly perplexed 
 the aftronomers, till the connexion of it with Jupiter's 
 change of diftnnce was remarked by Mr Roemcr, a Da- 
 nifli aflronomer, in 1674. As foon as this gentleman 
 took notice of this connexion, he concluded that the re- 
 tardation of the eclipfc was owing to the time employed 
 by the .light in coming to us. The fatellite, now e- 
 clipfed, continued to be feen, till the lajl reflected light 
 reached .us, and, when the ftream of light ceafed, the 
 fatellite difappeared, or was eclipfed. When it has pafT- 
 ed through the fliadow, and is again illuminated, it is 
 not feen at that Inftant by a fp^ftator almoft four hun- 
 dred millions of miles ofF — it does not reappear to him, 
 till the firji reflected light reaches him. . It is not till 
 about forty minutes after being reilluminated by the Sun, 
 that the firft refle£l:ed light from the fatellite reaches the 
 Earth when Jupiter is in quadrature, and about thirty- 
 two minutes when he is in oppofition. 
 
 This ingenious inference of Mr Roemer was doubted 
 for fome time ; but moft of the eminent philofophers a- 
 greed with him. It became more probable, as the mo- 
 tions of the fatellites were more accurately defined ; and 
 it received complete confirmation by Dr Bradley difco- 
 vering another, and very different confequence of the 
 progreflive motion of light from the fixed ftars and pla- 
 nets. This will be confidered afterwards ; and, in the 
 mean time, it is evinced that light, or the caufe of vi- 
 Gon, is propagated in time, and requires about i6y mi^ 
 H h 2 nute§ 
 
^44 ASTRONOMICAL PHENOMENA. 
 
 nutes to move along the diameter of the Sun's orbit, or 
 about 8' ii" to come from the Sun to us, moving about 
 200,000 miles in a fecond. Some imagine vifion to be 
 produced by the undulation of an elaftic medium, as 
 found is produced by the undulation of air. Others 
 imagine light to be emitted from the luminous body, as 
 a flream of water from the difperfcr of a vi^atering-pan. 
 Whichever of thefe be the cafe, Light novi^ becomes a 
 proper fubjecSl of Mechanical difcuffion ; and we may 
 now fpeculate about its motions, and the forces which 
 produce and regulate them. 
 
 379. Saturn is alfo obferved to be accompanied by 
 feven fatellites, which circulate round him in ellipfes, 
 having Saturn in the focus. They defcribe areas pro- 
 portional to the timeSj and the fquares of the periodic 
 times are proportional to the cubes of their mean dif-. 
 tances. 
 
 380. Befides this numerous band of fatellites, Sa- 
 turn is alfo accompanied by a vafl arch or ring of co- 
 herent matter, which furrounds him, at a great diftance. 
 Its diameter is about 208,000 miles, and its breadth a- 
 bout 40,000. It is flat, and extremely thin ; and as it 
 fhines only by receding the Sun's hght, we do not fee 
 it wlien its edge is turned towards us. Late obfervation 
 has fnewn it to be two rings, in the fame plane, and al- 
 moft united. But that they are feparated, is demon- 
 ftrated by a flar being (o^n through the interval between 
 
 them= 
 
SATELLITES OF THE PLANETS. 245 
 
 rliem. Its plane makes an angle of 29° or 30° with that 
 of Saturn's orbit; and when Saturn is in 11* 20°, or 
 5S 20®, the plane of the ring paHcs through the Sun, 
 and refleds no light to us. 
 
 381. In 1787, Dr Herfchel difcovcred two fatellites 
 attending the Georgian planet; and in 1798, he difco- 
 vered four more. Their diftances and their periodic 
 times obferve the laws of Kepler ; but the pofition of 
 their orbits is peculiarly interefling. InPtead of revolving 
 in the order of the figns, in planes not deviating far from 
 the ecliptic, their orbits are almofl, if not precifely per- 
 pendicular to it ; fo that it cannot be faid that they move 
 either in the order of the iigns, or in the oppofite. 
 
 382. Thus do they prefent a nevv-- problem in Phy- 
 fical Aftronomy, in order to afcertain the Sun's influence 
 on their motions — the interfe^tion of their nodes, and 
 the other difturbances of their motions round the planet. 
 
 383. They alfo fiiew the miilake of the Cofmogo- 
 nifls, who would willingly afcribe the general tendency 
 of the planetary motions from weft to eaft along the 
 ecliptic to the influence of fome general mechanical im- 
 pulfion, inftru£ting us how the world may be made as 
 we fee it. Thefe perpendicular orbits are incompatible 
 with the fuppofed influence. 
 
 Of 
 
^6 ASTRONOMICAL PHENOMENA. 
 
 Of the Rotation of the Heavenly Bodies, 
 
 384. In 161 1, Scheiner, profefibr at Ingolfladt, ob- 
 fcrved fpots on the diik of the Sun, which come into 
 view on the eaftern hmb, move acrofs his dillc in parallel 
 circles, difappear on the weftern limb, and, after fome 
 time, again appear on the eaftern limb, and repeat the 
 fame motions. Hence it is inferred that the Sun re- 
 volves from weft to eaft in the fpace of 25^ 14" 12^, 
 round an axis inclined to the plane of the ecliptic 7^ de- 
 grees, and having the afcending node of his equator in 
 longitude 2* 10°. 
 
 Philofophers have formed various opinions concerning 
 the nature of thefe fpots. The moft probable is, that 
 the Sun confifts of a dark nucleus, furrounded by a lu- 
 minous covering, and that the nucleus is fometimes laid 
 bare in particular places. For the general appearance of 
 a fpot during its revolution is like fig. 43. 
 
 385. A feries of moft interefting obfervations has 
 been lately made by Dr Herfchel, by the help of his 
 great telefcopes. Thefe obfervations are recorded in the 
 Philofophical Tranfa6lions for the years i8ci and 1802. 
 They lead to very curious conclufions refpe6ling the pe- 
 culiar conftitution of the Sun. It would feem that the 
 Sun is im^mediately furrounded by an atmofphere, heavy 
 and tranfparent, like our air. This reaches to the height 
 of feveral thcufand niiles. On this atmofphere feems to 
 
 float 
 
CONSTITUTION OF THE SUN. 247 
 
 f.oat a ftrritum of fliinlng clouds, alfo fome thoufands of 
 miles in thicknefs. It Is not clear however that this 
 cloudy flratum ililnes by its native light. There is above 
 it, at fome diftance, another ftratum of matter, of moft: 
 dazzling fplendor. It would feem that it is this alone 
 which illuminates the whole planetary fydem, and alfo 
 the clouds below it. This refplendent ftratum is not 
 equally fo, but moft luminous in irregular lines or ridges, 
 which cover the whole difk like a very clofe brilliant 
 network. Something of this appearance v/as noticed by 
 Mr James Short in 1748, while obferving a total ecllpfe 
 of the Sun, and is mentioned in the Philofophical Tranf- 
 «.£^Ions. Som.e operation of nature in this folar atmo- 
 fphere feems to produce an upward motion in it, like a 
 biafl, which caufes both the clouds and the dazzling 
 flratum to remove from the fpot, making a fort of hole 
 in die luminous ftrata, fo that we can fee through them, 
 dov/n to the dark nucleus of the Sun. Dr Herfchel has 
 obferved tha-. this change, and this denudation of the 
 jiucleus, is much more frequent in fome particular 
 places of the Sun's difk. He has alfo obferved a fmail 
 bit of fliining cloud come in at one fide of an opening, 
 and, in a fliort time, move acrofs it, and difappear on 
 the other {ide of the opening ; and he thinks that thefe 
 moving clouds are confiderably below the great cloudy 
 Itratum. 
 
 386. Dr Herfchel is difpofed to think that the up- 
 per refplendent flratum never fhines on the nucleus j not 
 
 even 
 
248 ASTRONOMICAL PHENOMENil. 
 
 even when an opening has been made in the ftratum of 
 clouds. For he remarks that the upper ftratum is al- 
 ways much more driven afide by what produces the 
 opening than the clouds are ; fo that even the moft ob- 
 lique rays from the fplendid ftratum do not go through, 
 being intercepted by the border of clouds which imme- 
 diately furround the opening. 
 
 387. From Dr Herfchel's defcription oi" this won= 
 derful obje6lj we are almoft led to believe that the fur- 
 face of the Sun may not be fcorched with intolerable 
 and dePtrudive heat. It not unfrequently happens that 
 we have very cold weather in fummer, when the fky is 
 overcaft with thick clouds, impenetrable by the direct 
 rays of the Sun. The curious obfervations of Count 
 Rumford of the manner in which heat is moft copioufly 
 communicated through fluid fubftances, concur with 
 what we knew before, to ftiew us that even an intenfe 
 heat, communicated by radiation to the upper furface of 
 the fliining clouds by the dazzling ftratum above them, 
 may never reach far down through their thicknefs* 
 With much more confidence may we affirm that it 
 would never warm the tranfparent atmofphere below 
 thofe clouds, nor fcorch the firm furface of the Sun. It 
 is far from being improbable therefore, that the furface 
 may not be uninhabitable, even by creatures like our- 
 felves. If fo, there is prefented to our view a fcene of 
 habitation 13,000 times bigger than the furface of this 
 Earth, and about 50 times greater than thofe of all the 
 planets added together. 
 
 388. 
 
ROTATION OF JUPITER. 249 
 
 388. Similar obfervations, firfl made by Dr Hooke 
 in 1664, on fpots in the dilk of Jupiter, fhow tUat he 
 revoh'es from weft to eaft in ^^ 56', round an axis in- 
 clined to the plane of his orbit 2^°. It is alfo obferved 
 diat his equatoreal diameter li to his axis nearly as 14 
 to 13- 
 
 389. There are fome remarkable circumftances in 
 the rotation of this planet. The fpots, by whofe change 
 of place on the dilk we judge of the rotation, are not 
 permanent, any more than thofe obferved on the Sun's 
 difk. We muft therefore conclude that, either the fur- 
 face of the planet is fubje61: to very confiderable varia- 
 tions of brightnefs, or that Jupiter is furrounded by a 
 cloudy atmofphere. The laft is, of itfelf, the moft pro- 
 bable ; and it becomes ftill more fo from another cir- 
 cumftance. There is a certain part of the planet that is 
 fenfibly brighter than the reft, and fometimes remarkably 
 fo. It is known to be one and the fame part by its fi- 
 tuation. This fpot turns round in fomewhat lefs time 
 than the reft. That is, if a dark fpot remains during 
 feveral revolutions, it is found to have feparated a little 
 from this bright fpot, to the left hand, that is, to the 
 wcftward. There is a minute or two of difference be- 
 tween the rotation of Jupiter, as deduced from the fuc- 
 ceflive appearances of the bright fpot, and that deduced 
 from obfervations made on the others. 
 
 390. Thefe circumftances lead us to imagine that 
 Jupiter is really covered with a cloudy atmofphere, and 
 
 I i tliat 
 
2^6 AiJTRdNOMICAL I^HENOM^Ni'. 
 
 that this has a flow motion from eafl to weft relatire id 
 the fm-face of the planet. The ftriped appeatances, 
 called Belts or Zones, are undoubtedly the effect of 
 a difference of climate. They are difpofed with a cer- 
 tain regularity, generally occupying a complete round of 
 hi» furface. Mr Schroeter, who has minutely ftudied 
 their appearances for a long tradl of time, and with ex- 
 cellent glaffes, fays that the changes in the atmofphere 
 are very anomalous, and often very fuddert and exten- 
 five ; In fhort, there feems almoft the fame unfettled 
 weather as on this globe. He does not imagine that 
 we ever fee the real furface of Jupiter -, and even the 
 bright fpot which fo firmly maintains its fituation, is 
 thought by Schroeter to be in the atmofphere. The 
 general current of the clouds is from eaft to weft, like 
 our trade winds, but they often move in other direcStions. 
 The motion is alfo frequently too rapid to be thought the 
 transference of an individual fubftance ; it more refembles 
 the rapid propagation of lome fliort-lived change in the 
 ftate of the atmofphere, as we often obferve in a thunder 
 ftorm, Tlie axis of rotation is almoft perpendicular ta 
 the plane of the orbit, fo that the days and nights are 
 always equal. 
 
 391. The rotation of Mars, firft obferved by Hooke 
 and Caffmi in 1666, is ftill more remarkable than that 
 of Jupiter. The furface of th^ planet is generally o£ 
 unequal brightnefs, and fomething like a permanent fiv- 
 gurc may be obferved in it, by which' we guefs at the 
 
 ^ tinwJ 
 
ROTATION or MARS, 2^1 
 
 ume of the rotation. But tlic figure is fo ill defined, 
 and fo fubjeft to confiderable changes, that it was long 
 before allronomers could be certain of a rotation, fo as 
 to afcertain the time. Dr Herfchel has been at much 
 pains to do this with accuracy, and, by comparing many 
 fucceilive ;ipparitions of the fame objefts, he has found 
 that the time of a revolution is 24 hours and 40 minutes, 
 round an axis inclined to the plane of the ecliptic in an 
 angle of nearly 60 degrees, but making an angle of 
 61° 18' with his own orbit. 
 
 392. It is midfummer-day in Mars when he is in 
 long. 11^ 19° from our vernal equinox. As the planet 
 is of a very oblate form, and probably hollow, there 
 may be a confiderable preceffion of his equinoctial points, 
 by a change in the direction of his axis, 
 
 393. Being fo much inclined to the ecliptic, the 
 poles of Mars come into fight in the courfe of a revolu- 
 tion. When either pole comes firll into view, it is ob- 
 ferved to be remarkably brighter than the relt of the 
 diik. This brightnefs gradually dlminiflies, and is ge- 
 nerally altogether gone, before tliis pole goes out of fight 
 by the change of the planet's pofition. The other poje 
 jiow comes into view, and exhibits fimilar appearances. 
 
 394. This appearance of Mars greatly refembles 
 what our own globe will exhibit to a fpedator placed 
 on Venus or Mercury. The fnows in the colder cli- 
 
 l i 2 mates 
 
252 ' ASTRONOMICAL PHENOMENA. 
 
 mates diminlfh during fummer, and are renewed in tlie 
 enfuing winter. The appearances in Mars may eithev 
 be owing to fnows, or to denfe clouds, which con- 
 denfe on his circumpolar regions during his winter, and 
 are diflipated in fummer. Dr Herfchel remarks that the 
 atmofphere of Mars extends to a very fenfible diftance 
 from his dilk. 
 
 395. Obfervers are not agreed as to the time of 
 the rotation of Venus. Some think that fhe turns round 
 her axis in 23'', and otiiers make it 23 days and 8 hours. 
 The uncertainty is owing to the very fmall time allowed 
 for obferyation, Venus never being feen for more thaw 
 three hours at a time, fo that the change of appearance 
 that we obferve day after day may either be a part of a 
 flow rotation, or rnore than a complete rotation made in 
 a (hort time. Indeed no diftin6l fpots have been ob- 
 ferved in her difk fmpe the time of the elder CafTmi, 
 about the middle of the feventeenth century. Dr Her- 
 fchel has always obferved her covered with an impene- 
 trable cloud, as white as fnow, and without any variety 
 of appearance* 
 
 396. The Moon turnvS round her axis in the courfe 
 of a periodic month, fo that one face is always pre^ 
 fented to our view. There is indeed a very fmall li- 
 5RATI0N, as it is called, by which we occafionally fee a 
 little variation, fo that the fpot which occupies the very 
 CiCntre of the difk, when the Moon is in apogee and in 
 
 perigee, 
 
ROTATION OF SATURN. 253 
 
 perigee, flilfts a little to one fide and a little up or down. 
 This arifes from the perfetl uniformity of her rotation, 
 and tlie unequal motion in her orbit. As the greateft 
 equation of her orbital motion amounts to little more 
 than 5°, this caufes the central fpot to fliift about -^^ of 
 her diameter to one fide, and, returning again to the 
 centre, to fliift as far to the other fide. She turns al- 
 ways the fame face to the other focus of her elliptical 
 orbit round the Earth, becaufe her angular motion round 
 that point is almoft perfectly equable. 
 
 397. It has been difcovered by Dr Herfchel that 
 Saturn turns round his axis in 10^ i&, and that his ring 
 turns round the fame axis in lo*^ 324'. This axis is in- 
 clined to the ecliptic in an angle of 60° nearly, and the 
 InterfetStion of the rinig and ecliptic is in the line pafling 
 through long. 5^ 20° and 11' 2o^ We fee it very open 
 when Saturn is in long. 2^ 20°, or 8^ 20° j and its length 
 is then double of its apparent breadth. It is then mid- 
 fummer and midwinter on Saturn. When S?.turn is in 
 the line of its nodes, it difappears, becaufe its plane 
 pafles through the Sun, and its edge is too thin to be 
 vifible. It fliines only by reflecting the Sun's light. For 
 we fometimes fee the fliadow of Saturn on it, and fome- 
 times its fhadow on Saturn. It will be very open in 
 1 81 1. Juft now (1803) it is extremely flender, and it 
 difappeared for a while in the month of June. Its dia- 
 meter is above 200,000 miles, almofl half of that of the 
 Moon's orbit round the Earth, 
 
 39% 
 
254 ASTRONOMICAL PHENOMENA. 
 
 398. No rotation can be obferved in Mercury, on 
 account of his apparent minutenefs ; nor is any obferve4 
 in the Georgian planet for the fame reafon, 
 
 399. Many philofophers have imagined that the 
 Earth revolves round its axis in 23" 56' 4" from weft to 
 eaft : and that this is the caufe of the obferved diurnal 
 motion of the heavens, which is therefore only an ap- 
 pearance. It muft be acknowledged that the appear- 
 ances will be the fame, and that we muft be infenfible of 
 the motion, There are alfo many circumftances which 
 render this rotation very probable. 
 
 400. I. All the celeftial motions will be rendered 
 Incomparably more moderate and fimple. If the hea-» 
 vens really turn round t:he Earth in 23*" 56' 4", the motion 
 of the Sun, or of any of the planets, is fwifter than any 
 motion of which we have any meafure ; and this to a 
 degree almoft beyond conception. The motion of the 
 8un would be 20,060 times fwifter than that of a can-, 
 non ball. That of the Georgian planet will be twenty 
 times greater than this. If the Earth turns round its 
 axis, the fwifteft motion necefl'ary for the appearances 
 is that of the Earth's equator, v/hich does not exceed 
 that of a cannon ball. 
 
 The motions alfo become incomparably fimpler. For 
 the combination of diurnal motion with the proper mo- 
 tion of the planets makes it vaftly more complex, and 
 imnofhble tp account for on any mechanical principles. 
 
 This. 
 
DIURNAL ROTATION OF-THE EARTH. 2^^ 
 
 This diurnal motion nuift vary, in all the planets, by their 
 change of declination, being about j- flower when they 
 are near the tropics. Yet we cannot conceive that any 
 phyfical relation can fubfid between the orbital motion 
 of a planet and the pofition of the Earth's equator, fufiV- 
 cient for producing fuch a change in the planet's mo-- 
 tion. Befides, the axis of diurnal revolution is far from 
 being the fame juft now and in the time of Hipparchus. 
 Jull now, it pafTes near the ftar in the extremity of the 
 tail of the little Bear. When Hipparchus obferved the 
 heavens, it pafled near the fnout of the Camelopard. 
 It is to the bft degree improbable that every object in 
 the univerfe has changed its motion in this manner. It 
 fnufl be fuppofed that all have changed their motions 
 in different degrees, yet all in a certain precife order, 
 without any connexion or mutual dependence that wc 
 can conceive. 
 
 401. 2. There is no withholding the belief that the 
 Sun was intended to be a fource of light and genial 
 warmth to the organized beings which occupy the fur- 
 face of our globe. How much more fimply, eafily, and 
 beautifully, this is elTecled by the Earth's rotation, and 
 how much more agreeably to the known occonomy of 
 nature 1 
 
 402. 3. This rotation would be analogous to what 
 is obferved in the Sua and mofl of the planets. 
 
 403. 
 
256 ASTRONOMICAL PHENOMENA. 
 
 403. 4. We obferve phenomena on our globe that 
 are iiecefTary confequences of rotation, but cannot be 
 accounted for without it. We know that the equatoreal 
 regions are about twenty miles higher than the circum- 
 polar ; yet the waters of the ocean do not quit this ele- 
 vation, and retire and inundate the poles. This may be 
 prevented by a proper degree of rotation. It may be 
 fo fwift, that the waters would all flow toward the equa- 
 tor, and inundat-e the torrid zone ; nay, fo fwift, that eve- 
 ry thing loofe would be thrown off, as we fee the water 
 difperfed from a twirled mop. Now, a very fimple cal- 
 culation will fliew us t-hat a rotation in 23^ 56' is pre- 
 cifely what will balance the tendency of the waters to 
 flow from the elevated equator towards the poles, and 
 will keep it uniformly fpread over the whole fpheroid. 
 We alfo obferve that a lump of matter of any kind 
 weighs more (by a fpring fteelyard) at Spitzbergen than 
 at Quito, and that the diminution of gravity is precifely 
 what would arife from th^fuppofed rotation, viz. -y^y-- 
 
 There are arguments which give the moft convincing 
 demonftration of the Earth's rotation. 
 
 404. I . Did the heavens turn round the Earth, as 
 has long been believed, it is almoft certain that no zo- 
 diacal fixed ftar could be feen by us. For it is highly 
 probable that light is an emiflion of matter from the lu- 
 minous body. If this be the cafe, fuch is the diftance 
 of any fixed flar A (fig. 44.) that, when its velocity 
 A C is compounded with the velocity of light emitted in 
 
 any 
 
DIURNAL ROTATIOTN OF THE EAKTH. 257 
 
 any direclion A B, or A b, it would produce a motion in 
 a direction AD, or A J, which would never reach the 
 Earth, or which might chance to reach it, but with a velo- 
 city infinitely below the known velocity of light •, and, 
 in any hypothefis concerning the nature of light, the velo- 
 city of the light by which we fee the eircumpolar flars 
 muft greatly exceed that by which we fee the equatoreal 
 Itars. All this is contrary to obfervation. 
 
 a. The fliadow of Jupiter alfo fhould deviate greatly 
 from the line drawn from the Sun to Jupiter, juft as 
 we fee a (hip's vane deviate from the dire6tion of the 
 wind, when fhe is failing briikly acrofs that direction. 
 If the diurnal revolution is a real motion, when Jupiter 
 is in oppofition, his firfl fi^tellite m.ufl be feen to come 
 from behind his difk, and, after appearing for about 
 1^ 10', muft be eelipfed. This is alfo contrary to ob- 
 fervation ; for the fatellites are eelipfed precifely when 
 they come into that line, whereas it fhould happen more 
 than an hour after. 
 
 405. We muft therefore conclude that the Earth 
 revolves round its axis from weft to eaft in 23^ 56' .4''^ 
 We muft further conclude, from* the agreement of the 
 ancient and modern latitudes of places, that the axis o£ 
 the Earth is the fame as formerly ; brut thafit changes its 
 pofitbOR, as we obferve in a top who fe. motion is nearly 
 fpent. This change of pofition is. feeti:.by the ftiifting of 
 the equinodial points. As thefe rliake a tour of thja-- 
 ecliptic in: 25972 yeai's, tlxe pole of die eoitaitQ?,' i»:eepieg^^ 
 ;o K k always- 
 
2^8, ASTRONOMICAL PHENOMENA. 
 
 always perpendicular to Its plane, mufl defcrlbe a circle 
 round the pole of the ecliptic, diftant from it 23° 28' 10", 
 the inclination of the equator to the ecliptic. It will be 
 feen, in due time, that this motion of the Earth's axis, 
 which appeared a myftery even to Copernicus, Tycho 
 Brahe and Kepler, is a neceflTary confequence of the ge- 
 neral power of nature by which the whole aflemblage is 
 held together ; and the detection of this confequence is 
 the moft illuftrious fpecimen of the fagacity of the dif- 
 eoverer, Sir Ifaac Newton. 
 
 Of the Solar Syjlem. 
 
 406. We have feen (372.) that the planets are always 
 found in the circumferences of ellipfes, which have the 
 Sun in their common focus, while the Sun moves in an 
 ellipfe round the Earth. The motion of any planet is 
 compounded of any motion which it has in refped: of 
 the Sun, and any motion which the Sun has in refpecSt of 
 the Earth. Therefore (92. 93.) the appearances of the 
 planetary motions will be the fame as we have defcribed, 
 if we fuppofe the Sun to be at reft, and give the Earth a 
 motion round the Sun, equal and oppofite to what the 
 Sun has been thought to have round the Earth. 
 
 In the fecond part of that article concerning relative 
 motion, it was ftiewn that the relative motion, or change 
 of motion, of the body B, as feen from A, is equal and 
 oppofite to that of A feen from B. In the prefent cafe, 
 tlie diftance of the Sun from the Earth is equal to that 
 
 of 
 
PROOFS OF THE MOTION OF THF EARTH. 2^() 
 
 of the Earth from the Sun. The pofitioii or bearing Is 
 the cppofite. When the Earth is in Aries or Taurus, 
 the Sun will be foen in Libra or Scorpio. When the 
 Earth is in the tropic of Capricorn, the Sun will appear 
 in that of Cancer, and her north pole will be turned to- 
 ward the Sun ; fo that the northern hemifphere will have 
 longer days than nights. In fliort, the gradual variation 
 of the feafons will be the fame in both cafes, if the 
 Earth's axis keeps the fame pofition during its revolution 
 round the Sun. It muft do fo, if there be no force to 
 change its pofition j and we fee that the axes of the other 
 planets retain their pofition. 
 
 407. Then, with refpecl: to the planets, the appear- 
 ances of dire(Sl and retrograde motion, with points of 
 ftation, will alfo be the fame as if the Sun revolved round 
 the Earth. That this may be more evident, it muft be 
 obferved that our judgement of a planet's fituation is 
 precifely fimilar to that of a mariner who fees a fhip's 
 light in a dark night. He fets it by the compafs. If he 
 fees it due north, and a few minutes after, fees it a 
 little to the weftward of north, he imagines that the 
 (hip has really gone a little weftward Yet this might 
 have happened, had both been failing due eaft, provided 
 that the (hip of the fpectator had been failing fafter. It 
 is juft the fame in the planetary motions. If we give 
 the Earth the motion that was afcribed to the Sun, the 
 real velocity of the Earth will be more than double of 
 the velocity of Jupiter. Now fuppofe, according to the 
 K k 2 old 
 
260 ASTRONOMICAL PHENOMENA. 
 
 old hypothefis, the Earth at T (fig. 40.) and the Sun 
 at cc. Suppofe Jupiter in oppofition. Then we muft 
 place the centre of his epicycle in A, and make A a 
 equal to T a. Jupiter is in ^, and his bearing and dif- 
 tance from the Earth is T a, nearly ^ of T A. Six 
 weeks after, the Sun is in /S ; the centre of Jupiter's epi- 
 cycle is in B. Draw B b equal and parallel to T /3, and 
 h is now the place of Jupiter, and T d is now his bear- 
 ing and diftance. He has changed his bearing to the 
 right hand, or weftward on the ecliptic j and his change 
 of pofition is had by meafuring the angle aT b. His 
 longitude on the ecliptic is diminiflied by this number of 
 degrees. 
 
 408. Now let the Sun be at T, according to the 
 new hypothefis, and let A B E L be Jupiter's orbit round 
 the Sun. Let Jupiter be in oppofition to the Sun. We 
 muft place Jupiter in A, and the Earth in ?, fo as to 
 have the Sun and Jupiter in oppofition. It is evident 
 that Jupiter's bearing and diftance from the Earth are the 
 fame as in the former hypothefis. For A a being ' equal 
 to g T, we have g A, the diftance of Jupiter from the 
 Earth, equal to T^ of the former hypothefis. Six v/eeks 
 after, the Earth is at <?, and Jupiter at B. Join (p B, 
 and draw cp N parallel to T A. It is evident that the 
 diftance <p B of Jupiter from the Earth, is equal to the 
 diftance T ^ of the former conftruflion. Alfo the angle 
 H <p B, which is Jupiter's change of bearing, (by the 
 :*ftro,nomer's conipafs, the ecliptic), is equal to the angk 
 
i6i 
 
 d T h of the former conftru^lion. Jupiter therefore, In- 
 fteacl of moving to the left hand, has moved to the right, 
 or weftward, and has diminiflied his ecliptical bearing or 
 Jongitude by the degrees in the angle N' (^ B. 
 
 409. In the^fame manner may the apparent motion 
 of Jupiter be afcertained for every fituation of the H irth 
 2nd Jupiter ; and it will be found that, in every cafe, 
 the line correfponding to ^ B is equal and parallel to the 
 line correfponding to T^ ; thus yC is eqi^l and parallel 
 to T <: ; ;i D is equal and parallel to T d, &c. 
 
 The apparent motions of the planets are therefore 
 precifely the fame in either hypothelis, fo that we are 
 left to follow either opinion, as it appears belt fupported 
 by other arguments. 
 
 410. Accordingly, it has been the opinion of fomc 
 phiiofophers, both in ancient and modern times, that the 
 Earth is a planet, revolving round the Sun placed in the 
 focus of its elliptical orbit, and that it is accompanied 
 by the Moon, in the fame manner as Jupiter and Saturn 
 are by their fatellites. 
 
 The following are the reafons for preferring this opi- 
 nion to that contained in the 371st and 373d articles, 
 which equally explains all the phenomena hitherto men- 
 tioned, and is more confident with our iirfl judge- 
 ments. 
 
 411. I. The celeflial motions become incomparably 
 .Taore fimple, and free of thofe looped contortions which 
 
 , . mufl 
 
252 ASTRONOMICAL PHENOMENA. : 
 
 muft be fuppofed in the other cafe, and which are ex- 
 tremely improbable, and incompatible with what we 
 know of the laws of motion. 
 
 412. 2. This opinion Is alfo more reafonable, on 
 account of the extreme minutenefs of the Earth, when 
 compared with the immenfe bulk of the Sun, Jupiter, 
 and Saturn ; and becaufe the Sun is the fource of light 
 and heat to all the planets. 
 
 The reafons adduced in this and the preceding article 
 were all that could offer themfelves to the philofophers 
 of antiquity. They had not the telefcope, and the fa- 
 tellites were therefore unknown. They had no know- 
 ledge of the powers of nature by which the planetary 
 motions are produced and regulated ; their knowledge of 
 dynamical fcience was extremely fcanty. Yet Pythago- 
 ras, Philolaus, Apollonius, Anaxagoras, and others, main- 
 tained this opinion. But they had few followers in an 
 opinion fo different from our habitual thoughts, and 
 for which they could only offer fome reafons founded 
 on certain notions of propriety or fuitablenefs. But, as 
 men became more converfant, in modern times, with 
 the mechanical arts, every thing conne61:ed with the 
 motion of bodies became more familiar, and was better 
 underftood, and we had lefs hefitation in adopting fenti- 
 ments unlike the firft and mod familiar fuggeftions of 
 fenfe. Other arguments now offered themfelves. 
 
 413. 3. If the Earth turns round the Sun, then the 
 analogy between the fquares of the periodic times and 
 
 the 
 
2(53 
 
 the cubes of the diftances, will obtain in all the bodies 
 which circulate round a common centre ; whereas this 
 will not be the cafe with refpedl to the Sun and IMoon, 
 if both turn round the Earth. 
 
 414. 4. It is thought that the motion of the Sun 
 round the Earth is inconfiflent with the difcoveries which 
 have been made concerning the forces which operate in 
 the planetary motions. 
 
 We have feen, by article 230, combined with the 
 third law of motion, that neither can the Sun revolve 
 round the Earth at reft, nor the Earth round the Sun at 
 reft, but that both muft revolve round their common 
 centre of pofition. It is difcovered that the quantity of 
 matter in the Sun is more than 300,000 times that of 
 the matter in the Earth. Therefore the centre of por- 
 tion of thefe two bodies muft be almoft in the centre of 
 the Sun. Nay, if all the planets v/ere on one fide of 
 the Sun, the cominon centre would be very near his 
 ticntre. 
 
 415. But, perhaps, this argument Is not of the 
 great weight tliat is fuppofed. The difcovery of the- 
 proportion of thefe quantities of matter feems to de- 
 pend on its being previoully eftabllftied that the Sun is 
 in, or near, the centre of pofition of the whole affem- 
 blage. It muft be ov/ned, however, that the perfe<fl 
 harmony of all the comparative meafures of the quan- 
 tities of nratter of the Sun and planets, deduced from 
 
 fources 
 
1^4 ASTRONOMICAL PHENOMENA. 
 
 fources mdependent of each other, renders their accu* 
 racy almofl unqueftionable. 
 
 416. 5. It is inconteftably proved by obfervation. 
 A motion has been difcovered in all the fixed ftars, which 
 arifes from a combination of the motion of light with 
 the motion of the Earth in its orbit. 
 
 Suppofe a fhower of hail falling during a perfect 
 calm, and therefore falling perpendicularly. Were it re- 
 quired to hold a long tube in fuch a pofition that a hail- 
 ftone fhall fall through it without touching either fide, 
 it is plain that the tube muft be held perpendicular. 
 Suppofe now that the tube is faftened to the arm of a 
 gin, fuch as thofe employed in raifing coals from the 
 pit, and that it is carried round, with a velocity that is 
 equal to that of the falling hail. It is now evident 
 that a perpendicular tube will not do. The hailftones 
 will all ftrike on the hindmoft fide of the tube. The 
 tube muft be put into the direction of the relative motion 
 of the hailftones. Now, it was demonftrated in § 93, 
 that this is the diagonal of a parallelogram, one fide of 
 which is the real motion of the hail, and t\iQ other is 
 equal, but oppofite, to the motion of the tube. There- 
 fore if the tube be inclined forijuardy at an angle of 
 45°, the experiment will fucceed, becaufe the tan- 
 gent of this angle is equal to the radius ; and, while the 
 hailftone falls two feet, the tube advances two, and tlie 
 hailftone will pafs along the tube without touching it. 
 
 In the very fame manner, if the Earth be at reft, 
 
 and 
 
ABERRATION OF THE STARS. ^26^ 
 
 niid we would view a ftar near the pole of the ecliptic, 
 the teleicopc muft h-i pointed diredly at the ftar. But 
 if the Earth be in motion round the Sun, the telefcop.e 
 mud be pointed a little forward, that the light may come 
 along the axis of the tube. The proportion of the ve- 
 locity of light to the fuppofed velocity of the Earth in 
 its orbit is nearly that of x 0,000 to i. Therefore the 
 telefcope mult lean about 20" forward. 
 
 Half a year after this, let the fame Ttar be viewed 
 again. The telefcope muft again be pointed 20" a-head 
 of the true pofition of the ftar : but this is in the op- 
 pofite direftion to the former deviation of the telefcope, 
 becaufe the Earth, being now in the oppofite part of its 
 orbit, is moving, the other way. Therefore the pofition 
 of the ftar muft appear to have changed 40" in the fix 
 months. 
 
 It is eafy to (hew that the confcquence of this is, 
 thai every ftar muft appear to have 40'' more longitude 
 when it is on our meridian at niidnight, than when it is 
 on the meridian at mid-day. The e'Je£i of this compo- 
 fition of motions which is moft fufceptible of accurate 
 examination is the following. Let the declination of 
 Ibme ftar near the pole of the eclipt; : be obferved at the 
 time of the ec^uinoxes. It will be found to have 40" 
 more declination in the autumnal than in the vernal e- 
 quinox, if the obferver be in latitude 66° 30'; and not 
 much lefs if lie be in. the latitude of X-ondon. Alfo 
 every ft^r m the heavens fliould appear to defcrlbe a little 
 eiJipie, whofe longer axis is 40''. 
 
 ■ " L 1 417- 
 
Itf^ iSTAONOMICAL PH^NOMENl* 
 
 417. Now this is a6lually obferved, and was dKcc^ 
 rered by Dr Bradley about the year 172(5. It is called 
 the ABERRATION OF THE FIXED STARS, and is One of 
 the moft curious, and mod important difcoveries of the 
 eighteenth century. It is important, by fumifhing an 
 incontrovertible proof that the Earth is a planet, re- 
 volving, hke the others, round the Sun. It is alfo im- 
 portant, by (hewing that the light of the fixed ftaris 
 moves with the fame velocity ^frith the light of the Sun, 
 which illuminates our fyflem* 
 
 418. This arrangement of the planets is called the 
 CoPERNiCAN SYSTEM, having been revived and eftablifh- 
 ed by Copernicus, reprefented in fig. A, The other opi- 
 nion, mentioned (371.), which equally explains the ge- 
 neral phenomena, was maintained by Longomontanus. 
 
 419. Account of the Ptolemaic, Egyptian, and 
 Tychonic fyftems (fig. B, C, D.) * 
 
 420. The Copernican fyftem is now univerfally ad- 
 mitted •, and it is fully eftablifhed, i. That the planets 
 
 and 
 
 * In the preceding pages, no notice has been taken of the 
 latitude of the planets, and the obfenations by which it may 
 be afcertained. What is delivered here is not to be confidered 
 as a treatife of the celeftial motions ; nothing was inferted 
 but what was necefTary for enabling the reader to judge of the 
 •evidences for the progreflivc and other motions of the heavenly 
 
 bodies^ 
 
COPERNICAN SYSTEM. 2t*J 
 
 and the comets defcribe round the Sun areas proportional 
 to the times ; and that the Moon, and the fatellites of 
 Jupiter and Saturn, defcribe round the Earth, Jupiter, 
 and Saturn, areas proportional to the times. 2. That 
 the orbits defcribed by thofe bodie^J are ellipfes, having 
 the Sun, or the primary planet, in one focus. 3. That 
 the fquares of the periodic times of thofe bodies which 
 revolve round a common centre are proportional to the 
 cubes of their mean diflances from that centre. Thefe 
 three propofitions are called the laws of keplkr. 
 
 421. There is however an objection to this account 
 of the planetary motions, which has been thought for- 
 midable. Suppofe a telefcope pointed in a diredion 
 perpendicular to the plane of the Earth's orbit, and car- 
 ried round the Sun in this pofition. Its axis, produced 
 to the ftarry firmament, fhould trace out a figure pre- 
 cifely equal and fimilar to the orbit, and we fliould be 
 able to mark it among the ftars round the pole of thi? 
 ecliptic. But, if this be tried, we find that we are al- 
 ways looking at the fame point, which always remains 
 the centre of the Uttle elilpfe which is the effect of the 
 aberration of light. 
 
 This objection was made, even in the fchools of 
 
 Greece, to Aridarchus of Samos, when he ufed his ut- 
 
 L 1 2 moil 
 
 bodies, from which we are to infer the nature of thofe forces 
 by which they are continually regulated. The motion of re- 
 volution, from which the inference is made, is in one plane, 
 aad is elliptical. This fu^ice; for the purppfe of phiJofophy, 
 
2^5 ASTRONOMICAL PHENOMENA- 
 
 moft endeavours to bring into credit the later opinion oi 
 Pythagoras, placing the Sun -in the centre of the fyftem. 
 And the anfwer given by Ariflarchus h the only one tha>t 
 we can give at the prefent day. » 
 
 422. The only anfwer that can be given to this is, 
 that the diftance of the fixed ftars is fo great, that £ 
 figure of near 200 miUions of miles diameter is not a 
 fenfible obje<Sl. This, incredible as it may feem, has 
 nothing in it of abfurdity. We know that their diftance 
 is immenfe. The comet of 1680 goes 150^ times farther 
 from the Sun than we are, and we muf^ fuppofe it much 
 farther from the neareft ftar, that it may not be affe^led 
 by it in its motion round our Sun. Suppofe it only twice 
 as far, the Earth's orbit traced among the ft^rs would 
 appear only half the diameter of the Sun. We have tele- 
 fcopes which magnify the diameter of obje£ls 1200 times. 
 Yet a fixed ftar is not magnified by them, in the fmallefl 
 degree. That is, though we were only at the i2oodth 
 part of our prefent diftance from it, it would appear no 
 bigger. The more perfect the telefcope is, the ftars sppear 
 the fmaller. We need not be furprifed therefore that 
 obfervation fhews no parallax of the fixed (lars, not 
 even i". Yet a parallax of i" puts the object 206,000 
 times farther oiF than the Sun. But fpace is v/ithoiU 
 bounds, and we have no reafon to thinly that our view 
 comprehends the whole creation. On the contrary, it is 
 more probable that we fee but an inconfiderable part of 
 the fcene on which the perfections of the Creator and 
 Governor of the univerfe are difplayed. 
 
 Of 
 
ev THE COMKTS. 20^ 
 
 Of the Comets, 
 
 423. There are fometimes feen in the heavens cer- 
 tain bodies, accompanied by a train of faint light, which 
 has occafioned them to be called comets. Their appear- 
 ance and motions are extremely various ; and the only 
 general remarks that can be made on them are, tliat" the 
 train, or tail, is generally fmall on the firft' appearance 
 of a comet, gradually lengthens as the comet comes into 
 the neighbourhood of the Sun, and again diminlfhes as 
 it retires to a diftance. Alio the tail is aWays extended 
 \r\ a direcSliou nearly oppofite to the Sun. 
 
 424. The opinionbpf philofophers concerning comets 
 llave been very diiFerent. Sir Ifaac Newton firft fliowed 
 that they are a part of "Cix^ folar fyflem, revolving round 
 the Sun in trajectories, nearly parabolical, having the Sun 
 in the focus. Dr Halley computed the motions of feveral 
 comets, and, among them, found fome which had precifely 
 the fame trajectory. He therefore concluded, that thefe 
 were different appearances of one comet, and that the 
 path of a comet is a very eccnitiic ellipfe, having the 
 Sun in one focus. The apparition of tlie comet of 1082 
 in 1759, which was predicted by Halley, has given his 
 opinion the moft complete confirmation. 
 
 425. Comets are therefore planets, refembling the 
 others in the laws of their motion, revolving 'round the 
 &UU in eiljpfes, defcribing areas proportional to the times, 
 
 and 
 
27© ASTRONOMICAL PHENOMENA. 
 
 and having the fquares of their periodic times propor* 
 tional to the cubes of their mean diftances from the Sun. 
 They differ from the planets in the great variety in the 
 pofition of their orbits, and in this, that many of them 
 have their courfe in anUcedentia Jignorurju 
 
 42$. Their number is very great ; but there are 
 but few with the elements of whofe motions we are well 
 acquainted. The comet of 1680 came very near to the 
 Sun on the nth of December, its diftance not exceeding 
 his femidiameter. When in its aphelion, it will be al- 
 moft 150 times farther from the Sun than the Earth is. 
 Our ideas of the extent of the fo]ar fyftem are thus greatly 
 enlarged. 
 
 427. No fatisfa^lory knowledge has been acquired 
 concerning the caufe of that train of light which accom- 
 panies the comets. Some philofophers imagine that it 
 is the rarer atmofphere of the comet, impelled by the 
 Sun's rays. Others imagine, that it is the atmofphere 
 of the comet, rifing in the folar atmofphere by its fpecific 
 levity. Others imagine, that it is a phenomenon of the 
 fame kind with the aurora borealis, and that this Earth 
 would appear like a comet to a fpedtator placed on ano» 
 ther planet. Confult Newton's Principia ; — a DifTerta* 
 tion, by ProfeiTor Hamilton of Trinity College, Dublin ; 
 -»-a DifTertation, by ^Ir Winthorpe of New Jerfey, &;c. j 
 both in the Philofophical Tranfa<ftions, 
 
 PHYSI* 
 
PHYSICAL ASTRONOMY. 
 
 428. It 19 hoped that the preceding account of the 
 celeftial phenomena has given the attentive ftudent a 
 diftind^ conception of the nature of that evidence which 
 Kepler had for the truth of the three general h£is dif- 
 covered by him in all the motions, and for the truth of 
 thofe feeming deviations from Kepler's laws which were 
 fo happily reconciled with them by Sir Ifaac Newton, by 
 (hewing that thefe deviations are examples of mutual 
 deflections of the celeftial bodies towards one another. 
 Several phenomena were occafionally noticed, although 
 not immediately fubfervient to this purpofe. Thefe are 
 the chief obje£ls of our fubfequent attempts to explain. 
 The account given of the kind of obfervation by which 
 the different motions were proved to be what has been 
 affirmed of them, has been exceedingly fhort and flight, 
 on the prefumption that the young aftronomer will ftudy 
 the celeftial phenomenology in the detail, as delivered by 
 Gregory, Keill, and other authors of reputation. This 
 ftudy will terminate in the fulleft conviction of the vali- 
 dity of the evidence for the truth of the Copernican fyf- 
 tem cf th« Sun and planets ; and in a minute acquaintance 
 
 with 
 
PHYSICAL ASTRONOMY. 
 
 428. It 19 hoped that the preceding account of the 
 celeftial phenomena has given the attentive ftudent a 
 diftindl conception of the nature of that evidence which 
 Kepler had for the truth of the three general fafts dif- 
 covered by him in all the motions, and for the truth of 
 thofe feeming deviations from Kepler's laws which were 
 fo happily reconciled with them by Sir Ifaac Newton, by 
 fhewing that thefe deviations are examples of mutual 
 deflections of the celeftial bodies towards one another. 
 Several phenomena were occafionally noticed, although 
 not immediately fubfervient to this purpofe. Thefe are 
 the chief objects of our fubfequent attempts to explain. 
 The account given of the kind of obfervation by which 
 the different motions were proved to be what has been 
 affirmed of them, has been exceedingly Ihort and flight, 
 on the prefumption that the young aftronomer will ftudy 
 the celeftial phenomenology in the detail, as delivered by 
 Gregory, Keill, and other authors of reputation. This 
 ftudy will terminate in the fulleft convi£lIon of the vali- 
 dity of the evidence for the truth of the Copernican fyf- 
 tem cf th« Sun and planets ; and in a minute acquaintance 
 
 with 
 
Ifl PHYSICAL ASTRdN^OMY. 
 
 with all tliofe peculimtics of motion that didinguifh th{? 
 individuals of the magnificent aflemblage. 
 
 We are now in a condition to invefligate the particu- 
 lar chara<fl;ers of thofe extenfive powers of nature, thofe 
 mechanical afFeclions of matter, which caufe the chferved 
 deviations from that uniform rectilineal motion which 
 would have been obferved in every body, had it been un- 
 der no mechanical influence. And we ihall alfo be able 
 to explain or account for the diftmguiflnng peculiarities 
 of motion which charadlerife the individuals of the fyf-- 
 tern, if we fhall fo far fucceed in our firfl inveftigation as 
 fo {hew that no other force operates in the fyftem, and 
 that thefe peculiarities are only particular and accurately 
 narrated cafes of the three general laws, precifely con- 
 formable to their legitimate confequences. * 
 
 In 
 
 * I think it neceffaty here to forewarn the well-informed 
 mathematician, if any fuch fhall honour thefe pages with a 
 perufal, that he will be difappointed if he look for any thing 
 profound, or cuiious, or new, in what follows. My fole aim 
 is to affifl: the ignorant in the elements of phyfical aftronomy ; 
 ,and I mean to infert nothing but what feems to mc to be ele- 
 mentary in the Newtonian philofophy. This ftudy requires 
 (I think) a few more fteps than are ufually given in the ele- 
 anentary pubUcations of this country. Thefe performances 
 generally leave the iludent too fcantily prepared for reading 
 the valuable works on this fubjeft, unlefs by a very obftinate 
 and fatiguing ftudy* They are deterred by the great difficul- 
 ties 
 
IN GENERAL. 273 
 
 tn our firfl Invefligation, we mufl: aOirm the forces to 
 be fuch as are Indicated by the motions, in the manner 
 agreed on In the general docl:rines of dynamics. That 
 Is, the kind and the intenfity of the force mufl be- in- 
 ferred from the direcl:ion and the magnitude of the change 
 which we confider as its efFect. 
 
 In all this procefs, it is plain that we confider the 
 heavenly bodies as confiding of matter that has tlie fame 
 mechanical properties with the bodies which are daily in 
 our hands. We are not at liberty .to imagine that the 
 celeftial matter has any other properties than what is in- 
 dicated by the motions, otherwife we have no explana* 
 
 tlon, 
 
 ties thus occaiioned in the beginning ; and, proceeding no 
 further, they never tafte the great pleafure afforded by this 
 noble fcience. I wifh to render it accefTible to all who have 
 learned Euclid's Element?, and the leading properties of the 
 three conic feclions, I have preferred the geometrical to 
 the algebraical manner of exprefTmg the quantities under con- 
 fideration. Frequently both methods are fymbolical ; but, 
 even in this cafe, the geometrical fymbol, by prefenting a pie* 
 ture of the thing, gives an objc^ of eafier recolledion, and 
 more exprefTive of its nature, than an algebraical formula ; 
 and in phyfical aftronomy, the geometrical figure is often not 
 a fymbol, but the very quantity under examination. It is 
 from the experience of my own ftudies that I am ir.duccd to 
 prefer this method, fully aware, however, that its advantages 
 are rellricled to mere elem.entary iullruftion, and that no very 
 great progrefs will b? made in the more recondite parts of 
 
 Mm. phylkal 
 
^74' l^HYSICAL ASTKONOMT. 
 
 tion, and may as well reft contented with the fimple nar- 
 ration of the fads. The conftant practice, in all at- 
 tempts to explain a natural appearance, is to try to find 
 a clafs of familiar phenomena which refemble it *, and if 
 we fucceed, we account it to be one of the number, and 
 we reft fatisfied with this as a fufficient explanation. Ac- 
 cordingly, this is the way that philofophers, both in an- 
 cient and modern times, have proceeded in their attempt 
 to difcover the caufes of the planetary motions. 
 
 429. I. Nothing is more familiar to our experience 
 than bodies carried round fixed centres by means of folid 
 matter connecl:ing the bodies with the centre, in one way 
 or another. This was the firft attempt to explain the 
 
 planetary 
 
 phyfical aftronomy without employing the algebraic along with 
 the geometrical analyfis.. 
 
 I fear that I fhall frequently be thought prolix and inele- 
 gant. But I beg that it may be remembered for whom thefe 
 pages are wTitten- — for mere beginners in the ftudy. I wifh 
 to leave no difEculty in the way that I can remove. If I have 
 failed in this — operam perdidi et oleum. But I hope that I 
 may enable an attentive ftudent to read Newton's lunar theory 
 with fome rehfh, and a perception of its beauty. If fo, my 
 favourite point is gained,— the ftudent will go forward. 
 
 The two articles which occupy fo much at the clofe of 
 this fubjed, are not fo far purfued in our elementary books ; 
 yet what is here inferted are only the elements of the fubjedl j 
 and without this inftrudion, we can have no conceptioa oC 
 them that is of any ufe. 
 
EXPLANATION BY SOLID ORBS. 275 
 
 planetary motions of which we have any account. Eu- 
 doxus and Callippus, many ages before oui ?era, taught 
 that all the ftars in the firmament are iO many lucid 
 points or bodies, adhering to the infide of a vafl material 
 concave fphere, which turned round ihe Earth placed in 
 the centre in twenty-four hours. It was called the crys- 
 talline ORB or Sphere. 
 
 But this will not explain the eafterly motion of the 
 Sun and Moon, unlefs we fuppofe them endowed with 
 fome felf-moving power, by which they can creep flowly 
 «aftward along the furface of the cryftalline orb ; far lefs 
 will it account for the Moon fometimes hiding the Sun 
 from us. Thefe philofophers were therefore obliged to 
 fay that there were other fpheres, or rather fpherical 
 fhells, tranfparent, like valt glafs globes, one within ano- 
 ther, and all having a common centre. The Sun and 
 the Moon were fuppofed to be attached to the furface of 
 thofe globes. The fphere which carried the Moon was 
 the fmalleft, immediately furrounding the Earth. The 
 fphere of the Sun was much larger, but ilill left a vafl 
 fpace between it and the fphere of the fixed ftars, which 
 contained all. 
 
 This machinery may make a fhift to carry round the 
 Moon, the Sun, and the ftars, in a way fomewhat like 
 what we behold. But the planets gave the philofophers 
 much trouble, in order to explain their retrograde and 
 diredt motions, and ftationary points, &c. To move Ju- 
 piter in a way refembling what we behold, they fuppofed 
 the fhell of his fphere to be of vaft thicknefs, and in its 
 M m 2 folid 
 
276 PHYSICAL ASTR-ONOMY. 
 
 folid matter they lodged a fmall tranfparent fphere, in 
 the furface of which Jupiter was fixed. Tliis fphere 
 turned round in the hollow made for it in the thick fhell 
 of the deferent fphere, and, as ail was tranfparent, exhi^ 
 bited Jupiter mo\dng to the wcflward, w;;en his epifphere 
 brought him toward us^ and to the eaft, when it carried 
 hiin round toward the outer furface of the deferent fliell. 
 jNIeanwhile, the great deferent globe was moving flowly 
 eaftward, or rather was turning more llowly wellward 
 than the fphere of the flars. 
 
 No doubt, this mechanifm will produce round-about 
 motions, and ftations and retrogradations, &c. This, 
 however, is only a very grofs outline of the planetary 
 motions. But ths Sun's unequable motion could net be 
 reprefented without fuppofing the Earth out of the cen- 
 tre of rotation of his fphere. This was accordingly fup- 
 pofed — and it was an eafy fuppofition. But the mo- 
 tion of Jupiter in relation to the centre of his epi- 
 cycle mufl be fimilar to the Sun's motion in relation 
 to the Earth (361.); but a folid fphere, turning in u 
 hollow which exatlly fits it, can only turn round its 
 centre. This is evident. Therefore the inequality of 
 Jupiter's eplcyclical motion cannot be reprefented by this 
 mechanifm. The deferent fphere may be eccentric, but 
 the epicycle cannot. This oblig-ed thofe engineers to 
 give Jupiter a fecondary epicycle much fmaller than the 
 epicycle which produced his retrogradations and flations. 
 It jnoved in a hollow lodgement made for it in the folid 
 matter of the epicycle, jufl as this moved in a hollow in 
 aIic folid matter of the d'-fcrenf globe. 
 
EXPLANATION BY SOLID ORBS. 277 
 
 Even this would not corrcfpond witli tolerable exadl- 
 nofs with the obferved tenor of Jupiter's motion -, other 
 epicycles were added, to tally with every improvement 
 made on the equation of the apparent motion, till the 
 •whole fpace was almoil crammed full of folid matter ; 
 and after all thefe efforts, fome mathematicians affirmed 
 that there are motions in the heavens that are neither 
 uniform nor circular, nor can be compounded of fuch 
 motions. If fo, this fpherical machinery is impoilible. 
 In modern times, Tycho Bralie proved beyond all con- 
 tradiction that the comet of 1574 pafied through all thofe 
 fpheres, and therefore their exiftence was a mere fiftion. 
 
 One (hould think the whole of this contrivanpe fo 
 artlefs and rude, that we wonder that it ever obtained 
 the lead credit ; yet was it adopted by the prince of an- 
 cient philofophers, — by Ariftotle ; and his authority gave 
 it pofreffion of all the fchools till modern times. 
 
 But -where, all this while, is the mover of all this 
 machinery ? Ariflotle taught that each globe v/as con- 
 ducted, or turned round its axis, by a peculiar genius 
 or daemon. This was worthy of the reft ; and when 
 fuch affertions are called explanations^ nothing in nature 
 need remain unexplained. We muft however do Hip- 
 parchus and Ptolemiy the juftice to fay that they never 
 adopted this hypothefis of Eudoxus and Callippus ; they 
 did not fpeculate about the caufes, but only endeavoured 
 to afcertain the motions j and their epicycle and deferent 
 circles are given by them merely as fteps of mathema- 
 tical contemplation, and in order to have fome principle 
 
 to 
 
^78 PHYSICAL ASTRONOMY. 
 
 to diretH: their calculation, juft as we demonftrate the 
 parabolic path of a cannon ball by compounding a uni- 
 form motion in the line of diredion with a uniformly- 
 accelerated motion in the vertical line. There is no fuch 
 compofition, but the motion of the ball is the fame as if 
 there were. 
 
 430. 2. A much more feafible attempt was made 
 by Cleanthes, another philofopher of Greece, to affign 
 the caufes of the planetary motions. He obferved that 
 bodies are eafily carried round in whirlpools or vortices 
 of water. He taught that the celeflial fpaces are filled 
 with an ethereal fluid, which is in continual motion 
 round the Earth, and that it carried the Sun and planets 
 round with it. But a flight examination of this fpecious 
 hypothefis fliewed that it was much more difficult to 
 form a notion of the vortices, fo as to correfpond with 
 the obferved motions, than to fludy the motions them- 
 felves. It therefore gave no explanation. Yet this very 
 hypothefis was revived in modern times, and was main- 
 tained by two of the mofi: eminent mathematicians and 
 philofophers of Europe, namely, by Des Cartes and Leib- 
 nitz ; and, for a long while, it was acquiefced in by all. 
 
 We mufl conflantly keep in mind that an explana- 
 tion always means to fhew that the fubje£l: in queftion 
 is an example of fomething that we clearly underftand. 
 Whatever is the avowed property of that more familiar 
 fubje6b, mufl therefore be admitted in the ufe made of 
 it for explanation. We explain the fplitting of glafs by 
 
 heat. 
 
EXPLANATION BY VORTICES- 2/9 
 
 Keatj by fhewing that the known and avowed effecls of 
 heat make the glalJs fwell on one fide to a certain de- 
 gree, v\'Ith a certam known force ; and we fliew that the 
 tenacity of the other fide of the glafs, which is not 
 fwelled by the heat, is not able to refifl this force v»'hich 
 is pulling it afundcr ; it mud therefore give way. In 
 fliort, we fliew the fpHtting to be one of the ordinary 
 effefts of heat, which operates here as it operates in all 
 other cafes. 
 
 Now, if we take this method, we find that the ef- 
 fecls of a vortex or whirl in a fluid jire totally unlike 
 tlie planetary motions, and that we cannot afcribe them 
 to the vortical motion of the sether, without giving it 
 laws of motion unlike every thing obferved in all the 
 iiuids that we know j nay, in contradi£tion of all thofe 
 laws of mechanics which are admitted by the very pa- 
 trons of the hypothefis. To give this fluid properties 
 unknown in all others, is abfurd ; we had better give 
 tliofe properties to the planets themfelves. The facl is, 
 that thefe two philofophers had not taken the trouble to 
 think about the matter, or to inquire what motions of a 
 vortex of fluid are poflible, and what are not, or what 
 effects will be produced by fuch vortices as are poffible. 
 They had not thought of any means of moving the fluid it- 
 felf, or for preferving it in motion ; they contented them- 
 felves (at leaft this was the cafe with Des Cartes) with 
 merely throwing out the general facT:, that bodies may 
 be carried round by a vortex. It is to Sir Ifiac Newton 
 tJiat, we are indebted for all that we know of vortical 
 
 motion. 
 
28o PHYSICAL ASTRONOMY. 
 
 motion. In examining this hypotheCs of Des Cartes, 
 whicli had fupreme authority among the philofophers at 
 that time, he found it necefTary to inquire into the manner 
 m which a vortex may be produced, and the conftitution 
 of the vortex which refuhs from the mode of its pro- 
 duclion. This led him, by neceflary fteps, to difcover 
 what forms of vertical miction are pofTible, vidiat are 
 permanent, and the variations to which the others are 
 fubjeti:. In the fecond book of his Mathematical Prin- 
 ciples of Natural Philofophy, he has given the refult of 
 this examination ; and it contains a beautiful fyftem of 
 mechanical doctrine, concerning the mutual action of 
 the filaments of fluid m.atter, by which they modify each 
 other's motion. The refuit of the whole was a complete 
 refutation of this hypothefis as an explanation of the 
 planetary motions, (hewing that the legitimate confe- 
 quences of a vortical motion are altogether ynlike the 
 planetary motions, nay, are incompatible with them. It is 
 quite enough, in this place, for proving the infufficiency 
 cf the hypothefis, to obferve that it muft explain the 
 motion of the comets as well as that of the planets. If 
 Mars be carried round the Sun by a fluid vortex, fo is 
 the comet which appeared in 1682 and 1759. This^ 
 comet cam^e from an immenfe diilance, in the north- 
 ern quarter of the heavens, into our neighbourhood, paff- 
 ing through the vortices of all the planets, defcribing its 
 very eccentric cllipfe with the mod perfect regularity. 
 Now, it is abfolutely impoffible that, in one and the 
 fame place, there can be pafling a dream of the vortex 
 
 cf 
 
EXPLANATION BY VORTICES. dSl 
 
 ■Of -3 planet, and a ftream of the cometary vorteic, having 
 a dire£i:ion and a velocity fo very different. It Is Incon- 
 ceivable that thefe two ftreams of fluid {hall have force 
 enough, one of them to drag a planet along with it, and 
 the other tg drag a comet, and yet that the particles of the 
 one ftream ihal I not difturb the motion of thofe of the 
 other in tlie fir.alieft degree : even the infinitely rare 
 Vapour which formed the tail of the comet v/as not in 
 the Icaft deranged by the motion of the planetary vor- 
 tices through which it pafled. All this is inconceivable 
 and abfurd. 
 
 It is a pity that the account given by Newton of vor- 
 tical motions appealed on fuch an occafion ; for this 
 limited the attention of his readers to this particular 
 employment of it, whiqh purpofe being completely an- 
 fwered in another way, this argument became unnecef- 
 fary, and was not looked into. But it contains much 
 valtia-ble information, of great fervice in all problems of 
 hydraulics. Many eonfequences of the mutual action 
 ©f the fluid filaments produce important changes on the 
 motion of the whole ; fo that till thefe are underftood 
 and taken Into the account, we cannot give an anfwer 
 to very fimple, yet important queflions. This is the 
 caufe why this branch of mechanical philofophy is in fo 
 jmperfefl a flate, although it is one of the moft im» 
 portant» 
 
 431. 3. Many of the ancient phllofophers, ftruck 
 -^•Ith the order, regularity, and harflionious cooperation 
 
 N n • of 
 
2>2? PHYSICAL ASTRONOMY. 
 
 of the planetary motions, imagined that they were con> 
 dueled by intelligent minds. Ariftotlc's way of conceiv- 
 ing this has been already mentioned. The fame doftrine 
 has been revived, in fome refpec^:, in modern times. 
 Leibnitz animates every particle of matter, when he 
 gives his Monads a perception of their fituation with 
 refpeft to every other monad, and a motion in confe- 
 quence of this perception. This, and the elemental mind 
 afcribed by Lord Monboddo to every thing that begins 
 motion, do not feem to diifer much from the oT'rn^ -^vxni 
 of Ariftotle •, nor do they differ from what all the world 
 dillinguifhes by the name oi force. 
 
 This dodlrine cannot be called a hypothefis ; it Is ra- 
 ther a definition, or a mifnomer, giving the name Mind 
 to what exhibits none of thofe phenomena by which we 
 diftinguifli mind. No end beneficial to the agent is 
 gained by the motion of the planet. It may be beneficial 
 to its inhabitants — But fhould we think more highly of 
 die mind of an animal when it is covered with vermin ? — 
 Nor does this do£trine give the faialleft explanation of 
 the planetary motions. We rauil explain the motions 
 by ftudying them,. in order to difcover the laws by which 
 the action of their caufe is regulated : this is juft the 
 way that we learn the nature of any mechanical force. 
 Accordingly, 
 
 432. 4. Many philofophers, both in ancient and 
 modem times, imagined that the planets were defledled 
 frem uniform re^ilineal motion by forces fimilar to what 
 
 we 
 
EXPLANATION BY ATTRACYlON. 283 
 
 we obfen'e in the motions of magnetical and ele£lncal 
 bodies, or in the motion of common heavy bodie5, where 
 one body feems to influence the motion of another at 
 a diftance from it, without any intervening impul- 
 fion. It is thus that a ftone is bent continually from 
 the line of its direction towards the Earth. In the 
 fame manner, an iron ball, rolling along a level table, 
 will be turned afide toward a magnet^ and, by properly 
 adjufling the diftance and the velocity, the ball may be 
 made to revolve round the pole of the magnet. Many 
 of the ancients faid that the curvilineal motions of the 
 planets were produced by tendencies to one another, or to 
 a common centre. Among the moderns, Fermat is the 
 firft who faid in precife terms that the weight of a bxxiy 
 is the fum of the tendencies of each particle to every 
 particle of the Earth. Kepler faid ft ill more exprefsly, 
 that if there be fuppofed two bodies, placed out of the 
 reach of all external forces, and at perfect liberty to 
 move, they would approach each other, with velocities 
 inverfely proportional to their quantities of matter. The 
 Moon {fays he) and the Earth mutually attract each o- 
 ther, and are prevented from meeting by their revolu- 
 tion round their common centre of attraftion. And he 
 fays that the tides of the ocean are the effects of the 
 Moon^s attraction, heaping up the waters immediately 
 under her. Then, adopting the opinion of our country- 
 man, Dr Gilbert of Colchefter, that the Earth is a great 
 magnet, he explains how this mutual attraction will pro- 
 duce a deflection into a curvilineal path, and a4ds, ' Veritath 
 * in me Jit amor an gloria, loquantur dogmata rnea, qu£ pic- 
 N n 2 [ rague 
 
284 FHYSICAL ASTROhOJiir. 
 
 * raque ah aliis accepta fero. Totam ajlroriomiani Qopsr*' 
 
 * tiici hypcthefibus de mimdo^ Tychonis vero Bra.ha obferva'* 
 
 * tionibtiSy denique Gultelmi Gilberii ^?igU philofophla magr 
 *• mtic/£ i?iiedifico» ' epit. astr. copern. 
 
 433. The moft exprefs furmife to this purpofe is 
 that of Dr Robert Hooke, oiie of the moft ardent and 
 ingenious ftudents of nature in that bufy period. At a 
 meeting of the Royal Society, on May 3. 1666, he exr 
 preiTed himfelf in the following manner, . 
 
 '' I v*iU explain a fyflem of the world very diiierent 
 *^ from any yet received j and it is founded on the three- 
 *' following pofiuQns. 
 
 . " I. That all the heavenly bodies have not only a gra- 
 .** vitaiion of their parts to their own proper centre, but 
 <* that they alfo mutually attra£l each Other within their 
 ** fpheres of action. 
 
 " 2. That all bodies having a nmple motion, wiij 
 **. continue to move in a ftraight line, unlefs continually 
 *' deflected from it by fome extraneous forie, caiifing 
 ^* them to defcribe a circle, an ellipfe, or fome other curve. 
 
 '' 3. That th>s attraction is fo much the greater ^^ 
 " the bodies are nearer. As to the proportion in whi<Jh 
 f' thofe forces diminifli by an inercafe of diitancCj I ovfh 
 '* (fays he) I have not difcovered it> 'although I have made 
 " fome experiments to this purpofe. J leave this to other&j 
 *' who have time and knowledge fuflicient for the tii&,''': 
 - . . Xtiia is a very precife enunciation of a proper philoib- 
 phical theory. The phenomenon, the. change of motion ^ 
 
 ' . i-e 
 
>-a HOOKE's THhORY^ . ^8^ 
 
 /s Cohfidered as the mark and measure of a chaRg^ing 
 force, and his audience is referred to Experience for the 
 nature of this force. He had before this exhibited to the 
 Society a very pretty experiment contrived on thefe prin^ 
 ciples. A ball fufpended by a long^ thread from the 
 (^eiling, was made to "fwing round another ball laid on a 
 table immediately, below the point of fufpenfion. When 
 the pufli given to the pendulum was nicely adjuftcd to its 
 deviation from the perpendicular, it defcribed a perfe<Sfe 
 circle round tlie bull on the table. "But when the puflj 
 was very great, or* very fmall, it 'defcribed an ellipfe, 
 having the other bail in its centre. Hooke fliewed rh^t 
 this was the operation of a deiieding force proportional 
 to the difl:ance from tlie other ball. He added, that al- 
 tliough this illuflratcd the planetary motions in fpme de- 
 gree, ' yet it was not fuitabie to their caufe. For the pla- 
 i^iets defcribe ellipfes having the Sun, not in the centre, 
 but in the focus. Therefore they are not retained by a 
 force proportional to their dlftance from the Sun. This 
 was ftri£l: reafoning, frOm good pinciples. It is worthy 
 of remark, that in this clear, and candid, and modeft 
 expofition of a ravional theorVj he anticipated the difco- 
 veries of Newton, as he anticipated, with equal diftindl- 
 nefs and precifion, the difcovcries of Lavoifier, a philofo- 
 pher inferior perhaps only to Newton. 
 
 Thus we fee that many had noticed certain points of 
 refemblance between the celeftial motions and the mo- 
 tions of magnets and heavy bodies. But thefe obfervers 
 >.t the remark regain barren in their hands, becaufe they 
 
 had 
 
I8f^ ^ PHYSICAL ASTRONOMT. 
 
 had neither examined with fufficient attention the celeflial 
 motions, which they attempted to explain, nor had they 
 formed to themfelves any precife notions of the motions 
 from which they hoped to derive an explanation. 
 
 ' 434. At lafl a genius arofe, fully qualified both by 
 talents and difpofition, for thofc arduous tafks. I fpeak 
 of Sir Ifaac Newton. This ornament, this boaft of our 
 nature, had a moil acute and penetrating mind, accom- 
 panied by the foundeft judgment, with a modefl and 
 proper diffidence in his own underftanding. He had a 
 patience in inveftigation, which I believe is yet without 
 an equal, and was convinced that this was the only com- 
 penfation attainable for the imperfe£lion of human un- 
 derftanding, and that when exercifed in profecuting the 
 conje£tu^es of a curious mind, it would not fail of giv^ 
 ing him all the information that we are warranted to 
 hope for. Although only 24 years of age, Mr Newton 
 had already given the moft illullrious fpecimen of his abi- 
 lity to promote the knowledge of nature, in his curious 
 ^ifcoveries concerning light and colours. Thcfe were the 
 refult of the mofl unwearied patience, in making experi- 
 ments of tlie moft delicate kind, and the m^oft acute pe- 
 netration in Separating the refulting phenomena from 
 each other, and the cleared and moft precife logic in 
 reafoning from them •, and they terminated in forming a 
 body of fcience which gave a total change to all the no- 
 tions ofphilofopliers on this fubjeff. Yet this body of op- 
 ^j<;al fcience was nothing but a f;iir narration of the fa6ls 
 
 prefented 
 
SPECULATIONS OF NEWTON. 287 
 
 prefentcd to his view. Not a fingle fuppofitlon ©r con-r 
 je(fi:ure is to be found in it, nor reafoning on any thing 
 not immediately before the eye ; and all its fcience con- 
 fifted in the judicious claflification. This had brought to 
 light certain general laws, which comprehended all the 
 re It. Young Newton faw that this was fure ground, 
 and that a theory, fo founded, could never be ihaken. 
 He was determined therefore to proceed in no other way 
 in all his future fpeculations, well knowing that the fair 
 exhibition of a law of nature is a difcovery, and all the 
 difcovery to which our limited powers will ever admit 
 us. For he felt in its full force the importance of that 
 maxim fo warmly inculcated by Lord Bacon, that no- 
 thing Is to be received as proved in the iludy of nature 
 that is not logically inferred from an obferved facl 5 that 
 accurate obfervation of phenomiena muft precede all 
 theory; and that the only admifiible theory is a proof 
 that the phenomenon under confideration is included in 
 fome general facl, or lav/ of nature. 
 
 435. Retired to his country houfe, to efcape the 
 plague which then raged at Cambridge where he ftudied, 
 and one day walking in liis garden, his thoughts were 
 turned to the caufes of the planetary motions. A con- 
 jeclure to this purpofe occurred to him. Adhering to 
 the Baconian maxim, he imm.ediately compared it with 
 the phenomena by calculation. But he was milled by a 
 falfe eftimation he had made of the bulk of -the Earths 
 His calculatign ihewed him. that his conje<^ure did not 
 
 igrfc 
 
agree with the phenomenon. Newton gave It up witlir 
 (Dttt hefitation '; yet' the difference was onlv about a fixth 
 or fevcnth part; and the conjecture, had it been con- 
 firmed by the calculation, was fuch as would have ac- 
 quired him great celebrity. What youth but Newton 
 could have refifted fuch a temptation ? But he thought 
 no more of it. 
 
 As he admired Des Cartes as the firft mathematician 
 of Europe, and as his. defire of underftanding the pla- 
 netary motions never quitted his mind, he ft2t himfelf to 
 examine, in his own ftri£l manner, the Cartefian theory, 
 which at this time was fupreme in the univerfities of Eu- 
 rope. He difeovered its nullity, but would never have 
 publiflied 3. refutation, hating controverfy above all things, 
 and being already made unhappy by the contefts to which 
 'his optical difcoveries had given occafion. His optical 
 difcoveries had recommended him to the Royal Society, 
 and he was now a member. There he learned the accu- 
 rate meafurement of the Earth by Picard, differing very 
 much from the eftimation by which he had made his cal- 
 culation in 1666 -y and he thought his conjedure now 
 more likely to be juft. He went home, took out his old 
 papers, and refumed his calculations. As they drew to 
 a clofe, he was fo much agitated, that he was obliged 
 to defire a friend to finifh them. His former con- 
 jefture was now found to agree with the phenomena 
 with the utmoft precifion. No wonder then that his 
 mind was agitated. He faw the revolution he was to 
 • . maltc 
 
SPECULATIONS OF NEWTON. 289 
 
 make in the opinions of men, and that he was to (land 
 dt the head of philofophers. 
 
 436. Newton now faw a grand fcene laid open be- 
 fore him ; and he was prepared for exploring it in the 
 completeft manner ; for, ere this time, he had invented 
 a fpecies of geometry tlrat feemed precifely made for this 
 refearch. Dr Hooke's difcourfe to the Society, and his 
 Ihewing that the pendulum was not a proper reprefenta- 
 tion of the planetary forces, was a fort of challenge to 
 him to find out that law of deflection which Hooke own- 
 ed himfelf unable to difcover. He therefore fet himfelf 
 ferioufly to work on the great problem, to " determine the 
 *^ motion of a body under the continual influence of a de- 
 '* fle<Sbing force. " There were found among his papers 
 many experiments on the force of magnets ; but this 
 does not feem to have detained him long. He began to 
 confider the motions of terreftrial bodies with an attention 
 that never had been bellowed on them before ; and in a 
 fhort time compofed twelve propofitions, which contained 
 the leading points of celeffcial mechanifm. Some years af- 
 ter, viz. in 1683, he communicated them to the Royal So- 
 ciety, and they were entered on record. But fo little was 
 Newton difpofed to court fame, that he never thought of 
 publifhing, till Dr Edmund Halley, the moll eminent 
 mathematician and philofopher in the kingdom, went to 
 vifit him at Cambridge, and never ceafed importuning 
 and entreating him, till he was prevailed on to bring his 
 whole thoughts on the fubjecl together, digefted into a 
 O o regular 
 
2^0 PHtSlCAL ASTRONOMl*. 
 
 regular fyftem of univerfal mechanics. Dr Halley was 
 even obliged to correft the manufcript, to get the figures 
 engraved, and, finally, to take charge of the printing 
 and publication. Newton employed but eighteen months 
 to compofe this immortal work. It was publifhed at 
 laft, in 1687, under the title of Mathematical Principles 
 of Natural Phihfophy^ and will be accounted the facred 
 oracles of natural philofophy as long as any knowledge 
 remains in Europe. 
 
 437. It is plain, that in this procefs of invefligatlon^ 
 in order to explain the planetary motions by means of 
 our knowledge of motions that are more familiar, New- 
 ton was obliged to fuppofe that the planets confift of 
 common matter, in which we infer the nature of the 
 moving caufe from the motions that we obferve. New- 
 ton's firfl: flep, therefore, was a fcrupulous obfervation 
 of the celeftial motions, knowing that any miftake with 
 regard to thefe muft bring with it a fimilar miftake with 
 regard to the natural power inferred from it. Every 
 force, and every degree of it, is merely a philofophical 
 interpretation of fome change of motion according to the 
 Copernican fyftem. The Earth is faid to gravitate to- 
 ward the Sun, becauf^s, and only becaufe flie defcribes 
 a curve line concave toward the Sun, and areas propor- 
 tional to the times. If this be not true, it is not true 
 that the Earth gravitates to the Sun. For this reafon, a 
 doubt was exprefled (4i5.)j whetlier the Newtonian dif- 
 coveries were ufed with propriety as arguments for the 
 truth of tlie Copernican fyftem. 
 
 Moft 
 
INVESTIGATION OF PLANETARY FORCE. 29I 
 
 Mod fortunately for fcience, tlie real motions of the 
 heavenly bodies had been at laft detecfled ; and the faga- 
 cious Kepler had reduced them all to three general fadls, 
 known by the name of the laws of Kepler. 
 
 438. The firft of thofe laws is, that all the planets 
 move round the Sun in fuch a inafuier that the line drawn 
 from a planet to tl:>e Sun pajfes over or defcr'bij (verrit, 
 /weeps) areas proportional to the times of the motion. 
 
 Hence Newton made his firft and great inference, 
 that the defecfion of each planet is the aElion of a force 
 always direfted toward the Sun (zip.)? that is, fuch, that 
 if the planet were ftopped, and then let go, it would 
 move toward the Sun in a ftraight line, with a motion 
 continually accelerated, juft as we obferve a fhone fall 
 toward the Earth. Subfequent obfervation has fhewn 
 this obfervation to be much more extenfive than Kepler 
 had any notion of; for it comprehends above ninety co- 
 mets, which have been accurately obferved. A fimilar 
 action or force is obferved to conne«Si: the Moon with this 
 Earth, four fatellites with Jupiter, feven with Saturn, 
 and fix with HerfcheFs planet, all of which defcribe 
 round the central body areas proportional to the times. 
 Newton afcribed all thefe deflexions to the aftion of a 
 mechanical force, on the very fame authority with wliich 
 we afcribe the defle£l:ion of a bombfliell, or of a (lone, 
 from the line of projecStion to its weighty which all man- 
 kind confider as a force. He therefore faid that the 
 primary planets are retained in their paths round the Sun, 
 
 Q o 2 «r}(i 
 
2^2 PHYStCAL AST ilONOMt. 
 
 and the fatellltes i?i their paths rotifid their refpe^ive pri^ 
 tnaries, by a force tending toward the central body. But it 
 muft be noticed that this expreiTion afcertains nothing 
 but the direction of this force, but gives no hint as to its 
 manner of acting. It maybe the impulfe of a ftream 
 of fluid moving toward that centre ; or it may be the 
 attraction of the central body. It may be a tendency in- 
 herent in the planet — it may be the influence of fome 
 miniftring fpirit — but, whateyer it is, this is the direct 
 tion of its effetl,. 
 
 459. Having made this great ftep, by which the re» 
 iation of the planets to the Sun is eftabliflied, and the 
 Sun proved to be the great regulator of their motions, 
 Newton proceeded to inquire farther into the nature of 
 this deflefting force, of which nature he had difcovered 
 only one circumftance. He now endeavoured to difco- 
 ver what variation is made in this deflection by a change 
 cf diftance. If this follow any regular law, it will be 
 a material point afcertained. This can be difcovered 
 only by comparing the momentary deflections of a planet 
 in its different diflances from the Sun. The magnitude 
 or intenfity of the force muft be conceived as precifely 
 proportional to the magnitude of the defleCtion which 
 it produces in the fame time, juft as we meafure the 
 force of terreftrial gravity by the deflexion of fixteen 
 feet in a fecond, which we obferve, whether it be a 
 bombfhell flying three miles, or a pebble thrown to the 
 diftance of a few yards, or a ftone Amply dropped from 
 
 the 
 
INVESTIGATION OF PLANETARY FORCE. Jp^ 
 
 the hand. Hence we infer that gravity is every where 
 the fame. We mufl reafon in the {lime way concerning 
 the planetary defleflions in the different parts of their or- 
 bits. 
 
 Kepler's fecond law, with the afliftance of the firft, 
 enabled Newton to make this comparifon. This fecond 
 general fajft is, that each planet defcribes an ellipfe^ having 
 the Sun in one focus. Therefore, to learn the proportion 
 of the momentary defledlions in different points of the 
 ellipfe, we have only to know the proportion of the 
 arches defcribed in equal fmall, mom.ents of time. This 
 we may learn by drawing a pair of lines from the Sun to 
 different parts of the ellipfe, fo that each pair of lines 
 fiiall comprehend equal areas. The arches on which 
 thefe areas (land muft be defcribed in equal times ; and 
 the proportion of their linear defie£lions from the tan- 
 gents mufl be taken for the proportion of the defled^ing 
 forces which produced them. To make thofe equal areas, 
 we rrqjfl know the precife form of the ellipfe, and we 
 xnull know the geometrical properties of this figure, that 
 we may know the proportion of thofe linear deflec- 
 tions. * 
 
 440. 
 
 * Some of thofe properties are not to be found among the 
 elementary propofitions. For this reafon, a few propofitions> 
 containing the properties frequently appealed to in aftrono- 
 mical difcuffions, are put into the hands of the fludents, and 
 they are requeftcd to read them with care. Without this in- 
 formation, 
 
294 PHYSICAL ASTRONOMY. 
 
 440. 27»<? force by nvhich a planet defcrihes areas prO' 
 portional to the times round the focus of its elliptical orbit is 
 as the fquare of its dfiance from the focus ^ inverfely. 
 
 Let F be the deflecting force in the aphelion A (fig, 
 45.) and f tlie force in any intermediate point P. Let 
 V and V be the velocities in A and P, and C and c be 
 the defleclive chords of the equicurve circles in thofe 
 points. 
 
 Then, by the dynamical propofition in art. 210, we 
 
 have F if— -p- : — y or = W : v^C But, when areas 
 V-' c 
 
 are defcribed proportional to the times, the velocity in 
 A is to that in P inverfely as the perpendiculars drawn 
 from F to the tangents in A and P (102.) F A is per- 
 pendicular to the tangent in A, and F N is perpendicu- 
 
 c C 
 
 lar to the tansjent P N. Therefore F ; f= ^ , ■ : ^^^, 
 ft J FA* FN = 
 
 = FN* Xr:FA*xC, 
 
 But it is fliewn (ElHpfe, § 4.) that C, the deflective 
 chord at A, is equal to L the principal parameter of the 
 ellipfe. It was alfo fhewn (Ellipfe, § 9.) that P O is 
 half the defleaive chord at P, and {§ 8.) that P R is half 
 the principal parameter L. Moreover, the triangles F N P 
 and P Q O and P Q R are fimilar, and therefore FN : FP 
 = PQ:PO. ButPO:PQ=:PQ:PR. Therefore 
 PO:PR=PO*:PO\ Therefore FN' : FP' = PR :PO, 
 
 and 
 
 formation, no confident knowledge can be acquired of that 
 noble collection of demonftrative truths taught by our iiluftri* 
 ous countryman. 
 
LAW OF PLANETARY FORCE. 295: 
 
 and FN* xPO = FP^xPR, and FN^ x 2PO = FP* 
 X 2PR, that is, FN'x^ = FPxL. 
 
 Therefore F :/= F P' X L : F A* X L, = FP' : F A% 
 
 that is, inverfely as the fquare of tlie diftance from F. 
 
 441. This propofition may be demonftrated more 
 briefly, and perhaps more palpably, as follows : 
 
 It was fhewn (Ellipfe, § 10. Cor.) that if P/> be a 
 very minute arch, and pr he perpendicular to the radius 
 vedorPF, then q p, the linear defledion from the tan- 
 gent is, ultimately, in the proportion of p r*. But, be- 
 caufe equal areas are defcribed in equal times, the ele- 
 mentary triangle PF/> is a conflant quantity, when the 
 moments are fuppofed equal, and therefore p r is inverfe- 
 ly as P F, and p r' inverfely as P F^ Therefore qp is 
 inverfely as P F', or the momentary defledion from the 
 tangent is inverfely as the fquare of PF, the diftance 
 from the focus. Now, the momentary defledlion is the 
 meafure of the defle£ling force, and the force is inverfe- 
 ly as the fquare of the diftance from the focus. 
 
 Here then is exhibited all that we know of that pro- 
 perty or mechanical affection of the mafles of matter 
 which compofe the folar fyftcm. Each is under the 
 continual influence of a force directed toward the Sun, 
 urging the planet in that dire£tion ; and this force is va- 
 riable in its intenfity, being more intenfe as the planet 
 €omes nearer to the Sun ; and this change is in the in- 
 verfe duplicate ratio of its diftance from the Sun. It 
 will free us entirely from many mctaphyucal objedions 
 
 which 
 
ipo PHYSICAL ASTRONOMY. 
 
 which have been made to this inference, if, inftead of 
 faying that the planets manifeft fuch a variable tendency 
 toward the Sun, we content ourfelves with fimply affirm- 
 ing the facfi:, that the planets are continually defledted to- 
 ward the Sun, and and that the momentary defle6lions 
 are in the inverfe duplicate ratio of the diftances from, 
 him. 
 
 442. We mufl affirm the fame thing of the force* 
 which retain the fateilites in their elliptical orbits round 
 their primary planets. For they alfo defcribe ellipfes 
 having the primary planet in the focus •, and we muft 
 alfo include the Halleyan comet, which fhewed, by its 
 reapparltion in 1759, that it defcribes an ellipfe having 
 the Sun in the focus. If the other comets be alfo car- 
 ried round in eccentric ellipfes, we muft draw the fame 
 conclufion. Nay, Ihould they defcribe parabolas or hy- 
 perbolas having the Sun in the focus, we fhould flill 
 find that they are retained by a force inverfely propor- 
 tional to the fquare of the diftance. This is demonftrated 
 in precifely the fame manner as in the cafe of elliptical 
 motion, namely, by comparing the linear deflecSlions cor- 
 refponding to equal elementary fediors of the parabola 
 or hyperbola. Thefe are defcribed in equal times, and 
 the linear deflections are proper meafures of the deflecl- 
 ing forces. We (hall find in both of thofe curves q p 
 proportional to p /•\ It is the common property of the 
 i:onic fe£lions referred to a focus. 
 
 It is moft probable that the comets defcribe very ec- 
 centric 
 
•r4:) 
 
 :i 
 
Law of planetary tORCE. 297 
 
 c^nttlc elHpfes. But we get fight of them only when 
 they come near to the Sun, within the orbit of Saturn. 
 None has yet been obferved as far off as that planet. 
 The vifible portion of their orbits fenfibly coincides with 
 3 parabola or hyperbola having the fame focus ; and their 
 motion, computed on this fuppofition, agrees with ob- 
 fervation. The computation in the parabola is very eafy, 
 and can then be transferred to an ellipfe by an ingenious 
 theorem of Dr Halley's in his Jjlronomy of Comets. M. 
 Lambert of Berlin has greatly Amplified the whole pro- 
 cefs. The (Indent will find much valuable information 
 on this fubje£t in M*Laurin's Treatife of Fluxiofis, The 
 chapters on curvature and its variations, are fcarcely dif- 
 tinguifhable from propofitions on curvilineal motion and 
 deflecting forces. Indeed, fince all that we know of a 
 deflecting force is the defledlion which we afcribe to it, 
 the employment of the word force in fuch difcuflfions is 
 little more than an abbreviation of language. 
 
 This propofition being, by its fervices in explaining 
 the phenomena of nature, the mofl: valuable mechanical 
 theorem ever given to the v/orld, we may believe 
 that much attention has been given to it, and that many 
 methods of demonftrating it have been ofi^ered to the 
 choice of mathematicians, the authors claimhig fome 
 merit in facilitating or improving the inveiiigation, 
 Newton's demonitration Is very fliort, but is a good deal 
 incumbered with compofition of ratios, and an arithme- 
 tical or algebraical turn of cxpreflion frequently mixed 
 
 P p vvicb. 
 
tAW OF PLANETARY fORCE. ^97 
 
 c-enttic ellipfes. But we get figlit of them only when 
 they come near to the Sun, within the orbit of Saturn. 
 None has yet been obferved as far olF as that planet. 
 The vifible portion of their orbits fenfibly coincides with 
 3 parabola or hyperbola having the fame focus ; and their 
 motion, computed on this fuppofition, agrees with ob- 
 fervation. The computation in the parabola is very eafy, 
 and can then be transferred to an ellipfe by an ingenious 
 theorem of Dr Halley's in his AJlronomy of Comets. M. 
 Lambert of Berlin has greatly fimplified the whole pro- 
 cefs. The ftudent will find much valuable information 
 on this fubje£l in M*Latirin's Treaiife of Fluxions, The 
 chapters on curvature and its variations, are fcarcely dif- 
 tinguifhable from propofitions on curviiineal motion and 
 deflecting forces. Indeed, fmce all that we know of a 
 deflefting force is the defle£lion which we afcribe to it, 
 the employment of the word force in fuch difcuITions is 
 little more than an abbreviation of language. 
 
 This propofition being, by its fervices in explaining 
 the phenomena of nature, the mod valuable mechanical 
 theorem ever given to the world, we may believe 
 that much attention has been given to it, and that many 
 methods of demonftrating it have been offered to the 
 choice of mathematicians, the authors claiming fome 
 merit in facilitating or improving the inveiiigation, 
 Newton's demonitration is very fliort, but is a good deal 
 incumbered with compofition of ratios, and an arithme- 
 tical or algebraical turn of expreflion frequently mixed 
 
 P p wiub. 
 
IpS PHYSICAL ASTRONOMT. 
 
 with ideas purely geometrical. Newton was obliged to 
 comprefs into it feme properties of the conic fedlions 
 which were not very familiar at that time, becaufe not 
 of frequent ufe : they are now familiar to every ftudent, 
 making part of the treatlfes of conic fe<!^ions. By re- 
 ferring to thefe, the fucceeding authors gave their de- 
 monftrations the appearance of greater fimplicity and 
 elegance. But Newton gives another demonftration in 
 the fecond and third editions of the Prindpia, employing 
 the defleiStive chord of the equicurve circle precifely in 
 the way employed in our -text. This mode of demon- 
 ftration has been varied a little, by employing the radius 
 of curvature, inftead of the chord palling through^the 
 centre of forces. The theorems given by M. De Moivre 
 were the firft in this way, and are very general, and very 
 elegant. Thofe of Jo. Bernoulli, Hermann, and KelU, 
 fcarcely differ from them, and none of them all is pre- 
 ferable to Newton's now mentioned, either for generaHty, 
 fimplicity, or elegance. 
 
 443. It remains now to inquire whether there be 
 any analogy between the forces which retain the differ- 
 ent planets in their refpedtive orbits. It is highly pro- 
 bable that there is, feeing they all refpe^t the Sun. But 
 it is by no means certain. Different bodies exhibit very 
 different laws of action. Thofe of magnetifm, ele£l:ri- 
 city, and cohefion, are extremely different ; and the che- 
 mical affinities, confidered as the effects of attradtive and 
 
 repulfivc 
 
LAW OF PLANETARY FORCE. 299 
 
 repulfive forces, are as varicus as the fubftances them- 
 felves. As we know nothing of the conftitution of the 
 heavenly bodies, we cannot, a priori, fay that it is not fo 
 here. Perhaps the planets are defleded by the impulfion 
 of a fluid in motion, or are thruft toward the Sun by 
 an elaftic aether, denfer and more elaflic as we recede 
 from the Sun. The Sun may be a magnet, and at the 
 fame time ele£b:Ical. The Sun fo conftituted would adl 
 on a magnetical planet both by magnetical and eledrical 
 attraction, while another planet is afFeded only by his 
 electricity. A thoui^uid fuch fuppofitions may be form- 
 ed, all very poffible. Newton therefore could not leave 
 this queflion undecided. 
 
 Various means of deciding it are oiFered to us by the 
 phenomena. The motion of the comets, and particularly 
 of the Halleyan comet^ feems to decide it at once. This 
 comet came from a diftance, far beyond the remoteft of 
 the known planets, and came nearer to the Sun than 
 Venus. Therefore we are entitled to fay, that a force 
 inverfely as the fquare of the diftance from the Sun, ex- 
 tends without interruption through the whole planetary 
 fpaces. But farther, if we calculate the deflection ac- 
 tually obferved in the Halleyan comet, when it was at 
 the fame diftance from the Sun as any of the planets, 
 we fhall find it to be precifely the fam.e with the deflec- 
 tion of that planet. There can remain no doubt there- 
 fore that it is one and the fame force which deflects 
 both the comet and the planet. 
 
 ^ P * J^ui 
 
3C« PHYSICAL ASTRONOMY. 
 
 But Newton could not employ this argument. The 
 motions of the comets were altogether unknown, and 
 probably would have remained fo, had he not difcovered 
 the famenefs of the planetary force through its whole fcenc 
 of influence. The hO: is, that Newton's firft conjectures 
 about the law of the folar force were founded on much 
 eafier obfervations. 
 
 Kepler's third law is, that the Jquares of the periodic 
 times of the pla?iets are in the fame proportion iv'ith the cubes 
 of their mean dijlances from the Stw, Thus, Mars is 
 nearly four times as far from the Sun as Mercury, and 
 his period is nearly eight times that of Mercury— -Now 
 43 = 64, = S\ 
 
 The planets defcribe figures which differ very little from 
 circles, whofe radii are thofe mean diftances. If they de- 
 fcribed circles, it would have been very eafy to afcertain the 
 proportion of the centripetal forces. For, by art. 216, we 
 
 hady== -. Now, in the planetary motions, we have 
 
 f- ==zdK Therefore, in this cafe, f ~ ~ or ~ -~ ^ that 
 
 ' d^ ' d"- 
 
 is, the forces which regulate the motions of the planets 
 at their mean diftances are inverfely as the fquares of 
 thofe diftances. 
 
 It was this notion (by no means precife) of the pla- 
 netary force, which had firft occupied the thoughts of 
 young Newton, while yet a ftudent at college — and, on 
 no better authority than this, had he fuppofed that a 
 fimilar analogy would be obfervcd between the deficdipn 
 
 of 
 
LAW OF PLANETARY FORCE. 3OC 
 
 of the Moon and that of a cannon ball. His dirappoint- 
 ment, occafioned by his erron(5ous cftimation of the bulk 
 of this Earth, and his horror at the thoughts of any fuch 
 controverfies as his optical difcoveries had engaged him 
 in, feem to have made him refolve to keep thefe thoughts 
 to himfelf. But when Picard's meafure of the Earth 
 had removed his caufe of mlfliake, and he faw that the 
 analogy did really hold v/ith refpect to tlie force reach- 
 ing from the Earth to the Moon ; he then thought it 
 worth his while to fludy the fubje^l feriouily, and 
 to inveftigate the defledion in the arch of an el'ipfe. 
 His ftudy terminated in the propofition demonilrated 
 above, — doubtlefs, to his great delight. He was no 
 longer contented with the vague guefs which he had 
 made as to the proportion of the forces which defleded 
 the different planets. The orbit of Mars, and ftill more, 
 the orbit of Mercury, is too eccentric to be confidered 
 i a circle. Befides, at the mean diftances, the radius 
 ve6lor is not perpendicular to the curve, as it is in a circle. 
 He was now in a condition to compare the fnnukaneous 
 deflections of any two planets, in any part of their or- 
 bits. This he has done. In the fifteenth propofition* of 
 the firfh book of the Prindpia, he demonftrates that if 
 the forces actuating the different planets are in the in- 
 verfe duplicate ratio of the diflances-from the Sun, then 
 the fquares of the periodic times muft be as the cubes 
 of the mean diftances. — This being a matter of obferva- 
 tlon, it follows, converfely,, that the forces are in this 
 Inyerfe duplicate ratio of tlie diftances. 
 
 Thng 
 
302 PHYSICAL ASTRONOMY. 
 
 Thus was his darling obje£l attained. But, as this 
 fifteenth propofition has feme intricacy, it is not fo clear 
 as we fhould wifh in an elementary courfe like ours. 
 The fame truth may be eafily made appear in the follow- 
 ing manner. 
 
 444. If a planet^ when at its mean dijlance from the 
 SuHy be projected in a direction perpendicular to the radius 
 veBor^ with the fame velocity which it has in that point of 
 its orbit, it will defcribe a circle round the Sun in the fame 
 time that it defcribes the ellipfe^ 
 
 Let ABPD (fig. 46.) be the elliptical orbit, having 
 the Sun in the focus S. Let A P, B D, be the two axes,^ 
 C the centre, A the aphelion, P the perihelion, and 
 B, D, the two fituations of mean diflance. About S 
 defcribe the circle B D M. Let B K and B N be very 
 fmail equal arches of the circle and the ellipfe, and let 
 BEbeonehalf of BS. 
 
 B M, the double of B S, is the defledive chord of 
 ' the circle of curvature in the point B of the orbit (ellipfe, 
 #9.), and B E is -^ of that chord. Therefore (212.) 
 the velocity in B is that which the force in B would ge- 
 nerate by uniformly impelling the planet along B E. But 
 a body projected with this velocity in the dire<£lion B K 
 will defcribe the circle B K M D. (106 — 212.) 
 
 The arches B K and B N, being equal, and defcribcd 
 with equal velocities, will be defcribcd in equal times. 
 'The triangles B K S^ B N S, having equal bafes B K and 
 
 BN, 
 
LAW OP PLANETATIY FORCE. 30]^ 
 
 B N, are proportional to their altitudes B S and B C (for 
 the elementary arch B N may be confidered as coincid- 
 ing with the tangent in B, and B C is perpendicular to 
 this tangent). But, becaufe B S is equal to C A, the 
 area of the circle B M D is to that of the ellipfe A B P D 
 as A C to B C, that is, as B S to B C, that is, as the 
 triangle BKS to thie triangle B N S. Thefe triangles 
 are therefore (imilar portions of the whole areas, and 
 therefore, fince they are defcribed in equal times, the 
 circle B M D and the ellipfe A B P D will alfo be de- 
 fcribed in equal times. 
 
 Thus it appears tiiat Newton's firft conjecture was 
 perfectly juft. For if the planets, inflead of defcribing 
 their elliptical orbits, were defcribing circles at the fame 
 diftances, and in the fame times, they would do it by ' 
 the influence of tlie fame forces. Therefore fmce, in 
 this cafe, we fliould have ^* == ^5, the forces will Ije 
 proportional to d' inverfely. 
 
 445. We now fee that the forces which retain the differ- 
 ent planets in their orbits are not different forces, but that 
 all are under the influence of one force, which extends 
 from the Sun in every direction, and decreafes in Inten- 
 fity as the fquare of the diflance from the Sun increafes. 
 The intenfity at any particular diflance is the fame, in 
 whatever direction the diflance is taken. Although the 
 planetary courfes do not depart far from our ecliptic, 
 the influence of the regulating force is by no means con- 
 
 fmed 
 
5C'4 PHYSICAL ASTRONOMr. 
 
 fined to that neighbourhood. Comets have been feen 
 which rife ahnoft perpendicular to the ecHptic ; and 
 their orbits or trajectories -occupy all quarters of the 
 heavens. 
 
 This relation, in which they all iland to the Sun, 
 may juftly be called a cofmical relation, depending on 
 their mutual conflitution, which appears to be the fame 
 in them all. As this force refpecls tlie Sun, it may be 
 called a solar force, in the fame fenfe as we ufe the 
 term magnettcal force. All perfons unaffected by pecu- 
 liar philofophical notions, conceive magnetifm diilinclly 
 enough by calling it AtiraElio?i. For, whatever it is, its 
 ciFedls refemble thofe of attraftion. If we conceive the 
 magnetical phenomena as effects of a tendency toward 
 the magnet, inherent in the iron, v/e may conceive the 
 planetary deflexions as produced in the fame way ; but 
 this alfo indicates a famenefs in the conftitution of all the 
 planets. Or we may afcribe the defle6l:ions to the im- 
 pulfions or prefTure of an aether ; but this alfo indicates 
 a famenefs of conftitution over the w^hole fyftem. 
 
 Thus, whatever notion we entertain of what we have 
 called a folar or a planetary force (and the obferved law 
 of aftion limits us to no exclufive manner of conceiving 
 it), we fee a power of nature, whether extrinfic, like the 
 adion of a fluid, or intrinfic, like tendencies or attrac- 
 tions, which fit the Sun and planets for a particular pur- 
 pofe, giving them a cofmical relation, and iav/s of ac- 
 tion, 
 
 f ■ quai 
 
LAW OF PLANETARY FORCE. 30^ 
 
 < qiias dttni primordia rerum 
 
 * Paf}geretf omnipar^ris kges violare Creator 
 
 * Noluitf eter?iique operis fundamina jixif, 
 
 * Sol folio refiii^jii adfejuhet cmma prono 
 
 * Tendere defcenfu^ mc reEio tramite currus 
 
 * Sidereos patitur vaftum per inane moveri, 
 
 ^ Sed rapiti immotiSf fe centra f ftngula gyris, * 
 
 Halley* 
 
 446. It is flill more interefling to remark that th^ 
 iatellites obferve the fame law of a6i:ion. For, in the 
 little fyftems of a planet and its fatellites, we obferve the 
 fame analogy between the diftances and periodic times. 
 In (hort, a centripetal force in the inverfe duplicate ratio 
 of the diftance feems to be the bond by which all is held 
 together^. 
 
 447. As the analogy obferved by Kepler between 
 the diftances of the revolving bodies and the periods of 
 their revolutions, led Newton to the difcovery of the law 
 «!)f planetary deflection^ fo, this law being eftabli{lied> 
 we are led to the fecoml and third fa£l obferVed by Kep- 
 ler as its neceflary confequences. It appears that the 
 periodic time of a planet under the influence of a force 
 inverfely as the fquare of the diftance, depends on it$ 
 mean diftance alone, and will be the fame^ whether the 
 planet defcribe a circle or an ellipfe having any degree 
 whatever of eccentricity. This, as was already obferved, 
 >5 the fifteenth propofition of the firft book of Newton'^ 
 
 Q q Principle. 
 
26€ PHYSICAL ASTRONOMY. 
 
 Priiicipia. Suppofe the ihorter axis B D of the ellipiV 
 ABPD (fig. 47.) to diminifli continually, the longer 
 axis AP reir.almng the fame. As the extremity B of 
 the invariable line B"S moves from B toward C, the ex- 
 trem.ity lS" will move toward P, and when B coincides 
 with C, S will coincide with P, and the ellipfe is chang- 
 ed into a flraight line P A, whofe length is twice the 
 mean diftance SB. 
 
 In all the fucceflive ellipfes produced by this gradual 
 diminution of CB, the periodic time rem.ains unchanged. 
 Juft before the perfe(9: coincidence of B with C, the el- 
 lipfe may be conceived as undiftinguifhable from the line 
 P A. The revolution in this ellipfe is undiflinguifhabTe 
 from the afcent of the body from the perihelion P to the 
 aphelion A, a:nd the fubfequent defeent from A to P. 
 Therefore a body under the influence of the central force 
 will defcend from A to P in half the time of the revolu- 
 tion in the -ellipfe A D P B A. Therefore the time of 
 defcending fi-om any diftance B S is half the period of 
 a body revolvirtg at half that diftance from the Sun. 
 By fuch means we can tell the time in which any planet 
 would fall to the Sun. Multiply the half of the time of 
 a revolution by the fquare root of the cube of i, that is, 
 by the fquare root of ^ ; the produ£l is the time of de- 
 feent. Or divide the time of half a revolution by the 
 fquare root of the cube of 2, that is, by the fquare root 
 of 8, that is, by 2,82847 J ^^> which is the fhorteft pro- 
 cefs, multiply the time of a revolution by the decimal 
 0.1767765 
 
 Mercur? 
 
30? 
 
 d. 
 
 h. 
 
 15 
 
 13 
 
 39 
 
 ^7 
 
 64 
 
 10 
 
 121 
 
 
 
 290 
 
 
 
 798 
 
 
 
 406 
 
 
 
 4 
 
 21 
 
 •THE SATELLITES TEND TO THE SUN. 
 
 Mercury will fall to the Sun in - 
 
 Venus -----.. 
 
 The Earth 
 
 Mars 
 
 Jupiter ------- 
 
 Saturn - ----.- 
 
 Georgian planet - - - - - 
 
 The Moon to this ^Earth - - - - 
 
 Cor. The fquares of the times of falling to the Sun 
 are as th^ cubes of the diftanccs fron; him. 
 
 448. So far did Ney/ton proceed in hh reafoiiings 
 from the obfervations of Kepler. But there remained 
 many important queft.ions to be decided, in which thofe 
 obfervations offered no direct help. 
 
 It appeared improbable that the folar force fiiculd 
 no.t affecl: the fecondary planets. It has been dcmon- 
 ftrated (252.) that if a body P (fig. 29.) revolve round 
 anotlier body S, defcribing areas proportional to the times, 
 while S revolves round fome other body, or is aiji'dled 
 by fom.e external force, P is not only acted on by a cen^ 
 tral force directed to S, but is alfo affected by every ac- 
 celerating f9rce which a6ts on S. 
 
 While, therefore, the Moon xlefcribes areas propor- 
 tional to the times round the Earth, it is not only dcr- 
 He£ted towaird the Earth, but it is alfo defle6ted as much 
 3« the Earth is toward the Sun. For the Moon acconi- 
 Q q 2 panlecJ 
 
508 FHTSICAL ASTHONOMT. 
 
 panics the Earth in all its motions. The fame thing mult 
 be affirmed concerning the fatellites attending the other 
 planets. 
 
 And thus has Newton eftablifhed a fourth propol^^ 
 tion, namely. 
 
 The force by which a fecofidary planet is trtpde to accotn^". 
 pany the primary in its orbit round the Sun is continually, 
 directed to the Sun, and is inverjely cu the fquare of the 
 diflance from him. For, as the primary changes its diflance 
 from the Sun, the force by which it is retained in its 
 orbit varies in this inverfe duplicate ratio of the diftance. 
 Therefore the force which caufes the fecondary planet x.o 
 accompany its primary miifl vary in the fame proportion^ 
 . in order to produce the fame change in its motion that 
 is produced in that of the primary. And, further, fmce 
 the force which retains Jupiter in his orbit is to that 
 which retains the Earth as the fquare of the Earth's dif- 
 lance is to that of Jupiter's diflance, the forces by which 
 their refpeftive fatellites are m.ade to accompany them 
 niufl vary in the fame proportion. 
 
 Thus, all the bodies of the folar fyflem are continually 
 urged by a force dire<£led to the Sun, and decreafing as 
 the fquare of the diflance from him increafes. 
 
 449. Newton remarked, that in all the changes 
 of motion obfervable in our fublunary world, the 
 changes in the a£ling bodies are equal and oppofite. 
 In all impullions, one body is obferved to lofe as much 
 motion as the other gains. All magn^tical and electri- 
 cal 
 
HEClPROCiL ACTION OF THE stN, &C. 30^ 
 
 pal attractions and repulfioiis are mutual. Every acfllon 
 feems to be accompanied by an equal reaClion in the 
 oppofite direction. He even imagined that it may be 
 proved, from abftraCl principles, that it mufl be fo. He 
 therefore affirmed that this law obtained alfo in the ce- 
 leftial motions, and that not only were the planets con- 
 tinually impelled toward the Sun, but alfo that the Sun 
 was impelled toward the planets. The doubts which 
 may be entertained concerning the authority of this law 
 of motion have been noticed already. At prefent, we 
 are to notice the fa(5ls which the celeftial motions furnifh 
 in fupport of Sir Ifaac Newton's aflertion. 
 
 450. Diriiflions have been given (294.) how to cal- 
 culate the Sftn's place for any given moment. When 
 the aftronomers had obtained inllruments of nice con- 
 ftru(Si:ion, and had imprgved the art of obferving, there 
 was found an irregularity in this calculation, which had 
 an evident relation to the Moon. At new Moon, the 
 obfervations correfponded exa£lly with the Sun's calcu^ 
 lated place 5 but feven or eight days after, the Sun is 
 obferved to be about 8" or 10" to the eaftv/ard of his 
 calculated place, when the Moon is in her firft quadra- 
 ture, and he is obferved as much to the well ward whei^ 
 fhe is in the laft quadrature. In intermediate fituations, 
 the error is obferved to increafe in the proportion of the 
 fme of the Moon's diflance from conjunction or oppofi- 
 tion. 
 
 Things muft be fo, if it be true that the defied ion of 
 
 the 
 
3IO 
 
 PHYSICAL ASTRONOMY. 
 
 the Moon toward the Earth is accompanied with an er 
 qual deflecSlion of the Earth toward the Moon. For 
 (230.) the Moon will not revolve round the Earth, but 
 the Earth and Moon will revolve round their common 
 centre of pofition. When the Moon is in her firft qua- 
 4rature, her pofition may be reprefented by M (fig. 48.) 
 while the Earth is at E, and their common centre is at 
 A. A fpedlator in A will fee the Sun S in his cal- 
 culated place B. But the fpeclator in the Earth E fees 
 the Sun in C, to the left hand, or eaftward of B. The 
 interval B C meafures the angle B S C, or A S E, fub^ 
 tended at the Sun by the diilance E A of the common 
 centre of the Earth and Moon from the centre of the 
 Earth. At new Moon, A, E, and S, are in a flraight line, fo 
 that B and C coincide. At the iait quadrature, the Moon 
 js at w, the Earth at f, and the common centre at a. 
 Now the Sun is feen at c, S" or 10" to the weftwara of 
 his calculated place. This correction has been pointed 
 out by Newton, but it was not obferved at the firil, ow- 
 ing to its being blended with the Sun's horizontal pa- 
 rallax which had not been taken into account. But it 
 was foon recognifed, and it now makes an article a- 
 mong the various equations ufed in calculating the Sun's 
 place. 
 
 Here^ then, is a plain proof of a mutual action and 
 jreaCtion of the Earfh and Moon. For, fnice they revolve 
 round a common centre, the Earth is unqueflionably dc- 
 iie£i:ed into the curve line by the aCtion of a force di- 
 •^•ccled towards the Moon. But ^-e have a much better 
 
 proof. 
 
MUTUAL ACTION OF SUN AND PLANETS. 31: 
 
 proof. The waters of tlie ocean are obferved every day 
 to heap up on that part of our globe which is under the 
 Moon. In this fituationj the weight of the water is di- 
 minifhed by the attra<flion of the Moon, and it requires 
 a greater elevation, or a greater quantity, to compenfate 
 for the diminiilied weight. On the other hand, we fee 
 the waters abftra^led from all thofe parts which ha:ve 
 the ^Toon in the horizon. Kepler, after afTerting, in 
 very pofitive terms, that tlie Earth and Moon would run 
 together, imd are prevented by a mutual circulation round 
 their common centre, adduces the tides as a proof. 
 
 45:1. As the art of obfervatlon continued to improve, 
 aftronomers were able to remark abundant proofs of the 
 tendency of the Sun toward the planets. When the 
 great planets Jupiter and Saturn are in quadrature witk 
 the Earth, to the right hand of the line drawn from the 
 Earth to the Sun's calculated place, the Sun is then ob- 
 ferved to fhift to the left of that line, keeping always on 
 the oppofite fide of the common centre of pofition. Tliefe 
 deviations are indeed very minute, becaufe the Sun is 
 vaflly more maffive than all the planets collected into 
 one lump. But in favourable iituations of thefe planets, 
 they are perfectly fenfible, and have been calculated ; 
 and they mujl be taken into account in every calculation 
 of the Sun*s place, in order to have it with the accuracy 
 that is now attainable. It muft be granted that this ac- 
 curacy, atSlually attained by means of thofe corredtions, 
 and unattainable without them, is a pofitive proof of 
 
 this 
 
MUTUAL ACTION OF sUN AND PLANETS. 31: 
 
 proof. The waters of the ocean are obferved every day 
 to heap up on that part of our globe which is under the 
 Moon. In this fituation, the weight of the water is di- 
 minifhed by the attraclion of the Moon, and it requires 
 a greater elevation, or a greater quantity, to compenfate 
 for the diminiilied weight. On the other hand, we fee 
 the waters abftra6led from all thofe parts which harve 
 the ^Toon in the horizon. Kepler, after afierting, in 
 very pofitive terms, that tlie Earth and Moon would run 
 together, iuid are prevented by a mutual circulation round 
 their common centre, adduces the tides as a proof. 
 
 45^1. As the art of obfervation continued to improve, 
 aftronomers were able to remark abundant proofs of the 
 tendency of the Sun toward the planets. When the 
 great planets Jupiter and Saturn are in quadrature v/itk 
 the Earth, to the right hand of the line drawn from the 
 Earth to the Sun's calculated place, the Sun is then ob- 
 ferved to fhift to the left of that line, keeping always on 
 the oppofite fide of the common centre of pofition. Tliefe 
 deviations are indeed very minute, becaufe the Sun is 
 vaflly more maffive than all the planets collected into 
 one lump. But in favourable iituations of thefe planets, 
 they are perfectly fenfible, and have been calculated ; 
 and they mujl be taken into account in every calculation 
 of the Sun*s place, in order to have it with the accuracy 
 that is now attainable. It muft be granted that this ac- 
 curacy, adlually attained by means of thofe corrections, 
 and unattainable without them, is a pofitive proof of 
 
 this 
 
%tt PHTSICAL ASTRONOMY*. 
 
 this mutual deflexion of the Sun toward the planets. 
 The quantity correfponding to one planet is too fmall, of 
 itfelf, to be very diftindlly obferved ; but, by occafionally 
 combining with others of the fame kind, the fum be- 
 comes very fenfible, and fufceptible of meafure. It 
 {ometimes amounts to 38 feconds, and mud never be o- 
 mitted in the calculations fubfervient to the finding the 
 longitude of a fhip at fea. Philofophy, in this inftance, 
 is greatly indebted to the arts. And flie has liberally re- 
 paid the fervice. 
 
 452. Here It is worthy of remark, that had the Sun 
 been much fmailer than he is, fo that he would have mov- 
 ed much further from the common centre, and would have 
 been much more agitated by the tendencies to the differ- 
 ent planets, it is probable that v/e never fhould 'have ac- 
 quired any diflin6l or ufeful knowledge of the fyflem^ 
 For we now fee that Kepler's laws cannot be flridlly 
 true ; yet it was thofe laws alone that fuggelled the 
 thought, and furniftied to young Newton the means 
 of inveiligation. The analogy of the periodic times and 
 dlftances is accurate, only with refpedt to the common 
 centre, but not with refpeft to the Sun. But the great 
 mafs of the Sun occafions this common centre to be 
 generally within his furface, and it is never diftant from 
 it I of his diameter. Therefore this third law of Kepler 
 is fo nearly exa£l; in refpeft of the Sun, that the art of 
 obfervation, in Newton's lifetime, could not have found 
 any errors* The penetrating eye of Newton however 
 
 immediately 
 
THE PLANETS TEND TO E/CH OTHER. 313 
 
 Immediately perceived his own good fortuiTc, and his 
 ^rror in fuppofmg KepkVs laws accurately true. But 
 this was not enough for his philofophy ; he was deter- 
 mined that it fiiould narrate nothing but truth. With 
 great ingenuity, and elegance of method, he demonftrutes 
 that his mechanical inferences from Kepler's laws are 
 flill ftricbly true, and that his own law of planetary force 
 is exa£^, although the centre of revolution is not the 
 centre of the Sun. All the difference refpecls the ab*- 
 folute magnitude of the periodic times in relation to the 
 magnitude of the force. This he demonftrates in a fe- 
 ries of propolitions, of which our $231. is the chief. 
 
 453. New^^ton proceeds flill further in his inveftiga- 
 tion of the extent 01 the influence of this planetary force, 
 and fays that all the planets imttually tend toward each 
 other. It does not appear how this opinion arofe in his 
 mind. There are abundance of phenomena, however, 
 of eafy obfervation, .which make it very evident. It was 
 probably a conje6lure, fuggefted by obferving this reci- 
 procal adtioh between the Earth and Moon. But he im- 
 mediately followed ir into its confequences, and pointe4 
 them out to the aftronomers. They are very important, 
 and explain many phenomena whicii had hitherto great- 
 ly perplexed the aftronomers. 
 
 Suppofe Jupiter and Mars to be in conjunction, lying 
 in the fame line from the Sun. As Mars revolves much 
 quicker than Jupiter, he gets before him, but, being at* 
 ti"a£led by Jupiter, his motion is rctarded--and Jupiter, 
 
 R r \^^x% 
 
314 PHYSICAL. ASTRONOMY. 
 
 being attracted by Mars, is accelerated. On the contrary, 
 before Mars arrives at conjuncftion with Jupiter, Mars is 
 accelerated, and Jupiter is retarded. Further, the attrac- 
 tion of Mars by Jupiter muft dimiiiifti the tendency of 
 Mars to the Sun, or mufl: acSt in oppofition to the attrac- 
 tion of the Sun ; therefore the curvature of Mars's orbit 
 in that place mufl be diminifhed. On the conti'ary, the 
 tendency of Jupiter to Mars, acting in the fame direc- 
 tion as his tendency to -the Sun, muft increafe the cur- 
 vature of that part of Jupiter's orbit. If Jupiter be at 
 this time advancing to his aphelion, this increase of cur- 
 vature will fooner bend the line of bis m.otion from an ob- 
 tufe into a right angle with the radius ve6lor. Therefore 
 his aphelion will be fooner attained, and it will appear to 
 have fhifted to the weftward. For the oppofite reafons, 
 the apfides of Mars will feem to fhift to tlie eaftward. 
 There are other fituations of thefe planets where the 
 contrary effeds will happofi. In each revolution, each 
 planet will be alternately accelerated twice, and twice, 
 retarded, and the apfjdes of the exterior planet will con- 
 tinually recede, and that of the interior will advance. 
 It is obvious that this difturbance of the motion of a 
 planet by its defieftion to another^j though probably very 
 minute, yet-being continued for a tra6i: of time, its accu- 
 mulated refult niay become very fcnfible. Thefe changes 
 tire all fufceptible of accurate calculation^ as we Ihall 
 afterv/ards explain particularly. 
 
 This muft be confidered as a convincing proof of the 
 mutual adion of the heavenly bodies, and it adds frefii 
 
 lufi^re 
 
MUTUAf. ACTION OF THE SUN, &c; 3I5 
 
 iuftre.to the penetration and genius of Newton, who 
 made thefe afleitions independent of obfervation, point- 
 ing out to aftronomers the fure means of perfeding their 
 knowledge of the celeftial motions. 
 
 454. Here therefore we have eftabllfiied a fifth pro- 
 pofiticn in phyfical aftronomy, namely, that all the bodies 
 in the folar fyjl^m tend mutually toward one another y iv'ith 
 forces which vary in the inverfe duplicate ratio of the dif- 
 tances. 
 
 It did not fatisfy Newton that he merely pointed out 
 the grofs efFe£l of this mutual tendency. He gave aftro- 
 nomers the means of inveftigating and afcertaining its 
 intenfity, and its variation by a Variation of diftance. 
 The efFe£l of the Earth's tendency to Jupiter during any 
 length of time, may be computed by means of Newton's 
 dynamical propofitions, contained in the firfl book of his 
 Principia, particularly by the 39th. Of thefe we have 
 given a proper feledion in the general dodrines of Dy- 
 namics. 
 
 455. But the inquifitive mind of Newton did not 
 ilop here. He was anxious to learn whether Hhis plane- 
 tary tendency had any refemblance or relation to forces 
 v/lth which we are more familiarly acquainted. Of this 
 kind are magnetifm and gravity. He was the more in- 
 cited to this inveftigation by the conje6lures on this fub- 
 jedi which had arifen in the mind of Kepler. Thi$ 
 great allronomer had been much 'taken with the difco- 
 
 R K ^ ver/ 
 
510 PHYSICAL ASTRONdMr. 
 
 ^ery juft pubHffied by Dr Gilbert of ColcKefter, ftating; 
 tliat this Earth is a great magnet, and he was difpofed to 
 afcribe the revolution o-f the Moon to the magnetical in- 
 fluence of the Earth. It appears from Kewton's papers^ 
 that he had made a grecrt many experiments for difco- 
 Bering the law of magnetic a<flion. But he had found 
 k fo dependant on circumflances of form and (ituation^ 
 and fo changeable by time, that it feemed fufccptible 
 of no comparifon wfth the folar force ; and he foon gave 
 It up. He was more fuccefsfui in- tracing the refem- 
 hlances obfervable in the plienomena of common gra- 
 vity. It ha^ been already remarked (435.)? that, very- 
 early in life, fie had conjectured -that it was the fame 
 with the folar force ; and that after he had formed the 
 opiniofi that -the folar force varied in the inverfe dupli- 
 cate ratio of the diftance, he put his ccnje<flure to the 
 left, by comparing the fall of a ftone with the deflec-^ 
 tion of te Moon. The diftance of the Moon is efti- 
 matcd to be 60 femidiameters of the Earth. Therefore,, 
 if gravity and the lunar defle£ling force be the fame> 
 the ftone fliould deflect as much in one fecond as the 
 Moon does in a minute. For we may, without any fen- 
 fible error, fuppofe that the lunar force a6ts uniformly 
 during one minute. If fo, the linear deflections muft 
 be as the fquares of the times, llie defledlion in a mi- 
 nute muft be 60 >< 60 times, or 3-600 times the deflec- 
 tion in a fecond. But, according to the law of plane- 
 tary force, the defle£lion at the Earth's furface muft be 
 60 X 60, or 3600 times the defledion at the Moon~ 
 
 Now,. 
 
CraVitation of the moon. jry 
 
 Now, ill a fccond, a flone falls i6 feet and an inch. 
 Therefore tlie Moon fhould defledi: i6 feet and an inch in 
 a minute from the tangent of her orbit. Newton caku- 
 lated the verfed fine of the arch defcribed by the Moon 
 in a minute, to a radius equal to 60 femidiameters of 
 the Earth. He found it only about 13^ feet, and he 
 gave over any farther inquiry. But he had haftily fup- 
 pofed a degree to contain 60 miles, not attending to the 
 difference between a geographical mile, or 6otli of a 
 degree, and an Englifii ftatute mile. A degree contaiiis 
 69^ fuch rhiles •, fo that he had made the Moon's orbit, 
 and Confequently her defie6^ion, too fmall in the fame 
 proportion. If vre increafe the calculated deflexion iit 
 this proportion, it comes out exa^ly i6/x j and the con- 
 jecture is fully eftablifhed. 
 
 "When Picard's accurate meafure of the Earth had 
 enabled Newton to confirm his former conje6ture con- 
 cerning the identity of the planetary force and terreftrial 
 gravity, he again made the calculation and comparifon 
 i/i the moft fcrupulcus manner. For we now fee that fe- 
 Veral circumftances fnufl be taken into the account, wluch 
 he had omitted in his firii computation from Picard's 
 meafunJ of the Earth. The fall in a feeond is not th$ 
 €xa£l: meafure of terreftrial gravity. A ftone would fall 
 farther, were it no! that its gravity is-.diminiflied by th« 
 Earth's totation. It is alfo diminiflied by the action o£ 
 the Sun and Moon, and by the weight of the air which 
 the ftone difplaces. • Ail thefe diminutions of the accele- 
 fating fore^ of gravity are Aifceptible of exacl ealcula- 
 
 tio». 
 
3l8 PHYSICAL ASTRONOMY. 
 
 tion, and were accordingly calcnlated by Newton, and 
 the amount added to tlie obferved acceleration of a fall- 
 ing body. In tlie next place, the real radius of the 
 Moon's orbit niuil be reckoned only from the common 
 centre of tlie Earth and Moon. And then the force de- 
 duced from this deflection muil be increafed m the fub- 
 duplicate ratio of the matter in the Earth to the matter 
 in the Earth and Moon added together (231.) All this 
 has been done, and tlie refuit eojncides precifely witli 
 obfervation. 
 
 This may be demonflrated in another way. We can 
 tell in what time a body would revolve round the Earth, 
 clofe to its furfacc. For we muft. have f proportional 
 to ^^ It will be found to be 84 minutes and 34 feconds. 
 Then we know the arch defcribed in one fecond, and 
 can calculate its defledion from the, tangent. We fhall 
 iind it i6~t ^eet, the fame with that produced by com- 
 mon gravity. 
 
 456. TerreJIrial gravity^ therefore, or that force which 
 caufes bodies to fall^ or to prefs on their fupports, is only a 
 particular exaviple of that univerfal tendency^ by ivhich ail 
 the hsdics of the folarfy fern are retained in their orbits. 
 
 \s^Q muft nov/ extend to thofe bodies the other fymp- 
 toms of common gravity. It is by gravity that water ar- 
 ranges itfeif into a level furface, that is, a furface which 
 jftiakes a part of the great fphere of the ocean. The 
 •weight of tliis water keeps it together, in a round form. 
 We muft afcribe the globular forms of the Sun and pla- 
 nets 
 
GRAVITY. — GRAVITATION. 319 
 
 nets to a finiilar operation. A body on their furface will 
 prefs it as a heavy body prelTes the ground. Dr Hooke 
 remarks that all the protuberances on the furface of the 
 Moon are of forms confiftent with a gravity toward its 
 centre. They are generally floping, and, though in fome 
 places very rugged and precipitous, yet nowhere over- 
 hang, or have any fnape that would not (land on the 
 ground. The more rugged parts are moll evidently mat- 
 ter which has been thrown up by volcanic explofion, and 
 have fallen down again by tlieir lunar gravity. 
 
 457. That property by which bodies are heavy is 
 called GRAVITY, heaviness— the being heavy; and the 
 /flj? that it moves toward the Earth, may be called gra- 
 vitation. While it falls, or preiTes on its fupports, it 
 may be faid to gravitate, to give indication of its being 
 gravis or heavy. In this fenfe the planets gravitate to 
 the Sun, and the fecondary planets to their primaries, 
 and, in ihort, everybody in the folar fyftem to every o- 
 ther body. By the verb to gravitate, nothing is meant 
 but the fa6l, that they either actually approach, or ma- 
 nifefl, by a very fenfible prefTure, tendencies to approach 
 the body to which they are faid to gravitate. The verb, 
 or the noun, fhould not be confidered as the expreiTion of 
 any quality or property, but merely of a phenomenon, ^ 
 facb or event in nature. *- ^ 
 
 458. But this deviation from uniform re^lilirieal mo- 
 pen is (^dilfidered as an (f^^, and it is of importance to 
 
 difcQver 
 
^50 PHYSICAL ASTRONOMY. 
 
 difcover the cnufc. Now, in the moft familiar inftance, 
 the fall or preflure of a heavy body, we afcribe the fall, 
 or preflure indicating the tendency to fall, to its heavi- 
 nefs. But we have no other notion of this heavinefs than 
 the very thing which we afcribe to it as an eifetl. The 
 feeling the heavinefs of the piece of lead that lies in our 
 hand, is the ftttn of all that ive know about it. But we 
 confider this heavinefs as a property of all terreftrial mat- 
 ter, becaufe all bodies give fome of thofe appearances 
 which we confider as indications of it. All move toward 
 the Earth if not fupported, and all prefs on the fupport. 
 The feeling of preflure which a heavy body excites might 
 be confidered as its chara6leriftic phenomenon j for it 
 is this feeling that makes us think it a force — we muil 
 oppofe our force to it ; but we cannot diftinguifh it 
 from the feeling of any other equal preflure. It is moft 
 diftinguifhable as the caufe of motion, as a moving or ac- 
 celerating force. In faort, we know nothing of gravity 
 but the phenomena, which we confider, not as gravity, 
 but as its indication. It is, like every other force — an 
 unknown quality. 
 
 The weight of a body (hould be diftinguifhed from its 
 Gravity or heavinefs, and the term fliould be referved for 
 expreflTing the mcofure of the united gravitation of all the 
 mater in the body. This is indeed the proper fenfe of the 
 term weight— pondus. In ordinary bufmefs, we meafure 
 the weights of bodies by means of knov.-n units of weight. 
 A piece of lead is fiiid to be of twenty pounds weight, 
 r'hen it balances twenty pieces of matter, each of which 
 
is ALL MATTER E<V[TALLY HEAVY. 32I 
 
 is a pound ; but we frequently meafure it by means of 
 other preimres, as when we judge of it by the divifioa 
 to which it draws the fcale of a fpring fteelyard. 
 
 459. We eilimr.te the quantity of matter in a 
 body by its weight, and f.iy t!'?;- there is nineteen 
 tuTies as maich matter in a cubic foot of gold as there 
 is in a cubic foot of water. This evidently prefup- 
 pofes that all matter is heavy, and equally heavy— ti\-\t 
 every primitive atom of matter is -equally heavy, But 
 this feems to be more than we are entitled to fiiy, 
 without fome pofitive proof. There is nothing incon- 
 ceivable or abfurd in fuppofing one atom to be twice or 
 thrice as heavy as another. As- gravity is a contingent 
 quality of matter, its abfolute ftrength or force is alfo 
 contingent and arbitrary. We can conceive an atom to 
 have no v/eight. Nay, we can as clearly conceive an atom 
 of matter to be endowed with a tendency upwards as with 
 a tendency downwards. Accordingly, during the preva- 
 lence of the Stahlian doftrine of combufhion, that mat- 
 ter which imparts inflammability to bodies was fup-- 
 pofed to be not only without weight, but pofitively light, 
 and to diminifn the weight of the other ingredients with 
 which it was combined in a combuftible body. In this 
 way, the abettors of that doctrine accounted for the in- 
 creafe of weight obfervabie when a body is burnt. 
 
 There is nothing abfurd or unreafonable in iiii this ; 
 
 and had we no'other indication of gravity but its pref- 
 
 fure, we do not fee how this quellion can be decided.- 
 
 "Bat gravity is not only a prelTmg povv^er, but alfo a 
 
 S i moving; 
 
^22 PHYSICAL ASTRONOMY. 
 
 moving or accelerating power. If a body confifled of a 
 thoufand atoms of gravitating matter, and as many atoms 
 of matter which does not gravitate, and if the gravity of 
 each atom exerted the prefliire of one grain, this body 
 would weigh a thoufand grains, either by a balance or a 
 fpring fteelyard, yet it contains two thoufand atoms of 
 matter. But take another body of the fame weiglit, but 
 confining wholly of gravitating atoms ; drop thefe two 
 bodies at once from the hand — the laft mentioned will 
 fall 1 6 feet in the hrit fecond — the other will fall only 
 8 feet. For in both there is the fame moving force , 
 therefore the fame quantity of motion will be produced 
 in both bodies ; that is, the produfts of the quantities of 
 matter by the velocities generated will be the fame. 
 Therefore the velocity acquired by the miixed body will 
 be one half of that acquired in the fame time by the 
 fimple body. The phenomenon will be what was alTert- 
 ed, one will fall i6 and the other only 8 feet. 
 
 This will be ftill more forcibly conceived, if we take 
 two bodies a and b, each containing icoo atoms of gravi- 
 tating matter, and attach a to another body r, containing 
 looo atoms which do not gravitate. Now, unlefs we 
 fuppcfe c moveable and arreftable by a thought or a 
 word, we can have no hefitation in faying that the mafs 
 a •{- c will fail with half the velocity of b. 
 
 We fee therefore that the accelerating power alone of 
 gravity enables us to decide the queflion, ' whether all 
 terreilrial matter gravitates,' and gravitates alike. We 
 have only to try whether all terreftrial bodies fall equally 
 
IS ALL MATTER EQUALLY HEAVY. 323 
 
 ■far in the fame time, or receive an equal increment of 
 velocity in the fame time. This teft of the matter did 
 not efcape the penetrating genius of young Newton. 
 He made experiments on every kind of fubltance, me- 
 tals, ftones, woods, grain, falts, animal fubllances, &c. 
 and made them in a way fufceptible of the utmoft ac- 
 curacy, as we fhall fee afterwards. The refult was, that 
 all thefe fubftances were equally accelerated ; and, on 
 this authority, Newton thought himfelf entitled to fay 
 
 that ALL TERRESTRIAL MATTER IS EQUALLY HEAVY. 
 
 This however may be difputed. For it is plain that 
 if all bodies contain a?i equal proportion of gravitating and 
 nongravitating matter, they will be equally accelerated ; 
 nay, the unequal gravitation of different fubftances, and 
 even pofitive levity, may be fo compenfated by the pro- 
 portion of thofe different kinds of matter, that the total 
 gravitation may ftill be proportional to the whole quan- 
 tity of matter. 
 
 But, till we have fome authority for faying that there 
 is a difference in the gravitation of difi'erent atoms, the 
 juft rules of philofophical difcuffion oblige us to believe 
 tliat all gravitate alike. This is corroborated by the uni- 
 verfality of the law of mutual and equal reaction. This 
 is next to demonftration that the primitive atoms are a- 
 like in every refpe(£l, and therefore in their gravitation. 
 
 We are entitle<:l therefore to fay that all terreftrial 
 matter is equally heavy, and that the weight of a body 
 is the meafure of the united gravitation of every atom, 
 and therefore is a meafure of, or is proportional to, the 
 quantity of matter contained in it. 
 
 S f .2 4^0^ 
 
324 PHYSICAL ASTRONOMY. 
 
 460. Newton naturally, and juftly, extended the af- 
 .finnation to the planets and to the Sun. But here arifee 
 a queflion, at once nice and important. The law of 
 gravitation, fo often mentioned,, is exhibited in the mur 
 tual deflexions of great mi; lies of matter. Thefe de- 
 flections are in the inverfe duplicate ratio 01 the diflances 
 between the centres of the maffes. Are we warranted 
 by this obfervation to fay that this is alfo the law of ac- 
 tion between every atom of one body and every atom 
 of another ? .Can we fay in general that the law of cor^ 
 pufcular action is the fame with that of maffes, refult- 
 ing from the combined action of each atom on each ? 
 We are afliired by experience that it is not. For w^e 
 obferve that, in magnets, the law of action (that is, the 
 relation fubfifting between the diftances and the inten- 
 fities of force) is different in almoft every different mag-- 
 net, and feems to depend in a great meafure on their 
 form. 
 
 Newton was too cautious, and too good a logician, 
 to advance fuch a proportion without proof ; and there- 
 fore, confining himfelf to the fmgle cafe of fpherical 
 and fpheroidal bodies, the forms in which we obferve 
 the planetary mafies to be compa(3:ed, he inquired what 
 fenfible action between the maffes will refult from an 
 action between their particles inverfely proportional to 
 the fquare of their diflances. 
 
 Let ALBM, aibm [fig. 49.) be two fpherical fur- 
 faces, of which C is the common centre, and let the 
 fpace between them be filled with gravitating matter^ 
 
 uniformly 
 
GRAVITATION TO A SPHERE. 325 
 
 iHiiformly denfe. Let /> be a particle placed any where 
 within this fpherical fliell, to every particle of which it 
 gravitates with a force inverfely as the fqiiare of its dif- 
 tance from it. This particle will have no tendency to 
 move in any direction, becaufe its gravitation in any one 
 direftion is exa6lly balanced by an equal gravitation in 
 the oppofite dire^lion. 
 
 "Draw through p the two ftraight lines d p t^ ep^, 
 making a veryTmali angle at p. This may reprefent the 
 fection of a very flender "double cone dp e^ ^p g, having 
 p for the common vertex, and d e, ^ & for the diameters 
 of the circular bafes. The gravitation of p to the mat- 
 ter in the bafe a e is equal to its gravitation to the mat- 
 ter in the bafe ^g. For the number of particles In dg 
 is to the number in ^ g as the furface of the bafe d e to 
 that of the bafe ^ £, that is, as de' to ^i^, that is, as /> J* 
 to /^% that is, as the gravitation to a particle in ^g to 
 the gravitation to a particle in d e. Therefore the whole 
 gravitation to the matter in d e is the fame with the 
 whole gravitation to the matter in ^e — fmce it is alfo in 
 the oppofite direftion, the particle p is in equilibrio. 
 The fame thing may be demonftrated of the gravitation 
 to the matter in ^ r and in s /, and, in like manner, of 
 the gravitation to the matter in the fecftions of the cones 
 dp e, ^p i by any other concentric furface. Confequent- 
 ly, the gravitation to the whole matter contained in the 
 folid dqre is equal to the gravitation to the whole mat- 
 ter in the* folid '^tsi, and the particle p is ftill in e- 
 
 Now, 
 
623 PHYSICAL ASTRONOMY. 
 
 Now, fince the lines dp t, ep 3 may be drawn in any 
 dire£lion, and thus be made to occupy the whole fphere, 
 it is evident that the gravitation of p is balanced in every 
 direcftion, and therefore it has no tendency to move in 
 any direction in confequence of this gravitation to the 
 fpherical fliell of matter comprehended between the fur- 
 faces A L B M and alb m. 
 
 It is alfo evident that this holds true with refpe^l; to 
 all the matter comprehended between A L B M and the 
 concentric furface p nv paffing through p ; in fliort, p 
 is in equilibrio in its gravitation to all the matter more 
 remote than itfelf from the centre of the fphere, and ap- 
 pears as if it did not gravitate at all to any matter more 
 remote from the centre. 
 
 461. We have fuppofed the fpherical fliell to be uni- 
 formly denfe. But p will ftill be in equilibrio, although 
 the fliell be made up of concentric flrata of different den- 
 fity, provided that each ftratum be uniformly denfe. For, 
 fhould we fuppofe that, in the fpace comprehended be- 
 tween A L B M and p n v, there occurs a furface a I b 7n 
 of a different denfity from all the reft, the gravitation to 
 the intercepted portions q r and j- t are equal, becaufe 
 thefe portions are of equal denfity, and are proportional 
 to p f and p J* inverfely. The propofition may there- 
 fore be exprefled in the following very general terms. 
 <* A particle placed any where ivithin a fpherical Jhell of 
 ** gravitating matter^ of equal denfity at all equal d fiances 
 ** from the centre y nvill be in equilibrio ^ and vjill have no 
 *' tendenry to move in any direBion. " 
 
 jR.emnrL 
 
GRAVITATION TO A PYRAMID. 327 
 
 Remark — The equality of the gra- Itation to the fur- 
 face ed and to the furface e^ is afRrmed, becaufe the 
 numbers of particles in the two furfaces are inverfely as 
 the gravitations tov^^ards one in each. For the very fame 
 reafon, the gravitations to the furfaces e d, and q r, and 
 / /, are all equal. Hence may be derived an elementary 
 propofition, w^hich is of great ufe in all inquiries of this 
 kind ; — namely, 
 
 462. If a cone or pyramid dp e, of uniform gravi- 
 tating matter, be divided by parallel feftions de, qr^ &c. 
 the gravitation off a particle p in the vertex to each of 
 thofe fections is the fame, and the gravitations to the 
 folids pqr^ pdey q d e r, &c. are proportional to their 
 lengths p q, p d, q d, Sec, The firfi: part of this propo- 
 fition is already demonftrated. Now, conceive the cone 
 to be thus divided into innumerable fiices of equal thick- 
 nefs. It is plain that the gravitation to each of thefe is 
 the fame, and therefore the gravitation to the folid qpr 
 is to the gravitation to the folid q de r as the number of 
 fiices in the firft to the number in the fecond, that is, as 
 p qy the length of the firfb, to q d, the length of the fe- 
 cond. 
 
 The cone dp e was fuppofed extremely llender. This 
 was not neceflary for the demonftration of the particular 
 cafe, where all the fe^lions were parallel. But in this 
 elementary propofition, the angle at p is fuppofed fmaller 
 than any afligned angle, tliat the cone or pyramid may 
 be confidered as one of the elements into vv'hich we may 
 
 refolve 
 
328 PHYSICAL ASTRONOMY. 
 
 refolve a body of any form. In this refolutlon, the b'afes 
 are fuppofed, if not otherwife exprefsly ftated, to be pa- 
 rallel, and perpendicular to the axes j indeed they are 
 fuppofed to be portions x r, y f, 2 s, &c. of fpherical 
 furfaces, having their centr-es in p. The fmall portions 
 9i r q^ y e d, z i ^, &c. are held as infignificant, vanifhing 
 in the ultimate ratios of the whole folids. 
 
 It is eafy alfo to fee that the equilibrium of p is not 
 limited to the cafe of a fpherical fliell, but v/ill hold true 
 of any body compofed of parallel flrata, or ftrata fo form- 
 ed that the lines p cl, p^ are cut in the fame proportion 
 by the feclions d e^ q r, &c. In a fpheroidal fhell, for 
 example, v/hofe inner and outer furfaces are fimilar,*'and 
 {jmiilariy pofited fpheroids, the particle/) will be in equili- 
 brio any where witliin it, becaufe in this cafe, the lines 
 p ^ and ;/ e are equal ; fo are the lines p g and d, the 
 lines / ^ and r e, the lines s i and q d, &c. In moft 
 cafes, however, there is but one fituation of the particle p 
 that will infure this equilibrium. But we may, at the 
 fame time, infer the following very ufeful propofition. 
 
 463. If there be two folids perfecily fimilar^ and of the 
 fame uniform de?fity\ the gravitation to each of thefe folids 
 by a particle fimilarly placed on or in each^ is proportional to 
 any homologous lines of the folids. 
 
 For, the folids being fmiilar, they may be refolved 
 into the fame number of fimilar pyramids fimilarly placed 
 in the folids. The gravitations to each of any corref- 
 ponding pair of pyramids are proportional to the lengths^ 
 
 of 
 
GliAtlTATION TO A SfHEkfi. 329 
 
 of thofe pyramids. Thcfe lengths liave the fame pro- 
 portion in every correfponding pair. Therefore the abfo- 
 late gravitations to the whole pyramids of one fohd has 
 the fame ratio to the abfolute gravitation to the whole 
 pyramids of the other folid. And, fince the folids are 
 fimilar, and the particles are at the fimilarly placed ver- 
 texes of all the fimilar and fimilarly placed pyramids, the 
 gravitation compounded of the abfolute gravitations to 
 the pyramids of one folid has the fame ratio to the gravi- 
 tation fimilarly compounded of the abfolute gravitations 
 to the pyramids of the other. 
 
 464. The gravitation of an external particle to afphf'^ 
 rical furfacey fiell^ or entire fphere^ which is equally denfe 
 at all equal dijlances from the centre^ is the fame as if the 
 luhole matter ivere colleEied in its centre. 
 
 Let A L B M (fig. 49.) reprefent fuch a fphere, and 
 let P be the external particle. Draw P A C B through 
 the centre C of the fphere, and crofs it by L C M at right 
 angles. Draw two right lines P D, P E, containing a very 
 fmall angle at P, and cutting the great circle A L B M 
 in D, E, D', E'. About P as a centre, with the diflancc 
 P C, defcribe the arch Qdm^ cutting DP in ^, and E P 
 In e. About the fame centre defcribe the arc DO. 
 Draw dYy eO parallel to A B, and cutting L C in f 
 and^. Draw CK perpendicular fo P D, and ^H, D^, 
 and Ft^ perpendicular to A B. Join CD and CF. 
 
 N'ow let the figure be fuppofed to turn round the 
 axis P B. The' femicircumference ALB will generate 
 
 Tt a 
 
33^-' PHYSICAL ASTRONOMY. 
 
 a complete fplierlcal furface. The arch C d 7n will ge"-^ 
 nerate another fpherical furface, having P for its centre* 
 The fmall arches D E, d e, Y G will generate rings or 
 zones of thofe fpherical furfaces. D O will alfo gene- 
 rate a zone of a furface having P for its centre, fg and 
 F I wall generate zones of flat circular furfaces. 
 
 It is evident that the zones generated by DE and 
 D O (which we may call the zones D E and DO), hav- 
 ing the fame radius D ?, are to each other as their re- 
 fpe6live breadths D E and D O. In like manner, the 
 zones generated by d <?, fg, F I, F G, being all at the 
 fame defiance from the axis A B, are alfo as their refpec- 
 tive breadths d e^ fg,. F I, F G. But the zone D O is 
 to the zone d e as P D^ to P ^^ For D O is to de as 
 P D to P ^, and the radius of rotation D ^ is to the ra- 
 dius d H, alfo as P D to F d. The circumferences de- 
 fcribed by D O and de are therefore in the fame pro- 
 portion of P D to P d. Therefore the zones, being as 
 their breadths and as their circumferences jointly, are as 
 P D^ and P d\ 
 
 CK and dH, being the fines of the fame arch C^, 
 are equal. Therefore K D and/F, the halves of chords 
 equally diftant from the centre, are alfo equal. There- 
 fore the triangles C D K and C Vf are equal and fimilar. 
 But CDK is fimilar to EDO. For the right angles 
 P D O and C D E are equal. Taking away the common 
 angle C D O, the remainders CDK and EDO are e- 
 qual. In Hke manner, C Ff and G F I are fimilar, and 
 therefore (fince CDK and CF/ are fimilar) the ele- 
 mentary 
 
GRAVITATION TO A SPHERE. 33 J 
 
 rnentary triangles EDO and G F I are fimilar,' and 
 DO:DE = FI:FG. 
 
 The abfolute gravitation or tendency of P to the 
 zone D O is equal to its abfolute gravitation to the zone 
 diy becaufe the number of particles of the firft is to 
 the number in the laft in P D^ to P ^% that is, in- 
 verfely as the gravitation to a particle in the firft to the 
 gravitation to a particle in the lail. Therefore let c 
 exprefs the circumference of a circle whofe radius is i. 
 The furface of the zone generated by D O v^^ill be 
 D O X r X E) ^, and the gravitation to it vi^ill be 
 
 p^ , to which ^rji y 01^ p-Qi 
 
 is equal. This exprefles the abfolute gravitation to the 
 zone generated by D O, this gravitation being exerted in 
 the diredion P D. 
 
 But it is evident that the tendency ^f P, arifmg from 
 its gravitation to every particle in the zone, muft be in 
 the direction P C. The oblique gravitation muft there- 
 fore be eftimated in the dire6lioH jP C, ajid muft (178.) 
 be reduced, in the proportion of P ^ to P H. It is 
 plain that V d '.VH^z d e ifg^ becaufe d e and fg are 
 perpendicular to P ^ and P H. Therefore the reduced 
 or central gravitation of P to the zone generated by D O 
 
 will be exprefled by '^-^^—^-p- -. 
 
 But the gravitation to the zone generated by D O is 
 to the gravitation to the zone generated by D E as DO 
 to D E, that is, as F I (or fg) to F G. Therefore the 
 ^enti'al gravitation to the zone generated by D E will be 
 
 T t 2 exprelTed 
 
332 PHYSICAL ASTRONOMY. 
 
 exprelTed by — —^^L^— . Now F G X f X ^H is 
 
 the value of the furface of the zone generated by F G, 
 And if all this matter were collefted in C, the gravita- 
 
 r r. • 111 ^, F G X r X ^H , . 
 
 tion 01 r to It would be exactly TTpi ' ^"^ '* 
 
 would be in the direction P C. Hence it follows that 
 the central gravitation of P to the zone generated by 
 D E, is the fame as its gravitation to all the m.atter in 
 the zone generated by F G, if that matter were placed 
 in C. 
 
 "What has been demonftrated. refpe£l:ing the arch D E 
 is true of every portion of the circumference. Each has a 
 fubftitute F G, which being placed in the centre C, the 
 gravitation of P is the fame. If P T touch the fphere in 
 T, every portion of the archv TLB will have its fubfti- 
 tute in the quadrant L B, and every part of the arch A T 
 has its fubftitute in the quadrant A T Jj, as is eafily feen. 
 And hence it follows that the gravitation to a particle P 
 to a fpherical furface ALB INI is the fiuiie as if all the 
 matter of that furface were colle£led in its centre. 
 
 "We fee alfo that the gravitation to the furface gene- 
 rated by the rotation of A T round A B is equal to the 
 gravitation to the furface generated by T L B, which ig 
 ^nuch larger, but more remote. 
 
 What we have now demonftrated with refpe£t to the 
 furface generated by the femicircle A L B is equally true 
 with regard to the furface generated by any concentric 
 femicircle, fuch as alb. It is true, therefore, in regard 
 to the ftiell comprehended between thofe furfaces j for 
 
 Ais 
 
GRAVITATION TO A SPHERE. 333 
 
 ihis fhell may be refolved into innumerable concentric 
 ilrata, and the propofition may be alHrmed with rcfpech 
 to each of them, and therefore with refpeft to the whole. 
 And this will dill be true if the whole fphere be thus 
 occupied. 
 
 Laflly, it follows that the propofition is flill true, af- 
 though thofe (trata fliould differ in denfity, provided that 
 each ftratum is uniformly denfe in every part. 
 
 It may therefore be affirmed in the moil general 
 terms, that a particle P, placed without a fphericai fur- 
 face, (liell, or entire fphere, equally denfe at equal dif- 
 tances from the centre, tends to the centre with the fame 
 force as if the whole matter of the furface, fnell, or 
 fphere, were colledled there. 
 
 This will be found to be a very imiportant propofition, 
 greatly affilling us in the explanation of abflrufe pheno- 
 mena in other departments of natural philofophy. 
 
 465. The gravitation cf an external particle to a 
 fphericai furjace^ Jhell^ or entires fphere^ of uniform defi- 
 fity at Cftual diflances from the centre, is as the quantity 
 
 cf matter in that body, direBly, and as the fquare of the 
 d fiance from its centre, inverfely. 
 
 For, if ail the matter were coIle£led in its centre, the 
 gravitation would be the fame, and it would then vary in 
 the inverfe duplicate ratio of the diftance. 
 
 466. Cor. 1 . Particles placed on the furface of fpheres 
 of equal denfity gravitate to the centres of thofe fpheres 
 with forces proportional to the radii of the fpheres. 
 
 For 
 
334 PHtSICAL ASTRONOMY. 
 
 For the quantities of matter are as the cubes of the 
 radii. Therefore the gravitation ^ is as — , that is, as d. 
 This is a particular cafe of Prop. 463. 
 
 467. Cor. 2. The fame thing holds true, if the 
 diftance of the external particles from the centres of the 
 fpheres are as the diameters or radii of the fpheres. 
 
 468. Cor. 3. If a particle be placed within the fur- 
 face of a fphere of uniform denfity, its gravitation, at 
 different diftances from the centre, w\\\ be as thofe dif- 
 tances. For it will not be affe£led by any matter of the 
 fphere that is more remote from the centre (463.) ; and 
 its gravitation to what is lefs remote is as its diftance 
 
 from the centre, by the laft corollary. 
 
 469. ^he mutual gravitation of tiuo fpheres of un form 
 denfity in their concentric frata is in the inverfe duplicate 
 ratio of the diftance hetiveen their centres. 
 
 For the gravitation of each particle in the fphere A 
 to the fphere B is the fame as if all the matter in B 
 were coUedled at its centre. Suppofe it fo placed. The 
 gravitation of B to A will be the fame as if all the mat- 
 ter in A were collected in its centre. Therefore it will 
 be as d"- inverfely. But the gravitation of A to B i^ 
 equal to that of B to A. Therefore, &c. 
 
 470. The abfolute gravitation of two fpheres whofe 
 quantities of matter are a and b^ and d the diftance of 
 
 their 
 
UNIVERSAL GRAVITATION. 335. 
 
 their centres, is — .— . For the tendency of one par- 
 ticle of a to /', being the aggregate of its tendencies to 
 every particle of hy is -j^. Therefore the tendency of 
 
 the whole of « to 5 muft be —rr • -^^^^ ^^ tendency 
 oi b "io a is equal to that of a to b. 
 
 471. This confequence of 'a mutual gravitation be- 
 tween particles proportional to — > is agreeable to what 
 
 is obfeiTed in the folar fyftem. The planets are very 
 nearly fpherical, and they are obferved to gravitate mu- 
 tually in this proportion of the diitance between their 
 centres. This mutual aftion of two fpheres could not 
 refult from any other law of a<£lion between the par- 
 ticles. Therefore we conclude that the particles of gra- 
 vitating matter of which the planets are formed gravi- 
 tate to each other according to this law, and that the 
 obferved gravitation of the planets is the united efFe6b 
 of the gravitation of each particle to each. There is 
 juft one other cafe, in which the law of corpufcular ac- 
 tion is the fame with the law of aftion between the 
 maffes ; and this is when the mutual aci:ion of the cor- 
 pufcles is as their diftance direftly. But no fuch law is 
 obferved in all the phenomena of nature. 
 
 The general inference drawn by Sir Ifaac Newton 
 from the phenomena, may be thus exprefled : Every par- 
 ticle of matter gravitates to every ether particle of matter 
 
 vj'ith 
 
33^ PHYSICAL ASTRONOMT. 
 
 with a force inverfely proportional to the fqunre of the dif- 
 tance from It. Hence this do6lrine has been called the 
 
 DOCTRINE OF UNIVERSAL GRAVITATION. 
 
 The defcriptlon of a conic feclion round the focus 
 fully proves that this law of tlie diftances is the law 
 competent to all the gravitating particles. But, whether 
 all particles gravitate, and gravitate alike, is not demon- 
 ftrated. The analogy between the diftance of the dif-^ 
 ferent planets and their periodic times only proves that 
 the total gravitation of the different planets is in the 
 fame proportion with their quantity of matter. For the 
 force obferved by us, and found to be in the inverfe 
 duplicate ratio of the diftance of the planet, is the ac- 
 celerating force of gravity, being meafured by the ac» 
 celeration which it produces in the different planets* 
 But If one half of a planet be matter which does not 
 gravitate, and the other half gravitates twice as much as 
 the matter of another planet, thefe two planets will (till 
 have their periods and diftances agreeable to Kepler's 
 third law. But, fmce no phenomenon indicates any In- 
 equality In the gravitation of different fubftances, it is 
 proper to admit its perfect: equality, and to conclude with 
 Sir Ifaac Nev/ton. 
 
 472. The general confequence of this do£lrine 1% 
 that any two bodies, at perfect liberty to move, fliould 
 approach each other. This may be made the fubjecl of 
 experiment, in order to fee whether the mutual ten- 
 dencies of the planets arife fron; that of their particles. 
 
 For 
 
UNIVERSAL GRAVITATION. 337 
 
 "For it muft flill be remembered that although this con- 
 llitution of the particles will produce this appearance, it 
 may arife from fome other caufe. 
 
 . Such experiments have accordingly been made. Bo- 
 Hies have been fufpended very nicely, and they have b§en 
 obferved to approach each other. But a more careful 
 examination of all circumftanees has fliewn that mod of 
 iliofe mutual approaches have arifen from other caufes; 
 Several philofophers of reputation have therefore refufed 
 to admit a mutual gravitation as a phenomenon compe- 
 tent to all matter. 
 
 But no fuch approach fliould be obferved in the ex- 
 periments now alluded to: The mutual approach of two 
 fpheres A and B, at the diftance D Of their centres, 
 muft be to the approach to the Earth E at the dillaiKe d 
 
 A X B A X E 
 
 of their centres in the proportion of — T^r"" ^° — j z — > 
 
 B E :• . 
 
 that is, of c^ to -^— . Therefore, if a particle be placed 
 
 at the furface of a golden fphere one foot in diameter,' 
 its gravitation to the Earth muft be more than ten mil- 
 lions of times greater than its graLvitation to the gold. 
 For the diameter of the Earth is nearly forty millions of 
 feet, and the denfity of gold is nearly four times the 
 mean denfity of the Earth. And therefore, in a fecond, 
 it would approach lefs than the ten millionth part of id 
 feet — a quantity altogether infenfible. 
 
 If we could employ in thefe experiments bodies of 
 
 fufhcient magnitude, a fenfible efFed might be expeded; 
 
 Suppofe T (fig. 50.) to be a ball of equal denfity with 
 
 U u the 
 
^•^a PHYSICAL ASTRONOMY. 
 
 the Earth, and two geographical miles in diameter, and 
 let the particle B be at its furface. Its gravity to T will 
 be to its gravitation to the Earth nearly as i to 2300, 
 and therefore, if fufpended like a plummet, it would cer- 
 tainly deviate i' from the perpendicular. A mountain 
 two miles high, and hemifpherical, rifmg in a level 
 country, would produce the fame deviation of the 
 plummet. 
 
 474. Accordingly, fuch deviation of a plumb line 
 has been obferved. Firft by the French academicians- 
 employed to m.eafure a degree of the meridian in Peru. 
 Having placed their obfervatories on the north ai*d fouth- 
 fides of the vail moufitain Chimboracao, they found 
 that the plummets of their quadrants were deflected to- 
 ward the mountain. Of this they could accurately judge, 
 by means of the ftars wliich they faw through the te- 
 lefcope of their quadrant, when they were pointed ver- 
 tically by means of the plummet. 
 
 Thus, if the plummets take the pofitlons A B, CD 
 (fig. 51.), inftead of hanging in the verticals AF and 
 C H, a ftar I, will feem to have the zenith diftances e I, 
 1; I, inftead of E I, G I, which it ought to have ; and the 
 diftance F H on the Earth's furface will feem the mea«* 
 fiire of the difference of latitude eg, whereas it corre- 
 fponds to E G. The meafure of a degree including the 
 fpace F H, and eftiraated by the declination of a ftar I, 
 will be too ihort, and the meafure of a degree termin- 
 ating either at F or H will be- too long, when the fpace 
 FH i« excluded. 
 
 Confiderablc 
 
UNIVERSAL GRAVITATION. 339 
 
 Confiderable doubts remaining aii to the inferences 
 <irawn from this obfervation, the philofophers were very 
 defirous of having it repeated. For this reafon, our 
 Sovereign, George III., ever zealous to promote true 
 feience, fent the Royal aftronomer Dr Maficelyne to Scot- 
 land, to make this experiment on the north and fouth 
 fides of Shihallien, a lofty and folid mountain in Perth- 
 fhire. The deviation toward the mountain on each fide 
 exceeded 7" *, thus confirming, beyond doubt, the noble 
 difcovery of our illuftrious countryman. 
 
 Perhaps a very fenfible efFed: might be obferved at 
 Annapolis-Royal in Nova Scotia, from the vaft addition 
 of matter brought on the coaft twice every day by the 
 tides. . The water rifes there above a hundred feet at 
 fpring-tide. If a leaden pipe, a few hundred feet long, 
 were laid on the level beach at right angles with the 
 coaft, and a glafs pipe fet upright at each end, and the 
 whole filled with water j the water will rife at the 
 outer end, and fink at the end next the land, as the tide 
 rifcs. Such an alternate change of level would give the 
 moft fatisfa^tory evidence. Perhaps the effe6t might be 
 fenfible on a very long plummet, or even a nice fpirit 
 kvel. 
 
 475. A very fine and fatisfa£tory examination was 
 made in 1788 by Mr H. Cavendifli. Two leaden balls 
 were faftened to the ends of a flender deal rod, which 
 was fufpended horizontally at its middle by a fine wire. 
 This arm, after ofcillating fome time horizontally by the 
 U U 2 twifting 
 
34<^ VHISICAL ASTRONOMY. 
 
 twifting and untwifting of the wire, came to reft in a 
 certain pofition. Two great mafles of*Icad were now 
 brought within a proper diitance of the two fufpended 
 balls, and their approach produced a deviation of the 
 arms from the points of reft. By the extent of this de- 
 viation, and by the times of the ofcillations when the 
 great mafles were withdrawn, the proportion was dif- 
 covered between the elafticity of the wire and the gra- 
 vitation of the balls to the great mafles ; and a medium 
 of all the obfervations was taken. 
 
 By thefe experiments, the mutual gravitation of 
 terreftrial matter, even at confiderable diftances, was moft 
 evincingly demonftrated 5 and it was legitimately deduced 
 from them that the medium denfity of the Earth was 
 more than five times the denfity of water. Thefe cu- 
 rious and valuable experiments are narrated in the Phi» 
 iofophical Tranfa^lions for 1798. 
 
 476. The oblate form of the Earth alfo afl^ords an- 
 other proof that gravity is directed, not to any Angular 
 point within the Earth, but that its direftion is the com- 
 bined cffc€k of a gravitation to every particle of matter. 
 Were gravity direded to the centre, by any peculiar 
 virtue of that point, then, as the rotation takes away 
 T--^ of the gravity at the equator, the equatorial parts of 
 a fluid fphere muft rife one half of this, or yi-g-, before 
 all is in equilibrio. 
 
 For, fuppofe CN and CQ^(fig. 33.) to be two canals 
 Teaching from ^e pole and from the equator to th^ 
 
 centre. 
 
UNIVERSAL GRAVITATION. 34| 
 
 tentre. Since the diminution of gravity at Q^is ob- 
 ferved to be ^-^^, and the gravitation of every particle in 
 CQ^is diminiflied by rotation in proportion to its dif- 
 tance from the axis of rotation, the diminution occasion- 
 ed in the w^eight of the v^rhole canal v^riU be one half of 
 the diminution it would fuftain if the vircight of every 
 particle were as much diminifhed as that of the particle 
 Q^is. Therefore the canal preffes lefs on the centre bv 
 -j-4t> ^"^ mull be lengthened fo much before it will ba-. 
 lance N C, which fuftains no diminution of weight. 
 Every other canal parallel to C Q^Cuftains a fimilar lofs 
 of weight, and muft be fimilarly compenfated. This will 
 produce an elliptical fpheroidal form. 
 
 But the equatoreal parts of our globe ^re much more 
 elevated than this ; not lefs than ^\-^. The reafon is 
 this. When the rotation of the Earth has raifed the 
 equatoreal points ^4^, the plummet, which at a (fig. 33,) 
 would have hung in tlie direction a D, tangent to the 
 evoiute A B D F, i$ attracted fidewife by the protuberant 
 matter toward the equator. But the furface of the ocean 
 muft ftill be fuch that the plummet is perpendicular to 
 it. Tlierefore it cannot retain the elliptical form pro- 
 duced by the rotation alone, but fwells ftill more at the 
 equator j and this ftill increafes the deviation of the 
 plummet. This muft go on, till a new equilibrium is 
 produced by a new figure. This will be confidered af- 
 terwards. No more is mentioned at prefent than what 
 is neceirary for fhewing that the protuberance produced 
 by the rotation caufcs, by its attriidion, the plummet to 
 
 deviate 
 
342 PHYSICAL ASTRONOMY. 
 
 deviate from the pofitlon which it had acquired in coHif 
 fequence of the fame rotation. 
 
 477. By fuch indu(f^ion, and fuch reafoning, is efta- 
 blifhed the do6lrine of univerfal gravitation, a doctrine 
 which is placed beyond the reach of controverfy, and 
 has immortalized the fame of its illufhrious inventor. 
 
 Sir Ifaac Newton has been fuppofed by many to have 
 afligned this mutual gravitation, or, as he fometimes calls 
 it, this attraction, as a property inherent in matter, and 
 as the caufe of the celeftial phenomena ; and for this rea- 
 fon, he has been accufed of introducing the occult qua-. 
 lities of the peripatetics into philofophy. Nay, many ac- 
 cufe him of introducing into philofophy a manifell ab^ 
 furdity, namely, that a body can aft where it is not pre- 
 fent. This, they fay, is equivalent with faying that the 
 Sun attracts the planets, or that any body aCls on ano- 
 ther that is at a diflance from it. 
 
 Both of thofe accufations are unjuft. Newton, in no 
 place of that work which contains the do£lrine of uni- 
 verfal gravitation, that is, in his Mathematical Principles 
 of Natural Philofophy ^ attempts to explain the general phe- 
 nomena of the folar fyftem from the principle of univer- 
 fal gravitation. On the contrary, it is in thofe general 
 phenomena that he difcovers it. The only difcovery tq 
 which he profefles to have any claim is, i/?, the matter 
 of faft, that every body in the folar fyflem is continually 
 defleCled toward every other body in it, and that the 
 defle£lion of any individual body A toward any other 
 
 body 
 
GRAVITATION NOT AN OCCULT C^IALITY. 343 
 
 bo<iy B is obftrvcd to be in the proportion of the quan- 
 tity of matter in B diredly, and of the fquare of the dif- 
 tance A B inverfely ; and, 2dlyy that the falling of ter- 
 reftrial bodies is juil a particular example of this univer- 
 fal defledion. He empJoys this difcovery to explain phe- 
 nomena that are more particular 5 and all the explanation 
 that he gives of thefe is the ihev/jng that they are modi- 
 fied cafes of this general phenomenon, of which he knows 
 no explanation but the mere defcription. Newton was 
 not more eminent for mathematical genius, and pene- 
 trating judgement, than for logical accuracy. He ufes 
 the word gravitation as the expreflion, not of a quality, 
 but of a fac^ j not of a caufe, but of an event. Having 
 eftabliihed this fa61: beyond the power of controverfy, 
 by an induction fufficiently copious, nay without a fingle 
 exception, he explains the more particular phenomena, by 
 iliewing with what modifications, arifing from the cir- 
 cumllances of the cafe, they are included in the general 
 fact of mutual deflexion ; and, finally^ as all changes 
 of motion are conceived by us as the effe£ls of force, he 
 fays that there is a deflecting force continually acting on 
 every particle of matter in the folar fyftem, and that thifr 
 deflecting force is what we call weight, heavinefs. Few 
 perfons think themfelves chargeable with abfurdity, or 
 with the abetting of occult qualities, when they reallv 
 confider the heavinefs of a body as one of its properties. 
 So far from being occult, it feems one of the moil ma- 
 nifeft. It is not the heavinefs of this body that is the 
 occult quality ; it is the caufe of this heavinefs. In thus. 
 
 confiderin^. 
 
744 PHYSICAL AsT!lONO:VlV. 
 
 confidering gravity as competent to all matter, Newton 
 does nothing that is not done by others, when they afcribe 
 impulllvenefs or inertia to matter. Without fcruple, they 
 fay that impulfivenefs is an univerfal property of matter. 
 Impulfivenefs and heavinefs are on precifely the fame 
 footing — mere phenomena ; and the moft general pheno- 
 mena, that we knov/. We know none more general than 
 impulfivenefs, fo as to include it, and thus enable us to 
 explain it. Nor do we know any that includes the phe- 
 nomena of univerfal defleclion, with all the modifications 
 of the heavinefs of matter. Whether one of thefe can 
 explain the other is a different queftion, and will be con- 
 fidered on another oCcafion, when we fnall fee with how 
 little juftice philofophers have refufed all a6tion at a dif-- 
 fance. 
 
 But it would feem that there Is fome peculiarity iri 
 this explanation of the planetary motions which hinder^' 
 it from giving entire fatisfaclion to the mind. If this bef 
 the cafe, it is principally Owing to miftake ; to carelefsly 
 imputing to Newton views which he did not entertain. 
 His do(ftrine of univerfal gravitation dofes not attempt to 
 explain /jQiu the operating caufe retards the Moon's mo- 
 tion in the nrft and third quarters of a lunation ; it merely 
 narrates in what direction, and with what velocity this 
 change is produced ; or rather, it fhews how the Moon's 
 defledion toward the Earth, joined to her deflexion to-' 
 Ward the Sun, both of which are matters of fa6l, confti- 
 tute this /ee?ni/?g irregularity of motion which we confider 
 2S a difturbance. But with refpedi to the operating- caufe 
 
MEvUHANICAL EXPLANATIONS OF GRAVITY. 345 
 
 of this general defle£lion, and the manner in which it 
 produces its efFeil, fo as to explain that etFe£l, Newton 
 is altogether filent. He was as anxious- as atiy p'erfori 
 not to be thought to afcribe inherent gravity to matter, 
 or to aiTerl: that a body could a(St on another at a diftance, 
 without foHie mechanical intervention. In a letter to 
 Dr Bentley he exprefles this anxiety in the ftrongeft 
 terms. It is difficult to know Newton's precife meaning 
 by the word aciion. In very ftri^l language,, it is nbfurd 
 to fay that matter a£l:s at all, — in contacft, or at a diftance. 
 But, if one ihould afTert that the condition of a particle a 
 cannot depend on another particle b at a diftance from 
 it, hardly any perfon v/ill fay that he makes this aflertion 
 from a clear perception of the abfurdity of the con- 
 trary proportion. Should a perfon fay that the mere 
 prefence of the particle b is a fufiicient reafon for a ap- 
 proaching it, it v/ill be difficult to prove the aiTertion to 
 be abfurd. 
 
 478. Such, however, has been the general opinion 
 of philofophers ; and numberlefs attempts have been 
 made to thruft in fome material agent iA all the cafes of 
 feeming a6lion at a diftance. Hence the hypothefes of 
 magnetical and electrical atmofpheres ; hence the vor- 
 texes of Des Cartes, and the celeftial machinery of Eu- 
 doxus and Callippus. 
 
 Of all thofe attempts, perhaps the moft rafh and un- 
 
 juftifiable is. that of Leibnitz, pilbllfhed in the Leipzig 
 
 A£\:s i6a9, two years after the publication of Newton's 
 
 X y. Piincipia, 
 
34^ l^KYSICAL ASTRONOMT. 
 
 Principla, and of the review of it in thofe very a£tS', 
 It may be called rafli, becaufe it trufted too much to the 
 deference which his own countrymen had hitherto fliewn 
 for his opinions. In this attempt to account for the el- 
 liptical motion of the planets, Leibnitz pays no regard 
 to the acknowledged laws of motion. He affumes as 
 principles of explanation, motions totally repugnant to 
 thofe laws, and motions and tendencies incongruous and 
 contradiftory to each other. And then, by the help of 
 geometrical and analytical errors, which compenfate each 
 other, he makes out a ftrange co-nclufion, which he calls 
 a demonflration of the law of planetary gravitation ; 
 and fays that he fees that this theorem is known to Mr 
 Newton, but that he cannot tell how he has arrived 
 at the knowledge of it. This is fomething very remark- 
 able. Newton's procefs is fufficiently pointed out in the 
 ^Ba Eruditoruniy which M. Leibnitz acknowledges that 
 he had feen. A copy of the Pri/icipia was fent to him, 
 by order of the Royal Society, in lefs than two months 
 after the publication. — It was foon known over all Eu- 
 rope. 
 
 It is without the leafl foundation that the partifans 
 of M. Leibnitz give him any fliare in the difcovery of the 
 law of gravitation. None of them has ventured to quote 
 this diflertation as a propofition juftly proved, nor to de- 
 fend it againfl: the obje£lions of Dr Gregory and Dr 
 Keill. M. Leibnitz's remarks on Dr Gregory's criti- 
 cifm were not admitted into the Acia JSruditorum, though 
 under the management of his particular friends. In Oc- 
 tober 
 
HYPOTHESES OF LEIBNITZ AND 'NEWTON. 347 
 
 tober 1706 they inferted an extract from a letter, con- 
 taining fome of tliofe remarks ; — if poflible, they are more 
 abfurd and incongruous than the original diflertation. 
 
 It is worth while, as a piece of amufement, to read 
 the account of this diflertation by Dr Gregory in his 
 Aftronomy, and the obfervations by Dr Keill in the 
 Journal Liter aire de la Hayey Auguft 1 7 14» 
 
 479. Sir Ifaac Newton has alfo fliewn fome difpo- 
 fition to account for the planetary defleftion by the ac- 
 tion of an elaftic asther. The general notion of the at- 
 tempt; is this. The fpace occupied by the folar fyftem 
 is fuppofed to be filled with an elaftic fluid, incompa- 
 rably more fubtile and more elaftic than our air. It is 
 fuppofed to be of greater and greater deniity as we re- 
 cede from the Sun, and in general, from all bodies. In 
 confequence of this, Newton thinks that a planet placed 
 any where in it will be impelled from a denfer into a 
 rarer part of th^ jether, and in this manner have its 
 courfe incurvated toward the Sun. 
 
 But, without making any remarks on the impoflibility 
 of conceiving this operation with any diftin^tnefs that 
 can entitle the hypothefis to be called an explanation^ it 
 need only be obferved that it is, in its firft conception, 
 quite unfit for anfwering the very purpofe for vi^hich it 
 is employed, namely, to avoid the abfurdity of bodies 
 acting on others at a diftance. For, unlefs this be al- 
 lowed^ an tether of different denfity and elafticity in its 
 'different ftrata cannot exift. It m.uft either be uniform^ 
 
348 PHYSICAL ASTHONOMY. 
 
 Jy dcnfe and elaftlc throughout, or there mufl exift a 
 repulfive force operating between very diflant particles — 
 perhaps extending its influence as far as the folar influr 
 ence extends — nay, elafticity without an acli.on e, difanti, 
 even between the adjoining' particles, is inconceivable. 
 What is meant by elafticity ? Surely fuch a conftitution 
 of the aficmblage of particles as makes them recede 
 from each other j and the abfurdity is as great at the dif- 
 tance of the millionth part of a hair^s breadth as at the 
 diftance of a milHon of leagues. If we attempt to evade 
 this, by faying that the particles are \\\ contaft, and are 
 elaftic, we mull grant that they are comprelTible, and 
 are really comprelTed, otherwife they are not exerting 
 any elaftic force; therefore they are dimpled, and can 
 no more conftitute a fluid than fo many blov/n bladders 
 compreiTed in a box. 
 
 The laft attempt of this kind that fiiall be mentioned 
 is that of M. Le Sage of Geneva, put into a better Ihape 
 by M. Prevot, in a Memoir publilhed by the Academy 
 cf Berlin, under the name of Lucrece Neutonien. This 
 philofopher fuppofes that through every point of fpace 
 there is continually paiTmg a ftream of azther in ^vcry 
 direction, with immenfe rapidity. This will pvoduce no 
 eifecl: on a folitary body ; but if there are two, one of 
 them intercepts part of the ftream which would have 
 a£ted on the other. Therefore the bodies, being lefs 
 impelled on that fide which faces the other ^ will move 
 toward each other. Le Sage adds feme circumftances 
 refpeftjng the ftrudure of the bodies, which may give a 
 
INUTILITY OF HYPOTHESES. 349 
 
 iort of progrefTion in the intenfity of the impulfe, which 
 may produce a deileclion diminifliing 2.5 the diftance or 
 its • fquare increafes. But this hypothefis alfo requires 
 that we make light of the acknowledged law^s of motion, 
 it has other infuperable difliculties, and, fo far from af- 
 fording any explanation of the planetary motions, its moll 
 trifling circumflance is incomparably m.ore difficult to 
 comprehend, or even to conceive, than the moil intricate 
 phenomenon in aflronomy. * ■ 
 
 481. Indeed this difficulty obtains in every attempt 
 .of- the kind, it being neceilary to confider the combined 
 motion of millions of bodies, in order to explain the mo- 
 tion of one. But fuch hypothefes have a worfe fault 
 than their difficulty ; they tranfgrefs a great rule of phi- 
 lofophical difquifition, " never to admit as the caufe of 
 " a phenomenon any thing of which w^e do not know 
 ** the exiftence. " For, even if the legitimate confe- 
 quences of the hypothefis were agreeable to the pheno- 
 mena, this only flicws the poffibUHy of the theory, but 
 gives no explanation whatever. The hypothefis i« good, 
 only as far as it agrees with the phenomena ; we there- 
 fore underftand the phenomena "as far as we underlland 
 the explanation. The obferved laws of the phenomena 
 are as extenfive as our explanation, and the hypothefis is 
 ufelefs. But, alas, none of thofe hypothefes agree, in 
 their legitimate confequences, with the phenomena ; tlie^ 
 laws of motion mufl be thrown afide, in order to employ 
 them, and new laws muft be adopted. ■ This is unwife ; 
 
 it 
 
35© PHYSICAL ASTRONOAIY. 
 
 it were much better to give thofe pro re nata laws to 
 the planets themfelves. 
 
 Mr Cotes, a philofopher ^nd geometer of the firfl c- 
 minence, wrote a preface to the fecond edition of the 
 Principia, which was publiflied in 1 7 1 3 with many al- 
 terations and improvements by the author. In this pre- 
 face Mr Cotes gives an excellent account of the prin- 
 ciples of the Newtonian philofophy, and many very per- 
 tinent remarks on the maxim which made philofophers 
 fo adverfe to the admifTion of attracting and repelling 
 forces. Whatever may have been Newton's fentiments 
 in early life about the competency of an elaftic xther 
 to account for the planetary deflediionsj he certainly put 
 little value on it afterwards. For he never made any 
 ferious vSe of it for the explanation of any phenomenon 
 fufceptible of mathematical difcufTion. He had certainly 
 reje£led all fuch hypothefes, otherwife he never would 
 have permitted Mr Pemberton to prefix that preface of 
 Mr Cotes to an edition carried on under his own eye. 
 For in this preface the abfurdity of the hypothefis of an 
 ^elaftic xther is completely expofed, and it is declared to 
 be a contrivance altogether unworthy of a philofopher. 
 Yet, when Mr Cotes dietl foon after, Sir Ifaac Newton 
 fpoke of him in terms of the higheft refpecV. Alas, faid 
 he, rve have lojl Mr Cotes ; had he livedo ive Jhou/d foon 
 have learned Jcmething excellent^ 
 
 At prefent the mofl eminent philofophers and mathe- 
 maticians in Europe profefs the opinion of Mr Cotes, 
 a^d fee no validity in the philofophical maxim that bodies 
 
 cannot 
 
rs GRAVITY AN INHERENT PROPERTY ? 3,51 
 
 cannot acl at a diftance. M. dc la Place, the oxceller>t 
 commentator of Newton, and who has given the finifh- 
 ing llroke to the univerfallty of the influence of gravita- 
 tion on the planetary motions, by explaining, by this 
 principle, the fecular equation of the Moon, which had 
 reiifled the efforts of all the mathematicians, endeavours, 
 on the contrary, to prove that an action in the inverfe 
 duplicate ratio of the diftances refults from the very ef- 
 fence or exiftence of matter. Some remarks will be 
 made on this attempt of M. de la Place afterwards. But 
 at prefent we fliall find it much more conducive to our 
 purpofe to avoid altogether this metaphyfical queftiorr, 
 and ftritlly to follow the example of our illuftrious In- 
 ftru£lor, who clearly faw its abfolute infignificance for 
 increafing our knowledge of Nature. 
 
 Newton faw that any inquiry into the marjier of aEl^ 
 ing of the efficient caufe of the planetary deflections W25 
 altogether unneceffiry for acquiring a complete know- 
 ledge of all the phenomena depending on the law which 
 he had fo happily difcovered. Such was its perfedi fim- 
 plicity, that we wanted nothing but the affurance of its 
 conftancy — an affurance eftablifhed on the exquifite a- 
 greement of phenomena with every legitimate deduiSLioii 
 from the law. 
 
 Even Newton's perfpicacious mind did not fee the 
 number of important phenomena that were complete- 
 ly explained by it, and he thought that fome would 
 be found which required the admiffion of otlier prin- 
 ciples.' But the firft mathematicians of Eurppe haveac- 
 
 c^uired 
 
^52 PHYGICAt ASTRONOMY. 
 
 quired moil defcrved fame in the cultivation of this plii- 
 lofophy, and in their progrefs have found that there is 
 TxOt one appearance in the celeftial motions that is in- 
 confiftent with the Nevv- tonian law, and fcarcely a phe- 
 nomenon that requires any thing elfc for its complete ex- 
 planation. 
 
 Hitherto we'have been employed in the eftablifliment- 
 of a general law. AVe are now to fhew how the mo- 
 tions a6lualiy obferved in the individual members of the 
 folar fydem refult from, or are examples of the opera- 
 tion of the power called Gravity, and how its effe^ls 
 are modified, and made what we behold, by the circum- 
 ftances of the cafe. — To do this in detail would occupy 
 many volumes *, we muft content ourfelves with ad- 
 ducing one or two of the moft interefting examples. 
 The ftudent in this noble department af mechanical phi- 
 lofophy wjil derive great aififtance from Air Ai^Laurhi's 
 Accoufit of Sir Ifaac Neivton^s Dtfcoveries. Dr Pember' 
 torCs Vieiu of the Newtonian Philofophy has alfo confider- 
 able merit, an<l is peculiarly fitted for thofe \yho are lef* 
 habituated to mathematical difcuflion. The Cofmographia 
 of the Abbe Frifi is one of the moft valuable works extant 
 on this fubje<3:. This author gives a very compendious, 
 yet a clear and perfpicuous account of the Newtonian 
 doftrines, and of all the improvemients in the manner 
 of treating them which have refulted from the unremit- 
 ting labour of the great mathematicians in their afliduous. 
 cultivation of the Newtonian philofophy. He follows, 
 in general, the geometrical method, and his geometry is. 
 
 elegant^- 
 
TrtEORY O^ THE CELESTIAL MOTIONS. 353 
 
 efiegarit, and yet he exhibits (alfo with great neatnefs) all 
 the noted analytical procelTes by which this philofophy 
 has been brought into its prefent (late. 
 
 What now follows may be called an outline of 
 
 The Theory of the Celejilal Motions. 
 
 482. The firft general remark that arifes from th^ 
 eftabliiliment of univerfal and mutual gravitation is 
 that the common centre of the whole fyftem is not af- 
 fe£i:ed by it, and is either at refl, or, if in motion, this 
 thotion is produced by a force which is external to the 
 fyftem (98.)? and a£is equally and in the fame diredlioii, 
 on every body of the fyftem (229.) 
 
 483. A force has been difcovered pervading the 
 ^hoie fyftem, and determining or regulating the motions 
 of every individual body in it. The problem which na- 
 turally offers itfelf firft to our difcuflion is, to afcertain 
 *what will be the motion cf a bodyy projeBed from any given 
 point of the folar fyjleniy in any particular direBion, and ivith 
 any particular velocity — nvhat ivill he the form of its path^ 
 hoiv ivill it move in this path, and where will it be at any 
 infant we choofe to name* 
 
 Sir Ifaac has given, in the 41ft propofition of his firft 
 book, the folution of this problem, in the moft general 
 tetms, not limited to the obferved law of gravitation, but 
 eJEl^dcd to any conceivable relation between the dif- 
 
 Y y tances 
 
J54 FHYSICAL ASTRONO^tY. 
 
 tances and the intenfity of the force. This is, unquef- 
 tionably, the moft fubhme problem that can be propofed 
 in mechanical philofophy, and is well known by the name 
 
 of tlie INVERSE PROBLEM OF CENTRIPETAL FORCES. 
 
 But, in this extent, it is a problem of pure dynamics, 
 and does not make a part of phyfical ailronomy. Our 
 attention is limited to the centripetal force which con- 
 nects this part of the creation of God — a force inverfely 
 proportional to the fquare of the diftances. * It may be 
 Hated as follows. 
 
 Let a body P, (fi^. 52.) which gravitates to the Sun 
 in S, be proje6led in the diredlion P N, with the velo- 
 city which the gravitation at P to the Sun would gene- 
 rate in it by impelling it along P T, lefs than PS. 
 
 Draw P Q perpendicular to P N. Take P O equal 
 to twice P T, and draw O Q perpendicular to P Q, and 
 QR perpendicular to PS. Alfo draw P/, making the 
 ^"gl^ Q P -^ equal to Q P S. Join S Q, and produce 
 S Q till it meet P / in j. 
 
 The body will defcribe an ellipfis, which P N touches 
 in P, whofe foci are S and j-, and whofe principal para- 
 meter is twice P R. 
 
 For, draw S N perpendicular to P N, Make P O' = 
 2 P O or = 4 P T, and draw O' Q' perpendicular to 
 P O', and defcribe a circle pafling through P, O' and Q'. 
 It will touch P N, becaufe P O' Q' was made a right 
 angle, and therefore P Q' is the diameter of the circle. 
 
 We know that an ellipfe may be defcribed by a body 
 influenced by gravitation. This ellipfe may have S and x 
 
 for 
 
THEORY OF ELLIPTICAL MOTION. 355 
 
 for its foci, and PN for a tangent in P, becaufc the 
 angles are equal which P N makes with the two focal 
 lines. This being the cafe, we know that if P Q, O Q, 
 and Q R be drawn as directed in the foregoing conftruc- 
 tion, P O' Q' is the circle which has the fame curvature 
 with the ellipfe in P, wh©fe foci are S and /, and tangent 
 P N, and P T is 4 of the chord of curvature in P, and 
 P R is half the parameter of the ellipfe. Therefore 
 (212.) PT is the fpace along which the body muft be 
 uniformly impelled by the force in P, that it may ac^ 
 quire the velocity with which the body, a<^ually defcrib- 
 ing this ellipfe, palles through P. If this body, which 
 we fhall call A, thus revolves in an ellipfe, we fhould in- 
 fer that it is deflected toward S, by a force inverfely 
 proportional to the fquare of its diftance from S, and of 
 fuch magnitude in P, that it would generate the velocity 
 with which the body pafles through P, by uniformly im- 
 pelling it along P T. 
 
 Now, the other body (which we fliall call P) was ac- 
 tually projected in the direction P N, that is, in the di- 
 rection of A's motion, with the very velocity with which 
 A paffes through P in the fame direction, and it is un- 
 der the influence of a force precifely the fame that muft 
 have influenced A in the fame place. The two bodies 
 A and P are therefore in precifely the fame mechanical 
 condition ; in the fame place 5 moving in the fame direc- 
 tion ; with the fame velocity ; deflected by the fame in- 
 tenfity of force, acting in the fame direction. Their 
 pactions in the next moment cannot be different, and 
 Y y 2 t]iey 
 
35^ PHYSICAL ASTRONOMY, 
 
 they muft, at the end of the moment, be again in th| 
 fame condition j and this mud continue. A defcribes a 
 certain ellipfe ; P muft defcribe the fame ; for two mor 
 tions that are different cannot refult frpni thp fame force 
 aifting in the fame circumftances. 
 
 484. This demonflration is given by Sir Ifaac New- 
 ton in four lines, as a corollary from the proposition in 
 which he deduces the law of planetary defle£lion from 
 the motion in a conic feOiion. But it feemed neceifary 
 here to expand his procefs of reafoning a little, becaufe 
 the validity of the inference has been denied by Mr John 
 Bernoulli, one of the firft mathematicians of that age. 
 He even hinted that Nev/ton had taken that illogical me- 
 thod, becaufe he could not accommodate his 41ft pro- 
 pofition to the particular law of gravitation obferved in 
 the fyftem. And he claims to hinifelf the honour of 
 having the firft demonft rated that a centripetal force, in- 
 verfely as the fquare of tlie diftance, neceiTarily produces 
 a motion in a conic fec^tion. The argument by which 
 he fup ports this bold claim is very lingular, coming from 
 a confummate mathematician, who could not be i-^^norant 
 of its nullity ; fo that it was not a ferious argument, 
 but a trick to catch the uninformed. Newton, fays he, 
 might with equal propriety have inferred, from the de- 
 fcription of the logarithmic fpiral by a body influenced 
 by a force inverfely proportional to the cube of the dif- 
 tance, that a body fo defleded will defcribe the loga^ 
 rkhmic fpiral, whereas we know that it may defcribe the 
 
 hyperbolic 
 
INVERSE PROBLEM 01' CLNTRAI. rORCES. 357^ 
 
 hyperbolic fpiral. Not fatisficd with this triumph, he- 
 attacks Newton's procefs in his 4 ill or general propofi- 
 tion of central forces, faying that it is deduced from prin- 
 ciples foreign to tlie queflion ; and, after all, does not 
 exhibit the body in a ftate of continued motion, but 
 merely informs us where it will be found, and in what 
 condition, in any aligned moment. He concludes by 
 vaunting his own procefs as accomplirning all that can be 
 wanting in the problem. 
 
 Thefe alTertions are the mod unfounded and bold 
 vauntings of this vainglorious mathematician ; and his 
 own folution is a manifeft piagiarifm from the vTitings 
 of Newton, except in the method taken by him to demons- 
 ftrate the lemma which he as well as Newton premifes. 
 Newton's demonftration of this lemma is by the pureft 
 principles of free curvilineal motion ; and it is, in this 
 refpeclj a beautiful and original propofition. It m^ikes our 
 J 222. Bernoulli confiders it as fynonymous with mo-r 
 tioa on an inclined plane ; with which it has no analogy. 
 The folution of the great problem by Bernoulli is, in every 
 principle, and in every ftep, the fame wdth Newton's •, and 
 the only difference is, that Newton employs a geometrical, 
 and Bernoulli an algebraical expreflion of the proceeding. 
 Newton exhibits continued motion, whereas Bernoulli 
 employs the differential calculus, which ejjentially ex- 
 hibits only a fuccelTion of points of the path. It is 
 worth the Undent's while to read Dr Keill's Letter to 
 John Bernoulli, and liis examination of this boafted fo- 
 lution ot the celebrated problem. But it is ftill more 
 
 worth 
 
353 PHYSICAL ASTRONOMY. 
 
 •wortli his while to read Newton's folution, and the pro. 
 pofitions in M'Laurin's Fluxions and Hermann's Phoro- 
 nomia, which are immediately connected with this pro- 
 blem. This reading will greatly conduce to the forming 
 a good tafte in difquifitions of this kind. * 
 
 485. Our occupation at prefent is much more limit- 
 ed. We are chiefly interefted to fhew that gravitation 
 produces an elliptical motion, when the fpace P T, along 
 which the body mud be uniformly impelled by the force 
 as it exifts in P, in order to acquire the velocity of pro- 
 jection, is lefs than PS, But every ftep would have 
 been the fame, had we made P T equal to P S (as in 
 fig. 52. N° 2.) But we fhould then have found that 
 when the angle Q P j- is made equal to Q P S, the line 
 P s will be parallel to S O, fo that S Q will not interfe^: 
 it, and the path will not have another focus. It is a 
 parabola, of which PR is the principal parameter. 
 
 486. We {hall alfo find that if P T be made greater 
 than PS (as in fig. 52. N° 3.) the line P/ (making the 
 angles Q P S and § P J" eq\ial) will cut S O on the other 
 fide of S, fo that S and sXzre on the fame fide of O. 
 The path will be a hyperbola, of which P R is the prin- 
 cipal parameter. 
 
 487. 
 
 * The propofitions given by M. de Moivre in No. 352. 
 of the Philofophical Tranfaftions, and thofe by Dr Keill iu 
 No. 317. and 340. are peculiarly fimple and good. 
 
THEORY OF ELLIPTICAI MOTION 359 
 
 487. This rcftn£lion to the conic fedlions plainly 
 follows from the line P R, the third proportional to P O 
 and PQ, being the principal parameter, whether the path 
 be an eilipfe, parabola, hyperbola, or circle. * 
 
 It remains to point out the general circumftances of 
 •Jas elliptical motion, and their phyfical connexions. For 
 this purpofe, tlie following propofition is ufeful. 
 
 488. When a body defcribes any curve line BDP A 
 (hg. 53.} by means of a deflecting force directed to a 
 focus S, the angle S P N, which the radius vetSlor makes 
 with the direction of the motion, diminiflies, if the ve- 
 locity in the point P be lefs than what would enable the 
 body to defcribe a circle round S, and increafes, if the 
 velocity be greater. 
 
 If 
 
 * The only difliculty in the inference of a conic lection as 
 the neceflary path of a proje£lile influenced by a force in the 
 inverfe duphcate ratio of the dillance from the centre, has 
 arifen from the practice of the algebraic analyfls,of defining all 
 curve lines by the relation of an abfcifla to parallel ordinates. 
 But this is by no means neceflary ; and all curves which en- 
 clofe fpace, are as naturally referable to a focus, and definable 
 •by the relation between the radii and a circular arch. An 
 equation exprefling the focal chord of curvature is as difl:inc- 
 :Ue as the ufual equation, and leads us with eafe to the chief 
 properties of the figure. Therefore 
 
 Let S P, the given dillance, be a, and any indeterminate 
 'difliance be x. Let the perpendicular S N (alfo given by S P 
 
 and 
 
THEORY OF ELLIPTICAI MOTION 359 
 
 487. This rcftn£lion to the conic fecStions plainly 
 follows from the line P R, the third proportional to P O 
 and PQ, being the principal parameter, whether the path 
 be an eillpfe, parabola, hyperbola, or circle. * 
 
 It remains to point out the general circumftances of 
 •Jjis elliptical motion, and their phyfical connexions. For 
 this purpofe, tlie following proportion is ufeful. 
 
 488. When a body defcribes any curve line BDP A 
 (fig. 53.) by means of a deflecting force directed to a 
 focus S, the angle S P N, which the radius ve(Si:or makes 
 with the direction of the motion, diminiflies, if the ve- 
 locity in the point P be lefs than what would enable the 
 body to defcribe a circle round S, and increafes, if the 
 velocity be greater. 
 
 If 
 
 * The only difficulty in the inference of a conic fed;ion as 
 the neceflary path of a projectile influenced by a force in the 
 inverfe duphcate ratio of the diftance from the centre, has 
 arifen from the practice of the algebraic analyits,of defining all 
 curve lines by the relation of an abfcifTa to parallel ordinates. 
 But this is by no means necefTary ; and all curves which en- 
 clofe fpace, are as naturally referable to a focus, and definable 
 -by the relation between the radii and a circular arch. An 
 equation exprelTmg the focal chord of curvature is as diflinc- 
 :ive as the ufual equation, and leads us with eafe to the chief 
 properties of the figure. Therefore 
 
 Let S P, the given diftance, be a, and any indeterminate 
 'diftance be x. Let the perpendicular S N (alfo given by S P 
 
 and 
 
•^60 I'lTYSlCAL ASTRONbMt. 
 
 If the velocity of the body in P be lefs tban tliat 
 which might produce a circular motion round S, then 
 its path will coalefce with the nafcent arch P/* of a 
 circle whofe deiledlive chord of curvature is lefs than 
 2 P S (212.) Let its half be PO, lefs than PS, and let 
 Pj> be a very minute arch. Draw the tangents PN, p n^ 
 and the perpendiculars S N, S /r. V q perpendicular to 
 PN will meet pq perpendicular to pn (P p being eva- 
 nefcent) in q the centre of curvature. Draw pS and 
 
 It is evident that the angles V qp and V Op are ulti- 
 mately equal, as they ftand on the fame arch Pj^ of the 
 
 equicurve 
 
 and the given angle S P N) be /^, and let p be the perpendi- 
 cular and q the focal chord of curvature, correfponding to the 
 diftance x. Let 4 P T be = ^. Then (102. 210.) we have 
 I I _ I I 
 
 ¥~d'fq~'^'7'- 
 
 b^d ; p^q =z a^ : x^ 
 
 therefore q = -^, = -ix ^ 
 
 Let — J z= e then q zr. — -, which is an equation to a co- 
 
 nic fedion, of which e is the parameter, S the focus, and 
 P N a tangent in P. Now ^ is a given magnitude, becaufe 
 fl, h, d, are all given. Exprefllng the angle S P N by <p, we 
 have e z= d X fm.* (p. See alfo for the particular cafe of a 
 force proportional to — the differtations by Dr Jo. Keill in 
 the Philofophical TraiiiCatlions,. No. 317. and No. 340^. 
 
THEORY OF ELLIPTICAL MOTION. '^6l 
 
 equlcurve circle, and are, refpecSlively, the doubles of the 
 angles at the circumference. P ^-^ is evidently equal to 
 N S ;/. Therefore P 0/» is equal to N S «, and P S/ is 
 lefs than N S «. Therefore P S N is lefs than p S «, and 
 SPN IS greater than Sprt, Therefore the angle S P N 
 diminiflies when P O is lefs than P S, that is, when the 
 velocity in P is lefs than what would enable the centri- 
 petal force in P to retain the body in a circle round S. 
 
 On the other hand, if the velocity in P be greater 
 than what fuits a circular motion round S, it is plain that 
 P O will be greater than P S, and the angle P S ^ will 
 be greater than N S n, and the angle P S N greater than 
 p S //, and therefore the angle SPN will be lefs than 
 S p fly Sec* 
 
 489. Applying this obfervatlon to the cafe of ellip*- 
 tical motion, we get a more diftin^^ notion of its differ- 
 ent afFeclions, and their dependence on their phyfical 
 caufes. 
 
 In the half DAB (fig. 46.) of the ellipfe defcribed 
 by a planet round the Sun in its focus S, the middle 
 point of the deflective or focal chord of curvature lies 
 between the planet and the focus. Therefore, during 
 the whole motion from D to B, along the femiellipfe 
 DAB, the angle contained between the radius vector 
 and the line of the planet's motion is continually dimi- 
 nifhing. But during the motion in the femiellipfe, BPD, 
 the angle is continually increafing. It is therefore the 
 greateft poihble in D, and the fmalleft in B. 
 
 Z z Let 
 
3^2 I»HYSICAL ASTRONOMY. 
 
 Let tlie pknet fet out from its aphelion A, with its 
 due velocity, moving in the direction A F. The velocity 
 in A, being equal to that acquired by a uniform accele- 
 ration along ~ of the parameter, is vaftly lefs than what 
 would make it move in the circular arch A L, of which 
 S is the centre, and the planet mull fall within that circle. 
 Therefore its path will no longer be perpendicular to 
 the radius ve61:or, but mufl now make with it an angle 
 fomewhat acute. The centripetal force therefore is now 
 refolvable into two forces, one of which accelerates the 
 planet's motion, and the other incurvates its path. Its 
 direftion brings it nearer to the Sun. While in the 
 quadrant A F B, felie velocity is always lefs than what is 
 required for a circular motion. For, if from any point 
 F in this quadrant, F G be drawn perpendicular to the 
 tangent, meeting the tranfverfe axis in G, and if GH 
 be drawn perpendicular to the normal F G, H F is one 
 half of the focal chord of curvature, and H lies between 
 P and S. Now, it has been fliewn that when this is 
 the cafe, the angle S F « dimini flies, and, with it, the 
 ratio of S « to S F (this ratio is that of C B to the femi- 
 diameter CO, the conjugate of CF, (} 6. Ell.) Con- 
 fequently, there will be continually more and more of 
 the centripetal force employed in accelerating the mo- 
 tion, and lefs employed in incurvating the path, the firft 
 part being F n and the other S «. When the planet ar- 
 rives at B, the point H falls upon S, and the velocity is 
 precifely what would fuffice for a circular motion round 
 S, if the diredlion of the motion were perpendicular to 
 
 the 
 
THEORY OF ELLIPTICAL MOTION. 363 
 
 the radius vc£lor. But the direction of the motion 
 brings it ftill nearer to S. A great part of the centri- 
 petal force is ftill employed in accelerating the motion ; 
 and the moment the planet pafTes B, the velocity be- 
 comes greater than what might produce a circular mo- 
 tion round S. For H now lies beyond S from B. There- 
 fore the angle S B N, which is now in its fmalleft pof- 
 iible ftate, begins to open again ; and this diminiflies the 
 proportion of the centripetal force which accelerates the 
 motion, and increafes the proportion of the incurvating 
 force. The planet is, however, ftill accelerated, preferv- 
 ing the equable defcription of areas. The angle S B N 
 increafes with the increafmg velocity, and becomes a 
 right angle, when the planet arrives at its perihelion P. 
 
 It has been fhewn (Ellipfe, § 4.) that the chord PI 
 cut ofF from any diameter P A by the equicurve circle 
 Fa I, is equal to the parameter of that diameter. There- 
 fore the centre of this circle lies beyond S. The pla- 
 net, pafiing through P, is defcribing a nafcent arch of 
 this circle. Confequently, the curve which it is de- 
 fcribing paftes without a circle defcribed round S, and 
 the planet is now receding from the Sun. This is ufual- 
 ly accounted for, by faying that its velocity is now too 
 great for defcribing a circle round the Sun. And this 
 is true, when the intenfity of the deileding force is con- 
 fidered. But it has been thought difficult to account 
 for the planet now retiring from the Sun, in the peri- 
 helion, where the centripetal force is the greateft of all 
 —greater than what has already been able to bring it 
 Z z 7, contimially 
 
3^4 PHYSICAL ASTRONOMY. 
 
 continually nearer to the Sun. We are apt to expeft 
 tliat it will come ftill nearer. But the fad is, that the 
 planet, in pafling through P, is really moving fo that, 
 if the Sun were fuddenly transferred to o^ it would cir- 
 culate round it for ever. But, in defcribing the fmalleft 
 portion of the circle P ^ I, it goes without the circle 
 which has S for its centre, and its motion now makes an 
 obtufe angle with the radius veftor, although it is per- 
 pendicular to a radius drawn to o. There is now a por- 
 tion of the centripetal force employed in retarding the 
 motion of the planet, and its velocity is now diminilhed ; 
 and the angle of the radius vecflor and the path is now 
 increafed, by the fame degrees by which they had been 
 increafed and diminifhed during the approach to the Sun. 
 At D, the planet has the fame diilance from the Sun 
 that it had in B, and the fame velocity. The angle 
 S D "y is now as much greater than a right angle as S B N 
 was lefs j and at A, it is reduced to a right angle, and 
 the velocity is again the fame as the firil. In this way 
 the planet will revolve for evT?r. 
 
 It was fliewn in } 223. that in the curvilineal motion 
 of bodies by the action of a central force, the velocities 
 are inverfely as the perpendiculars from the centre of 
 forces on the lines of their direftions. In the perihe- 
 lion, the radius vector is perpendicular to the path. The 
 perihelion diftance may therefore be taken as the unit of 
 the fcale on which all the other velocities are mcafured. 
 The other velocities may therefore be confidered as frac- 
 tions of the perihelion velocity, which is the greatefl pf 
 
 In 
 
THEORY OF ELLIPTICAL MOTION. 365 
 
 In elliptical motions, the velocities in every point are 
 as the perpendicuL^rs ^rawn from the other focus on the 
 tangents in that point. For the perpendiculars on any 
 tangent drawn from the two foci are reciprocal. 
 
 490. Hence it appears that if a body fets out from 
 P, with the velocity acquired by uniform acceleration 
 along P S, and defcribes a parabola by means of a cen- 
 tripetal force directed to S, the velocity diminilhes with*- 
 out limit. For the perpendicular drawn from the focus 
 on a tangent to a parabola may be greater than any line 
 that can be affigned, if the point in the parabola b3 
 taken fufliciently remote from the vertex. 
 
 491. If the body fet out from P with a velocity ex- 
 ceeding what it would acquire by uniform acceleration 
 along P S, it Mnll defcribe a hyperbola, and its velocity will 
 diminifa continually. But it will never be lefs tlian i cer-. 
 tain determinable magnitude, to v/hich it continually ap- 
 proximates. For the perpendicular from the focus on 
 the tangent in the mod remote point of the hyperbola 
 that can be affigned, is ftill lefs than the perpendicular 
 to the affymptote, to which the tangent continually ap- 
 proaches. 
 
 But, when the velocity in the perihelion is lefs than 
 that acquired by uniform acceleration along PS, tliere 
 will always be a limit to its diminution by the recefi 
 from the centre of force. For the velocity being fo mo- 
 derate,' the path is more incurvated by the centripetal 
 
 force J 
 
266 PHYSICAL ASTRONOMY. 
 
 force ; fo th t the body is made to defcribe a curve 
 which has an upper apfis A, as well as a lower apfis P* 
 The body, after paffing through A at right angles to the 
 radius ve -^ or, is now accelerated, becaufe its path now 
 makes an acute angle with the radius ve£lor ; and thus 
 the velocity is again increafed. 
 
 492. The velocity in any point of the ellipfe de- 
 fcribed by a planet is to the velocity that would enable 
 the fame force to retain it in a circle at the fame dif- 
 tance, in the fubduplicate ratio of its diftance from the 
 upper focus f to the femitranfverfe axis. That is, call- 
 ing the elliptic velocity V, and the circular velocity v, 
 we have V^ : i;* = P / : C A. (fig. 53.) 
 
 For (488.) V*:i;^ = PO:PS. 
 
 But (Ellipfe 9.) it was fhewn that P O X C A was 
 equal to CK% =PSxP/. Therefore PO: PS = Pj:C A 
 andV^:'i;^ = P/:CA, 
 
 493. The angular motion in the ellipfe is to the an- 
 gular motion in a circle at the fame diftance, and by the 
 adtion of the fame force, in the fubduplicate ratio of 
 half the parameter to the diftance from S, 
 
 Take P/, a fmall arch of the ellipfe, and, with the 
 centre S, apd diftance SP, defcribe the circular arch 
 PzV, cutting S/> in 2. Make P/) to PV as the ve- 
 locity in the ellipfe to that in the circle. Then it is plain 
 that P z is to P V as the angular motion in the ellipfe 
 ii to the, angular motion in the circle. 
 
 The 
 
THEORY OF ELLIPTICAL MOTION. 367 
 
 The angle z P/ being the complement of N P S (be- 
 caufe NP may be confidered as coinciding with />P) it 
 is equal to N S P. Therefore 
 
 P 2= : P;>' = S N^ S P% = P Q^ : P O* 
 therefore Fz^ :Vp' = P P. : P O 
 but P/ :PV^ = PO :PS 
 
 therefore P 2= : P V^ = P R : P S. 
 Cor, The angular motion in the circle exceeds that 
 in the ellipfe, when the point R lies between P and S, 
 and falls (hort of it when R lies beyond S. They are 
 equal when P S is perpendicular to A C, or when the 
 true anomaly of the planet is 90°. For then R and S 
 coincide. Here the approach to S is moll rapid. 
 
 494. In any point of the ellipfe, the gravitation or 
 centripetal force is to that which would produce the fame 
 angular motion in a circle, at the fame diftance from the 
 Sun, as this diftance is to half the parameter, that is, as 
 P S to P R. 
 
 For, by the lafb propofition, when the forces in the 
 circle and ellipfe are the fame, the angular motion in the 
 circle was to that in the ellipfe as P V to P 2, which has 
 been fhewn to be as VlFS to VPR. Therefore, when 
 the angular velocity in the circle, and confequently the 
 real velocity, is changed from P V to P 2, in order that 
 it may be the fame with that in the ellipfe, the centripe- 
 tal force muft be changed in the proportion of P V to 
 P 2% that is, of P S to P R. Therefore the force which 
 retains the body in the ellipfe is to that which will retain 
 
 it 
 
^dt PHYSICAL ASTRONOMY, 
 
 it with tlie fame angular motion in a circle at that diftanctf 
 as PS to PR. 
 
 Thefe are the chief afFedions of a motion regulated 
 by a centripetal force in the inVerfe duplicate ratio of the 
 diftance from the centre of forces. The comparifon of 
 them with motions in a circle gives us, in moft cafes, 
 eafy means of ftating every change of angular motion, or 
 of approach to or recefs from the centre, by means of 
 any change of centripetal force, or of velocity. 
 
 Such changes frequently occur in the planetary fpaces ; 
 and the regular elliptical motion of any individual planet, 
 produced by its gravitation to the Sun, is continually di-. 
 ilurbed by its gravitation to the other planets. This di* 
 fturbance is proportional to the fquare of the diftance 
 from the diflurbing planet inverfely, and to the quantity 
 of matter in that planet diredlly. Therefore, before vi^e 
 can afcertain the difturbance of the EartVs motion, for 
 example, by the action of Jupiter, vi^e muft know the 
 proportion of the quantity of matter in Jupiter to that in 
 the Sun. This may feem a queftion beyond the reach of 
 human underftanding. But the Newtonian philofophy 
 furniflies us with infallible means for deciding it. 
 
 Of the Quantity of Matter in the 
 Sun and Planets. 
 
 Since it appears that the mutual tendency which we 
 have called Gravitation i^ competent to every particle of 
 
 matter^ 
 
QUANTITY OF MATTER IN A PLANET. 2^g 
 
 matter, and therefore the gravitation of a particle of mat- 
 ter to any mafs M'hatever is the fum 6r aggregate of its 
 gravitation to every atom of matter in that mafs, it fol- 
 lows that the gravitation to the Sun or to a plaiiet is 
 proportional to the quantity of matter in the Sun or the 
 planet* As the gravitation may thus be comp-ned, when 
 we know the quantity of matter, fo this may be com- 
 puted when we know the gravitation towards it. Hence 
 it is evident that we can afcertain the proportion of the 
 quantities of matter in any two bodies, if we know the 
 proportion of the gravitations toward them. 
 
 495. The tendency toward a body, of which m is 
 
 the quantity of matter and d the diftance, is =^ — . It 
 
 is this tendency which produces delle£lion from a itraight 
 Jine, and it is meafured by this defle£lion. Now this, 
 in the cafe of the planets, is meafured by the diftance 
 at which the revolution is performed, and the velocity of 
 that revolution. We found (224.) that this combination 
 
 is exprefTed by the proportional equation ^ == ~> where p 
 
 171 d 
 is the periodic time. Therefore we have 17 == — j and, 
 
 confequently, m = —. 
 
 By this means we can compare the <juantity of mat- 
 ter in all fuch bodies as have others revolving round 
 them. Thus, we may compare the Sun with the Earth, 
 by comparing the Moon's gravitation to the Earth with 
 ^e Earth's gravitation to the Sun. It will be convenient 
 3 A' to 
 
37® 
 
 PHYSICAL ASTRONOMY. 
 
 to confider the Earth as the unit in this comparilbn with 
 
 the other bodies of the fyftem. 
 
 The Sun's diftance in miles is - - - 93726900 
 The Moon's diftance ------ 240144 
 
 The Eartli's revolution (fydereal) days - 365,25 
 The moon's fydereal revolution (days) - 27,322 
 
 Therefore 937^^i^?M„^^ =. 332669. 
 
 240144^ X 3^5^25 
 
 But this muft be increafed by about ^-^^ becaufe the gravi- 
 tation to the Earth i^ ftated beyond its real value by the 
 fuppofition that the revolution of the Moon is performed 
 round the centre of the Earth, whereas it is really per- 
 formed round their common centre (231.) Thus increaf- 
 ed, the Sun's quantity of matter may be eftimated at . 
 337422 times that of this Earth. 
 
 It muft be obferved that this computation is not of 
 very great accuracy. It depends on the diftance of the 
 Sun ; and any miftake in this is accompanied by a fimilar 
 miftake, but in a triplicate proportion. Now our efti- 
 mation of the Sun's diftance depends entirely on the 
 Sun's horizontal parallax, as meafured by means of the 
 tranfits of Venus. The error of ~ of a fecond in this 
 parallax, (which is only about 8",7 or 8",8) will induce 
 an error of -^-^ of the wliole. 
 
 In like manner, v/e compare Jupiter with the Earth, 
 by comparing the gravitation of the firft fatellite with 
 that of the Moon. This makes Jupiter about 313 times 
 more maflive than the Earth. 
 
 The quantity of matter in Saturn deduced from the 
 
 revolution 
 
MASSrS OF THE PLANETS. 371 
 
 revolution of his fecond Caffinian fatellite, is about 103 
 times that of the Earth. 
 
 Herfchel's planet contains about 17 times as much 
 matter as our globe, as we learn by the revolution of its 
 firft fatellite. 
 
 We have no fuch means for obtaining a knowledge 
 of the quantity of matter in Venus, Mars, or Mercury. 
 Thefe are therefore only gueiTed at, by means of certain 
 phylical confiderations which afford fome data for an 
 opinion. Venus is thought to be about ~% of the Earth, 
 Mars about -^y and Mercury about ■^^. But thefe are 
 very vague guefles. We judge of the Moon's quantity 
 of matter with fome more confidence, by comparing the 
 influence of the Sun and Moon on the tides, and on the 
 preceffion of the equinoxes. The Moon is fuppofed a- 
 bout -^^ of the Earth. 
 
 From this comparifon it will appear that the Sun con* 
 tains nearly 800 times as much matter as all the planets 
 combined into one mafs. Therefore the gravitation to 
 the Sun fo much exceeds that of any one planet to an^ 
 other, that their mutual diilurbances are but inconfi- 
 derable. 
 
 496. The proportion of the quantities of matter, dif- 
 covered by this procefs of reafoning, is very diJferent 
 from what we fhould have deduced from the obferved 
 bulk of the different bodies. Thus, Saturn's diameter 
 being al^out ten times that of the Earth, we fhould have 
 iaferred that he contained a thoufarjd times as much 
 3 A 2 matter^ 
 
372 PHYSICAL ASTRGNOMT. 
 
 matter, whereas he contains only about roj or 104. 
 We muft therefore conclude that the denfities of the Suri 
 and planets are very different. Still taking the Earth as 
 the unit of the fcale, and combining the ratios of the 
 bulks and the quantities of matter, we may fay that the 
 denfity of the Sun is - _ - - 0,25 
 
 - Venus - - - r - 1,27 
 Earth ----- i 
 Mars - - - - - 0,73 
 Jupiter ----- 0,292 
 Saturn - - - - " - o, 1 84 
 Georgian Planet - - - 0,212 
 It appears by this ftatement that the denfity of the 
 planets is lefs as they are more remote from the centre 
 of revolution. Herfchel's planet is an exception j but a 
 fmall change on his apparent diameter, not exceeding 
 half a fecond, will perfedly reconcile them, 
 
 497. Knowing the quantity of matter, and the diar 
 meter of the bodies of the fyftem, we can eafily tell the 
 accelerative force of gravity a6ting on a body at their 
 furfaces by" article 465, that is, what velocity gravity 
 will generate in a fecond of time, or how far a body 
 will fall in a fecond. lu like manner, we can tell the 
 prefTure occafioned by the v/eight or heavinefs of a body, 
 as this may be meafured by the fcale of a fpring fteel- 
 yard, graduated by additions of equal known preffures. 
 It cannot be meafured by a balance, which only compares 
 ^ne mafs of equally heavy matter with another. 
 
 Thus, 
 
WEIGHT ON THE PLANETS. 373 
 
 Thus, the fpace fallen through, and the apparent 
 weight of a lump of matter, by a fpring fteelyard, will 
 be 
 
 
 Falli?i i". 
 
 TVeight, 
 
 At the furface of tlie Sun 
 
 451 feet. 
 
 28,2 
 
 Earth 
 
 16,09 
 
 I 
 
 Jupiter 
 
 41,64 
 
 2,6 
 
 Saturn 
 
 i4>4 
 
 0,89 
 
 Herfchel 
 
 18,7 
 
 1,16 
 
 Of the Mutual Bijliirbances of the 
 Planetary Motions, 
 
 498. The queftions v/hich occur in this department 
 of the ftudy are generally of the moft delicate nature, 
 and require the moft fcrupulous attention to a variety of 
 circumftances. It is not enough to know the dire£lion 
 and intenfity of the difturbing force in every point of a 
 planet's motion. We muft be able to collecl into one 
 aggregate the minute and almoft imperceptible changes 
 that have accumulated through perhaps a long tra£l: • of 
 time, during which the forces are continually changing, 
 both in diredion and in intenfity, and are frequently 
 combined with other forces. This requires the conftant 
 employment of the inverfe method of fluxions, which is 
 by far the moft difficult department of the higher geo- 
 metry, and is ftill in an imperfed" ftate. Thefe pro- 
 i^lems have been exclufively the employment of the moft 
 
 eminent 
 
374 PHYSICAL ASTRONOMY. 
 
 eminent mathematicians of Europe, the only perfons who 
 are in a condition to improve the Newtonian philofophy ; 
 and the refuh of their labours has fliewn, in the cleared 
 manner, its fupreme excellence, and total diffimiHtude to 
 all the phyficai theories which have occupied the atten- 
 tion of philofophers before the days of the admired in- 
 ventor. For the feeming anomalies that are obferved in 
 the folar fyftem are, all of them, the confequences of 
 the univerfal operation of one fimple force, without the 
 interference of any other, and are all fufceptible of the 
 moft precife meafurement and comparifon with obferva- 
 tion ; fo that what we choofe to call anomalies, irregu- 
 larities, and difturbances, are as much the refult of the 
 general pervading principle as the elliptical motions, of 
 which they are regarded as the difturbances. * 
 
 It is in this part of the ftudy alio in which the pene- 
 trating and inventive genius of Newton appear moft con- 
 fpicuoully. The firft law of Kepler, the equable de- 
 fcription of areas, led the way to all the reft, and made 
 the detection of the law of planetary force a much ea- 
 fier talk. But the moft difcriminating attention was ne- 
 ceiTary for feparating from each other the deviations from 
 fimple elliptical motion which refult from the mutual gra- 
 vitation of the planets, and a confummate knowledge of dy- 
 namics for computing and fummingup all thofe deviations. 
 The fcience was yet to create j and it is chiefly to this 
 that the firft book of Newton's great work is dedicated. 
 He has given the moft beautiful fpecimen of the invef- 
 tigation in his theory of the lunar inequalities. Tq every 
 
 one 
 
DISTURBANCES Or ELLIPTICAL MOTION. 37.5 
 
 one who has acquired a juft tafte in mathematical com- 
 pofition, that theory will be confidered as one of the 
 mofl elegant and pieafuig performances ever exhibited to 
 the public. It is true, that it is but a commencement 
 of a moft delicate and difficult invefligation, which has 
 been carried to fucceffive degrees of much greater improve- 
 mentj by the unceafing labours of the firft mathema- 
 ticians. But in Newton''s work are to be found all the 
 helps for the profecution of it, and the firfl application 
 of his new geometry, contrived on purpofe •, and all the 
 fteps of the procefs, and the methods of proceeding, are 
 pointed out — all of Newton's invention, fua mathefi facem 
 P'i\rferente. 
 
 It mud be farther remarked that the knowledge of 
 the anomalies of the planetary motions is of the greateft 
 importance. Without a very advanced (late of it, it 
 w^ould have been impofnble to conflru61: accurate tables 
 of the lunar motions. But, by the application of this 
 theory, Mayer has conftrudled tables fo accurate, that 
 by obferving the diftance of the Moon from a properly 
 feledted ftar, the longitude may be found at fea with 
 an exa£tnefs quite fufficient for navigation. This method 
 is now univerfally pra£lifed on board of our Eaft India 
 fhlps. This requires fuch accurate theory and tables of 
 the Moon's motion, that we muft at all times be able to 
 determine her place within the 30th part of her own 
 diameter. Yet the Moon is fubjedl to more anomalies 
 than any other body in the folar fyftem. 
 
 But the ftudy Is no lefs valuable to the fpeculatlve 
 
 philofopher. 
 
^^5 PHYSICAL ASTRONOMY. 
 
 philofopher. Few things are more pleafing than the 
 being able to trace order and harmony in the midft 
 of feeming confufion and derangement. No where, in 
 the wide range of fpeculation, is order more complete- 
 Jy efFecled. All the feeming diforder terminates in 
 the deteclion of a clafs of fubordinate motions, which 
 have regular periods of increafe and diminution, never 
 arifing to - a magnitude that makes any confiderable 
 change in the fimple eUiptical motions ; fo that, finally, 
 the fclar fyilem feems calculated for almoft eternal du- 
 ration, without fuftaining any deviation from its prefent 
 ftate that will be perceived by any befides aftronomers. 
 The difplay of wifdom, in the fele£l:ion of this law of 
 mutual aftion, and in accommodating it to the various 
 circumilances which contribute to this duration and con- 
 ftancy, is furely one of the moft engaging obje6LS th2rt 
 can attract the attention of mankind. 
 
 In this elementary courfe of inflruftion, we cannot 
 give a detail of the mutual difturbances of the planet- 
 ary motions. Yet there are points, both- in refpe6b of 
 do6l:rine and of method, which may be called element- 
 ary, in relation to this particular fubje£t. It is proper to 
 confider thefe with feme attention. 
 
 499. The regularity of the motions of a planet A 
 round the Sun would not be diflurbed by the gravitation 
 of both to another planet B, if the Sun and the planet 
 A gravitate to B with equal force, and in the fame or 
 in a parallel dire£lion (98.) The difturbance arifes en- 
 tirely 
 
blSTURBANCE OF PLANETAtlY MOTIONS. 377 
 
 tirely from the inequality and the obliquity of the gravi- 
 tations of the Sun and of the planet A to B. The man- 
 ner in which thefe diflurbances may be confidcred, and 
 the grounds of computation, will be more clearly under- 
 ftood by an example. 
 
 Let S (fig. 54.) reprefent the Sun, E the Earth, and 
 J the planet Jupiter. Let it be farther fuppofed (which 
 may be done without any great error) that the Earth and 
 Jupiter defcribe concentric circles round the Sun, and 
 that the Sun contains 1000 times as much matter as Ju- 
 piter. Make J S to E A as the fquare of E J to the 
 fquare of S J. Then, if we take S J to reprefent the 
 gravitation of the Sun to Jupiter, it is plain that E A will 
 reprefent the gravitation of the Earth, placed in E, to 
 Jupiter. Draw E B, parallel and equal to JS, and com- 
 plete the parallelogram E B A D. 1 he force with which 
 Jupiter deranges the motion of the Earth round tlie Sun 
 will be reprefented by E D. 
 
 For the force E A is equivalent to the combined forces 
 E B and E D. But if the Sun and Earth were impelled 
 only by the equal and parallel forces S J and E B adding 
 on every particle of each, it is plain that their relative 
 motions would not be affe£led (98.) It is only by the im- 
 pulfion arifmg from the force E D tliat their relative fu 
 tuations will fuftain any derangement, 
 
 500. This derangement is of two kinds, afFeding 
 either the gravitation of the Earth to the Sun, or her an- 
 gular motion round him. Let ED be confidcred as the 
 3 ^ diagonal 
 
jyS^ PHYSICAL ASTRONOMT. 
 
 diagonal of a redtangle E F D G, EG lying in the direc- 
 tion of the radius S E, and E F being in the dire6lion 
 of the tangen^ to the Earth's orbit. It is plain that the 
 force EG affects the Earth's gravitation to the Sun, 
 while E F afFe£ls the motion round him. As E G is in 
 the direction of the radius, it has no tendency to acce- 
 lerate or retard her motion round the Sun. E F, on the 
 other hand, does not affect the gravitation, but the mo- 
 tion in the curve only. 
 
 This diflurbing force E D varies, both in direftioa 
 and magnitude, by a variation in the Earth's pofition ii> 
 relation to the Sun and Jupiter. Thus, in fig. A, which 
 reprefents the Earth as almoft arrived at the conjuncElion 
 with Jupiter, having Jupiter near his oppofition to the 
 Sun, the force E G greatly diminiflies the Earth's gravi- 
 tation to the Sun, and the force E F accelerates her mo- 
 tion round him in the order of the letters E C P O Q. 
 In fig. B, the force E G ftiil diminiflies the Earth's gra- 
 vitation to the Sun, but E F retards her motion from 
 O to Q. In fig. C, E G increafes the Earth's gravita- 
 tion to the Sun, and E F accelerates her motion round 
 him. It appears very plainly that the motiqn round the 
 Sun is accelerated in the quadrants O C and P O, and 
 is retarded in the quadrants CP and O Q. We may 
 alfo fee that tlie gravitation to the Sun is increafed in the 
 neighbourhood of the points P and Q, but is diminifhed 
 in the neighbourhood of C and O, and that there is an 
 intermediate point' in each quadrant where the gravita- 
 tion fuffer* no change. The greateft (diminution of the 
 
 Earth'8 
 
DiSTURBANCE OF THE PLANETARY MOTIONS. ^19 
 
 Earth's gravitation to the Sun muft be in C, when Jupi- 
 ter Is neareft to the Earth, in the time of his oppoiition 
 to the Sun. 
 
 We alfo fee very plainly how all thefe dlflurbing 
 forces may be precifely determined,- depending on the 
 proportion of E I to E S and to S I. Nor is the con- 
 ftru£lion reftritied to circular orbits. Each orbit is to 
 be confidered in its true figure, and the parallelogram 
 E G D F is not always a re^langle, but has the fide E F 
 lying in the direction cf the tangent. But we believe 
 that the computation is found to be fuiBciently exa£l 
 without confidering the parallelogram E G D F as ob- 
 lique. The eccentricity of Jupiter's orbit muft not be 
 negledted becaufe it amounts to a fourth part of the 
 Earth's diftance from the Sun- 
 
 We have taken the Sun's gravitation to Jupiter as the 
 fcale on which the difturbing forces are meafured ; but 
 this was for the greater facility of comparing the dlflurb- 
 ing forces with each other. But they muft be compared 
 with the Earth's gravitation to the Sun, in order to learn 
 their efFe£t on her motions. It will be exacl; enough for 
 the prefent purpofe of merely explaining the method, to 
 fuppofe Jupiter's mean diitance five times the Earth's 
 from the Sun, and that tlie quantity of matter in the Sua 
 is 1000 times that of Jupiter. Therefore the Eartli's 
 gravitation to the Sun muft be 25000 times greater than 
 to Jupiter, when the Earth is about P or Q, When the 
 Earth is at C, her gravitation to Jupiter is increafed -121 
 the proportion of 4' to 5% and it is now t^^oo ^^ her 
 •- 3 B 2 gravitation. 
 
%S0 PfiTSICAL ASTRONOMY. : 
 
 gravitation to the Sun. When the Earth is in O, ife^ 
 gravitation to Jupiter is ^^^^^ of her gravitation to the 
 Sun. 
 
 But we ^re not to imagine that when the Eafth is 
 at Cj her motion relative to the Sun is afFe£^ed in the 
 fame manner as if -rg^rro S>^ her gravitation were taken 
 away. For we muft recoIjeO: that the Sun alfo gravi- 
 tates to Jupiter, or is deflected toward him, and there- 
 fore toward the Earth at C. The diminution of the re- 
 lative gravitation of the Earth is not to be meafured by 
 E A, but by E G. All the difturbing- forces E G and 
 E F, correfponding to every pofition of the Earth and 
 Jupiter, muil be confidered as fractions .of S J, the mea- 
 i\ire taken for the mean gravitation to Jupiter. This is 
 ■xrido- of the Earth's gravitation to the Sun. 
 
 Meafuring in this. way, we fliail find that when the 
 Earth is at P or Q her gravitation to the Sun is increafed 
 by x^o-Q-o • ^^^ P S ^V Q S will, in this cafe, come 
 in the place of E G in fig. C, and there will be no fuch 
 force as E F. At C the Earth's gravitation is diminifhed 
 
 To be able to afcertain the magnitude of the difturb- 
 ing force in the different fituations of the Earth is but a 
 very fmall part of the tafk. It only gives us the mo- 
 mentary i'mpulfion. We muft afcertain the accumulated 
 efFe6b of the aftion during a certain time, or along a 
 certain portion of the orbit of the diilurbed. planet. This 
 is the celebrated problem of three bodies^ as it is called, 
 which has employed the utmoft effort? of the great- ma- 
 tKemtijcians ever fince the time that it firft appeared in 
 
 Newton's 
 
COMPUTATION or THE DISTURBANCE. 3S1 
 
 Newton's lunar theory. It cnn only be folved by ap- 
 proximation ; and even this folution, except In fo me very 
 particular cafes, is of the utmofl difficulty, which fliews, 
 by the way, the folly of all who pretend to explain the? 
 motions of the planets by the impulfions of 'fluids, -wh^Q 
 not three, but millions of particles are atllng at once. 
 
 We Iiave to afc^rtain, in the iirft place, the accumu- 
 lated effect of the acceleration and retardation of the an- 
 gular motion of the Earth round the Sun. The generaji 
 procefs is one of the two following. 
 
 ly?, Suppofe it required to determine how far the 
 attrad^ion of Jupiter has made the Earth overpafs the 
 quadrantal arch OC of her annual orbit. The arch is 
 fuppofed to be unfolded into a ftraight • line, and divided 
 into minute portions, defcribed In equal times. At each 
 point of 'divifion is erected a perpendiGuiar ordinate -equal 
 to the accelerating difturbing force E F correfpoiiding 'to 
 that point. A curve line is drawn through the extremlr. 
 ties of thofe ordinates. The unfolded arch heing conii- 
 dered as the reprefentation oi the time, and the ordinares 
 as the accelerating forces, it is plain that the area 
 will reprefent the acquired velocity (70.) Now let ano- 
 ther figure be conftrucled, having an abfcifla to repre- 
 fent the time of' the motion. But the ordinates muft 
 iiow be made prdportional to the areas of the laft figure. 
 It is plain, from article' 50, that the area of this new fi- 
 gure will reprefent, or be proportional to the fpaces de- 
 fcribed, in confequence of the adlion of the difturbing 
 force ; and therefore it will exprefs, nearly, the addition' 
 
 to 
 
3«2 PHYSICAL ASTRONOMY. 
 
 to the fpace defcribed by the undlfturbed planet, or the 
 diminution, if the accelerations have been exceeded by 
 the retardations. 
 
 The other method is to make the unfolded arch the 
 fpace defcribed, and the ordinates the accelerations, as 
 before. The area now reprefents the augmentation of 
 the fquare of the velocity (75.) A fecond figure is now 
 conftruO:ed, having the fame abfcifla now reprefenting 
 the time. The ordinates are made proportional to the 
 fquare roots of the areas of the firfl figure, and they will 
 therefore reprefent the velocities. The areas of this new 
 figure will reprefent the fpaces, as in the firfl procefs, to 
 be added to the arch defcribed by the undiflurbed planet, 
 or fubtra£led from it. 
 
 501. All this being a taflc of the utmofl labour and 
 difficulty, the ingenuity of the mathematicians has been 
 exercifed in facilitating the procefs. The penetrating eye 
 of Newton perceived a path which feemed to lead di- 
 re£tly to the defired point. All the lines which reprefent 
 the diflurbing forces are lines connefted with circular 
 ■arches, and therefore with the circular motion of the 
 planet. The main diflurbing force E D is a fun6lion of 
 the angle of commutation C S E, and EF and EG are 
 the fine and cofine of the angle DEC Newton, in his 
 lunar theory, has given moft elegant examples of the 
 fummation of all the fucceffive lines E F that are drawn 
 to every point of the arch. Sometimes he finds the 
 fums or accumulate adlions of the forces exprefTcd by 
 
 the 
 
COMPUTATION OP THE DISTUi^BANCfi. 2^3 
 
 the fine of an arch ; fometimes by the tangent -, by a 
 fegment of the circular area, &c. &c. &c. Eulel*, 
 D'AIembert, De la Grange, Simpfon, and other illuf- 
 trious cultivators of this philofophy, have immenfely im- 
 proved the methods pointed out and exemplified by Nev/- 
 ton, and, by more convenient reprefentations- of the 
 forces than this elementary view will admit, have at laft 
 made the whole procefs tolerably eafy and plain. But it 
 is ftill only fit for adepts in the art of fymboUcal analyfis. 
 Their procefTes are in general fo recondite and abftrufe 
 that the analyft lofes all conception, either of motions or 
 of forces, and his mind is altogether occupied with the 
 fymbols of mathematical reafoning. 
 
 502. The fecond part of the tafk, the afcertaining 
 the accumulated efFedl: of the force E G, is, in general, 
 much more difficult. It includes both the changes made 
 on the radius vedor S E, and the change made in the 
 curvature of the orbit. The department of mathematical 
 fcience immediately fubfervient to this purpofe, is in a 
 more imperfe<Sl ftate than the quadrature of curves. The 
 procefs is carried on, almoft entirely by means of con- 
 verging feriefes. We cannot add any thing here that 
 tends to make it plainer. The lunar theory of Newton, 
 ■with the commentary of Le Seur and Jacquier, com- 
 monly called the 'Jefuits'' Commentary^ gives very good 
 examples of the methods which mull be followed in this 
 procefs. We mufl refer to the works of Euler, Clairaut, 
 Simpfon, and De la Place, on the perturbations of Ju« 
 
 piter 
 
38^4 PHYSICAL ASTRONOMY^. 
 
 pitor and Saturn, &c. and content ourfelves with merely 
 pointing out fome of the more general and obvious con- 
 fequsnces of this mutual a£lion of the planets. La Lande 
 has given in his aftronomy a very good fynopfis of the' 
 nioft. approved method. In the Tra6ls Ph^ical and Ma-* 
 thematical^ by Dr Matthew Stewart, and in his EiTay on 
 the Dillance of the Sun, are fome beautiful fpecimens o£ 
 the geometrical folutions of thefe problems, 
 
 503. When we confidering the motion of an in-* 
 ferior planet, difturbed by its gravitation to a fuperiof 
 planet, we fee that the inferior planet is retarded in the 
 quadrants C P and O O, and accelerated in the qua-- 
 drants P O and O C of its fynodical period. Its orbit 
 is more incurvated in the vicinity of the points P and O, 
 and its curvature is diminiflied in the vicinity of the 
 points O and C, and moft of all in the vicinity of C in 
 the line of conjunction with the fuperior planet. There- 
 fore, if the aphelion and perihelion of the inferior planet 
 fhould chance to be near the line J C S O of the fynodi- 
 cal motion, thefe points will feem to fhift forward. For, 
 the gravitation of the inferior planet to the Sun being 
 diminifhed, it will not be able fo foon to bend its path 
 to a right angle with the radius vector. On the other 
 hand, fhould the apfides of the inferior orbit be near th? 
 line P S ^, the increafe of the inferior planet*s gravita- 
 tion to the Sun muft fooner produce this effe6l, and it 
 will arrive fooner at its aphelion or perihelion, or thofe 
 points will feem to corae w€flward and to meet it. And 
 
 thus. 
 
:^.9S>^. 
 
 
IPLANETARY DISTURBANCES. 385 
 
 thus, in every fynodical revolution, tlie apfides of the in- 
 ferior planet will tw^ice advance and twice retreat, as if 
 the elliptical orbit iliifted a little to the eaftward or weft- 
 Avard. But, as the diminution of the iinferior planet's 
 gravitation to the Sun is much greater when it is in the 
 line C S O than the augmentation of it when in the line 
 5* S Q, the advances of the apfides, in the courfe of a 
 fynodical period will exceed the retreats, and, on the 
 whol<*> tliey will advance. 
 
 The perturbations of the motion of a fuperior planet 
 by its gravitation to an inferior, are In general oppofite, 
 both in kind and in dire(5tion, to thofe of the inferior 
 planet. Therefore, in general, their apfides retreat. 
 
 All thefe derangements, or deviations from the fimple 
 elliptical motion, are diftin^tly obferved In the heavens j 
 and the calculated effe£l: on each planet correfponds with 
 what is obferved, with all the precifion that can be wi{h- 
 ed for. It is evident that this calculation muft be ex- 
 tremely complicated, and that the effe6l depends not or>- 
 ly on the refpe^live pofitions, but alfo on the quantities 
 of matter of the different planets. For thefe reafons, as 
 Jupiter and Saturn are much larger than any of the o- 
 ther planets, thefe anomalies are chiefly owing to thefe 
 two planets. The apfides of all the planets are obferved 
 to advance, except thofe of Saturn, which fejifibly re- 
 treat, chiefly by the a6lion of Jupiter. The apfides of 
 the planet difcovered by Dr Herfchel doubtlefs retreats 
 confid^rably, by the ad ion of the great planets Jupiter 
 tmd Saturn. It might be imagined that the vaft number 
 
 3 C cf 
 
JPLANETARY DISTURBANCES. 385 
 
 thus, in every fynodicai revolution, the apfidcs of the in- 
 ferior phuiet V ill twice advance and twice retreat, as if 
 the elliptical orbit lliifted a little to the eaftward or weft- 
 Avard. But, as the diminution of the inferior planet's 
 gravitation to the Sun is much greater when it is in the 
 Hne C S O than the augmentation of it when in the line 
 P S Q, the advances of the apfides, in the courfe of a 
 fynodicai period will exceed the retreats, and, on the 
 \\'hol<», they will advance. 
 
 The perturbations of the motion of a fuperior planet 
 by its gravitation to an inferior, are in general cppofite, 
 both in kind and in dire(flion, to thofe of the inferior 
 planet. Therefore, in general, their apfides retreat. 
 
 All thefe derangements, ox deviations from ths fimple 
 elliptical motion, are -diftin^tly obferved in the heavens j 
 and the calculated efFe61: on each planet correfponds with 
 what is obferved, with all the preciuon that can be wifh- 
 ed for. It is evident that this calculation mud be ex- 
 tremely complicated, and that the efFed: depends not on*- 
 ly on the refpe^tive pofitions, but alfo on the quantities 
 of matter of the different planets. For thefe reafons, as 
 Jupiter and Saturn are much larger than any of the o- 
 ther planets, thele anomalies are chiefly owing to thefe 
 two planets. The apfides of all the planets are obferved 
 to advance, except thofe of Saturn, which fejifibly re- 
 treat, chiefly by the adion of Jupiter. The apfides of 
 the planet difcovered by Dr Herfchel doubtlefs retreats 
 confid^rably, by the ad ion of the great planets Jupiter 
 nnd Saturn. It might be imagined that the vafl number 
 
 3 C of 
 
^96 PHYSICAL ASTRONOMy. 
 
 ©f comets, which are almoft conftantly without the or- 
 bits of the planets, would caufe a general advance of all 
 the apfides. But thcfe bodies are fo far off, and pro- 
 bably contain fo little matter, that their action is in* 
 'fenfible. 
 
 504. The alternate accelerations and retardations of 
 the planets Mercury, Venus, the Earth, and Mars, in 
 confequence of their mutual gravitations, and their gra- 
 vitations to Jupiter, nearly compenfate each other in e- 
 very revolution ♦, and no effects of them remain after a 
 long tra^t: of time, except an advance of their apfides. 
 But there are peculiarities in the orbits of Jupiter and 
 Saturn, which occafion very fenfible accumulations, and 
 have given confiderable trouble to tlie aftronomers in dif- 
 covering their caufes. The period of Saturn's revolution 
 round the Sun increafes very fenfibly, each being about 
 7 hours longer than the preceding. On the contrary, 
 the period of Jupiter is obferved to diminifli about half 
 as much, that is, about li hours in each revolution. 
 
 This is owing to the particular pofition of the aphe- 
 lions of thofe two planets. Let ABPC (fig. 5^.) be 
 the elliptical orbit of Jupiter, A being the aphelion and 
 P the perihehon. Suppofe the orbit a bp c of Saturn to 
 be a circle, having the Sun S in the centre, and let Sa- 
 turn be fuppofed to be in a. Then, becaufe Jupiter em- 
 ploys more time (about 140 days) in moving from A to 
 C tlwn in moving from C to P, he mud retard the mo- 
 tion of Saturn more than he accelerates him, and Jupiter 
 
 mull 
 
DISTURBANCES OF SATURN AND JUPITER. 387^ 
 
 mud be more accelerated by Saturn than he Is retarded. 
 The contrary muft happen if Saturn be in the oppofite 
 part p of his orbit. After a tra£l of feme revolutions, 
 all muft be compenfated, becaufe there will be as many 
 oppofitions of Saturn to the Sun on one fide of the 
 tranfverfe diameter of Jupiter's orbit as on the other. 
 
 But if the orbit of Saturn be an ellipfe, as in fig. 5 c;. 
 B, and if the aphelion a be 90 degrees more advanced 
 iii the order of the figns than the aphelion A of Jupiter, 
 it is plain that there will be more oppofitions of Saturn 
 while Jupiter is moving over the femiellipfe A C P, than 
 while he moves over the femiellipfe P B A, for Saturn is 
 about 400 days longer in the portion b a c of his orbit 5 
 and therefore Saturn will, on the ^hole, be retarded, and 
 Jupiter accelerated. 
 
 Now, it is a fa^t that the aphelion of Saturn Is 70 
 degrees more advanced on the ecliptic than that of Ju- 
 piter. Therefore thefe changes muft happen, and tlie 
 retardations of Saturn muft exceed the accelerations. 
 They do fo, nearly in the proportion of 3^*3 to 352. 
 This excefs will continue for about 2000 years, when 
 the angle AS/) will be 90 degrees complete. It will 
 then begin to decreafe, and will continue decrealing for 
 i6coo years, after which Saturn will be accelerated, and 
 Jupiter will be retarded. The prefent retardation of Sa- 
 turn is about 2', or a day's motion, in a century, and the 
 concomitant acceleration of Jupiter is about half as much. 
 (See Mm- Acad. Par, 1746.) 
 
 M. de la Place has happily fucceedcd in account.- 
 • 3 C 2 ing 
 
5&8 PHYSIC iiL ASTROXOMY. 
 
 ing for feveral irregukritles in this gradual change ot the; 
 mean motions of thefe two planets, which had confiderably 
 perplexed the aftronomers in their attempts to afcertaitv 
 their periods and their maximum by mere obfervation. 
 Thefe were accompanied by an evident change in the 
 elliptical equations of the orbit, indicating a change of 
 eccentricity. M. de la Place has fnewn that all are 
 precife confequences of univerfal gravitation, and depend 
 on the near equality of five times the angular motion of 
 Saturn to twice that of Jupiter, while the deviation from 
 perfect equality of thofe two motions introduces a varia- 
 tion in thefe irregularities, wliich has a very long period 
 (about 877 years). He has at lail given an equation, 
 which exprefTes the motions with fuch accuracy, that the 
 calculated place agrees with the modern obfervations, 
 and with the moft ancient, without au error exceed? 
 ing 2'. (See Mem, Acad Far^ i/^S-) 
 
 505. In confequence of the mutual gravitation of 
 the planets, the node of the difturbed planet retreats on 
 the orbit of the difturbing planet. Thus, let EK (fig. 56.) 
 be the plane of the dillurbing planet's orbit, and let Ji B 
 be the path of the other planet, approaching to the node 
 N. As the difturbing planet is fomewhere in the plane 
 EK, its attraction for A tends to make A approach 
 that plane. We may fuppofe the oblique attraction re- 
 folved into two forces, one of which is parallel to E K, 
 and the other perpendicular to it. Let tliis iafl: be fach 
 that^ in the time that the planet A, if not dillurbedj 
 
MOTION OF THE ?LANETART KODES. 3^^^ 
 
 would move from A to B, the perpendicular force would 
 caufe it to defcrlbe the fmall fpnce AC. By the com- 
 bm^d a£i:ion of this force AC with the motion A B, the 
 planet defcribes the diagonal A D, and croilcs the plane 
 EK in the point «. Thus the notie has iliiftcd from N 
 to Tiy m a diredliion contrary to that of the planet^s mo- 
 tion. The planet now proceeds in the line n o^ getting 
 to the otlier fuie of the plane E K. The attraction of 
 the diilurbing planet now becomes oblique again to the 
 plane^ and is partly employed in drawing A (now in a) 
 toward the plane. Let this part of the attraction be a- 
 gain reprefented by a fmall fpace a c. This, compound- 
 ed with the progreirive motion a by produces a motion in 
 the diagonal a r/, as if the planet had come, not from «,. 
 but from N', a point flill more to the weftward, Tlie 
 node feems again to have fliifted in aitfecedentiu ftgnomm. 
 And thus it appears that, botli in approaching the node, 
 and in quitting the node, the node itfelf fliifts its place, 
 in a direction contrary to that of the motion of the dif- 
 turbed planet. 
 
 It is farther obfervable that the inclination of the dif. • 
 turbed orbit increafes while the planet approaches the 
 ncde, and diminiihes during the fubfequent recefs from 
 it. The original inclination A N E becomes A n E, which 
 is greater than A N E. The angle A ;z E or ^ // K is af- 
 terwards changed into a N'K, W"hi^h is Jefs than a n K. 
 
 In this manner we percci^^e that when a planet, hav- 
 ing croiT^d the ecliptic, procc^eds on the other fide of 
 it^ the node recedes, -that i'',' the planet moves as if it 
 
 had 
 
39© PHTSICAL ASTRONOMY. 
 
 had come from a node fituated farther weft on the eclip- 
 tic ; and all the while, the inclination of the orbit to the 
 ecliptic is diminifhing. When the planet has got 90* 
 eaftward from the node which it quitted, it is at the 
 greateft diftance from the ecliptic, and, in its farther 
 progrefs, it approaches the oppofite node. Its path now 
 bends more and more toward the ecliptic, and the in- 
 clination of its orbit to the ecliptic increafes, and it 
 crolTes the ecliptic again, in a point confiderably to the 
 weftward of the point where it crofled it before. 
 
 The confequence of diis modification of the mutual 
 action of the planets is, that the nodes of all their orbits 
 in the ecliptic recede on the ecliptic, except the node of 
 Jupiter's orbit J J (fig. 57.)> which advances on the e- 
 cliptic EK, by retreating on the orbit S S of Saturn, 
 from which Jupiter fuffers the greatell difturbance *. 
 
 506. We have hitherto confidered the ecliptic as a 
 permanent circle of the heavens. But It now appears 
 that the Earth mull be attracted out of that plane by the 
 
 other 
 
 * As this motion of the nodes, and that of the apfides 
 formerly mentioned, become fenfible by continual accumula- 
 tion, and as they are equally fufceptible of accurate medfure 
 and comparifon as the greater gravitations which retain the 
 revolving bodies in their orbits, Mr Machin, profefTor of af- 
 trononiy at Grefham College, propofed them as the fitteft 
 phenomena for informing us of the dillance of the Sun, D^ 
 
 Matthew 
 
PI. L'>- 
 
 -I^.3S>o- 
 
 \i 
 
 It 
 
 n 
 
 Y 
 
 ie 
 
 I 
 
 'ir 
 
 F:l(^..',7 
 
 C)l J3 
 
 ! r 
 
Change of the ecliptic. 3^1 
 
 other planets. As we refer every phenomenon to the e- 
 cllptic by its latitude and longitude in relation to the ap*- 
 parent path of the Sun, it is plain that this deviation of 
 the Sun from a fixed plane, muft change the latitude of 
 all the ftars. The change is fo very fmall, however, that 
 it never would have been perceived, had it not been 
 pointed out to the aftronomers by Newton, as neceflarily 
 following from the univerfal gravitation of matter. The 
 ecliptic (or rather the Sun's path) has a fmall irregular 
 motion round two points fituated about 7I degrees weft- 
 ward from our equinodlial points. 
 
 507. The comets appear to be very greatly deranged 
 in their motions by their gravitation to the planets. The 
 Halleyan comet has been repeatedly fo difturbed, by paf- 
 fing near to Jupiter, that its periods were very confider- 
 ably altered by this action. A comet, obferved in 1770 
 by Lexel, Profperin, and other accurate aftronomers, has 
 been fo much deranged in its motions, that its orbit has 
 been totally changed. Its mean diftance, period, and pe- 
 rihelion diftance, calculated from good obfervations, 
 
 which 
 
 Matthew Stewart made a trial of this method, employing 
 chiefly the motion of the lunar apogee, and has deduced a 
 much greater diftance than what can be fairly deduced from 
 the tranfit of Venus. Notwithftanding fome overfights in the 
 fummations there given of the diflurbing forces, the conclu- 
 fion feems unexceptionable, and the Sun's diftance is, in all 
 probability, not kfs than iiQ or 115 millions of miles. 
 
ti(l 
 
 ec 
 
 ea 
 
 pr( 
 be 
 
 cli 
 
 cr<| 
 we 
 
 ac 
 
 in! 
 Ju 
 cli 
 frc 
 
 th 
 
 tic 
 
 1 
 
 tr 
 pi 
 
Change of the ecliptic. 3^1 
 
 other planets. As we refer every phenomenon to the e- 
 cllptlc by its latitude and longitude in relation to the ap^ 
 parent path of the Sun, it is plain that this deviation of 
 the Sun from a fixed plane, muft change the latitude of 
 all the ftars. The change is fo very fmall, however, that 
 it never would have been perceived, had it not been 
 pointed out to the aftronomers by Newton, as neceflarily 
 following from the univerfal gravitation of matter. The 
 ecliptic (or rather the Sun's path) has a fmall irregular 
 motion round two points fituated about 7I degrees weft- 
 ward from our equinodlial points. 
 
 507. The comets appear to be very greatly deranged 
 in their motions by their gravitation to the planets. The 
 Halleyan comet has been repeatedly fo difturbed, by paf- 
 fing near to Jupiter, that its periods were very confider- 
 ably altered by this action. A comet, obferved in 1770 
 by Lexel, Profperin, and other accurate aftronomers, has 
 been fo much deranged in its motions, that its orbit has 
 been totally changed. Its mean diftance, period, and pe- 
 rihelion diftance, calculated from good obfervations, 
 
 which 
 
 Matthew Stewart made a trial of this method, employing 
 chiefly the motion of the lunar apogee, and has deduced a 
 much greater diftance than what can be fairly deduced from 
 the tranfit of Venus. Notwithftanding fome overfights in the 
 fummations there given of the dillurbing forces, the conclu- 
 fion feems unexceptionable, and the Sun's diftance is, in all 
 probability, not kfs than iiQ or 115 million* of miles. 
 
^2 PHYSICAL ASTRONOMT. 
 
 which had been continued during three months, aglreed 
 with all the obfervations within i' of a degree. In its 
 aphehon, it is a fniail liiatter more remote than Jupiter, 
 and mull have been fo near him in 1 767 (about -^^ of 
 its diftance from the Sun) that its gravitation to Jupiter 
 muft have been thrice as great as that to the Sun. More- 
 over, in its revolution following this appearance in 1770, 
 namely on the 23d of Auguft 1777, it mufi: have come 
 vaftly nearer to Jupiter, and its gravitation to Jupiter 
 muft have exceeded its gravitation to the Sun m.ore than 
 200 times. No wonder then that it has been diverted 
 into quite a dilTerent path, and that afhronomers cannot 
 tell what is become of it. And this, by the way, fug* 
 g^fts fome fingular and momentous reflections. The 
 number of the comets is certainly great, and their courfes 
 are unknown. They may frequently come near the pla* 
 nets. The comet of 1 764 has one of its nodes very clofe 
 to tlie Earth's orbit, and it is very poffible that the Earth 
 and it may chance to be in that part of their refpedtive 
 orbits at the fame time. The effect of fuch vicinity muft 
 be very remarkable, probably producing fuch tides as 
 would deftroy mofl of the habitable furface. But, as its 
 continuance in that great proximity muft be very momen- 
 tary, by reafon of its great velocity, the effeCt may not be 
 fo great. When the comet of 1770 was fo near to Ju- 
 piter, it was /« aphelioy moving Ilowly, and therefore 
 may have continued fome confiderable time there. Yet 
 It does not appear that it produced any derangement in 
 ^e motion of his fatellites. We muft therefore con- 
 clude 
 
LUNAR INEQUALITIES.- 393 
 
 elude that either the comet did not continue in the path 
 that was fuppofed, or that it contained only a very fmall 
 quantity of matter, being perhaps little more than a 
 denfe vapour. Many circumftances in the appearance 
 of comets countenance this opinion of their nature. As 
 they retire to very great difi:ances from the Sun, and in 
 that remote fituation move very flowly, they may greatly 
 difturb each other's motion. It is therefore a reafonable 
 conjetlure of Sir Ifaac Nevi^ton that the comet of 1680, 
 at its next approach to the Sun, may really fall into hinx 
 altogether. 
 
 Of the Lunar Inequalities, 
 
 508. Of all the heavenly bodies, the Moon has at- 
 tracted the greateit notice, and her motions have been 
 the mod fcrupuloufly examined : and it may be added, 
 that of them all flie has been the moil refraftory. It is 
 but within thefe few years paft that we have been able 
 to afcertain her motions with the precifion attained in 
 the cafes of the other planets. Not that her apparent 
 path is contorted, like thofe of Mercury and Venus, run- 
 ning into loops and knots, but becaufe the orbit is conti- 
 nually fl-.ifting its place and changing its form \ and her real 
 motion^ in it are accelerated, retarded, and deflecled, in 
 a great variety of ways. While the afcertaining the 
 place of Jupiter or Saturn requires the employment of 
 five or fix equations, the Moon requires at leaft forty to 
 3D attain 
 
394 PHYSICAL ASTRON'OMr. 
 
 attain the fime exacflnefs. The correftions introduced 
 by thofe equations are fo various, both in their magni- 
 tude and in their periods, and have, of confequence, been 
 fo blended and compHcated together, that it furpafled 
 the power of obfervation to difcover the greateft part of 
 them, becaufe we did not know the occafions which 
 made them neceffary, or the phyfical connexion which 
 they had with the afpe£ts of the other bodies of the fo- 
 lar fyftem. Only fuch as arofe to a confpicuous magni- 
 tude, and had an evident relation to the lituation of the 
 Sun, were filhed out from among the reft, 
 
 509. From all this complication and embarraflment 
 the difcovery of univerfal gravitation has freed us. We 
 have only to follow this into its confequences, as modi- 
 fied by the particular fituation of tl^e Moon, and we get 
 an equation, which w«/? be made, in order to determine 
 a deviation from fimple elliptical motion that ?fjufi refult 
 from the a£i;ion of the Sun. This alone, followed re- 
 gularly into all its confequences, gives, all the great equa- 
 tions which the fagacity of obfervers had difcovered, and 
 a multitude of other corre<fl:ions, which no fagacity could 
 ever have dete£ted. 
 
 Difcimus hinc tandein qua caufi a}'ge?itea Phoebe 
 PaJJibtis baud aquis eat, cur fuhdita nulli 
 HaBenus ajlronomo^ numerorum frana recufat 
 Ohvia confpicimuSy nubem pollente matheft. 
 We have feen (232.) that fmce the Moon accompa- 
 nies the Earth in it« revolution round the Sun, we muft 
 
 Gonclude 
 
LUNAR INECiUALlTIES. 29S 
 
 conclude that fhe Is under the influence of that force 
 which deflects the Earth into that revolution. If, in 
 every inftant, the Moon were impelled by precifely the 
 fame force which then impels the Earth, and if this force 
 Vere alCo in the fame direction, the Moon's motion re- 
 lative to the Earth would not fuflain any change (98.) 
 She would defcribe an accurate ellipfe having the Earth 
 in the focus, and would defcribe areas proportional to 
 the times. But neither of thefe conditions are agreeable 
 to the real ftate of things. The Moon is fometimes 
 nearer to the Sun, and fometimes more remote from him 
 than the Earth is, and is therefore more or lefs attract'- 
 ed by him j and though the diltances of both from the 
 Sun are fometimes equal (as when the Moon is in qua- 
 drature) the direcSlion of her gravitation to the Sun i$ 
 then confiderably diiFerent from that of the Earth's gra- 
 vitation to him. 
 
 Thefe circumftanees change confiderably all her mo- 
 tions relative to the Earth. But, fince the planetary 
 force follows the precife inverfe duplicate ratio of the 
 diftances, we can tell what its intenfity is in every po- 
 Ction of the Moon, in what direction it a^ts, and what 
 deviation it will produce during any interval of time. 
 We may proceed in the following manner. 
 
 510. Let S (fig. 59.) reprefent the Sun, E the Earth, 
 
 moving in the arch A E B. Let the Moon be fuppofed 
 
 to defcribe round the Earth the circle CBOA. Join 
 
 E S arid M S, and let S M cut the Earth's orbit in N. 
 
 3 D 7, Laftiy, 
 
Jp6 PHYSICAL ASTRONOMY. 
 
 Laflly, Let E S be taken as the menfure of the Earth's 
 gravitation to the Sun, and as the fcale on which we 
 eflimate the difturbing forces. 
 
 To learn the magnitude and direction of the force 
 which difturbs the Moon's motion when (lie is in any 
 point M of her orbit, gravitating to the Sun in the di- 
 rection MSj we muil inflitute the following- an»tlogy 
 M S^ : E S' = E S : M G. Then it is evident that if the 
 Moon's gravitation to the Sun be reprefented by ES 
 when fhe is in the points A or B, equally diftant with 
 the Earth, M G will reprefent her gravitation to the Sun 
 when (he is in M ^ for it is to E S in the inverfe dupli- 
 cate ratio of the diftances from him. 
 
 Now this force M G, being neither equal to E S, 
 nor in the fame dire£lion, muft change or difturb the 
 Moon's motion relative to the Earth. We may fuppofe 
 M G to refult from the combined atiion of two forces 
 M F and M H (that is, M G may be the diagonal of a 
 parallelogram M FGH), of which one, MF, is parallel 
 and equal to E S. Were the Earth and Moon urged by 
 the forces E S -and M F only, their relative motions would 
 not be afFeded (98.) Therefore MH alone difturbs this 
 relative motion, and may be taken for its indication and 
 meafure. 
 
 The difturbing force may be othcrwife reprefented, 
 by varying the conditions on which the parallelogram 
 M F G H is formed. It may be formed on the fuppofi- 
 tion that one fide of the parallelogram ftiali have the di- 
 jre£tion M E» And this is perhaps the beft way of re- 
 
 folvine 
 
LUNAR IIJEqUAI.ITIES. 397 
 
 folving M G for the purpofes of calculation, and accord- 
 ingly has been moft generally employed by the great geo- 
 meters who have cultivated this theory. Bat the me- 
 thod followed in this outline was thought more elemen- 
 tary, and moft illuftrative of the effefts. 
 
 The magnitude and direction of this diiturbing force 
 depends on the form of the parallelogram M F G H, 
 ■and confequently on the pro'portion of M F and M G, 
 and on their relative pofitions. We may obtain an caly 
 expreffion of the force M H by the confi deration that the 
 rate of increafe of M S' is double of the rate of increafe 
 of MS. When a line increafes by a very fmall addi- 
 tion, the ratio of the increment of the line to the line is 
 but the h ulf of that of the fquare to the fquare. Thus, 
 let the line MS be fuppofed 100, and ES loi, differ- 
 ing by one part in a hundred. We have M S"* = loooo, 
 and ES- =: 10201, differing by very nearly tv^^o parts in 
 a hundred ; the error of this fuppofition being only one 
 part in ten thoufand. Suppofe M S =: 1000, and ES 
 = 1 00 1, differing by one part in a thoufand. Then 
 M S^ = loooooo, and ES^'zr 1002001, differing from 
 M S^ by two parts in a thoufand very nearly, the er- 
 ror of ths fuppofition being only one part in a million, 
 &c. &c. 
 
 Now the greatefl: difference that can occur between 
 E S and M S is at new and full Moon, when the Moon 
 is in C or O. In this cafe E C is nearly the 390th part 
 v£ E S, and we have E S* ; O S* = 390= : 391% or 
 ^ 390 : 392,o2<5 5 and therefore, in fuppofmg E S^ to OS* 
 
 as 
 
39^ PHYSICAL ASTRONOMY. 
 
 as 390 to 392, we commit an error of no inore than 
 ^^ of y-J-Y, that is TT^T3-> viz. lefs than one part in 
 fifteen thoufand, in the moft unfavourable circumftances. 
 Therefore the diiFerence between N S (or E S) and M G 
 may be fuppofed equal to MD, without any fenfible 
 ^rror, that is, to the double of N M, the difference of 
 NS and MS. Therefore MG--NS = 2MN very 
 nearly, and MG^ — MS, that is, SG = 3MN very 
 nearly. We may alfo take MI for M H without any 
 fenfible error, and may fuppofe E I = 3 M N. For the 
 lines M F, IP, H G, being equal and parallel, and S P 
 nearly coinciding with S G, from which it never deviate* 
 more than 9', E I will nearly coincide with EH, = S G, 
 = 3 M N nearly. 
 
 511. Thefe confiderations will give us a very fimple 
 manner of reprefenting and meafuring the difturbing 
 force in every pofition of the Moon, which will have no 
 error that can be of any fignificance. Moreover, any 
 error that inheres in it, is completely compenfated by an 
 equal error of an oppofite kind in another point of the 
 orbit. Therefore 
 
 Let us fuppofe that the portion of the Earth's path 
 round the Sun fenfibly coincides with the ftraight line 
 AB (fig. 60.) perpendicular to the line OCS, paffing 
 through the Sun, and called the line of the stzigies, as 
 A B is called the line of the quadratures. Let M D 
 crofs A B at right angles, and produce it to R fo that 
 M P = 3 M N. Join R E, and draw M I parallel to it. 
 
 MI 
 
LUNAR INEQUALITIES. 399 
 
 M I will, in all caffS, have the pofitlon and magnitude 
 corrcfponding to the difturbing force. 
 
 Or, more fimply, make E I = 3 M N, taking the point 
 I on the fame fide of AB with M, and draw MI. 
 M I is the difturbing force. 
 
 512. This force M I may be refolved into two, viz. 
 ML, having the direction of the Moon's motion, and 
 M K, perpendicular to her motion, that is, M K lying 
 in the dire<Slion of the radius vedlor M E, and M L hav- 
 ing the direcftion of the tangent. The force M L affe£ls 
 the Moon's angular motion round the Earth, either ac- 
 celerating or retarding it, while the force M K either 
 augments or diminifhes her gravitation to the Earth. 
 
 The diflurbing force M I may alfo be refolved into 
 M R' = 3 M N, and R' I, or ME; that is, into a force 
 always proportional to M N, and in that direction, and 
 another force in the direction of the Moon's gravitation 
 to the Earth. This is ufeful on another occafion. 
 
 513. "When the Moon is in quadrature, the point I 
 coincides with E, becaufe there is no MN. In this 
 cafe, therefore, the force ML does not exift, and MK 
 coincides with M E. The diflurbing force M I is now 
 wholly employed in augmenting the Moon's gravitation to 
 the Earth. The gravitations of the Earth and Moon to the 
 Sun are equal, but not parallel. If E S exprefles the 
 magnitude of the Moon's' gravitation to the Sun, then 
 ME will exprefs (on the fame Ycale) the augmentation 
 
 in 
 
^OO PHYSICAL ASTRONOMY. 
 
 in quadratures of the Moon's gravitation to tiie Eattfi, 
 occafioned by the obliquity of the Sun's action. It i» 
 convenient to take this quadrature augment of the Moon's 
 gravitation to the Earth as the unit of the fcale on which 
 all the diflurbing forces are meafured, and to calculate 
 what fraiflion of her whole gravitation it amounts to. 
 
 514. Let G exprefs the Moon's gravitation to the 
 Sun, g her gravitation to the Earth, and g' the increafe 
 of this gravitation. Alfo let y and m be the length of a 
 fydereal year and of a fydereal month. In order to learn 
 in what proportion the Moon's gravitation to the Earth 
 is affected by the difturbing force, it will be convenient 
 to know Vv'hat proportion its increment in quadrature has 
 to the whole gravitation. We may therefore inftitute 
 the following proportions. 
 
 __ D , ^ ^ E^S . EB ^, 
 
 g' :G=: E B : E S. Therefore 
 
 ESxEB EBxES , , 
 
 The 
 
 ^ ES EB 390 I _ 
 
 * ; — r- = H — — ^- : r> = 2,1833 : i 
 
 f m- l6s,2.S^' 27,322* ^^ 
 
 very nearly. Thus we fee that the Moon's gravitation to the 
 
 Sun is more than twice her gravitation to the Earth, The 
 
 confequence of this is, that even when the Moon is in con- 
 
 junftion, at new Moon, between the Earth and the Sun, her 
 
 path in abfolute fpace is -concave toward the Sun;^ and convex 
 
 towarci 
 
LUNAR INEQUALITIES. 40I 
 
 The Pvioon's mean gravitation to the Earth is thcre- 
 .ore to its increment in the quadratures by the action of 
 rhe Sun, in the duplicate ratio of the Earth's period round 
 the* Sun to the lunar period round the Earth. This is 
 very nearly in the proportion of 179 to i. Her gravi- 
 tittion Is increafed, when in quadrature, about xt-j-* This 
 will diminifli the chord of curvature and increafe the 
 curvature in the fame proportion. 
 
 515. In order to fee what change it fuftains in any. 
 other pofition of the Moon, fuch as M, join E D, and 
 
 draw 
 
 toward the Earth. Everl there fiie i§ defiefted, not toward 
 the Earth, but toward the Sun. This is a very curious, and 
 feemingly paradoxical alTertion. But nothing is better efta- 
 bhihed. The tracing the Moon's motion in abfohite fpace is 
 the completeil demonftration of it. It is not a looped curve, 
 as one, at firfl thinking, would imagine, but a line always 
 concave toward the Sun. Indeed fcarcely any things can be 
 more unhke than the real motions of tlie Moon are to what 
 we firil imagine them to be. At new Moon, (he appears to 
 l)e moving to the left, and we fee her gradually pafling the 
 }lars, leaving them' to the right ; and, calculating from the 
 diil:ance 240000 miles, and the angular motion, about half a 
 degree in an hour, we fliould fay that flie is moving to the 
 left at the rate of 38 miles in a minute. But the faCi is that 
 ihe is then moving to the right at the rate of 11 00 miles in a 
 minute. But as t&e Earthy from whence we view her," is 
 moving ^t the rate of 11 46 miles in a minute, the Moon is 
 left behind. 
 
 3E 
 
402 « PBYSICAL ASTRONOMl.^ 
 
 draw DQ perpendicular to EM. It is plain that D Q 
 IS the fine of the angle D E Q, which is twice the angle 
 OEO or CEM, that is, twice the Moon's diftance 
 from the nearefl: fyzigy. O E is the cofine of the fame 
 angle. The triangles M D O and E I K are fimilar. 
 E I is equal to i^ M D. Therefore E K = H M Q, 
 = i|ME -f i-|EO, ufing the fign 4- when D E //z is 
 lefs than 90°, or C E M is lefs than 45°, and the fign — 
 when CEM is greater than 45°. Therefore MK = 
 iME+iiEQ. Therefore, if ^ME be equal to 
 li E Q, that is, if M E be = 3 E Q, M K is reduced 
 to nothing, or the force M I is then perpendicular to the 
 radius vector, or is a tangent to the circle. The angle 
 C E M, or the arch G M, has then its fecant E I equal 
 to thrice its cofme M N. This arch is 54° 44'. There 
 are therefore four points in the circular orbit diilant 
 54"^ 44' from the line of the fyzlgies, where the Moon's 
 gravitation to the Earth is not afFe6led by the a61ioii of 
 tlie Sun. If the arch CM exceed this, the point K 
 will lie within the orbit, as in fig. 60. 2. indicating an 
 augmentation of the Moon's gravitation to the Earth. 
 
 At B, iiEQ=i|EM, and therefore liEQ — 
 -I E M = E M, as before. 
 
 516. At O and at C, i| E O + i E M = 2 E M. 
 Therefore, in the fyzigies, the diminution of the Moon's 
 gravitation to the Earth is double of the augmentation 
 of it in quadratures, or it is ^-^ of her gravitation, to 
 '^e Earth* 
 
 517^ 
 
LUNAR TirEORY. 4O3 
 
 51'/. With refpe£l: to the force ML, it is evidently 
 = li D O or li of the fine of twice the Moon's diftance 
 from oppofitiou or conjunction. It augments from the 
 fyzigy to the o61ant, where it is a maximum, and from 
 thence it diminiflies to nothing in the quadrature. In its 
 maximum ftate, it is about xlo o^ tJie Moon's gravitation 
 to the Earth. 
 
 518. It appears, by conftrucling the figure for the 
 different pofitions of the Moon in the courfe of a luna- 
 tion, that this force M L retards the Moon's motion 
 round the Earth in the firft and tliird quarters C A and. 
 O B, but accelerates her motion in tlie fecond and lall 
 quarters A O and B C. Thus, in lig. 60, M L leads 
 from M in a direction oppofite to that of the Moon's 
 motion eaftward from her conjunction at C to her firft 
 quadrature in A. In fig. 60. 3. ML lies in the direc- 
 tion of her -motion ; and it is plain that M L will be 
 fimilarly fituated in the quadrants C A and O B, as alfo 
 in the quadrants A O and B C. 
 
 All tliefe diflurbing forces depend on the proportian 
 of E B to E S. Therefore, wdiilo E S remains tlie fiime, 
 the diflurbing forces will change in the fame proportion 
 with the Moon's diftance from tlie Earth. 
 
 519. But let us fuppofc that ES changes in th^e 
 courfe of the Earth's motion in her elliptical orbit. Then, 
 did the Sun continue to att with the fame force as be- 
 iore^ ftili the diflurbing force would change in the pro- 
 
 3 E 2 portion 
 
404 PHYSICAL ASTRONOI.ir. 
 
 portion of ES, becoming fmaller as ES becomes greater, 
 becaufe the proportion of EB to ES becomes finaJkr. 
 But, when E S increafes, the gravitation to the Sun di- 
 miniflies in the dupHcatc ratio of ES. Therefore the 
 difturbing force varies in the inverfc proporiion of E S^, 
 
 . E B 
 
 and, in general, is =^ -^ „ , . Therefore, as the Earth 
 
 is nearer to the Sun about -^^ i" January than in July, it 
 follows that in January all the difturbing forces will be 
 nearly ^~ greater than in July. 
 
 What has now been faid mufl; fuffice for an account 
 of the forces which difturb the Moon's motion in the dif- 
 ferent parts of a circular orbit round the Earth. The 
 fame forces operate on the Moon revolving in her true 
 elliptical orbit, but varying, with the Moon's diftance 
 from the Earth. They operate in the fame manner, pro- 
 ducing, not the fame motions, but the fam.e changes of 
 motion. 
 
 520. It would feem now that it is not a very difii- 
 cult matter to compute the motion raid the place of the 
 Moon for any particular moment. But it is one of the 
 moft difficult problems that have employed the t.ilents of 
 the firfl mathematicians of Europe. Sir Ifaac Newton 
 has treated this fubje6l with his ufual fuperiority, in his 
 Principles of Natural Philofophy, and in the fcparate 
 Effay on the Lunar Theory. But he only began the fub- 
 je6i:, and contented himfelf with .marking the prhicipal 
 topics of inveftigation, pointing out the roads that wert^ 
 
LUNAR THEORY. 405 
 
 tQ be held in each, and furnifhlng us with the mathema- 
 tics and tlie methods which were to be followed. In all 
 thefe particulars, great improvements have been made by 
 Euler, D^Alembcrt, Clairaut, and Mayer of Gottingen. 
 This laft gentleman, by a moft fagacious examination and 
 comparifon of the data furniflied by obfervation, and a 
 judicious employment of the phyfical principles of Sir 
 Ifaac Newton, has conilructed equations fo exactly fitted 
 to the various clrcum.ftances of the cafe, that he has made 
 his lunar tables correfpond with obfervation, both the 
 moft ancient and the moft recent, to a degree of exacl- 
 nefs that is -not exceeded in any tables of the primary 
 planets, and far furpaihng any other tables of the lunar 
 motions. 
 
 "We can, ^viih propriety, only make fome very gene- 
 ral obfervations on the effeds of tlie continued action of 
 the diilurbing forces. 
 
 521. In the fyzigies and quadrature, the combined 
 force, arifmg from the IMoon's natural gravitation to the 
 Earth and the Sun's diilurbing force, is direded to the 
 Earth. Therefore the Moon will, notwithftanding the 
 diilurbing force, continue to defcribe areas proportional 
 to the times. But as foon as the Moon quits thofe fta- 
 ticxis, the tangential force M L begins to operate, and the 
 combined force is no longer directed precifcly to the 
 Earth, In the octants, where the tangential force is at 
 Its maximum, it qaufes the combined force to deviat^e 
 
 about 
 
4o6 PHYSICAL ASTRONOMY. 
 
 about half a degree from the radius ve6lor, and therefore 
 confiderably affecls the angular motion. 
 
 Let the Moon fet out from the fecond or fourth oc- 
 tant, with her mean angular velocity. Therefore M L, 
 then at its maximum, increafes continually this velocity, 
 which augments, till the Moon comes to a fyzigy. Here 
 the accelerating force ends, and a retarding force begins 
 to a6^, and the motion is now retarded by the fame de- 
 grees by which it M-as accelerated juil before. At the 
 next o£lant, the fum of the retardations from the fyzigy 
 is juil equal to the fum of the accelerations from the 
 preceding oclant. The velocity of the Moon is now 
 reduced to its mean ftate. But her place is more ad- 
 vanced by 37' than it would have been, had the Moon 
 not been affected by the Sun, but had moved from the 
 fyzigy with her mean velocity. Proceeding in her courfe 
 from this o6tant, the retardation continues, and in the 
 quadrature the velocity is reduced to its iov/eft ilate ; 
 but here the accelerating force begins again, and reilores 
 ihe velocity to its mean ftate in the next ociant. 
 
 Thus, it appears tliat in rhe octants, the velocity is 
 always in its medium ftate, attains a maximum in pall- 
 ing through a fyzigy, and i^ the leaft poiTible in qua- 
 drature. In tlie firft and third oclant, the Moon is 37' 
 eaft, or a-head, of her mean place -, and in the fecond 
 and fourth, is as much to the weft ward cf it ; and in the 
 fyzigies and quadratures her mean and true places are 
 the fame. Thus, when her velocity differs, moft from 
 its i;iedium ftate, her calculated and obfervcd places 
 
 are 
 
LUNAR THKORY — VARIATION. 4O7 
 
 are the fame, and where her velocity lias attained its 
 mean (late, her calculated and obfcrved places ditlbr moit 
 widely. This is the cafe with all aflronomical equa- 
 tions. The motions are computed lirll in their mean 
 Itate ; and when the changing caufes increafe to a maxi- 
 mum, and then diminifh to nothing, the efFe6i:, which if^ 
 a change of place, has attained its maximum by conti- 
 nual addition or dedu£lion. 
 
 522. This alternate increafe and diminution of the 
 Moon's angular motion in the courfe of a lunation was 
 firft difcovercd, or at leaft diftinguifhed from the other 
 irregularities of her motion, by Tycho Brahe, and by 
 him called the Equation of variation. The dedudion 
 of it from the principle of unlverfal gravitation by Sir 
 Ifaac Newton is the mod elegant and perfpicuous fpeci- 
 men of mechanical inveftigation that is to be feen. The 
 addrefs which he has (hewn in giving fenfible reprefen- 
 tations and meafures of the momentary actions, and of 
 their accumulated refults, in ail parts of the orbit-, are 
 peculiarly pleafmg to all perfoiis of a mathematical tafte, 
 and are fo appofite and plain, that the inveftigation be- 
 comes highly inftruiSlive to a beginner in this part of the 
 higher mathematics. The late Dr Matliew Stewart, in 
 his Tnicfs Fhyfical and Maihemaiicaly following Newton's 
 example, has given fomc very beautiful examples of the 
 lame method. ' 
 
 523. We have hitherto confidered the IMocn's orbk 
 as circular, and muil now in(|uire whetherits form will 
 
 fulTer 
 
4C8 PHYSICAL ASTRONOMY. 
 
 fufFer any change. %Ye may expecl that it wlil, fince 
 we fee a very great diilurbing force diminiiiiing its ter- 
 rellrial gravity in the fyzigies, and increafmg it in the 
 quadratures. Let us fuppofe the Moon to fet out from 
 a point 35° 16' fnort of a quadrature. The force INI K, 
 which we may call a centripetal force, begins to act, in- 
 creafing the deflecting force. This muft render the or- 
 bit more incurvated in that part^ and this change will 
 be continued through the whole of the arch extending 
 35° 16' on each fide of the quadrature. At 35'' 16' eaft 
 of a quadrature, the gravity recovers its mean (tate , but 
 the path at this point now makes an acute angle with the 
 radius vector, which brings the Moon nearer to the 
 Earth in paffmg through the point of conjunction or op- 
 pofition. Through the whole of the arch V v, extend- 
 ing 54° 44' on each fide of the fyzigies, the Moon's gra- 
 vitation is greatly diminiflied ; and therefore her orbit in 
 this place is flattened, or made lefs curve than the circle, 
 till at V, 54*^ 44' caft of the fyzigy, the Moon's gravity 
 recovers its mean (late, and the orbit its mean cur- 
 vature. 
 
 5 24. In this manner, the orbit, from being circular, 
 becomes of an oval form, moft incurvated* at A and Bj 
 and lead fo at O and C, and having its longeft diameter 
 lying in the quadratures ; not exadlly however in thoie 
 points, on account of the variation of velocity v/hich we 
 have fhewn to be greateft in the fecond and fourth qua- 
 drants. The longell diameter lies a fmall m.atter fliort 
 
 of 
 
LUNAk THEORY. ^Op 
 
 of the points A and B, tiiat is, to the weftward of them. 
 Sir Ifaac Newton has determined tiie proportion of the 
 two diameters of this oval, viz. A B =r 70 and O C = 69. 
 It may fecm fl range that tlie Moon comes neareft to the 
 Earth wlien her gravity is moll diminiflied ; but this is 
 owing to the incurvation of the orbit in the neighbour- 
 hood of the quadratures. 
 
 c^^. The Moon's orbit is not a circle, but an ellip- 
 fis, having the Earth in one of the foci. Still, however, 
 the above alTertions will apply, by always conceiving a 
 circle defcribed through the Moon's place in the real 
 orbit. But we muft now inquire whether this orbit alfo 
 fufFers any change of form by the a£tion of the Sun. 
 
 Let us fuppofe that the line of the apfides coincides 
 with the line of fyzigies, and that the Moon Is in apogee. 
 Her gravitation to the Earth is diminiflied in conjunc- 
 tion and oppofition, fo that, when her gravitation in pe- 
 rigee is compared with her gravitation in apogee, the 
 gravitations differ more than in the Inverfe duplicate 
 ratio of the diftance. The natural forces in perigee and 
 apogee are inverfely as tlie fquares of tlie diftance. If 
 the diminutions by the Sun's action were alfo inverfely 
 as the fquare of the diftance, the remaining gravitations 
 would be in the fame proportion ftill. But this is far 
 from being the cafe here j for the diminutions are direct- 
 ly as the diftance, and the greateft quantity is taken from 
 the fmalleft force. Therefore the forces thus diminiflied 
 .muft dilfer' In a greater proportion than before, that is, 
 
 3 1^^ ;u 
 
410 PHYSICAL ASTRONO^rY. 
 
 in a greater ratio than the inverfc of the rtjuare of the 
 diftances. * 
 
 Let the Moon come from the apogee of this cUilurb- 
 ed orbit. Did her gravity increafe in the due proportion, 
 file would come to the proper perigee. But it incrcafes in 
 a greater proportion, and will bring the Moon nearer to 
 the focus ; that is, the orbit will become more eccentric, 
 and its elliptical equation will increafe along with the 
 eccentricity. Similar efFedts will refult in the Moon's 
 motion from perigee to apogee. Her apogean gravity 
 being too much diminifhed, fhe will go farther off, and 
 thus the eccentricity and the equation of the orbit will 
 be increafed. Suppofe the Moon to change when in 
 apogee, and that v/e calculate her place feven days after, 
 when fhe fliould be in the vicinity of the quadrature. 
 We apply her elliptical equation (about 6° 20') to her 
 mean motion. If we compare this calculation with her 
 
 real 
 
 * Thus, let the following perigee and apogee diftances bt 
 compared, and the correfponding gravitations with their dimi- 
 nutions and remainders. 
 
 Perigee. Mean. Apogee. 
 
 Diftances 8 10 12 
 
 Gravitations - - - - 144 100 64 
 
 Diminutions - - - - 2 "2^ 3 
 
 Remaining gravities - - 142 97^ 6r 
 
 Now 12*: 8^= 142:63,11. Therefore 142 is to 61 in a much 
 greater ratio than the inverfe of the fquare of the diftance-. 
 
LUNAR THEORY — EVECTION. 4 1 1 
 
 real place, we fliall find the true place alnioil: 2° behind 
 the calculation. We iliouKl find, in like manner, that 
 in the lafl quadrature, Iier calculated place, by means of 
 tlie ordinary equation of the orbit, is more than 2° be- 
 hind the true or obferved place- The orbit has become 
 more eccentric, and the motion in it more un-jquable, 
 and requires a greater equation. This may rife to 7° 40', 
 inflead of (f 20', which correfponds to the mean form of 
 the orbit. 
 
 But let us next fuppofe that the apfides of the orbit , 
 lie in the quadratures, where the Moon's gravitation to 
 the Earth is increafed by the aclion of the Sun. Were 
 it increafed in the inverfe duplicate ratio of the diftances, 
 the new gravities would ftill be in this duphcate pro- 
 portion. But, in the prefent cafe, the greateft addition 
 will be made to the fmalleil force. The apogee and 
 perigee gravities therefore will not differ fufficiently ; and 
 the Moon, fetting out from the apogee in one quadra- 
 ture, will not, on her arrival at the oppofite quadrature, 
 come fo near the Earth as (he othery/ife would h?.ve 
 done. Or, fhould (he fet out from her perigee in one 
 , quadrature, (lie will not go far enough from the Earth 
 in the oppofite quadrature \ that is, the eccentricity of 
 the orbit will, in both cafes, be diminiOied, and, along 
 with it, the equation correfpondlng. Our calculations 
 for her place in the adjacent oppofition or conjunction, 
 made Vsith the ordinary orbital equation,, will be faulty, 
 $nd the errors will be of the oppofite kind to the former. 
 3 E 2 U^hii 
 
412 PHYSICAL ASTRCNO'MY. 
 
 The equation neceifary in the prefent cafe will not cx^ 
 ceed 5° 3'. 
 
 In all intermediate pofitions of the npfides, fimilar a- 
 nomalies will be obferved, verging to the one or the 
 other extreme, according to the pofitlon of the line of 
 the apfides. The equation pro expediendo calculo^ by Dr 
 Halley, contains the corredlions which nniil: be made on 
 the equation of the orbit, in order to bring it into the 
 (late which correfponds with the prefent eccentricity of 
 the orbit, depending on the Sun^s pofitlon in relation to 
 its tranfverfe axis, 
 
 52(5. All thefe anomalies arc dlllinCtly obferved, a- 
 greeing with the dedu(ftions from the effefts of univerfal 
 gravitation with the utmoft precifion. The anomaly it- 
 felt was difcovered by Ptolemy, and the difcovery is the 
 greateft mark of his penetration and fagacity, becaufe it 
 is extremely difficult to find the periods and the changes 
 of this corredion, and it had efcaped the obfervation 
 of Hipparchus and the other eminent aftronomers at 
 Alexandria during three hundred years of continued ob- 
 fervation. Ptolemy called it the Equation of evec tion, 
 becaufe he explained it by a certain fhifting of the orbit. 
 His explanation, or rather his hypothcfis for diredling 
 his calculation, is moft mgenlous and refined, but is thti 
 leaft compatible with other phenomena of any of Ptolc- 
 iny's contrivances, 
 
 527. The deduction of this anomaly from its phyfi- 
 cal principles was a far more intricate and dlilicult talk 
 
 than 
 
LUNAR THF.ORY. 413 
 
 tlian the variation which equation had ftirniflied. It is 
 however accompliflied bv Newton in the completed 
 manner. 
 
 It Is ail interefling cafe of the great fvohlevi of three 
 bodies J which has employed, and continues to employ, 
 the talents and beft efforts of tlie great mathematicians^ 
 In , Mr Macliin gave a pretty theorem, which feem- 
 
 ed to promife great alTiftance in the folution of this pro- 
 blem. Newton had demonftrated that a body, defiedled 
 by a centripetal force directed to a fixed point, m.oved 
 fo that the radius vector defcribcd areas proportional to 
 I he times. Mr Machin demonfi. rated that if deflected by 
 forces directed to tv/o fixed points, the triangle connect- 
 ing it with them (which may be called the plaJin veEIrix) 
 alfo defcribed folids proportional to the times. Little 
 help has been gotten from it. The equations founded 
 on it, or to which it leads, are of inextricable com- 
 plexity. 
 
 528. Not only the form, but alfo the pofition of the 
 lunar orbit, nmfl fuffer a change by the a61ion of the 
 Sun, It was demonftrated (226.) that if gravity de- 
 creafed fafter than in the proportion of --:-, the npfides of 
 an orbit will advance, but will retreat, if .the gravitation 
 decreafe at a flower rate. Now, we have {^tn that 
 Vv'hile the Moon Is within 54° 44' of the fyzigies, the 
 gravity is diminiflied in a greater proportion than that of 
 
 -^— . Therefore the apfides which lie In this part of the 
 
 fy nodical 
 
414 PHYSICAL ASTRONOMY. 
 
 fynodlcal revolution mufl advance. For the oppofite 
 reafons, while they lie within 35^^ 16' of the quadra* 
 tures, they mufh recede. But, fince the diminution in 
 fyzigy is double of the augmentation in quadrature, and 
 is continued through a much greater portion of the or- 
 bit, the apfides muft, in the courfe of a complete luna- 
 tion, advance more than they recede, or, on the whole, 
 they mull advance. They mud advance mod, and re- 
 cede lead, when near the fyzigies J becaufe at this tims 
 the diminution of gravity by the diiturbing force bears 
 the greated proportion to the natural diminution of gra- 
 vity correfponding to the elliptical motion, and becaufe 
 the augmentation in quadrature will then bear the fmall- 
 ed proportion to it, becaufe the conjugate axis of the 
 ellipfe is in the line of quadrature. 
 
 The contrary mud happen when the apfides are near 
 the quadratures, and it will be found that in this cafe 
 the recefs will exceed the progrefs. In the octants, the 
 motion of the apfides in coufcqucnUa is equal to tliclr 
 mean motion ; but their place is mod didant from their 
 true place, the difference being the accumulated fum of 
 •the variations. 
 
 But, fince in the courfe of a complete revolution of 
 tlie Earth and Moon round the Sun, the apfides take 
 every pofition with refpe6l to the line of the fyzigies, 
 they will, on the whole, advance. Their mean progrefs 
 is about three degrees in each revolution. 
 
 £^29. It has been obferved, already, that the invedi- 
 cation jof the cirects of the force ;^IK is much more 
 
 dilTicuU 
 
LUNAR THEORY. ^I^ 
 
 tllfficult than that of the effeas of the force M L. ThitJ 
 laft, only treating of acceleration and retardation, rarely 
 employs more than tlie direft method of fluxions, and 
 the finding of the fimpler fluents which are exprefled 
 by circular arches and their concomitant Hues. But the 
 very elementary part of this fecond inveftigaticn eno-ages 
 us at once in the ftudy of curvature and the variation of 
 curvature ; and its fimpleft procefs requires infniite fe- 
 riefes, and the higher orders of fluxions. Sir Ifaac New- 
 ton has not confidered this queftion in the fame fyilema- 
 tic manner that he has treated the other, but has gene- 
 rally arrived at his conclufions by more circuitous helps, 
 fuggefted by circumftances peculiar to the cafe, and not 
 fo capable of a general application. He has not even 
 given us the fteps by which he arrived at fome of his 
 conclufions. His excellent commentators Le Seur and 
 Jaquier have, with much addrefs, fupplied us with this in- 
 formation. But all that they have done has been very par- 
 ticular and limited. The determination of the motion 
 of the lunar apogee by the theory of gravity is found to 
 be only one half of what is really obferved. This was 
 very foon remarked by Mr Machin, but without beinjr 
 able to amend it •, and it remained, for many years, a fort 
 of blot on the doctrine of univerfal gravitation. 
 
 530. As the Newtonian mathematics continued to 
 improve by the united labours of the firft geniufes of 
 Europe, this invefligation received fuccefiive improve- 
 ments alfo. At laft, M. Clairaut, about the year 1743, 
 
 confidered 
 
4l6 PHY.-ICAL ASTRONOMY*. 
 
 confidered the probit^m of tliefe bodies, muturdly gravi- 
 tating, in general terms. But, finding it beyond tnc 
 reacli of our attainments in geometry, unlcfc; confider- 
 ably limited, he con lined his attention to a cafe whicli 
 fuited the interefting cafe of the lunar motions. He fup- 
 pbfed one of the three bodies immerifely larger than the 
 other two, and at a very great diftance from them ; and 
 the fmalleft of the others revolving round the third in 
 an ellipie little different from a circle ; and limited his 
 attention to the dijhirbajices only of this motion. — With 
 this limitation, he foived the problem of the lunar theory, 
 and conflrucbcd tables of the Moon's motion. But he 
 too found the motion of the apogee only one half of 
 what is obferved. — Euler, and D'Alembert, and Simpfon, 
 had the fame refult ; and mathematicians began to fuf- 
 pe£t that fome other force, befides . that of a gravitation 
 inverfely as the fquare of the diftance, had fome ihare 
 in thefe motions. 
 
 At laft, M. Clairaut difcovered the fource of all their 
 miilakes and their trouble. A t^rm had been omitted, 
 which had a great influence in this particular circum- 
 ftance, but depended on fome of the other anomalies of 
 the IVToon, with which he had not fufpefted any con- 
 nexion. He found that the dillurbances, which he was 
 confidering as relating to the Moon's motion in the 
 fnnple ellipfe, fliould have been confidered as relating 
 to the orbit already afFefted by the other inequalities. 
 When this was done, he found that the motion of the 
 apogee, deduced from the a^Hiion of the Sun, was pre- 
 
 cifelv 
 
LUNAR THEORY. 417 
 
 cifely what Is obferved to obtain. Eulcr and D'Alem- 
 bert, who were employed in the fame invelligation, ac- 
 ceded without fcruple' to M. Clairaut's improvement of 
 his analyfis ; and all are now fatisfied with refpecSt to the 
 competency of the principle of univerfal gravitation to 
 the explanation of all thefe phenomena of the lunar 
 motions. 
 
 531. In the whole of the preceding invelligation, 
 we have confidered the difturbing force of the Sun as 
 afting in the plane of the Moon*s orbit, or we have con- 
 fidered that orbit as coinciding with the plane of the 
 ecliptic. But the Moon's orbit is inclined to the plane 
 of the ecliptic nearly 5°, and therefore the Sun is feldom 
 in its plane. His a£lion muft generally have a tendency 
 to draw the Moon out of the plane in which flie is then 
 moving, and thus to change the inclination of the Moon's 
 orbit to the ecliptic. 
 
 But this oblique force may always be refolved into 
 two others, one of which (hall be in the plane of the 
 orbit, and the other perpendicular to it. The firft will 
 be the difturbing force already confidered in all its mo- 
 difications. We muft now confider the cfFe£l of the 
 other. * 
 
 532- 
 
 * It is very difficult to give fuch a leprefentation of the 
 lunar orbit, inclined to the plane of the ecliptic, that the lines 
 which reprefent the different aff'eftions of the difturbing forct 
 
 3 G may 
 
418 PHYSi'CAL ASTRONOMY. 
 
 532. Let ACBO (fig. 61.) be" the moon's orbie 
 cutting the ecliptic in the line N N' of the nodes, the 
 half N M A N', being raifed above the ecliptic, and the 
 other half N B O N' being below it. llie clotted cir- 
 cle is the orbit, turned on the line N N' till it coin- 
 cide with the plane of the ecliptic. C, O, A and B 
 are, as formerly, the points of fyzigy and quadrature. 
 Let the Moon be in M. Let A E B be the interfedion 
 of a plane perpendicular to the ecliptic. Draw M n 
 perpendicular to the plane A E B, and therefore parallel 
 to the ecliptic, and to O C. Take E I equal to 3 M «, 
 and join ML M I is the Sun's difturbing force (5ii.)> 
 and E M meafures the augmentation of the Moon's gra- 
 vitation when in quadrature. It is plain that M I is in 
 a plane pafling through E S, and interfe£ting the lunar 
 orbit in the line M E, and the ecliptic in the line E L 
 M 1 therefore does not lie in the plane of the lunar orbitj 
 nor in that of the ecliptic, but is between them both. 
 The force M I may therefore be conceived as refolvable 
 into two forces, one of which lies in the Moon's orbit, 
 and the other is perpendicular to it. This refolution will 
 be efFefted, if we draw 1 1 upward from the ecliptic, 
 till it meet the plane of the lunar orbit perpendicularly in /. 
 
 Now 
 
 may appear detached from the planes of the orbit and eclip- 
 tic, and thus enable us to perceive the efficiency of them, and 
 the nature of the effed produced. The moft attentive confi- 
 deration by the reader is neceffary for giving him a diftin^ 
 ''Otion of thefe circumftances. 
 
LUNAR THEORY. 4T9 
 
 Now join M /, and complete the parallelogram M z I w, 
 having M I for its diagonal. The force M I is equiva- 
 lent to M /■ lying in the plane of the Moon's orbit, and 
 M ;// perpendicular to it. By the force M i the Moon 
 is accelerated or retarded, and has her gravitation to the 
 Earth augmented or diminifhed, while the force M m 
 draws the Moon out of the plane N C M ; or that plane 
 is made to fhift its pofition, fo that its interfe£lion N N' 
 iliifts its place a little. The inclination of the orbit to 
 the ecliptic alfo is aiFe£led. Let a plane I i G be drawn 
 through I / perpendicular to the line N N' of tlie nodes. 
 The Hne E G .is perpendicular to this plane, and there- 
 fore to the Hnes G I and G /. Alfo I z G is a right 
 angle, becaufe I i was drawn perpendicular to the plane 
 M / G E. 
 
 Now, if E M be confidered as the radius of the tables^ 
 M n is the fine of the Moon's diflance from quadrature. 
 Call this q. Then E I = 3 ^. Alfo making E I radius, 
 I G is the fme of the node's dillance from the line of 
 fyzlgy. Call this s. Alfo, I G being made radius, I i 
 or M m is the fine of the inclination of the orbit to the 
 ecliptic. Call this /. 
 
 Therefore we have EM:EI=rR:3^ 
 EI:IG = R:s 
 IG:M;/7 = R:/ 
 Therefore E M : M m = R^ : 2 q s i 
 
 and M« = 3EMx^j^'. 
 
 Thus we have obtained an exprefhon of the force 
 M m, which tends to change the pofition and inclination 
 3 G 2 of 
 
420 PHYSICAL ASTRONOMY. 
 
 of the orbit. From this expreflion we may draw feveral 
 conclufions which indicate its different effcfts. 
 
 Cor. I. This force vaniflies, that is, there is no fuch 
 force when the INIoon is in quadrature. For then q, or 
 the Hne M «, is nothing. Now g being one of the nu- 
 
 J I 
 
 merical factors of the numerator of the fraction ^ry-j the 
 
 fraction itfelf has no value. We eafily perceive the phy- 
 fical caufe of the evanefcence of the force M ;;? when 
 M comes into the line of quadrature. VvHien tliis hap- 
 pens, the whole difturbing force has the direction A E, 
 the then radius ve£tor, and is in the plane of the orbit. 
 There is no fuch force as M m in this fituation of things, 
 the difturbing force being wholly employed in augment- 
 ing the Moon's gravitation to the Earth. 
 
 2. The force M w vaniflies alfo when the nodes are 
 in the fyzigy. For there, the fa£tor s in the numerator 
 vanifhes. We perceive the phyfical reafon of this alfo. 
 For, when the nodes are in the fyzigies, the Sun is in 
 the plane of the orbit -, or this plane, if produced, pafles 
 through the Sun. In fuch cafe, the difturbing force h 
 in the plane of the orbit, and can have no part, M ;// 
 a£ting out of that plane. 
 
 3. The chief varieties of the force M m depend how- 
 ever on s, the fine of the node's diftance from fyzigy. 
 For in every revolution, g goes through the fame feries 
 of fucceflive values, and / remains nearly the fiime in all 
 revolutions. Therefore the circumftance which will moft 
 diftinguifli the different lunations is the fituation of the 
 node. 
 
LUNAR THEORY. 4^1 
 
 534. This force bends the Moon's path toivard the 
 ecliptic, when the points M and I arc on the fiime fide 
 of the line of the nodes, but bends it awayyrow the eclip- 
 tic when N lies between I and M. This circumftance 
 kept firmly in mind, and confidered with care, will ex- 
 plain all the deviations occafioned by the force M J7i. 
 Thus, in the fituation of the nodes reprefented in the 
 figure, let the Moon fet out from conjun61:ion in C, 
 moving in the arch C M A O. All the way from C to 
 A, the difturbing force M I is below the elevated half 
 N M N' of the Moon's orbit betv/een it and tlie ecliptic, 
 and therefore the force M m pulls the Moon out of the 
 plane of her orbit toward the ecliptic. The fame thing 
 happens during the Moon's motion from N to C. This 
 will appear by conftru6^ing the fame kind of parallelo- 
 gram on the dbgonal M I drawn from any point between 
 N and C. 
 
 When the Moon has pafled the quadrature A, and is 
 In M', the force M' V is both above the ecliptic, and 
 above the elevated half of the Moon's orbit. This will 
 appear by drawing M'^ perpendicular to E N', and join- 
 ing g r. The line M'^- is in the orbit, and g V is in 
 the ecliptic, and the triangle M'^ T ftands elevated, and 
 nearly perpendicular on both planes, fo that WV is 
 above them both. In this cafe, the force M' m' in pull- 
 ing the Moon out of the plane of her orbit, feparates her 
 from it on that fide which is molt remote from the eclip- 
 tic ; that is, caufes the path to approach more obliquely 
 to the ediptic. The figure 61. B will illullrate this. 
 
 NT 
 
42Z PHYSICAL ASTRONOMY. 
 
 N' r is the ecliptic, and M' N' is the orbit, both feen edge- 
 wife, as they would appear to an eye placed in f, (fig-. 6i.) 
 in the line N N' produced beyond the orbit. The diflurb- 
 ing force, a£ting in the direction M' I', may be refolvcd 
 into Mp in the dire£lion of the orbit plane, and M' m' 
 perpendicular to it. The part M' w', being compounded 
 with the fimultaneous motion M' q^ compofes a motion 
 M'r, which interfecfts the ecliptic in n. "When M' in 
 fig. 6i. gets to M'', the path is again bent toward the 
 ecliptic, and continues fo all the way from N' to B, where 
 it begins to acl in the fame manner as in M' between A 
 and N'. 
 
 535. By the a£lion of this lateral force, the orbit 
 muft be continually fhifting its pofition, and its interfcc- 
 tion with the ecliptic ; or, to fpeak more accurately, the 
 Moon is made to move in a line which does not lie all in 
 one plane. In imagination, we conceive an orbital material 
 line, fomewhat like a hoop, of an elliptical ihape, all in one 
 plane, pafhng through the Earth, and, inflead of conceiving 
 the Moon to quit this hoop, we fuppcfe the hoop itfelf 
 to fliift its pofition, fo that the arch in which the Moon 
 is in any moment takes the direction of the Moon's mo- 
 tion in that moment. Its intcrfedion with the ecliptic 
 (perhaps at a confiderable diflance from the point occu- 
 pied by the Moon) iliifts accordingly. This hoop may 
 be conceived as having an axis, perpendicular to its plane, 
 pafhng through the Earth. This axis will incline to one 
 (idQ from the pole of the ecliptic about five degrees, 
 
 and, 
 
22. 
 
MOTION OF LUNAR NODES. 423 
 
 anJ, as the line NN' of the nodes fliifts round the eclip- 
 tic, the extremity of this axis will defcribe a circle round 
 the pole of tlie ecliptic, diftant from it about 5° all 
 round, juft as the axis of the Earth defcribes a circle 
 round the pole of the ecliptic, difbant from it about 23^ 
 degrees. 
 
 536. When the Moon's path is bent toward the 
 ecliptic, fhe muft crofs it fooner than flie would other- 
 wife have done. The node will appear to meet the 
 Moon, that is, to fhifc to the weflvv^ard, /;; antecedentut 
 fignormny or to recede. But if her path be bent more 
 away from the ecliptic, ihc mud proceed farther befoira 
 file crofs it, and the nodes will Ihift in confequentUj that 
 is, will advance. 
 
 Ccr. I. Therefore, if the nodes have the fi tuatlon 
 reprefented in the figure, in the fecond and fourth qua- 
 drant, the nodes murt; retreat while the Moon defcribes 
 the arch N C A, or tlie arch N'OB, that is, while fhe 
 pafles from a node to the next quadrature. But, while 
 the Moon defcribes the arch AN'^ or the arch BN, the 
 force which pulls the Moon from the plane of the orbit, 
 caufes her to pafs the points N' or N before flie reach 
 the ecliptic, and the node therefore advances, while the 
 Moon moves from quadrature to a node. 
 
 It is plain, that the contrary mufl happen when the 
 nodes are fituated in the firfl and third quadrants. They 
 will advance while the Mocn proceeds from a node to 
 the next quadrature, and recede while ib.Q proceeds from 
 a quadrature to the next node. 
 
 a-. 
 
MOTION OF LUNAR NODES. 423 
 
 anJ, as the line N N' of the nodes fliifts round the eclip- 
 tic, the extremity of this axis will defcribe a circle round 
 the pole of tlie ecliptic, diftant from it about 5° all 
 round, juft as the axis of the Earth defcribes a circle 
 round the pole of the ecliptic, diftant from it about 23^ 
 degrees. 
 
 536. When the Moon's path is bent toward the 
 ecliptic, fhe muft crofs it fooner than flie would other- 
 wife have done. The node will appear to meet the 
 Moon, that is, to fhift to the weflward, in antecedentLt 
 figjiorinn^ or to recede. But if her path be bent more 
 away from the ecliptic, ihe mud proceed farther before 
 file crofs it, and the nodes will fhift In confequentUj that 
 is, will advance. 
 
 Ccr. I. Therefore, if the nodes have the fiturction 
 reprefented in the figure, in the fecond and fourth qua- 
 drant, the nodes mufi retreat while the Moon defcribes 
 the arch N C A, or tlie arch N'OB, that is, while fhe 
 pafles from a node to the next quadrature. But, while 
 the Moon defcribes the arch AN'^ or the arch BN, the 
 force which pulls the Moon from the plane of the orbit, 
 caufes her to pafs the points N' or N before flie reach 
 the ecliptic, and the node therefore advances, \vhile the 
 Moon moves from quadrature to a node. 
 
 It is plain, that the contrary muft happen when the 
 nodes are fituated in the firft and third quadrants. They 
 will advance v/hile the Mccn proceeds from a node to 
 the next quadrature, and recede while fhe proceeds from 
 '^ quadrature to the next node. 
 
424 PHYSICAL ASTRONOMY. 
 
 Cor. 2. Ill each fy nodical revolution of the Moon, 
 the nodes, on the whole, retreat. For, to take the ex- 
 ample reprefented in the figure, all the while that the Moon 
 moves from N to A, the line M 1 lies between the orbit 
 and ecliptic, and the path is continually inclining more 
 and more towards it, and, confequently, the nodes are all 
 this while receding. They advance while the Moon 
 moves from A to N'. They retreat while fhe moves 
 from N' to B, and advance while flie proceeds from 
 B' to N. The time therefore during w^hich the nodes 
 recede exceeds that during which they advance. There 
 will be the fame difference or excels of the regrefs of the 
 nodes when they are fituated in the angle C E A. 
 
 It is evident that the excefs of the arch N C A above 
 the arch B N or A N', is double of the diftance N C of 
 the node from fyzigy. Therefore the retreat or wefter- 
 ly motion of the nodes will gradually Increafe as they 
 pafs from fyzigy to quadrature, and again decreafe as the 
 node pafles from quadrature to the fyzigy. 
 
 Cor. 3. When the nodes are in the quadratures, the 
 lateral force M m is the greatell polTible through the 
 whole revolution, becaufe the fadlor j- in the formula 
 
 ^-y is then equal to radius. In the fyzigies it is 
 
 nothing. 
 
 The nodes make a complete revolution in 6803d 2** 
 55' 18", but with great inequality, as appears from what 
 has been faid in the preceding paragraphs. The exad: 
 determination of their motions is to be feen in Newton's 
 Frincipia^ B. III. Prop. 32. j and it is a very beautiful ex- 
 ample 
 
luVar ine(:vualitie$. 425 
 
 ample of dynamical analyfis. The principal equation a« 
 mounts to 1° 37' 45" at its maximum, and in other fitua- 
 (ions, it is proportional to the fine of twice tlie arch N C. 
 The annual regrefs, computed according to the principles 
 of the theory, does not diiFer two minutes of a degre6 
 from what is aftually obfervcd in the heavens. This 
 wonderful coincidence is the great boaft of the doftrine 
 of univerfal gravitation. At the fame time, the perufal 
 of Newton's inveftigation will fliev/ that fuch agreement 
 is not the obvicus refult of the happy fimplicity of the 
 great regulating power j we Oiall there fee many ab- 
 ftrufe and delicate circumftances, which muft be confi- 
 dered and taken into the account before we can obtain a 
 true ftatement. 
 
 This motion of the nodes is accompanied by a varia- 
 tion of the inclination of the orbit to the ecliptic. The 
 inclination increafes, when the Moon is drawn from the 
 ecliptic while leaving a node, or toward it in approach- 
 ing a node* It is diminifhed, when the Moon is drawn 
 toward the ecliptic when leaving a node, or from it in 
 approaching a node. Therefore, when the nodes are 
 fituated in the firll and third quadrants, the inclination 
 increafes while the Moon palTes from a node to the next 
 ■quadrature, but it diminiflies till flie is 90° from the 
 node, and then increafes till Hie reaches the other node. 
 Therefore, in each revolution, the inclination is incrcaf- 
 ed, and becomes continually greater, while the node re- 
 cedes from the quadrature to the fyzigy ; and it is the 
 greateft poffible when the nodes are in the line of the 
 3 ^i fyzigies. 
 
42(^ PHiSlCAL ASTRONOMT, 
 
 fyzigies, Jind it is then nearly 5° 18' 30". Wlicn the 
 iiodeiT are fituated In the fccond and fourth cjuadrantSy 
 the inclination of the orbit diminlflies wliile flic Moon 
 paflcs from the node to tlie 90th degree j it is increafed- 
 from thenee to the quadrature, and then dimlniilies till 
 the Moon reaches the other node. While the nodes are 
 thus fituated, the inclination diminiilies in every revolu- 
 tion, and is the Icaft of all when the node is in quadra- 
 ture, and the Moon in fyzigy, being then nearly 4° 58', 
 and it gradually increafes again till the nodes reach the 
 line of fyzigy. Whil^ the nodes are in the quadratures, 
 or in the fyzigies, the inclination is not fenfibly changed 
 during that revolution. 
 
 Such are the general efFecls of the lateral force M mv 
 that appear on a flight confideration of the circumftances 
 of the cafe, A more particular account of them caniwt 
 be given in this outhne of the fcience. Wemayjuil 
 add, that the deductions from the general principle agree 
 precifely with obfervation. The mathematical inveftiga- 
 tion not only points out the periods of the different ine- 
 qualities, and their relation, to the refpeftive pofitions of 
 the Sun and Moon, but alfo determines the abfolute mag- 
 nitude to which each of them rifes. The only quantity 
 deduced from mere obfcrvation is the mean inclination 
 of the Moon's orbit. The time of the complete revolu- 
 tion of the nodes, and the magnitude and law of vari- 
 ation of this motion, and the change of inclination, with 
 all its varieties, are deduced from the theory of univer- 
 fal gravitation. 
 
 S^9^ 
 
ANNUAL EQUATION OF THE MOON. 427 
 
 539. There is anot])cr cafe of this problem which 
 IS confiderably dlflcrent, namely, the fatellites of Dr 
 Herfchcl's planet, the planes of whofc orbits are nearly 
 perpendicular to the orbit of the planet. This problem 
 offers fome curious cafes, which deferve the attention of 
 the mechanician ; but as they intcrefl us merely as ob- 
 jects of curiofity, they have not yet been confidercd. 
 
 c;4C. There is flill another confiderable derangement 
 of the lunar motions by the aclion of the fun. We Iiave 
 feen that, in quadrature, the Moon's gravitation to the 
 Earth is augmented ^y^, and tlpt in fyzigy it is dimi- 
 niOied -r-TTr* Taking tlie whole fynodical revolution to- 
 gether, this i^ e<]uivalent, nearly, to a diminution of 
 
 ~r"' ^^* TTT* That is to fay, iji confequcnce of the 
 Sun's a£lion, the general gravitation of the Moon to the 
 Earth is -y^^ Icfs than if tlie Sun were away. If the Sun 
 were away, ^lerefore, the Moon's gravitation v/ould be 
 TTT g^"c^ter' than her prcfent mean gravitation. , The 
 confequence would be, that tlie Moon would come nearer 
 to the Earth. As this would be done without any change 
 on her velocity, and a,s flie now will be retained in a 
 fmaller orbit, ih-e will defcribe it in a proportionally lefs 
 time ; and we can compute cxaclly Iiov/ near (he would 
 come before this increafed gravitation will be balanced by 
 the velocity (224.) We mufl conclude from this, that the 
 mean diftance and the mean period of the Moon whicli 
 we obferve, are greater than her natural diflance and pe- 
 riod, 
 
 3 H 2 From 
 
428 PHYSICAL ASTRONOMY. 
 
 From this It is plain that if any thing Hiall iiicreaie or 
 diminilh the adlion of the Sun, it mull equally increafe 
 or diminifli the diftance which the Moon aflumes from the 
 Earth, and the time of her revolution at that diftance. 
 
 Now there a£lually is fuch a change in the Sun's ac- 
 tion. When the Earth is /// penhclio^ in the beginning 
 of January, fhe is nearer the iSun than in July by i part 
 in 30 J confequently the ratio of E M to E S is incrcafcd 
 by 3^, or in the ratio of 30 to 31. But her gravitation 
 (and confequently the Moon's) to the Sun is increafed ~^y 
 or in the ratio of 30 to 32. Therefore the difturbing force 
 is increafed by i part in 10 nearly. The Moon mud there- 
 fore retire farther from the Earth i part in 1790. She 
 muft defcribe a larger orbit, and employ a greater time. 
 
 We can compute exasftly what is the extent of this 
 change. The fydereal period of the Moon is 27^ 7*^ 43'> 
 or 39343'. This muft be increafed ttfoj becaufe the 
 Moon retains the fame velocity in the enlarged orbit. 
 This will make the period 39365', which exceeds the 0- 
 ther 22'. The obferved difference between a Junation in Ja- 
 nuary and one in July fomewhat exceeds 25'. This, when 
 reduced in the proportion of the fynodical to the perio- 
 dical revolution, agrees with this mechanical conclufioiv 
 with great exaQ:nefs, when the computation is made M^ith 
 due attention to every circumftance that can affe£l the 
 conclufion. For it muft be remarked that the computa- 
 tion here given proceeds on the legitimacy of affuming a 
 general diminution of yf-g of the Moon's gravitation as 
 equivalent to the variable change of gravity that really 
 
 takes 
 
SECULAR EQUATION OF THE MOON. 429 
 
 takes place. In the particular circumllances of the cafe, 
 this is very nearly exaft. The true method is to take 
 tlie average of all the difturbing forces M K through 
 the quadrant, multiplying each by the time of its a6lion. 
 And, here, Euler makes a fagacious remark, that, if the 
 diameter of the Moon's orbit had exceeded its prefent 
 magnitude in a very confiderable proportion, it would 
 jfcarcely have been pofTible to affign the period in which 
 fhe would have revolved round the Earth ; and the great- 
 eft part of the methods by which the problem has been 
 folved could not have been employed. 
 
 541. There ftill remains an anomaly of the lunar 
 motions that has greatly puzzled the cultivators of pliyfi- 
 cal aftronomy. Dr Halley, when comparing the ancient 
 Chaldean obfervations with thofe of modern times, in or- 
 der to obtain an accurate meafure of the period of the 
 Moon's revolution, found that fome obfervations made by 
 the Arabian aftronomers, in the eighth and ninth centuries, 
 did not agree with this meafure. When the lunar period was 
 deduced from a comparifon of the Chaldean obfervations 
 with the Arabian, the period was fenfibly greater than 
 what was deduced from a comparifon of the Arabian and 
 the modern obfervations •, fo that the Moon's mean mo- 
 tion feems to have accelerated a little. This conclulion 
 \vas confirmed by breaking each of thefe long intervals 
 into parts. When the Chaldean and Alexandrian obfer- 
 vations were compared, they gave a longer period than 
 the Ale.vndriaii compared with the Arabian of the eighth 
 
 century ; 
 
430 PHYSICAL ASTRONOMY. 
 
 century ; and tliis laft period exceeded what is deduced 
 from a comparifon of the Arabian with the modern ob- 
 fervations ; and even the comparifon of the modern ob- 
 fervations with each other Hiews a continued diminution. 
 This conje<n.ure was received by the mechanical philofo- 
 phers with hefitation, becaufe no reafon could be afligned 
 for the acceleration ; and the more that the Newtonian 
 philofophy has been cultivated, the more confidently did 
 it appear that the mean diflances and periods could fuf- 
 tain no change from the mutual action of the planets. 
 Nay, M. de la Grange has at laft demonftrated that, in 
 the folar fyftem as it exifts, this is ftri(ftly tr^e, as to any 
 change that will be permanent : all is periodical and com- 
 penfator)-. Yet, as obfervatron alfo improved, this acce- 
 leration of the Moon's mean motion became undeniable 
 and confpicuous, and it is now admitted by every aftro- 
 nomer, at the rate of about ii" in a century, and her 
 change of longitude increafes in the duplicate ratio of the 
 times. 
 
 Various attem.pts have been made to account for this 
 acceleration. It was imagined by fcvcral that it was ow- 
 ing to the refiftance of the cclcftial fpaees, which, by dc- 
 minifliing the progrefTive velocity of the Moon, caufed 
 her to fall within her preceding orbit, approaching the 
 Earth continually in a fort of elliptical fpiral. But the 
 free motion of the tails of comets, tlie rare matter of 
 which fcems to meet with no fenfible rcfi (lance, rendered 
 this explanation unfatisfadtory. Others were difpofed to 
 ^ think that gravity did not operate inftantaneoufiy through 
 
 the 
 
SECULAR EQUATION OF THE MOON. 43 X 
 
 the wliole extent of its influence. The application of 
 this principle did not feem to bo obvious, nor its effects 
 to be very clear or definite. 
 
 At hft, M. de la Place difcovered the caufe of 
 this perplexing fa£l ; and in a differtation read to the 
 Royal Academy of Sciences in 1785, he fliews tliat the 
 acceleration of the Moon's mean motion nccelTarily urifes 
 from a (mail change in the eccentricity of the Earth's 
 orbit round the Sun, which is now diminiiliing, and will 
 continue to diminifli for many centuries, by the mutual 
 gravitation of the planets. He was led to the difcovery 
 by obferving in the feries which exprefles the increafe of 
 the lunar period by tJie dillurbing force of the Sun (a 
 feries formed of fines and cofines of the Moon's angular 
 motion and their multiples) a term equal to ^4t ^^ ^^^^ 
 angular motion multiplied by the fquare of the eccentri- 
 city of the Earth's orbit. Confequently, when this ec- 
 centricity becomes fmaller, the natural period of the Moon 
 is lefs enlarged by the Sun's aclion, and therefore, if tlie 
 Earth's eccentricity continue to diminifli, fa will the lu- 
 nar period, and this iu a duplicate proportion. Without 
 entering into the difcufGon of this analyfis, which is a- 
 bundantly complicated, we may fee the general effect of 
 a diminution of the Earth's eccentricity in this manner. 
 'X'he ratio of the cube of the mean diftance of the Earth 
 from the Sun to the cube of her perihelion diftance is 
 greater than the ratio of the cube of her aphelion diftance 
 to that of the mean diftance. Elence it follows that the 
 increafe of the mean luiiar period, during the fmaller 
 
 diftances 
 
43^ PHYSICAL ASTRONOMY. 
 
 diftances of the Earth from the Sun, is greater than its 
 diminution, during her greater diftances j and the fum of 
 ail the lunations, during a complete revolution of the 
 Earth, exceeds the fum of the lunations that would have 
 happened in the fame time, had the Earth remained at her 
 mean diftance from the Sun. Therefore, as the Earth's 
 eccentricity diminiflies, the lunar period alfo diminiflies, 
 approximating more and more to her period, undifturbed 
 by the change in the Sun's action. M. de la Place find? 
 the diminution in a century = ii'^,i35, which diifers 
 little from that alTumed by Mayer from a comparifon of 
 obfervations. This centurial change of angular velocity 
 mufb produce a change in the fpace defcribed, that is, in 
 the Moon's longitude, in the duplicate proportion of the 
 time, as in any uniformly accelerated motion. There- 
 fore ii",i35 multiplied by the fquare of the number of 
 centuries forward or backward, will give the correction 
 of the Moon's longitude computed by the prefent tables. 
 La Place finds that, in going back to the Chaldean ob- 
 fervations, we mud employ another term (nearly -^^ of a 
 fecond) multiplied by the cube of the number of centu- 
 ries. With thefe corredtions, the computation of the 
 Moon's place agrees with all obferv'ations, ancient and 
 modern, with moll wonderful accuracy j fo that there 
 no longer remains any phenomenon in the fyftem vi^hich 
 is not deducible from the Newtonian gravitation. 
 
 542. We fliould, before concluding this account of 
 the perturbations of the planetary motions, pay fome at- 
 tention 
 
iNEqUALlTlES OF THE SATELLITES. 433 
 
 tentioii to the motions of the other fecondary pLinetsi 
 and particularly of Jupiter's fateliites, feeing tliat the 
 exa(^ knowledge of their morions is almoft as conducive 
 to the improvement of navigation and geography as that 
 of the lunar motions. But there is no room for this dif- 
 culTion, and we rnuil refer to the dillertations of War- 
 gentin, Profperin, La Place, and others, who have flu- 
 died the operation of phyfical caufes on thofe little pla- 
 nets with great affiduity and judgement, and with the 
 grcateft fuccefs. The little fyftem of Jupiter arid his 
 fateliites has been of immenfe fervice to the philofophi- 
 cal ftudy of the whole folar fyftem. Their motions are 
 fo rapid, that, in the courfe of a few years, many fyno- 
 dical periods are accompliflied, in which the perturba- 
 tions arifmg from their mutual a£lions return again In 
 the fame order. Nay, fuch fynodical periods have been 
 obferved as bring the whole fyftem again into the fame 
 relative fituation of its different bodies. And, in cafes 
 where this is not accurately accomplifned, the deficiency 
 introduces a fmail difference between the perturbations qf 
 any period and the correfponding perturbations of the 
 preceding one ; by which me-^ns another and much longer 
 period is indicated, in w:hieh this difference goes through 
 all its varieties, fwelling to a maximum and again dim*i- 
 niflijng to nothing* Thus the 'fyftem of Jupiter and liis 
 fateliites, as a fort " of epitome of the great folar fyPcem, 
 has fuggefted to the ftgacipus philofopher the proper wajt 
 of fludying the great fyftem, namely, by loohing oin for 
 Cmilar periods in its anonialies, an^ by boldly cifTerting 
 
 3 I the 
 
434' PHYSICAL ASTRONOMY. ' 
 
 the reality of fuch correfponding equations as can be- 
 fhewn to refulf from the operation of univerfal gravita- 
 tion. The fa6i: is, that we have nov^^ the mofl demon- 
 itrative knowledge of many fuch periods and equations, 
 which could not be deduced from the obfervations of 
 many thoufand years. 
 
 In the courfe of this inveftigation, M, de la Grange 
 has made an important obfervation, which he has demon- 
 ftrated in the molt incontrovertible manner, namely, that 
 it neceffarily refults from the fmali eccentricity of the pla- 
 netary orbits — ^their fmall inclination to each other — the* 
 jmmenfe bulk of the Sun — and from the planets all mov- 
 ing in one direction — that all the perturbations that are ob- 
 ferved, nay all that can exijl in this fyilem, are periodical, 
 and are compenfated in oppofite points of every period. 
 He fhevi^s alfo that the greateft perturbations are fo mo- 
 derate, that none but an aftronomer will obferve any dif- 
 ference between this perturbed ftate and the mean ftate 
 of the fyftem. The mean diftances and the mean pe- 
 riods remain for ever the fame. In fhort, the whole af- 
 femblage will continue, almofl to eternity, in a ftate fit 
 for its prefent purpofes, and not difiiinguiftiable from its 
 prefent ftate, except by the prying eye of an aftronomer. 
 
 Cold, we think, muft be the heart that is not affefted 
 by this mark of beneficent wifdom in the Contriver of 
 tlie magnificent fabric, fo manifeft in felefting for its 
 fconnedling principle a power fo admirably fitted for con- 
 tinuing to anfwer the purpofes of its firft formation. 
 And he muft be little fufceptible of moral impreflion who 
 
OF THE FIGURES OF THE PLANETS. 435 
 
 docs not feci himfelf highly obliged to the Being who has 
 made him capable of perceiving this difplay of wifdom, 
 and has attached to this perception fentiments fo pleafing 
 and delightful. The extreme fimplicity of the conftitu- 
 tion of the folar fyftem is perhaps the moil remarkable 
 feature of its beauty. To this circumftance are we in- 
 debted for the pleafure afforded by the contemplation. 
 For it is this alone that has allowed our limited under- 
 ftanding to acquire fuch a comprehenfive body of well- 
 founded knowledge, far exceeding, both in extent and 
 in accuracy, any thing attained in other paths of philofo- 
 phical refearch. But we have not yet feen all the capa- 
 bilities of this wonderful power of nature. Let us there- 
 fore ftiil foUov/ our excellent leader in a new path of 
 inveitigation. 
 
 Of the Figures of the Plaitets, 
 
 544. Sir Ifaac Newton, having fo happily explained 
 all the phenomena of progrejjlve motion exhibited by the 
 heavenly bodies, by fhewing that they are all, without 
 exception, modified examples of deflecSlion towards one 
 another, in the inverfe dupHcate ratio of the diflances, 
 was induced to examine the other motions obferved in 
 fome of thofe bodies, to fee what m.odification thefe mo- 
 tions ;"eceived by the influence of univerfal gravitatioiii 
 The Sun, and feveral planets turn round their axes. The 
 iludy of celeftial mechanifm is not complete, till we fee 
 3 I 2 whethes" 
 
43<5 PHYSICAL ASTRONOMY. 
 
 whether tliis kind of motion is in any way influenced by 
 gravitation^ 
 
 It does not appear, at firft confideration, that there 
 can be any grea.t myftery in the mere rotation of a body 
 round its axis. It feems to be one of the fimplefl me- 
 clianical queftions. But the fa(ft is juft the oppofite. 
 Before the rotative motion that we obferve in our Earth 
 call be fecured-, in the way in which we fee it aclually 
 performed, adjuftments are neceffary, which are very ab- 
 ftrufe, and required all the fagacity of Newton to dif- 
 covor and appreciate ; and it is acknowledged that this is 
 *he department of phyGcal ailronomy where his acute- 
 nefs of difcernment appears the moft remarkable. V It is 
 alfo the clafs of phenomxcna in v/Iiich the effects of uni- 
 vcrfal gravitation are mofc convincingly feen. For this 
 reafon, feme more notice yviU be taken of the rotation of 
 the planets, and of its confequences, than is ufuahy done 
 in our elementary treatifes. But, a? in the other depart- 
 ments, fo here, it is only the more fimple and -general 
 fads that can be confidered. To go a very fmall flep 
 beyond thefe, engages us at once in the mofi: difficult pro-^ 
 blems, which have occupied and ftill occupy the firft ma- 
 thematicians of Europe, and require all the refources of 
 their fciencCo Such difcuffion, however, would be un- 
 fultable here. But without fome attempt of this kind, 
 we muft rcn^ain ignorant of tlie mechaninn of fome plie- 
 tnomena, more familiar and important than many of thofe 
 ^p/hich we have already difcuifed. 
 
 ■When a body turns round an axis, eacli particle de- 
 
 fcribes 
 
CONSEQUENCES Or ROTATION, 437 
 
 I'ciib'js a circle, to which this axis is perpendicular. Nov/ 
 we know that a particle of matter cannot defcribe a circle, 
 unlefs fome dcilecling force retain it in the periphery. 
 In coherent niafTes, this retaining force is fupplied by the 
 cohefion. But even this is a limited thing. A ftone 
 may be fo brifkly whirled about in a fling that tlie cord 
 will break. GrindftOnes are fometimes whirled about in 
 .our manuf^clures with fuch rapidity that they fplit, and 
 the pieces fly ofF with prodigious force. If matters be 
 lying loofe on the furface of a revolving planet, their gra- 
 vitation may be infufficient to retain them in that velo- 
 city of rotation. In every cafe, the force which actually 
 retains fuch loofe bodies on the furface can be found only 
 in their weight , and part of it is thus expended, and 
 they continue to prefs the ground only with the remain- 
 der. If the velocity of rotation be increafed to a certain 
 degree, it may require the whole weight of the body for 
 its fupply. If the velocity dill increafe, the body is not 
 retained, but thrown oiF. If this Earth turned round in 
 84 minutes, things lying on the equator might remain 
 tliere •, but they would not prefs the ground, nor ftretch 
 the thread of a plummet. For this is precifely the time 
 in which a planet would circulate round the Earth, clofe 
 to the furface, moving about 17 tinies falter than a 
 camwn ball. The weight of the body, defleding it 16 
 feet in a fecond, juft keeps it in the circumference of a 
 circle clofe to the furface of the Earth. The Earth, 
 turning as faft, will have the planet always immediately 
 above the fame point of its furface ; and the planet will 
 
 nfot 
 
43 5 THYSICAL ASTRONOMY. 
 
 not appear to have any weight, becaufe it will not de- 
 fcend, but keep hovering over the fame fpot. If the ro- 
 tation were ftiil fwifter, every thing would be thrown 
 off, as we fee water flirted from a mop brifkly whirled 
 round. 
 
 545. As things are really adjufted, this does not 
 happen. But yet there is a certain meafurable part of 
 the weight of any body expended in keeping it at reft, 
 in the place where it lies loofe. At the equator, a body 
 lying on the ground defcribes, in one fecond, an arch of 
 1528 feet nearly. This deviates from the tangent nearly 
 ToV of ^^ inch. This is very nearly -^^ pari of 1 6x5 
 ieety the fpace through which gravity, or its heavinefs, 
 would caufe a fbone to fall in that time. Hence we muft 
 infer that the centrifugal tendency arifmg from rotation 
 is ™-g- of the fenfible weight of a body on the equator, 
 and 2-w of its real weight. Were . this body therefore 
 taken to the pole, it would manifeil a greater heavinefs. 
 If, at the e<iuator, it drew out the fcale of a fpring fteel- 
 yard to the divifion 288, it would draw it to 289 at the 
 pole. 
 
 545. M. Richer, a French mathematician, going to 
 Cayenne in 1672, was directed to make fome aftronomi- 
 cal obfervations there, and was provided with a pendu- 
 Jum clock for this purpofe. He found that his clock, 
 which had been carefully adjufted to mean time at Paris, 
 loft above two minutes every day, and he was obliged to 
 
 iliorten 
 
FlCrtJRE OF THE EARTH. ip^^ 
 
 lliorten the pendulum -^-^ of an inch before it kept right 
 time. Hence he concluded that a heavy body dropped 
 at Cayenne would not fall 193 inches in a fecond. It 
 would fall only about ipa-f. Richer immediately wrote 
 an account of this very fingular diminution of gravity., 
 it was feouted by almoil all the philofophers of Europe, 
 but has been confirmed by many repetitions of the expe- 
 r4ment. Here then is a dire6l proof that the heavinefs 
 ©f a body, whether confidered as a mere preflure, or as 
 an accelerating force, is employed, and in part expended,, 
 in keeping bodies united to a whirling planet. 
 
 547. Thefe confiderations are not nev/. Even in 
 ancient tifnes, men of refiecSlion entertained fuch thoughts. 
 The celebrated Rom.an general Polybius, one of the moll 
 intelligent philofophers of antiquity, is quoted by Strabo, 
 as faying that in confequence of the Earth's rotation, 
 every body was made lighter, and that the globe itfelf 
 ivv-elled out in the middle. [Were it not fo, fays he, the 
 waters of the ocean would ail run to the ihores of the 
 torrid zone, and leave the polar regions dry. Dr Hooke 
 is the firft modern philofopher who profeffed this opi- 
 nion. Mr Huyghens, however, is the firft who gave it 
 the proper attention. Occupied at the time of Richer's 
 remark Vv^ith his pendulum clocks, he took great intereft 
 in this obfervation at Cayenne, and inf^antly perceived 
 the true caufe of the retardation of Richer's clock. He 
 perceived that pendulums mull vibrate more flowly, in 
 proportion as their fituation removes tliem. farther fi-om 
 
 the 
 
;3f40 PHYSICAL ASTRONOMY. 
 
 the axis of the Earth ; and he alTigncd the, proport'ion ct 
 the retardation in different places. 
 
 548. Refuming this fubje£l fome time after, It oc- 
 curred to him, that unlefs the Earth be protuberant ail 
 around the equator, the ocean muft overflow the landsy 
 increafing in depth till the height of the water compen- 
 fated for its diminifhed gravity. He confiders the con- 
 dition of the water in a canal reaching from the furface 
 of tlie equator to the centre of the Earth (fuppofe the 
 canal C O, fig. 33.) and there communicating with a canal 
 C N reaching from the centre to the pole. The water 
 in the laft muil retain all its natural gravity, becaufe its par- 
 ticles do not defcribe circles round the axis. But every par- 
 ticle in the column C O reaching to t?ie furface of the e- 
 quator mud have its weight diminifhed in proportion to 
 • its diftaiice from the centre of the globe. Therefore the 
 whole diminution will be the fame as if each particle loft 
 half as much as the outermoft particle lofes. This k 
 very plain. Therefore thefe two columns cannot balan-ce 
 each other at the centre, unlefs the equatoreal column be 
 loHger than the polar column by -^l-^ (for the extremity 
 of tliis column lofes —^ of its weight by the centrifugal 
 force employed in the rotation). 
 
 Being an excellent and zealous geometer, this fubjecJ 
 feemed to merit his ferious ftudy, and he inveftigated the 
 form that the ocean muft acquire fo as to be /';/ eqitili- 
 brjo. This he did by inquiring what will be the pofition 
 ©f a plummet in any latitude. This he knew muft be 
 
 perpendicular 
 
FIGURE or THE EARTH. 44t 
 
 perpendicular to the furface of ftlU water. On the fup- 
 pofitlon of gravity directed to the centre of the Earth, 
 and equal at all diftances from that centre, he conftrufl- 
 ed tlie meridional curve, which fliould in every point have 
 the tangent perpendicular to the direction of a plummet 
 (ietermined by him on thefe principles. 
 
 549. At this very timej another circumftance gave a 
 peculiar intereft to this queftionof the figure of the Earth. 
 The magnificent proje£l of meafuring the whole arch of 
 the meridian which pafles through France was then carry- 
 ing on. (See $ 267.) It feemed to refult from the compa- 
 rifon of the lengths of the different portions of this arch, 
 that the degrees increafed as they were more foutherly. 
 This made the academicians employed in the meafure- 
 ment conclude that the Earth was of an egg-like fhape. 
 This was quite incompatible with the reafoning of Mr 
 Huyghens. The contell was carried on for a long while 
 with great pertinacity, and fome of the firft mathema- 
 ticians of the age abetted the opinion of thofe astrono- 
 mers, and the honour of France was made a party in the 
 difpute. The opinion of Mr Huyghens, the greateft or- 
 nament of their academy, could not prevail ; indeed his 
 inferences were fuch, in fome refpe^ls, that even the 
 impartial mathematicians were diiTatisfied with them. 
 The form which he afligned to the meridian was very re- 
 markable, confining of two paraboloidal curves, which 
 had their vertex in the poles, and their branches inter- 
 fetled e^ach other at the equator, there forming an angu- 
 
 3 K lar 
 
442 PHYSICAL ASTRONOMY* 
 
 lar ridge, elevated about (i^ven miles above the inrcribed 
 fphere. No fuch ridge had been obferved by the navi- 
 gators of that age, who had often croiTed the equator. 
 Nor had any perfon on fliore at the line obferved that 
 two plummets near each other were not parallel, but fen- 
 fibly approached each other. All this was unHke the or- 
 dinary gradations of nature, in which we obferve nothing 
 abrupt. 
 
 550. Wliile this queflion was fo keenly agitated in 
 France, Mr Newton was engaged in the fpeculations 
 which have immortalized his name, and it was to him an 
 Interefting thing to know what form of a whirling planet 
 was compatible with an equilibrium of all the forces 
 which aft on its parts. He therefore took the queftlon up 
 in its mod fimple form. He fuppofed the planet com- 
 pletely fluid, and therefore every particle is at liberty to 
 change its place, if it be not in perfect equilibrium. 
 The particles all attract one another with a force in the 
 inverfe duplicate ratio of the diftance, and they are at 
 the fame time a£luated by a centrifugal tendency, in con- 
 fequence of the rotation ; or, to exprefs it more accu» 
 rately, part of thofe mutual attraftlons is employed in 
 keeping the particles in their different circles of rotation. 
 He demonftrated that this was poffible, if the globe have 
 the form of an elliptical fpheroid, compreffed at the poles, 
 and protuberant at the equator ^-^ part of the axis. He 
 alfo pointed out the phenomena by which this may be 
 afcertained, namely, the variation of gravity as we re- 
 cede 
 
FIGURE OF THE EARTH. 443 
 
 cede from the equator to the poles, fliewing that the in- 
 crements of fenfible gravity are as the fquarcs of the 
 fines of the latitude. This can cafily be decided by ex- 
 periments with nice pendulum clocks. He fliewcd alfo 
 that the remaining gravity, on different parts of the Earth's 
 furface, is inverfely proportional to the diftance from the 
 centre, when eflimated in the direction of the centre, 
 &c. &c. His demonftration of the precife elliptical form 
 confifts in proving two things : i ft, That on this fuppo- 
 fition, gravity is always perpendicular to the furface of 
 the fpheroid : 2d, That all reftilincal canals leading from 
 the centre to the furface will balance one another. There^ 
 fore the ocean will maintain its form. 
 
 It was fome time before the philofophy of Newton 
 could prevail in France over the hypothefis of the French 
 philofopher Des Cartes ; and the great mathematician 
 Bernoulli endeavoured to fhew that the oblong form of 
 the Earth which had been demonftrated (he fays) by the 
 meafurement of the degrees, was the effe^l of the pref- 
 fure of the vortices in which the Earth was carried 
 about. 
 
 551. Mr Hermann, a mathematician of moft re* 
 fpe61:able talents, took another view of the queftion of 
 the figure of the Earth. Newton had d.emonftrated in 
 the moft convincing manner that particles gravitated to 
 the centre of fimiiar folids, or portions of a folid, with 
 forces proportional to their diftances from the centre. 
 Hermann availed himfelf of this, and of another thcoren; 
 3 K 2 of 
 
444 PHYSICAL ASTRONOMY. 
 
 of Newton founded on it, viz. that fuperficial gravity in 
 different latitudes Is invcrfely as the diftance from tlie 
 centre. * But he obferved that Newton had by no means 
 demonftrated the elliptical form, but had merely afTumed 
 it, or, as it were, guefled at it. This is indeed true, 
 and his application is made by means of the vulgar rule 
 of falfe pofition. Hermann therefore fct himfelf to in- 
 quire what form a fluid will aflume when turning round 
 an axis, its particles fituated in the fame diameter gravi- 
 tating to the centre proportionally to their diftance, yet 
 exhibiting a fuperficial gravity in different parts inverfely 
 as the diftance from the centre. He found it to be an 
 elllpfe, with fuch a protuberancy, that the r. lius of the 
 equator is to the femiaxis in the fubduplicate ratio of the 
 primitive equatoreal gravity to the remaining equatoreal 
 gravity. This gives the fame proportion of the axes 
 which had been alTigned by Huyghens, though accom- 
 panied by a very different form. He then inverted his 
 procefs, and demonftrated the perpendicularity of gravity 
 to the furface, the equilibrium of canals, and fome other 
 conditions that appeared indlfpen fable j and he found all 
 right. This confirmed him in his theory, and he found 
 fault with t)x D. Gregory, the commentator of Newton^^ 
 for adhering to Newton*s form of the elllpfe. He defied 
 them to point out any fault in his own demonftration of 
 
 the 
 
 * * Both of thefe propofitions arc eafily infeired from 
 Art. 463, and need not be particularly infilled on in tins 
 place, for reafons which will foon appear. 
 
FIGURE OF THE EARTH. 445 
 
 ^hc eillptlcal figure, and confideretl this as fuflicicnt for 
 proving the inaccuracy of the Newtonian conje6lure, for 
 it could get no higher name. 
 
 552. By very flow degrees, the French academi- 
 cians began to acknowledge the comprefTed form of the 
 Earth, and to reexamine their obfervations, by which it 
 had feemed that the degrees increafed to the fouthward. 
 They now afFeclcd to find that their meafuremelit had 
 been good, but that fome circumftances had been over- 
 looked in the calculations, which fliould have been taken 
 into the account. But they were not aware that they 
 were now vindicating the goodnefs of their inftruments 
 and of their eyefight at the expence of their judge- 
 ment. 
 
 All thefe things made the problem of the fi^re of 
 the Earth extremely interefting to the great mathemati- 
 cal philofophers. Newton took no part in the further 
 difcuffion, being fatisfied with the evidence which he 
 had for his own determination of the precife fpecles of 
 the terraqueous fpherold. His philofophy gradually ac- 
 quired the afcendancy ; but the comparifon made of the 
 degrees of the meridian argued a fmaller ellipticity than 
 he had afligned to the Earth, on the fuppofition of uni- 
 form denfity and primitive fluidity. He had however 
 fufficiently pointed out the varieties of ellipticity which 
 might arife from a diifercnce of denfity in the interior 
 j)arts. Thefe were acquiefced in, and the mathemati- 
 cians fpeculatcd on the ways by which the obfervations 
 
 and 
 
446 PHYSICAL ASTRONOMY. 
 
 find the theory of univerfal gravitation might be adapted 
 CO each other. But, all this while, the original problem 
 was confidered as too difficult to be treated in any cafe 
 remarkably deviating from a fphere, and even this cafe 
 was folved by ^^ewton and his followers only in an in- 
 direct manner. 
 
 553. The firft perfon who attempted a dire£t gene- 
 ral folution was Mr James Stirling. In 1735 he commu- 
 nicated to the Royal Society of London two elegant pro- 
 pofitions (but without demonftration), which determine 
 the form of a homogeneous fpheroid turning round its 
 axis, and which, w'hen applied to the particular cafe of 
 the Earth, perfectly coincided with Newton's determi- 
 nation. In 1737 Mr Clairaut communicated to our Royal 
 Society, and alfo to the Royal Academy at Paris, very 
 elaborate and elegant performances on the fame fubje<5t, 
 which he afterwards enlarged in a feparate publication. 
 This is the completeft work on the fubjed, and is full 
 of the mofl curious and valuable refearch, in which are 
 difculTed all the circumftances which can afFe£l the quef- 
 tion. It is alfo remarkable for an example of candour 
 very rare among rivals in literary fame. The author, in 
 extending his m'emoire to a more complete work, quits 
 his own method of inveftigation, though remarkable for 
 its perfpicuity and neatnefs, for that of another mathe- 
 matician, becaufe it was fuperior •, and this with unaf- 
 fefted acknowledgement of its fuperiority. The refults 
 of Clairaut's theory perfectly coincide with the Kewto- 
 
 niau 
 
Figure ot^ the earth. 447 
 
 hian theory, making the equatoreal diameter to the polar 
 diameter as 231 to 230, though it is agreed by all the 
 mathematicians that Newton's method had a chance of 
 being inaccurate. So true is the faying of Daniel Ber- 
 noulli, when treating this fubjecl in his theory of the 
 the tides, " The fagacity of that great man [Neiuto?i) faiu 
 ** clearly through a m'lji wljat others can fcarcely difcover 
 ** through a micro/cope. " 
 
 > Mr Stirling had fald that the revolving figure was not 
 an accurate elliptical fpheroid, but approached infinitely 
 near to it. Mr Clairaut's folutlons, in mofl cafes, fup- 
 pofe the fpheroid very nearly a fphere, or fuppofe lines 
 and angles equal which are only very nearly fo. With- 
 out this allowance, the treatment of the problem feemed 
 impra<^icable. This made Mr Stirling's aflertion more 
 credited ; and we apprehend that it became the general 
 opinion that the folutions obtainable in our prefent ftate 
 of mathematical knowledge were only approximations, 
 exad indeed, to any degree that we pleafe, in the cafes 
 exhibited in the figures of the planets, but ftill they were 
 but approximations. 
 
 554. But in 1740, Mr M*LaurIn, in a diflertation 
 on the tides, which fliared the prize given by the Aca- 
 demy of Paris, demonftrated, in all the rigour and ele- 
 gance of ancient geometry, that an homogeneous ellipti- 
 cal fpheroid, of any eccentricity ivhatever, if turning in a 
 proper time round its axis, will for ever preferve its 
 form. He gave the rule for inveftigatlng this form, and 
 
 the 
 
44^ PHYSICAL ASTRONOMY. 
 
 the ratio of Its axes. His final propofitions to tliis puf- 
 pofe are the fame that Mr Stirling had communicated 
 without demonftrationj This performance was much ad- 
 mired, and fettled all doubts about the figure of a homo- 
 geneous fpheroid turning round its axis. It is indeed 
 equally remarkable for its fimplicity, its pcrfpicuity and 
 its elegance. Mr M*Laurin had no occafion to profccute 
 the fubje<5l beyond this fimple cafe. Proceeding on his 
 fundamental propofitions, the mathematical philofophers 
 have made many important additions to the theory. But 
 It ftill prefents many queftions of mod difficult folutlon, 
 yet intimately conne£led with the phenomena of the folai 
 fyftem. 
 
 In this elementary outline of phyfical afironomy, we 
 cannot difcufs thofe things in detail. But it would be 
 a capital defe£l not to include the ge?ieral theory of the 
 figure of planets which turn round their axes. No more, 
 however, will be attempted than to fhew that a homo- 
 geneous elliptical fpheroid will anfwer all the condi- 
 tions that are required, and to give a general notion of 
 the change which a variable denfity will produce In this 
 figure. * 
 
 The 
 
 * The ftudent will confult, with advantage, the original 
 dilTertations of Mr Clairaut and Mr M*Laurin, and tlie great 
 additions made by the laft in his valuable work on Fluxions. 
 The Cofmographta of Frtftus alfo contains a very excellent epi- 
 tome of all that has been done before his time j and the Mc- 
 
 char.lque 
 
riG?)RE OF THE EARTH. 449 
 
 The following lemma from Mr IV'PLaurin mufl be 
 
 pre mi fed. 
 
 ^^^. Let AEBQ and aebq (fig. 64. No. i.) be 
 two concentric and fmiilar ellipfes, having their fhorter 
 axes A B and a b coinciding. Let P « L toiich the in- 
 terior ellipfe in the extremity a of the fliorter axis, to 
 which let P K, a chord of the exterior ellipfe be parallel, 
 and therefore equal. Let the chords af and ^^ of the 
 interior ellipfe make equal angles witli the axis, and join 
 their extremities by the chord y^ perpendicular to it in /. 
 Draw P F and P G parallel to af and ag, and draw 
 F H and P I perpendicular to P K. 
 
 Then, P F together with P G are equal to twice a z, 
 when P F and P G He on different fides of P K. But 
 if they are on the fame fide (as P F' and P G') then 
 2a I is equal to the difference of P F' and P G'. 
 
 Draw K k parallel to P G or ^ gy and therefore e- 
 qual to P Fj being equally inclined to K P. Draw the 
 diameter MCz, bifecling the ordinates KZ', P G, and 
 a gy in iUy J-, and z, and cutting PK in ;/. 
 
 By fimilarity of triangles, we have 
 
 K ?« : K « = P i : P ?i, ~ a z : a Cy z=. a g'.ab. 
 
 Therefore 
 
 chaniqite Cekjle of La Place contains fome very curious and 
 recondite additions. A work of F. Bofcoinch on the Figure 
 of the Earth has peculiar merit. This author, by employing 
 geometrical exprcfiions of the acting forces, wherever it can be 
 done, gives us very clear ideas of the fubjed. 
 
450 PHYSICAL ASTRONOMt. 
 
 Therefore Km -{- T s :Kri -\~V n =.ag :a I?, 
 and K X' (or P F) + P G : 2 P K = 2 ^^ : 2 ^ h 
 
 and P F + P G : 2 ^^ = 2 P K : 2 .7 ^ ; 
 
 Jlnd, by fimilarity of triangles, we have 
 
 PH +PI:2«i=:2PK:2^^. 
 But 2 P It = 2 <? ^. Therefore P H + P I = 2 .7 f, and 
 PP — PH'=2«/'. 
 
 t;^6. Let the two planes AG^B (lig. 63.) AE^B, 
 intcrfe£ling in the line A B, and containing a very fmall 
 nngle G A E, be fuppofed to compreliend a thin elementa- 
 ry wedge or llice of a folid confi fling of gravitating matter. 
 If two planes GPE, FPD, (landing perpendicularly on 
 the plane A D ^ B, contain a very fmall angle E P D, they 
 will comprehend a flender, or elementary pyramid of this 
 flice, having its vert-ex in P, and a quadrilateral bafe 
 GEDF. If two other planes g p e^ fpdy be drawn 
 from another point /, refpeclively parallel to the planes 
 G P E, fp dy they will comprehend another pyramid, 
 having its fides paralloi to thofe of the other, and con-- 
 taining equal angles, and the ekmentary pyramids F P E, 
 fp e^ may therefore be confidered as fimilar. The bafe 
 gedf is not indeed always parallel and fmiilar to G E D F. 
 But for each of them may be fubflituted fpherical fur- 
 faces, having their centres in P and in />, and then they 
 will be fimilar. 
 
 The gravitation of a particle P to the pyramid GPD 
 is to the gravitation oi p to the pyramid gpd as any 
 
 fide 
 
FIGURE OF THE EARTH. 45 I 
 
 fid« P D of tlie one to the homologous fide p d oi the 
 other. This is evident, by what was fliewn in J 462. 
 
 The fame proportion will hold when the abfolute gx?.- 
 vltation in the dire<£lion of the axis of the pyramid is efli- 
 mated in any other direction, fuch as P ;/;. For, draw- 
 ing p n parallel to P ;//, and the perpendiculars D m, 
 d Tiy it is plain that the ratio P D : p d =. V ?n: p ti} 
 -^ D /« : d n. 
 
 This propofition is of moft extenfive ufe. For we 
 thus eftimate the gravitation of a particle to any folid, by 
 refolving it into elementary pyramids ; and having found 
 the gravitation to each, and reduced them all to one di- 
 rection, the aggregate of the reduced forces is the whole 
 gravitation of the particle eftimated in that direction. 
 The application of this is greatly. expedited by the fo.k 
 lowing theorem. 
 
 K,^%. Two particles fimilarly fituated in refpe<rt of 
 fimilar folids, that is to fay, lituated in fnnilar points of 
 homologous lines, have their whole gravitations propor- 
 tional to any homologous lines of the folids. 
 
 For, we can draw through the two particles flraight 
 lines nmilarly pofited in refpecl of the folids, and then 
 draw |>lanes pafling through thofe lines, and through fi- 
 milar points of the folids. The fe61:ions of the folids 
 made by thofe two planes mufl: be fimilar, for they arr- 
 fimilarly placed in fimilar folids. We can then draw o- 
 ther planes through the fame two ftraight lines, contain- 
 ■Jpg with the former planes very fmall equal angles. Tiie 
 3 L 2 fcction^ 
 
452 PHYSICAL ASTRONOMY. 
 
 fe6tions of thefe two planes will alfo be fimllar, and there 
 will be comprehended between them and the two former 
 planes fimilar llices of the two folids. 
 
 We can now divide the flices into two fericfes of fi- 
 milar pyramids, by drawing planes fuch as G P E, gp e^ 
 and FPD, y/>^, of fig. 63. the points P and p being 
 fuppofed in different lines, related to each of the two 
 folids. By the reafonings employed in the laft propofi- 
 tion," it appears that when the whole of each 11 ice is oc- 
 cupied by fuch pyramids, the gravitations to the corref- 
 ponding pyramids are all in one proportion. Therefore 
 the gravitation compounded of them all is in the fame 
 proportion. As the whole of each of the two fimilar 
 flices may be thus occupied by feriefes of fimilar and fi- 
 milarly fituated pyramids, fo the whole of each of the 
 two fimilar folids may be occupied by fimilar llices, con- 
 fifting of fuch pyramids. And as the compound gravi- 
 tations to thofe llices are fimilarly formed, they are not 
 only in the proportion of the liomologous lines of the 
 folids, but they are alfo in fimilar diredlions. Therefore, 
 finally, the gravitations compounded of thefe compound 
 gravitations are fimalarly compounded, and are in the 
 fame proportion as any homologous lines of the folids. 
 
 Thefe things being premifed, we proceed to confider 
 the particular cafe of elliptical fpheroids. 
 
 559. Let AEBQ, aebq (fig. 64.) be concentric 
 and fimilar elUpfes, which, by rotation round their fliorter 
 
 axis 
 
FtGURE OF THE EARTH. 4^3 
 
 axis AabV>, generate Hmilar concentric fpherolds. We 
 may notice tlie following particulars^ 
 
 560. {a) A particle r, on the furface of the inte- 
 rior fpheroid, has no tendency to move in any direction 
 in confequence of its gravitation to the matter contained 
 between the furfaces of the exterior and interior fphe- 
 roids. For, drawing through r - the ftraight line P r / G, 
 it is an ordinate to feme diameter C M, which bifefts it 
 in s. The part r t comprehended by the interior fphe- 
 roid is alfo an ordinate to the fame diameter and is bi- 
 fe£led in /. Therefore P r is equal to t G. Now r 
 may be conceived as at the vertex of two fimilar cones 
 or pyramids, on the common axis P r G. By what was 
 demonftrated in art. 462. & 557> it appears that the gra- 
 vitation of r to the matter of the cone or pyramid whofe 
 axis is r P is equal and oppofite to the gravitation to the 
 matter contained in the fnijlum of the fimilar cone or 
 pyramid, whofe axis is t G. As this is true, in what- 
 ever direction P r G be drawn through r, it follows that 
 r is in equUibrio in every diredtion, or, it has no ten- 
 dency to move in any direction. 
 
 561. (b) The gravitations of two particles P and p 
 (fig. 64. No. 2.) fituated in one diameter PC, are pro- 
 portional to their diftances PC, pQ, from the centre. 
 For the gravitation of p is the fame as if all the matter 
 between the furfaces AEBO and aebq v/ere av, :iy 
 (by the laft article), and thus P and p are fimilarly firu- 
 
 ated 
 
454 PHYSICAL ASTRONOMT. 
 
 .ited on fimilar follds •, and P C and p C are homolo- 
 gous lines of thofe folids ; and the propofition is true, by 
 
 ? 5:58. 
 
 562. [c] All particles equally diilarat from the plane 
 of the equator gravitate towards that plane with equal 
 forces. 
 
 Let P be the particle (fig. 64. No. i.) and P^ a line 
 perpendicular to the axis, and parallel to the equator 
 E Q. Let P ^ be perpendicular to the equator. Let 
 a eh q be the fe<2:ion of a concentric and fimilar fphe- 
 roid, having its axis a h coinciding with A B. Drawing 
 any ordinate fg to the diameter ah oi the interior el- 
 lipfe, join ^y and a g^ and draw PF and PG parallel 
 to af and ag^ and therefore making equal angles with 
 P ^C. Let f g cut a h in /, and draw F H, G I, per- 
 pendicular to P L 
 
 The lines P F and P G may be confidered as the 
 axes of two very ilender pyramids, comprehended be* 
 tween the plane of the figure and another plane interfed;- 
 ing it in the line P ^z L and making with it a very mi-, 
 nute angle. Thefe pyramids are conftltuted according 
 to the conditions defcribed in art. 556. The lines af^ 
 ag are, in like manner, the axes of two pyramids, whofe 
 fides are parallel to thofe of P F and P G. The gravi- 
 tation of P to the matter contained in the pyramids P F 
 and P G, and the gravitation of a to the pyramids af 
 and a g^ are as the lines P F, P G, af^ and ag^ refpec- 
 tively. Thefe gravitations, eftimatcd in the diredlion 
 
fiGURE OF THE EARTH. 45^ 
 
 P ^, ^ C, perpendicular to the equator, are as the Hues 
 P H, PI, a /, a /, refpedively. Now it has been (liewn 
 (555.) that P H 4- P I are equal to ^ / + ^ /. Therefore 
 the gravitations of P t(f this pair of pyramids, when efli- 
 matcd perpendicularly to the equator, is equal to the 
 gravitation of a to the corrcfponding pyramids lying on 
 the interior ellipfe a e b q. 
 
 It is evident that by carrying the ordinate y^' along 
 the whole diameter from b to a, the lines nfy a gy will 
 diverge more and more (always equally) from o b and 
 the pyramids of which thefe lines are the axes, will thus 
 occupy the whole furface of the interior ellipfe. And tlie 
 pyramids on the axes P F and P G, will, in like man-- 
 ner, occupy the whole of the exterior ellipfe. It is alfo 
 evident that the v/hole gravitation of P, eftiniated in the 
 dire£lion P dj arifmg from the combined gravitation 3, to 
 every pair of pyramids ellimated in the fame direiflion, 
 is equal to the whole gravitation of «, ariling from the 
 com.bined gravitation to every correfponding pair of py« 
 ramids. That is, the gravitation of P in the diretStion 
 P d to the whole of the matter contained hi the elemen- 
 tary (lice of the fpherold comprehended between the two 
 planes which interfe^t in the line P«L, is equal to the 
 gravitation of fl to the matter contained in that part of 
 the fame flice which lies within the interior fpheroid. 
 
 But this is not confined to that Hice which has the 
 ellipfe A E B Q for one of its bounding planes. Let the 
 fpheroid be cut by any other plane pafling through the 
 line P«L, It is known Uikt this fe^ion"alf<3 is ^n el- 
 
 liuie, 
 
45^ PHYSICAL ASTRONOMY. 
 
 lipfe, and that it is concentric with and fimilar to the 
 ellipfe formed by the interfeclion of this plane with the 
 interior fpheroid a e b q. They are concentric fimilar el- 
 lip fes, although not hmilar to the generating ellipfes 
 A E B O and a e h q. Upon this fecStion may another 
 flice be formed by means of another fcclion through 
 P ^ L, a little more oblique to the generating ellipfe 
 A E B Q. And the folidity of this fedion may, in like 
 manner, be occupied by pyramids conftituted according 
 to the conditions mentioned in art. 558. 
 
 From what has been demonftrated, it appears that the 
 gravitation of P to the whole matter of'' this flice, efti- 
 mated in the dire£lion perpendicular to P ^ L, is equal 
 to the gravitation of a to the matter in the portion of 
 this flice contained in the interior fpheroid. 
 
 Hence it follows that when thefe flices are taken in 
 every direction through- the line P <7 L, they will occupy 
 the whole fpheroid, and that the gravitation of P to the 
 matter in the whole folid, eftimated perpendicularly to 
 P <2 L, is equal to the gravitation of a to the matter that 
 is contained in the interior fpheroid, eftimated in the 
 fame manner. 
 
 This gravitation will certainly be in the dire6lion per- 
 pendicular to the plane of the equator of the two fphe- 
 roids. For the flices which compofe tlie folid, all pafling 
 through the generating ellipfe A E B Q, may be taken in 
 pairs, each pair confifliing of equal and fimilar flices, 
 equally inclined to the plane of the generating ellipfe. 
 The gravitaiions to each flice of a pair are equal, and 
 
 equally 
 
FIGURE OF THE PLANETS. 457 
 
 equally Inclined to the plane A E B Q. Therefore they 
 Compofe a gravitation in the direction which bife^s the 
 angle contained by the flices, that is, in the direction 
 of the plane A E B O, and parallel to its axis A B, or 
 perpendicular to the equator. 
 
 From all this it follows, that the gravitation of P to 
 the whole fpheroid, when eflimated in the direction P d 
 perpendicular to the plane of its equator, is equal to the 
 gravitation of a to the interior fpheroid a e b q^ which is 
 evidently in the fame direction, being directed to the 
 centre Ci 
 
 In like manner, the gravitation of another particle P' 
 (In the line P a; L), in a dire6lion perpendicular to the 
 equator of the fpheroid, is equal to the gravitation of a 
 to the interior fpheroid a eh q\ for P' may be conceived 
 as on the furface of a concentric and fimiiar fpheroid. 
 "When thus fituated. It is not affected by the matter in 
 the fpheroidal ftratum without it, and therefore Its gra- 
 vitation is to be eftimated in the fame way with that of 
 the particle P. Confequently the gravitation of P and 
 of P', eftimated in a direction perpendicular to the equa- 
 tor, are equal, each being equal to the central gravita- 
 tion of a to the fpheroid a eh q. Therefore all parti- 
 cles equidiftant from the equator gravitate equally to- 
 ward it. 
 
 563. {d) By reafoning in the fame manner, we 
 prove that the gravitation of a particle P in the direc- 
 tion P/7, -perpendicular to the axis A B, is equal to the 
 3 M gravitation 
 
4S^ PHYSICAL ASTRON'OMt. 
 
 gravitation of the particle r/ to the concentric flniiiar 
 fpheroid d a q /^ ^ and therefore all particles equidiftaut 
 from the axis gravitate equaliy in a direiElion perpendi- 
 cular to it. 
 
 564. (e) The gravitation of a particle to the fphe- 
 roid, eftimated in a dire£l:ion perpendicular to the equa- 
 tor, or perpendicular to the axis, is proportional to its 
 diftance from the equator, or from the axis. For the 
 gravitation of P in the dircQion P ^ is equal to the gra- 
 vitation of a to the fplieroid a e h q. But the gravita- 
 ilon of a to the fpheroid a e b q, \\ to the gravitation of 
 AtoAEBOas^CtoAC rc,^8.> Therefore the 
 gravitation of P in the direcLicii V d is to the gravitation 
 of A to the fpheroid A E B Q as aC to A C, or as P ^ 
 to A G ; and the fame may be proved of any other par- 
 ticle. The gravitation of A is to the gravitation of any 
 particle as the diftance AC is to the diftance of that 
 particle. All particles therefore gravitate towards the 
 equator proportionally to tlieir dillances from it. 
 
 In the fame manner, it is demonftrated that the gra- 
 vitation of E to the fpheroid in the direftion E C per- 
 pendicular to the axis, is to the gravitation of any particle 
 P in the fame dire61:ion as E C to P a^ the diftance of 
 that particle from, the axis. 
 
 Therefore, &c. 
 
 S^S* (/) ^^ ^-'^^ "O'^ "^^^ ^^ afcertain the direc- 
 tion and intenfity of the compound or abfolute gravita- 
 
 toon of any particle P. 
 
 For 
 
flGURE OF TflE PLANETS. 459 
 
 For tlil^ purpofe let A reprefent the gravitation of the 
 particle A in the pole, and E the gravitation of a particle 
 E on the furface of the equator j alfo let the force with 
 which P is urged in the dirccflion P d be exprefled by 
 the fymbol /, P dy and let f, P a exprefs its tcndencj 
 in the diredion P a. Wc have 
 
 f,Vd:K:=zVd:AC 
 and A : E = A : E 
 
 and E :/, P ^ = E C : P ti. Therefore 
 /, P ^ :/, P ^ = P ^ X A X E C : A C X E X P^, 
 Now make dC : dv = A X E C ; E x A C, and draw 
 P V. We have now/, F d :f,V a ~? d X d C : P axdv, 
 = F d X 'P a : F a X d V, z:: F d : d V. F is therefore 
 urged by two forces, in the diredions P d and P a, and 
 thefe forces are in the proportion of Fd and dv. There- 
 fore the compound force lias the direction P v. 
 
 Moreover, this compound force is to the gravity at 
 the pole, or the gravitation of the particle A^ as P ^' to 
 A C. For the force P v is to the force P d as P ^' to 
 P d ', and the force P d is to A as F d to A C. There- 
 fore the force F v is to A as P v to A C. 
 
 In like maimer, it may be compared with the force 
 at E. Make ^v C : ^ // - E x C A : A x C E. We flialJ 
 then have /, Fa :f, F d =z F a : a 1/ -/ and the force in 
 the direction P a, wlien compounded with that in the 
 diredion P dy form a force in the direction P //, and 
 having to the force at E the proporiion of P 7* to E C. 
 
 Thus have we obtained the dire6lion of gravitation 
 
 for any individual particle on the furface, and its magni- 
 
 3 ^'i 2. um 
 
4^0 PHYSICAL ASTRONOMr. 
 
 tudc when compared with the forces at A and at E, which 
 are fuppofed known. 
 
 566. (g) )3ut it is neceffary to have the meafure of 
 the accumulated force or pieflure occafioned by the gra- 
 vitation of a column or row of particles. 
 
 Draw the tangent E T, and take any portion of it, 
 fuch as E T, to reprefent the gravitation of the parti- 
 cle E. Join C T, cutting the perpendicular d ^ in ^. 
 Since the gravitations of particles in one diameter are 
 as their diftances from the centre (561.) d^ will exprefs 
 the gravitation of a particle d. Thus, the gravitation of 
 the whole column E C will be reprefented by the area 
 of the triangle CET, and the gravitation of the part 
 E d, or the preffure exerted by it at d, is reprefented by 
 the area ET^ d. We may alfo conveniently exprefs the 
 
 E X E C 
 
 preflure of the column E C at C by , and, in 
 
 A X A C 
 
 like manner, — exprelTes the weight of the co- 
 lumn A C, or the preffure exerted by it at C. 
 
 Should we exprefs the gravitation of E by a line ET 
 equal to E C, the weight of the whole column E C 
 
 E C^ 
 
 would be expreiTed by — , and that of the portion 
 
 EC'-—dC' , . ,Ed X dO _^ 
 E« by , or by it» equal ~ . We 
 
 fee alfo that v/hatever value we afhgn to the force E, the 
 gravitations or prefiures of the columns E C and E d are 
 proportional to EC% and EC — ^/C% or to EC' and 
 }ld X d O. This remark will be foequently referred to. 
 
FIGURE OF THE PLANETS. 461 
 
 567. From thefe obfervations it appears that the two 
 columns A C and E C will exert equal or unequal pref- 
 fures at the centre C, according to the adjuftment of the 
 forces in the direction of the axis, and perpendicular to 
 the axis. If the ellipfe do not turn round an axis, then, 
 in order that the fluid in the columns A C and E C may 
 prefs equally at C, we mult have AxACr=ExEC, 
 or A C : E C = E : A. The gravitation at the pole mufl 
 be to that at the equator as the radius of the equator to 
 the femiaxis. But we fliall find, on examination, that 
 fuch a proportion of the gravitations at A and E cannot 
 refult folely from the mutual gravitation of the particles 
 of a homogeneous fpheroid, and that this fpheroid, if 
 fluid, and at reft, cannot preferve its form. 
 
 568. The fix preceding articles afccrtain the mecha* 
 nical ftate of a particle placed any where in a homogene- 
 ous fpheroid, inafmuch as it is aflFecled folely by the 
 mutual gravitation to all the other particles. We arc 
 now to inquire what conditions of form and gravitating 
 force will produce an exa6t equihbrium in every particle 
 of an elliptical fpheroid of gravitating fluid when turn- 
 ing round its axis. For this purpofe, it k neceflary, in 
 the firft place, that the diredion of gravity, affeded by 
 the centrifugal force of rotation, be every where perpen- 
 dicular to the furface of the fpheroid, otherwife the wa- 
 ters would flow off toward that quarter to which gravity 
 inclines. Secondly, all canals reaching from the centre 
 ,tQ the furface muft balance at the centre, otherwife the 
 
 preponderating 
 
t 
 
 4^2 PHYSICAL ASTRONOMY. 
 
 preponderating column will fubfide, and prefs up the 
 other, and the form of the furface will change. And, 
 laflly, any particle of the whole mafs muft be in equili- 
 briof being equally prefled in every direction. Thefe 
 three conditions feem fufficient for infuring the equili- 
 brium of the whole. 
 
 5(59. Thefe conditions will be fecured in an ellipti- 
 cal fluid fpheroid of uniform denfity turning round its 
 axis, if the gravity at the pole be to the equatoreal gravity^ 
 diminijljed by the centrifugal force arifing from the rotation^ 
 as the radius of the equator to the femiaxis. 
 
 We iliall firft demonftrate that in this cafe gravity 
 will be every where perpendicular to the fpheroidal fur- 
 face. 
 
 Let p exprefs the polar gravity, e the primitive equa- 
 toreal gravity, and c the centrifugal force at the furface 
 of the equator, and let e — <:, = J", be the fenfible gravity 
 remaining at the equator. Then, by hypothefis, we have 
 /» : J = C E : C A. Confidering the ftate of any indivi- 
 dual particle P on the furface of the fpheroid, we per- 
 ceive that that part of its compound gravitation which is 
 in a dire6lion perpendicular to the plane of the equator 
 is not affected by the rotation. It ftill is therefore to the 
 force p at the pole as P^ to AC (564.) But the other 
 conilituent of the whole gravitation of P, which is efti- 
 rnated perpendicular to the axis, is diminiflied by the 
 centrifugal force of rotation, and this diminution is in 
 proportion to its di (lance frorj. the axis; that is, in pro- 
 per tioi: 
 
riGURE CF THE PLANETS* 463 
 
 J)ortion to this primitive conftltuent of its whole gravita- 
 tion. Therefore its remaining gi-avity in a dire 61 ion per- 
 pendicular to the axis is dill in the proportion of its dif- 
 tance from it. And this is the cafe with every individual 
 particle. Each particle therefore may Hill be confidered as 
 urged only by two forces, one of which is perpendicular 
 to the equator and proportional to its diflance from it, 
 and the other is perpendicular to the axis and propor- 
 tional to its diflance from it. Therefore, if v/e draw a 
 line "P V Uy (o that d C may be to d v as /> x E C to 
 J X A Cj Pi' will be the direction of the compound 
 force of gravity at P, as afFe61;ed by the rotation. 
 
 But, by hypothefis /> : / = E C : A C 5 therefore 
 p X EC:/ X AC=:Ee: AC% and EOtAC'zrc'C 
 idvy =V u'.V V. But (Elllpfe 7.) if P u be to P v as 
 E CMo A C% the line V vu is perpendicular to the tan- 
 gent to the ellipfe In the point P, and therefore to the 
 fpheroidal furface, or to the furface of the ftill ocean. 
 
 Thus, then, the firfh condition is fecured, and the fu- 
 perficial waters of the ocean will have no tendency to 
 move in any direction. Having therefore afcertained a 
 fuitable direBmi of the affected gravitation of P, we may 
 next inquire Into its intenfity. 
 
 570. The fenfible gravity of any fuperficial particle 
 P is every where to the poUr gravity as the line P u (tlie 
 normal terminating in the axie) to the radius of meri- 
 dional curvature at the pole 5 and it is to the fenfible gra» 
 vity at the 'equator as tlie portion P^' of the fame normal 
 
 ter'Tiinatii:ig 
 
464 PHYSICAL ASTRONOMY. 
 
 terminating in the equator is to the radius of meridional 
 curvature at the equator. For it was fliewn (565.) to be 
 to the force at E as P « to E C. If, therefore, the ra- 
 dius of the equator be taken as the meafure of the gra- 
 vitation there, P u will meafure the fenfible gravitation 
 at P. And fince the ultimate (ituation of the point //, 
 when P is at the pole, is the centre of curvature of the 
 ellipfe at A, the radius of curvature there will meafure 
 tlie polar gravity. That is, the fenfible gravity at the 
 equator is to the gravity at the pole, as the radius of the 
 equator to the radius of polar curvature. By a perfe£tly 
 fimilar procefs of reafoning, it is proved that if the gra- 
 vity at the pole be meafured by A C, the gravity at P 
 is meafured by P ^', and at the equator by the radius of 
 curvature of the ellipfe in E. 
 
 571. Cor, I . The fenfible gravity in every point P of 
 the furface is reciprocally as the perpendicular C t from 
 the centre on the tangent in that point. For every where 
 in the ellipfe, C ^ X P // = C E% and C / X P -y = C A% 
 as is well known. 
 
 572. Cor. 2. The central gravity of every fuperfi- 
 cial particle P, that is, its abfolute gravity V u or P -y 
 eftimated in the dire6tion P C, is inverfely proportional 
 to its diftance from the centre, that is, the central gra- 
 vity at P is to the central gravity at E as E C to P G, 
 and to the polar gravity as A C to P C. For, if the gra- 
 vity P !> be reduced to the diredion P C by drawing vo 
 perpendicular to C P, Vq will meafure this central gra- 
 vity. 
 
FI&URH OF THE PLANETS. 465 
 
 VXy. Now, it is well known that P X P C is every 
 wJiere = A C" ; and, in like manner, P « X P C = E C*. 
 Therefore P c, or P *^, are every where reciprocally as 
 PC. 
 
 Henc« it follows that the fenfible increment of gra- 
 vity in proceeding from the equator to the pole is very 
 nearly as tlie fquare of the fine of the latitude ; for, with- 
 out entering on a more curious inveftigation, it is plain 
 that the increments of gra%'ity, when fo minute in com- 
 parifon witli the whole gravity, are very nearly as the 
 <lecrements of the diftance. Now, in a fpheroid very 
 little compreiTed, thefe decrements are in that proportion. 
 It may be demonllrated that in the latitude where fm.* 
 =r i-, namely, lat. 35° 16', the gravity is the fame as to 
 a perfect fphere of the fame capacity, having for its ra- 
 tlins the femidiameter of the ellipfe in that point. It is 
 alfo a dillinguifliing property of this latitude that, if this 
 femidiameter be produced, the gravitation of a particle, 
 at any diftance in this dire<flion, is the fame as to a per- 
 fecSl fphere of the fame capacity. This is not the cafe 
 in any other dire cl ion. 
 
 573. Cor. 3. Laftly, the force efllmated in the di- 
 rection P^/ is to the force in the diretlion P^ as EC* X 
 P^ to AC* X P«. For we had (564.)/, PJ:/, Va 
 = A X E C X P ^/ : E X A C X P ^^, which, by fubitif 
 tuting p and J- for A and E, it becomes / X E C X 
 P^:j^ X AC xP^, =EC* XP^: AC* XP^. 
 
 Hitherto we have confidered only the particles on the 
 3 N furfuce 
 
456 PHYSICAL''ASTRONOMt. 
 
 furfacc of the fpherold. But we mud know the concIi-» 
 tion of a particle any M-hcre within it, 
 
 574. A particle /', in any internal point of a dia- 
 meter, ha? its fcnfible gravity in the direction perpendi- 
 cular to tlie furface of a concentric and fimilar fpheroid 
 pafling through the particle. For the gravity at/ is com- 
 pounded of forces perpendicular to the axis and to the 
 equator, iind proportional to the diftances from them, 
 and therefore proportional to the fimilar forces adling on 
 the particle P (558.) Therefore the compound force of 
 p will be parallel, and in the fame proportion, to the 
 compound force Pi; of P, and mull therefore be per- 
 pendicular to the tangent of the furface in p. It is as 
 pv\ 
 
 575. Ccr.^ Hence we mull Infer that if there were 
 a cavern at />, containing w^ater, the furface of this ftill 
 water would be a part of the fpheroidal furface aebq. 
 Should this cavern extend all the way to e or ^, the wa- 
 ter (hould arrange itfelf according to this furface \ or, if 
 er p be a pipe or conduit, the water in it fhould be ftill, 
 except fo far as it is aife<Sled by the prelTures of the co- 
 lumns A a and P / and E e (thefe preflures will be 
 proved to be equal). 
 
 It would feem, from thefe premifes, that if the el- 
 liptical fpheroid confift of different iluids, which do not 
 mix, and which differ in denfity, they will be difpofed 
 in concentric funilar elliptical ftrata, fo that their bound- 
 ing 
 
FIGURE OF THE PLANlnS. 467 
 
 ing furfaees fliall be fimilar. The proof of this feems 
 the fame with what is received for a demon ft ration of tlie 
 horizontal furface of the boundary between water and 
 oil contained in a velTel. Accordingly, this has been fup- 
 pofed by many refpe6table writervS, as a thing that needed 
 no other proof. But thi-s is by no means tlie cafe. It 
 can be ftri£lly demonftrated tliat the dcnfer fluids occupy 
 the loweft place, and that the ftrata become lefs and lefs 
 eccentric as we approach the centre, where the ultimate 
 evanefcent figure may be denominated a fpherical point. 
 It may be fecn, even at prefent, that they cannot be fmii- 
 lar, unlefs homogeneous. For, without this condition, 
 it cannot be generally demonftrated that the gravitation 
 of a particle p to the ecjuato^, and to the axis, is as the 
 diftance from them, which is the foundation of ;iJ.I the 
 fubfequent demonftrations. 
 
 576. In the next place, all redilineal columns, ex- 
 tending from the centre to the furf^ice, will balance in 
 the centre. For, drawing vo^ i>' g' perpendicular to PC, 
 it is plain that P and p 0' reprefcnt the gravities of P 
 and /) eilimated in the direction PC. Now l^cipo'zr. 
 V C : p C, Therefore the gravitation of the whole co- 
 
 P (5 X P C 
 
 lurnn, or tlie prefliire on C, is rcprefcnted by — 
 
 (566.) Now, in the ellipfe Vo X PC = CA% a con- 
 ftaiit quantity. Therefore the preffure of every column 
 at C is the fame. In lik^ manner, the prefTure of the 
 columns, C^ and Ca are equal, and therefore alfo the 
 3 N 2 preflures 
 
4^8 PHYSICAL ASTRONOM'if. 
 
 prefTures of P/, E f, and A a, at py e, and- ^, are aD 
 equal. 
 
 577, Lallly, any particle of the fluid Is ccjually 
 prefled in every direction, and if the whole were fluid, 
 would be /;/ equilihrio, and remain at reft. 
 
 To prove this, let V p (fig. 64. 3.) be a column reach- 
 ing from P to the furface, and taken in any direction, 
 but, firft, in one of the meridional planes, of which A B 
 is the axis, and E Q the interfection by the equatoreal 
 plane. In the tangent A a take A a equal to E C, and 
 A u equal to A C. Draw aQe and a C g to the tangent 
 E £ at the equator. It is evident that E ^ :;r A C, and 
 E g = E C. Through p and P draw the lines p L l, 
 N P z, parallel to E C, and the lines /) N (p, I P 5 parallel 
 to A B. Draw alio IKk parallel to E C. 
 
 Since, by hypothefis, the whole forces at A and E 
 are inverfely as A C and EC, A a and E e are as the 
 forces acting at A and E. Confequently, the weights of 
 the columns F D, L Z, and K L, will be reprefentcd by 
 the areas Vfd D, L Iz Z, and KkIL {^66.) 
 
 All the preflures or forces which att on the particles 
 of the column p P may be refolved into forces acting pa- 
 rallel to A C, and forces acting parallel to E C, and the 
 force atting on each particle is as its diftance from the 
 axis to which it is directed ('564.) Therefore the whole 
 force with which the column p P is prelTed in the direc- 
 tion A C is to the force Mith which the column O P is 
 prefixed in the lame direction, as the number of particles 
 
 i.U 
 

 Zi^^^f^'^^ 
 
 „ ^"^^1. ■ 
 
 / 
 
 (^--- '■-■^y'L: 
 
 ^V^HOx 
 
 
 
 .^^^^\- 
 
 \ 
 
 
 ■■ \/ J 
 
 xv^ 
 
 
 
 
 '^-^^A. 
 
 c^ 
 
 
 1 
 
 '■ 
 
 F Lg / 04 
 
ricuRK or TiiF. ri.ANr,T«;. 469 
 
 in pV to the num])ci- in () ?, tli.vf is, as pV to O P. 
 But there is only a part of this force employed in prelhng 
 the particles in the dire(Si:if)ii of the canal. Another part 
 merely prefles the fluid to the iide of the cairal p P. Draw 
 O g perpendicular to p P. The force a£ling in the di- 
 rection A C on any parti-cle in /> P is to Its efficacy in 
 the direction / P as OP to i' P, that is, as /> P to O P 
 Therefore, the prefllire which the particle P fuflains in 
 the direction p P, from the action of all the particles in 
 p P in the direction A C, is precifely equal to the prcf- 
 fure it fuftains from the a6tion of the column O P, act- 
 ing in the fame direction A C. But it has been fliewn 
 (566.) that the preiTure of OP in the direction A C is 
 precifely the fame with the weight of the column L Z, 
 which weight is reprefented by the area L /s Z. 
 
 In the very fame manner, the whole prefllire on P 
 in the direction p P arifing from the preflure of eacli of 
 the particles in /> P In the direction E C, is precifely the 
 fame with the preflure on P, arifmg from the preiTure of 
 t]ie 'column NP in this direction E C, that is, it is equal 
 to the weight of the column F D, which is reprefented 
 by the area ¥fdT). 
 
 Bccaufe E j is equal to E C, we have F <p ^ D = 
 
 , = -^- , ='^ . And m 
 
 T^ T I ^ X ^ ' Tl ^ ^ 
 
 like manner, Kx a L = . Bnt p O X O m : 
 
 2 
 
 I O X O / = E C^ : A C% and therefore 
 
 F <P ^ D : K ;-. A L = E C : A C* 
 but K>caL:K>^'/L = AC :EC 
 
 mi 
 
ricrRK OF TiiF. rr.ANr.T<;. 469 
 
 in /) P to the iiiim])cr in () ?, tli.vf is, uS pV to O P. 
 But there Is only a part of this force employed in prelFnig 
 the particles in the direction of the canal. Another part 
 merely prefles the fluid to the iide of the cairal p P. Draw 
 O g perpendicular to p P. The force afting in the di- 
 rection A C on any particle In /» P is to its efiicacy iu 
 the direction / P as OP to ^ P, that is, as pV to O P 
 Therefore, the prelTure which the particle P fuftalns In 
 the direction p P, from the action of all the particles in 
 p P in the direction A C, is precifely equal to the pref- 
 fure it fuftains from the aclion of the column O P, act- 
 ing in the fame direftion A C. But It lias been flicwn 
 (566.) that the preiTure of O P in the direclion AC is 
 precifely the fajne with the weight of the column L Z, 
 which weight is reprefented by the area h /zZ. 
 
 In the very fame manner, the whole prcfllire on P 
 in the direcl:ion p P arifing from the prelTure of each of 
 the particles in / P In the direction E C, is precifely the 
 fame with the preflure on P, arifing from the prcfTurc of 
 tLe 'column NP in this direction E C, that is, it is equal 
 to the weight of the column F D, which is reprefented 
 by the area YfdD. 
 
 Bccaufe E g is equal to E C, we have F <p 5 D = 
 CF^ — CD^ J.p^ — LCy _pOxOm 
 
 222 
 
 And 
 
 m 
 
 like manner, K » a L = . But p O X O in : 
 
 2 
 
 I O X O / =: E C^ : A C% and therefore 
 
 F ^ ^ P : K X. A L = E C^ : A C* 
 but KxaL:KZ'/L = AC:EC 
 
47© "I'HYSICAL ASTRONOAiy. 
 
 niid 7fd D:F^5D=:AC:EC, therefore 
 
 YfdD : K Z /L = E e X A e : A e X E C% 
 
 that is, in the ratio of cquaUty. Now the area K ^ / L 
 reprefents the weight of the column K L, or the preflure 
 exerted in the diredion A C by the column I O. 
 
 Thus it appears that when the forces a6ling on the 
 particles in the column p P are eftimated in the direQion 
 of the canal, the preflure exerted on the particle P is 
 equal to the united preflures of the columns O P and I O 
 acting in the dire£lion A C, that is, to the preflure of 
 the fluids in -the canal I P in its own diretlion. * There- 
 fore the fluid in the canal I P will balance the fluid in 
 the canal p P, and the particle P will have no tendency 
 to move in either direction. And, fmce this is equally 
 true, whatever may be the dire£lion of the canal P/, or 
 P TTj it follows tliat tlie particle P is equally prefl"ed in 
 every direction in the plane of the figure, and would re- 
 main at reft, if the whole fpheroid were fluid. 
 
 But now let the canal l?p be in a pl.uie diF ent from, 
 a meridional plane (as in fig. 64. 4.) In whatever direc- 
 tion 
 
 * The fl-udent muft not confound this with a compofltion 
 of two prefliires or forces II P and O P, ccmpofing a prcfl'ure 
 or force p P. There is no fuch compcfuion in the prefent 
 cafe. It is only meant that the prefliye in the diredlion p P 
 arifing from the gravitation of the particles in tlie car.?.l, is the 
 fame, in refpetl of magnitude, Vvith the prefllire in f!:e direc- 
 tion I P, arifing from the gravitation of the fluid in I P. 
 
riGUREOF THE PLANETS. 47I 
 
 tion V p is dlfpofccl, a plane may be made to pafs tli^ugh 
 it, perpendicular to the plane E ^' O ^ of the equator of 
 the fpheroid. Let p\qi c be this plane. Its feclion 
 with the fpheroid will be an ellipfe, fimilar to the gene- 
 rating ellipfe A E B O , as is well known. Let the rae-» 
 ridional fe6lion A E B Q pafs through the point P of the 
 canal p P. It will cut the feclion e\q i in a line I P i 
 perpendicular to its interfeftion eq with the equator of 
 the fpheroid, and therefore parallel to the axis a c b oi 
 the fedion, if it do not coincide with this axis. Let 
 C D E be the femidiameter of the generating elllpfe 
 which pafles through the interfeftion D of I z and c q\ 
 and draw P Z parallel to D C, and P z parallel to ^ ^ 
 cutting a cb in 2, and join 2; Z and c C. It is plain 
 that the plane paffing through the axis A B af the fphe- 
 roid and the axis ab oi the fe6lion e\qi is perpendi- 
 cular to that fetflion (for it bife(9:s e qy which is a chord 
 of the equatoreal circle L ^ Q ^), and that the planes 
 D ^ C and P s Z are parallel, and the angles at c and z 
 right angles. 
 
 Let us now confider the forces wMith. a£l on the par- 
 ticles of fluid in the canal p P. They are, as before, all 
 refolvable into two, one of them parallel to A C, aad 
 the other perpendicular tO' it. Thus, the particle P ia 
 urged by a force in the direction P D parallel to A C, 
 and proportional to its diflance P D from the equator of 
 "he fpheroid. It is alfo urged by a force in the direQion 
 P Z perpendicular to A C, and proportional to its diftance 
 P Z. This' force P Z may be refolved into P z and 2 Z. 
 
 The 
 
J{JZ PHYSICAL ASTRO^^OMY* 
 
 The iiorce z Z remains tlie fame, for all the particles in 
 the .vaiial / P, s Z being equal to c C. But the force 
 P 2 h always proportional to the diilance of the particle 
 in the canal /> P from the axis ac I? of the feclion ilq i. 
 It is dfo to the axipetal force in the direction P Z as P z 
 to PZ. 
 
 Moreover, it has been fliewn (573.) that the force in 
 the direction P Z is to the force in the direction PD in 
 the ratio of A C" x P Z to E C* X P D, that is (on ac- 
 count of the fimilarity of the fe6lions A E B Q and aehq)^ 
 as « f* X P Z to e c^ x P D. Therefore the force in the 
 direction P s is to the force in the direction P D as ac^ 
 X P 2 to ^ 6* X P D. Wherefore, fince from thefe ele- 
 ments it has been proved already that the whole preilure 
 on P in the canal p P, lying in the plane A E B Q, is 
 equal to the preffure of the canal I P, it follows that the 
 prelTure of the canal/ P, lying in the plane aeh q is alio 
 equal to the prelliire of the canal I P. 
 
 Thus it now appears that the particle P is urged in 
 every direction with the fame force by the fluid in any 
 rectilineal canal whatever reaching to the furface. It is 
 therefore in equillbrio ; and, as it is taken at random, in 
 ajiy part of tlie fpheroid, the whole fluid fpheroid is hi 
 equUibr'io. 
 
 We alfo fee that the whole force with which any par- 
 ticle P is preffed in any direction whatever is to the 
 preflure at the centre C as the re£tangle IP/ to A C^ 
 For that is the proportion of the preflure of the canal I P 
 
 to 
 
FIGURE OF THE PLANETS. 473 
 
 to that of the canal A C ; and all canals termmating m 
 the centre exert equal preflures. 
 
 578. It is now demonflratecl that a mafs of unlforrftly 
 denfe matter, influenced in every particle by gravitation, 
 and fo conllituted that an equilibrium of force on every 
 particle is neceiTary for the maintenance of its form, may 
 exift, with a motion of rotation, in the form of an ellip- 
 tical fpheroid, if there be a proper adjuftment between 
 the proportion of the two axes and the time of the rota- 
 tion. Whatever may be the proportion of the axes of an 
 oblate fpheroid, there is a rapidity of rotation which vi^ill 
 induce that proportion between the undiminiflied gravity 
 at the pole and the diminifhcd gravity on the furface of 
 the equator, which is required for the prefervation of that 
 form. But it has not been proved that a fluid fphere, 
 when fet in motion round its axis, muft aflume the form 
 of an elliptical fpheroid, but only that this is a poflible 
 form. This was all that Newton aimed at, and his proof 
 is not free from reafonable objections. The great matlie- 
 maticians fince the days of Newton have done little more. 
 They have not determined the figure that a fluid fphere, 
 or a nucleus covered with a fluid, tnuj} afl'ume when fet 
 in motion round its axis. * But they have added to the 
 number of conditions that muft be implemented, in order 
 to produce another kind of affurance that an elliptical 
 
 fpheroid 
 
 * Montucla fays (Vol. IV.) that M. le Gendre has de- 
 monllrat^d that an elliptical fpheroid is the only poflible form 
 for a homogeneous fluid turning round its axis, 
 
474 PHYSICAL ASTR-ONOMY, 
 
 fpheroid will anfwcr the purpofe, and by this Ihnitalioii' 
 have greatly increafed the dllliculty of the queflion. M, 
 Clairaut, who has carried his fcruples farther than the 
 reft, requires, befidcs the tiiree conditions which have 
 been fhewn to confifl with tlie permanence of the eUip- 
 tical form, that it alfo be demo-nftrated, i/;w. That a ca- 
 nal of any form whatever niuil every where be hi equili* 
 brio-: 2do, That a canal of any fhape, reaching, from one 
 j^rt of the furface, through the mafs, or along the fur- 
 face, to any other part, fhall exert no force at its extre- 
 mities : 3//'?, That a canal of any form, returning into- 
 itfelf, fliall be /;/ eauiiibrlo through its whole extent. 
 
 579. I apprehend that in the cafe of uniform denfity,. 
 ail thefe conditions are involved in the propofition In art. 
 (577.) For we can fuppofe the canal /> P of fig. 64. 
 N° 4. to communicate with the canal P ^. It has been 
 fhewn that they are in equilihrio in P. The canal 4 /3 
 may branch ofF from P ^. Thefe are /// equilihrio in 
 the point 4. The canal 3 oc may branch off at 3, and 
 they will be flill in equilihrio i and the canal 2 1 will 
 be in equilihrio with all the foregoing. Now thefe 
 points of derivation may be multiplied, till the polygonal 
 canal j!) P 4 3 2 i becomes a canal of continual curvature 
 of any form. In the next place, this canal exerts no force 
 at either end. For the equilihrium is proved in every 
 ftate of the canal p P — it may be as fhort as we pleafe — 
 it may be evanefcent, and actually ceafe to have any 
 length, without any interruption of the equilibrium. 
 
 Therefore, 
 
HGURE OF THE PLANETS. 475 
 
 Thercf-^re, there is no force exerted at its extremity to 
 difturb the form of the furface. It may be obferved that 
 this very circumftance proves that the direftioii of gra- 
 vity is perpendicular to the furface. And it muft be ob- 
 ferved that the perpendicularity of gravity to the furface 
 is not employed in demonltrating this propofition. The 
 whole refts on the propofitions in art. 02* 563. and 564, 
 both of which we owe to Mr M*Laurin. 
 
 580. Having now demonftrated the competency of the 
 elliptical fpherold for tlie rotation of a planet, we proceed 
 to inveftigate the precife proportion of diameters which is 
 required for any propofed rotation. For example, What 
 protuberancy of the equator will difFufe the ocean of this 
 Earth uniformly, confidently with a rotation in 23^^ 
 ^6' 04", the planet being uniformly denfe ? 
 
 Let p and e exprefs the primitive gravity of a particle 
 placed at the pole and at the furface of the equator, arifing 
 folely from the gravitation to every particle in the fphe- 
 roid, and let c reprefent the centrifugal tendency at the 
 furface of the equator, arifmg from the rotation. We 
 fliall have an elliptical fpherold of a permanent form, If 
 AC be to EC as e — c is to / {s^9') We muft there- 
 fore find, firft of all, what isnhe proportion of / to ^ re- 
 'fulting from any proportion of A C to EC. 
 
 To accomplifh this in general terms with precifion, ap- 
 peared fo difficult a talk, even to Newton, that he avoid- 
 ^d it, and took an indirect method, which his fngacity 
 3 O 2 ili^wed 
 
47^ PHYSICAL ASTRONOMY. 
 
 fliewed him to be perfectly fafe ; and even this was ditR* 
 cult. It Is in the complete folution of this problem that 
 the genius of M^Laurin has fliewn itfelf moft remarkable 
 both for acutenefs and for geometrical elegance. It is not 
 €xeeeded (in the opinion of the firft mathematicians) * 
 by any thing of Arcliimedes or Apollonius. For this 
 reafon, it is to be regreted that vre have not room for the 
 feries of beautiful propofitions that are ncccllary in his 
 method. We muft take a iliorter courfe, limited in- 
 deed to fpheroids of very frnall eccentricity (whereas 
 the method of M'Laurin extends to any degree of eccen^ 
 ijicity), but, with this limitation, perfectly exact, and 
 abundantly eafy and fimple. It is, in its chief fteps, the 
 method followed by M. Bofcoyich. ♦ 
 
 580. Let AEBQ (fig. 65.) reprefent the terrjeilnal 
 fpheroid, nearly fpherical, and let A <? B ^ and )La Qb 
 reprefent the infcribed and circumfcribed fpheres. With 
 the axis and parameter A B defcribe the parabola A F G, 
 drawing the ordinates B D F, E C H, &c. Defcribe 
 alfo the curve line A I L G, fuch, that we have, in every 
 point of it, AB:AD = DF:Pl5 AB : A C =r C H ; 
 CL, &c. 
 
 Our firft aim fliall be to find an expreffion and value 
 of the polar gravity. We may conceive the fpheroid as 
 a fphere, on which there is fpread the redundant matter 
 contained between jlie fpherical and the fpheroidal fur- 
 faces, 
 
 * See gofTuet Bj/fr des MathematiqueSf 
 
FIGURE OF THE TLANETS. 477 
 
 faces. We know the gravitation of tlie polar particle A 
 to the fphere, and now vi^ant to have the meafure of its 
 gravitation to this redundant matter. Suppofe the figure 
 to tnrn round the axis A B. The femiellipfis A E B will 
 generate a fpheroidal furface ; the femicircle A ^ 3 will 
 generate a fpherlcal furface, and the intercepted portions 
 P/), E ^, Sec, of the ordinates will generate flat rings of 
 the redundant matter. As the deviation from a fphere is 
 fuppofed very fmall (E^* not exceeding the 5oodth part of 
 E Q), we may fuppofe, without any fenfible error, th?i.t 
 Ap IS the diflance of A from the vrhole of the ring ge- 
 nerated by P/>. 
 
 Proceeding on this aflumptlon, we fay that the gravi- 
 tation of A to the rings generated by P^, E e, &c. is 
 proportional to the portions FI, HL, &c. of the cor- 
 refponding ordinates D F, C H, &c., and that the gra- 
 vitation of A to the whole redundant matter may be ex- 
 prefled by the furface A F H G L I A comprehended be- 
 tween the lines A F H G and AIL G. 
 
 For, the abfolute gravitation of A to the ring P/ is 
 diredly as the furface of the ring, and inverfely as the 
 fquare of its diflance from A. Now, the furface of the 
 ring is as its breadth, and its circumference jointly. Its 
 breadth P/, and alfo its circumference, being propor- 
 tional to Dp, the furface is proportional to T)p\ The 
 
 abfolute gravitation is therefore proportional to — ^^ 
 
 Ap''* 
 
 This may be refoiyed into forces in the dire6lions A D 
 3nd Dp. The force in the diredion Dp is balanced 
 
 by 
 
478 PHYSICAL ASTmONOiVlY. 
 
 by an equal force on the other fide of the axis. There- 
 fore, to' have the gravitation in the direction of the axis, 
 the value of the abfolute gravitation in the dire6i:ion Ap 
 muft be reduced in the proportion of Kp to A D. It 
 
 , , ^ Dp' X AD Dp' X AD 
 therefore becomes -^i^j^^ =' ^, . or, 
 
 ,.,• 1 r f Di'xADxA/ ^ ^ 
 which IS the lame thmg, — ■^— -. But 
 
 A/)* = ABxAD, and A/)♦ = AB'xAD^ Alfo 
 D/' = A D X D B. Therefore the value laft found be- 
 
 ADxDBxADxA/) , . , . , 
 
 comes • AB^ x AD^ ' ^^ ^"^ ' 
 
 or the fame thing with Xl\~ Since A B* is a 
 
 conftant quantity, the gravitation in tlie dire£lion AC 
 to the ring generated by Fp is proportional to D B X 
 A/. 
 
 It is very obvious that D F, C H, B G, &;c. are re- 
 fpedlively equal to Ap, Ae, A B, &;c. Therefore the 
 gravitation to the matter in the ring generated by P^ 
 is proportional to D B X D F. 
 
 Now, by the conftruftiou of the curve line A L C, 
 we have AB:AD = DF:DI 
 
 therefore AB:DB = DF:IF 
 and ABxIF = DBxDF 
 
 Therefore, fmce A B is conftant, I F is proportional to 
 D B X D F, that is, to the gravitation to the ring gene- 
 rated by V p. Therefore the gravitation to the whole 
 redundant matter may be reprcfented by the fpace 
 A H G L A. 
 
PI CURE OF THE PLANET 3. 479", 
 
 Let T be the periphery of a circle of which the ra- 
 dius is I. The circumference of that generated by E<? 
 will be TT X Cef and its furface =7rXCfXEf, and 
 
 ,., .. . . ^ X C e X E e 
 
 tnc ablolute gravitation to it is — , or 
 
 TrxCeXY.e , . TrXEe 
 
 r-7T^ , that IS, 7-?^- • i his, when reduced 
 
 2 A L." 2 A C 
 
 to the direftion A C, becomes r r-rr—3 that is, 
 
 ' 2 Atf X AC ' * 
 
 A — > or T — ; . And becauXe A ^* = 2 A C% 
 
 2 A^ 2 A^- ' 
 
 and LHrriCH, =^Af, the reduced gravitation be- 
 
 ^ X E^ 
 
 comes — —^ X L H. 
 2 A C 
 
 This being the meafure or reprefentative of the gra- 
 vitation to the material furface or ring generated by E e, 
 the gravitation to the whole redundant matter contained 
 between the fpheroid and the infcribed fphere will be re- 
 
 prefented by . ^^ multiplied by the fpace compre- 
 
 2 A V* 
 
 liended between the curve lines A F G and A L G. 
 We muft find the value of this fpace. 
 
 The parabolic fpace A H G B A Is known to be 
 = 4 A B X B G, = I A B\ The fquare of D I is pro- 
 portional to the cube of B D. For, by the con{lru6lion 
 of the curve A B' : A D' = D F' : D P, and D P = 
 
 A D' X D F' _ AD* DP _ AD' . n - AD^ 
 AB^ ' ~ AB ^ AB' ~ AB ' "" XB' 
 
 J. 
 Therefore D I is proportional to AD^, and the area 
 
 ABGLA is=iABxBG, =fAB^ Take this 
 
 from the parabolic area -f- A B% and there remains 
 
 t^tAB^ 
 
4^6 fHt^ICAL ASTRONOMT. 
 
 /^ A B» for the value of A L G H A. This is equal 
 to if A C\ 
 
 No'^, the gravitation of A to the redundant matter 
 
 was fliewn tobe^ALGHAx — r-r^r- This now 
 
 2 AC" 
 
 becomes if AC X "^ ^^ , or -j-V ^ X E^. Such is 
 
 2 A v-- 
 
 the gravitation of a particle in the fole of the fpheroid to 
 the redundant matter fpread over the infcribed fphere. 
 
 The gravitation of a particle fituated on the furface 
 of the equator to the fame redundant matter is not quite 
 fo obvious as the polar gravity, but may be had with the 
 fame accuracy, by means of the following confidera- 
 tions. 
 
 581. Let A B ^ ^ (fig. 66.) reprefent an oblate fphe- 
 roid, formed by rotation round the fhorter axis B ^ of 
 the generating elllpfe, and viewed by an eye fituated in the 
 plane of its equator. Let A E ^ ^ be the circumfcribe4 
 fphere. This fpheroid is deficient from the fphere by 
 two menifcufes or cups, generated by the rotation of the 
 lunula A E ^ B A and Aeab A. 
 
 Now fuppofe the fame generating ellipfe ^ A B « ^ A to 
 turn round its longer axis A a. It Vv'ill generate an ob- 
 long fpheroid, touching the oblate fpheroid in the whole 
 circumference of one elliptical meridian, viz. the meri- 
 dian A B ^ ^ A which pafies through the poles A and a 
 of this oblong fpheroid. It touches the equator of the 
 oblate fpheroid only in the points A and a, and has the 
 
 diameter 
 
FIGURE or THE PLANETS. 48! 
 
 diameter A^ for its nxis. This oblong fpheroid is other- 
 wife wholly within the oblate fpheroid, leaving between 
 their furfaces two mcnifcufes of an oblong form. This 
 may be better conceived by firft fuppofing that both the 
 fpheroids and alfo the circumfcr-bed fphcre arc cut by a 
 plane P Ggp, perpendicular to the axis A ^ of the ob- 
 long fpheroid, and to the plane of the equator of the ob- 
 late fpheroid. Now fuppofe that the whole Egure mak(?s 
 the quarter of a turn round the axis B /^ of the oblate 
 fpheroid, fo that the pole n of the oblong fpheroid comes 
 quite in front, and is at C, the eye of the fpeclator be- 
 ing in the axis produced. The equator of the oblong 
 fpheroid will now appear a circle OBo h O, touching the 
 oblate fpheroid in its poles B and b. The feftion of the 
 plane IP p with the circumfcribed fphere will now appear 
 as a circle P' Kp' r. Its feftion with the oblate fpheroid 
 will a|)pear an ellipfe R G' r g' fimilar to the generating 
 ellipfe A B « ^, as is well known. And its fe£lion with 
 the oblong fpheroid will now appear a circle I G' / g' pa- 
 rallel to its equator OBob. V p is equal to P'/', and 
 G ^ to G' «-'. Thus it appears that as every fe£lion of 
 the oblate fpheroid is deficient from the concomitant fec- 
 tion of the circumfcribed fphere by the want of tvv'-o lu- 
 nulae R P> G' and Kp'rg^ fo it exceeds the conco- 
 mitant feclion of the obiong fpheroid by two lunulas 
 G' R ^' I and G' r g' i. It is alfo plain that if thefe 
 fpheroids differ very little from perfe£l fpheres, as when 
 E B does not exceed ~o of E ^, the deficiency of each 
 fedlion G g from^ the concomitant fe£bion of the circum- 
 3 P f.^ribed 
 
48i PHYSICAL ASTRONOMY. 
 
 fcribed fpliere is very nearly equal to its excefs above the 
 concomitant feclion of the infcribed oblong fpheroid* 
 It may fafely be confidered as equal to one half of the- 
 fpace contained between the circles on the diameters V' p 
 and; G'^', * in the fame way that we confidered the lu- 
 nula APEB^/)A of fig. 65. as one half of the fpace 
 contained between the feniicircles A e B and a^b. 
 
 From this view of the figure, it appears that the gra- 
 vitation of a particle a in the equator of the oblate fphe- 
 roid to the two cups or menifcufes R V r G' and R/ rg'y 
 by which the oblate fpheroid is lefs than the circum- 
 fcribed fph^re, may be computed by the very fame method 
 that we employed in the lafl proportion. But, inftead 
 of computing (as in lait propofition) the gravitation of a 
 to the ring generated by the revolution pf P G (fig. 66.)y 
 that is, to the furface contained between the two circles 
 RP'r/>' and IG'ig^ v.'e muft ernploy only the two 
 lunula RP'rG'R and R/r^'R. In this way, we 
 may account the gravitation to the deficient matter (or 
 the deficiency of gravitation) to be one half of the 
 quantity determined by that propofition, and therefore' 
 = -s^ 'f X E ^ of fig. 65. The laft propofition gave us- 
 the gravitation to all the matter by which the fpheroid 
 ejtceeded the infcribed fphere. The prefent propofition 
 
 gives 
 
 * For the circumfcribed circle is to the ellipfe as the el- 
 lipfe to the infcribed circle. When the extremes differ fo 
 little, the geometrical and arkhn)etical mean will differ but 
 iRJfenfibly. 
 
FIGURE OF THE PLANETS. 483 
 
 gives the gravitation to all the matter by whicli it fiills 
 fliort of the circumfcribed fphere. 
 
 582. We can nov/ afccrtain the primitive gravita- 
 tion at the pole and at the equator, by adding or fub- 
 tracling the quantities now found to or from the gravi- 
 tation to the fpheres. Let r be the radius of the fphere, 
 and cT r the circumference of a great circle. The dia- 
 
 meter is 2 r. The area of a great circle is , and 
 
 ijhe whole furface of the fphere is t ^ r% and its folrd 
 contents is y 9r r^. Therefore, fmce the gravitation to a 
 fphere of uniform denfity is the fame as if all its matter 
 were collected in its centre, and is as the quantity of 
 ipatter directly, and as the fquare of the diftance r in- 
 verfely, the gravitation to a fphere will be proportional 
 
 to y — — , that is, to y TT r. * 
 
 Now 
 
 * I beg leave to mention here a circumllance which ihould 
 have been taken notice of in art. 464, when the lirll principles 
 of fpherical attractions were eftabhfhed. It was fhewn that 
 the gravitation of the particle P to the fpherical furface gene- 
 rated by the rotation of the arch A D' T is equal to its gra- 
 vitation to the furface generated by the rotation of B D T. 
 Therefore if P be Infinitely near to A, fo that the furface 
 generated by A D' T rnay be confidered as a point or fingle 
 particle, the gravitation to that particle is equal to the gravi- 
 tation to all ^he reft of the furface ; that is, it is one ha!f of 
 3 P 2 the 
 
464 PHYSICAL ASTRONOMY. 
 
 Now kt AEBQ (iig. 65.) be an oblate fpberoid, 
 whofe poles are A and 13, The gravity of a particle A 
 to the fphcre whofe radius is AC is y ^r X AC, = ^ ''** 
 X E C — -J- ^ X E ^, or y ;r X E C — i-^ ^ X E f . Add 
 to this its gravitation -,-^j- tt x E f, to the redundant 
 luatter. The fum is evidently .^- ;r x E C — -iV '^ Xi 
 
 The gravitation of the particle E on the furface 01 
 tjie equator to a fphere v/hofe radius is EC is y^ X EC. 
 From this fubtra61: its deficiency of gravitation, viz. 
 4t ^ X E ^, and there remains the equatoreal primitive 
 gravity =|;rx EC — ^:rx E^. 
 
 Therefore, in this fpheroid, the polar gravity is to 
 the equatoreal gravity as | tt X E C — Vy tt x E ^ to 
 ^^-ttxEC — ^TTxE^, or (dividing all by 4 ^) as 
 EC — y E ^ to EC — y E £>, or (becaufe E ^ is fup- 
 pofed to be very fmali in comparifon with EC) as EC 
 to EC — -J- E e. In general terms, let g reprefent the 
 mean gravity, p the polar, and e the equatoreal gravity, 
 r tlie radius of the infcribed fphere, and x the elevation 
 E e of the equator above the infcribed fphere. We have, 
 for a general exprefiion of this proportion of the primi-. 
 
 tive 
 
 the whole gravitation. If we fuppofe P and A to coincide, 
 that is, make P or.e of the particles of the furface, its gra- 
 vitation to the fpherical furface will be only one half of what 
 it was when it was without the furface ; ajid if we fuppofe 
 P adjoining to A internally, it will exhibit no gravitation at 
 
FiCrRE OF THE PLANETS. 485 
 
 tive gravitations, p : e z= r -\- ^- .\ : r, or (becaufe .v is 
 very Imall in compariibn with r), p : e = r * r — ^ x. 
 This lail is generally the moll convenient, and it is exa£l, 
 if r be taken for the equatoreal radius. 
 
 583. Had the fpheroid been prolate (oblong) the 
 fame reafoning would have given us p : e = r : r '^- j. x. 
 
 I may add here that the gravitation at the pole of an 
 oblong fpheroid, the gravitation at the equator of an ob- 
 late fpheroid (having the fame axes) and the gravitation 
 -to the circumfcribcd fphere, on any point of its furface, 
 are proportional, refpe6i:ively, to \ r -{- S^ x ; | r -f- i .v j 
 and y ^ 4- y A^ * 
 
 584. It now appears, as was formerly hinted (567.) that 
 we cannot have an elliptical fpheroid of uniform denfity, 
 
 in 
 
 * Many quellions occur, in which we want the gravitation 
 of a particle P' fituated in the diredlion of any diameter C P 
 (fig. 6^.) Draw the conjugate diameter CM, and fuppofe 
 the fpheroid cut by a plane palBng through C M perpendicu- 
 lar to the plane of the figure. This fedlion will be an ellipfe, 
 cf which the femiaxes are C M and CE (= r -^ x). A 
 circle whofe radius is the mean proportional between CM and 
 C K has the fame area with this fe^lion, and the gravitation 
 to this circle will be the fam.e (from a particle placed in the 
 a>:is) with the graNatation to this feftion. Therefore, as the 
 angle PCM is very nearly a right angle, the gravitation of 
 
 F 
 
^8(5 PHYSICAL ASTRGNOMT. 
 
 in whicli the gravitation at the pole is to that at the e- 
 quator as the equatoreal radius to the polar radius. This 
 \70uld make p :e-=: r -.r — ac, a ratio five times greater 
 than that which refults from a gravitation proportional 
 
 Thus have we obtained, with fiifficient accuracy, the 
 ratio of polar and equatoreal gravity, unafFe£led by any 
 external force, and we are now in a condition to tell 
 what velocity of rotation will fo diminifh the equatoreal 
 gravitation that the remaining gravity there fhall be to the 
 polar gravity as A C to E C. 
 
 585. Let c be taken to reprefent the centrifugal 
 tendency generated at the furface of the equator by the 
 rotation of the planet round its axis, and let the other 
 fymbols be retained. The fenfible gravity at the equator 
 is € — c, the polar gravity/), and the excefs of the equa- 
 toreal radius above the femiaxis r is x. 
 
 We have {hewn (582.) that the primitive gravities at 
 the pole and the equator are in the ratio of r to r — f ;i^ 
 
 or, 
 
 P to the fpheroid will be the fame with its polar (or axicular) 
 gravitation to another fpheroid, whofe polar femiaxis is P C, 
 and whofe equatoreal radius is the mean proportional between 
 "C M and C E* This is eafily computed. If the arch P E 
 be 35° 16', a fphere having the radius PC has the fame ca- 
 pacity with the fpheroid A E B Q (when E ^ is very fmall).. 
 Hence follows what was faid in tlie note on art. 572. 
 
riGURE OF THE EARTIt. ^J 
 
 w, (becaufe ;v is a very fmall part of r), in the ratio o£ 
 
 r + f .V to r. That is, r:r -J^- }^ x = e \ p. This gives 
 
 e X 
 p — e -\ . Therefore the ratio of the ftnfihle equato- 
 
 real gravity to the gravity at the pole is^ — c:^-j-— , 
 
 e X 
 or, very nearly, e \e -\- \- c. Therefore we mufl have, 
 
 for a revolving fphere of fmall eccentricitv, 
 
 £ X 
 
 e :e -\ — — -X-c^rir + x 
 
 5 ^ 
 
 and 
 
 e X 
 e : — -^ c zz r :x 
 
 confequently 
 
 ex 
 
 e X ■=: \- re 
 
 and 
 
 ex A£ X 
 
 ex or z=:r c 
 
 5 5 
 
 and 
 
 4 ^ ;v =r 5 r r, and a; = -^ 
 
 /\e 
 
 X c c 
 and the elllpticity - = ^ , that is, 
 ^ r ^e 
 
 Four times the primitive gravity at the equator ts to 
 
 five times the centrifugal force of rotation as the fcmiaxis to 
 
 the elevation of the equator above the infcrihed fphere, 
 
 586. It is a matter of obfervation tliat the dimi|r 
 nution of equatoreal gravity by the Earth's rotation in 
 ss^^ ^a 4" is nearly ^\^, Therefore 4 X 289 : 5 = r : 
 X = 23if : I, very nearly. This is the ratio deduced by 
 Newton iji his indireft, and feemingly incurious, method. 
 That method has been much criticifed by his fcholars, as 
 if it could be fuppofed that Newton was ignorant that the 
 
 proportionality 
 
4S8 PHYSICAL ASTRONOMt. 
 
 proportionality employed by him, in a rough way, was 
 not nccejfafily involved in the nature of the thing. But 
 Newton knew that, in the prefent cafe, the error, if any, 
 muft be altogether infignificant. He did not demonftrate, 
 but afTumed as granted, that the form is elliptical, or that 
 an elliptical form is competent to the purpofe. His jufh- 
 nefs of thought has Ven fo repeatedly verified in many 
 cafes as abilrufe as this, that it is unreafonable to afcribe 
 it to conjecture, and it fliould rather, as by Dan. Ber- 
 noulli, be afcribed to his penetration and fagacity. He 
 had fo many new wonders to communicate, that he had 
 not time for all the lemmas that were requifite for enab- 
 ling inferior minds to trace his fleps of inveftigation. 
 
 587. When confiderlng the aftronomical phenomena, 
 fome notice Vx^as taken of the attempts which have been 
 made to decide this matter by obfervation alone, by mea- 
 furing degrees of the meridian in diiFerent latitudes. 
 
 But fuch irregularity is to be feen among the mea- 
 fures of a degree, that the queftion is ftill undecided by 
 this method. All that can be made evident by the com- 
 parifon is that the Earth is oblate, and much more oblate 
 ^an the ellipfe of Mr Hermann •, and that the medium de- 
 du£^ion approaches much nearer to the Newtonian form. 
 When we recolledt that the error of one fecond in the 
 eftimation of the latitude induces an error of more than 
 thirty yards in the meafure of the degree, and that the 
 form of this globe is to be learned, not from the lengths 
 of the degrees, bttt from the differences of thofe lengths. 
 
P.JSf^j. 
 
..... r 
 
FIGURE OF THE EARTH. 489 
 
 it muft be clear that, unlefs the lengths, and the celeilial 
 arc correfponding, can be afcertained with great pracifion 
 indeed, our inference of the variation of curvature muft 
 be very vague and uncertain. The perufal of any page 
 of the daily obfervations in the obfervatory of Paris will 
 fliew that errors of 5" in declination are not uncommon, 
 and errors of 2" arc very frequent indeed. * So many cir- 
 cumftances may alfo afFe£l the meafure of the terreftrial 
 arc, that there is too much left to the judgement and choice 
 of the obferver, in drawing his conclufions. The hiftory 
 of the firfl meafurement of the French meridian by Caf- 
 fmi and La Hire is a proof of this. The degrees feem- 
 ed to increafe to the fouthward — the obfervations were 
 affirmed to be excellent — and for fome time the Earth was 
 held to be an oblong fpherold. Philofophy prevailed, and 
 this was allowed to be impo^ible ; — yet the obfervations 
 were ftill held to be faultlefs, and the blame was laid on 
 the negle£l of circumllances which fliould have been con- 
 fidered. It was afterwards found that the deduced mea- 
 
 fares 
 
 * I mention particularly the daily obfervations of the Pa- 
 rifian Obfervatory, becaufe the French allronomers are dif- 
 pofed to reft the queflion on the obfervations of their own 
 academicians, who have certainly furpafTed all the aftronomers 
 of Europe in the extent of their meafurement of degrees, I 
 fee no reafon for giving their obfervations made in diftant places 
 a greater accuracy than what is to be found in the Royal Ob- 
 iEervatory, with capital inftruments, fixed up in the moil folid 
 aianner. 
 
 3Q 
 
FIGURE OF THE EARTH. 489 
 
 it muft be clear that, unlefs the lengths, and the celeftial 
 arc corrcrpondlng, can be afcertained with great precifion 
 indeed, our inference of the variation of curvature muft 
 be very vague and uncertain. Tlie perufal of any page 
 of the daily obfen^ations in the obfervatory of Paris will 
 fliew that errors of 5" in declination are not uncommon, 
 and errors of 2" arc very frequent indeed. * So many cir- 
 cumftances may alfo aite6l the meafure of the terreftrial 
 nrc, that there is too much left to the judgement and choke 
 of the obferv'er, in drawing his conclufions. The hiftory 
 of the firft meafurement of the French meridian by Caf- 
 fmi and La Hire is a proof of this. The degrees feem- 
 ed to increafe to the fouthward — the obfcrv-ations were 
 affirmed to be excellent — and for fome time the Earth was 
 held to be an oblong fpheroid. Philofophy prevailed, and 
 this was allowed to be impoyfible ; — yet the obfervations 
 were ftill held to be faultlefs, and the blame was laid on 
 the negledl of circumilances which fliould have been con- 
 fidered. It was afterwards found that the deduced mea- 
 
 fures 
 
 * I mention particularly the daily obfervations of the Pa- 
 riiian Obfervatory, becaufe the French allronomers are dif- 
 pofed to reft the queflion on the obfervations of their own 
 academicians, who have certainly furpafTed all the aftronomers 
 of Europe in the extent of their meafurement of degrees. I 
 fee no reafon for giving their obfervations made in diftant places 
 a greater accuracy than what is to be found in the Royal Ob- 
 jiervatory, with capital inflruments, fixed up ia the moll folid 
 wanner- 
 
 3Q 
 
490 PHYSICAL ASTRONOMY. 
 
 fures did not agree with feme others of unqueftidnable 
 authority, but would agree with them if the corrections 
 were left out ; — they were left out, and the obfervations 
 declared excellent, becaufe agreeable to the doftrine of 
 gravitation. * 
 
 588. The theory of univerfal gravitation affords ano- 
 ther means of determining the form of the terraqueous 
 globe dire61:ly from obfcrvation.- Mr Stirling fays, very 
 juflly, that the diminution of gravity deducible from the 
 remark of M. Richer,, and confirmed by many fimilar 
 obfervations,. gives an incontellible proof, both of the 
 rotation of the Earth, and of its oblate figure. It could 
 not be of an oblate figure, and have the ocean uniformly 
 
 diflributed, 
 
 * They were reconciled with the dodlrine of gravitation 
 by attributing the enlargement of the fouthern degrees to the 
 action of the Pyrenean mountains, and thofe in the fouth of 
 France, upon the plummets. But it appears clearly, by the 
 examination of thefe obfervations by Profefibr Celfuis, that the 
 obfervations were very incoiTedl, and fome of them very inju- 
 dicioufly contrived (See Phil. Tranf. N° 457.\.and 386.) 
 The palpable inaccuracies gave fuch latitude for adjuftment 
 that it was eafy for the ingenious Mr Mairan to combine them 
 in fuch a manner as to deduce from them inferences in fupport 
 of opinions altogether contradiftory of thofe of the academy. 
 Have we not a remarkable example of the doubtfulnefs of 
 fuch meafures, in the meafurement of the Lapland degree J 
 It is found to be almoll 200 fathoms too long. 
 
FIGURE OF THE EARTH. ^p-'l 
 
 diftributed, without turning round its axis ; and it could 
 not turn round its axis without inundating; the equator, 
 unlefs it have an oblate form, accompanied with dimi- 
 nifhed equatoreal gravity. By the Newtonian theory, 
 ^he increments of gravity as w^e approach the poles are 
 in the dupHcate ratio of the fines of the latitude. The 
 increm.ents of the length of a feconds pendulum will 
 have the fame proportion. Nothing can be afcertained 
 ^y obfervation with greater accuracy than this. For the 
 London artifts can make clocks which do not vairy one 
 •fecond from mean motion in three or four days. We 
 need not meafure the change in the length of the pen- 
 dulum, a very delicate talk — but the change of its rate of 
 vibration by a change of place, which is eafily done ; and 
 w^e can thus afcertain the force of gravity without an er- 
 ror of one part in 86400. This furpaffes all that can be 
 done in the meafurement of an angle. Accordingly, the 
 ellipticities deduced from the experiments with pendulums 
 are vaflly more confident with each other, and it were 
 to be wiflied that thefe-experiments were more repeated. 
 "We have but very few of them. 
 
 589. Yet even thefe experiments are not without 
 anomalies. Since, from the nature of the experiment, 
 we cannot afcribe thefe to errors of obfervation, and the 
 doctrine- of univerfal gravitation is eflablifhed on too 
 broad a foundation to be called in queftion for thefe ano- 
 malies, philofophers think it more reafonablc to attribute? 
 ;the anomalies to local irregularity in terreflrial gravity. 
 
 3 Q 2 If. 
 
4p2 PHYSICAL ASTRONOMY. 
 
 If, in one place, the perululum is above a great mafs of 
 folid and denfe rock, perhaps abounding in metals, and, 
 in anotlier place, has below it a deep ocean, or a deep 
 and extenfive ftratunj of light fand or earth, we {hould 
 certainly look for a retardation of the pendulum in the 
 latter fituation. The French academicians compared the 
 vibrations of the fame pendulum on the fea-fhore in 
 Peru, and near the top of a very lofty mountain, and 
 they obferved that the retardation of its motion in the lat- 
 ter fituation was not fo great as the removal from the 
 centre required, according to the Newtonian tlieory, viz. 
 in the proportion of the diftance (the gravity being in the 
 inverfe duplicate proportion). * But it fhould not be fo 
 much retarded. The pendulum was not raifed aloft 'in 
 the air, but was on the top of a great mountain, to which, 
 as well as to the reft of the globe, its gravitation was di- 
 rected. Some obfervations were reported to have been 
 made in Switzerland, wdiich fliewed a greater gravitation 
 on the fummit of a mountain than in the adjacent val- 
 lics ; and much was built on this by the partizans of vor- 
 tices. 
 
 * The length of a pendulum vibrating feconds was found 
 to be 439,2 1 French lines on the fea-fnoie at Lima j when 
 reduced to time at Quito, 1466 fathoms higher, it was 438,88 ; 
 and on Pichinka, elevated 2434 fathoms, it was 438,69. Had 
 gravity diminiflied in the inverfe duplicate^ratio of the diilancite, 
 the pendulum at Quito fhould have been 438. oo, and at Pi- 
 chinka it fhould have been 438,55. 
 
riGURE OF THE EARTH. 493 
 
 tices. But, after due inquiry, the obfervations were found 
 to be altogether fictitious. It may juft be noticed here, 
 that fome of the anomahes in the experiments with pen- 
 dulums may have proceeded from magnetifm. The clocks 
 employed on thofe occafions probably had gridiron pen- 
 duium.s, having five or feven iron rods, of no inconfider- 
 able weight. We know, for certain, that the lower end 
 of iuch rods acquires a very diflinft magnetifm by mere 
 upright pofition. This may be confiderable enough, efpe- 
 cialiy in the circumpolar regions, to affe£t the vibration, 
 and it is therefore advifeabie to employ a pendulum hav- 
 ing no iron in its compofition. 
 
 Although the dedu6i;ion of the form of this globe 
 from obfervations on the variations of gravity is expofed 
 to tlie fame caufe of error which affects the pofition of 
 the plummet, occafioning errors in the meafure of a de- 
 gree, yet the errors in the variations of gravity are incom- 
 parably lefs. What would caufe an error of a whole 
 mile in the meafure of a degree will not produce the ^g- 
 part of this error in the difference of gravity. 
 
 590. Thefe obfervations naturally lead to other re- 
 fl eel ions. Newton's determination of the form of the 
 terraqueous globe, is really the form of a homogeneous 
 and fluid or perfedly flexible fpheroid. But will this 
 be the form of a globe, conftituted as ours in all proba- 
 bility is, of beds or layers of different fubftances, whofe 
 dcnfity probably increafes as they are farther down ? 
 
 This is a very pertinent and momentous quellion. 
 
 But 
 
494 PHYSICAL ASTRONOMY, 
 
 But this outline of mechanical philofophy will not adniit 
 of a difculTion of the many cafes which may reafonably 
 be propofed for folution. All that can, with propriety, 
 be attempted here is to give a general notion of the 
 change of form that will be induced by a varying den- 
 fity. And even in this, our attention muft be confined to 
 fome fimple and probable cafe. "We fhall therefore fuppofe 
 the denfity to increafe as we penetrate deeper, and this in 
 fuch fort, that at any one depth the denfity is uniform. 
 •It is highly improbable that the internal conftitution of 
 this globe is altogether irregular. 
 
 591. We fhall therefore fuppofe a fphere of folid 
 matter, equally denfe at equal diftances from the centre, 
 and covered with a lefs denfe fluid ; and we fhall fup- 
 pofe that the whole h-is a form fuitable to the velocity 
 of its rotation. It is this form that we are to find out. 
 With this view, let us fuppofe that all the matter, by 
 which the folid globe or nucleus is denfer than the fluid, 
 is collected in the centre. We have feen that this v/ill 
 make no change in the gravitation of any particle of the 
 incumbent fluid. Thus, we have a folid glbbe, covered 
 with a fluid of the fame denfity, and, befides the mutual 
 gravitation of the particles of the fluid, we have a force 
 of the fame nature ailing on every one of them, direct- 
 ed to the central redundant matter. Now, let the 
 globe liquefy or diflblve. This can induce no change of 
 force on any particle of the fluid. Let us then deter- 
 mine the form of the now fluid fpheroid, which will 
 
 ' mainta::: 
 
FIGURE OF THE EARTM. 4^5 
 
 maintain itfelf in rotation. This being determrned, let 
 the globe again become folid. The remaining fluid v/iiL 
 not change its form, becaufe no. change is inauced on 
 tiie force acliing on any parcicle of the fluid. Call this 
 Hypothefis A. 
 
 592. In order -to determine tliis flate of equUihrium, 
 <ix the form which infures it, which is the chief diiH- 
 cuhy, let us fcrm another hypothefis B, differing from 
 A cniy in this circumftance, that tlie matter colleded in the 
 centre, inftead. of attrading the particles of the incumbent 
 fluid with a force decreafing in tlie inverfe duplicate ratio 
 of their diftances, attracts them with a force increafing in 
 the direct ratio of their diilances, keeping the fame inten- 
 fity at the diftance of the pole as in liypothefis A. Thia 
 ficliitious hypothefis, fimilar to Hermann's, is chofen, .be- 
 caufe a mafs fo conftituted will maintain the form of an 
 accurate elliptical fpheroid^ by a proper adjuflment of 
 the proportion of its axis to the velocity of its rotation. 
 This will eafily appear. For we have already ieen that 
 the mutual gravitation of the particles of the elliptical 
 fluid fpheroid, produces, in each particle, a force which 
 may be refolved into two forces, one of them perpendi- 
 cukr to the axis, and proportional to the diftance from, 
 it, and the other perpendicular to the equator, and pro- 
 portional to the diftance from its plane. There is now 
 by hypothefis B fuperadded, on each particle, a force 
 proportional to its diftance from the centre, and direded 
 to the centre. This may alfo be refolved into a force 
 
 perpendicular 
 
4f;(> PHYSICAL ASTRONOMt. 
 
 perpendicular to the axis, and another perpendicular fa 
 the equator, and proportional to the diftances from them. 
 Therefore tlie whole combined forces a6ling on each 
 particle may be thus refolved into two forces in thofe 
 directions and in thofe proportions. Therefore a mafs 
 fo comlituted will maintain its elliptical form, provided 
 that the velocity of its rotation be fuch that the whole 
 forces at the pole and the equator are inverfely as the 
 axes of the generating ellipfe. We are to afcertain this 
 form, or this required magnitude of the centrifugal force. 
 Having done this, we fhall reftore to the accumulated 
 central matter its natural gravitation, or its aClion on 
 the fluid in the inverfe duplicate ratio of the diftances, 
 and then fee what change muft be made on the form of 
 the fpheroid in order to reftore the equilibrium, 
 
 593. Let B A ^^ (fig. 67.) be the fi£titious elliptical 
 fpheroid of liypothefis B. Let B E ^ f be the infcribed 
 fphere. Take E G, perpendicular to C E, to reprefent 
 the force of gravitation of a particle in E to the cen- 
 tral matter, correfponding to the diftance CE or C B. 
 Draw C G. Draw alfo A I perpendicular to C A, meet- 
 ing C G in L Defcribe the curve G L R, whofe ordi- 
 
 pates G E, L A, R M, 8cc. are proportional to ^^^^ , 
 
 TTT-ii rlvp' ^^' Thefe ordinates will exprefs the 
 gravitations of the particles E, A, M, &c. to the central 
 anatter by hypothefis- A. 
 
 In hypothefis A, the gravitation of A is reprefented 
 
 by 
 
FISURE OF THE EARTH. 497 
 
 hy A L, but in hypothefis B It is reprefentcd by A I. 
 For in hypothefis B the gravitations to this matter are as 
 the diftances. E G is the gravitation of E in both hy- 
 pothefes. NoMT, E G : A L = C A' : C E% but E G : A £ 
 = C E :C A. — In hypothefis A the weight of the column 
 A E Is reprefented by the fpace A L G E, but by A I G E 
 in hypothefis B. If therefore the fpheroid of hypothefis 
 B wd.s in equilibrioy w^iile turning round its axis, the 
 eqnilihrium is deftroyed by merely changing the force 
 ailing on the column E A. There is a lofs of prefiure 
 or weight fuftained by the column E A. This may be 
 exprefled by the fpace L G I, the difference between the 
 tvi'o areas EGIA and EGLA. But the equilibrium 
 may be reftored by adding a column of fluid AM, 
 whofe weight A L R M fiiall be equal to L G I, which 
 
 , LIxAE 
 
 IS very nearly i= . 
 
 In order to find the height of this column, produce 
 G E on tlie other fide of E, and make E F to E G as 
 the denfity of the fluid to the denfity by which the nu- 
 cleus exceeded it. E F will be to E G as the gravitation 
 of a particle in E to the globe (now of the fame denfity 
 with the fluid) is to its gravitation to the redundant mat- 
 ter colle£led in the centre. Now, take DE to repre- 
 fent the gravitation of E to the fluid contained in the 
 concentric fpheroid E /3 ^ /3, which is fomewhat lefs 
 than its gravitation to the fphere E B ^ ^. Draw C D N. 
 Then A N reprefents the gravitation of A to the whole 
 fluid fpheroid, by § 558. In like manner, NI is the u- 
 3 R cited 
 
49B PHYSICAL AsrRONOMT. 
 
 nited gravitation of A to both the fluid and the central 
 matter, in the fame hypothefis. But in hypothefis A, 
 this gravitation is reprefented by N L. 
 
 Let N O rcprefent the centrifugal force afFe£ling the 
 particle A, taken in due proportion to N A or NL, its 
 whole gravitation in hypothefis A. Dravi^ C K O. D K 
 will be the centrifugal force at E. The fpace O K G I 
 will exprefs the whole fenCble weight of the fluid in 
 A E, according to hypothefis B, and O K G L will ex- 
 exprefs the fame, according to hypothefis A. L G I is 
 the difference, Lo be compenfated by means of a due 
 addition A M. 
 
 ^rhis addition may be defined by the quadrature of 
 the fpaces G E A L and G L I. But it will be abund- 
 antly exa£l to fuppofc that G L R fenfibly coincides with 
 a (Iraight line, and then to proceed in this manner. "We 
 have, by the nature of the curve G L R, 
 
 AL:EG = ED:AC* 
 
 Alfo AH, orEG:AI = EC:AC 
 
 Therefore AL : AI = EO : AD. 
 
 Now, when a line changes by a very fmall quan- 
 tity, the variation of a line proportional to its cube is 
 thrice as great as that of the line proportional tp the 
 root. H I is the quantity proportional to E A the in- 
 crement of the root EC. I L is proportional to the va- 
 riation of the cube, and is tlierefore very nearly equal to 
 thrice H I. 
 
 Therefore 
 
FIGURE OF THE EARTH. 499 
 
 Therefore fmcc E G : H I = E C : A E, we may 
 ftate EG:LI = EC:3AE, 
 
 or 3 E G : L I = E C : A E. 
 
 Now, O O L R may be confidered as equal to Q R 
 X A M, or as equal to K G X A M, and L G I may be 
 confidered as equal to LI x iAE, and 2KG X AM 
 = LIxAE. 
 
 Therefore 2KG:AE = LI:AM 
 but EC:AE = 3EG:LI 
 
 therefore 2 K G X E C : A E' = 3 E G : A M 
 
 and .2KG::|^=3EG:AM 
 
 and 2 KG : 3 E G = 4^ : AM 
 
 ^ EC 
 
 That is, twice the fenfiblc gravity at the equator is te 
 thrice the gravitation to -the central matter as a third 
 proportional to radius and the elevation of the equator 
 is to the addition necelTary for producing the equilibriitm 
 required in hypothefis A. 
 
 This addition may be more readily conceived by- 
 means of a conffcru61:ion. Make AE:Ef=:2KG: 
 3 E G. Draw e a parallel to E A, and draw C e m, cut- 
 ting A N in tn. Then a m is the addition that muft be 
 made to the column AC. A fimilar addition muft be 
 made to every diameter CT, making 2KG:3EG = 
 
 TV 
 
 •FT^ ' T /, and the v/hole will be in equilibria, 
 
 594. This determination of the ellipticity will equal- 
 ly fuit thofe cafes where the fluid is fuppofed denfer than 
 
 ^ R 2 the 
 
50O PHYSICAL ASTRONOMT. 
 
 the folid nucleus, or where there is a central hollo}^''. 
 For E G may be taken negatively, as if a quantity o£ 
 matter v/ere placed in the centre acting with a repelling 
 or centrifugal force on the iiuid. This is reprefented on 
 the other fide of the axis B b. The fpace g i I in this 
 cafe is negative, and indicates a diminution of the co- 
 lumn a r, in order to reflore the eqiiUihrium. 
 
 595. It is evident that the figure refulting from this 
 conftruclion is not an accurate ellipfe. For, in the el- 
 3iple, T/ would be in a conftant ratio to VT, whereas 
 it is as VT^ by our conftruilion. But it is alfo evident 
 that in the cafes of fmaii deviation from perfect fphe- 
 jicity, the change of figure from the accurate ellipfe of 
 hypothefis B is very fmall. The greateil deviation hap- 
 pens when E f is a maximum. It can never be fen- 
 libly greater in proportion to A E than i- of A E is in 
 proportion to E C, unlefs the centrifugal force F D be 
 very great in compari(bn of the gravity D E. In the 
 cafe of the Earth, where E A is nearly -^yo of E C, if 
 we fuppofe the mean denfity of the Earth to be five 
 times that of fea water, a jn will not exceed ttttt^ ^^ 
 EC, or^4-Tof EA. ^ ' 
 
 596. We are not to imagine that, fince central mat- 
 ter requires an addition A M to the fpheroid, a greater 
 denfity in the interior parts of this globe requires a great* 
 er cquatoreal protuberancy than if all were homogene- 
 ous ; for it is juft the contrary. The fpheroid to which 
 
 the 
 
FlOfURE OF THE EARTH. 50I 
 
 tlie addition muft be made Is not the figure fulted to a 
 homogeneous mafs, but a fi6litious figure employed as a 
 ftep to facihtate inveftigatlon. We muft therefore de- 
 fine its ellipticity, that we may know the fliape refulting 
 from the final adiuftmcnt. 
 
 Let f be the denfity of the fluid, and « the denfity 
 of the nucleus, and let n — f be = 7, fo that q corref- 
 poiids with E G of our conftru6lion, and expreffes the 
 redundant central matter (or the central deficiency of 
 matter, when the fluid is denfer than the nucleus). Let 
 B C or E C be r, A E be a*, and let g be the mean gra- 
 vity (primitive), and c the centrifugal force at A. Laft- 
 ly, let TT be the circumference when the radius of the 
 circle is r. 
 
 The gravitation of B to the fluid fpheroid is \ ^/r 
 (582.)> a"d its gravitation to the central matter is y ^r^r r. 
 The fum of thefe, or the whole gravitation of B, is \ 
 T 71 r. This may be taken for the mean gravitation on 
 every point of the fpheroidal furface'. 
 
 But the whole gravitation of B differs confiderably 
 from that of A. 
 
 \mo. C A, or C E, is to -J- AE as the primitive gravity 
 of B to the fpheroid is to its excefs above the gravitation 
 (primitive) of A to the fame, (582.) That is, r \ \ x 
 z=. y Trfr : tV ^f^i and ^V "^/^ expreffes this excefs. 
 
 2do, In hypothefis B, we have CE to CA as the 
 gravitation of B or E to the central matter is to the gra- 
 vitation of A to the fame. Therefore C E is to E A as 
 the gravitation of E to this matter is to the excefs of A*s 
 
 gravitation 
 
5^-2 
 
 PHYSICAL ASTRONOMY, 
 
 gravitation to the fame. This excefs of A's gravitatiori 
 is expreiTed by 4- x 5- .v, {or r : x = y -^ q r : j tt q x. 
 
 ^tio. Without any fenfible error, we may (late the 
 Tatio of ^ to <: as the ratio of the whole gravitation of 
 A to the centrifugal tendency excited in A by the ro- 
 tation. Therefore gr : c = ^Trft r : , and this cen- 
 
 trifugal tendency of the particle A is . This is 
 
 D o 
 
 what is exprefled by N O in our conftru6lion. 
 
 The whole difference between the gravitations of B 
 
 and A is therefore -^r ^/^ — yTrqx -\ . The 
 
 D o 
 
 gravitation of B is to this difference as -f- ^r ;z r to tV ^f^ 
 
 Q. '7T ft f C 
 
 — -y^S'^H or (dividing all by y'?rn) as r to 
 
 D o 
 
 f X qx .cr 
 5« n g ' 
 
 Now the equilibrium of rotation requires that the 
 whole polar force be to the fenfible gravitation at the 
 -equator as the radius of the equator to the femiaxis 
 (569.) Therefore we muft make the radius of the equa- 
 tor to its excefs above the femiaxis as the polar gravita- 
 tion to its excefs above the fenfible equatoreal gravitation. 
 
 That is r : ^ = r : *^ — ^ > and therefore x = 
 
 5 « n g 
 
 f X X C f C V Q X f X 
 
 ^- j . Hence we have — = ;c -j- ^ -^ — . 
 
 ^n n g g n $n 
 
 Butj? = «~/. Therefore— zz^c-f — -'^^-^, 
 
 g fi n ^n 
 
 Wherefore 
 
FIGURE OF THE EARTHT. 50^ 
 
 Wherefore a? = ^^-., = 1 :-, which. 
 
 c c f 
 
 is more conveniently exprefTed in this form x = - — X 
 The fpecies, or ellipticity of the fpheroid is 
 
 5 ^^ — 3/ 
 
 X 
 
 r' 2g 5^^ — 3./' 
 
 Such then is the elHptical fpheroid of hypothefis B , 
 and we faw that, in refpe6t of form, it is fcarcely diftin- 
 guifliable from the figure which the mafs will have when 
 the fidlitious force of the central matter gives place to 
 tlie natural force of the denfe fpherical nucleus. This 
 is true at leaft in all the cafes where the centrifugal 
 force is very fmall in comparifon with the mean gravita-> 
 tion. 
 
 We muft therefore take fbme notice of the influ- 
 ence which the variations of denfity rhay have on the 
 form of this fpheroid. We may learn this by attending 
 
 to the formula 
 
 X c c n . 
 — = ^— X • 
 
 r 2g 5 '' — 3/ 
 The value of this formula depends chiefly on the fra£lion 
 n 
 
 s^ — if' 
 
 597. If the denfity of the interior parts be immenfely 
 greater than that of the furrounding fluid, the value of this 
 
 fraction becomes nearly f, and - becomes nearly = — , 
 
 and the ellipfe nearly the fame with what Hermann af- 
 fignedto a homogeneous fluid fpheroid. 
 
|»04 PHYSICAL ASTRONOMT. 
 
 If ;z =: c /; then ■ ^= -— ; and, in the cafe of 
 
 S^ — Zf 22 
 
 the Earth, - would be nearly == — --:, makhiff an equa- 
 r •' 508,0 ° ^ 
 
 toreal elevation of nearly 7 miles. 
 
 troS. If n =: /, the fradlion :, becomes -|-> 
 
 and - = — J which we have already fhewn to be fuitable 
 
 to a homogeneous fpheroid, with which this is equiva- 
 lent. The protuberance or ellipticity in this cafe is to 
 that v/hen the nucleus is incomparably denfer than the 
 fluid in the proportion of 5 to 2. This is the greateft 
 ellipticity that can obtain when the fluid is not denfer 
 than the nucleus* 
 
 Between thefe two extremes, all other values of the 
 formula are competent to homogeneous fpheroids of gra- 
 vitating fluids, covering a fpherical nucleus of greater 
 denfity, either uniformly denfe or confiding of concentric 
 fpherical ftrata, each of which is uniformly denfe. 
 
 From this view of the extreme cafes. We may infer 
 in general, that as the incumbent fluid becomes rarer in 
 proportion to the nucleus, the ellipticity diminifhes. M. 
 Bernoulli (Daniel), mifled by a gratuitous aflumption, 
 fays in his theory of the tides that the ellipticity produced 
 in the aereal fluid which furrounds this globe will be 
 800 times greater than that of the folid nucleus ; but 
 this is a miftake, which a jufter aflTumption of data would 
 have prevented. The aereal fpheroid will be fenfibly 
 
 Jefs oblate than the nucleus. 
 
 It 
 
I^IGURE OF THE EARTH. 505 
 
 It was faid that the value of the formula depended 
 
 chiefly on the fradion ^. But it depends alfo 
 
 5 « — 3/ 
 
 C c 
 on the fra£lion ~ , increafmg or diminifliing as c In- 
 
 creafes or diminifhes, or as g dimiiiifhes or increafes. 
 
 It mufi: alfo be remarked that the theorem - = ^- for 
 
 a homogeneous fpheroid was deduced from the fuppofi- 
 tion that the eccentricity is very fmall (See § 5S0. 585.) 
 When the rotation is very rapid, there is another form 
 of an elliptical fpheroid, which is in that kind of equi- 
 libriumj which, if it be difturbed, will not be recovered, 
 but the eccentricity will increafe with great rapidity, till 
 the whole diflipates in a round flat (heet. But within 
 this limit, there is a kind of {lability in the equUihrluin^ 
 by which it is recovered when it is difturbed. If the 
 rotation be too rapid, the fpheroid becomes more oblate, 
 and the fluids which accumulate about the equator, hav- 
 ing lefs velocity than that circle, retard the motion. This 
 goes on however fome time, till the true jQiape is over- 
 paiTed, and then the accumulation relaxes. The motion 
 is now too flow for this accumulation, and the waters 
 flow back again toward the poles. Thus an ofcillation is 
 produced by the di'fturbance, and this is gradually dimi- 
 nlHied by Che mutual adhelion of the waters, and by fric- 
 tion, and tilings foon terminate in the refumption of the 
 proper form. 
 
 599. When the denfity of the nucleus is lefs than 
 that of the fluid, the varieties which jrefult in the form 
 
 3 S from 
 
505 PHYSICAL ASTRONOMY. 
 
 from a variation in the denfity of the fluid are mucli 
 greater, and more remarkable. Some of them are even 
 paradoxical. Cafes, for example, may be put, (when 
 the ratio of n to / differs but very little from that of 3 
 to 5), where a very fmall centrifugal force, or very flo\r 
 rotation, fhali produce a very great protuberance, and, 
 on the contrary, a very rapid rotation may coiifift with 
 an oblong form like an egg. But thefe are very fingular 
 cafes, and of little ufe in the explanation of the pheno- 
 mena a0:ually exhibited in the folar fyflem. The erjui/i- 
 hriu'/n which obtains in fuch cafes may be called a tot- 
 tering equUibriumy which, v/hen once dillurbed, will not 
 be again recovered, but the difTipation of the fluid will 
 immediately follow with accelerated fpeed. Some cafes 
 will be confidered, on another occafion, where there is a 
 deficiency of matter in the centre, or even a hollow. 
 
 600. The chief diftin£lion betv/een the cafes of a 
 nucleus covered with an equally denfe fluid, and a denfe 
 nucleus covered with a rarer fluid, confifts in tlie diflx?r- 
 ence between the polar and equatoreal gravities ; for we - 
 fee that the diflference in fhape is inconfiderable. It has 
 ^been (hewn already that, in the homogenous fpheroid of 
 fmall eccentricity, the excefs of the polar gravity above. 
 
 tjie fenfible equatoreal gravity is nearly equal to — ^ (for 
 
 rx^x = g : ^—] . When, in addition to this, we take 
 into account the diminution c, produced by rotation, we 
 
 iiave -— + i: for the whole difference between the pO' 
 
FIGURE OF THE EARTH. $07 
 
 iar and the fenfible equatoreal gravity. But, in a homo- 
 
 geneous fpheroid, we have tc = . Therefore the ex- 
 
 cefs of polar gravity in a homogeneous revolving fphe- 
 
 rold is --he or ^— . We may diftinguiih this excefs 
 
 4 4 
 
 ill the homogeneous fpheroid by the fymbol E. 
 
 601. But, in hypothefis B, the equilibrium of rota- 
 tion requires that r be to x as ^ to — - , and the excefs 
 
 of polar gravity in this hypothefis is ~ . But we have 
 
 alfo (een that in this hypothefis, - = — X 
 
 Therefore the excefs of polar gravity in this hypothefis 
 
 is ^— X 7.. Let this excefs be dilliniruifhed by 
 
 2 5« — 3/ 
 
 the fymbol f. 
 
 602. The excefs of polar gravity muft be greater 
 than this in hypothefis A. For, in that hypothefis the 
 equatoreal gravity to the fluid part of the fpheroid is al= 
 ready fmaller. And this fmaller gravity is not fo much 
 increafed by the natural gravitation to tlie central matter, 
 in the inverfe dupUcate ratio of the diftance, as it was in- 
 creafed by the fictitious gravity to the fame matter, in 
 the direft ratio of the diftances. The fecond of the 
 three diftinctions noticed in § 596. between the gravi- 
 tations of B and A was — — . This muft now be 
 
 ?i 
 
 "2. Q X 
 
 clianged into -f- — ^— , as may eafily be deduced from 
 3^2 § 593, 
 
5e8 PHYSICAL ASTRONOMY. 
 
 § 593, where — — is reprefented by H I in fig. 6j, 
 
 and the excefs, forming the compenfa'tion for hypothefis 
 A is reprefented by H L, nearly double of H I, and in 
 |:he opppfite dire£lion, diminifliing the grayitation of A. 
 
 The difference of thefe two ftates is -^—y by which thq 
 
 ' • n ■* 
 
 tendency of A to the central matter in hypothefis A falls 
 Ihort of what it was in hypothefis B. Therefore, 'as 
 
 *'" — f- — is to ^-^ , fo is the excefs £ to a quan- 
 
 ^ f^i n g n 
 
 tity £'5 >yhich mud be added to e, in order to produce 
 the difference of gravities ^, conform.able to the ftatement 
 of hypothefis A. Now, in hypothefis B, we had .v = 
 
 -^-^ — + — J ^nd we may, without fcruple, fuppofe 
 
 ^ the fame in hypothefis A. Therefore « : t' =r a^ ; 
 
 n n 71 71 '^ 
 
 5 ^ .. ^ .. 3 ^^ — 3 / _ 5 ^' v/ 3 « 
 
 f 
 
 :>./ 
 
 ■ 2 ^ 5^~-3/^ « ' ^ 5^ — 3/' 
 
 Add to this 2, which is ~ X — - — ^5 and we ob- 
 
 2 5^^-7-3/ 
 tain for the excefs e of polar gravity in hypothefis A 
 
 = If X 4^ — 3/ 
 
 2 5^^ — 3/* 
 
 603. Let us npv*r compare this excefs of polar gra- 
 vity above the fenfible equatoreal gravity in the three 
 hypothefes : ift^ A, fuited to the fluid furrounding a 
 fphericaj i^i^cleu^ of greater denfity : 2^, B, fuited tq 
 the fame fluid, furrounding a central nucleus which at- 
 iracls with a force proportional to the diftance : and, 3^, 
 
 -■ c, 
 
FIGURE GF THE EARTH. 509 
 
 C, firited to a homogeneous fluid fpheroid, or enclofmg 
 a fphcrical nucleus of equal doiifity. Tliefvi cxccfies 
 
 are 
 
 5 C 71 
 
 B ^ X 
 
 2 5-^ — 3./ 
 
 C if, or -ii- X 5-!L=Ji:. 
 4 4 5'^ — 3/ 
 
 It is evident that the fum of A and B is — v 
 
 2, 
 
 5 ^^ •" 3 / ^vhich is double of C, or ^^ X ^^^~^^^.. 
 S'^ — Zf 4 5 '^-^3/ 
 
 and therefore C is the arithmetical mean between them. 
 
 re A. n '— Q /* 
 
 Now we have feen that ^— X ~ ^^— .expreffes 
 
 die ratio of the excefs of polar gravity to the mean gra- 
 
 z c 
 vity in the hypothefis A. We have alfo {^^Vi that •^— 
 
 X may fafely be taken as the value of the el- 
 
 5" — 3/ ^ 
 
 iipticity in the fame hypothefis. It is not perfectly <»x- 
 
 acl, but the deviation is altogether infenfibie in a cafe* 
 like that of the Earth, where the rotation and the ec- 
 centricity are fo moderate. And, laffcly, we have i^^w 
 that the fame fraction that exprefles the ratio of the ex- 
 cefs of polar gravity to mean gravity, in a homogeneous 
 fpheroid, alfo expreifes its ellipticity, and that twice this 
 fraction is equal to the fum of the other two. 
 
 604. Hence may be derived a beautiful theorem, 
 fu:ft given by M. Ciairaut, that the fraciion exprejfmg 
 twife the ellipticity cf a hcriiogeneous revolving fpheroid is ike 
 
 fum 
 
510 PHYSICAL ASTRONOMY. 
 
 fuin of two fraclionSf one of which expre/fef the ratio of 
 the excefs of polar gravity to mean gravity^ and the other 
 exprejfes the ellipticity of any fpheroid of f mall eccefitricityy 
 ivhlch confijls of a fluid covering a denfer fpherical nucleus. 
 If therefore any other phenomena give us, in the 
 cafe of a revolving fpheroid, the proportion of polar and 
 cquatoreai gravities, we can find its ellipticity, by fub- 
 tra£^ing the fraction expreifing the ratio of the excefs of 
 polar gravity to ths mean gravity from twice the ellip- 
 ticity of a homogeneous fpheroid. Thus, in the cafe of 
 the Earth, twice the ellipticity of the homogeneous fphe- 
 roid is TTT* -^ medium of feven comparifons of the 
 rate of pendulums gives the proportion of the excefs of 
 polar gravity above the mean gravity = -^y o • ^^ ^^^ 
 fra£i:ion be fubtra£led from -^\-^^ it leaves — p- for the 
 medium ellipticity of the Earth. Of thefe feven expe- 
 riments, five are fcarcely different in the refult. Of the 
 other two, one gives an ellipticity not exceeding ~ j- . 
 The agreement in general is incomparably greater than 
 in the forms deduced from the comparifons of degrees of 
 the meridian. All the comparifons that have been pub- 
 lifhed concur in giving a ccnfiderably fmaller eccentricity 
 to the terraqueous fpheroid than fuits a homogeneous 
 maH^, aild which is ufually called Newton's determina- 
 tio-^.. It is indeed his determination, on the fuppofition of 
 homogeneity ; but he exprefsly fays that a different denfity 
 in the interior parts will induce a different form, and he 
 points out fome fuppofititious cafes, not indeed very pro- 
 bable, where the form will be different. Newton has 
 not conceived this fubje(n; w^th his ufual fagacity, and 
 
 has 
 
FIGURE OF THE EARTH. ^tt 
 
 has made fome inferences that are certainly inconliftent 
 with his law of gravitation. 
 
 That the protuberancy of the terreflrial equator is 
 certainly lefs than ^-y-j- proves the interior parts to be of 
 a greater mean dcnflty than the exterior, and even gives 
 us fome means for determining how much they exceed in 
 
 denfity. For, by making the fradion ^— X ^. 
 
 =: T-^-Q, as indicated by the experiments with pendu- 
 lums, we can find the value of w. 
 
 605. The length of the feconds pendulum is the 
 meafure of the accelerating force of gravity. Therefore 
 let / be this length at the equator, and / -j- J the length 
 
 at the pole. We have — X — -4^= tj whence 
 
 2^ 5«-3/ / 
 
 An — -if 1 z,d —. . . . , 
 
 ~—r= — ^. This equation, when properly treat- 
 
 ed, 2;ives -. = — ^~ ■ — -,, &c. &c.* 
 
 ^ J 1Q c L — i o g a 
 
 The fame principles may be applied to any other pla- 
 net as well as to this Earth. Thus, we can tell what 
 portion of the equatoreal gravity of Jupiter is expended 
 in keeping bodies on his furface, by comparing the time 
 
 of 
 
 * We have information very lately of the meafurement of 
 a degree, by Major Lambton in the Myfore in India, with 
 excellent inftruments. It lies in lat. 12° 32', and its length is 
 60494 Britifti fathoms. We are alfo informed by Mr Melan- 
 derhielm of the Svvedifh academy that the meafure of the de- 
 gree in Lapland by Maupertuis is found to be 208 toifes too 
 great. This was fufpeded. 
 
5I2J PHYSICAL ASTRONOMY. 
 
 of his rotation with the period of one of his fatellit^s..^ 
 We find that the centrifugal force at his equator is -^-^ 
 
 of the whole gravity, and from the equation = x, 
 
 we (hould infer that if Jupiter be a homogeneous fluid 
 or flexible fpheroid, his equatoreal diameter will exceed 
 his polar axis nearly lo parts in 113, which Is not very 
 ilifl^erent from what we obfcrve ; fo much however as to 
 authorife us to conclude that his denfity is greater near 
 the centre than on his furface. 
 
 Thefe obfervations mufl fuffice as an account of this 
 fubjeft. Many circumftances, of great efFecl:, are omit- 
 ted, that the confideration might be reduced to fuch fim- 
 plicity as to be difcufl^ed without the aid of the higher 
 geometry. The (Indent who wilhes for more complete 
 information mud confiilt the elaborate performances of 
 -Euler, Clairaut, D'Alembert, and La Place. The dif- 
 fertation of Th. Simpfon on the fame fubje6t is excellent. 
 The diflTertation of F. Bofcovich will be of great fervic^ 
 %% thofe who are lefs verfant in the fluxionary calculus, 
 that author having every where endeavoured to reduce 
 things to a geometrical conflru^liorf. To thefe I would 
 add the Cofmographia of Frifius, as a very mafterly per- 
 formance on this part of his fubjedl:. 
 
 It were delireable that another element were added to 
 the problem, by fuppofing the planet to confift of co- 
 herent flexible matter. It is apprehended that this would 
 give it a form more applicable to the actual flate of 
 things. If a planet confifl: of fuch matter, ducSlile like- 
 melted glafs, the fhape which rotatioi^ combined with gra* 
 
 vitatioa 
 
FIGURE OF THE PLANETS. 51^ 
 
 vitation and this kind of cohefion, would induce, will 
 be confiderably different from what we have been con- 
 fideringi and fufceptible of great variety, according to 
 the thicknefs of the fhell of which it is fuppofed to con- 
 fift. The form of fuch a fliell will have the chief in- 
 fluence on the form which will be alTumed by an ocean 
 ©r atmofphere which may furround it. If the globe of 
 Mars be as eccenttic as the late obfervations indicate it to 
 be, it is very probable that it is hollow, with no great 
 thicknefs. For the Centrifugal force muft be exceeding- 
 ly fmalL 
 
 606. The. mod fingular example of this phenome- 
 non that is exhibited in the folar fyftem, is the vaft arch 
 or ring which furrounds tlie planet Saturn, and turns 
 round its axis with moll aftonifhing rapidity. It is above 
 200000 miles in diameter, and makes a complete rota- 
 tion in ten hours and thirty-two minutes. A point on its 
 furface moves at the rate of iooo|- miles in a minute, or 
 nearly 17 miles in one beat of the clock, which is 58 
 times as fwift as the Earth's equator. 
 
 M. La Place has made the mechanifm of this mo- 
 tion a fubje£l of his examination, and has profecuted it 
 with great zeal and much ingenuity. He thinks that 
 the permanent flate of the ring, in its period of rota- 
 tion, may be explained, en the fuppofitlon that Its parts 
 are without connexion, revolving round the planet like 
 fo many fatellites, fo that it may be confidered as a 
 vapour. It appears to me that this is not at all probable- 
 
 ? T He 
 
jr4 -PHYSICAL ASTRONO^IT» 
 
 He fays that the obferved inequalities in the circle of* th^ 
 ring are neceilary for keeping it from coalefcing with 
 the planet. Such inequalities feem incompatible with 
 its own conftitution, being inconfiftent with the equili- 
 brium of forces amo^ng incoherent bodies. Befides, as 
 he fuppofes no cohefion in tt, any inequalities in the con- 
 ftitutixDil of its different parts cannot influence the gene- 
 ral motion of the whole in the manner he fuppofes^ but 
 merely by an inequality of gravitation. The effe6l of 
 this, it is apprehended, would be to deftroy the perma- 
 nency of its conftruiftion, without fecuring, as he ima* 
 gines, the fteadinefs of its pofition. But this feems to 
 be the point which he is eager to eftablifh ; and he finds, 
 in the numerous lift of poiTibilities, conditions which 
 bring things within his general equation for the equili" 
 hriiim of revolving fpheroids *, but the equation is fo 
 very general, and the conditions are fa many, and fo im- 
 plicated, tliat there is reafbn to fear that, in fome cir- 
 cumftances, the equilibrium is of that kind that has no 
 ftability, but, if diflurbed in the fmalleft degree^ is de- 
 ftroyed altogether, being like the equilibrium of a needle 
 poifed upright on its point. There is a flronger objec- 
 tion to M. La Place's explanation. He is certainly mif- 
 taken in thinking that the period of the rotation of the 
 ring is that which a fatellitc would have at the fame 
 diflance. The fecond CalTmian fatellite revolves in 65^ 
 44', and its diftance is 56,2 (the -elongation in feconds). 
 Now 65^ '44' I* : 10' 32^1' = 56,2^ : 16,45. This is the 
 diflance at which a fatellite would revolve in lo** 32'. 
 
 It 
 
GONSTITWTION OF SATURN's RING. $\ $ 
 
 It muft be fomewhat lefs than this, on account of the 
 oblate figure of the planet. Yet even this is lefs than 
 the radius of the very inmoll edge of the ring. The ra- 
 dius of the outer edge is not lefs than 22i, and that of 
 its middle is ^o. 
 , It is a much more probable fuppofition (for we can 
 only fuppofe) that the ring confiils of coherent matter. 
 It has been reprefented as fupporting itfelf like an arch; 
 but this is lefs admiflible than La Place's opinion. The 
 rapidity of rotation is fuch as would immediately fcatter 
 the arch, as water is flirted about from a mop. The 
 ring mull cohere, and even cohere with confiderable 
 force, in order to counteract the centrifugal force, which 
 confiderably exceeds its weight. If this be admitted, 
 and furely it is the mod obvious and natural opinion, 
 there will be no diihculty aridng from the velocity of 
 rotation or the irregularity of its parts. M. La Place 
 might eafily pleafe liis fancy by contriving a mechanifm 
 for its motion. We may fuppofe that It is a vifcid fub- 
 ftance like melted glafs. If matter of this conilitution, 
 covering the equator of a planet, turn round its axis too 
 fwiftly, the vifcid matter will be thrown off, retaining 
 its velocity of rotation. It will therefore expand into a 
 ring, and will remove from the planet, till the velocity 
 of its equatoreal motion correlpond with its diameter 
 and its curvature. However fmaii we fuppofe the co- 
 hefive or vifcid force, it v/Ill caufe this ring to flop at a 
 dimenfion fmaller than the orbit of a planet moving with 
 the fame velocity. — ^Thefe fecm to be legitimate confe^ 
 quences of what we know of coherent matter, and they 
 
 3 T 2 greatly 
 
5l5 I^HTSICAL ASTR0>W5MT. 
 
 greatly refemble what we fee in Saturn^s rmg. This 
 conftitution of the ring is alfo well fitted for admitting 
 thofe Irregularities which are indicated by the fpots on 
 the ring, and which M. La Place employs witli fo much 
 ingenuity for keeping the ring in fuch a pofition that 
 the planet always occupies its centre. This is a very 
 curious circumftance, when confidered attentively, and 
 its importance is far from being obvious. The planet 
 and the ring are quite feparate. The planet is moving 
 in an orbit round the Sun. The ring accompanies the 
 planet in all the iiTegularities of its motion, and has it 
 always in the middle. This ingenious mathematician 
 gives ftrong reafons for thinking that, if the ring wer^ 
 perfectly circular and uniform, although it is pojfthle tQ 
 place Saturn exactly in its centre, yet the fmalleft dif- 
 turbance by a fatellite or paihng comet would be the 
 .beginning of a derangement, which would rapidly in- 
 creafe, and, after a very fhort time, Saturn would be in 
 conta6l with the inner edge of the ring, never more to 
 feparate from it. But if the ring is not uniform, but 
 more mafTive on one fide of the centre than on the o- 
 ther, then the planet and the ring may revolve round a 
 common centre, very near, but not coinciding with the 
 centre of the ring. He alfo maintains that the oblate 
 form of the planet is another circumftance abfoluteiy ne- 
 celTary for the ftability of the ring. The redundancy 
 of the equator, and flatnefs of the ring, fit thefe two 
 bodies for afting on each other like two magnets, {o as 
 X^ adjuft each other's motions. 
 
 The 
 
FIGITRE OF THE MeCyN. ^ I J 
 
 The whole of this analyfis of th<? mechaiiifin of Sa- 
 turn's ring is of the moft intricate kind, and is carried 
 on by the author by calculus alone, fo as not to be in- 
 llruQive to any but very learned and expert analyfts. 
 Several points of it however might have been treated 
 more familiarly. But, after all, it mufl: reft entirely oji 
 the truth of the conjectures or aflumptions made for 
 procuring the poffible application of the fundamental 
 equations. 
 
 607. The Moon prefents to the refleCling mind a 
 phenomenon that is curious and interefting. She always 
 prefents the fame face to the Earth, and her appearance 
 juft now perfeftly correfponds v/lth the oldeft accounts 
 we have of the fpots on her difk. Thefe indeed are not 
 cf very ancient date, as they cannot be anterior to the 
 telefcope. But this is enough to fliew that the Moon 
 turns round Iier a:5fis in precifely the fame time that fhe 
 revolves round the Earth. Such a precife coincidence 
 i-3 very remarkable, and naturally induces the mind to 
 fpeculate about tlie caufe cf it. Newton afcribed it to 
 an oblong oval figure, more denfe, or at leaft heavier, at 
 one cad than at the other. This he thought might o- 
 perate on the Moon fomewhat in the way that gravity 
 operates on a pendulum. He defines this figure in Pro- 
 pofition 38. B. III. J and as the eccentricity, or any devia- 
 tion of its centre of gravity from that of its figure, is ex- 
 tremely fmall, the ijIs difpofieus, by which one diameter 
 is diredcd towards the Earth, is alfo very minute, and 
 
 its 
 
ri^ PHYSICAL ASTRONOMY. 
 
 its operation muft be too flow to keep one face fteadily 
 turned to the Earth, in oppofitlon to the momentum of 
 rotation round the axis, feven or eight days being all the 
 time that is allowed for producing this efFe6l. There- 
 fore we obferve what is called the Lihration of the Moon, 
 arifing from the uniform rotation of the Moon, com- 
 bined with her unequable orbital motion. One diame- 
 ter of the Moon is always turned to the upper focus of 
 her orbit, becaufe her angular motion round that focus 
 is almoft perfedly uniform, and therefore correfponds 
 with her uniform rotation. But that diameter which is 
 towards us when the Moon is in her apogee or perigee, 
 deviates from the Earth almoft fix degrees when flie is 
 in quadrature. But although, in the fliort fpace of eight 
 days, the pendulous force of the Moon cannot prevent 
 this deviation altogether, it undoubtedly leflens it. It is 
 faid to produce another effect. If the origirml projec- 
 tion of the Moon in the tangent of her orbit did not 
 precifely, but very nearly, correfpond w-ith the rotation 
 imprefled at the fame tim.e, this pendulous tendency 
 would, in the courfe of many ages, gradually leflen the 
 difference, and at laft make the rotation pcrfe£lly com- 
 jnenfurate with the orbital revolution. 
 
 But we apprehend that this conclufion cannot be ad- 
 mitted. For, in whatever v/ay we fuppofe this arrang- 
 ing force to operate, if it has been able, in the courfe of 
 ages, to do away fome fmall primitive difference between 
 the velocity of rotation and the velocity of revolution, 
 <tniuft ccrt.nnly liave been able to annihilate a mucU ■ 
 
 ^ fmaller 
 
FIGURE OF THE UOCSff, 519 
 
 fihaller difference in the pofitlon of the Moon's figure^ 
 namely, the obUquity of the axis to the plane of the 
 orbit. * It deviates about i or 2 degrees from the per- 
 pendicular, and it firmly retains this obliquity of pofi- 
 tion ; and no obfervation can difcover any deviation from 
 perfect- paralielifm of the axis in all fituations. It fure- 
 ly requires much lefs action of the direfting force to 
 produce this change in the pofition of the axis, than to 
 overcome even a very fmall difference in angular mo- 
 tion, becaufe this laft difference accumulates, and makes 
 a great difference of longitude. 
 
 Thfefe confiderations feem to prove that the conftanfi' 
 appearance of one and the fame part of the Moon's fur- 
 face has not been produced by the caufe fufpe(5ted by 
 Newton. The coincidence has more probably been ori- 
 ginal. We have no reafon to doubt that the fame con- 
 fummate fi!:ill that is manifeft in every part of the fyf- 
 tem, in which o-ery thing has an accurate adjuflment, 
 pondere et menfura, alfo made the prim.itive revolution 
 rotation of the Moon that which we now behold and ad- 
 mire. 
 
 * The axis round which die rotation of the Moon is 
 performed is inclined to the plane of the ecliptic in an angle 
 of 88^ °, and it is inchned to the plane of the lunar orbit 
 ^2^. It is always fituated in the plane palTing through the 
 poles of the ecliptic and of the lunar orbit. It therefore de- 
 viates about i^ from the axis of the ecliptic, and 7 from that 
 of the Moon's orbit. The defcending node of the Moon's 
 cc[uator coincides with the afcending node of her orbit. 
 
^2G pMTsicAl Astronomy. 
 
 mire. The manifeft fubferviency to great and good pur- 
 pofes, in every thing that we in fome meafure under- 
 ftand, leaves us no room to imagine that this adjuftment 
 of the lunar motions is not equally proper. 
 
 608. Philofophers have fpeculated about the nature 
 of that body of faintly fhlning riiatter in which the Sun 
 feems immerged, and is called the zodiacal lights becauf^ 
 it lies in the zodiac. It is rarely perceptible in this cli- 
 mate, yet may fometimes be feen In a clear night in Fe- 
 bruary and March, appearing ill the \veft, a little to the 
 north of where the Sun fet, like a beam of faint yeU 
 lowilh grey light, flanting toward the north, and ex- 
 tending, in a pointed or leaf fhape, about eight or ten 
 degrees* The appearance is nearly what would be ex- 
 hibited by a fliining or refle^blng atmofphere furround^ 
 ing the Sun, and extending, in the plane of the ecliptic, 
 at lead as far as the orbit of Mercury, but of fmall thick- 
 nefs, the whole being flat like a cake or dilk, whofe 
 breadth is at lead ten times its thicknefs in the middle. 
 
 This has been tlie fubjedl: of fpeculation to the mc- 
 thanical philofophers. It is fomething^ conne<Sted witl: 
 the Sun. We have no knowledge of any conne^lingr 
 principle but gravitation. But fmiple gravitation would 
 gather this atmofphere into a globular fhape, whereas it 
 is a very oblate dilk or lens. Gravitation, combined with 
 a proper revolution of the particles round the Sun, might 
 throw the vapour into this form ; and the bbje6l of the 
 fpeculation is to alT.gn the rotation that is fuitabie to it. 
 
 If 
 
OF THE ZODIACAL LlCMt. ^H 
 
 tf the zodiacal light be produced by the refle£l:ion of an 
 jtmofphere that is retained by gravity alone, without any 
 mutunl adhefion of its particles, it cannot have the form 
 that we obferve. The greateft proportion that the eqiia- 
 toreal diameter can have to the polar is that of 3 to 2 ; 
 for, beyond that, the centrifugal force would more than 
 balance its gravitation, and it would diffipate. A very 
 fi:rong adhefiort is necelTary for giving fo oblate »a form 
 as we obferve in the zodiacal light. Combined with this, 
 it may indeed expand to any degree, by rapidly whirling 
 about, as v/e fee in the manufacture of crown-glafs. But 
 hov/ is this whirling given to the folar atmofphere ? It may 
 get it by thb rnere action of the furface of the Sun, in the 
 manner defcribed by Newton in his account of the produc- 
 tion of the Cartefian vortices. The furface drags round 
 what is in conta£l: with it. This ftratum a6ls on the 
 liext, and communicates to it part of its own motion. 
 This goes on from ftratum to ftratum, till the outermoft 
 ftratum begins' to move alfo. All this wliile, each inte- 
 rior ftratum is circulating more fwiftly than the one im- 
 mediately without it. Therefore they are ftill a6^i.ng on 
 one another. It is very evident that a permanent ftate 
 is not acquired, till all turn round in the fame time with 
 the Sun's body. This circumftance limits the poffib.le 
 expanfion of an atmofphere that does not cohere. It 
 cannot exceed the orbit of a planet which would revolve 
 round the Suti in that time. But the zodiacal light ex- 
 tends much farther. •- '^^"- 
 
 The difcoveries of Df Herfchel on the furface of the 
 3 U Sun, 
 
5-22^ PHYSICAL ASTRONOMY. 
 
 Sun, If confirmed by future obfervation, render this pro- 
 duction of the zodiacal light inconceivable. For mo- 
 tions and changes are obferved there, which fhew a per- 
 fect freedom, not conflrained by the adhefion of any fu- 
 .perior ftrata. This would give a conftant wefterly mo- 
 tion on the furface of the Sun. 
 
 The difficulty in accounting for this phenomenon Is 
 greatly increafed by the faft that when a comet pafles 
 through this atmofphere, the tail of the comet is not per- 
 ceptibly affe6led by it. The comet o£ 1 743 gave a very 
 good opportunity of obferving this. It was not attended 
 to j but the defcriptions that are given of the appearances 
 of that comet fliew clearly that the tail was (as ufual) 
 directed almoil ftraight upward from the Sun, and there- 
 fore it mixed with this vapour, or whatever it may be, 
 without any mutual dilturbance. 
 
 It appears therefore, on the whole, that we are yet 
 ignorant of the nature and mechanifm of the zodiacal 
 light. 
 
 609. Before concluding this fubjedt, it is not Im- 
 proper to take fome notice of an obfervation to which 
 great importance has been attached by a certain clafs of 
 philofophers. We fhall find it demonftrated in its pro- 
 per place, that when the force which impels a firm body 
 forward a£ts in a dire£tion which pafles through its centre 
 of gravity, it merely impels it forward. The body moves 
 in that diredion, and every particle moves alike, fo that, 
 during its progrefs, the body preferves the fame attitude 
 
 (fo 
 
ROTATIVE AND ORBITAL MOTION. 523 
 
 ^ro to fpeak). Taking any tranfverfe line of the body for 
 n diameter, we exprefs the circumftance by faying that 
 this diameter keeps parallel to itfelf, that is, all its fuc- 
 cefTive pofirions are parallel to its firft pofition. But, 
 when the moving force afts in a line which paffes on 
 one fide of the centre of the body, the body not only 
 advances in the direction of the force, but alfo changes 
 its attitude, by turning round an axis. This is eafily 
 {een and underftood in fome fimple cafes. Thus, if a 
 beam of timber, floating on water, be pufhed or pulled 
 in the middle, at right angles to its length, it will move 
 in that diredlion, keeping parallel to its firft pofition. 
 But, if it be pufhed or pulled in the fame direction, ap- 
 plying the force to a point fituated at the third of its 
 length, that end is moft affected (as we fhall fee fully 
 ^emonftrated) and advances fafteft, while the remote 
 end is left a little behind. In this particular cafe, the 
 initial motion of all the parts of the beam is the fame as 
 if the remote end were held faft for an inftant. If the 
 impulfe has been nearer to one end than y of the length, 
 the remote end will, in the Jirji injlanty even move a 
 little backward. We fhall be able to ftate precifely the 
 relation that will be obferved between the progreflive mo- 
 tion and the rotation, and to fay how far the centre of the 
 body will proceed while it makes one turn round the 
 axis. We fhall demonftrate that this axis, round which 
 the body turns, always pafles through its centre of gra- 
 vity in a certain determined direction. 
 
 It very rarely happens that the diredlion of the im- 
 3 U 2 _ Polling 
 
524 PHYSICAL ASTRONOMY. 
 
 pelling force pufles exactly through the centre of a body-; 
 and accordingly we very rarely obfervc a body mov- 
 ing forward in free fpace without rotation. A ilone 
 thrown from the hand never does. A bonib-iliell, ov 
 a cannon bullet, Iras commonly a very rapid motloii 
 of rotation, which greatly deranges its intended direc- 
 tion. 
 
 The fpequiative philofophers who wiih to explain ail 
 the celeflial motions mechanically, think that they ex- 
 plain the rotation of the planets, and all the phenomena 
 dependnig on it, by fayir4g that one and the fame force 
 produced the revolution round the Sun, and the rotation 
 round the axis ; and produced thofe motions, becaufe the 
 direclion of the primitive impulfe did not pafs precifely 
 through the centre of the planet. They even Ihew by 
 calculation the diftance between the centre and the line 
 of dire6i:ion of the impelling force. Thus, they flie-w.: 
 that the point of impulfion on this Earth is diftant from 
 its centre -^i-- of its diameter- 
 
 Having thus accounted, as they imagine, for the 
 Earth's rotation, they fay that this rotation caufes the 
 Earth to fwell out all around the equator, and t]iey affign 
 the precife eccentricity that the fpheroid m.uft acquire. 
 They then fhew that the aftion of the Sun and Moon 
 on this equatoreal protuberance deranges the rotation, 
 lb that the axis does not remain parallel to itfelf, and 
 produces the phenomenon called the preceflion of the equi- 
 noxes. And thus all is explained mechanically. An4 
 en this explanation a conjedure is founded, which leads 
 
 to 
 
ROTATION COMBINED WITH REVOLUTION. 525 
 
 ^to very magnificent conceptions of the vifible univerfe. 
 The Sun turns round an axis. Analogy ihould lead us 
 to afcribe this to the fame caufe— to the aftion of a 
 force whofe dire<Slion does not pafs through his centre. 
 If fo, the Sun has alfo a progreflive motion through the 
 boundlefs fpace, carrying all the planets and comets along 
 witli him, juft as we obferve Jupiter and Saturn carrying 
 their (^Uellites round their annual orbits. 
 
 This is, for the moft part, perfectly juft. A planet 
 turns round its axis and advances, and therefore the force 
 which refults from the adiual compofition of all the forces 
 which cooperated in producing both motions, does not 
 pafs through the centre of the pl.met, but precifely at 
 the diftance afligned by thefe gentlemen. But there is 
 nothing of explanation in all this. From the manner in 
 which the remark and its application are made, we are 
 milled in our conception of the fact, and the imagina- 
 tion immediately fuggefts a fmgle forcey fuch as we are 
 accuftomed to apply in our operations, acting in one 
 precife line, and therefore on one point of the body. 
 It is this fimplification of conception alone which gives 
 the remark the appearance of explanation. A mathema- 
 tician may thus give an explanation of a firft rate fhip 
 I of war turning to windward, by (lie wing how a rope 
 I may be attached to the (hip, and how this rope may 
 be pulled, fo as to make her defcribe the very line 
 Tne moves in. But the feaman knows that this is no 
 explanation, and that he produced this motion of the 
 fiiip by various manoeuvres of the fails and rudder. The 
 
 onlv 
 
5^5 PHTSICAL A-STRONOMY. 
 
 only explanation that could be given, correfponding to 
 the natural fuggeftion by this remark, would be the 
 ihewing fome general h£k in the fyftem, in which this 
 fingie force may be found that muft thus impel the pla- 
 nets eccentrically, ard thus urge them into revolution 
 and rotation at once, as they would be urged by a ftroke 
 from fome other planet or comet. With refpeft to this 
 Earth, there is not the lead appeaiTtnce of the etFecl 
 which muft have been produced on it, had it been urged 
 into motion by a fnigle force applied to one point. The 
 force has been applied alike to every particle •, there is 
 no appearance of any fuch general force competent to 
 the production of fuch motions. Nay, did we clearly 
 perceive the exiftence of fuch a force, we fliould be as 
 far from an explanation as ever. It is not enough that 
 Jupiter receives an impulfe v/hich imprefles both the 
 progreffive and rotative motion. His four fatellites muft 
 receive, each feparately, an impulfe of a certain precife 
 intenfity, and in a certain precife dire£l:ion, very different 
 m each, and which cannot be deduced from any thing 
 that we know of matter and motion. No principle of 
 general influence has been contrived by the / zealous pa- 
 trons of this fyftem (for it is a fyftem) that gives the 
 fmalleft fatisfadion even to themfelves, and they are 
 obliged to reft fatisfied with expreffing their hopes that 
 it may yet be accompliflied. 
 
 But fuppofe that an expert mechanician fhould fhew 
 liow the planets, fatellites, and comets may be fo placed 
 that an impulfe may at once be given to them all, precifely 
 
 competeiit 
 
ROTATION COMBINED WITH REVOLUTION. ^27 
 
 competent to the production of the very motions that 
 we obferve, which motions will now be maintained for 
 ever by the 'univerfal operation of gravity. We fhould 
 certainly admire his fagacity and his knowledge of na- 
 ture. But we flill wonder as much as ever at the nice 
 adjuftment of all this to ends which have evidently all the 
 excellence that order and fymmetry can give, while miany 
 of them are indifpenfably fubfervient to purpofes which 
 we cannot help thinking good. The fuggeftion of pur- 
 pofe and final eaufes is as ftrong as ever. It is no more 
 eluded than it would be, fhould any man perfe£lly ex- 
 plain the making of a watch wheel, by {hewing that it 
 was the neceflary refult of the fhape and hardnefs of the 
 fifes and drills and chizels employed, and the intenfity 
 and diicClion of the forces by which thofe tools were 
 moved ; and having done all this, fhould fay that he had 
 accounted for the nice and fuitable form of the wheel 
 as a part of a watch. And, with refpeft to the fubfe- 
 quent oblate form of the planet fet in rotation, the me- 
 chanical explanation of this is incompatible with the 
 luppofition that the revolution and rotation are the ef- 
 feds of one fimple force. The oblate form, if acquired 
 by rotation, requires primitive fluidity, which is incom- 
 patible with the operation of one fimple force as the 
 primitive mover. There is no proof whatever that this 
 Earth was originally fluid ; it is not nearly fo oblate as 
 primitive fluidity requires ; yet its form is fo nicely ad- 
 jufted to its rotation, that the thin film of water on it 
 is diflributed with perfedl uniformity. "We are obliged 
 
 to 
 
528 Pi4YSICAL Ai^TRONOMY. 
 
 to grant that a form has been originally given it fuitahle 
 to its deftination, and we enjoy the advantages of thi 
 exquifite adjuftment. 
 
 I acknowledge that the influence Of final caufes haJi 
 been frequently and egregioufly mifapplied, and that 
 thefe ignorant and precipitate attempts to explain phe- 
 nomena, or to account for them, and even fometimes to 
 authenticate them, have certainly obftru61:ed the progrefs 
 of true fcience. But what gift of God has not been 
 thus abufed ? A true philofopher will never be fo re- 
 gardlefs of logic as to adduce final caufes as arguments 
 for the reality of any fadl ; but neither will he have fuch 
 a horror at the appearances of wifdom, as to fhun look- 
 ing at them. And we apprehend that unlefs fom^ 
 < Frigidus obfliterit circum pracord'ia fanguis^ ' 
 it is not in any man's power to hinder himfelf from per- 
 ceiving and wondering at them. Surely 
 
 ' ^To look thro* nature up to Nature' i God, * 
 cannot be an unpleafant talk to a heart endowed with an 
 ordinary fiiare of fenfibility ; and the face of nature, ex- 
 prefBng the Supreme Mind which gives animation to its 
 features, is an objecl more pleafmg than the mere work- 
 ings of blind matter and motion. 
 
 But enough of this. — We fliall clofe this fubje<Sl: of 
 planetary figures by flightly noticing, for the prefent, a 
 confequence of the oblate form perceptible in all the 
 planets which turn round their axes j in the explanation 
 of which the penetration of Newton'i intellect is emi- 
 nently confpicuous. 
 
 6io. 
 
pkkCESSlON OF THE EQUINOXES. j'^p 
 
 610. In § 584, and feveral following paragraphs, we 
 explained the eiFe£t:s arifing from the inclination of the 
 Moon's orbit round the Earth to the plane of the Earth^s 
 orbit round the Sun. We faw, for example, that when 
 the interfe£lion of the two planes is in the line A B 
 (fig. 61.) of quadrature, the Moon is perpetually drawn 
 out of that plane, and her path is continually bent down 
 toward the ecliptic, during her moving along the femi- 
 circle A C B, and ihe defcribes another path A r ^, crofT- 
 ing the ecliptic in ^, nearer to A than B is. In the o- 
 ther half of her orbit, the fame deviation is continued, 
 and the Moon again croiTes the ecliptic before fhe come 
 to A, croffes her laft path near to r, and the ecliptic a 
 third time at d, and fo on continually. Hence arifes the 
 retrograde motion of the nodes of the lunar orbit. We 
 fhewed that this obtains, in a greater or lefs degree, in 
 every pofition of the nodes, except when they are in the 
 line of fyzigy. 
 
 What is true of one moon, would be true of any 
 number : It would be true, were there a complete ring 
 of moons furrounding the Earth, not adhering to one 
 another. We faw that the inclination of the orbit is 
 continually changing, being greatefl: when the nodes are 
 in the line of the fyzigies, and fmallell when they are 
 in quadrature. Now, if we apply this to a ring 6£ 
 moons, we fhall find that it will never be a ring that Is 
 all in one plane, except v/hen the nodes are in the fyzigies, 
 and at all other times will be warped, or out of fliape. 
 Now, let- the moons all cohere, and the ring become 
 
 3 X ftiiFj 
 
53© PHYSICAL ASTRONOMY. 
 
 lliff ; and let this happen when its nodes are in fyzigy. 
 It will turn round without difturbance of this fort. 
 But this pofition of the nodes of the ring foon changes, 
 by the Sun's change of relative fituation, and now all 
 the derangements begin again. The ring can no longer 
 go out of fliapc or warp, becaufe we may fuppofe it in- 
 flexible. But, as in the courfe of any one revolution of 
 the Moon round the Earth, the inclination of the or- 
 bit would cither be increafed, on the whole, or dimi- 
 nilhed, on the whole, and the nodes would, on the 
 whole, recede, this effe£l mud be obferved in the ring. 
 When the nodes are fo fituated that, in the courfe of one 
 revolution of a fmgle Moon, the inclination will be more 
 increafed in one part than it is diminiflied in another, 
 the oppofite actions on the different parts of a coherent 
 and inflexible ring will deftroy each other, as far as they 
 are equal, and the excefs only will be perceived on the 
 whole ring. Hence we can infer, with great confidence, 
 that from the time that the nodes of the ring are in fy- 
 zigy to the time they are in quadrature, the inclination 
 of the ring of moons will be continually diminifhing ; 
 wdll be leaft of all when the Sun is in quadrature with 
 the line of the nodes ; and will increafe again to a maxi- 
 mum, when the Sun again gets into the line of the nodes, 
 that is, when the nodes are in the line of the fyzigies. 
 But the inertia of the ring will caufe it to continue any 
 motion that is accumulated in it till it be deftroyed by 
 contrary forces. Hence, the times of the maximum and 
 miniinum of inclination will be confiderably different 
 
 fro»> 
 
^-RECESSION OF THE ECVUIN^OXES. 5>I 
 
 from what is now ftated. This will be attended to by 
 and by. 
 
 For the fame reafon, the nodes of tlie ring will con- 
 tinually recede -, and this retrograde motion will be moll 
 remarkable when the nodes are in quadrature, or the Sun 
 in quadrature with the Kne of the nodes ; and will gra- 
 dually become lefs remarkable, as the nodes approach 
 the line of the fyzigies, where the retrograde motion 
 -ivill be the leaft poffibie, or rather ceafes altogether. 
 
 All thefe things may be diftinftly perceived, by ftea- 
 dily confidering the manner of acting of the difturbin^^r 
 force. This fteady contemplation however is necefTary, 
 as fome of the effefts are very unexpected. 
 
 Suppofe now that this ring contrails in its dimen- 
 fions. The difturbing force, and all its efFecls, muft 'di- 
 minifh in the fame proportion as the diameter of the 
 ring diminifhes. But they will continue the fame in 
 kind as before. The inclinatbn will increafe till the Sun 
 comes into the line of the nodes, and diminifli till he 
 gets into quadrature with them. Suppofe the ring to 
 contract till almoft in contact with the Earth's furface. 
 The recefs of the nodes, inltead of being almoft three 
 degrees in a month, will now be only three minutes, and 
 the change of inclination in three months will now be- 
 only about five feconds. 
 
 Suppofe the ring to contraft ilill more, and to cohere 
 with the Earth. This will make a great change. Tlie ten- 
 dency of the ring to change its inclination, and to change its 
 iji^erfcclion with the ecliptic, ftill continues. But it can- 
 3X2 up: 
 
532 PHYSICAL ASTRONOMY-. 
 
 not now produce the eiicCt, without dragging with it 
 the whole mafs of the Earth. But the Earth is at per- 
 fe6l liberty in empty fpace, and being retained by no-^ 
 thing, yields to every impulfe, and therefore yields to 
 this aclion of the ring. 
 
 Now, there is fuch a ring furrounding the Earthy 
 having precifely this tendency. T he Earth may be con- 
 fidered as a fphere, on which there is fpread a quantity 
 of redundant matter which makes it fpheroidal. The 
 gravitation of this redundant matter to the Sun fuflains 
 all thofe difturbing forces which acl on the inflexible 
 ring of moons ; and it will be proved, in its proper 
 place, that the efte£\ in changing the pofition of the 
 globe is j- of what it would be, if all this redundant 
 matter were accumulated on the eqaator. It will alfo 
 appear that the force by which every particle of it is 
 urged to or from the plane of the ecliptic, is as its dif- 
 tance from that plane. Indeed, this appears already, be- 
 caufe all the difturbing forces aifiing on the particles of 
 this ring are finailar, both iif dire(Slion and proportion, to 
 thofe which we fne wed to influence the Moon in the 
 fimilar fituations of her monthly courfe round, the Earth. 
 Similar efFecSls will therefore be produced. 
 
 Let us now fee what thofe. efiecls will be.— The Iu» 
 nar nodes continually recede ; fo will tlie nodes of this 
 equatoreal ring, that is, fo will the nodes of the equa- 
 tor, or its interfe^Stion with the ccjiptic. But the inter- 
 fe£lions of the equator with the ecliptic are what we 
 call the Equincdial Points. The plane of the Earth's 
 
 equator. 
 
PRECESSION OF THE EQUINOXES. .533 
 
 equator, being produced to the (lurry heavens, Interfecls 
 that feemhigly concave fphere in a great circle, which 
 may be traced out among the ftars, and marked on a 
 celeflial globe. Did the Earth's equator always keep the 
 fame pofition, this circle of the heavens would always 
 pafs through tlie fame ilars, and cut the ecliptic in the 
 fame two oppofite points. When the Sun comes to one 
 of thofe points, the Earth turning round under him, e- 
 very point of irs equator has him in the zenith in fuc- 
 ceflion ; and all the inhabitants of the Earth fee him 
 rife and fet due eaft and wed, • and have the day and 
 night of the fame length. But, in the courfe of a year, 
 the a£lion of the Sun on the protuberance of our equator 
 deranges it from its former pofition, in fuch a manner 
 that each of its interfeclions with the ecliptic is a little 
 to the weftward of its former place in the ecliptic, fo 
 that the Sun comes to the interfe£lion about 20' before 
 he reaches the interfe^lion of the preceding year. This 
 anticipation of the equal divifion of day and night is there- 
 fore called the precession of the equinoxes. 
 
 The axis of diurnal revolution is perpendicular to the 
 plane of the equator, and mud therefore change its po- 
 fition alfo. If the inclination of the equator to the eclip- 
 tic were always the fame (231 degrees), the pole of the 
 diurnal revolution of the heavens (that is, the point of 
 the heavens in which the Earth's axis would meet the 
 concave) would keep at the fame diftance of 23^ degrees 
 from the pole of the ecliptic, and would therefore always 
 be found in the circumference of a circle, of which the 
 
 pole 
 
'534 I'HYSICAL ASTRONOMY. 
 
 pole oF the ecliptic Is the centre. The meridian which 
 pafles through the poles of the ecliptic and equator muft 
 always be perpendicular to the meridian which pafles 
 through the equinoclial points, and therefore, as thefe 
 ihift to the weftward, the pole of the equator muft alfo 
 ihift to the weftward, on the circumference of the circle 
 above mentioned. 
 
 But we have feen that the ring of redundant matter 
 does not preferve the fame inclination to the ecliptic. It 
 is moft inclined to it when the Sun is in the nodes, and 
 fmalleft when he is in quadrature with refpedl to them. 
 ' Therefore the obliquity of the equator and ecliptic fliould 
 be greateft on the days of the equinoxes, and fmalleft 
 when the Sun is in the folftitial points. The Earth's 
 axis fhould twice in the year incline downward toward 
 the ecliptic, and tv/ice, in the intervals, fhould raife itfelf 
 "up again to its greateft elevation. 
 
 Something greatly refembling this feries of motions 
 may be obferved in a child's humming top, when fet a 
 fpinning on Its pivot. An equatoreal circle may be drav.'n 
 on this top, and a circular hole, a little bigger than the 
 top, may be cut in a bit of ftifF paper. When the top is 
 fpinning very fteadily, let the paper be held fo that half 
 of the top is above it, the equator aimoft touching the 
 fides of the hole. When the whirling motion abates, 
 the top begins to ftagger a little. Its equator no longer 
 coincides with the rim of the hole In the paper, but in- 
 terfedls It in two oppofite points. Thefe interfe£iions 
 will be obferved to fnift round the v/hole circumferehee 
 
 9f 
 
PRECESSION OF THE E(^INOXES. 53^ 
 
 of the hole, as the axis of the top veers round. The 
 axis becomes continually more oblique, without any pe- 
 riods of recovering its former pofition, and, in this re- 
 fyeQ: only the phenomena differ from thofe of the pre- 
 cefTion. 
 
 It u-as affirmed that the obliquity of the equator is 
 greateft at the equinoxes, and fmallefl at the folftices* 
 This would be the cafe, did the redundant ring inftant- 
 ly attain the pofition w^hich makes an equilibrium of ac- 
 tion. But this cannot be ; chiefly for this reafon, that 
 it muft drag along vrith it the whole infcribed fphere. 
 During the motion from the equinox to the next folftice, 
 the Earth's equator has been urged toward the ecliptic, 
 and it muft approach it with an accelerated motion. 
 Suppofe, at the inftant of the folftice, all aftion of the 
 Sun to ceafe ; this motion of the terreftrial globe would 
 not ceafe, but would go on for ever, equably. But the* 
 Sun's aftion continuing, and now tending to raife the 
 equator again from the ecliptic, it checks the contrary- 
 motion of the globe, and, at length, annihilates it alto- 
 gether ; and then the effed of the elevating force begins 
 to appear, and the equator rifes again from the ecliptic. 
 When the Sun is in the equinox, the elevation of the e- 
 quator fhould be greateft ; but, as it arrived at this pofi- 
 tion with an accelerated motion, it continues to rife 
 (with a retarded motion) till the continuance of the Sun's 
 depreffing force puts an end to this rifmg ; and now the 
 cffeft of the depreffing force begins to appear. For 
 rhefe reafons, it happens tliat the greateft obliquity of 
 
 the 
 
 L 
 
§3^ PHYSrCAL ASTRONOM"?. 
 
 the equator to the ecliptic is not on the days of the e- 
 quinoxes, but about fix weeks after, viz. about the firlf 
 of May and November ; and the fmalleft obliquity is not 
 at midfummer and midvv'inter, but about the beginning 
 of February and of Auguft. 
 
 And thus, we find that the fame principle of univer- 
 fal gravitation, which produces - the elliptical motion of 
 tlie planets, the inequalities of their fatellites, and deter- 
 mines the (hape of fuch as turn round their axes, alfo 
 explains tliis moft remarkable motion, which had baffled 
 all the attempts of philofophers to account for — a mo- 
 tion, which feemed to the ancients to afFe6l the whole 
 hoft of heaven ; and when Copernicus fliewed that it 
 was only an appearance in the heavens, and proceeded 
 from a real fmall motion of the Earth's axis, it gave him 
 more trouble to conceive this motion with diflincStnefs, 
 than all the others. All thefe things — obvta confpicimus* 
 nubem pellente mathefi, 
 
 6 1 1 . Such is the method which Sir Ifaac Newton, 
 the fagacious difcoverer of this mechanifm, has taken to 
 give us a notion of it* Nothing can be more clear and 
 familiar in general. He has even fubje£l:ed his expla- 
 nation to the fevere tefl of calculation. The forces are 
 known, both in quantity and direction. Therefore the 
 effects mud be fuch as legitimately flow from thofe 
 forces. When we confider what a minute portion of 
 the globe is a£led upon, and how much inert matter 
 is to be moved by the force which afFe£ls fo fmall a 
 
 portion, 
 
PRECESSIOK OF THE ECtUINOXES. 537 
 
 portion, we muil expe£l very feeble cfFeds. All the 
 change that the action of the Sun produces on the in- 
 clination of the equator amounts only to the fraftion of 
 a fecond, and is therefore quite infenfible. The change 
 in the pofition of the equinoxes is more confpicuous, 
 becaufe it accumulates, amounting to about 9" annually^ 
 by Newton's calculation. We {hall take notice of this 
 calculation at another time, and at prefent (hall only 
 obferve that this motion of the equinox is but a fmall 
 part of the preceflion a^lually obferved. 'This is about 
 50y" annually. It would therefore feem that the theory 
 and obfervatiori do not agree, and that the preceflion of 
 the equinoxes is by no means explained by it. 
 
 612. It muft be remarked that we have only giveii 
 an account of the efFe£t refulting from the unequal gra- 
 vitation of the terreftrial matter to the Sun. But it 
 gravitates alfo to the Moon. Moreover, the inequality 
 of this gravitation (on which inequality the difturbance 
 depends) is vaftly greater. The Moon being almoft 400 
 times nearer than the Sun, the gravitation to a pound 
 of lunar matter is almoft 640,000,000 times greater 
 than to as much folar matter. When the calculation is 
 made from proper data, (in which Newton was con- 
 fiderably miftaken) the effect of the lunar a<Slion muft 
 very confiderably exceed that of the Sun. He wa3 
 miftaken, in refpe£l to the quantity of matter in the 
 Sun and in tjie Moon. The tranfit of Venus, and th$ 
 obfervations which .have been made on the tides, have 
 3 y brought; 
 
53? PHYSICAL ASTROXOMT. 
 
 brought us much nearer the truth in both thefe rcfpe^t?. 
 When the calculation is made on fuch aflumptions of the 
 matter in the Sun and Moon as are beft fupported by 
 obfervation, we find that the annual preceflion occafion- 
 ed by the Sun's aftion on the equatorcal protuberance 
 is about 14" or 15", and that produced by the Moon is- 
 about' 35''. The precelTion really cbferved is about 
 5^0", and the agreement is abundantly exaft. It muft 
 be farther remarked that this agreement is no longer in- 
 ferred from a due proportioning of the whole obferved 
 precefTion between the Sun aiKl the Moon, as we were 
 formerly obliged to do ; but each Ihare is an independent 
 thing, calculated without any rtrference to the whole 
 precellion. It is thus only that the phenomenon may be 
 affirmed to be truly explained. 
 
 ^^ "^n. For this demonftration we are indebted to D? 
 Bradley. His difcovery of vrhat is now called the 
 NUTATION of the Earth's axis, gave us a precife mea- 
 flire of the lunar action which removed every doubt. 
 It therefore muft be confrdered here. 
 
 The adlion of the luminaries on the Earth's equator, by 
 which the pofition of it is deranged, depends on the magni- 
 tude of the angle w^hich the equator makes with''ihe line 
 joining the Earth with the difturbing body. The Sun is ne- 
 vermore than 23I degrees from the equator. But when the 
 Modn's afcending node is in the vernal equinoxy fhe may 
 deviate nearly -Ip degrees from it. And when the node 
 is in the autumnal equinox, i}ie cannot go more than 17 
 * ^ -^ i degrees 
 
NUTATION OF THE EARTH's AXIS. 539 
 
 degrees from It. Thus, the a6lion of the Sun is, froiu 
 year to year, the fame. But as, in 19 years, the Moon's 
 nodes take all fituations, tlie aftion of the Moon is very 
 variable. It was one of the elFeclis of this variation that 
 Bradley difcovered. While the Earth's equator continued 
 to open farther and farther from the line joining the Earth 
 with the Moon, the axis of the Earth was gradually de- 
 preiled towards the ecliptic, and the diminution of its 
 inclination at lad amounted to 18 feconds. Dr Bradley 
 faw this by its increafing the declination of a ftar properly 
 iltuated. After nine years, when the Moon was in 
 fuch a fituation that (he never went more than 1 7" from 
 the Earth's equator, the fame ftar had 18" lefs declina- 
 tion. 
 
 614. This change in the inclination of the Earth's e- 
 quator is accompanied with a change in the precelTion of 
 ■the equinoxes. This muft increafe as the equator is more 
 open when viewed from the Moon. In the year in 
 v/hich the lunar afcending node is in the vicinity of the 
 vernal equinox, the preceffion is more than 58^5 and it 
 is but 43" when the node is near the autumnal equinox. 
 Thefe are very confpicuous changes, and of cafy obfer- 
 vation, although long unnoticed, while blended with o- 
 ther anomalies equally unknown. 
 
 Few difcoveries in aftronomy have been of more for- 
 
 vice to the fcience than this of the nutation, and that of 
 
 aberration, both by Dr Bradley. For till they were 
 
 Jsuown, there was an anomaly, which might fomctimes 
 
 3^2 amount 
 
540 PHYSICAL ASTRONOMY. 
 
 amount to 58" (tlie fum of nutation and aberration), and 
 afFe£ted every motion and every obfervatlon. No theory 
 of any planet could be freed from this uncertainty. Buti 
 now, we can give to every phenomenon its own proper 
 motions, with ail the accuracy that modem iiiftruments 
 can attain. Without thefe two difcoveries, we could 
 not have brought the folution of the great nautical pro- 
 blem of the longitude to any degree of perfe<!^ion, be- 
 caufe we could not render either the folar or lunar tables 
 perfe£l. The changes in the pofition of the Earth's axis 
 by nutation, and the concomitant equation of the pre- 
 ceffion, by recurring In the moft regular manner, have 
 given us the moft exaft meafure of the changes in the 
 Moon's aftlon ; and therefore gave an incontrovertible 
 meafure of her whole a£i:ion, becaufe the proportion be- 
 tween the variation and the whole action was diftiniSlly 
 known. 
 
 This not only completes the prafticai folution of the 
 problem, but gives the moft unqueftionable proof of the 
 foundnefs of the theory, fhewing that the oblate form of 
 the EartPi is the caufe of this nutation of its axis, and 
 eftablifhing the univerfal and mutual attra(9:ion of all 
 matter. It ftiews with what confidence we may proceed, 
 in following this law of gravitation into ail its confe- 
 quences, and that we may predi£^, without any chance 
 of miftake, what will be the eife«5l of any combination 
 of circumftances that can be mentioned. And it furely 
 fnews, in the moft confpicuous manner, the penetration 
 and fagacity of Newton, who gave encouragement to a 
 furmife fo fmgular and fo unlike all the ufual queftions of 
 
 ■progreffiye 
 
PRECESSION AND NUTATION. 54 1 
 
 progrcfTive motion, even In all their varieties. Yet this 
 ?nofl: recondite and delicate fpeculation was one of his 
 early thoughts, and Is one of the twelve propofitlons which 
 he read to the Royal Society. 
 
 6j^, It mufh be acknowledged however that thif 
 manner of exhibiting the theory of the preceflion of the 
 equinoxes is not complete, or even accurate in the felec- 
 tlon of the phyfical circumftances on which the proof 
 proceeds. It Is merely a popular way of leading the 
 mind to the view of a£lIons, which are indeed of the 
 fame kind with thofe actually concurring in the produc- 
 tion of the effect. But it is not a narration of the real 
 a£l:Ions. Nor are the eiFe£ls of thofe that are employed 
 elllmated according to their real manner of a£l:ing. Tlie 
 whole is rather a fiirewd guefs, in which Newton's great 
 penetration enabled him to catch at a very remote ana- 
 logy between the libration of the Moon and the wavering 
 motion of the Earth's axis. We are not in a condition 
 in this part of the courfe to treat this queftlon in the pro- 
 per manner. We muft firft underftand the properties of 
 the Icf er as a mechanical power, and the operation of the 
 connecting forces of firm or rigid bodies. What we 
 have faid will fuffice however for giving a diftincl enough 
 conception of thp general efFefts of the a^lion of remote 
 bodies on a fpheroidal planet turning round its axis. * It 
 
 Is 
 
 * To thofe who v.^iih to ftudy this very curious and diffi- 
 ^^ult problem, I fhouid recommend the folutioa given by Fri- 
 
 iius 
 
^^2 PHYSICAL ASTROKOMY. : 
 
 is fcarcely neceflary to add that the other planets cannot 
 fenfibly influence the motion of the Earth's axis. Their 
 accumulated aiftion may add about y of a fecond to the 
 annual preceflion of the equinoxes. 
 
 The planets Mars, Jupiter, and Saturn, being vaflly 
 more oblate than the Earth, muft be more expofed to 
 this derangement of the rotative motion. Jupiter and 
 Saturn, having fo many fatellites, which take various po- 
 fitions round the planet, the problem becomes immenfely 
 complicated. But the fmall inclination of the equator, 
 and the great mafs of the planet, and its very rapid ro- 
 tation, muft greatly diminifh the eiFe<Sl: we are now con- 
 Yidering. Mars, being fmall, turning flowly, and yet 
 being very oblate, muft fuftain a greater degree of this 
 derangement ; and if Mars had a fatellite, we might ex- 
 pe£l fuch a change in the pofition of his axis as fhould 
 become very fenfible, even at this diftance. 
 
 The ring of Saturn muft be fubjeft to fimilar diilurbr 
 ances, and muft have a retrogradation of its interfeclioa 
 
 with 
 
 fius in the fecond part of his Cofmographia, as the nioft per- 
 fpicuous of any that I am acquainted with. The elaborate 
 performance of Mr Walmefely, Euler, D'Alembert, and La 
 Grange, are acceflible only to expert analyfls. The efTay by 
 T. Simpfon in the Philofophical Tranfaftions, Vol, L. is re- 
 markable for its fimplicity, hut, by employing the fymbollcal 
 or algebraic anal3''fis, the ftudent is not fo much aided by the 
 conftant acconipaniment of phyfural ideas, as in the geome- 
 trical method of Frifius- 
 
ELEMENTS OF CERES AND PALLAS. 543- 
 
 With the plane of the orbit. Had v/e nothing to confider 
 but the ring itfolf, it would be a very eafy problem 
 to determine the motion of its nodes. But the proxi- 
 mity, and the oblate form, of the pbnet, and, above all, 
 tlie complicated action of the fatellites, make it next to 
 unmanageable. It has not been attempted, that I know 
 of. It may (I think) be deduced, from the Greenwich 
 obfervations fmce 1750, that the node^ retreat on the 
 orbit of Saturn about 34' or 36'' in a century, and that 
 their longitude in i8oi was 5^ 17° 13' and 11^ 17° 13'. 
 This may be received as more exa<Sl than the determina- 
 tion given in art. 380. 
 
 I laid, in art. 370, that we ha^'e feen too rittle of the 
 motions of Ceres and Pallas to announce the elements 
 of their theories with any thing like precifion. But^ 
 that they may not be altogether omitted,, the following 
 may be received as of mod audiority. 
 
 a 
 
 Mean diftaruce ----- 276723 
 
 Eccentricity to m. d. i. - - 
 
 Long, aphelion - - - - 
 
 Period (fydereal) in days 
 
 Mean long. Jan. 1 804. - - 
 
 Inclin. orbit - - - - - 
 
 Long, node .> - - - - 
 
 Thefe bodies prefent fome very fmgular circumftances 
 to our ftudy ; their diftances and periods being almoft the 
 fame, and their longitudes at prefent diifering very little. 
 They differ confiderably m eccentricity^ the place of the 
 
 uodej 
 
 Ceres. 
 
 Pallas, 
 
 2767231 
 
 2767123 
 
 V 0,079 
 
 0,2463 
 
 r 4.26.44 
 
 4.I.7.— 
 
 ^ 1682,25 
 
 1681,22 
 
 10.11.59 
 
 9.29.53 
 
 —.10.37 
 
 —•34.39 
 
 2.21. 7 
 
 5.22.27 
 
^44 PHYSICAL ASTRONOMr. 
 
 node, and the Inclination of their orbits. They muft be 
 greatly difturbed by each other, and by Jupiter, and it 
 will bfe loilg before -we fhall obtain exact elements. 
 
 With thefe obfervations I might conclude the dif- 
 cufTion of the mechanifm of the folar fyltem. The facts 
 obferved in the appearances of the comets are too few 
 to authorife' me to add any thing to what has been al- 
 ready faid concarning them. I refer to Newton's Prin- 
 cipia for an account of that great philofopher's conjec- 
 tures concerning the luminous train which generally at- 
 tends them, acknowledging that I do not think thefe 
 conjeftures well fupported by the eftabliflied laws of 
 motion. Dr Winthorp has given, in the 57th volume 
 of the Phil. Tranf. a geometrical explanation of the me- 
 chanifm of this phenomenon that is ingenious and ele- 
 gant, but founded on a hypothefis which I think inad- 
 miffible. 
 
 616, No notice has yet been taken of the relations^ 
 of the folar fyftem to the reft of the vifible hoft of 
 heaven, and we have, hitherto, only confidered the ftarry 
 heavens* as affording us a number of fixed points, by 
 which we may eftimate the motions of the bodies which 
 compofe our fyftem. It will not therefore be unaccept- 
 able fhould I now lay before the reader feme reflections, 
 which naturally arife in the mind of any perfon who has 
 been much occupied in the preceding refearches and fpe- 
 culations, and which lead the thoughts into a fcene of 
 cpnteftiplation faj exceeding in itiagnificence any thing 
 
 yet 
 
GENERAL REFLECTIONS. 54^ 
 
 yet laid before the reader. As they are of a mlfcella- 
 neous nature^ and not fufceptible of much arrangement, 
 I fhall not pretend to mark them by any diftindtions, but 
 ihall take them as they naturally offer themfelves. 
 
 The fitnefs for almoft eternal duration, fo confpicu- 
 ous in the conftitution of the folar fyftem, cannot but 
 fuggeft the higheft ideas of the intelligence of the Great 
 Artifl:. No doubt thefe conceptions will be very ob- 
 fcure, and very inadequate. But we fhall find that the 
 farther we advance in our knowledge of the phenomena, 
 we fhall fee the more to admire, and the more numerous 
 difplays of great wifdom, power, and kind intentions. 
 
 It is not therefore fearful fuperflition, but the cheer- 
 ful anticipation of a good heart, which will make a ftu- 
 dent of nature even endeavour to form to himfelf ftill 
 higher notions of the attributes of the Divine Mind. 
 He cannot do this in a dired manner. All he can do is 
 to abftraft all notions of imperfection, whether in power, 
 j(kill, or benevolent intentions, and he will fuppofe the 
 Author of the univerfe to be infinitely powerful, wife 
 tnd good. 
 
 It is impoflible to flop the flights of a fpeculative 
 mind, warmed by fuch pleafing notions. Such a mind 
 will form to itfelf notions of what is mod excellent in 
 the defigns which a perfeft being may form, and it finds 
 itfelf under a fort of necelTity of believing that the Di- 
 vine Mind will really form fuch defigns. This romantic 
 wandering has given rife to many ftrange theological o- 
 piuions. Not doubting (at leafl in the moment of en- 
 3 Z thufiafm} 
 
54^ PHYSICAL ^STRONOMYy 
 
 thufiafm) that we can judge of what is moft excellent, 
 we take it for granted that this creature of our heated 
 imaglnatibn muft alfo appear moil excellent to the Su- 
 preme Mind. From this principle, theologians have ven- 
 tured to lay down the laws by which God himfelf muft 
 regulate his actions. No wonder that, on fo fanciful a 
 foundation as our capacity to judge of what is moft ex- 
 cellent, have been ere<fl:ed the moft extravagant fabrics, and 
 that, in the exuberance of religious zeal, the Author of 
 Al has been defcribed as the moft limited Agent in the 
 univerfe, forced, in every adion, to regulate himfelf by 
 our poor and imperfect notions of what is excellent. 
 "We, who vanifli from tlie fight, at the diftance of a 
 neighbouring hili — whofe greateft works are invifible 
 from the Moon — whofe whole habitation is not vifible to 
 a fpe6lator in Saturn — fliall fuch creatures pretend to 
 judge of what is fupremely excellent ? 
 
 Let us not pretend even to guefs at the fpecific laws 
 by which the condu^ of the Divinity muft be diredled^ 
 except in fo far as it has pleafed him to declare them to 
 us. We fliall purfue the only fafe road in this fpecu- 
 lation, if we endeavour to- difcover the^ laws by which 
 hiij vifible and comprehenfible works are actually con- 
 dudlcd. The more we difcover of thefe, the more do 
 we find to fill us with admiration and aftonifliment. 
 The only fpeculations in which we can indulge, without 
 the continual danger of going aftray, are thofe which 
 enlarge our notions of the fcene on which it has pleafed 
 the Almighty to difplay his perfe(^iori8. This will be 
 
 the 
 
GENERAL REFLECTIONS. 547 
 
 the undoubted efFeft of enlarging the field of our own 
 obfervatlon. After examining this lower world, and o'b- 
 ferving the nice and infinitely various adjuftments of 
 means to ends here below, we may extend our obferva- 
 tion beyond this globe. Then fliall we find that, as far 
 as our knowledge can carry us, there is the fame art, 
 and the fame production of good effects by beautifully 
 contrived means. We have lately difcovered a new pla- 
 net, far removed beyond the formerly imagined bounds 
 of the planetary world. This difcovery fhews us that if 
 there are thoufands more, they may be for ever hid from 
 our eyes by their immenfe diftance. Yet there we find 
 the fame care taken that their condition fhall be perma- 
 nent. They are influenced by a force directed to the 
 Sun, and inverfely as the fquare of the dlllance from 
 him ; and they defcribe eUipfes. This planet is alfo ac- 
 companied by fatellites, doubtlefs rendf »ing to the pri- 
 mary and its inhabitants ferviees fimilar to what this 
 Earth receives from the Ivloon. All the comets of 
 whofe motions we have any precife knowledge, are e- 
 equally fecured ; none feems to defcribe a parabola or 
 hyperbola, fo as to quit the Sun for ever. 
 
 This mark of an intention that this noble fabric fliall 
 .continue for ever to declare itfelf the work of an Al- 
 mighty and Kind Hand, naturally carries forward the 
 mind into that unbounded fj^ace, of which our folar 
 fyftem occupies fo inconfiderable a portion. The mind 
 revolts at the thought that this is fludded with ilars for 
 •^o other purpofe than to affifl the aftronomer in his conir- 
 3 Z 2 put.uions. 
 
548 PHTSICAL ASTRQNOMk. 
 
 putations, and to furnlfli a gay fpectacle to the unthlnk-? 
 ing multitude. We fee nothing here below, or in our 
 fyflem, which anfwers but one folitary purpofe, and we 
 require that a pofitive reafon fliall be given for limiting 
 the Hoft of Heaven to fo ignoble an office. As fuch has 
 not been given, we indulge ourfelves in the pleafing 
 thought that the ftars make a part of the univerfe, no lefs 
 important in purpofe than great in extent. We are jufti- 
 fiable, by what we in fome meafure underftand, in fup- 
 pofing each ftar a fun, the centre of a planetary fyftem, 
 full of enjoyment hke our own, and fo conflruited as to 
 laft for ever. 
 
 When the philofopher indulges himfelf in thofe amaz-*. 
 ing, but pleafing thoughts, he muH: regulate his fpecu- 
 lations by analogies and refemblances to things more fa- 
 miliarly known to him. We muft fuppofe thofe fyftems 
 to refemble our own, and that they are kept together by 
 a gravitation in the inverfe duplicate ratio of the dif- 
 tances. For we know that this alone will infure perma- 
 nancy and good order. 
 
 But in fo doing, we extend the influence of gravity 
 to diflances inconceivably greater than any tliat we have 
 yet confidered, and we come at laft to believe that gra- 
 vitation is the bond of connexion which unites the moft 
 diftant bodies of the vifible univerfe, rendering the whole 
 one great machine, for ever operating the mod magnifi- 
 cent purpofes, worthy of its All-Perfe£t Creator. And, 
 when we fee that fuch a connexion is neceflary for this 
 end, we are apt to imagine that gravity is ejfential to or 
 
 indifpenfabie 
 
IS GRAVITY OF INDEFINITE "EICTENT ? 54^ 
 
 Indlfpenfable In that matter that is to be moulded Into a 
 world. 
 
 But let not our ignorance miilead us, nor let us mea- 
 fure every thing by that fmall fcale which God has en- 
 abled us to ufe, unlefs we can fee feme circumftances of 
 refemblance in the appearances, which may juftify the 
 application. 
 
 * A frame of material nature of any kind cannot be 
 conceived by the mind, without fuppofmg that the matter 
 of which it confifls is influenced by fome aflive powers, 
 conftituting the relations between its different parts. 
 Were there only the mere inert materials of a world, it 
 would hardly be better than a chaos, although moulded 
 into fymmetrical forms, unlefs the fpirit of it5 author 
 were to animate thofe dead malfes, fo as to bring forth 
 change, and order, and beauty. Our illuftrious New- 
 ton 
 
 * For many of the thoughts in what follows, the reader 
 Is indebted to a very ingenious pamphlet, pubhflied by Cad- 
 del & Davies in 17771 entitled, Thoughts on General Grav'tta- 
 tion. It is much to be regreted that the author has not availed 
 himfelf of the fuccefsful refearchcs of aftronomers fince that 
 time, and profecuted his excellent hints. If it be the per- 
 formance of the perfon whom I fuppofe to be the author, I have 
 fuch an opinion of his acutenefs, and of his juftnefs of thought, 
 that I take this opportunity of requefting him to turn his at- 
 tention afrefh to the fubjed. His advantages, from his pre- 
 fent fituation and connexions, are precious, and fliould not be 
 loll. 
 
550 PHYSICAL ASTRONOMY. 
 
 ton therefore fays, with great propriety, that the bufi- 
 nefs of a true philofophy is to inveftigate thofe a£live 
 powers, by which the courfe of natural events, to a very 
 great extent at leaft, is perpetually governed. Philofo- 
 phifing with this view, he difcovered the law of unlver- 
 fal gravitation, and has thus given the brighteft fpeci- 
 men of the powers of human underftanding. 
 
 The notion of fomething like gravity feems infepa- 
 rable from our conception of any eftablifhed order of 
 things. For unlefs fome principle of general union ob- 
 tain among the parts of matter, we can have no concep- 
 tion of the very firft formation of the individuals of 
 which a world may be compofed. 
 
 But general gravitation, or that power by which the 
 dijlant bodies belonging to any fyftera are connected, 
 and a6t on one another, does not feem fo indifpenfably 
 neceflary to the very being of the fyftem, as particular 
 gravity is to the being of any individual in it. We can- 
 not difcern any abfurdity in the fuppofition of bodies, 
 fuch as the planets, fo fituated with refpeO: to another 
 great body, fuch as the Sun^ as to receive from it fui't- 
 able degrees of light and heat, withaut their having any 
 tendency to approach the Sun, or each other. But then, 
 how far fuch limitation of gravity may be a pofTible 
 thing, or how far its indefinite extenfjon in every di- 
 rection may be involved in its very nature, we cannot 
 tell, until we are able to confider gravity as an effe£t, 
 and to deduce the laws of its operation from our know^ 
 ledge of its caufe. 
 
 That 
 
li GRAVITATION OF INDEFINITE EXTENT ? 55 1 
 
 That the influence of gravity extends into the bound* 
 lefs void, to the greateft aflignablc diftance, feems to 
 be almolt the hinge of the Newtonian philofophy. At 
 leaft, there is nothing that warrants any limit to its ac* 
 tion. Fatlier Bofcovich indeed fliews that all the phe- 
 nomena may be what they are, without this as a necef- 
 fary confequence. But he is plainly induced to bring 
 forward the limitation in order to avoid what has been 
 thought a neceflary confequence of the indefinite exten- 
 fion of gravity ; and what he olFers is a mere poflibility. 
 
 Now, if fuch extenfion of gravitation be infeparable, 
 in fa£l, from its nature, then, if all the bodies of ou? 
 fyftem are at reft in abfolute fpaee, no fooner does the 
 influence of general gravitation go abroad into the fyf- 
 tem, than all the planets and comets muft begin to ap* 
 proach the Sun, and, in a very fmall number of days, 
 the whole of the folar fyftem muft fall into the Sun, 
 and be deftroyed. 
 
 But, that this fair order may be preferved, and ac- 
 commodajted to this extended influence of gravity, which 
 appears fo elTential to the conftitution of the feveral parts 
 of the fyftem, we fee a moft fimple and efFeftual pre- 
 vention, by the introdu£tion of prcjeBlle forces, and pro^ 
 grejfive motion. For upon tliefe being now combined, 
 and properly adjufted with the variation of gravity, the 
 planets are made to revolve round the Sun in ftated 
 €Ourfes, by which their continual approach to the Sun 
 and to one another is prevented, and the adjuftment is 
 made with fuch exquifite propriety, that the perfect or- 
 der 
 
552 PHYilCAL ASTRONOMY. 
 
 der of things is almoft unchangeable. This adjuftment 
 is no lefs manifell: in the fubordinate fyftems of a pri- 
 mary planet and its fatellites, which are not only regu- 
 lar in their own orbital motions, but are the conftant 
 attendants of their primaries in their revolution round 
 the Sun. 
 
 In this view of the fubjeft, forafmuch as gravity 
 feems eflential to the conftitution of all the great bodies 
 of the fyftem, and in fo far as its indefinite extenfion 
 may be infeparable from its nature, it appears that perio- 
 dical motion mufl be neceflary for the permanency and 
 order of every fyftem of worlds whatever. 
 
 But here a thought is fuggefted which obvioufly leads 
 to a new and a very grand conception of the univerfe. 
 If periodical motion be thus neceflary for the preferva- 
 tion of a fmall aiTemblage of bodies, and if Newton's 
 law prefent to us the whole hoft of heaven as one great 
 aflembiage affe£l:ed by gravitation, we muft ftill have re- 
 courfe to periodical motion, in order to fecure the eftab- 
 lifhment of this grand univerfal fyftem. For if there be 
 no bounds to the influence of gravitation, and if all the 
 ftars be fo many funs, the centres 'of as many fyftems 
 (as is moft reafonable to believe) the" immenfity of their 
 diftance cannot fatisfy us for their being long aBle to 
 remain in any fettled order. Thofe tliat are fituated to- 
 wards the confines of this magnificent creation muft for- 
 fake their ftations, and, with an approach, continually 
 accelerated, muft move onwards to the centre of. gene- 
 ral 
 
is GRAVITATION OF INDEFINITE EXTENT? 553 
 
 tal gravitation, and, after a feries of ages, the whole 
 glory of nature mult end in a univerfal wreck* 
 
 As the fyflcm of Jupiter and his f^itelHtes is but an 
 epitome of the great folar fylleni to which he belongs, 
 may not this, in its turn, be a faint reprefentation of 
 tliat grand fyftem of the univerfe, round whofe centre 
 this San, with his attending planets, and an inconceive- 
 able mukitude of like fyilems, do in reality revolve ac- 
 cording to the law of gravitation ? Now, will our anti- 
 cipation of diforder and ruin be changed into the con- 
 templation of a countlefs number of nicely adjufted 
 motions, all proclaiming the fuftaining hand of God. 
 
 This is indeed a grand, and almoft overpowering 
 thought ; yet juftified both by reafon and analogy. The 
 grandeur however of this univerfal fyllem only opens 
 upon us by degrees. If it refemble our folar fyltem in 
 conftru£lion, what an inconceivable difplay of creation 
 is fuggefted, wjien we turn our thoughts towards that 
 place which the motions of fo many revolving fyflems 
 are made to refpeft ! Here may be an unthought of 
 univerfe of itfelf, an example of material creation, which 
 muft individually exceed all the other parts, though add- 
 ed into one amount. As our Sun is almoft four thou- 
 fand times bigger than all his attendants put together, 
 it is not unreafonable to fuppofe the fame thing here. 
 It is not necelTary that this central body (hould be vi- 
 fible. The great ufe of it is not to illuminate, but to 
 govern the motions of all the reft. We know, how- 
 ever, that the exigence of fuch a central body is not 
 
 4 A neceflary. 
 
^^4 PHYSICAL ASTRONOlvlY. 
 
 neceifary. Two bodies, although not very unequal, msy- 
 be projected with fuch velocities, and in fuch dire£tions, 
 that they wiH revolve for ever round their common 
 centre of pofitioti and gravitation. But fuch a fyftem 
 could hardly maintain any regularity of motion when a 
 third body is added. . It may indeed be faid that the 
 fame tranfcendent wifdom, M'hich has fo exquifitcly ad- 
 apted all the c'ircumfiances of our fyftem, may fo ad- 
 juft the motions of an itnmenfe number of bodies, that 
 their difturbing a£lions fliall accurately compenfate each 
 ether. But ftill, the beautiful nmplicity that is manifefl 
 in what we fee and undtrfland, feems to warrant a like 
 limplicity in this great fyftem, and therefore renders the 
 exiftence of fuch a great central Regulator of the move- 
 ments of all, the moft probable fuppofition. 
 
 Sobei* reafon will not be difpofed to revolt at fo glo- 
 rious an extenfion of the w orks of God, however much 
 it may overpower our feeble conceptions. Nay this ana- 
 logy acquires additional weight and authority even from 
 the tranfcendent nature of the univerfe to which it di- 
 recfls our thoughts. Nothing lefs magnificent feems fuit- 
 able to a Being of infinite perfecl:ions. 
 
 But we are not left to mere conje61:ure in fupport of , 
 this conception of a great univerfe, connected by mu- 
 tual powers. There are circumft.ances of analogy which 
 tend greatly to perfuade us of the reality of our conjec- 
 ture — circumftances which feeni to indicate a connexion 
 among the moft diftant objects of the creation vifible 
 from our habitation. The light by which the fixed ftars 
 
 ^ are 
 
CONJECTURES RESPECTING! THE FIXED STARS. 5^^^ 
 
 are feen is tlie fame with that by wliich we behold our 
 SuTi and his attending planets. It moves with the fame 
 velocity, as we difcover by comparing the aberration of 
 tlie fixed ftars with the eclipfes of Jupiter's fatellites. 
 it is refra£led and reflefted according to the fame laws. 
 It confifts of the hme colours. No opinion can be form- 
 ed therefore of the folar light, which muit not alfo be 
 adopted with refpecl to the light of the fixed ftars. The 
 jiiedium of vifion mult be a6ted on in the fame manner 
 by both, whether we fuppofe it the undulation of an 
 rether, or the cmiilion of matter from the luminous body. 
 In either cafe, a mechanical connexion obtains between 
 thofe bodies, however diftant, and our fyftem. Such a 
 connexion in mechafiical properties induces us to fuppofe 
 that gravitation, which we know reaches to a dillance 
 which exceeds all our diflincl: conceptions, extends alfo to 
 the fixed ftars. 
 
 If this be really the cafe, motion muft enfue, even in 
 producing the final ruin of the vifible univerfe ; and pe- 
 riodic motion is indifpenfably necefiary for its perma- 
 nency. 
 
 If all the fixed ftars, and our Sun, were equal, and 
 placed at equal diflances, in the angles of regular fo- 
 lids, their mutual ruinous approach could hardly be per- 
 ceived. For in every moment, they would ilili have the 
 fame relative pofitions, and an increafe of brightnefs 
 is all that could enfue after many ages. But if they 
 were irregularly placed, and unequal, their relative pofi- 
 tions would change, with an accelerated niorion, and 
 4 A 2 this 
 
^^6 PHYSICAL ASTRONOMY. 
 
 this change might become fenfible after a long courfe of 
 ages. If they have periodical motions, fuited to the per- 
 manency of the grand fydem of the miivcrfe, the changes 
 of place may be much more fenfible *, and if we fuppofe 
 that their difference in brilUancy is owing to the differences 
 in their diilance from us, we may expc^'^t that thefe 
 changes will be mofl fenfible in the brightell flars. 
 
 Fa£l:s are not wanting to prove that fuch changes 
 really obtain in the relative pofitions of the fixed ftars. 
 This was firft obferved by that great aflronomer, mathe- 
 matician and philofopher, Dr Halley. He found, after 
 comparing the obfervations of Ariftillus, Tmiochares and 
 Ptolemy with thofe of our days, that feveral of tl^ 
 brighter ftars had changed their fituation remarkably (See 
 Phil. Tranf. N° 355.) Aldebaran has moved to the 
 fouth about 35'. Syrius has moved fouth about 42', and 
 Arfturus, alfo to the fouth, about 33'. The eaftern 
 flioulder of Orion has moved northward about 61'. Ob- 
 fervations in modern times fhew that Arifturus has moved 
 in 78 years about 3' 3". This is a very fenfible quantity, 
 and is eafily obferved, by means of the fmall ftar h irr 
 its immediate neighbourhood. (See Phil. Tranf. LXIII, 
 alfo 1748. ; and Memi. Par. 1755.) Syrius in like man- 
 ner increafes its latitude about 2' in a century (Mem. 
 Par. 1758.) Aldebaran moves very irregularly. The 
 bright ftar in Aqiiila has changed its latitude 36' fince 
 the time of Ptolemy, and 3' fincc^ the time of Tycho. 
 This is eafily feen by its continual feparation from tlie 
 fmall ftar ^. 
 
 Thefe 
 
MOTIONS IN THE STARRY HEAVENS. 557 
 
 Thefe motions feem to indicate a motion in our fyf- 
 tem. Mod of the ftars have moved tov^ard the fouth. 
 The ftars in the northern quarters feem to widen their 
 relative pofitions, while thofe in the fouth feem to con- 
 traft their diftances. Dr Herfchel thinks that a compa- 
 rifon of all thefe changes indicates a motion of our Sun 
 with his attending planets toward the conftellation Her- 
 cules (Phil. Tranf. 1788.) A learned and ingenious 
 friend thinks it not impoflible to difcover this motion by 
 means of the aberration of the ftars. Suppofe the Sun 
 and planets to be moving toward the Pole-ftar, and that his 
 motion is 1 00 times greater than that of the Earth in her or- 
 bit (a very moderate fuppofition, when we compare the or- 
 bital motion of the Earth with that of the Moon), every 
 equatoreal ftar will appear about 34' north of its true 
 place, when viewed through a common telefcope, but 
 only 23' when viewed through a telefcope filled with 
 water. The declination of every fuch ftar will be ii' 
 lefs through a water telefcope than through a common 
 telefcope. Stars out of the equator will have their decli- 
 nation diminifhed by a water telefcope 11' X cof. declin. 
 
 In 1 761, the ingenious Mr Lambert publiftied his 
 Letters on Cofmology (in the Gennan language), in which 
 he has confidered this fubje£t with much attention and 
 ingenuity. He treats of the motion of the Sun round 
 a central body — of fyftems cf fyftem.s, or milky ways, car- 
 ried round an immenfe body — of fyftems of fueh galaxies 
 ' — and of the great central body of the univerfe. In thefe 
 
 fpeculatioris 
 
558 PHYSICAL ASTRONOMY. 
 
 fpeculations he Infers much from final caufts, and is of- 
 ten ingenloufly romantic. But Lambert was alfo a true 
 juduclive philofopher, and makes no aflertion with con-- 
 fidence that is not fupported by good analogies. The 
 rotation of the Sun is a ftrong ground of belief to Mr 
 Lambert that he has alfo a progrefTive motion. 
 
 Tobias Mayer of Gottingen fpeaks in the fame man- 
 ner, in fome of his diflertations publifhed after his death 
 by Lichtenberg. Sec alfo Bailli^s Account of Modern 
 jyirommy, Vol. IL 664, 689. Mayer of Manheim has 
 alfo publiflied tlwughts to this effe£l. See Comment, 
 Jcad. Falatin. IV. Prevoft, Mem, Berlin 1 78 1 . Mitchel 
 ThiU Tranf, LVIL 252. 
 
 The gravitation to the fixed flars can produce no fen- 
 fible difturbances of the motions of our fyftem. This 
 gravitation mufh be inconceivably minute, by reafon of 
 the immenfe diftance ; and, as they are in all quarters of 
 the heavens, they will nearly compenfate each other's 
 a£lion ; and the extent of our fyftem being but as a 
 point, in comparifon with the diftance of tlie neareft 
 ftar, the gravitation to that ftar in all tlie parts of our' 
 fyftem muft be fo nearly equal and parallel, that (98.) no 
 fenfible derangement can be efi'e£led, even after ages of 
 ages. 
 
 As a further circumftance of analogy with a periodi- 
 cal motion in the whole vifible univerfe, we may adduce 
 the remarkable periodical changes of brilliancy that are 
 obferved in many of the fixed ftars. 
 
 This 
 
PtRIODIC CHANGES IN THE STARS. 559 
 
 This was firft obferved (I think) in a ftar of the con- 
 ftellatlon Hyiira. Montanari had obferved it in 1670, 
 and left fonie account of it in his papers, which Maraldi 
 took notice of. Maraldi, after long fearching in vain, 
 found it in 1704, and law fcveral alternations of its 
 brightncfs and dimnefs, but without being able to- afcer- 
 tain their period. It was long loft again, till Mr Edward 
 Pigot found it in 1786. He determined its period to be 
 404 days. Since that time, tliis gentleman, and his fa- 
 ther, with a Mr Goodricke, have given more attention 
 to this department of aftronomy, and their example has 
 been followed by other aitronomers. Mr Pigot has given 
 us, in Phil. Trafjf, 1786, a lift of a great number of 
 iiars (above fifty) in which fuch periodical changes have 
 been obferved, and has given particular determinations 
 of twelve or thirteen, afcertalning their periods with pre- 
 cifion. The whole is followed by fome very curious re- 
 fledllons. 
 
 Of thefe ftars, one of the moft remarkable is % Cyg- 
 ni, having a period of 415-'- days. See PhiL Tranf, 
 N° 343. ♦, alfo Man. Acad. Paris, ^1^9i ^1S9' 
 
 Another remarkable ftar is Ceti, having a period of 
 334 days. (See Phi/. Tranf. N^ 134. 346. ; Metn, Par. 
 1719.) 
 
 There is another fucli, clofe to y Cygni. 
 
 The double ftar ^ Lyra: exhibits very iingular appear- 
 ances, the fouthernmoft fometimes appearing double, 
 and fometimes accompanied by more little ftars. Gri- 
 
 fchoi? 
 
^6o PHYSICAL ASTRONOMY. 
 
 fchofF of Berlin is pofitive that it has planets moving 
 round it. 
 
 Some of thofe ftars have very ihort periods. The 
 mod remarkable is Algol, in the head of Medufa. Its 
 period is 2'^ 20*" 49', in which its changes are very irre- 
 gular, although perfeftly alike in every period. Its or- 
 dhiary appearance is that of a (tar of the fecond magni- 
 tude. It fufFers, for about 3^- hours, a reduction to the 
 appearance of a ftar of the fourth or fifth magnitude. 
 
 Mr Goodricke obferved fimilar variations in the flar 
 ^ Cephei. During 5^* S^ 37' it is a ftar of the fifth mag- 
 nitude. For Id 13^ it is of the fecond or third. It di- 
 miniflies during id 18*^ ; remains 36 hours in its faintefl 
 ftate, and regains its brilliancy in 13^ more {Phi/. 
 Tranf. 1786.) 
 
 Mr Pigot obferved the ftar n Antinoi to maintain its 
 utmoft brilliancy during 44 hours, and then gradually to 
 fade during 62 hours, and, after remaining 30 hours of 
 the fifth magnitude, it regains its greateft brilliancy in 
 36 hours {Phil. Trarif. 1786.) 
 
 Whatever may be the caufe of thefe alternations, 
 they are furely very analagous to M^hat,we obferve in our 
 iyftem, the individuals of which, by varying their pofi- 
 tions, and turning their different fides toward us, exhi- 
 bit alternations of a fimilar kind ; as, for example, the 
 apparition and difparition of Saturn's ring.- Thefe cir- 
 cumftances, therefore, encourage us to fuppofe a fimi- 
 lariiy of conftitution in our fyftem to the reft of the 
 
 heavenly 
 
INDICATIONS 0> DICAT. 561 
 
 heavenly Hoft, and render it more probable that all are 
 conne£led by one general bond, and are regulated by 
 fimilar lav/s. Nothing is fo likely for conftituting thii 
 connexion as gravitation, and its combination with pro- 
 jectile force and periodic motion tends to fecure the per- 
 manency of the whole. 
 
 But I muft at the fame time obferve that fuch ap- 
 pearances in the heavens make it evident that, notwith- 
 llanding the wife proviiion made for maintaining that 
 order and utility which we behold in our fyftem, the day 
 may come * when the heavens fhall pafs away like a 
 "* fcroU that is folded up, when the flars in heaven fhall 
 * fail, and the Snn fliall ceafe to give his light. ' The 
 fuftaining hand of God is flill necefiary, and the prefent 
 order and harmony which he has enabled us to under- 
 ftand and to admire, is wholly dependent on his will, 
 and its duration is one of the unfearchable meafures of 
 his providence. What is become of that dazzling ftar, 
 furpa fling Venus in brightnefs, M-hich flione out all at 
 once in November 1572, and determined Tycho Brahe 
 to become an aftronomer ? He did not Tee it at half 
 an hour paft five, as he was crolFmg fome fields in going 
 to his laboratory. But, returning about ten, he came to 
 a crowd of country folks who were flaring at fomething 
 behind him. Looking round, he faw this wonderful ob- 
 ject. It was fo bright that his flafF had a fhadow. It 
 was of a dazzling white, with a little of a bluiih tinge. 
 Jn this ftate it continued about three weeks, and then be- 
 
 4 B came 
 
5^2 PHYSICAL ASTRONOMY. 
 
 came yellowlfh and lefs brilliant. Its brilliancy climinifh- 
 ed faft after this, and it became more ruddy, like glow- 
 ing embers. Gradually fading, it was wholly inviiible 
 after fifteen months. 
 
 A fimilar phenomenon is faid to have caufcd Hip- 
 parchus to devote himfelf to aftronomy, and to hi^ 
 vail project of a catalogue of the ftars, that pofterlty 
 might know whether any changes happened in the hea- 
 vens. And, in 1604, another fuch phenomenon, though 
 much lefs remarkable, engaged for fome time the atten- 
 tion of aftronomers. Nor are thefe all the examples of 
 the perifliable nature of the heavenly bodies. Several 
 ftars in the catalogues of Hipparchus, of Ulugh Beigh, 
 of Tycho Brahe, and even of Flamftead, are no more to 
 be feen. They are gone, and have left no trace. 
 
 Should we now turn our eyes to objedls that are 
 nearer us, we fhall fee the fame marks of change. When 
 the Moon is viev/ed through a good telefcope, magnify- 
 ing about 150 times, we fee her whole furface occupied 
 by volcanic craters ; fome of them of prodigious magni- 
 tude. Some of them give the moft unqv.eftionable marks 
 of feveral fucceflive eruptions, each deftroying in part 
 the crater of a former eruption. The precipitous and 
 craggy appearance of the brims of thofe craters is pre- 
 cifely fuch as would be produced by the ejetSiion of rocky 
 matter. In fhort, it is impofiible, after fuch a view of 
 the Moon, to doubt of her being greatly changed from 
 her primitive ftate. 
 
 Evea 
 
INDICATIONS OF DECAY; 563 
 
 Even the Sun himfelf, the fource of light, and heat, 
 and Hfe, to the whole fyftem, is not free from fuch 
 changes. 
 
 If we now look round us, and examine with judici- 
 ous attention our own habitation, we fee the moft incon- 
 trovertible marks of great and general changes over the 
 whole face of the Earth. Befides the flow degradation 
 by the adion of the winds and rains, by which the foil 
 is gradually waflied away from the high lands, and car- 
 ried by the rivers into the bed of the ocean, leaving the 
 Alpine fummits dripped to the very bone, we cannot 
 fee the face of any rock or crag, or any deep gully, which 
 does not point out much more remarkable changes. 
 Thefe are not confined to fuch as are plainly owing to 
 the horrid operations of volcanoes, but are univerfal. 
 Except a few mountains, where we cannot confidently 
 fay that they are factitious, and which for no better rea- 
 fon we call primitive, there is nothing to be feen but 
 ruins and convulfions. What is now an elevated moun- 
 tain has moft evidently been at the bottom of the fea, 
 and, previous to its being there, has been habitable fur- 
 face. 
 
 It is very true that all our knowledge on this fubje£l: 
 is merely fuperficial. The higheft mountains, and deepefl 
 excavations, do not bear fo great a proportion to the globe 
 as the thicknefs of paper that covers a terreftrial globe bears 
 to the bulk of that philofophical toy. We have no au- 
 thority from any thing that we have feen, for forming 
 4 B 2 any 
 
564. PHYSICAL ASTRONOMY. 
 
 any judgement concerning the internal conllltution of the 
 Earth. But we fee enough to convince us that it bears 
 no marks of eternal duration, or of exifling as it is, by its 
 own energy. No ! — all is perifhable — all requires the 
 fuflaining hand of God, and is fubje£^ to the unfearch- 
 able defigns of its Author and Preferver. 
 
 There is yet another clafs of obje£ls in the heavens, 
 of M^hich I have taken no notice. They are called ne- 
 bul;e, or nebulous stars. They have not the fparkling 
 briUiancy that diftinguifhes the ilars, and they are of a 
 fenfible diameter, and a determinate fliape. Many of 
 them, when viewed through telefcopcs, are clufters of 
 ftars, which the naked eye cannot diftinguiili. The moft 
 remarkable of thefe is in the conftellation Cancer, and is 
 known by the name Prafepe. Ptolemy mentions it, and 
 another in the right eye of Sagittarius. Another may 
 be feen in the head of Orion. Many fmall clullers have 
 been difcovered by the help of glafTes. The whole ga- 
 laxy is nothing elfe. 
 
 But there is another kind, in which the fined tele- 
 fcopes have difcovered no clufterlng ftars. Moft of them 
 have a ftar in or near the middle, furrounded with a pale 
 light, which is brighteft in the middle, and grows more 
 faint toward the circumference. This circumference is 
 diftin£l, or well defined, and is not always round. One 
 or two nebulae have the form of a luminous difk, with a 
 hole in the middle like a milftone. They are of various 
 colours, white, yellow, rofe-coloured, &c. Dr Her- 
 fchel, in feveral of the late volumes of the Philofophi- 
 
 cal 
 
NEBULOUS STARS, 565 
 
 cal Tranfacflions, has given us tlie places of a vafl num- 
 ber of nebula, with curious defcriptions of their pecu- 
 liar appearances, and a feries of moft ir.genious and in- 
 tereiling refledions on their nature and conftitution. 
 His Thoughts on the Struclure of the Heavens are full of 
 moll curious fpcculation, and fliould be read by every 
 philofopher. 
 
 When we refie^l: that thefe fmgular objects are not, 
 hke the fixed liars, brilliant points, which become foiailer 
 when feen through finer telefcopes, but have a fenfible, 
 and meafureable diameter, fometimes exceeding 2' \ and 
 when we alfo recoiled: that a ball of 200,090,000 miles 
 -i^^iameter, which would fill the whole orbit of the 
 ■'^Earth round the Sun, would not fubtend an angle of 
 two feconds when taken to the nearefl fixed ftar, what 
 muft we think of thefe nebulx ? One of them Is cer- 
 tainly fome thoufands of times bigger than the EartVs 
 orbit. Although our fineft telefcopes cannot feparate it 
 into flars, it is ftill probable that it is a clufler. It is 
 not unreafonable to think, with Dr Herfchel, that this 
 objed, M^hich requires a telefcope to find it out, will 
 appear to a fpedator in its centre much the fame as the 
 vifible heavens do to us, and that this ftarry heaven, 
 which, to us, appears fo magnihcent, is but a nebulous 
 ftar to a fpedator placed in that nebula. 
 
 The human mind is almofl overpowered by fuch a 
 thought. When the foul is filled with fuch conceptions 
 of the extent of created nature, we can fcarcely avoid 
 exclaiming, * Lord, what then is man that thou ait 
 
 * mindful 
 
^6^ PHYSICAL ASTRONOMY. 
 
 < mindful of him ! * Under fuch imprelTions, David 
 fhrunk into nothing, and feared that he (liould be for- 
 gotten amongft fo many great objects of the Divine at- 
 tention. His comfort, and ground of rehcf from this 
 deje6ling thought, are remarkable. * But, ' fays he, 
 
 * thou haft made man but a little lovirer than the angels, 
 
 * and haft crowned him with glory and honour. ' David 
 correcfled himfelf, by calling to mind how high he ftood in 
 the fcale of God's w^orks. He recognifed his own divine 
 original, and his alliance to the Author of all. Now^ 
 cheered, and delighted, he cries out, * Lord, how glori- 
 ous is thy name ! ' 
 
 There remains yet another phenomenon, which is 
 very evidently connected with the mechanifm of the fo- 
 lar fyftem, and is in itfelf both curious and important. 
 I mean the tides of our ocean. Although it appears im- 
 proper to call this an aftronomical phenomenon, yet, as 
 it is moft evidently conne6led wath the pofition of the 
 Sun and Moon, we muft attribute this connexion in fa£i 
 to a natural connexion in the way of caufe and efFe£l» 
 
 Of the Tides. 
 
 617. It is a very remarkable operation of nature 
 that we obferve on the fhores of the ocean, when, in the 
 calmeft weather, and moft ferene fky, the vaft body of 
 waters that bathe our coafts advances on our fhores, in- 
 undating 
 
OF THE TIDES. $6 J 
 
 nndating all the flat i;inds, rifing to a confidcrable height, 
 and then as gradually retiring again to the bed of the ocean ; 
 and ail this without the appearance of any caufe to im- 
 pel the waters to our Hiorcs, and again to draw them off. 
 Twice every day is this repeated. In many places, this 
 motion of the waters is even tremendous, the fea advance 
 ing, even in the calmeft weather, with a high furge, 
 rolling along the flats with refiftlefs violence, and rifmg 
 to the height of many fathoms. In the bay of Fundy, 
 it comes on with a prodigious noife, in one vaft wave, 
 that is fecn thirty miles off; and the waters rife loo and 
 1 20 feet in the harbour of Annapolis-Royal. At the 
 mouth of the Severn, the flood alfo comes up in one 
 head, about ten feet hi^h, bringing certain deftru£tion 
 to any fmall craft that has been unfortunately left by the 
 ebbing waters on the flats ; and as it paffes the mouth of 
 the Avon, it feuds up that fmall river a vafh body of 
 water, rifmg forty, or fifty feet at BrifloL 
 
 Such an appearance forcibly calls the attention of 
 thinking men, and excites the greateft curiofity to difco- 
 ver the caufe. Accordingly, it has been the objecl of re- 
 fearch to all who would be thought philofophers. We 
 .find very little however on the fubjetl; in the writings of 
 the Greeks. The Greeks indeed had no opportunity of 
 knowing much about the ebbing and flowing of the fea, 
 as this phenomenon is fcarcely perceptible on the fhores 
 of the Mediterranean and its adjoining feas. The Per- 
 fian expedition of Alexander gave them the only oppor- 
 tunity they ever had, and his army was aflonifned at 
 
 finding 
 
568 PHYSICAL ASTRONOMY. 
 
 finding the fliips left on the dry flats when the fea re^ 
 tired. Yet Alexander's preceptor Ariitotle, the prince 
 of Greek philofophers, fhews little curiofity about the 
 tides, and is contented with barely mentioning them, 
 and faying that the tides are moft remarkable in great 
 feas. 
 
 618. When we fearch after the caufe of any recur- 
 ring event, we naturally look about for recurring conco- 
 mitant circumftances -, and when we find any that gene- 
 rally accompany it, we cannot help inferring fome con- 
 nexion. All nations feem to have remarked that the 
 flood-tide always comes on our coafts as the Moon moves 
 acrofs the heavens, and comes to its greateft height when 
 the Moon is in one particular pofition, generally in the 
 ibuth-weft. They have alfo remarked that the tides are 
 moft remarkable about the time of new Moon, and be- 
 come more moderate by degrees every day, as the Moon 
 drav/s near the quadrature, after which they gradually 
 increafe till about the time of full Moon, when they are 
 nearly of their greateft height. They novv; leflen every 
 day as they did before^ and are loweft about the laft qua- 
 drature, after which they increafe daily, and, at the nexc 
 new Moon are a third time at the higheft. 
 
 Thefe circumftances of concomitancy have been no- 
 ticed by all nations, even the moft uncultivated ; and all 
 feem to have concurred in afcribing the ebbing and flow- 
 ing of the fea to the Moon, as the efficient caufe, or, at 
 leaft, as the occafion, of this phencaienon, altliough 
 
 without 
 
or THE TIDES IN GENERAL. S'^9 
 
 N\-ithout any comprehenfion, and often without any 
 thought, in what manner, or by what powers of nature, 
 this or that pofition of the Moon fliould be accompanied 
 by the tide of flood or of ebb. 
 
 Although this accompaniment has been every where 
 remarked, it is liable to fo many and fo great irregulari- 
 ties, by winds, by freflies, by the change of feafons, and 
 other caufes, that hardly any two fucceeding tides are ob- 
 ferved to correfpond with a precife pofition of the Moon. 
 The only way therefore to acquire a knowledge of the 
 connexion that may be ufeful, either to the philofopher 
 or to the citizen, is to multiply obfervations to fuch a 
 number, that every fource of irregularity may have its 
 period of operation, and be difcovered by the return of 
 the period. The inhabitants of the fea-coafts, and par- 
 ticularly the fifhermen, were moft anxioufly interefted in 
 this refearch. 
 
 619. Accordingly, it was not long after the conquells 
 of the Romans had given them poiTeflion of the coafts of 
 the ocean, before they learned the chief circumftances or 
 laws according to which the phenomena of the tides 
 proceed. Pliny fays that they had their fource in the 
 Sun and the Moon. It had been inferred from the gradual 
 change of the tides between new Moon and the quadra- 
 ture, that the Sun was not unconcerned in the operation. 
 Pytheas, a Greek merchant, and no mean philofopher, 
 relident at Marfeilies, the oldell Grecian colony, had often 
 i?epn in Britain, at the tin mines in Cornwall and its ad- 
 
 4 C jacent 
 
S'JO PHYSICAL ASTRONOMY. 
 
 jacent Idands. He had obferved the phenomena witti 
 
 great fagaclty, and had coUe£led the obfervations of the 
 
 natives. Plutarch and PHny mention thefe obfervations 
 
 of Pytheas, fome of them very delicate, and, the vidiole 
 
 taken together, containing ahuoft all that vt-as known of 
 
 the fubje^t, till the difcoveries of Sir Ifaac Newton taught 
 
 the philofophers what to look for in their inquiries into 
 
 the nature of the tides, and how to clafs the phenomena. 
 
 Pytheas had not only obferved that the tides gradually 
 
 abated from the times of new and full Moon to the time 
 
 of the quadratures, and then increafed again, but had 
 
 alfo remarked that this vulgar obfervation was not exa£l, 
 
 but that the greateft tide was always two days after new 
 
 or full Moon, and the fmalleft was as long after the 
 
 quadratures. He alfo corrected the common obfervation 
 
 of the tides falling later every day, by obferving that this 
 
 retardation of the tides was much greater when the 
 
 Moon was in quadrature than when new or full. The 
 
 tide-day, about the time of new and full Moon, is really 
 
 fhorter by 50' than at the time of her quadrature. 
 
 620. This variation in the interval of the tides is 
 called the priming or the lagging of the tides, accord- 
 ing as we refer them to lunar or folar time. Pytheaa 
 probably learned much of this nicety of obfervation 
 from the Cornifh fifliermen. By -Elian's accounts, they 
 had nets extended along fhore for feveral miles, and were 
 therefore much interefted in this matter. 
 
 621- 
 
OF THE TIDES IN GENERAL. 571 
 
 621, Many obfervations on the fcrles of phenomena 
 whlcii completes a period of the tides are to be found 
 in the books of hydrography, and the inn:ru6lions for 
 mariners, to whom the ex2.Ci knowledge of the courfe of 
 the tides is of the utmoft importance. But we never 
 had any good colleftion of obfervations, from which the 
 laws of their progrefs could be learned, till the Academy 
 of Paris procured an order from government to the offi- 
 cers at the ports of Bred and Rochefort, to keep a regi- 
 (ter of all the phenomena, and report it to the Academy, 
 A regifter of obfervations was accordingly continued for 
 fix years, without interruption, at both ports, and the 
 obfervations were publiihed, forming the moll complete 
 feries that is to be met with in any department of fci- 
 ence, aftronomy alone excepted. The younger Callini 
 undertook the examination of thefe reglllers, in order to 
 deduce from them the general laws of the tides. This 
 tafk he executed with confiderable fuccefs *, and the gene- 
 ral rules which he has given contain a much better ar- 
 rangement of all the phenomena, their periods and 
 changes, than any thing that had yet appeared. Indeed 
 there had fcarcely any thing been added to the vague ex- 
 perience of illiterate pilots and iilhermen, except two 
 differtations by Wallis and Fiamftcad, publifhed in the 
 Philofophical Tranfatlions. 
 
 622. It is not likely, notwithftanding this excellent 
 colleilion of obfervations, that our knowledge would have 
 proceeded much farther, had not Newton demonftrated 
 
 4 C 2 that 
 
57a PHYSICAL ASTRONOMT. 
 
 that a feries of phenomena perfectly refembling the tides 
 refuited from the mutual attra<£lion of all matter. Thefe 
 confequences pointed out to thofe Interefted in the know- 
 ledge of the tides what viciflitudes or changes to look 
 for — what to look for as the natural or regular feries — 
 what they are to confider as mere anomalies— what pe- 
 riods to expert in the different variations — and whether 
 there are not periods which comprehend the more obvi- 
 ous periods of the tides, diftinguifhing one period from 
 another. As foon as this clue was obtained, eveiy thing 
 was laid open, and without it, the .labyrinth was almoft 
 inextricable ; for in the variations of the tides there are 
 periods In which the changes are very confiderable j and 
 thefe periods continually crofs each other, fo that a tide 
 w^hich (liould be great;, confidered as a certain tide of 
 one period, fhould be fmall, confidered as a certain tide 
 of another period. When it arrives, It is neither a great 
 nor a fmall tide, but it prevents both periods from offering 
 themfelves to the mere obferver. The tides afford a 
 very ftrong example of the great Importance of a theory 
 for dire£ilng even our obfei"vations. Aided by the New- 
 tonian theory, we have difcovered many periods', in which 
 the tides fuffer gradual changes, both in their hour and 
 in their height, which commonly are fo implicated with 
 one another, that they never would have been difcovered 
 without this monitor, whereas now, we can predict them 
 all. 
 
 623. The phenomena of the tides arc, in general, 
 the following. 
 
GEKERAL PHENOMENA OF THE TIDES. $13 
 
 I. The wajters of the ocean rife, from a medium 
 height to that of high water, and again ebb away from 
 the fliores, falhng nearly as much below that medium 
 flate, and then rife again in a fucceeding tide of flood, 
 and again make high water. The interval between two 
 fucceeding high waters is about 12*" 25', the half of the 
 time of the Moon's daily circuit round the Earth, fo that 
 we have two tides of flood and two ebb tides in every 
 24'' 50'. f his is the fhortefl period of phenomena ob- 
 ferved in the tides. The gradual fubfidence of the wa- 
 ters is fuch that the diminutions of the height are nearly 
 as the fquares of the times from high water. The fame 
 may be faid of the fubfequent rife of the waters in tha 
 next flood. The tim.e of low water is nearly half way 
 between the two hours of high water ; not indeed exactly, 
 it being obferved at Breft and Rochefort that the flood 
 tide commonly takes ten minutes lefs than the ebb tide. 
 
 624. As the different phenomena of the tides are 
 chiefly diftinguifliable by the periods, or intervals of time 
 in which they recur, it will be convenient to mark thofe 
 periods by different names. Therefore, let the time of 
 the apparent diurnal revolution of the Moon, viz. 24^ 50', 
 be called A lunar day, and the 24th part of it be called 
 A lunar hour. To this interval almoft all the vicilTi- 
 tudes of the tides are mofl conveniently referred. Let 
 the name tide day be given to the interval between two 
 high waters, or two low waters, fucceeding each otlier 
 \vith the Moon nearly in the fame pofition. This inter- 
 val 
 
74 PHYSICAL ASTRONOMY. 
 
 val compreKends two complete tides, one of the full 
 Teas happening when tlie Moon is above the horizon, and 
 the next, when flie is under the horizon. We fliall alfo 
 find it convenient to diflingulfli thefe tides, by calhng the 
 iirft the superior tide, and the other the inferior 
 TIDE. At new Moon they may be called the Morning 
 and Evening tides. 
 
 625. 2. It is not only obferved that we always have 
 high water when the Moon is on fome particular point 
 of the compafs (S. W. nearly) but alfo that the height 
 of full fea from day to day has an evident reference to 
 the phafes of the Moon. At Breft, the higheft tide is 
 always about a day and a half after full or change. If 
 it fhould happen that high water falls at the very time 
 of new or full Moon, the third full fea after tliat one 
 is the higheft of all. This is called the spring-tide. 
 Each fucceeding full fea is lefs than the preceding, till 
 we come to the third full fea after the Moon's quadra- 
 ture. This is the loweft tide of all, and it is called 
 NEAP-TIDE. After this, the tides again increafe, till the 
 next full or new Moon, the third after which is again 
 the greateft tide, 
 
 626. The higher the tide of flood rifes, the lower 
 does the ebb tide generally fink on that day. The total 
 magnitude of the tide is eftimated by taking the differ- 
 ence between high and low water. As this is continu- 
 ally varying, the bed way of computing its magnitude 
 
 feems 
 
Ceneral laws of the tides. 5^j 
 
 feems to be, to take the half fum of two fiicccedhig tides. 
 This muft always give us a mean value for the tide whofe 
 full fea was in the middle. The medium fpring>tldc at 
 Brefl is about nineteen feet, and the neap-tide is about 
 nine. 
 
 Here then we have a period of phenomena, the time 
 of which is half of a lunar month. This period com- 
 prehends the mofh important changes, both in rcfpe£l of 
 magnitude, and of the hours of high and low water, and 
 feveral modifications of both of thofe circumflances, 
 fuch as the daily difFerence in height, or in time. 
 
 627. 3. There is another period, of nearly twice 
 the fame duration, which greatly modifies all thofe lead- 
 ing circumftances. This period has a reference to the 
 diftance of the Moon, and therefore depends on the 
 Moon's revolution in her orbit. All the phenomena are 
 increafed when the Moon is nearer to the Earth. There- 
 fore the higheft fpring-tide is obferved wdien the INIooa 
 is in perigeo, and the next fpring-tide is the fmalleft, be- 
 caufe the Moon is then nearly in apegeo. This will 
 make a difFerence of 2^ feet from the medium height of 
 fpring tide at Breft, and therefore occafion a difFerence of 
 ^~ between the greatefb and the lead. It is evident thaf 
 as the perigean and apogean fituation of tlie Moon may 
 happen in every part of a lunation, the equation for the 
 height of tide depending upon this circum.ftance may 
 often run counter to the equation correfponding to the 
 
 regular 
 
576' PHYSICAL ASTRONOMY. 
 
 regular monthly feries of tides, and will ieemingly de- 
 ilroy their regularity. 
 
 628. 4. The variation in the Sun's diftance alfo af- 
 fe£ls the tides, but not nearly fo much as thofe in the 
 diftance of the Moon. In our winter, the fprlng-tides are 
 greater than in fummer, and the neap-tides are fmaller. 
 
 629. 5. The declination, both of the Sun and 
 Moon, aftedls the tides remarkably ; but the effects are 
 too intricate to be diftindlly feen, till we perceive the 
 caufes on which they depend. 
 
 630. 6. All the phenomena are alfo modified by 
 the latitude of the place of obfervation ; and fome phe- 
 nomena occur in the high latitudes, which are not feen at 
 all w^hen the place of obfervation is on the equator. In 
 particular, when the obferver is in north latitude, and 
 the Moon has north declination, that tide in M^hich the 
 Moon is above the horizon is greater than the other tide 
 of the fame day, when the Moon is below the horizon. 
 It will be the contrary, if either the obferver or the Moon 
 (but not both) have fouth declination. If the polar 
 diftance of the obferver be equal to the Moon's declina- 
 tion, he will fee but one tide in the day, containing 
 twelve hours flood and twelve hours ebb. 
 
 631. 7. To all this it muft be added, that local 
 circumftances of fituation alter all the phenomena re- 
 markably. 
 
THEORY OF THE TIDES. «^77 
 
 markably, fo as frequently to leave fcarcely any circum- 
 ftances of refemblancc, except the order and periods in 
 which the various phenomena follow one another. 
 
 We muft now endeavour to account for thefe remark- 
 able movements and viciflitudes in the waters of the 
 ocean. 
 
 632. Since the phenomena of the planetary motions 
 demonftrate that every particle of matter in this globe gra- 
 vitates to the Sun, and fince they are at various diftances 
 from his centre, it is evident that they gravitate une- 
 qually, and that, from this inequality, there muft arife 
 a difturbance of that equilibrium which terreftrial gravi- 
 tation alone might produce. If this globe be fuppofed 
 either perfectly fluid and homogeneous, or to conlift of 
 a fplterical nucleus covered with a fluid, it is clear that 
 the fluid muft aflume a perfeftly fpherical form, and that in 
 this form alone, every particle will be in equilibrio. But 
 when we add to the forces now a6ling on the waters 
 of the ocean their unequal gravitation to the Sun, this 
 equilibrium is difturbed, and the ocean cannot remain in 
 this form. We may apply to the particles of the ocean 
 every thing that we formerly faid of the gravitation of 
 the Moon to the Sun in the different points of her orbit ; 
 and^the fame conftruflion in fig. 59, that gave us a re- 
 prefentation and meafure of the forces which deranged 
 the lunar motions, may be employed for giving us a no- 
 tion of the manner in which tlie particles of water in 
 the ocean are afFcded. The circle O B C A may re- 
 
 4 ^ prefent 
 
578 PHYSICAL ASTR0N0-M7. 
 
 prefent the watery fphere, and M any particle of the 
 water. The central particle E gravitates to the Sun 
 with a force which may be represented by E S. The 
 gravitation of the particle M mufl be meafured by M G. 
 This force M G may be conceived as compounded of 
 MF, equal and parallel to ES, and of M H. The 
 force M F occafions no alteration in the gravitation of 
 M to the Earth, and M H is the only difturbing force. 
 We found that this conflrudlion may be greatly fimpli- 
 fied, and that M I may be fubftituted for M H without 
 any fenfible error, becaufe it never differs from it more 
 than yi^-. We therefore made E I, in fig. 60, =3MN, 
 and confidered M I as the difturbing force. This con- 
 ftru£lion is applicable to the prefent queftion, with much 
 greater accuracy, becaufe the radius of the Earth is but 
 the fixtieth part of that of the Moon's orbit. This reduces 
 the error to tttt^j ^ quantity altogether infenfible. 
 
 633. Therefore let OACB (fig. 68.) be the ter- 
 raqueous globe, and C S a line directed to the Sun, 
 and B E A the fecbion by that circle which Separates 
 the illuminated from the dark hemifphere. Let P be any 
 particle, whether on the Surface or within the mafs. 
 Let QPN be perpendicular to the plane B A. Make 
 E I rz 3 P N, and join PL P I is the difturbing force, 
 when the line E S is taken to reprefent the gravitation 
 of the particle E to the Sun. This force PI may be 
 conceived to be compounded of two forces P E and P Q. 
 
 FE 
 
THEORY OF THE TIDF.S. 579 
 
 P E tends to the centre of the Earth. P O tends from 
 
 •Si* 
 
 the plane B A, or toward the Sun. 
 
 If this conftruftion be made for every particle in the 
 fluid fphere, it is evident tliat all the forces P E balance 
 one another. Therefore they need not be confidered in 
 the prefent queftion. But the forces P O evidently di- 
 minifh the terreftrial gravitation of every particle. At 
 C the force P Q a£ls in direifl oppofition to the ter- 
 reftrial gravity of the particle. And, in the fituation P, 
 It diminiflies the gravity of the particle as efhimated in 
 the direction P N. There is therefore a force ailing in 
 the direction N P on every particle in the canal P N. 
 And this force is proportional to the diilance of the par- 
 ticle from the plane B A (far P O is always = 3 P N). 
 Therefore the v/ater in this canal cannot remain in its 
 former pofition, its equilibrium being now deflroyed. 
 This may be reftored, by adding to the column N P a 
 fmall portion Pj^, whofe v/eight may compenfate the di- 
 minution in the weight of the column N P. A fimilar 
 addition may be made to every fuch column perpendicu- 
 lar to the plane B E A. This being fuppofed, the 
 fpherical figure of the globe will be changed into that 
 of an elliptical fpheroid, having its axis in the line O C, 
 and Its poles in O and C ($69-) 
 
 Without making this addition to every column N P, 
 we may underftand how the equilibrium may be reftored 
 by the waters fubfiding all around the circle whofe fec- 
 tion is B A, and rifing on both fides of it. For it was 
 (liewn (564.) that in a fluid elliptical fpheroid of gravi- 
 4 D 2 tating 
 
580 PHYSICAL ASTRONOMY. 
 
 tating matter, the gravitation of any particle P to ail the 
 other particles may be rcfolved into two forces P N and 
 P M perpendicular to the plane B A and to the axis 
 O C, and proportional to P N and P M -, and that if the 
 forces be really in this proportion, the whole will be in 
 equilibrio, provided that the whole forces at the poles and 
 equator are inverfely as the diameters O C and B A. 
 Now this may be the cafe here. For the forces fuper- 
 added to the terreftrial gravitation of any particle are, 
 ly?, A force P E, proportional to P E. When this is 
 refolved into the directions P N and P M, the forces 
 arifrng in this refolutlon are as P N and P M, and there- 
 fore in the due proportion ; icly The force P Q, which 
 is alfo as P N. It is evident therefore that this mafs 
 may acquire fuch a protuberancy at O and C, that the 
 force at O (hall be to the force at B as B A to O C, or 
 as E A to EC. We are alfo taught in § 585. what 
 this protuberance muft be. It mud be fuch that four 
 times the mean gravity of a particle on the furface is to 
 five times the difturbing force at O or C as the diame- 
 ter B A is to the excefs of the diameter O C. This el- 
 lipticity is exprelled by the fame formula as in the former 
 
 cafe, VIZ. - = — , 
 
 E C — F 
 
 SS EC 
 
 634. Thus we have difcovered that, in confequence 
 of the unequal gravitation of the m.atter in the Earth to 
 the Sun, the waters will aflume the form of an oblong ellip- 
 tical fpheroid, having its axis diredlcd to the Sun, and its 
 
 poles 
 
THEORY OF THE TIDES. 58 1 
 
 poles in thofe points of the furface which have the Sun 
 in the zenith and nadir. There the waters are higheft 
 above tlie furface of a fphere of equal capacity. All 
 around the circumference B E A, the waters are below 
 tlie natural level. A fpedlator placed on this circumfer- 
 ence fees the Sun in the horizon. 
 
 We can tell exadly what this protuberance E O — E A 
 mud be, becaufe we know the proportions of all the 
 forces. Let W reprefent the terrePtrial gravitation, or 
 the weight of the particle C, and G the gravitation of 
 the fame particle to the Sun, and let F be the dillurb- 
 ing force acting on a particle at C or at O, and there- 
 fore = 3 C E. Let S and E be the quantity of matter 
 in the Sun and in the Eartli. 
 Then (fig. 59.) F ; G = 3 C E : C G 
 
 therefore F : W = 3 CEx^ ^ CG xE ^ 
 
 C S" C E* 
 
 C S^^ X C^ : cEl • ^^<^' ^^caufe CS- E S^ = E S : CG, 
 
 we have C S^^ x C G = E S^ X E S, = E S^ There- 
 
 "2 S E 
 fore F : W = |-g3 : ^g^ . Now E : S =: i : 338343, 
 
 and E C : E S = I : 23668. This will eive -|^ : =,— 
 ^ ^ £8* EC^ 
 
 = I : i277354i> = F : W. 
 
 Finally, 4W:5F = CE:CE — AE. We fliall 
 find this to be nearly 244- inches. 
 
 635. Such is the figure that this globe would aflinne, 
 had it been originally fluid, or a fpherical nucleus covered 
 
 with 
 
582 PHYSICAL ASTRONOMY. 
 
 with a fluid of equal denfity. The two fummlts of the 
 watery fpheroid would be raifed about two feet above 
 the equator or place of greateft depreflion. 
 
 But the Earth is an oblate fpheroid. If we fuppofe 
 it covered, to a moderate depth, with a fluid, the waters 
 would acquire a certain figure, which has been confidered 
 already. Let the difturbing force of the Sun a£l on this 
 figure. A change of figure muft be produced, and the 
 waters under the Sun, and thofe in the oppofite parts, 
 will be elevated above their natural furface, and the 
 ocean will be deprefled on the circumference B E A. It 
 is plain that this change of figure will be almofl: the fame 
 in every place as if the Earth were a fphere. For the 
 difference between the change produced by the Sun*s di- 
 fturbing force on the figure of the fluid fphere or fluid fphe- 
 roid, arifes folely from the difference in the gravitation of a 
 particle of water to the fphere and to the fpheroid. This 
 difference, in any part of the furface, is exceedingly fmall, 
 
 not being ~ of the whole gravitation. The differ- 
 ence therefore in the change produced by the Sun can- 
 not be • r of the whole change. Therefore, fince it 
 
 300* . .- 
 
 is from the proportion of the difturbing force to the force 
 
 of gravity that the ellipticity is determined, it follows 
 that the change of figure is, to all fenfe, the fame, whe- 
 ther the Earth be a fphere or a fpheroid whofe eccentri- 
 city is lefs than -^\^. 
 
 Let us fuppofe, for the prefent, that the watery fphe- 
 roid alM^ays has that form which produces an equilibrium. 
 
 in 
 
THEORY OF THE TIDES. 583 
 
 in all its particles. This cannot ever be the cafe, be- 
 caufe feme time muft elapfe before an accelerating force 
 can produce any finite change in the difpofition of the 
 waters. But the contemplation of this figure gives us 
 the mofl: diftin61: notion of the forces that are in action, 
 and of their efFe£l:s ; and we can afterwards ftate the 
 difference that mufl obtain becaufe the figure is not com- 
 pletely attained. 
 
 Suppofing it really attained, it follows that the ocean 
 will be moft elevated in thofe places which have the Sun 
 in the zenith or nadir, and mofl deprefTed in thofe places 
 where the Sun is feen in the horizon. Y^^hile the Earth 
 turns round its axis, the pole of the fpheroid keeps fliil 
 toward the Sun, as if the waters flood flill, and the folid 
 nucleus turned round under it. The phenomena may 
 perhaps be eafier conceived by fuppofing the Earth to re- 
 main at reft, and the Sun to revolve round it in 24 hours 
 from eaft to weft. The pole of the fpheroid follows 
 him, as the card of a mariner's compafs follows the mag- 
 net ; and a fpedator attached to one part of the nucleus 
 will fee all the vicifTitudes of the tide. Suppofe the Sun 
 in the equinox, and the obferver alfo on the Earth's equa- 
 tor, and the Sun juft rifing to him. The obferver is 
 then in the loweft part of the watery fpheroid. As 
 the Sun rifes above the horizon, the water alfo rifes ; and 
 when the Sun is in the zenith, the pole of the fpheroid 
 has now reached the obferver, and the water is two feet 
 deeper than it was at fun-rife. The Sun now approach- 
 ing the weftern horizon, and the pole of the ocean going 
 
 along 
 
584 PHYSICAL ASTRONOMY. 
 
 along with Kim, the obferver fees the water fubfule again, 
 and at fun-fet, it is at the fame level as at fun-rife. As 
 the Sun continues his courfe, though unfeen, the oppo- 
 fite pole of the ocean now advances from the eaft, and 
 the obferver fees the water rife again by the fame degrees 
 as in the morning, and attain the height of two feet at 
 midnight, and again fubfide to its loweft level at fix 
 o'clock in the following morning. 
 
 Thus, in 24 hours, he has two tides of flood and two 
 ebb tides ; high water at noon and midnight, and low 
 water at fix o'clock morning and evening. An obferver 
 not in the equator will fee the fame gradation of phe- 
 nomena, at the fame hours *, but the rife and fall of the 
 water will not be fo confiderable, becaufe the pole of the 
 fpheroid palTes his meridian at fome diftance from him. 
 If the fpeftator is in the pole of the Earth, he will fee 
 no change, becaufe he is always in the loweft part of 
 the watery fpheroid. 
 
 From this account of the fimplcfl; cafe, we may iiifer 
 that the depth of tlie water, or its change of depth, de- 
 pends entirely on tlie fhape of the fpheroid,^ and the place 
 of it occupied by the obferver. 
 
 636. To judge of this with accuracy, we mud: take no- 
 tice of fome properties of the ellipfe which forms the meri- 
 dian of the watery fpheroid. Let AE«0 (fig. 6^.) 
 reprefent this elliptical fpheroid, and let B E ^ O be the 
 infcribed fphere, and KGag the circumfcribed fphere. 
 Alfo let D F df be the fphere of equal capacity with the 
 
 fpheroid. 
 
THEORY OF THE TIDES. i^^:; 
 
 fplieroid. This will be the natural figure of the ocean, 
 uiidifturbed by the gravitation to the Sun. 
 
 In a fpheroiJ like tliis, fo little different from a fphere, 
 iliQ elevation A D of its fummit above the equally ca« 
 p.icious fphere is very nearly double of the depreffion F E 
 of its equator below the furface of that fphere. For 
 fpheres and fpheroids, being equal to y of the circum- 
 fcribing cylinders, are in the ratio compounded of the 
 ratio of their equators and the ratio of their axes. There- 
 fore, fnice the fphere D F df is equal to the fpheroid 
 AE^O, we have C F^ x C D zr C E' X C A, and 
 C E' : C F'- = C D : C A. Make CE:CF = CF:C^, 
 then CE:Ca; = CD:CA, and CE:Ea;~CD:DA, 
 and C E : C D = E a; : D A. Now C E does not diifer 
 fenfibly from C D (only eight inches in near 4000 miles), 
 therefore E x may be accounted equal to D A. But 
 E X is not fenfibly different from twice E F. Therefore 
 the proportion is manifeft. 
 
 637. In fuch an elliptical fpheroid, the elevation IL 
 of any point I above the infcribed fphere is proportion- 
 al to the fquare of the cofnie of its diftance from the 
 pole A, and the depreffion K I of this point below the 
 furface of the circumfcribed fphere is as the fquare of 
 the fine of its diilance from the pole A. Draw through 
 the point I, HIM perpendicular to C A, and I/)N 
 perpendicular to C E. The triangles C I N and / I L 
 are fimilar. 
 
 4 E Therefore 
 
53"6 JPH7SICAL ASTRONOMT. 
 
 Therefore pI:lL = C1:\N, = rad. : cof. I C A 
 
 but, by the elHpfe A B :/> I = A C : I N, = rad. : cof. I C ^ 
 
 therefore A B : I L := rad.^ : cof.^ I C A 
 
 and I L is always in the proportion of cof.% I C A, and 
 
 is = A B X cof.% I G A, radius being =: i. 
 
 In Hke manner H I : I K = C I : I M = rad. : fm. I C A. 
 
 and G E : H I = E C : I INI = rad. : fni. I C A 
 
 therefore G E : K I = rad.^ : Cmr I C A 
 
 and KI is = AB x fin.aeA. 
 
 638. We muil aifo know the elevations and depref- 
 fions in refpecl of the natural level of the undifturb- 
 ed ocean. This elevation for any point z is evidently 
 i / — ?w / = A B X cof.^ iCA—i-AB, =ABx 
 
 cof.* i C A — y, and the deprefhon nr of a point r is 
 kr — kn = AB X fin.^ rCA — |AB, =ABX 
 fm.^r C A — f. 
 
 It will be convenient to employ a fymbol for expreffing 
 the whole difference A B or G E between high and low 
 water produced by the a^lion of the Sun. Let it be ex^ 
 prefled by the fymbol S. Alfo let the angular diflance 
 from the fummit, or from the Sun's pface, be x. 
 The elevation mi is = S X cof.* a; — y S- 
 The depreffion « r is = S X fin.' .v — | S. 
 
 639. The fpheroid interfefts the equicapacious fphere 
 in a point fo fituated that S X cof.* x — -f S = 0, that 
 k^ where cof.* « =: -f . This is 54'' 44' from the pole 
 
 of 
 
THEOEY OF THE TIDES. 587 
 
 of the fpheroldj and 35° 16' from its equator, a fituation 
 that has feveral remarkable pliyfical properties. We have 
 already feen (572.) that on this part of the furface the gra- 
 vitation is the fame as if it were really a perfe6t fphere. 
 
 640. The ocean is made to aiTume an eccentric form, 
 not only by the unequal gravitation of its waters to the 
 Sun, but alfo by their much more unequal gravitation to 
 the Moon ; and, although her quantity of matter is very 
 fmall indeed, when compared w'lXh. the Sun, yet being 
 almoft 400 times nearer, the inequality of gravitation is 
 increafed aimoffc 400X400X400 times, and may there- 
 fore produce a fenfible efleft. * We cannot help pre- 
 fuming that it does, becaufe the viciiTitudes of the tides 
 have a moft diilinft reference to tlie pofition of the Moon. 
 Without going over the fame ground again, it is plain 
 that the waters will be accumulated under the Moon, 
 and in the oppofite part of the fpheroid, in the fame jnan- 
 
 ner as they are affeded by the Sun's aftion. 
 
 Therefore 
 
 * The diftance of the Sun being about 392 times that 
 of the Moon, and the quantity of matter in the Sun about 
 3 38000 times that in the Earth, if the quantity of matter in 
 the Moon were equal to that in the Earth, her accumulating 
 force would be 178 times greater than that of the Sun. We 
 fhall fee that it is nearly 2\ times greater. From which we 
 (hould infer that the quantity of matter in the Moon is nearly 
 ^V of that in the Earth. This feems the beft information 
 that we have on -this fubjed. 
 
 4 E ?' 
 
THEOEY OF THE TIDES. 587 
 
 of the fpherold, and 35° 16' from its equator, a fituation 
 that has feverjil remarkable phyfical properties. We have 
 already feen (572.) that on this part of the furface the gra- 
 vitation is the fame as if it were really a perfe£t fphere. 
 
 640. The ocean is made to afTume an eccentric form, 
 not only by the unequal gravitation of its waters to the 
 Sun, but alfo by their much more unequal gravitation to 
 the Moon ; and, although her quantity of matter is very 
 fmall indeed, when compared with the Sun, yet being 
 almoft 400 times nearer, the inequality of gravitation is 
 increafed almoft 400X400X400 times, and may there- 
 fore produce a fenfible eflotl. * We cannot help pre- 
 fuming that it does, becaufe the vicilTitudes of the tides 
 have a moft diftin£l reference to tlie pofition of the Moon. 
 Without going over tlie fame ground again, it is plain 
 that the waters will be accumulated under the Moon, 
 and in the oppofite part of the fpheroid, in the fame juan- 
 
 ner as they are aiFeded by the Sun's aftion. 
 
 Therefore 
 
 * The diftance of the Sun being about 392 times that 
 of the Moon, and the quantity of matter in the Sun about 
 338000 times that in the Earth, if the quantity of matter in 
 the Moon were equal to tliat in the Earth, her accumulating 
 force would be 178 times greater than that of the Sun. We 
 (hall fee that it is nearly 2\ times greater. From which we 
 Jhould infer that the quantity of matter in the Moon is nearly 
 ^V of that in the Earth. This feems the beft informatioB 
 i;bat we have on -this fubjecl. 
 
 4Ej 
 
^68 PHYSICAL ASTRONOMY* 
 
 Therefore let M reprefent the elevation of the pole of 
 the fphcroid above the equicapacious fphere that is pro- 
 duced by the unequal gravitation to the Moon, and let 
 y be the angular diftance of any part of this fpheroid 
 from its pole. We Ihall then have 
 
 The elevation of any point = M x cof/j — -f M. 
 
 The deprelhon ~ M x f^n/- y~-t IM. 
 
 641. In confequence of the fimultaneous gravitation 
 to both luminaries, the ocean muft aflunie a form differ- 
 ing from both of thefe regular fpheroids. It is a figure 
 of dilTicult inyeiligation, but all that we are concerned 
 in may be determined with fulhcient accuracy by means 
 of the following conliderations. 
 
 We have feen that the change of figure induced on 
 the fpheroidal ocean of the revolving globe is nearly the 
 fame as if it were induced on a perfect fphere. Much 
 more fecurely may we fay that the change of figure, in- 
 duced on the ocean already difturbed by the Sun, is the 
 fame that the Moon would have occahoned on the undi- 
 fturbed revolving fpheroid. We may therefore fuppofe,. 
 without any fenfible error, that the change produced in 
 any part of the ocean by the joint action of the two lu- 
 minaries is the fum or the difference of the change^ 
 which they would have produced feparately. 
 
 642. Therefore, fince the poles of both fpheroids 
 are in thofc parts of the ocean which have the Sim and 
 ihe Moon in the zenith, it follows that if >c be the ze- 
 nith 
 
THEORY OF THE TIDES. 589 
 
 mth diftance of the Sun from any place, and y the ze- 
 nith diltance of the Moon, the elevation of the waters 
 above the natural furface of the undifturbed ocean will 
 be S X cof.^ /v — I S 4- M X cof/ >• — -f M. And the 
 deprellion in any place will be S X fin.* x — y S -f 
 M X fin.- y — f M. This may be better exprefied a6 
 follows. 
 
 Elevation = S X cof.^ a; -f- M X cof.' y — | S 4- MT 
 Depreflion = S X fin.' ;v + M x Cm.^ y -— fS -f- M.' 
 
 643. Suppofe the Sun and Moon to be in the fame 
 part of the heavens. The folar and lunar tides will have 
 the fame axes, poles, and equator, the gravitations to 
 each confpiring to produce a great elevation at the com- 
 bined pole, and a great depreflion all round the common 
 equator. The elevation will be -f S -f M, and the de- 
 preflion will be y S + M. Therefore the elevation a- 
 bove the infcribed fphere (or rather the fpheroid fimilar 
 and fimilarly placed with the natural revolving fpheroid) 
 will be ^ + M, 
 
 644. Suppofe the Moon in quadrature in the line 
 E D M (fig. 70.) It is plain that one luminary tends to 
 produce an elevation above the equicapacious fphere 
 A O B C, in the point of the ocean A immediately un- 
 der it, where the other tends to produce a depreflion, 
 and therefore their forces counteract each other. Let 
 ^he Sun be in the line E S. 
 
 The 
 
59» PHYSICAL ASTRONOMY. 
 
 The elevation at S = S — | S + M, = ^ S — 4- M. 
 
 The depreflion at M = S — | S ~^Wy =r I S — | M. 
 The elevation at S above the infcribed fpheroid = S — M. 
 The elevation at M above the fame rr.M — S. 
 
 Hence it is evident that there will be high water at 
 M or at S, v/hen the Moon is in quadrature, according 
 as the accumulating force of the Moon exceeds or falls 
 ^ort of that of the Sun. Now, it is a matter of ob- 
 fervation, th^t when the Moon is in quadrature, it is high 
 water in the open feas under the Moon, and lov/ water 
 under the Sun, or nearly fo. This obfervation confirms 
 the conclufion drawn from the nutation of the Earth's 
 axis, that the difturbing force of the Moon exceeds that 
 of the Sun. This criterion has fome uncertainty, owing 
 to the operation of local circumftanceS) by which it hap- 
 pens that the fummit of the water is never fituated either 
 under the Sun or under the Moon. But even in this cafe, 
 we find that the high "W'^ter is referable to the Moon, 
 and not to the Sun. It is alv/ays fix hours of the day 
 later than the high water at full or change. This cor- 
 refponds with the elongation of the Moon fix hours to 
 the eaftward. The phenomena of the tides fhew fur- 
 ther that, at this time, the waters under the Sun are de- 
 prefTed below the natural furface of the ocean. This 
 ihews that M is more than twice as great as S. 
 
 64^. When the Moon has any other pofition befides 
 thefe two, the place of high water muft be fome inter- 
 mediate 
 
TliEORT OF THE TIDE?. ^ 59 T 
 
 n^cdiate pofition. It mud certainly be ia the great circle 
 paiTing through the fimultaneous places of the two luoii- 
 Haries. As the pUce and time of high and low water^ 
 and the magnitude of the elevation and deprelTion, are 
 the mofl intcrefting phenomena of the tides, they dial) 
 be the principal objects of our attention. 
 
 The place of high water is that where the fam of the 
 elevations produced by both luminaries above the natu- 
 tural furface of the ocean is a maximum. And the 
 place of low water, in the great circle paffing through 
 the Sun and Moon, is tliat where the depreffion below 
 the natural level of the ocean is a maximum. There- 
 fore, in order to have the place of high water we muft 
 
 find where S x cof.^ x + M x cof,' y — i- S + M is a 
 maximum. Or, fince f S -|- M is a conftant quantity, 
 we muft find where S X cof." .v + IvI X cof.* j is a 
 maximum. Now, accounting the tabular fines and co- 
 fijies as fractions of radius, = i, we have 
 Cof.* X = \ + I cof. 2X 
 and Cof.* ^ = -I -j- ^ cof. 2 y. 
 
 For let ABSD {iig. 71.) be a circle, and AS, BD 
 two diameters crofhng each otlier at ,right angles. De- 
 fcribe on the femidiameter CS the fmall circle CtnS/jy 
 having its centre in d. Let H C make any angle x with 
 C S, and let it interfe6l the fmall circle in /:. Draw J/6, 
 S /6, producing S h till it meet the exterior circle in S, 
 and join Ax, C /. Laftly, draw h and / r perpen- 
 dicular to C S. 
 
 S i' is perpendicular to C h^ and C S : C ^ = rad. : 
 
f^i PHYSICAL ASTRONOMY.' 
 
 cof. H C S, and C S : C ^ = R^ : cof/ H C S. The tngie 
 S C J- is evidently =2SCH = S^/6 and A r = 2 C <?. 
 Now if C S be = I ; C r = cof/ 2 at ; A r = i -f- cof, 
 2 X. Therefore C = i -{- i cof. 2 x. In Hke manner 
 Cof.*j); = I -{- i cof. 2_y. 
 
 r^., r n T vS S X cof. 2 ;c M 
 
 Ihereiore we mult have - 4- 4. — -f* 
 
 2 2 ^2 
 
 M X cof. 2 V . 
 
 ~ a maximum, or, negie<fting the conftant 
 
 S M 
 quantities -, — , and the conftant divifor 2, we muft 
 
 have S X cof. 2 .v -|- M X cof. 2 j; a maximum. 
 
 Let A B S D (fig. 71.) be now a great circle of the 
 Earth, paiTmg through thofc points S and M of its fur- 
 face which Jiave the Sun and the Moon in the zenith* 
 Draw the diameter S C A, and crofs it at right angles 
 by B C D. Let S d be to ^ ^ as the accumulating force of 
 the Moon to the accumulating force of the Sun, that is, as 
 M to S, which proportion we fuppofe known. Draw 
 C M in the dire61:ion of the Moon's place. It will cut 
 the fmali circle in fome point m. Join w a. Let H be 
 any point of the furface of the ocean. Draw C H, cut- ' 
 ting the fmall circle in h. Draw the diameter k d //. 
 Draw mt and ax perpendicular to h h', and ay pa- 
 rallel to h h'y and join m d. Alfo draw the chords m k 
 and in h\ 
 
 In this conf!;ru£l:ion, w d and d a reprefent M and 
 S, the angle M C H = j, and S C H = ^. It is farther 
 manifeft that the angle mdh=27nChy =iy, and that 
 nf ^ = M X cof. 2 y* In like manner /?>JS = 2HCS, 
 
THEORY OF THE TIDES. $93 
 
 _ 2.V, and d X := d a X cof. 2 ^', = S X cof. 2 a;. There- 
 fore / a: = S X cof. 2 a; -f M X cof. 2 y. Moreover t x 
 = a Vy and is a maximum when ay is a maximum. This 
 muit happen when ay coincides with a w, that is, when 
 /j d is parallel to a m. 
 
 Hence may be derived the following conftrudliion. 
 
 Let A M S (fig. 72.) be, as before, a great circle 
 w'hofe plane pafles tlirough the Sun and the Moon. Let 
 S and M be thofe points which have the Sun and the 
 Moon in the zenith. Defcribe, as before, the circle C;?zS, 
 cutting C M in ;«. Make S i/ : ^^ = M : S, and join 
 m a. Then, for the place of high water, draw the 
 diameter hdh' parallel to ma^ cutting the circle C f« S 
 in h. Draw C /?> H cutting the furface of the oceari 
 in H arid H\ Then H and H' are the places of high 
 water. Alfo draw C h\ cutting the furface of the o- 
 cean in L and L'. L and L' are the places of low 
 water in this circle. 
 
 For, drav/ing m t and a x perpendicular to h h'^ it 
 ' is plain that / a; =r M X cof. 2 j? -+- S X cof. 2 x. And 
 what was iuft now demonflrated fliews that / a^ is in its 
 maximum (late. Alfo, if the angle L C S = z/, and 
 L C M = z, it is evident that J;v = S X cof. ^ J;?, = S 
 X cof. //iS, =S X cof. 2//CS, = S X cof. 2LCS, 
 = S X cof. 2 // y and in like manner, / i = M x cof. 2 z ; 
 and therefore / at = S x eof. 2 // + M x cof. 2 z, and it 
 is a maximum. 
 
 It is plain, independent of this conftrudion, that the 
 places of high and Jow water are 90° afunder \ for the^ 
 
 4 F ^ two 
 
^p4 PHYSICAL ASTRONOMY. 
 
 two hemifpheres of the ocean mufl be fimllar and equa^is 
 and the equator muil be equidiftant from its poles. 
 
 648. Draw df perpendicular to ;w a. Then, if ^ S 
 be taken to reprefent the whole tide produced by the 
 Moon, that is, the whole difference in the height of high 
 and low water, m a will reprefent the compound tide at 
 H, or the difference between high and low water cor- 
 refponding to that fituation of the place H with refpedl 
 to the Sun and Moon, tnf will be the part of it pro- 
 duced by the Moon and af the pan produced by the 
 Sun. 
 
 For, the elevation at H above the natural level is S X 
 
 cof.*;e — \ + M X cof.^^' — -f, and the depreffion be- 
 low it at L is S X Im.^;^ — ^ -f M X iin/s— -i. But 
 fm.* u = cof/ Xy and fin.* 2 = cof.^ y Therefore the de- 
 preffion at L is S X cof." x — :^ + M x cof.';- — ^. The 
 fum of thefe makes the whole difference bct^veen high 
 and low water, or the whole tide. Therefore the tide is 
 
 = S X 2 COf.^ X I -f- M X 2 COf." ; I. But 2 Cof.' AT, 
 
 — I = cof. 2 Xy and 2 cof.* y — i = cof. 2 y. Therefore 
 the tide = S X cof. 2 ;v + M X cof. 2 y. Now it is 
 plain that mfz=. m d cof. d /«/, and that the angle d mf 
 zzmdhy z= 2mChy = 2 >'. Therefore ?n d x cof. d mf 
 = M X cof. 2 y. In like manner af=:Sx cof. 2 x. 
 
 The point a muft be within or without the circte 
 CmSy according as M is greater or lefs than S, that is, 
 according as the accumulating force of the Moon is greater 
 
 or 
 
THEORY OF THE TIDES. 595 
 
 or lefs than that of the Sun. It appears alfo that, in the 
 jfirfl cafe, H will be nearer to M, and in the fecond cafe, 
 it will be nearer to S. 
 
 Thus have we given a conftru£l:ion that feems to ex- 
 prefs all the phenomena of the tides, as they will occur to 
 a fpeclator placed in the circle pafling through thofe points 
 which have tlie Sun and Moon in the zenith. It marks the 
 diftanee of high water from thofe two places, and therefore, 
 if the luminaries are in the equator, it marks the time that 
 will eiapfe between the paflage of the Sun or Moon over 
 the meridian and the moment of high water. It alfo ex- 
 prefles the whole height of the tide of that day. And, 
 as the point H may be taken without any reference to 
 high water, we fliall then obtain the ftate of the tide for 
 that hour., when it is high water in its proper place H. 
 By confidering this conflru<^ion for the different relative 
 pofitions of the Sun and Moon, we fhall obtain a pretty 
 diftin£l: notion of the feries of phenomena which proceed 
 in regular order during a lunar month. 
 
 649. To obtain thij greater fimplicity in our firft 
 and moft general conclufions, we fhall firft fuppofe both 
 luminaries in the equator. Alfo, abftra6ting our atten- 
 tion from the annual motion of the Sun, we fhall confi- 
 der only the relative motion of the Moon in her fynodi- 
 cal revolution, dating the phenomena as they occur wherL 
 the Moon has got a certain number of degrees away from 
 the Sun ; and we fliall always fuppofe that the watery 
 ^kerjoid h^s attained the form fuited to its equilibrium 
 
 4 F 2 ia 
 
596 PHYSICAL ASTRONOMY. 
 
 ill that fituation of the two luminaries. The coiiclufions 
 will frequently differ much from common obfervation. 
 But we -fliall afterwards find their agreement very fatis- 
 fa£l:ory. The reader is therefore expec^led to go along 
 with the reafoning employed in this difcuflion, although 
 the conciufions may frequently furprife him, being very 
 different from his moft familiar obfervatlons. 
 
 650. - I. At new and full Moon, we fliall have high 
 water at noon, and at midnight, when the Sun and Moon 
 are on the meridian. For in this cafe C M, a ;«, C S, 
 d /', C H, all coincide. 
 
 651. 2. When the Moon is in quadrature in B, 
 the place of high water is alfo in B, under the Moon, 
 and this happens when the Moon is on the meridian. 
 For when M C is perpendicular to C S, the point m 
 coincides with C, am with ^ C, and dh with JC. 
 
 652. 3. While the Moon paffes from a fyzigy to 
 the next quadrature, the place of high water follows the. 
 Moon's place, keeping to the weilward of it. It over- 
 takes the Moon in the quadrature, gets to the eaftward of 
 the Moon (as it is reprefented at M^ H^ by the fiune 
 conllru61:ion), preceding her while flie palies forward to 
 the next fyzigy, in A, where it is overtaken by the 
 Moon's place. For v/hile M is in the quadrant S B, 
 or A D, the ponit h is in the arch S /;/. But when M 
 is in the quadrant B A or D S, ly is without or be- 
 yond 
 
THEORY OF TFIE TIDES. 597 
 
 \on(l the arch S 7;/* (counted eajlivard from 8). Tliere- 
 fore, during the firll and third quarters of the lunation, 
 we have high water after noon or midnight, hut before 
 tiie INIoon's fouthing. But in the fecond and fourth 
 quarters, it happens after the Moon's fouthing. 
 
 653. 4. Since the place of high v/ater coincides 
 with the Moon's place both in fvzigy and the following 
 quadrature, and in the interval is between her and the 
 Sun, it follows that it mud, during the firft and third 
 quarters, be gradually left behind, for a while, and then 
 muft gain on the Moon's place, and overtake her in qua- 
 drature. There muft therefore be a certain greateft dif- 
 tance between the place of the Moon and that of high 
 \vater, a certain maximum of the angle M C H. This 
 happens when H' C S is exa6tly 45°. For then h' d^ is 
 90^, 7/7' a is perpendicular to a S, and the angle a m' d 
 is a maximum. Now a m' d = m' d h\ z=l 2 y\ 
 
 654. When things are in this ftate, the motion of 
 high water, or its feparation from the Sun to the eaft- 
 ward, is equal to the Moon's eafterly motion. There- 
 fore, at new and full Moon, it muft be fiower, and at 
 tlie quadratures it muft be fwifter. Confequently, when 
 the Moon is in the ocSlant, 45° from the Sun, the inter- 
 val between two fucceflive fouthings of the Moon, M-hich 
 is always 24'' 5 o' nearly, muft be equal to the interval of 
 the two concomitant or fuperior high waters, and each 
 
 ide muft occupy \2^ 25', the half of a lunar day. But 
 
 at 
 
s 
 
 55^ PFTYSICAL ASTltONOMT. 
 
 at new or full Moon, the interval between the two fuc- 
 cefTive high waters muft be lefs than 12'* 25', and in the 
 quadratures it muft be more. 
 
 655. The tide day muft be equal to the lunar day 
 only when the high water is in the o£lants. It muft be 
 /liortet at new and full Moon, and while the Moon is 
 pafling from the fecond 0(9:ant to the third, and from th^ 
 fourth to the firft. And it muft exceed a lunar day 
 while the Moon pafles from the firft o6lant to the fecond, 
 and from the third to the fourth. The tide day is al- 
 ways greater than a folar day, or twenty-four hours. 
 For, while the Sun makes one round of the Earth, and 
 is again on the meridian, the Moon has got about 13* 
 eaft of him, or S M is nearly 13°, and S H is nearly 
 9°, fo that the Sun muft pafs the meridian about 35 or 
 36 minutes before it is high water. Such is the law of 
 the daily retardation called the priming or lagging of the 
 tides. At new and full Moon it is nearly 35', and at 
 the quadratures it is 85', fo that the tide day at new and 
 full Moon is 24^ 35', and in the quadratures it is 25'' 25" 
 nearly. 
 
 Our conftru<^ion gives us the means of afcertaining 
 this circumftance of the tides, or interval between two 
 fucceeding full feas, and it may be thus exprefled. 
 
 656. The fy nodical motion of the Moon is to the fy- 
 Tiodical motion of the high water as ma to T?iy. For, 
 take a point u very near to m. Draw u a and u d, and 
 
 draw 
 
THEORY OF TRE TlDEti 59<J> 
 
 draw d i parallel to a Uy and with the centte ay and 
 diftance a tiy defcribe the arch u Vy which may be con- 
 Cdered as a ftraighfe line perpendicular to ?« a. Then 
 u m and i b are refpe6tively equal to the motions of M 
 and H (though they fubtend twice the angles). The 
 angles ait ^', diim are equal, being right angles. There- 
 fore muv=z an dy -zz am dy and the triangles m u Vy 
 d tnfy are fmiilar, and tlie angles u a m, i d h are equal^ 
 and therefore 
 
 uv',ihz=.ma\hdy z=. ma '.m d 
 urn :uvz= 711 d". mf 
 
 therefore u m\ih zz. m a\ mf. 
 
 When m coincides with S, that is, at new or full 
 Moon, m a coincides with S a, and mf with S d. But 
 when m coincides with C, that is, in the quadratures^ 
 i« a coincides with C ay and mf with C d, 
 
 657. Hence it is eafy to fee that the retardation of 
 the tides at new and full Moon is to the retardation in 
 the quadratures as C ^ to S ^, that is, as M -f S to 
 M — S. 
 
 When the high water is in the oQant, ma \^ perpen- 
 dicular to S tty and therefore a and f coincide, and the 
 fynodical motion of the Moon and of high water are th^ 
 fame, as has been already obferved. 
 
 Let us now confider the elevations of the water, and 
 the magnitude of the tide, and its gradual variation in 
 the courfe of a lunation. This is reprefented by the line 
 
6oO PHYSICAL ASTRONOMY. 
 
 658. This ferles of changes Is very perceptible 112 
 our conftruftion. At new and full Moon, ;« a coincides' 
 with S a, and in the quadratures, it coincides with C a. 
 Therefore, the fpring-tide is to the neap-tide as Sa to 
 C ay that is, as M -f S to M — S. From new or full 
 Moon the tide gradually leiTehs to the time of the qua- 
 drature. We aifo fee that the Sun contributes to the 
 elevation by the part a f, till the high water is in the 
 oftants, for the point f lies between m and a. After 
 this, the aclion of the Sun diminifhes the elevation, the 
 point / then lying beyond a, 
 
 659. The momentary change in the height of the 
 whole tide, that is, in the difference between the high 
 and low water, is proportional to the fine of twice the 
 arch M H. It is meafured by df in our confl;ru6lion. 
 For, let m u be a given arch of the Moon's fynodical mo- 
 tion, fuch as a degree. Then m v is the difference be- 
 tween the tides »/ a and u a, correfponding to the con- 
 ftant arch of the Moon's momentary elongation from th^ , 
 Sun. The fimilarity of the triangles muv and m df 
 gives us mu :mv =1 m d : df. Now m u and m d are 
 conftant. Therefore m v is proportional to dfy and 
 m d : df~ rad. : fin. d inf, = fin. mdhy = fin. 2 M C H. 
 
 Hence it follows that the dimihution of the tides is 
 moft rapid when the high water is in the ocftants. This 
 will be found to be the difference between the twelfth 
 and thirteenth tides, counted from new or full Moon, 
 and between the feventh and eighth tides after the qua- 
 dratures. 
 
THEORY OF THE TIDES. 6ol 
 
 draturcs. If m u be taken = \ the Moon's daily elon- 
 gation from the Sun, which is 6° 30' nearly, the rule 
 will give, with fuilicient accuracy, \ the difference be- 
 tween the two fuperior or the two inferior tides imme- 
 diately fucceeding. It docs not give the difference be- 
 tween the two immediately fucceeding tides, becaufe 
 they are alternately greater and lefTer, as will appear after- 
 wards. 
 
 660, Having thus given a reprefentation to the eye 
 of the various circumftances of thefe phenomena in this 
 fimple cafe, it would be proper to fhew how all the dif- 
 ferent quantities fpoken of may be computed arithmeti- 
 cally. The firaplefl method for this, though perhaps not 
 the moft elegant, feems to be the following. 
 
 In the triangle m d a, the two fides m d and d a are 
 given, and the contained angle mday when the propor- 
 tion of the forces M and S, and the Moon's elongation 
 MCS are given. Let this angle m d a he called a. 
 Then make M + S : M — S ri tan., a : tan. i;. Then 
 
 a — b - a 4- If 
 V = , and X =z — . 
 
 For M + S :M'-Szz7nd + da : md — da, =: 
 
 i?2 a d 4- a m d mad — at?! d 2 a* + 2 V 
 
 tan. : tan. = tan. ^ 
 
 222 
 
 tan. 
 
 2 X — — 2 V 
 
 y = tan. X -^ y : tan. x — y = tan. a : tan. k 
 
 Now X -\- y -\- X — y =z 2 x and ;v -{- y — at— ^=:2j^. 
 
 a 4- 1 
 Therefore a -^ b =z 2 x and a — h = 2yy and x ~ - — , 
 
 •2 
 
 and V = » 
 
 ^ 2 
 
 4 G C6u, 
 
6oi PHYSICAL ASTRONOMY. 
 
 66i. It is of peculiar importance to know the greatefl 
 feparation of the high water from the Moon. This hap- 
 pens when the high water is in the ocSlant. In this fi- 
 
 tuation it is plain that ?;/ d ida, that is, M : S, = rad. : 
 
 g 
 {in, d/n' a, rr rad. :fm. 2/, and therefore fin.2y'=^. 
 
 Hence 2/ and / are found. 
 
 662. It is manifeft that the applicability of this con- 
 ilruftion to the explanation of the phenomena of the tides 
 depends chiefly on the proportion of S ^ to da, that is, 
 the proportion of the accumulating force of the Moon to 
 that of the Sun. This conftitutes the fpecies of the 
 triangle m d a, on v/hich every quantity depends. The 
 queftion now is. What is this proportion ? Did we 
 know the quantity of matter in the Moon, it would be 
 decided in a minute. The only obfervation that can give 
 us any information on this fubjeft is the nutation of the 
 Earth's axis. This gives at once the proportion of the 
 difturbing forces. But the quantities obferved, the de- 
 viation of the Earth's axis from its uniform conical mo- 
 tion round the pole of the ecliptic, and the equation of 
 the preceffion of the equino(5lial points, are much too 
 fmall for giving us any precife knowledge of this ratio. 
 
 Fortunately, the tides themfelves, by the modification 
 which their phenomena receive from the comparative 
 magnitude of the forces in queftion, give us means of 
 difcovering the ratio of S to M. The moft obvious cir- 
 cumftance of this nature is the magnitude of the fpring 
 and neap-tides. Accordingly, this wa§ employed by 
 
 Newton 
 
THEORY OF THE TIDES. 6o2 
 
 Newton in his theory of the tides. He coUecl'^vl a num- 
 ber of obfervations made at Briflol, and at Plymouth, 
 and, ftating the fpring-tide to the neap-tide as M -|- S 
 to M — S, he faid that the force of the Moon in raifing 
 the tide is to that of the Sun nearly as 4-5 to i. But it 
 was foon perceived that this was a very tincertain me- 
 thod. For there are fcarcely any two places where the 
 proportion between the fpring-tide and the neap-tide is 
 the fame, even though the places be very near each o- 
 ther. This extreme difcrepancy, while the proportion 
 was obferved to be invariable for any individual place, 
 fhewed that it was not the theory that was in fault, but 
 that the local circumftances of fituation were fuch as af- 
 fected very differently tides of different magnitudes, and 
 thus changed their proportion. It was not till the noble 
 collection of obfervations vi-as made at Breft and PvOche- 
 fort that the philofopher could afibrt and combine the 
 immenfe variety of heights and times of the tides, fo a«; 
 to throw them into claiTes to be compared with the af- 
 pe£ts of the Sun and Moon according to the Newtonian 
 theory. M, Caffini, and, after him, M.. Daniel Ber- 
 noulli, made this comparifon with great care and difcern- 
 ment ; and on the authority of this comparifon, M. Ber- 
 noulli has founded the theory and explanation contained 
 in his excellent Difiertation on the tides, which fhared 
 with M'Laurin and Eulcr the prize given by the Acade- 
 my of Paris in 1 740. 
 
 M. Bernoulli employs feveral circumftances of the 
 
 tides for afcertaining the ratio of M to S. lie cmplo)'^ 
 
 4 G- 2 thg 
 
6q4 physical astronomy. 
 
 the law of the retardation of the tides. This has great 
 advantages over the method employed by Newton. What- 
 ever are the obftrudiions or modifications of the tides, 
 they will operate equally, or nearly fo, on two tides that 
 are equal, or nearly equal. This is the cafe with two fuc- 
 ceeding tides of the fame kind. 
 
 . The Moon's mean motion from the Sun, in time, is 
 about 50^ minutes in a day. The fmalleft retardation, 
 in the vicinity of new and full Moon, i& nearly 35', 
 wanting 15^ of the Moon's retardation. Therefore, by 
 art. ^56, 
 
 M : S =; 35 : 15^, =: 5 : 2f nearly. 
 
 The longefl tide-day about the quadratures is 25 '^ 2^\ 
 exceeding a folar day S^'y and a lunar day 34I. There- 
 fore 
 
 M : S = 85 : 34i, =5 : 2yV nearly. 
 
 The proportion of M to S may alfo be inferred by a 
 diredl comparifon of the tid^e-day at new Moon and in, 
 the quadratures. 
 
 35 : 85 = M — S : M 4- S. Therefore 
 
 It may alfo be difcovered by obfendng the greateft 
 Reparation of the place of high water from that of the 
 Moon, or the elongation of the IVIoon when the tide-day 
 and the lunar day are equal. In this cafe y is obfcrvod 
 
 to be nearly 12° 30'. Therefore -^ = fin. 25% and 
 M : S = 5 : 27 nearly. 
 
 Thus it appears tlrat all thefe methods give nearly 
 
THEORY OF THE TIDES. 60J 
 
 the fame refult, and that we may adopt 5 to a as the 
 ratio of the two difturbing forces. This agrees extremely 
 well with the phenomena of nutation and preceffion. 
 
 Inftead of inferring the proportion of M to S from 
 the quantity of matter in the Moon, deduced from the 
 phenomena of nutation, as is affected by D'Alembert 
 and La Place, I am more difpofed to infer the mafs of 
 the Moon from this determination of M : S, confirmed 
 by fo many coincidences of different phenomena. Tak- 
 ing 5:2,13 as the mean of thofe determinations, and 
 employing the analogy in § 465, we obtain for the quan^ 
 tity of matter in tlie Moon nearly yV> the Earth be- 
 ing I. 
 
 If the forces of the two luminaries were equal, there 
 would be no high and low water in the day of quadra- 
 ture. There would be an elevation above the infcribed 
 
 fpheroid of j- M -f S all round the circumference of the 
 circle palTmg through the Sun and Moon, forming the 
 ocean intq an oblate fpheroid. 
 
 663. Since the gravitation to the Sun alone produces 
 %n elevation of 24^ inches, the gravitation to the Moon 
 will raife the waters 58 inches; the fpring-tide will be 
 24a +58, or 82i inches, and the neap-tide 33I Inches. 
 
 664. The proportion now adopted mull be confi- 
 dered as that correfponding to the mean intenflty of the 
 accumulating forces. But this proportion is by no means 
 conftant, by reafon of the variation m the diftanccs of 
 
 ^hp 
 
6g6 physical ASTRONOMt. 
 
 the 'luminaries. Calling the Sun's mean diflance i coo, 
 It is 983 in January and 1017 in July. The Moon's 
 mean diflance being 1000, {he is at the diflance 1055 
 'when in apogeo, and 945 when in perigeo. The a6lion 
 of the luminaries in producing a change of figure varies 
 in the inverfe triplicate ratio of their diflances (519.) 
 Therefore, if 2 and 5 are taken for the mean diflurb- 
 'ing forces of the Sun and Moon, we have the following 
 meafures of thofe forces. 
 
 Sun, Moo?t. 
 
 Apogean 1,901 4>258 
 
 Mean 2, — - 5,- — 
 
 . Perigean 2,105 $^9"^^ 
 
 Hence we fee that M : S may vary from 5>925 : 1,901 
 to 4,258 : 2,105, that is, nearly from 6:2 to 4:2. 
 The general exprefTion of the diflurbing force of the 
 
 Moon will be M ;= |- S X -^ X - where D and d ex- 
 
 prefs the mean diflances of the Sun and Moon, and A 
 and ^ any other fimultaneous diflances. 
 
 The folar force does not greatly vary, and need not 
 be much attended to in our computations fbr the tides. 
 But the change in the lunar a£lIon mufl not be negledl- 
 ed, as this greatly afFedls both the time and tjie height 
 of the tide. 
 
 665. Firfl, as to the times. 
 
 I. The tide-day following fpring-tlde is 24'' 27^' 
 when the Moon Is In perigeo, and 24^^ 33' when fhe is 
 in apogeo. 
 
THEORY OF THE TIDES. 6of 
 
 2. The tide-day following neap-tide is 25 •» 15' in 
 the firft cafe, and 25 '^ 40' in the fecond. 
 
 3 . The greateft interval between the Moon's fouthing 
 and high water (which happens in the otftants) is 39' 
 when the Moon is in perlgeo, and 61' when fhe is in 
 apogeo, ;; being 9° 45' and 15*^ 15'. 
 
 666. The height of the tide is ftill more affeded 
 by the Moon's change of diftance. 
 
 If the Moon is in perigeo, when new or full, the 
 fpring-tide will be eight feet, inftead of the mean fpring- 
 tide of feven feet. The very next fpring-tide will be 
 no more than fix feet, becaufe the Moon is then in apo- 
 geo. The neap-tides, which happen between thefe very 
 unequal tides, will be regular, the Moon being then in 
 quadrature, at her mean diftance. 
 
 But if the Moon change at her mean diftance, the 
 fpring-tide will be regular, but one neap-tide will be four 
 feet, and another only two feet. 
 
 We fee therefore that the regular monthly feries of 
 heights and times correfponding to our conftfudlion can 
 never be obferved, becaufe in the very fame, or nearly 
 the fame period, the Moon makes all the changes of dif- 
 tance which produce the effects above mentioned. As 
 the effect produced by the fame change of the Moon's 
 diftance is different according to the ftate of the tide 
 which it afFe£ls, it is by no means eafy to apply the e- 
 quation arifing from this caufc, 
 
 66'j. 
 
6qS physical astronomy. 
 
 667. As a fort of fynopfis of the whole of this de- 
 fcriptlon of the monthly feries of tides, the following 
 Table by D. Bernoulli will be of fomc* ufe. The firfl 
 column contains the Moon*s elongation S M (eaftward) 
 from the Sun, or from the point oppofite to the Sun, 
 in degrees. The fecond column contains the minutes 
 of folar time that the moment of high water precedes 
 or follows the Moon's fouthing. This correfponds to 
 the arch H M. The third column gives the arch S H, 
 or nearly the hour and minute of the day at the time 
 of high water ; and the fourth column contains the 
 height of the tide, as exprefled by the line tn a, the 
 fpace S^ being divided into 1000 parts, as the height 
 of a fpring-tide. Note that the elongation is fuppofed 
 to be that of the Moon at the time of her fouthing. 
 
 TABLE 
 
tHEORY OF THE TIDES. 
 
 609 
 
 
 TABLE I. 
 
 
 8M 
 
 HM 
 
 Hour. 
 
 ;/; a 
 
 
 Minutes. 
 
 
 
 
 
 — 
 
 — . — - 
 
 1000 
 
 10 
 
 20 
 
 22 '6 
 
 -.281 
 
 -.58 
 
 987 
 949 
 
 30 
 
 3^v '^ 
 
 i.28t 
 
 887 
 
 40 
 50 
 
 40 ^ 
 
 45 s 
 
 2.35 
 
 806 
 
 7^5 
 
 60 
 
 70 
 
 46"- ^ 
 
 40i 1 
 
 3-i3t 
 3-591 
 
 6ro 
 518- 
 
 80 
 
 'S §• 
 
 4.55 
 
 453 
 
 90 
 
 cq 
 
 6.— 
 
 429 
 
 100 
 
 > ^5 
 
 7- 5 
 
 453 
 
 no 
 
 S . 4oi 
 
 8. t 
 
 518 
 
 120 
 
 g- 46t 
 
 8.46t 
 
 610 
 
 130 
 
 § 45 
 
 9,25 
 
 715 
 
 140 
 
 
 
 9 40 
 
 10. — 
 
 806 
 
 150 
 
 "^^ CJ I ^ 
 
 10.31 
 
 887 
 
 160 
 
 g 22 
 
 II. 2 
 
 949 
 
 170 
 180 
 
 1 ''' 
 
 II. 3 1 
 12. — 
 
 987. 
 
 iOOO 
 
 668. It is proper here to notice a circumftance, of 
 very general obfervation, and which appears inconfiftent 
 with our conftruclion, which dates the high water of 
 neap-tides to happen when the Moon is on the meri- 
 dian. This mud nuke the high water of neap-tides fix 
 
 4 H hours 
 
tHEORy OF TBE TIDES. 
 
 609 
 
 
 T 
 
 ABL 
 
 E I. 
 
 
 SM 1 
 
 HM 
 
 Hour. 
 
 ;;; a 
 
 
 Minutes. 
 
 
 
 
 
 — 
 
 
 — . 
 
 1000 
 
 10 
 
 20 
 
 Ill 
 
 22 
 
 '6 
 
 -.281 
 
 -.58 
 
 987 
 949 
 
 30 
 
 3^4- 
 
 
 
 i.28t 
 
 S87 
 
 40 
 50 
 
 40 
 45 
 
 
 2. 
 
 2.35 
 
 806 
 7^5 
 
 ^0 
 
 70 
 
 46I- 
 
 404- 
 
 
 
 3.i3t 
 3'59i 
 
 (5io 
 518- 
 
 80 
 
 25 
 
 5' 
 
 4.55 
 
 453 
 
 90 
 
 — 
 
 CJ5 
 
 6.— 
 
 429 
 
 100 
 
 > 
 
 25 
 
 7- 5 
 
 453 
 
 no 
 
 
 4OT 
 
 8. t 
 
 518 
 
 120 
 
 CD 
 
 4^r 
 
 8.46t 
 
 610 
 
 130 
 
 140 
 
 
 
 
 45 
 40 
 
 9.25 
 10. — 
 
 715 
 806^ 
 
 150 
 
 
 31^^ 
 
 10.31 
 
 887 
 
 160 
 
 
 
 22 
 
 II. 2 
 
 949 
 
 170 
 180 
 
 0^ 
 
 11^ 
 
 II. 3 1 
 12. — 
 
 987. 
 
 1000 
 
 668. It is proper here to notice a circumftance, of 
 very general obfervation, and which appears inconfiftent 
 with our conftruction, which dates the high water of 
 neap-tides to happen when the Moon is on the meri- 
 dian. This ma(t make the high water of neap-tides fix 
 
 4 H hours 
 
^-XS PHYSICAL ASTRONOMY. 
 
 hours later than the high water of fpring-tideSj fuppofing 
 that to happen when the Sun and Moon are on the me- 
 ridian. But it is univerfally obferved that the high water 
 of tides in quadrature is only about five hours and ten 
 6t tweh'e minutes later than that of the tides in fyzigy. 
 
 This is owing to our not attending to another circum- 
 flance, namely, that the high water which happens in 
 fyzigy, and in quadrature, is not the high water of fpring 
 and of neap-tides, but the third before them. They 
 correfpond to a pofition of the Moon 19° weft ward of 
 the fyzigy or quadrature, as will be more particularly 
 noticed afterwards. At thefe times, the points of high 
 water are 13-1 weft of the fyzigy, and 29 weft of the 
 quadrature, as appears by our conftruclion. The lunar 
 hours correfponding to the interval are exactly 511 02', 
 which is n6^arly 5° 12' folar hours. 
 
 669. Hitherto we have confidered the phenomena 
 of the tides in their moft fimple ftate, by ftating the 
 Moon and the Sun in the equator. Yet this can never 
 happen. That is, we can never fee a monthly feries of 
 tides nearly correfponding with this fituation of the lumi- 
 naries. In the courfe of one month, the Sun may conti- 
 nue within fix degrees of the equator, but the Moon will 
 deviate from it, from 18 to 28 or 30 degrees. This will 
 greatly afFe61: the height of the tides, caufing them to 
 deviate from the feries exprefled by our conftrudion. It 
 Hill more afFe£ts the time, particularly of low water. 
 The phenomena depend primarily on the zenith diftances 
 
 of 
 
tkeoryofthetid.es. 6li 
 
 of the luminaries, and, wlien thefe are known, are accu- 
 rately^ expreiTed by the conflrucSlion. But the fe zenith 
 diftances depend both on the place of the luminaries in 
 the heavens, and on the latitude of tlie obferver. It is 
 difficult to point out the train of phenomena as they oc- 
 cur in any one place, becaufe the figure afllimed by the 
 waters, although its depth be eafily afcertained in any 
 fingle point, and for any one moment, is too complicated 
 to be explained by any general d<?fcription. Jt is not 
 an oblong elliptical fpheroid, fonned by revolution, ex- 
 cept in the very moment of new or full Moon. In other 
 relative fituations of the Sun and Moon, the ocean will 
 not have any feClion that is circular Its pol-es, and tlie 
 pofition^of its equator, are eafily determined. But tliis 
 equatoreal fection is not a circle, but approaches to an 
 elliptical form, and, in fome cafes, is an exa61: ellipfe. 
 The longer axis of this oval is in the plane parting 
 through the Sun and Moon, and its extremities are in 
 the points of low water for this circle, ^s determined by 
 our conflruclion. Its fliorter axis pailes through the 
 centre of the Earth, at right angles to the other, and its 
 extremities are the points of the loiuejl low ivaier. In 
 thefe tv/o points, the depreffion below the natural level 
 of the ocean is always the fame, namely, the fum of the 
 greateft depreffion produced by each luminary. It is 
 fnbjecled therefore only to the changes arifing from the 
 changes of diflance of the Sun and Moon. 
 
 Thus it appears that the furface of the ocean has ge- 
 nerally four poles, two of which are prolate or prutubc- 
 4 H 2 rant> 
 
(h2 PHYSICAL ASTRONOMY. 
 
 rant, and two of them are compreffed. This is mod re^ 
 markably the cafe when the Moon is in quadrature, and 
 there is then a ridge all round that fecllon which has the 
 Suii and Moon in its plane. The fccliion through the 
 four poles, upper and lower, is the place of high water 
 all over the Earth, and the fe6tion perpendicular to the 
 axis of this is the place of low water in all parts of the 
 Earth. 
 
 Hence it follows that when the luminaries are in the 
 plane of the Earth's equator, the two deprefied poles of 
 the watery fpheroid coincide with the poles of the Earth ; 
 and what we have faid of the times of high and low 
 water, and the other ilates of the tide, are exa^t in their 
 application. But the heights of the tides are diniiniflied 
 as we recede from the Earth's equator, in the proportion 
 of radius to the cciine of the latitude. In ail oilier fitua^ 
 tions of the Sun and Moon, the phenomena vary exceed- 
 ingly, and cannot eafdy be fhewn in a regular train. 
 The pofition of the high water fe^lion is often much in- 
 clined to the terreflrial meridians, fo that the interval be- 
 tween the traniit of the Moon and the tranfit of this fee-' 
 tion acrofs the meridian of places in the,iame meridian is 
 often very different. Thus, on midfummer day, fuppofe 
 the Moon in her laft quadrature, and in the node, there- 
 fore in the equator. The ridge which forms high v/ater 
 lies lb oblique to the meridians, that when the Moon ar- 
 rives at the meridian of London, the ridge of h^igh water 
 has pafTed London about two hours, and is now on the 
 north coaft of America. Hence it liappens that we have 
 
 no 
 
THECXRY OF THE TTDE3. 6j^ 
 
 210 fatisfa£lovy account of tlic times of high M^ater In dif- 
 ferent places, even though \vc fhould learn it for a par- 
 ticular day. The only way of forming a good guefs of 
 tlie ftate of the tides is to have a tcrreilrial globe before 
 us, and having marked the places of the luminaries, to 
 lap a tape round the globe, paihng thrcmgh thofe points, 
 and then to mark the place of high water on that line, 
 and crofs it with an arch at right angles. This is the 
 line of high watet. Or, a circular hcop may be made, 
 crofled by one femicircle. Place the circle fo as to pafs 
 through the places of the Sun and Moon, fetting the in- 
 terfeclion with the femicircle on the calculated place of 
 high water. The femicircle is now the line of high wa- 
 ter, and if this armilla be held in its prefent pofition, 
 while the globe turns once round within it, the fuccef- 
 fion of tide, or the regular hour of high water for every 
 part of the Earth v/ill then be feen, not very diftant from 
 the truth. 
 
 At prefent, in our endeavour to point out the chief 
 modifications of the tides w hich proceed from the decli- 
 nation of the luminaries, or the latitude of the place of 
 obfervation, we muft content ourfelves with an approxi- 
 mation, which {hall not r^ very far from the truth. It 
 will be fufiiciently exact, if we attend only to the Moon. 
 The eitecls of declination are not much affefted by the 
 Sun, becaufe the difTercnce bet^*Ten the declination of the 
 Moon and that of the pole of the ocean can never exceed 
 f.x or (even degrees. When the great circle paffnT^ 
 through the Sun and Moon is much inclined to the equa- 
 tor 
 
5r4 PHYSICAL ASTRONOMY. 
 
 tor (it may even be perpendicular to it), the luminaries 
 are very near each other, and the Moon's place hardly 
 deviates from the line of high water. At prefent we 
 fliall confider the lunar tide only. 
 
 670. Let NQSE (fig. 73.) reprefent the terra- 
 queous globe, N S being the axis, E O the equator, and 
 O the centre. Let the Moon be in the direction O M, 
 having the declination B Q. Let D be any point on 
 the furface of the Earth, and C D L its parallel of lati- 
 tude, and N D S its meridian. Let B' F b' f be the 
 elliptical furface of the ocean, having its poles B' and b' 
 m the line O M. Let /OF be its equator. 
 
 As the point D is carried along the parallel C D L, 
 it will pafs In fucceflion through all the Hates of the 
 tide, having high water when it is in C, and in L, and 
 low water when it gets into the interfeftion d of its pa- 
 rallel C L with the equator fd F of the watery fphe- 
 rold. Draw the meridian '^ dG through this interfec- 
 tion, cutting the terreftriai equator in G. Then the 
 arch O G, converted into lunar hours, will give the du- 
 ration of ebb of the fuperlor tide, and^GE rs the time 
 of the fubfequent flood of the inferior tide. It is evi- 
 dent that thefe are unequal, and that the whole tide 
 GQG, confiding of a flood-tide G Q and ebb-tide Q G, 
 while the Moon is above the horizon (which we called the 
 ftiperior tide), exceeds the duration of the whole inferior 
 zide G E G by four times G O (reckoned in lunar hours.) 
 
 If th^ fpheroid be fuppofed to touch the fphere, 
 
THEdRY OF THE TlDE^. 6l^ 
 
 111 / and F, then C c' is the height of the tide. At L, 
 the height of the tide is L L', and if the concentric 
 circle L,' q be defcribed, C q is the difference between 
 the fuperior and inferior tides. 
 
 From this con[lru6>ion we learn, in general, that 
 when the Moon has no declination, the duration of the 
 fup?rior and inferior tides of one day are equal, over all 
 the Earth. 
 
 671. 2. If the Moon has declination, the fuperior 
 tide will be of longer or of ihorter duration than the in- 
 ferior tide, according as the Moon's declination BO, 
 and the latitude C Q of the place of obfervation are of 
 the fame or of diirerent denominations. 
 
 672. 3. When the Moon's declination is equal to 
 the colatitude of the place of obfervation, or exceeds it, 
 that is, if B Q is equal to N 0, or exceeds it, there will be 
 only a fuperior or inferior tide in the courfe of a lunar 
 day. For in this cafe, the parallel of the place of obferva- 
 tion will pafs through fy or between N and y^ as i^ wz. ' 
 
 673. 4. The fine of the arch G O is = tan. lat. X 
 tan. declin. For rad. : cot. ^ O G =r tan. d G : fm. G O, 
 and fm. G O = tan. dG X cot. ^ O G. Now ^ G is tlie 
 latitude, and ^ O G is the coded. 
 
 674. The heights of the tides are affected in the 
 fame way by the declination of the Moon, and by the la- 
 
 y titude 
 
6l6 PIIYS1CA3L ASTRONOMY. 
 
 titudc of the place cf obfervation. The height of the 
 fuperior tide exceeds that of the inferior, if the Moon's 
 deciination is of the fame denomination with the latitude 
 of the place, and vice versa. Jt often happens that the 
 reverfe of tliis is uniformly obferved. Thus, at the Nore, 
 in the entry to the river Thames, the Inferior tide is 
 greater than the fuperior, when the Moon has north de- 
 clination, and vice versa. But this happens becaufe the 
 tide at the Nore is only the derivation of the great tide 
 whidi comes round tlie north of Scotland, ranges along 
 the eaftern coafhs of Britain, and the high water of a 
 Aiperior tide arrives at tlie Nore, while that of an inferior 
 tide is formed at the Orkney iflands, the Moon being un- 
 der the hciizon. 
 
 675. The height. of the tide in any place, occafioned 
 by tlie action of a fingle luminary, is as the fquare of the 
 cofine of the zenith or nadir dillance of that luminary. 
 Hence we derive the following conftru£l:ion, which will 
 exprefs. all the modifications of the luriar tide produced 
 by declination or latitude. It will not be far from the 
 truth, even for the compound tide, and it is perfecSlly ex- 
 act in the cafe cf fpring or neap-tides. 
 
 "With a radius C Q (fig. 74.) taken as the meafure 
 of the whole .elevation of a lunar tide, defcribe the circle 
 E P O /, to reprefent a terreftrial meridian, where P 
 and p are the poles, and E O the equator. Bife6l C P 
 in O, and round O defcribe the circle PBCD.. Let 
 M be that point of the meridian which has the Moon in 
 
 the 
 
Theory of the tides. 617 
 
 ihe zenith, and let Z be. the place of obfervation. Draw 
 the diameter Z C N, cutting the fmall circle in B, and 
 MCm cutting it in A. Draw A I parallel to E Q. 
 Draw the diameter B O D of the inner circle^ and draw 
 IK, GH, and AF perpendicular to B D. Laftly, 
 draw I D, I B, A D, A B, and C I M'^ cutting the me- 
 ridian in M'. 
 
 After half a diurnal revolution, the Moon comes into 
 the meridian at M', .and the angle M' C N is her diftance 
 from the nadir of the obferver. Tlie angle I C B is the 
 fupplemcnt of I CN, and is alfo the fupplcmeiit of IDB, 
 the oppofite angle of a quadrilateral in a circle. There- 
 fore I D B is equal to the Moon's nadir diftance. Alfo 
 A D B, being equal to A C B, is equal to the Moon's 
 zenith diftance. Therefore, accouiiting D B as the ra- 
 dius of the tables, D F and D K are as the fquares of 
 the cofmes of the Moon's zenith and nadir diftances ; 
 and fmce P C, or D B, was taken as the meafure of the 
 whole lunar tide, D F will be the elevation of high water 
 at the fituation Z of the obferver, when the Moon is 
 above his horizon, and D K is the height of the fubfe- 
 quent tide, when the Moon is under his horizon, or, 
 more accurately, it is the height of the tide feen at the 
 fam.e moment with D F, by a fpeftator at 2' in the fame 
 meridian and parallel. (For the Juhfequent tide, though 
 only twelve hours after, will be a little greater or lefs, 
 according as they are on the increafe or decreafe). D F, 
 then, and D K, are proportional to the heights of the 
 fuperior and inferior tides of that day. Moreover, as A I 
 4 I i« 
 
6l8 PHYSICAL ASTRONOMY, 
 
 is bife^led in G, F K is blfecled In H, and D H is tfic 
 arithmetical mean between the heights of the fuperior 
 and inferior tides. Accounting O C as the radius of the 
 tables, A G is the fine of the arch A C, which meafures 
 twice the angle M C Q, the Moon's decUnatiGn. O G 
 is the cofine of twice the Moon's declination. Alfo the 
 angle B O G is equal to twice the angle B C O, the la- 
 titude of the obferver. Therefore O H = cof. 2 decL 
 X cof. 2 lat., and D H = D O + O H, = M X 
 
 I 4- cof. 2 decl. d X cof. 2 lat. ^, . , ^ , 
 
 , 7^j-ij3 valux; of the me- 
 
 2 
 
 dium tide will be found of continual ufe* 
 
 This conftrucSlicn gives us very difilnci conceptions 
 of all the modifications of tlie height of a lunar tide^ 
 proceeding from the various declinations of the Moon,, 
 and the pofition of the obferver j and tlie height of the 
 compound tide may be had by repeating the conltru^llon 
 for the Sun, fubftitutlng the declination of the Sun for 
 that of the Moon, and S for M in the lail formula^ The 
 two elevations being added together, and i M •+■ S taken 
 from the fum, we have the height required. If it is a 
 fpring-tide that we calculate for, thess is fcarcely any 
 occafion for t^^o operations, becaafe the Sun cannot then 
 be more than fix degrees from the Moon, and tlie pole 
 of the fplieroid will ahiioil coincide with the Moon's 
 place. We may now draw fome inferences from this 
 reprefentation. 
 
 676. I. The greateft tides Ipppen when the Moon. 
 
 is 
 
THEORY OF THE TIDES. 619 
 
 i« in the zenith or nadir of the place of obfervation. 
 For as M approaches to Z, A and I approach to B and 
 D, and when they coincide, F coincides with B, and 
 tlie height of the fuperior tide is then = M. The me- 
 dium tide however diminiflies by this change, becaufe G 
 comes nearer to O, and confequently H comes alfo 
 nearer to O, and D H is diminillied. 
 
 If, on the other hand, the place of obfervation be 
 changed, Z approaching to M, the fuperior, inferior and 
 medium tides are all increafed. For in fuch cafe, D fe- 
 parates from I, and D K, D H, and D F are all en- 
 larged. 
 
 677. 2. If tlie Moon be in the equator, the fupe- 
 rior ard inferior tide^ are equal, and =: M k cof.^ lat. 
 For then A 2nd I coincide with C ; and F and K coa- 
 lefce in /- and D / = D B x cof/ B D C, = D B X cof.^' 
 
 zco. 
 
 673. 3. If the place of obfervation be in the equa- 
 tcr, the fuperior and inferior tides are equal every where, 
 and are = M x cof.*, declin. d . For B then coincides 
 with C ; the points F and K coincide with G 5 and P G 
 = P C X ccf.* C P A, = M X cof/ M C Q. 
 
 679. 4. The fuperior tides are greater or lefs than 
 
 the inferior tides, according as Z and M are on the fame 
 
 or on oppofite fides of the equator. For, by taking Q Z' 
 
 on the other Cde of the equator, equal to O Z, and 
 
 4 I 2 drawing 
 
^20 PHYSICAL ASTRONOMY.^ 
 
 drawing Z' C 2', cuttwig the fmall circle in ^, we fe« 
 that the figure is fimply reverfed. The magnitudes and 
 proportions of the tides are the fame in either cafe, but 
 the combination is inverted, and what belongs to a fupe- 
 rior tide in the one cafe belongs to an inferior tide in the 
 other. 
 
 680. 5. If the colatitude be equal to the Moon's 
 declination, or lefs than it, there will be no inferior 
 tide, or no fuperior tide, according as the latitude and 
 Moon's declination are of the fame or of different deno- 
 minations. For when P Z = M O, D coincides with I, 
 and K alfo coincides with I. Alfo when P Z is lefs 
 than M Q, D falls below I, and the point Z never 
 paffes through the equator of the watery fpheroid. The 
 low water mm' (fig. 73.) obferved in the parallel km is 
 only a lower part of the fame tide kk', of which the 
 high water is alfo obferved in the fame place. In fuch 
 fituations, the tides are very fmall, and are fubjecled to 
 iinguiar varieties which arife from the Moon's change of 
 declination and diflance. Such tides can be {een only in' 
 the circumpoiar regions. The inhabitants of Iceland no» 
 lice a period of nineteen years, in which their tides gra- 
 dually increafe and diminllli, and exhibit very fmgular 
 phenomena. This is undoubtedly owing to the revolu- 
 tion of the Moon's nodes, by v/liich her declination is 
 confiderably afxecfbed. That ifland is precifely in the 
 part of the ocean where the effedl of this is mo-ft remark- 
 able. A regifter kept there would be very uiftrudive •, 
 
 aii4 
 
THEORY OF THE TIDES. 621 
 
 mid it is to be hoped that this will be done, as in that 
 fequeftrated Thule, there is a zealous aftronomer, M. Lie- 
 Vog, furniflied with good inftruments, to whom this fe- 
 ries of obfervations has been recommended. 
 
 68 1. 6. At the very pole there is no daily tide* 
 But there is a gradual rife and fubfidence of the water 
 twice in a month, by the Moon's declining on both fides 
 of the equator. The water is lowed at the pole when 
 the Moon is in the equator, and it rifes about twenty-fix 
 Inches when the Moon is in the tropics. Alfo, when 
 lier afcending node is in the vernal equinox, and flie has 
 her greatefl declination, the water will be thirty inches 
 above its loweft itate, by the adlion of the Moon alone. 
 
 682. 7. The medium tide is, as has already been 
 
 , » , ^_ I -f- cof. 2 decl. d X cof. 2 lat. 
 
 obferved, = M x — . 
 
 2 
 
 As the Moon's declination never exceeds 30^, the co- 
 fine of twice her declination is always a pofitive quantity, 
 and never lefs than |. When the latitude is lefs than 
 45°, the cofine of twice the latitude is alfo pofitive, but 
 negative when the latitude exceeds 45°. Attending to 
 thefe circumflances, we may infer, 
 
 683. I. That the mean tides are equally afFe£led 
 by the northerly and foutherly declinations of the Moon. - 
 
 684. 2. If the latitude be exadly 45*^, the mean 
 tide is always the fame, and = | M. For in this cafe 
 
 BD 
 
iS2'X PHYSICAL ASTRONOMY. 
 
 B D is perpendicular to P C, and the point H always 
 coincides with O. This is the reafon why, on the coafts 
 of France and Spain, the tides are fo Httle aiFeded h^ 
 the deciinaticn of the luminaries. 
 
 6Z^. 3. When the latitude is lefs than 45°, the 
 mean-tides increafe as the decUnation of the Moon dimi- 
 3ii{hes. For cofm. 2 lat. being then a pofitive quantity, 
 the formula increafes v/hen the cofme of the declination 
 of the Moon increafes^ that is, it diminifnes wheii the 
 declination of the Moon increafes. As B O diminifhes, 
 G comes nearer to C, and H feparates from O towards 
 B, and D H increafes. 
 
 But if the latitude exceed 45"', the point H mud fall 
 between O and D, and the mean-tide will increafe as 
 tlie declination increafes. 
 
 686'. 5. If the latitude be — o, the point H coin- 
 cides. Vv'ith G, and the effecl of the Moon's detiination 
 33 then the moil fenfible. The mean-tide in this cafe is 
 ^^ I -}- cof. 2 deciin. (I 
 
 685. Every thing that has been determined here for 
 the lunar tide may eafily be accommodated to the high 
 and low water of the compound tide, by repeating the 
 computations with S in the place of M, as the conftant 
 coefficient. But, in general, it is almofl as exact as the 
 nature of the queflion will admit, to attend only to .the 
 
 lunar 
 
F'g 
 
 Ti- 
 
 
 y\l^ 
 
 ^AX\ 
 
 
 //'■ /]\ 
 
 X\ ^ 
 
 
 1 \ 1 )\ 
 
 \ / // 
 
 
 Ki\ 
 
 74 
 
 \ \ 
 
 
 1 ' ~ \/ 
 
 7^ 
 
 1 
 i 
 
 
 .^' 
 
 
 'C 
 
 
 J 
 
THEORY CF THE TIDES. 1^25 
 
 lunar tide. The declination of the real fammit of the 
 fpheroid, in this cafe, never differs from the declination of 
 the fummit of the lunar tide more tlian two degrees, and 
 the corre(ftion may be made at any time by a ♦little reflec- 
 tion on the fimultaneous pofition of the Sun. What 
 has been f\iid is itrifHy applicable to the fpring-tides. 
 
 M + S — tide X fm." ^O (fig. 73.) is the quantity to 
 be added to the tide found by the con(lru£lion. It is 
 exa^b in fpring-tides and very near the truth in all other 
 
 cafes. The Cm.' r/ O is = f! 1"^^ — . For fin. ^ O G : 
 
 coi. " decl. a 
 
 fin. J G O = fin. d G : fin. d O. 
 
 Such, then, are the more fimple and general confe- 
 quences of gravitation on the waters of our ocean, on the 
 fuppofition that the whole globe is covered with water, 
 and that the ocean always has the form which produces 
 a perfe(Sl: equilibrium of force in every particle. 
 
 686. But the globe is net fo covered, and it Is clear 
 that there muft be a very great extent of open fea, in or- 
 der to produce that elevation at the fummit of the fphe^ 
 roid which correfponds with the accumulating force of the 
 luminaries. A quadrant at lead of the elllpfe is necefiary 
 for giving the whole tide. With lefs than this, there 
 will not be enough of water to make up the fpheroid. 
 And, to produce the full daily viciffitude of high and 
 low water, this extent of fea mufl: be in longitude. Aa 
 equal extent in latitude may produce the greateft eleva- 
 tion ; but it will not produce the feries of heights that 
 
 fhould 
 
THEORY OF THE TIDES. tSl^ 
 
 lunar tide. The declination of the real fammit of the 
 fpheroid, in this cafe, never differs from the declination of 
 the fummit of the lunar tide niorc tlian two degrees, and 
 the correftion mny be made at any time by a ♦little reflec- 
 tion on the fimultaneous pofition of the Sun. What 
 has been faid is flridHy applicable to the fpring-tides. 
 
 iVl + S —tide X fin.' dO (fig. 73.) is the quantity to 
 be added to the tide found by the condruclion. It is 
 exaft in fpring-tides and very near the truth in all other 
 
 cafes. The fin.= ^ O is = -^^ — . For fin. dOG: 
 
 cof. ' decl. a 
 
 fin. J G O = fin. d G : fin. d O. 
 
 Such, then, are the more fimple and general confe- 
 quences of gravitation on the writers of our ocean, on the 
 fuppofition that the whole globe is covered with water, 
 and that the ocean always has the form which produces 
 a perfect equilibrium of force in every particle. 
 
 6S6. But the globe is net fo covered, and It is clear 
 that there muft be a very great extent of open fea, in or- 
 der to produce that elevation at the fummit of the fphe- 
 roid which correfponds with the accumulating force of the 
 luminaries. A quadrant at lead of the ellipfe Is necefi^ary 
 for giving the whole tide. With lefs than this, there 
 will not be enough of water to make up the fpheroid. 
 And, to produce the full daily viciflitude of high and 
 low water, this extent of fea muft be in longitude. An 
 equal extent in latitude may produce the greateft eleva- 
 tion ; but it will not produce the feries of heights that 
 
 fhould 
 
<^24 PHYSICAL ASTRONOMTi 
 
 fhould occur in the coiirfe of a lunar day. In confined, 
 feas of fmall extent, fuch as the Cafpian, the Euxine, the 
 Baltic, and the great lakes in North America, the tides 
 mufl be almoft infenfible. For it is evident that the 
 greateft difference of height on the fliore of fuch confined 
 feas. can be no more than the deflection from the tangent 
 of the arch of the fpheroid contained in that fea. This, 
 in the Cafpian Sea, cannot exceed feven inches ; a quan- 
 tity, fo fmall, that a flight breeze of wind, fetting ofF 
 fliore, will be fuflicient for preventing the accumulation, 
 and even for producing a depreffion. A moderate breeze, 
 blowing along the canal in St James's Park at London, 
 raifes the water two inches at one end, while it deprefTes 
 it as much at the other. The only confined feas of cori- 
 fiderable extent are the Mediterranean and the Red Sea. 
 The firfl has an extent of 40° in longitude, and the tides 
 there might be very fenfible, were it on the equator, but 
 being in lat. 35 nearly, the efFecls are lefTened in the 
 proportion of five to four. In fuch a fituation, the phe- 
 nomena are very different, both in regard to time and to 
 kind, from what they would be, if the Mediterranean- 
 were part of the open ocean. Its furface vjiW'he parallel 
 to what it would be in that cafe, but fiot the fame. This 
 will appear, by infpecllon of fig. 75, where in rp repre- 
 fents the natural level of the ocean, and M O repre- 
 fents the watery fpheroid, having its pole in M, and its 
 equator at O. S j may reprefent a tide pofl, fet up on 
 the fhore of Syria, at the eafl end of the Mediterranean, 
 and G i? a pofl fet up at the Gut of Gibraltar, which we 
 
 fhall 
 
THEORY OF THS TIDE?. 6i^ 
 
 {hall fuppofe at prefent to be dammed up. When the 
 Moon is over M, the waters of the INIediterranean aflume 
 the furface g r j-, parallel to the correfponding portion of 
 the elliptical furface Q o M, crofling the natural furface 
 at r, nearly in the middle of its length. Thus, on the 
 Syrian coaft, there is a confiderable elevation of the wa- 
 ters, and at Gibraltar, there is a confiderable depreflion. 
 In the middle of the length, the water is at its mean 
 height. The water of the Atlantic Ocean, an open and 
 extenfive fea, aflumes the furface of the equilibrated 
 fpherold, and it (lands confiderably higher on the outfide 
 of the dam, as is feen by G Oy than on the infide, as ex- 
 prelTed by G g. It is nearly low water within the Straits^ 
 while it is about 4- or 4- flood without. The water has 
 been ebbing for fome hours within the Straits, but flow- 
 ing for great part of the time without. As the Moon 
 moves weftward, toward Gibraltar, the water will begirt 
 to rife, but flowly, within the Straits, but it is flowing 
 very faft without. When the Moon gets to P, things 
 are reverfed. The fummit of the fpherold (It being fup- 
 pofed a fpring-tlde) is at P, and it is nearly high water 
 within the Straits, but has been ebbing for fome hours 
 w^ithout. It is low water on the coaft of Syria* All this 
 while, the water at r, in the middle of the Mediterra- 
 nean, has not altered Its height by any fenfible quantity. 
 It will be high water at one end of the Mediterranean, 
 and low water at the other, when the middle is in that 
 part of the general fpheroid where the furface makes the 
 xnofl unequal angles with the vertical This will be 
 
 4 K nearly 
 
626 PHYSICAL ASTRONOMY.' 
 
 nearly In the o6\:ants, and therefore about i| hours be-- 
 fore and after the Moon's fouthing (fuppofing it fpring- 
 tide). 
 
 Thefe obfer^^atlons greatly contribute to the explana- 
 tion of the fingular currents in the Straits of Gibraltar, 
 as they are defcribed by different authors. For although 
 the Mediterranean is not. (hut up, and altogether feparat- 
 ed from the Atlantic Ocean at Gibraltar, the communi- 
 cation is extremely fcanty, and by no means fufRcient for 
 allowing the tide of thj^ ocean to diffufe itfelf into this 
 bafon in a regular manner. Changes of tide, always dif- 
 ferent, and frequently quite oppoiite, are obferved on the 
 eaft and weft fide of the narrow neck which connects 
 the Rock with Spain j and the general tenor of thofe 
 changes has a very great analogy with what has now 
 been defcribed. The tides in the Mediterranean are 
 fmall, and therefore eafily affefted by winds. But 
 they are remarkably regular. TJiis may be expected. 
 For*as the collection or abftraCiion neceffary for pro- 
 ducing the change is but fmall, they are foon accom- 
 pUlhed. The regifters of the tides at Venice and fome 
 other ports in the Adriatic are furprifmgly conformable 
 to the theory. See Phil. Tranf. Vol. LXVII. 
 
 From this example, it is evident that great deviations 
 may be expelled in the obferved phenomena of the tides 
 from the imniediate refults of the fimple unobftrufted 
 theory, and yet the theory may be fully adequate to the 
 explanation of them, when the circumftances of local 
 fituation are properly confidered. 
 
 688. 
 
THEORY OF THE TIDE^. 027 
 
 688. The real (late of things is fuch, that there arc 
 very few parts of the ocean where the theory can be ap- 
 pHed witliout very great modifications. ' Perhaps the 
 great Pacinc Ocean is the only part of the temqueous 
 globe in which all the forces have room to operate. 
 When we confider the terreftrial globe as placed before 
 the a6ling luminaries, which have a relative motion round 
 it from eafl to wefl, and confider the accumulation of 
 the waters as keeping pace with them on the ocean, we 
 muft fee that the tides with which we are moft familiarly 
 acquainted, namely, thofe which vifit the weflern fhores 
 of Europe and Africa, and the eaftern fhores of America, 
 muft alfo be irregular, and be greatly diverfified by the 
 fituation of the coafts. The accumulation on our coafts 
 muft be in a great meafure fupplied by what comes from 
 the Indian and Ethiopic Ocean from the eaflward, and 
 what is brought, or kept back, from the South Sea ; and 
 the accumulation muft be diffufed, as from a colIe<ftion 
 coming round the Cape of Good Hope, and round Cape 
 Horn. Accordingly, the propagation of high water is 
 entirely confonant v/ith fuch a fuppofition. It is high 
 water at the Cape of Good Hope about three o'clock at 
 new and full moon, and it happens later and later, as we 
 proceed to the northward along the coaft of Africa ; 
 later and later fiiill as we follow it along the weft coafts 
 of Spain and France, till we get to the mouth of the 
 Englilh Channel. In fliort, the high water proceeds a- 
 long thofe fliores juft like the top of a wave, and it may 
 he followed, hour after hour^ to the different harbours 
 
 4 K 2 along 
 
^2^ PHYSICAL ASTROKOMY. 
 
 along the coaft. The fame wave continues its progrefs 
 northwards (for it feems to be the only fupply), part of 
 it going up St George^s Channel, part going northward 
 by the weft fide of Ireland, and a branch of it going up 
 the Englifti Channel, between this illand and France. 
 What goes up by the eaft and weft fides of Ireland unites, 
 and proceeds ftill northward, along the weftern coafts 
 and iflands of Scotland, and then difFufes itfelf to the 
 eaftward, toward Norway and Denmark, and, circling 
 round the eaftern coafts of Britain, comes fouthward, in 
 what is called the German Ocean, till it reaches Dover, 
 where it meets with the branch which went up the Eng- 
 lifh Channel. 
 
 689. It is remarkable that this northern tide, after 
 having made fuch a circuit, is more powerful than the 
 branch v/hich proceeds up the Englifti Channel. I^ 
 Teaches Dover about a quarter of an hour before the 
 fouthern tide, and forces it backwards for half an hour. 
 It muft alfo be remarked, that the tide which comes up 
 channel is not the fame with the tide which meets it 
 from the north, but is a whole tide earlier, if not two 
 tides. For the fpring-tide at Rye is a tide earlier than 
 the fpring-tide at the Nore. It even feems more nearly 
 two tides earlier, appearing the one as often as the other. 
 This may be better feen by tracing the hour of high 
 water from the Lizard up St George's Channel and along 
 the weft coafts of Scotland. Now it is very clear that 
 tjie fuperior tide at the Orkney iflands is fimultaneou3 
 
 ^ith 
 
THEORY OF THE TIDES. ^29 
 
 With the inferior tide at the mouth of the Thames. It 
 is therefore moft probable that tlie Orkney tide is at lead 
 one tide later than at the Lizard. The whole of this 
 tide is very anomalous, efpecially after getting to the Ork- 
 neys. It is a derivative from the great tide of the open 
 fea, which being verydiftant, is fubje£ted to the influence 
 of hard gales, at a diftance, and frequently unlike what 
 is going on upon our coafts. . 
 
 690. A fimilar progrefs of the fame high water from 
 the fouthward, is obferved along the eaftern fhores of 
 South America. But, after paffing Brazil and Surinam, 
 the Atlantic Ocean becomes fo wide that the effeO: of 
 this high water, as an adventitious thing fupplied from 
 the fouthv/ard, is not fo fenfible, becaufe the Atlantic 
 itfelf is now cxtcnfive enough to contribute greatly to 
 the formation of the regular fpheroid. But it contributes 
 chiefly by abftraclion of the waters from the American 
 fide, while the accumulation is forming on the European 
 fide of the Atlantic. By ftudying the fucceflive hours 
 of high water along tlie weftern coafts of Africa and Eu- 
 rope, it appears that it takes nearly two days, or between 
 four and five tides, to come from the Cape of Good 
 Hope to the mouth of the Englifh Channel. This re- 
 mark is of peculiar importance. 
 
 691. Few obfervations have, as yet, been made 
 public concerning the tides in the Great Pacific Ocean. 
 They muft exhibit phenomena confiderably different 
 
 from 
 
630 PHYSICAL ASTRONOMY; 
 
 from what are feen hi the Atlantic. The vafl: ftretch oF 
 uninterrupted coaft from Cape Horn to Cook's Straits, 
 prevents all fupply from the eaftward for making up the 
 fpheroid. So far as Vv^e have information, it appears that 
 the tides are very unlike the European tide?> tDl we get 
 40° or 50'' weft from the coaft of America. In the neigh- 
 bourhood of that coaft, there is fcarccly any inferior 
 tide. Even in the middle of the vaft Paciiic Ocean the 
 tides are very fmall, but abundantly regular. 
 
 692. The fetting of the tides is afreded, not only 
 by the form of the fliores, but alfo by the inequalities 
 which undoubtedly obtain in the bottom of the ocean. 
 A deep and long valley there will give a direction to the 
 waters which move along it, even although they far over- 
 top the higher parts on each fide, jufl as we obferve the 
 wind follow the courfe of the vallies. Tliis direction of 
 the undermoll waters afFe£l:s thofe that flow above them, 
 in confequence of the mutual adhefion of the filaments ; 
 and tlius the whole ilream is deflected from the direction 
 which it would have taken, had the ground been even. , 
 By fuch deflections the path is lengthened, and the time 
 cf its reaching a certain place is protracted , and this 
 produces other deviations from the calculations by the 
 fimple theory. 
 
 693. Thefe peculiarities in the bed or channel alfo 
 greatly aiTe^t the height of the tides. When a wave of 
 ■a certain magnitude enters a channel, it has a certrJu 
 
 quantity 
 
THEOPX.Y OF THE TIDE3. 6^t 
 
 cjuatitlty of motion, mcafurcd by the quantity of water 
 and Its velocity. If the channel, keeping the fime depth, 
 contra6l in its width, t]:»e water, keeping for a while Its 
 momentum, mufi: increafe its velocity, or its depth, or 
 both. And thus it may happen that, although the greateft 
 elevation produced by the joint aftion of the Sun and 
 Moon in the open fea does not exceed eight or nine feet, 
 the tide in fome fmgular fituations may mount confider- 
 ably liigher. It feems to be ov/ing to this that the high 
 water of the Atlantic Ocean, which at St Helena does 
 not exceed four or five feet, fetting in obliquely on the 
 coaft of North America, ranges along that coaft, in a 
 channel gradually narrowing, till it is flopped in the Bay 
 of Fundy as in a hook, and there it heaps up to an afto- 
 nifhlng degree. It fometimes rifes 120 feet in the har- 
 bour of Annapolis-Royal. Were it not that we fee in- 
 itances of as ilrange effects of a fudden check given to 
 the motion of water, we fhould be difpofed to think that 
 the theory is not adequate to the explanation of the phe- 
 nomena. But the extreme difparity that we may obferve 
 in places very near each other, and which derive their 
 tide from the very fame tide in the open fea, mufl con- 
 vince us that fuch anomalies do not impugn the general 
 principle, although we fhould never be able fully to ac- 
 count for the difcrepancy. 
 
 694. Nothing caufes fo much irregularity in the 
 tides as the reflection of the tide from fliore to fhore. 
 If a pendulum, while vibrating, receives little impulfes, 
 
 at 
 
^^2 PHYSICAL ASTRONOMY. 
 
 at Intervals that are always the fame, and very neai- 
 ly equal to its own vibrations, or even to an aliquot 
 part of them, the vibrations may be increafed to a great 
 magnitude after fome time, and then will gradually di- 
 minifh, and thus have periods of increafe and decrcafe. 
 So it happens in the undulation which conftitutes a tide. 
 The fituation of the coafts may be fuch, that the time 
 In which this undulation would, of itfelf, play backward 
 and forward from fhore to fhore, may be fo exactly fitted 
 to the recurring ȣlion of the Moon, that the fucceeding 
 impulfes, always added to the natural undulation, may 
 raife it to a height altogether difproportioned to what the 
 a61ion of the Moon can produce in open fea, where 
 the undulation diffufes itfelf to a vaft diftance. What 
 we fee in this way fliould fuflice for accounting for the 
 great height of the tides on the coafts of continents. 
 Dan. Bernoulli, juftly thinking that the obftru6lions of 
 various kinds to the movements of the ocean fhould make 
 the tides lefs than v/hat the unobftrucled forces are able 
 to produce, concluded, from the great tides a£l:ually ob- 
 ferved, and compared with the tides producible by the 
 Newtonian theory, that this theory was erroneous. He 
 thought it all derived from Newton's erroneous idea of 
 the proportion of the two axes of the terraqueous globe ; 
 which miftake refults from the fuppofition of primi- 
 tive fluidity, and uniform denfity. He inveftigates the 
 form of the Earth, accommodated to a nucleus of great 
 denfity, covered with a rarer fluid, and he thinks that 
 he has demonftrated that the height of the tide will be 
 
 in 
 
THEORY OF THE TIDES. 633 
 
 in proportion to the comparative denfity of this nucleus, 
 or the rarity of the fluid. This, fays he, alone can ac- 
 count for the tides that we really obferve •, and which, 
 great as they are, are certainly only a part of v/hat they 
 would be, were they not fo much obftru£led. This is 
 extremely fpecious, and, coming from an eminent ma- 
 thematician, has confiderable authority. But the problem 
 of the figure of the Earth has been examined with the 
 moffc fcrupulous attention, fmce the days of M. Ber- 
 noulli, by the firft mathematicians of Europe, who are 
 all perfe61:ly agreed in their dedudions, and confirm that 
 of Sir Ifaac Newton. They have alfo proved, and we 
 apprehend that it is fufEciently efhablifhed in art. 603, 
 that a denfer nucleus, inftead of making a greater tide, 
 will make it fmaller than if the whole globe be of one 
 denfity. The ground of Bernoulli's miftake has alfo been 
 clearly pointed out. There remains no other way of ac- 
 counting for the great tides but by caufes fuch as have now 
 been mentioned. When the tides in the open Pacific 
 Ocean never exceed three or four feet, we mufl be con- 
 vinced that the extravagant tides obferved on the coafts 
 of great continents are anomalies ; for there, the obftruc- 
 tions are certainly greater than in the open fea. We 
 mufb therefore look for an explanation in the motions and 
 coUifions of difturbed tides. Thefe anomalies therefore 
 bring no valid objection againfl the general theory. 
 
 695. There are fome fituations where it is eafy to 
 
 explain the deviations, and the explanation is inftru6live. 
 
 4 L Suppofe 
 
634 PHYSICAL A-STUONOMT. 
 
 Suppofe a great navigable river, running nearly in a me- 
 ridional direction, and falling into the fea in a fouthem 
 coaft. The high water of the ocean reaches the mouth 
 of this river (we may fuppofe) when the Sun and Moon 
 are together in the meridian. It is therefore a fpring- 
 tide high water at the mouth of the river at noon. This 
 checks the llream- at the mouth of the river, and caufes 
 it to deepen. This again checks the current farther up 
 the river, and it deepens there alfo, becaufe there is al- 
 ways tlie fame quantity of land water pouring; into it. 
 The dream is not perhaps flopped, but only retarded. 
 But'this cannot happen without its growing deeper. This 
 is propagated farther and farther up the ftream, and it 
 is perceived at. a great diflance up the river. But this 
 requires a confiderabie time. Our knowledge in hydrau- 
 lics is too imperfedt as yet to enable us to fay in what 
 number of hours this fenfible check,, indicated by the 
 fmaller velocity, and greater depth, will be propagat- 
 ed to a certain diflance. We may fuppofe it juft a 
 lunar day before it arrive at a certain wharf up th? 
 river. The Moon, at the end of the day, is again on 
 the meridian, as it was when it was a fprjng-tide at th^ 
 mouth of the river the day before. But, in this inter- 
 val, there has been another high water at the mouth of 
 the river, at the preceding midnight, and there has jufl 
 been a third high water, about fifteen minutes before 
 the Moon came to the meridian, and thirty-five minutes 
 after the Sun has pafled it. There muft have been two 
 low waters in the interval, at the mouth of the river. 
 
 Now„ 
 
THFORY OF THE TIDF.So ^3*5 
 
 Now, In the fame way that the tide of yeft' "relay noon 
 is propagated up the ftream, the tide of midnight has 
 alfo proceeded upAvards. A-^iid thus, there are three 
 coexiftent high waters in tlie river. One of them is a 
 fprlng-tide, and it is far up, at the wharf above men- 
 tioned. The fecond, or the midnight tide, m-uft be half 
 way up the river, and the third is at the mouth of the 
 river. And there muft be two low waters intervening. 
 The low water, thai 15, a flate of the river below its na- 
 tural level, is produced by the pafling low water of the 
 ocean, in the fame way that the high water was. For 
 when the ocean falls below its natural level at the mouth 
 of the river, it occafions a greater declivity of the iffuing 
 flream of the river. This muft augment its velocity — 
 this abftra6t:5 more water from the ftream above, and that 
 part alfo finks below its natural level, and gives a greater 
 declivity to the waters behind it, &c. And thus the 
 ftream is accelerated, and the depth is leftened, in fuc- 
 ceflion, in the fame way as the oppofite efFe61:s were pro- 
 duced. We have a low water at different wharfs in fue- 
 ^reffion, juft as we had -the high waters. 
 
 696. This ftate of things, which muft be famiharly 
 Icnown to all who have paid any attention to thefe mat- 
 ters, being feen in almoft every river which opens into a 
 tide way, gives us the moft diftinft notion of the me- 
 chanifm of the tides. The daily returning tide is nothing 
 but an undulation or wave, excited and maintained by 
 'the action of the Sun and Moon. It is a great miftafce 
 4 L 2 t» 
 
6^6 PriYSIGAL ASTRO>iOMY, 
 
 to imagine that we cannot have high water at London 
 Bridge (for example) unlefs the water be raifed to that 
 level all the way from the mouth of the Thames. In 
 many places that are far from, the fea, the ftream, at the 
 moment of high water, is down the river, and fometimes 
 it is confiderable. At Quebec, it runs downward at leaft 
 three miles per hour. Therefore the water is not heaped 
 up to the level j for there is no ftream without a decli- 
 vity. The harbour at Alloa in the river Forth is dry at 
 low water, and the bottom is about dx feet higher than 
 the higheft v/ater mark on the ftone pier at Leith. Yet 
 there are at Alloa tides of twenty, and even twenty-two 
 foet. All Leith would then be under water, if it flood 
 level from Alloa at the time of high water there. 
 
 After confidering a tide in this way, any perfon who 
 has remarked the very ftrange motions of a tide river, in 
 its various bendings and creeks, and tlie currents that 
 are frequently obferved in a direclion. oppofite to the 
 general ftream, will no longer expeel; that the phenomena 
 of the tides will be fuch as immediately refuit from the 
 regular operation of the folar and lunar forces. 
 
 697, There is yet another caufe of deyiatiorr, wliich 
 is perhaps more diffimilating than any local circumftances, 
 and the operation of which it is very difficult to ftate fa- 
 miliarly, and yet precifely. This is the inertia, as it is 
 called, of the waters. No finite change of place or of 
 velocity can be produced in an inflant by any acce- 
 lerating force. Time muft elapfe before a Hone can ac- 
 quire ?^ny meafurable velocity by falling. 
 
 Suppofb 
 
THEORY OF THE TIDES. 637 
 
 Suppofe the Earth fluid to the centre, and at reft, 
 without any external diflurbing force. The ocean will 
 form a perfect fphere. Let the Moon now a£l on it. 
 The v/aters will gradually rife immediately under the 
 Moon and in the oppofite part of the Earth, finking all 
 around the equator of the fpheroid. Each particle pro- 
 ceeds to its ultimate fituation with an accelerated motion, 
 becaufe, till then, the diflurbing force exceeds the ten- 
 dency of the water to fubfide. Therefore, when the 
 form is attained v.diich balances thofe forces, the motion 
 does not flop, juft a^ a pendulum docs not flop when it 
 reaches the loweft point of its arch of vibration. Sup- 
 pofe that the Moon ceafes to act at this inftant. The mo- 
 tion will {till go on, and the ocean will overpafs the ba- 
 lanced figure, but with a retarded motion, as the pen- 
 dulum rlfes on the other fide of the perpendicular. It 
 will (lop at a certain form, when all the former accele- 
 ration Is done away by the tendency of the water to fub- 
 fide. It now begins to fubfide at the poles of the fphe- 
 roid, and to rife at the equator, and after a certain time, it 
 becomes a perfed: fphere, that is, the ocean has its na- 
 tural figure. But it paiTes this figure as far on the other 
 fide, and makes a fiood where there was formerly an ebb ; 
 and it would now ofcillate for ever, alternately fwelling 
 and contracting at the points of fyzigy and quadrature. 
 If the Moon do not ceafe to ad, as was juft now fup- 
 pofed, there will ftill be ofcillations, but fomev/hat dif- 
 ferent from thofe now mentioned. The middle form, 
 on both fides of which it ofcillates in this cafe, is not 
 the perfe£l fphere, but the balanced fpheroid. 
 
638 PHYSICAL ASTRONOMY. 
 
 698. All this is on the fuppofition that there is n® 
 -obflru6lion. But the mutual adhefion of the filaments 
 of water will greatly check all thefe motions. The figure 
 will not be fo foon formed ; it will not be fo far over- 
 palled in the firft ofcillation ; the fecond ofcillation will 
 he lefs than the firft, the third will be lefs than the fe- 
 cond, and they will foon become infenfible. 
 
 But if it were poffible to provide a recurring force, 
 which fhould tend to raife the waters where they are al- 
 ready rifing, a^ld deprefs them where they are fubfiding, 
 and that would always renew thofe actions in the pro- 
 per time, it is plain that this force may be fuch as will 
 juft balance the obftru^lions com.petent to any particular 
 degree of ofcillation. Such a recurring force would juft 
 maintain this degree of ofcillation. Or tlie recurring 
 force may be greater than this. It will therefore increafe 
 the ofcillations, till the obftru£\;ions are alfo fo much in- 
 creafed that the force is balanced by them. Or it may 
 be lefs than what will balance the obftru^lions to the de- 
 gree of ofcillation excited. In this cafe the ofcillation 
 will decreafe, till its obftru^tions are no more than what 
 this force will balance. Or this recurring force may 
 come at improper intervals, fometimes_ tending to raife 
 the waters when they are fubfiding in the courfe of an 
 ofcillation, and deprefling them when they are rifing. 
 Such a force muft check and greatly derange the ofcilla- 
 tions ; deftroying them altogether, and creating new 
 ones, which it v/ill increafe for fome time, and then 
 check and deftroy them ; and will do this again and 
 
 '^gain. 
 
 Now 
 
THEORY OF THE TIDE*. &^' 
 
 Now there is fuch a recurring force. As the Eartk 
 turns round its axis, fuppofe the form of the balanced 
 fpheroid attained in the place immediately under the 
 Moon. This elevation; or pole is earned to the eailward 
 by the Earth, fuppofe into the pofition DOB (fig. ";6.\ 
 the Moon being in the lin« O M. Tlie pole of the wa- 
 tery fpheroid is no longer under the INIcon. The Moon 
 will therefore a6t on it fo as to change its figure, mak- 
 ing it fubfide in the remote quadrant B l? C, and rife a 
 little in the quadrant B ^ A. Thus its pole will come a 
 little nearer to the line O M. It is plain that if B is car- 
 ried farther eaflward, but within certain limits, the fitua- 
 tion of the particles will be ftill more unfuitable to the 
 lunar difturbing force, and its action on each to change 
 its pofition will be greater. The action upon them all 
 will therefore make a more rapid change in the pofition 
 of the pole of the difplaced fpheroid. It feems not im- 
 pofTible that this pole may be jufl fo far eaft, that the 
 changing forces may be able to caufe its pole to fliift its 
 pofition fifteen miles in one minute. If this be tlie cafe, 
 the pole of the fpheroid will keep precifely at its prefent 
 didance from the line O M. For, fince it would fhift 
 to the weflward fifteen miles in one minute by the ac- 
 tion of the Moon, and is carried fifteen miles to the 
 eaflward in that time by the rotation of the Earth, the 
 one motion jufl undoes the effect of the other. The 
 pole of the watery fpheroid is really made to (hift fifteen 
 miles to the weftward on the furface of the Earth, and 
 arrives at a place fifteen miles weft of its former place 
 
 on 
 
6*40 PHYSICAL ASTRONOMT. 
 
 the globe ; but this place of arrival is carried fifteen miles 
 to the ea ft ward ; it is therefore as far from the line O M 
 as before. 
 
 This may be illuftrated by a very fimple experiment, 
 where the operation of the adling forces is really very 
 like that of the lunar difturbing force. Suppofe a chain 
 or flexible rope A B C E D F laid over a pulley, and 
 hanging dovv-n in a bight, which Is a catenarean curve, 
 having the vertical line O D for its axis, and D for its 
 loweft point, which the geometers call its vertex. Let 
 the pulley be turned very llowly round its axis, in the 
 diredion ABC. The fide C E will defcend, and F A 
 will be taken up, every link of the chain moving in the 
 curve C E D F A. Every link is in the vertex D in its 
 turn, juft as every portion of the ocean is in the vertex 
 or pole of the fpheroid in its turn. Now let the pulley 
 turn round very brilkly. The chain will be obferved to 
 tilter its figure and pofition. O D will no longer be its 
 axis, nor D its vertex. It will now form a curve 
 C e dfK, lying to the left hand of C E D F A. Od will 
 be its new axis, and d will be its vertex. Gravity a61:s 
 in lines parallel to O D. The motions" in the diredlion 
 CE and FA nearly balance each other. But there is 
 a general motion of every Hnk of the hanging chain, by 
 which it is carried from E towards F. Did the chain 
 continue in the former catenarea, this force could not 
 be balanced. It therefore keeps fo much awry, in the 
 form Ce dj Ay that its tendency by gravity to return to 
 its former pofition is juft equal to the fum of all the mo- 
 tions 
 
TUrORY OF THE TIDES. 64I 
 
 '10113 in the links from E towards F. And it will fhew 
 this tendency by returning to that pofition, the moment 
 that the pulley gives over turning. The more rapidly 
 we turn the pulley round, the farther will the chain go 
 a fide before its attitude become permanent. 
 
 700. It furpaffes our mathematical knowledge to fay 
 with precifion how far eaftward the pole of the tide muft 
 be from the line of the Moon's diredion, even in the 
 fimple cafe which we have been confidering. The real 
 ftate of things is far more complicated. The Earth is 
 not fluid to the centre, but is a folid nucleus, on which 
 flows an ocean of very fmall depth. In the former cafe, 
 a very moderate motion of each particle of water is fuf- 
 ficient for making the accumulation in one place and the 
 depreflion in another. The particles do little more than 
 rife or fubfide vertically. But, in the cafe of a nucleus 
 covered with an ocean of fmall depth, a confiderable ho- 
 rizontal motion is required for bringing togetlier the quan- 
 tity of water wanted to make up the balanced fpheroid. 
 The obftrudions to fuch motion muil be great, both 
 fuch as arife from the mutual adhefion of the filaments 
 of water, and many that mull arife from fridion and 
 the inequalities of the bottom, and the configuration of 
 the fliores. In fome places, the force of the acting lu- 
 minaries may be able to caufe the pole of the fpheroid to 
 fliift its fituation as faft as the furface moves away, when 
 the angle MOB is 20 degrees. In other places, this 
 may not be tilJ it is 25°, and in another, 15^ may be e- 
 4 M iiough. 
 
642 PHYSICAL ASTRONQMY. 
 
 nough. But, ill every fituation, there will be r.n arrange-- 
 ment that will produce this permanent pofition of the 
 fummit. For when the obilrudions are great, the ba- 
 lanced form will not be nearly attained -, and when this is- 
 the cafe, the change producible on the pofition of a par- 
 ticle is more rapidly efFe61:ed, the forces being great, 
 or rather the refiftance arifing from gravity alone being 
 fmall. 
 
 701. The confequence of all this mufl be, in the 
 firft place, that that form which the ocean v/ould ulti- 
 mately affiime, did the Earth not turn round its axis, 
 will never be attained. As the waters approach to that 
 form, they are carried eaftward, into fituations where 
 the difturbing forces tend to deprefs them on one fide, 
 while they raife them on the other, caufmg a v/efterly 
 undulation, which keeps its fummit at nearly the fame 
 diftance from the line of the at^ing luminary's direftion. 
 This wefterly motion of the fummit of the undulation 
 does not necelTarily fuppofe a real transference of the 
 water to the weftward at the fame rate. It is more 
 like the motion of ordinary waves, in which yve fee a 
 bit of wood or other light body merely rife and fall with- 
 out any fenfible motion in the direction of the wave. 
 In no cafe whatever is the horizontal motion of the water 
 nearly equal to the motion of the fummit of the wave. 
 It refembles an ordinary wave alfo in tliis, that the rate 
 at which the fummit of the undulation advances in any 
 4ire£lion is very little affecSled by the height of the wave. 
 
 Our 
 
THEORY OF THE TIDES. 643 
 
 Our knowledge however in hydraulics has not yet enabled 
 us to fay with prcclfion what is the relation between the 
 height of the undulation and the rate of its advance. 
 
 702. Thus then it appears, in general, that the funi- 
 mit of the tide muft always be to the eaftward of the 
 place affigned to it by our (imple theory, and that expe- 
 rience alone can tell us how much. Experience is more 
 uniform in this refpe61: than one (liould expedl:. For it 
 is a matter of almoil univerfal experience that it is very 
 nearly 19 or 20 degrees. In a few places it is lefs, 
 and in many it is 5 or 6, or 7 degrees more. This is 
 inferred from obferving that the greateil and the fmallefl 
 of all the- tides do not happen on the very time of the 
 fyzigies and quadratures, but the third, and in loma 
 places, the fourth tide after. Subfequent obfervation has 
 .jfliewn that this is not peculiar to the fpring and neap- 
 tides, but obtains in all. At Breit (for example) the tide 
 which bears the mark of the augmentation ariling from 
 the Moon's proximity is not the tide feen while the Moon 
 is in perigeo, but the third after. In fliort, tlie whole 
 feries of monthly tides difagree with the fmiuitaneous 
 poution of the luminaries, but correfpond moil regularly 
 with their portions 37 or 38 hours before. 
 
 703. Another obfervation proper for this place is^, 
 that as different extent of fea, and different depth of 
 water, will and do occafion a difference in the time iu 
 which a great undulation may be propagated along it, it 
 
 4 ^^ 2 jpay 
 
644 PHYSICAL ASTRONOMY. 
 
 may happen that this time may fo correfpond with th« 
 repetition of all the agitating forces, that the adion of 
 to-day may fo confpire with the remaining undulation 
 of yefterday, as to incrcafe it by its reiterated impulfes, 
 to a degree vaftly greater than its original quantity. By 
 ejiving gentle impulfes in this way to a pendulum, in the 
 direcfiiion of its motion, its vibrations may be increafed 
 to fifty times their nrft fize. It is not neceflary, for this 
 cfFeft, that the return of the luminary into the favour- 
 able fituation be juft at the interval of the undulation. 
 It will do if it confpire with every fecond or third or 
 fourth undulation ; or, in general, if the amount of its 
 confpiring actions exceeds confiderably, and at no great 
 diftance of time, the amount of its oppofing aft ions. 
 In many cafes this cooperation will produce periods of 
 augmentation and diminution, and many feeming ano- 
 malies, which may greatly vary tlie phenomena. 
 
 ^04. A third obfervation that fhould be made here 
 is, that as the obftrudions to the motion of the ocean 
 arifing from the mutual adhefion and aftion of the fila- 
 ments are known to be fo very great, we have reafon to 
 believe that the change of form aftually produced is but 
 a moderate part of what the force can ultimately pro- 
 duce, and that none of the ofcillations are often repeated. 
 It is not probable that the repetitions of the great undu- 
 lations can m.uch exceed four or five. When experi- 
 ments are made on ftill water, we rarely fee a pure un- 
 dulation repeated (d often. Even in a fyphon of gfafs, 
 • where- 
 
' THEORY OF THE TIDES. 64$ 
 
 where ail difFufions of the undulating power is prevented, 
 they are rarely fenfible after the fifth or fixth. A gentle 
 fmooth undulation on the furface of a very fliallow bafon, 
 in tlje view of agitating the whole depth, will feldom be 
 repeated thrice. This is the form which mod refembles 
 a tide. 
 
 705. After this account of the many caufes of de- 
 viation from the motions affigned by our theory, many 
 of which are local, and reducible to no rule, it would 
 feem that this theory, which we have taken fo much 
 pains to eftabliili, is of no ufe, except that of giving us 
 a general and moll powerful argument for tlie univerfal 
 gravitation of matter. But this would be too hafty a 
 conclufion. We fhall find that a judicious confideration 
 of the different claffes of the phenomena of the tides 
 will fuggeft fuch relations among them, that by properly 
 combining them, we fhall not only perceive a very fatis- 
 fadlory agreement with the theory, but fhall alfo be able 
 to deduce fome important practical inferences frojn it. 
 
 706. Each of the different modifications of a tide 
 has its own period, and its peculiar magnitude. Where 
 the change made by the a£fing force is but fmall, and 
 the time in which it is effected is confiderable, we may 
 look for a confiderable conformity with the theory ; but, 
 on the other hand, if the change to be produced on the 
 tide is very great, and the time allowed to the forces for 
 effeding it is fmajl, it is equally reafonable to expe£i: 
 
 fenfible 
 
64^ PHYSICAL ASTRONOMY. 
 
 fenfible deviations. If this confideration be judicioully 
 applied, we ihall find a very fatisfa£lory conformity. 
 
 707. Of all the modifications of a tide, the greateft, 
 and the moft rapidly efFefted, is the difference between 
 the fuperior and inferior tides of the fame day. When 
 tlie Moon has great declination, the fuperior tide at Breft 
 may be three times greater than the fucceeding or infe- 
 rior tide. But the fact is, that they differ very little. 
 M. de la Place fays that they do not differ at all. We 
 cannot find out his authority. Having examined with 
 the moft fcrupulous attention more than 200 of the ob- 
 fervations at Breft and Rochefort and Port i'Orient, and 
 made the proper allowance for the diftances of the lumi- 
 naries, we can fay with confidence ihat this general af- 
 fertion of M. de la Place is not founded on the ob- 
 fervations that have been publiflied, and it does not agree 
 with what is obferved in the other ports of Europe. 
 There is always obferved a difference, agreeing with 
 theory in the proportions, and in the order of their fuc- 
 ceffion, although much fmaller. A very flight confidera- 
 tion will give us the reafon of the obferved difcrepancy. 
 It is not poffible to make two immediately fucceeding un- 
 dulations of inert water remarkably different from each 
 other. The great undulation, in retiring, caufes the wa- 
 ter to heap up to a greater height in the offmg ; and 
 this, in diffufing itfelf, muft make the next undulation 
 greater on the fliore. That this is the true account of 
 the matter is fully proved by obfprving that when the 
 
 theoretic 
 
THEORY OF THE TIDfiS. C^J 
 
 theoretic dilTerence between thofe two tides is very 
 fmall, it is as diftin£tly obferved in the harbours as when 
 it is great. Tliis is clearly ieen in the Erell obferva- 
 tions. 
 
 708. The abfolute magnitudes of the tides are great- 
 ly modified by local circumftances. In fome harbours 
 there is but a fmall difference between the fpring and 
 neap-tides, and in other harbours it is very great. But, 
 in either cafe, the fmall daily changes are obferved to 
 follow the proportion required by the theory with abun- 
 dant precifion. Counted half way from the fpring to the 
 neap-tides, the hourly fall of the tide is as the fquare of 
 the time from fpring-tide, except fo far as this may be 
 changed by the pofition of the Moon's perigee. In like 
 manner, the hourly increafe of the tides after neap-tide 
 is obferved to be as the fquares of the time from neap- 
 tide. 
 
 709. The priming and lagging of the tides corre- 
 fponds with the theory with fuch accuracy, that they 
 feem to be calculated from it, independent of obferva- 
 tion. There is nothing that feems lefs likely to be de- 
 ranged than this. Tides which differ very little from 
 each other, either as to magnitude or time, fiiould be 
 expected to follow one another juft as the forces re- 
 quire. There is indeed a deviation, very general, and 
 eafily accounted for. There is a fmall acceleration of 
 the tides from fpring-tide to neap-tide. This is un- 
 doubtedly 
 
648 PHYSICAL ASTRONOMT. 
 
 doubtedly owing to the ob{lru£l:ions. A fmaller tide be- 
 ing lefs able to overcome them, is fooner brought to its 
 maximum. The deviation however is very fmall, not 
 exceeding ■} of an hour, by vi^hich the neap-tide antici- 
 pates the theoretical time of its accomplil'hment. It 
 would rather appear at firfl fight that a fmall tide would 
 take a longer time of going up a river than a great one. 
 And it may be fo, although it be fooner high water, be- 
 caufe the defalcation from its height may fooner termi- 
 nate its rifing. There is no difference obferved in this 
 refpedl, when we compare the times of Iiigh water at 
 London Bridge and at the Buoy of the Nore. They 
 happen at the very fame time in both places, and there- 
 fore the fpring-tides and the neap-tides employ the fame 
 time in going up the river Thames. 
 
 710. This agreement of obfervation with theory is 
 moft fortunate j and indeed without it, it would fcarcely 
 have been poflible to make any practical ufe of the theory. 
 But now, if we note the exa£l: time of the high water 
 of fpring-tide for any harbour, and the exa£^ pofition of 
 the Sun and Moon at that time, we can eafijy make a 
 table of the monthly ferles for that port, by noticing 
 the difference of that time from our table, and making 
 the fame difference for every fucceeding phafis of the 
 tide. 
 
 71 T. But, in thus accommodating the theoretical 
 feries to any particular place, we muft avoid a miftake 
 
 commonly 
 
THEORY OF THE TIDES. 649 
 
 commonly made by the compofcrs of tide tables. They 
 give the hour of high water at full and change of the 
 Moon, and this is confidered as fpring-tide. But per- 
 haps there is no part in the world where that is the cafe. 
 It is ufually the third tide after full or change that is the 
 greateft of all, and the third tide after quadrature is, in 
 moft places, the fmalleft tide. Now it is with the great- 
 eft tide that our monthly feries commences. Therefore, 
 it is the hour of this tide that is to be taken for the 
 hour of the harbour. But, as winds, frefhes, and other 
 caufes, may afFe6l any individual tide, we muft take the 
 medium of many obfervations ; and we muft take care 
 that we do not confidcr as a fpring-tide one which is in- 
 deed the greateft, but chances to be enlarged by being a 
 perigean tide. 
 
 When thefe precautions are taken, and the tides of 
 one monthly feries marked, by applying the fame cor- 
 reftion to the hours in the third column of Bernoulli's 
 table (I.), it will be found to correfpond with obfervation 
 with fufficient accuracy for all purpofes. In making the 
 comparifon, it will be proper to take the medium be- 
 tween the fuperior and inferior tides of each day, both 
 with refpe£l to time and height, becaufe the difference 
 in thefe refpe6ts between thofe two tides never entirely 
 difappears, 
 
 712. The feries of changes which depend on the 
 
 change of the Moon's declination are of more intricate 
 
 comparifon, becaufe they are fo much iniplicated with 
 
 4N thft 
 
6^C PHYSICAL ASTRONOMY. 
 
 the changes depending on her diftance. But when freed 
 as much as poflible from this complication, and then efti- 
 mated by the medium between the fuperior and inferior 
 tide of the hme day, they agree extremely well with 
 the theoretical feries. 
 
 This, by the way, enables us to account for an ob- 
 fervation which would otherwife appear inconfiftent with 
 the theory, which affirms that the fuperior tide is greatefl 
 when the Moon is in the zenith (676.) The obfervation 
 is, that on the coafts of France and Spain the tides in- 
 creafe as the Moon is nearer to the equator. But it was 
 lliewn in the fame article, that in latitudes below 45®, 
 the medium tide increafes as the Moon's declination dimi- 
 nifhes. Bernoulli juftly obferves that the tides v/ith which 
 we are moft familiarly acquainted, and from which we 
 ■form all our rules^ muft be confxdered as derived from the 
 more perfe£l and regular tide formed in the wideft part 
 of the Atlantic ocean. Extenfive however as this may 
 be, it is too narrow for a complete quadrant of the fphe- 
 Toid. Therefore it will grow more and more perfedl: as 
 its pole advances to the middle of the ocean •, and the . 
 changes which happen on the bounding coafts, from 
 which the waters are drawn on all fides to make it up, 
 muft be vaftly more irregular, and will have but a par- 
 tial refemblance to it. They vvill however refemble it 
 in its chief features. This tide being formed in a con- 
 fiderably fouthern latitude, it becomes the more certain 
 that the medium tide will diminifti as the Moon's decli- 
 nation increafes. But although this fceming objection 
 
 occuvi 
 
^THEORY OF THE TiCES. 6^1 
 
 occurs on the French coafts, it is by no means the cafe 
 on ours, or more to the north. We always obferve the 
 fuperior tide to exceed the inferior, if the Moon have 
 north declination. 
 
 The fame agreement with theory is obfervable in the 
 folar tides, or in the elTeft of the Sun's declination. 
 This indeed is much fmaller, but is obfcrved by reafon 
 of its regularity. For although it is alfo complicated 
 with the effects of the Sun's change of diflance, this 
 effect having the fame period with his declination, one 
 equation may comprehend them both. M. Bernoulli's 
 obfervation, juft mentioned, tends to account for a very 
 general opinion, that the greateft tides are in the equi- 
 noxes. I obferve, however, that this opinion is far from 
 being well eftabliihed. Both Sturmy and Coleprefs fpeafc 
 of it as quite uncertain, and Wallis and Flamftead reject 
 it. It is agreed on all hands that our winter tides exceed 
 the fummer tides. This is thought to confirm that point 
 of the theory which makes the Sun's accumulating force 
 greater as his diflance diminiflies. I am doubtful of the 
 applicability of this principle, becaufe the approach of 
 the Sun caufes the Moon to recede, and her recefs is in 
 the triplicate ratio of the Sun's approach. Her accumu- 
 lating force is therefore ulminifhed in the fefquiplicate 
 ratio of the Sun's approach, and her influence qn the 
 phenomena of the tides exceeds the Sun's. 
 
 713. The changes arifmg from the Moon's change 
 
 of diftance are more confiderable than tliofe arifing from 
 
 4 N 2 her 
 
S^2 PHYSICAL ASTRONOMir. 
 
 her change of declination. By reafon of their implica- 
 tion with thofe changes, the comparifon becomes more 
 difficult. M. Bernoulli did not find it fo fadsfaclory. 
 They are, in general, much lefs than theory requires. 
 This is probably owing to the mutual effects of undula- 
 tions which fhouid differ very confiderably, but follow 
 each other too clofely. In M. de la Place's way of con- 
 fidering the phenomena (to be mientioned afterwards) the 
 diminution in magnitude is very accountable, and, in 
 other refpe^ts, the coiTefpondence is greatly improved. 
 When the Moon changes either in perigeo or apogeo, 
 the feries is confiderably deranged, becaufe the next 
 fpring-tide is form.ed in oppofite circumftances. Tlie de- 
 rangement is flill greater, when the Moon is in perigee 
 or apogee in the quadratures. The two adjoining fpring- 
 tides fhouid be regular, and the two neap-tides extremel|r 
 unequal. 
 
 714. We {hall firft confider the changes produced 
 on the times of full fea, and then the changes in the 
 height. M. Bernoulli has computed a table for both the 
 perigean and apogean diftance of the Moon, from which 
 it will appear what corre6lion muft be made on the re- 
 gular feries. It is computed precifely in the fame way 
 as the former, the only difference being in the magni- 
 tude of M and S, and we may imitate it by a conftrucr 
 lion fimilar to fig. 72. To make this table of eafier ufe, 
 ]^'I. Bernoulli introduces the important obfervation, that 
 ^he greateft tide is not, in any part of the world, the 
 
 tide 
 
THEORY OF THE TIDES. ^5J 
 
 tide which happens on the lay o^ r.^vr or full Moon, 
 nor even the firit or the feconu tide after j and that with 
 vefpeiSl to the Atlantic Ocean, and all its coifls, It is 
 very precifely the third tide. So that fiionld we have 
 high water in any port precifely at noon on the fuii i^r 
 change of the Moon, and on the firft day of the m.^iAth, 
 the greateft tide happens at midnight on tlie fecond day 
 of the month, or, exprefling it in the common way, It 
 is the tide which happens when the Moon is a day and a 
 half eld. The fummit of the fpheroid is therefore 19 
 or 20 degrees to the eaftw^rd of the Sun and Moon. 
 At this diftance, the tendency of the accumulating force* 
 of the Sun and Moon to complete the fpheroid, and to 
 bring its pole precifely under them, is juft balanced by 
 the tendency of the waters to fubfide. Therefore it i$ 
 raifed no higher, nor can it come nearer to the Sun and, 
 Moon, becaufe then the obliquity of the force is dimi- 
 niflied, on which the changing power depends. That 
 this is the true caufe, appears from this, that it is, in 
 like manner, on the third tide that all the changes are 
 perceived which correfpond to the declination of the 
 Moon, or her diilance from the Earth. Every thing 
 falls out therefore as if the luminaries were 19 or 20 de- 
 grees eaftward of where they are, having the pole of the 
 fpheroid in its theoretical fituation with refpe£t to this 
 fictitious fituation of the luminaries. But, in fuch a 
 cafe, were the Sun and Moon 20° farther eaftward, they 
 would pafs the meridian 80 minutes, or one hour and 2» 
 minutes later. Ther-efore i^ 20' is added to the hours 
 
 of 
 
^4 PHYSICAL ASTRONOMY. 
 
 of high water of the former table, calculated for the 
 mean diftance of the Moon from the Earth. Thus, on 
 the day of new Moon, we have not the fprhig-tide, 
 but the third tide before it, that is, the tide which 
 fliould happen when the Moon is 20° weft of the Sun, 
 or has the elongation 160°. This tide, in our former 
 table, happens at ii'^ 02'. Therefore add to this i^ 20', 
 and we have o" 22' for the hour of high water on the 
 day of full and change for a harbour which would other- 
 wife have high water when the Sun and Moon are on 
 the meridian. In this way, by adding !*» 20' to the hours 
 of high water in the former table for a pofition of the 
 luminaries 20° farther weft, it is accommodated to the 
 obferved elongation of the Moon, this elongation being 
 always fuppofed to be that of the Moon when fhe is on 
 the meridian. Such then is the following table of M. 
 Bernoulli. The firft column gives the Moon's elongation 
 from the Sun, or from the oppofite point of the heavens, 
 the Moon being then on the meridian. The fecond co- 
 lumn gives the hour of high water when the Moon is in 
 perigeo. The third column (which is the fame with the, 
 former table, with the addition of i^ 20') gives the hour 
 of high water when the Moon is at her mean diftance. 
 And the fourth column gives the hour when fhe is in 
 apogeo. 
 
 TABLE 
 
THEORY OF THE TIDES. 
 
 6iS 
 
 TABLE II. 
 
 dcc'e 
 
 d in 
 
 (J in 
 
 d in 
 
 d. 
 
 ,„1 
 
 d 
 
 inl 
 
 d 
 
 ml 
 
 Perigeo. 
 
 M.Dift. 
 
 Apogeo 
 
 Perigeo. 
 
 M.Difl. 
 
 Apogeo 
 
 o 
 
 -.18 
 
 -.2Z 
 
 -.27i 
 
 > 
 
 18 
 
 >22 
 
 > 
 
 27 
 
 
 
 
 / 2. 
 
 1-^ 
 
 
 •-n 
 
 
 *-n 
 
 
 
 
 
 
 
 
 
 i-f 
 
 
 10 
 
 -.491 
 
 -5ii 
 
 -54 
 
 
 
 9r 
 
 
 llf 
 
 
 
 14 
 
 20 
 
 1.20 
 
 1.20 
 
 1.20 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3® 
 
 1.50! 
 
 1.481 
 
 1.46 
 
 9T 
 
 t? 
 
 
 5:?^ 
 
 Hi 
 
 
 M 
 
 
 40 
 
 2.22 
 
 2.18 
 
 2.12i 
 
 18 
 
 
 
 
 22 
 
 
 
 27i 
 
 0^ 
 
 50 
 
 2.54 
 
 2.48I 
 
 2.40^ 
 
 26 
 
 
 
 3ii 
 
 
 
 393 
 
 C 
 
 60 
 
 3.27 
 
 3.20 
 
 3.10 
 
 33 
 
 g 
 
 40 
 
 g 
 
 50 
 
 2 
 
 70 
 
 4. 21 
 
 3-55 
 
 3-44 
 
 37 
 
 8 
 
 45 
 
 
 
 
 S6 
 
 8 
 
 80 
 
 4411 
 
 4.33I 
 
 4.22 
 
 38t 
 
 a." 
 
 46i 
 
 
 58 
 
 
 90 
 
 5.26I 
 
 5-191 
 
 5. 9i 
 
 33t 
 
 
 
 4oi 
 
 
 
 5oi 
 
 
 
 100 
 
 6.19 
 
 6.15 
 
 6.9 
 
 22 
 
 5". 
 
 5' 
 
 25 
 
 B- 
 
 31 
 
 5* 
 
 no 
 
 7.20 
 
 7.20 
 
 7.20 
 
 — 
 
 °? 
 
 — 
 
 °? 
 
 — 
 
 crq 
 
 120 
 
 8.21 
 
 8.25 
 
 8.31 
 
 > 
 
 21 
 
 > 
 
 25 
 
 > 
 
 31 
 
 130 
 
 p-'^s^- 
 
 9.20I 
 
 9-3oi 
 
 
 33 
 
 
 •-i 
 
 40 
 
 
 
 -1 
 
 5c 
 
 140 
 
 9-58i 
 
 10. 6i 
 
 ro.i8 
 
 
 38 
 
 3- 
 
 46 
 
 
 
 58 
 
 150 
 
 10.37^ 
 
 10.45 
 
 10.56 
 
 2 
 
 37 
 
 
 45 
 
 2 
 
 C/5 
 
 56 
 
 160 
 
 II. 13 
 
 11.20 
 
 1 1.30 
 
 CO 
 
 
 
 33 
 
 CO 
 
 o 
 
 40 
 
 C/2 
 
 
 
 5c 
 
 170 
 
 11.46 
 
 1 1.5 14 
 
 ri.59i 
 
 ?- 
 
 26 
 
 5- 
 
 31 
 
 ?t 
 
 39 
 
 180 
 
 —.18 
 
 — .22 
 
 —27^ 
 
 5* 
 
 18 
 
 ^ 
 
 22 
 
 5' 
 
 27 
 
 715. This table, though of confiderable fervice, be- 
 ing far preferable to the ufual tide tables, may fometimes 
 deviate a few minutes from the truth, becaufe it is cal- 
 culated on the fuppofition of the luminaries being in the 
 equator. But when they have confiderable declination, 
 
 the 
 
6^&, PHYSICAL ASTRONOMY. 
 
 the horary arch of the equator may differ two or thrj^e 
 degrees from the elongation. But all this error will be 
 avoided by reckor:ing the high water from the time of 
 the Moon's fouthing, whicli is always given in oar al- 
 manacks. This interval being alv/ays very fmall (ne- 
 ver 12°) the error will be infenflble. For this reafon, 
 the three other columns are added, exprefling the prim- 
 ing of the tides on the Moon's fouthing. 
 
 To accommodate this table to all the changes of the 
 Moon's declination v^-ould require more calculation than 
 all the reft. We Ihall come near enough to the truth, if 
 we leffen the minutes in the three hour-columns tV when 
 the Moon is in the equator, and increafe them as much 
 when fhe is in the tropic, and if we ufe them as they 
 iland when flie is in a middle fituation. 
 
 716. All that remains now, is to adjufl this general 
 table to the peculiar fituation of the port. Therefore, 
 colleft a great number of obfervations of the hour of 
 high water at full or chanri"? of the Moon. In making 
 this colledtion, note p.irticuhvly tlie hour on thofe days 
 where the Moon is new or full precifely at' noon ; for 
 this is tlie circumfti^nce necefiary for the truth of the 
 elongations in the niil colurni' of the table. A fmall 
 equation is neceflary i\ r correcting the obferved hour of 
 high water, when the ^yzigy is not at noon, becaufe in 
 this fituation of the lum.inarics, the tide lags 35' behind 
 the Sun in a day, as has been already fliewn. Suppofe 
 jhe lagging to he 36', this v.ill make the equation hi mi- 
 nute 
 
THEORY OF THE TIDES. 657 
 
 nute for every hour that the full or change has happened 
 before or after the noon of that day. This corre£tiou 
 tnufl be added to the obferved hoUr of high water, if the 
 fyzigy was before noon, and fubtracfled, if it happened 
 after noon. Or, if we choofe to refer the time of high 
 water to the Moon's fouthing, which, in general, is the 
 heft method, we muft add a minute to the time between 
 high fea and the Moon's fouthing for every hour and half 
 that the fyzigy is before noon, and fubtra6t it if the fy- 
 zigy has happened after noon. For the tides prime 15' 
 hi 24 hours* 
 
 717. Having thus obtained the medium hour of high 
 water at full and change of the Moon, note the differ- 
 ence of it from o^^ 22', and then make a table peculiar to 
 that port, by adding that difference to all the numbers of 
 the columns. The numbers of this table will give the hour 
 of high water correfponding to the Moon's elongation for 
 any other time. It will, however, always be more exact 
 to refer the time to the Moon's fouthing, for the reafons 
 already given. 
 
 By means of a table fo conftru6led, the time of high 
 water for the port, in any day of the lunation, may be de- 
 pended on to lefs than a quarter of an hour, except the 
 courfe of the tides be diflurbed by winds or frefnes, 
 which admit of no calculation.- It might be brought 
 nearer by a much more intricate calculation j but this is 
 altogether unnecelfary, on account of the irregularities 
 arifnig from thofe caufes. 
 
 4O It 
 
6^S' PHYSICAL ASTRONOMY. 
 
 It is not (o eafy to ftate in a feries the variations 
 which happen in the height of the tides by the MoonV 
 change of diftance, although they are greater than the 
 variations in the t}/?ies of high water. This is partly ow- 
 ing to- the great differences v/hich obtain in different ports 
 between the greateil and fmalleft tides, and partly from 
 the difficulty of expreffmg the variations in fuch a man- 
 ner as to be eafiiy underftood by thofe not familiar with 
 mathematicai computations. M. Bernoulli, whom we 
 have followed in all" the prafiical inferences from the 
 phyfical theory, iinagines that, notwithflanding the great 
 difproportion between the fpring and neap-tides in dif- 
 fei^nt places, and the differences in the abfolute magni- 
 tudes of both, the middle betv/een the higheil and loweft 
 daily variations will proceed in very nearly the fame way 
 as in theory. Inflead therefore of taking the values of 
 M and S as already eftabliflied, he takes the height of 
 fpring and neap-tides in any port as indicative of M -{- S- 
 and M — S for that port. Calling the fpring-tide A 
 and the neap-tide B, this principle will give us M = 
 
 — ~ — , and S = . From thefe values of M 
 
 2 2 
 
 and S he computes their apogean and perigean values, 
 
 and then conftrucls columns of the height of the tides, 
 apogean and perigean, in the fame manner a« the column 
 already computed for the mean diftance of the Moon, that 
 is, computing the parts mfznd af{Rg, 72.) of the whole 
 tide m a feparately. The fame may be done with in- 
 comparably lefs trouble by our conftru£llon (fig. 72.) and 
 
 - _ A + B ' A — B 
 the values M = , and S = . 
 
 Although 
 
THEORY OF THE TIDES. 6f 9 
 
 Although this is undoubtedly an approximation, and 
 perhaps all the accuracy that is attainable, it is not 
 founded on exaiSl phyfical principles. The local propor- 
 tion of A to B depends on circumftances peculiar to 
 the place ; and we have no affurance that the changes of 
 the lunar force will operate in the fame manner and pro- 
 portion on thefe two quantities, however different. We 
 are certain that it will not ; otherwife the proportion of 
 fpring and neap-tides would be the fame in all harbours, 
 however much the fprings may diiFer in different har- 
 bours. I com.pared Bernoulli's apogean and perigean 
 tides, in about twenty inftances, fele£led from the obfer- 
 vations at Breft and St Malo, where the abfolute quan- 
 tities differ very widely. I was furprifed, but not con- 
 vinced, by the agreement. I am however perfuaded 
 that the table is of great ufe, and have therefore inferted 
 it, as a model by which a table may eafily be computed 
 for any harbour, employing the fpring-tide and neap- 
 tide heights obferved in that harbour as the A and B 
 for that place. The table is, like the lafl, accommo- 
 dated to the eafterly deviation of the pole of the fphe- 
 roid from its theoretical place. 
 
 It appears from this table, and alfo from the lafl, 
 that the neap-tides are much more affefted by the ine- 
 quahties of the forces than the fpring- tides are. The 
 neap-tides vary from 70 to 128, and the fprings from 90 
 to 114. The firft is almoft doubled, the lafl is augment- 
 ed but J. 
 
 , 4O 2 TABLE 
 
666 
 
 PHYSICAL ASTRONOMY, 
 
 
 TABLE III. 
 
 
 a 
 S 
 
 o 
 
 HviGHT OF THE TIDE. 
 
 1 
 
 a in Perlgeo. | <X in M. Dill;. 
 
 (X in Apogeo. 
 
 o,99z\-f 0,156 
 
 o,88A-i-o,i2-> 
 
 o,79A-f 0,086 
 
 10 
 
 i,ioA-f-o,o4B 
 
 o,97A + o,o3B 
 
 o,87A-f 0,026 
 
 20 
 
 i,i4x\-f-OjOoB 
 
 i, 00 A -f 0,006 
 
 o,9oA-f 0,006 
 
 30 
 
 1,10 A -f- 0,046 
 
 0,97 A + 0,036 
 
 o,87A + o,c2>j 
 
 40 
 
 o,99A + o,i5B 
 
 o,88A-i-o,i2B 
 
 o,79A-j-o,o8i3 
 
 50 
 
 o,85A+o,32B 
 
 o,75A-ho,25B 
 
 o,68A-|-o,i86 
 
 60 
 
 o,67A + o,53B 
 
 o,59A-t-o,4i6 
 
 o,53A+o,296 
 
 70 
 
 o,46A-fo,75B 
 
 0,41 A+ 0,596 
 
 0,37 A + 0,4 [B 
 
 80 
 
 a,28A+c,96B 
 
 c,25A+o,756 
 
 o,23A+o,536 
 
 90 
 
 o,i3A-f 1,136 
 
 o,i2A + o,886 
 
 o,iiA4-o,626 
 
 100 
 
 0,03 A4- 1,246 
 
 o,o3A-fo,976 
 
 0,03 A +0,686 
 
 1 10 
 
 o,ooA-f i,28B 
 
 o,ooA+ 1,006 
 
 o,ooA+o,7o6 
 
 120 
 
 0,03 A+ 1,246 
 
 o,o3A + o,976 
 
 o,o3A+o,686 
 
 130 
 
 o,i3A4-i,i3B 
 
 o,i2A + o,888 
 
 o,iiA + o,626 
 
 140 
 
 o,28A-f.o,96B 
 
 o,25A-|-o,756 
 
 o,23A + o,53B 
 
 150 
 
 o,46A+o,75B 
 
 c,4iA+o,596 
 
 0,37^+0,416 
 
 160 
 
 o,67A-|-o,53B 
 
 0559A+o,4iB 
 
 o,53A+o,29B 
 
 170 
 
 0,85 A -I- 0,3 26 
 
 o,75A+o,256 
 
 . o,68A+o,i86 
 
 180 
 
 o,99A-f 0,156 
 
 o,88A4-o,i2B 
 
 0379A + o,o86 
 
 719. The attentive reader cannot but obferve that 
 all the tables of this monthly conftru6lion mitft be very 
 imperfeft, although their numbers are perfe611y accu- 
 rate, becaufe, in the courfe of a month, the declination 
 and diftance of the Moon vary, independently of each 
 
 other. 
 
THEORY OF THE TIDES. 66t 
 
 Other, through all their poflible magnitudes. The laft 
 table is the only one that is immediately applicable, by 
 interpolation. It would require feveral tables of the 
 fame extent, to give us a fet of equations, to be applied 
 to the original table of art. 66']. ; and the computation 
 would become as troublefome for this approximation as 
 the calculation of the exaci value, taking in every cir- 
 cumftance that can afFe^l: the que ft ion. For that calcu- 
 lation requires only the computation of two right-angled 
 fpherical triangles, preparatory to the calculation of the 
 place of high water. But, with all thefe imperfe£l;Ions, 
 M. Bernoulli's fecond table is much more exa£t than any 
 ^'Ide table yet publifhed, 
 
 Such, cu the whole, is the information furniflied by 
 the do£lrine of univerfal gravitation concerning this cu- 
 rious and important phenomenon. It is undoubtedly 
 the mod irrefragable argument that we have for the 
 iruth and unlvcrfality of this dodlrine, and at the fam.e 
 time for the fimpiicity of the whole conftitution of the 
 folar fyftem, fo far as it can be confidered mechanically. 
 No new principle is required for an operation of nature 
 fo unlike all the other phenomena in the fyllem. 
 
 720. The method which I have followed in the In- 
 veftigation is nearly the fame with that of its iiluftrious 
 difcoverer. We have contented ourfelves with (liewing 
 various feriefes of phenomena, which tally fo well with 
 the legitimate confequences of the theory, that the real 
 
 fource 
 
662 PHYSICAL ASTRONOMY. 
 
 •foiirce of them can no longer be doubted. And, not- 
 withftanding the various deviations from thofe confe- 
 <juences, arifing from other circumflances, we have ob- 
 tained praftical rules, v^hich njake the mariner pretty- 
 well acquainted v/ith the general courfe of the tides •, 
 fufficiently to put him on his guard againfl the dangers 
 he runs by grofsly niiftaking them, and even enabling 
 him to take advantage of the courfe of the tide for pro- 
 fecuting his voyage. Still, however, a great ftore of lo- 
 cal information is neceflary. For there are fome parts of 
 the ocean, where the tides follow an order extremely un- 
 like what we have defcribed. The bar of Tonquin in 
 China is one of the moll remarkable j and its chief pe- 
 culiarity confifls in its having but one tide in each lunar 
 day. It has been traced to the cooperation of two great 
 tides, coming from oppofite quarters, with almoft fix 
 hours of difference in the time of high water. The re- 
 fult of which is, that the compound tide is the excefs of 
 the one above the other, forming a high water when the 
 fum of both their elevations is a maximum. Dr Halley 
 has given a very difhin^l: explanation of this tide in N'^ i6z 
 of the Philofophical Tranfadtions. 
 
 721. A very different method of inveftigating this 
 and a fimilar phenomenon has been employed by the emi- 
 nent mathematicians D^Alembert and La Place, in which 
 M. La Place, wlio makes this a chief article of his Me- 
 chanique Celefte, deduces the whole dire£lly from the 
 interior mechanifm of hydroflatical undulations. His 
 main inferences perfedlly agree with thofe already deli- 
 vered^ 
 
THEORY OF THE TIDES. 66;^' 
 
 vercd. The method of Newton and Bernoulli has been- 
 pref erred here, becaufe by this means the connexion 
 with the operation of univerfal gravitation is much bet- 
 ter kept in fight. At the fame time La. Place's me- 
 tliod allows us, in fome cafes, to ibate the individual facfc 
 more nearly as it occurs, without confidering it as the 
 modification of another fact that is more general. But 
 it may be doubted, whether La Place has explained 
 all the variety of phenomena* His whole application is 
 limited by the data which furnifli the arbitrary quantities 
 in his equations. Thefe being wholly taken from the ob- 
 fervations in the ports of France and Spain, it may be 
 queftioned whether the famenefs, arifing from the lati- 
 tude being fo near 45°, may not have made the ingeni- 
 ous author fimplify too much his theory. He confiders 
 every clafs of phenomena as operations completely ac- 
 complifhed, and the ocean at the end of the action of 
 any one of the forces as in a flate of indifference, ready 
 for the free operation of the next. For example, the 
 equality of the fuperior and inferior tides of one day is 
 deduced by La Place immediately from the circumftance 
 of the ocean being of nearly an uniform depth, faying 
 that the fmall inferior tide is not affe^led by the great- 
 nefs of the preceding fuperior tide, becaufe the obftruc- 
 tions are fuch that all motions ceafe very foon, alrnoft 
 immediately after the force has ceafed to act. We doubt 
 the truth of the near uniformity of the fea's depth. The 
 unequal tides are confeffedly moil remarkable on the 
 coafts, where the depth is the mofl unequal. The other 
 
 principle^ 
 
6(j^ IPHYSICAL ASTRONOMY. 
 
 principle, that the effe^ls of primitive motions are all 
 obliterated, and therefore every tide is the completed 
 operation of the prefent force, is ftill more queftionable. 
 It is well known that the roll of a great ftorm in the 
 Bay of Bifcay is very fenfible indeed for three days. Of 
 this we have had repeated experience* The fuperficial 
 agitation of a ftorm (for it is no more) is nothing in 
 eomparifon with the huge uniform momentum of a tide 5 
 and the g re ate ft ftorm, even v/hile it blows, cannot raife 
 the tide three feet •, nor does it even then change what 
 we have called the tide, the difference between high and 
 low water *, it raifes or keeps dov/n both nearly alike. 
 Befides, how v/ill M. La Place account for the unde- 
 niable duration of every tide wave on the coafts of Eu-^ 
 tope and America for a day and a half? There can be 
 no queftion about this, becaufe the courfe of the tides 
 during a month is precifely conformable to it. The tide 
 which bears the mark of the perigean tide is not the 
 tide which happens when the JMoon is in perigeo, but 
 the third following that tide, juft as in the fprings and 
 neaps. In like manner, it is obferved at Breft, without 
 one exception for fix years, that the morning or fuperior 
 tide at nev/ Moon is fmaller than the inferior tide in 
 fummer. In winter it is the contrary, not, however, 
 with fuch conftant accuracy. Now, it fhould be juft the 
 contrary, if the tides obferved were the tides correfpond- 
 ing with the then ftate of the forces. But they are not. 
 They are tides correfponding with the ftate of the forces 
 thirty-fix hours before. (See Mem. Acad. Par. 1720, 
 
THtORY OF THE TIDES. 665 
 
 p. 206, duodecimo). It is the fame at full Moon, that 
 is, the morning tide In fummer is lefs than the evening 
 tide. The morning tide correfponding to the then ftate 
 of the forces is what we have called an inferior tide, the 
 Moon being then under the horizon, with fouth declina- 
 tion. The tide therefore fhould be greater than the 
 fubfequent or evening, or fuperior tide. But, like the lad 
 example, it is the tide correfponding to the forces in 
 a6lion thirty-fix hours before. Can we now deny that 
 the prefent ftate of the waters is affected by the a6tion 
 of forces which have ceafed thirty-fix hours ago ? and i£ 
 this be granted, it is impoffible that two tides immedi- 
 ately fucceeding can be very unequal. The contrary can 
 be fliewn in an experiment perfectly refembling the great 
 tides of the ocean. An apparatus, made for exhibiting 
 the appearance of a reciprocating fpring, was fo con- 
 ftru6led that one of its runnings was very fudden and 
 copious, and the next was moderate and flow. It emp- 
 tied into a fmall bafon, which communicated with a 
 long and narrow horizontal channel, fhut at the far end, 
 the bafon emptying itfelf by a fmall fpout on the oppo- 
 fite fide. Thus, two very unequal floods and ebbs pre- 
 fented themfelves at the mouth of this channel, and fent 
 a wave along it, which, at the firft, was very unequal. 
 But, when it was mixed with the returning wave from 
 the far end, they were foon brought to an apparent equa- 
 lity. The experiment appearing curious, it was profe- 
 cuted, by various changes of the apparatus ; and feveral 
 ciFeds tended very much to explain fome of the more 
 4 P iinguhx 
 
€66 PHYSICAL ASTRONOMY. 
 
 Cngular appearances of the tides. There is an examplie 
 of the continuance of former impreffions in the tides 
 amonjT the weftern iilands of Scotland, that confiderably 
 refembles the tide on the bar of Tonquin. The general 
 courfe of the flood round the little ifland of Berneray is 
 N. E. and that of the ebb is S W. But at a certain 
 time in the fpring, both flood and ebb run N. E. during 
 twelve hours, and the next flood and ebb run S. W. 
 The contrary happens in autumn. Yet in the ofHng, 
 the flood and ebb hold their regular courfes. This greatly 
 refembles the tide at Tonquin, and aifo the Grecian 
 Euripus. 
 
 722. The reader will recollect that we ftatedas our 
 opinion that, in confequence of the inertia of the wa- 
 ters, the pole of the ocean is always to the eaftward of 
 its theoretical place. For which reafon, the figure ac- 
 tually attained by the ocean is not a figure of equilibra- 
 tion. Did the Earth (land ftill, it would foon be brought 
 to its proper pofition, and completed to its due form. 
 Therefore, there is always a motion towards this com- 
 pletion : And this motion is ohjirucled. Hence we appre- 
 hend that there m.uft be a perpetual current of the wa- 
 ters, efpecially in the tropical regions, from eafl to weft. 
 We cannot fee how this can be avoided •, and we think 
 that it is eftablifhed as a matter of nautical obfervation. 
 In regard to the Atlantic, this feems to be a general opi- 
 nion of the navigators. There are two very excellent 
 journals of voyages from Stockholm to China, by Cap- 
 tain 
 
•n^CORY OF THE TIDE9. 66^ 
 
 tain Eckhart, in which there is a very frequent compa- 
 xifon of the (hip's reckoning with Umar obfervations and 
 the arrivals on known coafts, from which we cannot help 
 inferring the fame general current in the Indian and Ethi- 
 opic feas. It feems therVore to obtain over the whole. 
 The part of this current which difFufes itfelf into the At- 
 lantic is but fmall, it having a freer paflage ftraight for- 
 ward. But the part, thus difFufed produces the gulf 
 ftream, in its way along the American coafts, and efcapes 
 round the north capes of Europe and America. In all 
 probability, a foutherly current may be obferved in the 
 flraits which feparate America from the Afiatic conti- 
 nent. The whole amount of this motion cannot be con- 
 fiderable, but there muft be fome, if there be two cir- 
 cumpolar comm.unications between the great eaftern and 
 weftern divifions of the ocean. Without this, it mufh 
 be reduced to a reciprocating motion too intricate for in- 
 yeiligation. 
 
 723. There is another circumftance v/hich feems to 
 strengthen our confidence in the reality of this weflerly 
 current of the ocean. The gravity of the waters being 
 more diminifhed in conjund:ion and oppofition than it is 
 augmented in quadrature with the adding luminary, each 
 particle tends to recede from the centre, and to defcribe 
 a larger circle, employing a longer time. Here is a ten- 
 dency or mfus to a relative motion weflerly. Water, 
 being almoft perfectly fluid, will obey this tendency, and 
 in time acquire fuch a motion, were it not obftruded by 
 
 4.P 2 foliil 
 
^^8 PHYSICAL ASTRONOMY. 
 
 folid obflacles. But fome effeO: muft remain, too intri- 
 cate to admit any calculation, and perhaps not ultimately 
 fenfible. 
 
 724. If the height of the atmofphere be equal to the 
 radius of the Earth, we ihall have a tide in the air double 
 of that in the ocean. "When all the afFe^ling circum- 
 ftances are confidered, it appears that an ebb and flood 
 of the atmofphere may differ in elevation about 1 20 feet. 
 This might be fenfible by afFefting the barometer. True, 
 the gravity of the mercury is alfo diminifhed, but not fo 
 much as that of the more diflant air. But the height of 
 the atmofphere is too fmall to give rife to any fuch tides. 
 They cannot fenfibiy exceed thofe of the ocean, and this 
 cannot change the height of the mercury in the barome- 
 ter — o of an inch. ProfelTor Toaldo at Padua' kept a 
 regifter of the barometer for more than thirty years. He 
 has added into one fum all the mercurial heights obferved 
 at nev/ Moon. Another fum was made of all the heights 
 obferved in the quadratures ; another of the perigean 5 
 and another of the apogean heights, &c. &c. He thinks 
 that differences were obferved in thofe fums fuilicient for 
 proving the accumulation and comprefiion of the air by 
 its unequal gravitation to the Moon. Thus, the apogean 
 heights exceeded the perigean by 14 inches. The heights 
 in fyzigy exceeded thofe in quadrature by 1 1 inches. 
 (See Mem. Berlin 1777, and a book exprefslyon the fub- 
 
 But there is another efFe£l of this diflurbing force 
 
 which 
 
TRADE WINDS. ^6^ 
 
 which may be much more fenfible, namely, the general 
 wefterly current of the air. M. D'Alembert has invef- 
 tigated this with great care, and fingular addrefs, and has 
 proved that there muft be a wefterly current in the tro- 
 pical regions, at the rate of eight feet nearly in a fecond. 
 This is a very adequate caufe of the trade winds which 
 are obferved between the tropics. It is indeed increafed 
 by the rarefa£tion of the air occafioned by the heat of the 
 Sun, which expands the air heated by the ground, and 
 it is both raifed and diiFufed laterally. WJhen the Sun 
 has pafled the meridian a proper number of degrees, the 
 air mufi: now cool, and in cooling contract behind the 
 Sun. Air from the eaft comes in greater abundance 
 than from any other quarter to fupply the vacancy* 
 
 725. The dilK of Jupiter, when viewed through a 
 good telefcope, is diftinguifhable into zones, like a bit of 
 itriped fatin. Thefe zones, or belts, are of changeable 
 breadth and pofition, but all parallel to his equator. 
 Therefore they are not attached to his furface, but float 
 on it, as clouds float in our atmofphere. This Earth 
 will have fomew^hat of this appearance, if viewed from the 
 Moon. For each climate has a fcate of the Iky peculiar 
 in fome degree to itfelf in this refpe£t, and there mult 
 be a fort of famenefs in one climate all round the globe. 
 A feries of obfervations on a particular fpot of Jupiter's 
 furface demonitrate his rotation in 9^ ^6\ Spots have 
 been obferved in the belts, which have lafled fo long as 
 to make feveral revolutions before they were effaced. 
 
 They 
 
^7^ PHYSICAL ASTRO-NTOMT.' 
 
 They appear to require a minute or two more for their 
 rotation, and therefore have a wefterly motion relative to 
 the firm furface of the planet. This hov^ever cannot be 
 depended on from the time of their rotation. But a 
 few obfervations have been had of fpots in the vicinity of 
 the fixed fpot of his furface, and here the relative mo- 
 tion weftward was diftin6lly obferved. M. Schroetcr at 
 Manheim has obferved the atmofphere of Jupiter with 
 great care, and finds it exceedingly variable ; and fpots 
 are obfer^^ed to change their fituations with amazing ra- 
 pidity, with great irregularity, but molt commonly eaft- 
 ward. The motions and changes are fo rapid, and fo 
 extenfive, that we can fcarcely confider them as the 
 transference of matter from one place to another. They 
 more refemble the changes which happen in our atmo- 
 fphere, which are fometimes progreflive, over a great 
 tra£l of the country. The ftorm in 1772 was felt from 
 Siberia to America in fucceffion. The gale blew from 
 the weft, but the chemical operation which produced it 
 was in the oppoflte direction, being firft obferved in Si- 
 beria 5 three days afterward, it was felt at St Peterfburg ; 
 two days hfter this, at Berlin ; two days more, it was in 
 Britain ; and feven days after, it was felt in North Ame- 
 rica. Here then, vv^hile a fpe£lator on the Earth faw the 
 clouds moving to the eaftward, a fpedtator in the Moon 
 would fee the change of appearance proceed from eaft to 
 weft. The motions in the atmofphere of Jupiter muft 
 he very complicated, becaufe they are the joint operation 
 •f four fatellites. The inequality of gravitation to the 
 
 firft 
 
THEORY OF THE TIDES. &fl 
 
 firft fatelHte muft be very great. And as each fatellhe 
 produces a peculiar tide, the combination of all their ac- 
 tions muft be very intricate. We can draw no conclu- 
 fions from the variable fpots, becaufe their change o£ 
 place is no proof of the a£lual transference of matter. 
 
 Such a relative motion in our atmofpliere and in the 
 ocean may affect the rotation, retarding it, by its a£lioii 
 on the eaftern furface of every obftacle. Yet no change 
 is obferved. The year, and the periods of the planets, in 
 the time of Ptolemy are the fame with the prefent, that 
 is, contain the fame number of rotations of the Earth. 
 Perhaps a compenfation is maintained by tliis means for 
 the acceleration that fhould arife from the transference 
 of foil from the high land to the bottom of the fea, 
 where it is moving round the axis witli diminifhed velo- 
 city. 
 
 726. With this we conclude our account of phyfical 
 aftronomy, a department of natural philofophy which 
 fhould ever be cherifhed with peculiar affeftion by all 
 who think well of human nature. There is none in 
 which the accefs to well founded knowledge feems fo 
 effedtally barred againft us, and yet there is none in 
 which we have made fuch unqueftionable progrefs j none 
 in which we have acquired knowledge fo uncontrovertibly 
 fupported, or fo complete. How much therefore are we 
 indebted to the man who laid the magnificent fcene open 
 to our view, and who gave us the optics by which we 
 fan examine its molt exteufive, and its moft minute 
 
 parts I 
 
J^^2 PHYSICAL ASTRONOM'/. 
 
 parts ! For Newton not only taught us all that we kn6\V 
 of the celePuial mechanifm, but alfo gave us the mathe- 
 matics, without which it would have remained unfeen, 
 
 * TW Pater et verum Inventor, Tu patria nobis 
 
 * Suppeditas pracepta, tuifqiie ex inclyte chartis 
 
 * Floriferis ut apes in faliihus omnia libanty 
 
 * Omnia nos itidem depafdmur aiirea diBa 
 
 * Aurea, perpetiu femper dignijftma vitu, * 
 
 Lucretius. 
 For furely, the lefTons are precious by which we are 
 taught a fyftem of docftrine which cannot be iliaken, or 
 ffiare that fluft nation which has attached to all other fpe- 
 culations of curious man. But this cannot fail us, be- 
 caufe it is nothing but a well ordered narration of facets, 
 prefenting the events of nature to us in a way that at 
 once points out their fubordination, and mod of their re- 
 lations. "While the magnificence of the obje£ts com- 
 mands refpeft, and perhaps raifes our opinion of the ex- 
 cellence of hum.an reafon as high as is juilifiabls, we 
 fhould ever keep in m.ind that Newton's fuccefs was ow- 
 ing to the modefty of his procedure. He peremptorily 
 refilled all difpofition to fpeculate beyond, the province of 
 human intelleft, confcious that all attainable fcience con- 
 fided in carefuljy afcertaining nature's own laws, and 
 that every attemipt to explain an ultimate law of nature 
 by affigning its caufe is abfurd in itfeif, againft the ac- 
 knowledged laws of judgement, and will moil certainly 
 lead to error. It is only by following his example that 
 "we can hope for his fuccefs. 
 
 It 
 
GENERAL REFLECTIONS. 67^ 
 
 it is furely another great recommendation of this 
 branch of natural pliilofophy, that It is fo fimple. One 
 fingle agent, a force decrcafing as the fquare of the 
 diftance increafes, is, of itfelf, adequate to the produc- 
 tion of all the movements of the folar fyftem. If the 
 direction of the proje£^ion do not pafs through the centre 
 of gravity, the body will not only defcribe an ellipfe 
 round the central body, but will alfo turn round its axis. 
 By this rotation, the body will alter its form. But the 
 fame power enables it to alTume a new form, which is 
 perfedily fymmetrical, and is permanent. This new 
 form, however. In confequence of the unlverfality of 
 gravitation, induces a new motion in the body, by which 
 the poGtion of the axis is flowly changed, and the whole 
 hofl of heaven appears to the inhabitants of this Earth to 
 change Its motions. Laftly, if the revolving planet have 
 a covering of fluid matter, this fluid is thrown into cer- 
 tain regular undulations, which are produced and modi- 
 fied, by the fame power. 
 
 Thus we fee that, by following this fimple fa<^ of gra- 
 vitation of every particle of matter to every other par- 
 ticle, through all its complications, we find an explana- 
 tion of almoft every phenomenon of the folar fyftem that 
 has engaged the attention of the philofopher, and that 
 nothing more is needed for the explanation. Till 
 we were put on this track of invefligation, thefe dif- 
 ferent movements were folitary fa£ls ; and, being fo 
 extremely unlike, the wit of man would certainly 
 have attempted to explain them by caufcs equally difli- 
 
 4 Q mihr. 
 
674 PHYSICAL ASTROKOMY. 
 
 milar. The happy detection of this fimple and eafily 
 obferved principle, by a genius qualified for following it 
 into its various ccnfequences, has freed us from number- 
 lefs errors, into which v/e mufc have continually run 
 while pertinncioufly proceeding in an improper path. 
 But this detection has not merely faved us from errors, 
 but, wliich is mod remarkable, it h-^.s brought into view 
 many circumftances in the phenom.ena themfelves, many 
 peculiarities of motion, which would never have been 
 obferved by us, had we not gotten this monitor, pointing 
 out to us where to look for peculiarities. We fhould 
 never have been able to predial, with fuch wonderful 
 precifion, the complicated motions of fome of the pla- 
 nets, had Vv^e not had this key to all the equations by 
 which every deviation from regular elliptical motion is 
 exprefied. 
 
 On all thefe accounts^ phyfical aflronomy, or the 
 mechanifm of the celeftial m.otions, is a beautiful de- 
 partment of fcience. I do not know any body of doc- 
 trine fo comprehenfive, and yet fo exceedingly fimple j 
 and this confideration made me the more readily accede 
 to thofe reafons of fcientific propriety which point it 
 out as the firft article of a courfe of medianical philofo- 
 phy. Its hmplicity makes it eafy, and the exquifite 
 agreement with obfervation makes it a fine example of 
 the truth and competency of our dynamical doctrines. 
 
 727. But it has other recommendations, of a far greater 
 Talue, Nothing furely fo much engages a heart pcfTefled 
 
 of 
 
GENERAL RErLECTlONS. 67^ 
 
 ©f a proper fenfibility, as the (Contemplation of order and 
 harmony. No philofophy is requifite for bei-^g fufcepti- 
 ble of this imprelTion. We fee it influence the con(lu6t 
 of the molt uncultivated. What elfe does man aim at 
 in all the buftle of cultivated fociety ? N;iy, even the 
 favage makes fome rude aim at order and ornament. 
 
 But what we contemplate in the folar fyftem is 
 fomething more than mere order and fymmetry, fuch 
 as may be obferved in a fine f^:)ecimcn of cryllalhza- 
 tion. The order of the folar fyftem is made up of 
 many palpable fuhferviences, where we fee one thing 
 plainly done for the fake of another thing. And, to 
 render this ftill more interefting, a manifcft utility 
 appears in every circumftance of the conilitution of 
 the fyftem, as far as we underftand its applicability 
 to what we conceive to be ufeful purpofes. We 
 can mean nothing by utility but the fubferviency to 
 the enjoyments of fentient beings. Our opportunities 
 for obfervations of this kind are no doubt very limited, 
 confined to our own fublunary habitatton. But this cir- 
 cumfcribed fcene of obfervation is even crowded with 
 examples of utility. Surely it is unneceflary to recal our 
 attention to the numberlefs - adaptations of the fyflematic 
 connexion with the Sun and Moon to the continuance > 
 and the difFufion of the means of animal life and enjoy- 
 ment. As our knowledge of the celeflial phenomena is 
 enlarged, the probability becomes flronger that other 
 planets are alio flored with inhabitants who fhare with 
 4 Q 2 us 
 
6']6 PHYSICAL ASTRONOMY. 
 
 US the Creator's bounty. Their rotation, and the evi- 
 dent changes that we fee going on in their atrnofphereSj 
 fo much refemble what we experience here, that I ima- 
 gine that no man, who clearly conceives them, can ihut 
 out the thought that thefe planets are inhabited by fen- 
 tient beings. And there is nothing to forbid us from 
 fuppofing that there is the fame inexhauftible ilore of 
 fubordinate contrivance for their accommodation tliat we 
 fee here for living creatures in every fituation, with ap- 
 propriate forms, defires, and abilities. I fear not to 
 appeal to the heart of every man who has learned fo 
 much of the celefcial phenomiena, even the man v/ho 
 fcouts this opinion, whether he does not feel the difpo- 
 fition to entertain it. And I infift on it, that fom.e good 
 reafon is required for rejecting it. 
 
 728. When beholding all this, it is impoffible to 
 prevent the furmife, at leafc, of purpofe, defign, and 
 contrivance, from, nrifing in the mind. AVe may try to 
 fliut it out — We may be convinced, that to allege any 
 purpofe as an argum^ent for the reality of any difputed 
 faft, is againft the rules of good reafoning, and that 
 final caufes are improper topics of argument. But we 
 cannot hinder the anatomifl, who obferves the exquifite a- 
 daptation of every circumftance in the eye to the forming 
 and rendering vivid and diftin£t a pifture of external ob- 
 jects, from believing that the eye was made for feeing — 
 
 or the hand for handling. Neither can we prevent our 
 
 heart 
 
GENERAL REFLECTIONS. tSjf 
 
 lieart from fuggefling the thought of tranfccndent wif- 
 dom, when we contemplate the exqiufite fitnefs andadjuft- 
 ment whicli the mechanifm of the folar fyllem exhibits 
 in all its parts. 
 
 729. Newton was certainly thus aiTeclcd, when he 
 took a ccnfidcrate view of all his own cliicovcries, and 
 perceived the almoft eternal order and harnicny which re- 
 fults from the fimple and u.:mixed operation of univer- 
 fal gravitation. This fingle iu€t produces all this fair 
 order and utility. Newton was a mathematician, and 
 faw that the law of gravituion obferved in the fyflem is 
 the only one that can fecure the continuance of order. 
 He was a philofopher, and faw that it was a contingent 
 law of gravitation, and might have been otherwife. It 
 therefore appeared to Newton, as it would to any unpre- 
 judiced mind, a law of gravitation feledled as the moft 
 proper, out of many that were equally poffible ; it ap- 
 peared to be a choice, the acl of a mind, which com- 
 prehended the extent of its influence, and intended the 
 advantages of Its operation, being prompted by the de- 
 fire of giving happinefs to the works of almighty power. 
 
 Imprefled with fuch thoughts, Newton breaks out into 
 the following exclamation. * Elegantijfima hacce compages 
 ' ^ol'ts Planetarum et Cometarum^ non n'lfi confdio et dominh 
 
 * Entis cujufdam potentis et itttelligentis oriri potuit. Hac 
 ' omnia regit, tion ut anima mundiy fed lit tmiverforum 
 
 * Dominus mundorum, Et propter dominium Dominus 
 
 ' Deus, 
 
67J PHYSICAL ASTRONOMY. 
 
 * DcitSy UuvTOK^ccru^y diet folet. Deitas eft domwatio Dei, 
 ' non in corpus proprium, uti fentiimt quibus Deus ej] am* 
 
 * ma mundi, fed in fervos,^ &c. 
 
 Thefe were the efFufions of an afFeclionate heart, 
 fympathifing with the enjoyment of thofe who iliared 
 with him the advantages of their iltuation. Yet Newton 
 did not know the full extent of the harmony that he had 
 difcovered. He thought that, in the courfe of ages, things 
 would go into diforder, and need the reHoring hand of 
 God. But, as has been already obferved (543. )> I^e ^^ 
 Grange has demonftrated that no fuch diforder will hap- 
 pen. The greatell deviations from the moft regular mo- 
 tions will be almoft infenfible, and they are all periodical, 
 waneing to nothing, and again riCng to their fmall maxi- 
 mum. 
 
 ^30. Thefe are furely pleafing thoughts to a culti- 
 vated mind. It is not furprifing therefore that men of 
 affe<fl:ionate hearts fiiould too fondly indulge them, and 
 that they fhould fometimes be miftaken in their notions 
 of the purpofes anfwered by fome of the infinitely va- 
 ried and complicated phenomena of the uniyerfe. " And 
 it would be nothing but w^hat we have met with in other 
 paths of fpeculation, iliould we fee them confider a fub- 
 ferviency to this fancied purpofe as an argument that an 
 operation of nature is effecled in one way, and not in 
 another. In this vi^ay, the employment of final caufes 
 has fometimes obftruded the progrefs of knowledge,. 
 
 and 
 
GENERAL REFLECTIONS. 679 
 
 ami has been produ£^ive of error. But the impropriety 
 of this kind of argumentation proceeds chiefly from 
 the great chance of our being miftakcn with refpefb 
 to the aim of nature on the occafion. Could this be 
 properly eftabliflied as a fad, and could the fubferviency 
 of a precife mode of accompliihing a particular operation 
 be as clearly made out, I apprehend that, however un- 
 willing the logician may be to admit this as a good 
 rcafon, he cannot help feeling its great force. That 
 tlis is true, is plain from the rules of evidence that are 
 admitted in all courts ; where a purpofe being proved, 
 the fubfendency of a certain deed to that purpofe is al- 
 lowed to be evidence that this was the intention in the 
 cornmiflion of that deed. It is, however, very rarely 
 indeed that fuch argument can be ufed, or that it is 
 wanted, and it never fuperfedes the inveftigation of the 
 efficient caufe. 
 
 731. But fpeculative men have of late years fhewn 
 a wonderful hoftility to final caufes. Lord Bacon had 
 faid, more wittily than juflly, that all ufe of final caufes 
 ftiould be banifhed from philofophy, becaufe, * like Veflals, 
 * they produce nothing. * This is not hiftorically true ; 
 for much has been difcovered by refearches conducted 
 entirely by notions of final caufes. What other evidence 
 have we for all that we know concerning the nature of 
 man ? Is not this a part of the book of Nature, and fome 
 •f its moft beautiful pages ? We know them only by 
 
 the 
 
6'8o PHYSICAL ASTRONOMY. 
 
 the appearances of defign, that is, by the adaptations of 
 tilings in evident fubferviency to certain refults. Are 
 tiiere no fuch adaptations to be feen, except in the works 
 of man ? Nature is crowded with them on every hand, 
 and fome of her rnoft important operations have been 
 afcertained by atteiiding to them. Dr Harvey difcovered 
 the circulation of the blood in this very way. He faw that 
 the valves in the arteries and veins were conftrufted pre- 
 cifely like thofe of a double forcing pump, and that the 
 mufcles of the heart were aifo fitted for an alternate 
 fyftole and diaflole, fo correfponding to the ftruciure of 
 thofe valves, that the whole was fit for performing fuch 
 an office. With boldnefs therefore he afferted that the 
 beatings of th^ heart were the ftrokes of this pump ; and, 
 laying the heart o'f a living animal open to the view, he 
 had the pleafure of feeing the alternate expanfion and 
 contra dlions of its auricles and ventricles, exadUy as he 
 had expected. Here was a difcovery, as curious, as 
 great, as important^ as univerf^l gravitation. In precifely 
 the fame way have all the difcovcries in anatomy and 
 phyfiology been made. A nev/ objedt is feen. The dii- 
 coverer immidiateiy. examines its ilru6lure — vi^hy ? To 
 fee v/hat it can periorm ; and if he fees a number of 
 coadaptatiouo to a partieuiar purpofe, he does not hefi- 
 tate to fay, * tliis is its purpofe. ' He has often been mif- 
 taken ; but the miltakes have been gradually corrected — 
 how ? By difcovering what is the real ftrudure, and 
 what tlic thing is really fit for performing. The anato- 
 
 mift 
 
GENERAL REFLECTIONS. 6S I 
 
 mift never im.igliies that what he has difcovcred is of no 
 uie. * 
 
 732. So far therefore from banifhing the confidera- 
 tion of final caufes from our difculTions, it would look 
 more like philofophy, more like the love of true wifdom, 
 and it would tafte lefs of an idle curiofity, were we to 
 multiply our refearches in thofe departments of nature 
 where final caufes are the chief obje6ls of our atten- 
 tion — the ftru6lure and oeconomy of organifed bodies 
 in the animal and vegetable kingdoms. I cannot help 
 remarking, with regret, that of late years, the tafte 
 of naturalifts has greatly changed, and, in my hum-* 
 ble opinion, for the worfe. The ftudy of inert mat- 
 ter has fupplanted that of animal life. Chemiftry and 
 mineralogy are almbft the fole obje6ls of attention. Nay, 
 the ruins of nature, the fhattered relicks of a former 
 world, feems a more engaging object than the number- 
 lefs beauties that now adorn the prefent furface of our 
 globe. I acknowledge that, even in thofe inanimate 
 works, God has not left himfelf without a witnefs. Yet 
 
 furely 
 
 * I would earneitly recommend to my young readers fome 
 excellent remarks on the argument of final caufes (without 
 which Cicero thought that there ia no philofophy) in the 
 preface by the editor of Derham's Phyfico-Theology, pub- 
 lilhed at London in 1798. He there confiders the proper 
 province of this argument, its ufe, and incautious abufe, with 
 the greateft perfpicuity and judgement, 
 
 4R 
 
6S2 PHYSICAL ASTRONOMTo 
 
 furely we do not, in the bowels of the Eartli, nor even- 
 in the curious operations of chemical affinity, fee fo pal- 
 pably, or fo pleafantly, the incomp.rehenfible wifdom and 
 tli€ providential beneficence of the Father of all, as in 
 the animated objects. * 
 
 It is not eafy to account for it, and perhaps the ex- 
 planation M^ould not be very agreeable, why many na- 
 turalifts fo fadidrouily avoid fuch views of nature as tend 
 to lead the mind to the thoughts of its Author. We fee 
 them even anxious to weaken every argument for the ap- 
 pearance of defign in the canft;ru(£i:ion and operations of 
 nature. One fhould think, that, on the contrary, fuch 
 appearances would be moft welcome, and that nothing 
 would be more dreary and comfortlefs than the belief 
 that chance or fate rules all the events of nature. 
 
 733. I have been led into thefe reflections by read- 
 ing a paflage in M. de la Place's beautiful Synopfis of 
 the Newtonian Philofophy, publiflied by him in 1796, 
 
 under 
 
 * A naturalift repeats a fa)'ing of his own to the celebrated 
 cryftallographer Haiiy, * That, in future, the name of God 
 
 < would be as diftin6lly written on a ciyftal as it liad hitherto 
 
 < been feen m the heavens. ' This feems to me httle better 
 than declamation, if it be not irony. Haliy is the difcoverer 
 of the nccejjlty of the cryllalline forms ; and this philofopher 
 thinks himfelf the difcoverer of a fimilar neceflity in the ce- 
 leftial mechanifm. (See NichoJfon's Journal^ Odober i8o4> 
 ^ 87.) 
 
"REFLECTIONS OF LA PLACE. 683 
 
 under the title of Zyjlcme du Monde. In the whole of 
 this work, the author milTcs no opportunity of Icflening 
 the imprelhon that might be made by tlie peculiar fuit- 
 ablenefs of any circunvllance in the conilitution of the 
 iblar fyftem to render it a fccne of liabitation and enjoy- 
 ment to fentient beings, or which m-iglit lead the mind 
 to the notion of the fyflem's being contrived for any pur- 
 pofe whatever. He fonietimes, on the contrary, endea- 
 vours to ihew Iiow the alleged purpofe may be much 
 better accompliflied in fome other way. He labours to 
 leave a general impreflion on the mind, that the whole 
 frame is the necelTary refult of the primitive and effen- 
 tial properties of matter, and that it could not be any 
 thing but what it is. He indeed concludes, like the il- 
 luftrious Newton, v/ith a furvey of all that has been done 
 and difcovered, followed by fome rerle<Slions, fuggefted 
 (as he fays) by this furvey. 
 
 ' Aftronomy, ^ {lays M. de la Place, * in its prefent Rate, 
 
 * is unqueftionably tiie moft brilliant fpecimen of the pow- 
 
 * ers of the human underftanding. ' He does not how- 
 ever tell us how this is fo manifeft. He does not f^iy 
 that this objeci, which has engaged, and fo properly oc- 
 cupied this fme underllanding, has any thing to juftify 
 the choice, eitlier on account of its beautiful fymmetry, 
 or exquifite contrivance, or multifarious utility ; or, in 
 fiiort, that is an obj'.'dl: that is v/orth looking at. But 
 he gives us to underltand that agronomy has now tauglit 
 us how much we were miftaken, in thinking ourfelves 
 ,411 important part of the univerfe, for whofe accomm.o- 
 
 4 ^ ^ datipn 
 
^^4 PHYSICAL ASTRONOMY. 
 
 dation much has been clone, as if we were obJe£ls of pe- 
 culiar care. But we have been punifhed, fays he, for 
 thefe miftaken notions of felf-importance, by the foolifh 
 anxieties to which they have given rife, and by the fub- 
 jugation to which we have fubmitted, while under the 
 influence of thefe fuperftitious terrors. Miftaking our 
 relations to the reft of the univcrfe, fecial order has 
 been fuppofed to have other foundations than juftice and 
 truth, and an abominable maxim has been admitted, that 
 it was fometimes ufeful to deceive and to fubdue man- 
 kind, in order to fecure the happinefs of fociety. But 
 nature refumes her rights, and cruel experience has flieuii 
 that fhe will not allow thofe facred laws to be broken 
 with impunity. 
 
 734. I think it will require fome invefligation before 
 we can find out what connexion there is betv/een the 
 difcoveries of Sir Ifaac Newton and this myfterious de- 
 tection that M. de la Place has at laft deduced from the 
 furvey. It is communicated in the dark words of an 
 oracle, and we are left to interpret for ourfelves. I can 
 affix no meaning but this, that ignorance and felf-conceit 
 have made us imagine that this Earth is the' centre, and 
 the principal obje«Sl: of the univerfe, and that all that we 
 fee derives its value from its fubferviency to this Earth, 
 and to man its chief inhabitant. We fondly imagined 
 that we are the objeCls of peculiar care, — that it is for 
 us that the magnificent fpedlacle is difplayed, — and that 
 our fortunes are to be read in the ftarry heavens. But it" 
 
 is 
 
REFLECTIONS OF LA PLACE. 68^ 
 
 IS now demonftrated that this Earth, when compared, 
 even with fome fingle objects of our fyftem, is but like 
 a peppercorn. The whole fyftem is but as a point in 
 the univerfe. How infignificant then are we ! But we 
 have been juftly punifhed for our felf-conceit, by imagin- 
 ing that the ftars influence our fortunes, and have made 
 ourfelves the willing dupes of aftrologers and footh- 
 fayers. 
 
 Thus far I think that M. de la Place's words have 
 fome meaning, but, furely, very little importance ; noi* 
 did it call for any congratulatory addrefs to his contempo- 
 raries on their emancipation from fuch fears. It is more 
 than a century fmce all thoughts of the central fituation 
 and great bulk of the Earth, and of the influence of the 
 ftars on human affairs, have been exploded and forgot- 
 ten. 
 
 But the remaining part of the remarks, about focial 
 order, and truth, and juftice, and about deceiving and 
 enflaving mankind, in order to fecure their happinefs, is 
 more myfterious. * More is meant than meets the ear. ' 
 M. de la Place carefully abftains, through the whole of 
 this performance, from all reference to a Contriver, Crea- 
 tor, or Governor of the univerfe, particularly in the pre- 
 fent refle6lions, nvhich are fo pointedly contrajied with the 
 concluding reflexions of the great Newton, The oppo- 
 fition is fo remarkable, that it ftartles every reader who 
 has perufed the Principia. I cannot but fufpedl that M. 
 de la Place would here infmuate that the do6lrine of a 
 Deity, the Maker and Governor of this World, and of 
 
 his 
 
685 PHYSICAL ASTRONOMY. 
 
 his peculiar attention to the condud of men, is not 
 confillent with truth ; and that the fan£lions of religion, 
 which have long been venerated as the great fecurity of 
 fociety, are as little confident with juftice. The duties 
 which we are faid to owe to this Deity, and the terrors 
 of puniftiment in a future ftate of exiftence for the ne- 
 gleft of them, have enabled wicked men to cnilave the 
 world, fubje£^ing mankind to an opprefTive hierarchy, or 
 to fome temporal tyrant. The prlefthood has, in all ages 
 and nations, been the great fupport of the defpot's 
 throne. But now, man has refumed his natural rights. 
 The throne and the altar are overturnea, and truth and 
 juftice are the order of the day. 
 
 735. This is by no means a groundlefs interpreta- 
 tion of De la Place's words. He has given abundant 
 proofs of thefe being his fentiments. It accords com« 
 pletely Vvdth his anxious endeavours, on all occafions, to 
 flatten or deprefs every thing that has the appearance of 
 order, beauty, or fubferviency, and to refolve all into the 
 trrefiftible operation of the elTential properties cf matter. 
 
 736. Of all the marks of purpofe and of wife con- 
 trivance in the folar fyftem, the moft confpicuous is the 
 fele(£i:ion of a gravitation in the inverfe duplicate ratio of 
 the diftances. Till within thefe few eventful years, it 
 has been the profeffed admiration of philofophers of all 
 fefts. Even the materialiils have not always been on 
 their guard, nor taken care to fupprefs their wonder at 
 
 ^he 
 
IS THE LAW OF GRAVITY NRCFSSARY ? CS"] 
 
 the alinoft eternal duration and order which it fecures to 
 the folar fyftem. But M. de la Place annihilates at once 
 all the wifdom of this felec^ion, by faying that this law 
 of gravitation is efl'ential to all qualities that are diffufed 
 from a centre. It is the law of a£lion inherent in an 
 atom of matter in virtue of its mere exlftence. There- 
 fore it is no indication of purpofe, or mark of choice, or 
 example of wifdom. It cannot be otherwife. Matter 
 is what it is. 
 
 M. de la Place was aware that this ailerticn, fo con- 
 trary to a notion long and fondly entertained, would not 
 be admitted without fome unwillingnefs. Pie therefore 
 gives a demonflration of his propofition. He compares 
 the action of gravity at different diftances with the illu- 
 mination of a furface placed at different diftances from 
 the radiant point. Thus, let light, dilFufed from the 
 point A (fig. 77.) fliine through the hole B C D E, 
 which we fhall fuppofe an inch fquare, and let this light 
 be received on a furface hccle parallel to the hole, and 
 twice as far from A. We knov/ that it will illuminate 
 a furface of four fquare inches. Therefore, fmce all the 
 light which covers thefe four inches came through a hole of 
 one inch, the light in any part of the illuminated furface 
 is four times weaker than in the hole, where it is four 
 times denfer. In like manner, the intenfity, and efh- 
 ciency of any quality difFufed from A, and operating at 
 twice the diftance, muft be four times kfs or weaker ; 
 and at thrice the diftance it muft be nine times weaker, 
 &c. See. 
 
 737^ 
 
688 PHYSICAL ASTRONOMY. 
 
 737. But tliere is not the leafl fhadow of proof here, 
 nor any {imilarity, on which an argument may be found- 
 ed. We have no conception of any degrees or magni- 
 tude in the intenfity of any fuch quality as gravitation, 
 attraction, or repulfion, nor any meafure of them, ex- 
 cept the very efFe£t which we conceive them to produce. 
 At a double diftance, gravity will generate one fourth of 
 the velocity in the fame time. But this meafure of its 
 ftrength or weaknefs has no connexion whatever with 
 denfity, or figured magnitude, on which connexion the 
 whole argument is founded. What can be meant by a 
 double denfity of gravity ? What is this denfity ? It is 
 purely a geometrical notion, and in our endeavour to 
 conceive it with fome diftinftnefs, we find our thoughts 
 employed upon a certain determmed number of lines fpread- 
 ing every way from the radiant point, and paffing through 
 the hole B C D E at equal diitances among themfelves. 
 It is very true that the number of thofe lines which will 
 be intercepted by a given furface at twice the diftance 
 will be only one fourth of the number intercepted by the 
 fame furface at the fimple diftance. But I do not fee 
 how this can apply to the intenfity of a mechanical/ force, 
 unlefs we can confider this force as an efFe£l:, and can 
 Ihew the influence of each line in producing the efFe£t 
 which we call the force, and which we confider as the 
 caufe of the phenorhenon called gravitation. But if we 
 take this view of it, it is no longer an example of his 
 propofition — a force difFufed from a centre. For, in or- , 
 4er to have the efficiency inverfely as the fquare of the 
 
 diftance. 
 
THE LAW OF GRAVITY NOT NECESSARY. 689 
 
 diftance, it is meafured by the number of efficient lines 
 intercepted. Here it is plain that the efficiency of one 
 of thole lines is held to be equal at every diftance from 
 the centre. Such incongruity is mere nonfenfe. 
 
 This conception of a bundle of lines is the folc foun- 
 dation for any argument in the prefent cafe. La Place 
 indeed tries to avoid this by a different way of expref- 
 fing his example. A certain quantity of light, fays he, 
 goes through the hole. This is uniformly fpread over 
 four times the furface, and mull be four times thinner 
 fpread. But this, befides employing a gratuitous notion 
 of light, vi'hich may be refufed, involves the fam.e no- 
 tion of difcrete numerical quantity. If light be not con- 
 ceived to conlift of atoms, there can be no difference of 
 denfity ; and if we confider gravity in this way, we get 
 into the hypothecs of mechanical impulfion, and are no 
 longer confidering gravity as a primordial force or qua- 
 lity. 
 
 738. But this pretended demonftration is ftill more 
 deficient in metaphyfical accuracy. The propofition to 
 be demonftrated is, that the gravitation towards an atom 
 of matter is in the inverfe duplicate ratio of the diftance, 
 in ivhaiever point of /pace the gravitating atom is placed^ 
 But, if we take our proof of the ratio from the concep- 
 tion of thefe lines, and their denfity, we at once admit 
 that there are an infinity of fituations in which there is 
 no gravitation at all, namely, in the intervals of thefe 
 lines. The number of fituations in which the atom gra- 
 
 4 i) vitates 
 
THE LAW OF GRAVITY NOT NECESSARY. 689 
 
 diilance, it is meafured by the number of efficient lines 
 intercepted. Here it is plain that the efficiency of one 
 of thole lines is held to be equal at every diftance from 
 the centre. Such incongruity is mere nonfenfe. 
 
 This conception of a bundle of lines is the folo foun- 
 dation for any argument in the prefent cafe. La Place 
 indeed tries to avoid this by a different way of expref- 
 fmg his example. A certain quantity of light, fays he, 
 goes through the hole. This is uniformly fpread over 
 four times the furface, and mud be four times thinner 
 fpread. But this, befides employing a gratuitous notion 
 of light, vC'hich may be refufed, involves the fiime no- 
 tion of difcrete numerical quantity. If light be not con- 
 ceived to confift of atoms, there can be no difference of 
 denfity ; and if we confider gravity in this way, we get 
 into the hypothefis of mechanical impulfion, and are no 
 longer confidering gravity as a primordial force or qua- 
 lity. 
 
 738. But this pretended demonftration is flill more 
 deficient in metaphyfical accuracy. The propofition to 
 be demonftrated is, that the gravitation towards an atom 
 of matter is in the inverfe duplicate ratio of the diftance, 
 in ivhaiever point of f pace the gravitating atom is placed^ 
 But, if we take our proof of the ratio from t!ie concep- 
 tion of thefe lines, and their denfity, we at once admit 
 that there are an infinity of fituations in which there is 
 no gravitation at all, namely, in the intervals of thefe 
 lines. The number of fituations in which the atom gra- 
 
 4 b vitates 
 
dJp^ PHYSICAL ASTRONOMY. 
 
 vrtates is a mere nothing in comparifon with thofe in 
 which it does not. "VVe muft either fuppofe that both 
 the quaUty and the furface influenced by it are continu- 
 ous, uninterrupted, — or both muft be conceived as d'ljcreie 
 numerical quantities, the quality operating along a cer- 
 tarn ninnhcr cf lines, and the furface confifting of a cer- 
 tain number of points. We muft take one of thefe views. 
 But neither of them gives us any conception of a differ- 
 ent energy at different diftances. If the furface be con- 
 timiGuSy and the quality every ivhere operative, there can 
 be no difference of effeft, unlefs we at once admit that 
 the energy itfelf changes with the diftance. But this 
 change can have no relation to a change of denfity, a 
 thing altogether^ inconceivable in a continuous fubftance ; 
 —-where every place is full, there can be no more. On 
 the other hand, if the quality be exerted only along cer- 
 tain lines, and the furface only contain a certain number 
 cf points, we can find no ground for eftabliihing any 
 proportion. 
 
 739. The fimple and true ftate of the queftion is 
 this. Suppofe only two indivifible atoms, or two ma- 
 thematical points of fuch atoms, in the univerfe. If 
 thefe atoms be fuppofed to attract each other, nvherever 
 ihey are placed^ do we perceive any thing in our concep- 
 tion of this force that can enable us to fay that the at- 
 traction is equal or unequal, at different diftances ? For 
 my own part, I know nothing. The gravitation, and its 
 law of action, are mere phenomena, like the thing which 
 
 1 
 
THE LAW OF GRAVITY IS NOT NECESSARY. 6c)l 
 
 I call matter. This is equally unknown to mc. I merely 
 obferve certain relations, which have liitherto been con- 
 ftant, and I am led by the conftitution of my mind to 
 expe£l the continuation of thefe relations. My collec- 
 tion of fuch obfer\'ations is my knowledge of its nature. 
 This gravitation is one of them, and this is all that I 
 know about it, 
 
 74Q. The obferved relations may be fuch that they 
 involve certain confequences. This, in particular, has 
 confequences that cannot be difputed. If gravitation in 
 the ratio of — ^ be the primordial relation of all mat- 
 ter, and the fource of all others (which is a part of 
 La Place's fyfl:em), it is impoffible that a particle com- 
 pofed of fuch atom^ can a£l: with a force which dc^' 
 creafes more rapidly by an increafe of dillance. But 
 there are many phenomena which indicate a much more 
 rapid decreafe of force. Simple cohefion of folid bodies 
 is one of thefe. The expanfion of fome exploding com- 
 pofitions fhew the fame thing. We may add, that no 
 compofition of fuch atoms can form repelling particles, nor 
 give rife to many expanfive fluids, or indeed to any of the 
 ordinary phenomena of elaftic bodies. But thefe things 
 are not immediately before us, and we fhall have another 
 and a better opportunity of confidering many things con- 
 ne(ft:ed with this great queftion. 
 
 741. De la Place is not the firft perfon who has 
 
 attempted a demonftration of this propofition. Dr Da- 
 
 4 S 2 vi4 
 
692 PHYSICAL ASTRONOMY* 
 
 vid Gregory, in his valuable work on aftronom)^, has 
 done the fame thing, and nearly in the fame way with 
 La Place. Leibnitz, in that flrange letter to the editors 
 of the Leipzig Review, in which he anfwers feme of 
 Gregory's obje<^ions to his own theory of the celeftial 
 motions, meiitions an Italian profeflbr who gave tlie fame 
 argument, and affected to confider this ratio of planetary 
 force as known to him before Newton's difcovery. Leib- 
 nitz thinks the argument a very good one, becaufe, ma- 
 thematically fpeaking, it is the fame thing whether the 
 rays be illuminative or attractive. If this be not non- 
 fenfe, I do not know what is. — Several compilers of ele- 
 ments employ the fame argument. But nothing can be 
 lefs to the purpofe. Nothing can be more illogical than 
 to fpeak of demonflrating any primordial quality. New- 
 ton was furely more interefted in this quefcion than any 
 other perfon, and we may be certain that if he could 
 have fupported his difcovery of this law of gravitation 
 by ...any . argum^ent from higher principles, he moft cer- 
 tainly would have done it. But there is no trace of any 
 attempt of the kind among his writings ; doubtiefs be- 
 caufe he faw the folly of tlie attempt. 
 
 742. I truft that the reader will forgive me for tak- 
 ing up fo much of his time with this queilion. It feems 
 to me of primary importance. Charged as I am vAih 
 the inftruclion o£ youth — the future hopes of our coun- 
 try—it is my bounden duty to guard their minds from 
 
 r 
 
 cyery thing that I think hazardous. This is the more 
 
 incumbent 
 
THE LAW OF GRAVITY IS NOT NECESSARY. 693 
 
 incumbent on mc, when I fee natural pliilofophy calum- 
 niated, and accufed of lending her fupport to doftrines 
 which are the abhorrence of all the wife and good. I 
 cannot better difcharge this duty than by wiping oiF this 
 Ibain, with which careiefs ignorance, or atheiiHcal pervcr- 
 fion, has disfigured the fair features of philofophy. I 
 was grieved when I lirfi; faw M. de la Place, after hav- 
 ing fo beautifully epitomifed the philofophy of Sir Ifaac 
 Newton, conclude his performance with fuch a marked 
 and ungraceful parody on the clofing reflexions of our 
 illuftrious mafter j and, as I v/armly recommend this epi- 
 tome to my pupils, it became the more neceflary to take 
 notice of the reprehenfible peculiarities which occur in 
 different parts of the work ; and particularly of this pro- 
 pcfition, from which the materialifts feem to entertain 
 fuch hopes. Nor am I yet done v/ith it. A demonftra- 
 tion has been recently offered, in a work which profelTes 
 to explain the iTit'unate cotijlltuthn of matter^ and to ac- 
 count for all the phenomena of the univerfe. This will 
 come in my way when we fliall be employed in confider- 
 11 )g the force of cohefion. Till then, reqwefcat m puce. 
 
 It is fomewhat amufing to remark how the authority 
 of Sir Ifaac Newton has been eagerly catched at by the 
 atheiilicai fophifts to fupport their abjecl: do6lrines. While 
 fome hankering ■ remained in France for the Atomiftic 
 philofophy, and there was" any chance of bewildering the 
 imaginations, rnd mifleading the underftandings, of fuch 
 as wiflied to acquire a confident faith in the reveries of 
 i)emocritus and Epicurus, M. Diderot worked into a 
 
 better 
 
6g4 PHYSICAL ASTRONOMT. 
 
 better fhape the Hovenly performance of Roblnet, the 
 Sxjieme de la Nature^ and afFetled to deduce all his vi- 
 brations an«i vibratiuncles from the elaftic aetlier of Sir 
 Ifaac Newton, drefTmg up the fcheme with mathematical 
 theorems and corollaries. And thus, Newton, one of 
 the moft pious of mankind, was fet at the head of the 
 atheiflical fe61;. 
 
 But this mode, having had its day, is now palTed, 
 and is become obfolete — the tide has completely turned, 
 and the sether is no longer wanted. But the feft would 
 •not quit their hold of Sir Ifaac Newton. The do£lrine of 
 univerfal fate is now founded on Newton's great difco- 
 very of gravitation in the inverfe duplicate ratio of the 
 diftances. It is flill called the difcovery of the illuftri- 
 ous Engiifliman, and is pafled from hand to hand v/ith all 
 the authority of his name. 
 
 743. But furely to us, the fcholars of Newton, the 
 futility cf this attempt is abundantly manifeft. As the 
 worthy pupils of our accomplilhed teacher, v/e will join 
 with him in confidering univerfal gravitation as a noble 
 proof of the exiftence and fuperintendance of a Sup^reme 
 Mind, and a confpicuous mark of its tranfcendent wif- 
 dom. The difcovery of this relation between the par- 
 ticles of that matter of which the folar fyftem confiiis is 
 acknowledged, even by the materialifts, to have fet New- 
 ton at the head of philofophers. They muft therefore 
 grant that it has fomething in it of peculiar excellence. 
 Indeed *wdio.eyer is able to follow the flops of Newton 
 
 over 
 
COXCLUSIOK. 6^5 
 
 over the magnificent fcene, mud be affected as he was, 
 and muft pronounce ' all very good, ' M. de la Place 
 feems to think the lefs of man on account of the fmall- 
 nefs of his habitation. Is Abba Thule, King of Pelew, 
 a lefs noble creature than M. de la Place's Corsican 
 LIaster ? Or, does the fmallnefs of this globe Ihew 
 that little has been done for man ? — It is peculiarly de- 
 fending of remark, that we fee many contrivances in 
 this fyftem, which are of manifeft fubferviency to the 
 enjoyments of man, and which do not appear to have 
 any farther importance. Man is unqueflionably the 
 lord of this lower world, and all things are placed 
 •under his feet. But we fee nothing to which man is 
 cxclufively fubfervient — nothing that is fuperior to man 
 in excellence, fo far as we can judge of what is ex- 
 cellent — nothing but that wifdom^ that power, and that 
 beneficence, which feem to indicate and to chara^lerlfe 
 the Author and Condudlor of the whole; — and, I 
 may add, that it is not one of our fmalleft obligations 
 to the Author of Nature, that He has given us thofe 
 powers of mind which enable us to perceive and to be 
 delighted with the fight of this bright emanation of all 
 his perfeclions. 
 
 * Sancfius his animal, mentifque capacius alt^y 
 
 * Finxit in effigiem moderatitum cuticia Dcorum^ 
 
 * Pronaque cum fpeElent ammalia cater a terrain , 
 
 * Os homini fuhlime dedity caslumque tueri 
 
 * J^iffity et ereclos adfidera tollere vultus, ' 
 
 Ovid. 
 Aiiow 
 
^^6 PHYSICAL ASTRONOMY. 
 
 Allow me to conclude in the words of Dr Halle f* 
 
 * Talia mo7iJlrajitem mecum celebrate Camoenis, 
 
 * VoSy 6 ccellcoliim gaudentes neclare vefciy 
 
 * NeWtonum, claiift refera?Jtem Jcrinia Veriy 
 
 * Newtonum, Mufis char urn, cut peBore puro 
 
 * Phoebus adejly totoque tncejjlt Numine mentemy 
 
 * Nee fas ejt propius m:rtali attingere divos. ' 
 
 Halley. 
 
 END OF VOLUME FIRST. 
 
 Printtd by D. Willifon, Craig's Clofc, Edinburgh. 
 
RETURN 
 
 CIRCULATION DEPARTMENT 
 
 202 Main Library 
 
 LOAN PERIOD 1 
 HOME USE 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 
 
 1 -month loans may be renewed by calling 642-3405 
 
 6-month loans may be recharged by bringing books to Circulation Desk 
 
 Renewals and recharges may be made 4 days prior to due date 
 
 DUE AS STAMPED BELOW 
 
 RETO—Dtt — Tim 
 
 Hit CIRC 
 
 JUN 51987 
 
 ^^iO^i^c MAY 11198 
 
 UNIVERSITY OF CALIFORNIA, BERKELEY 
 FORM NO. DD6, 60m, 12/80 BERKELEY CA 94720 
 
 ®s 
 
 LD 21A-60m-10,'b5 
 (F7763sl0)476B 
 
 Uni>|ersity of California 
 Berkeley 
 
.■■r U.C. BERKELEY LIBRARIES WM 
 
 ^^ iinni 
 
 BDDlD23iS7 
 
 %.^