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SCIENTIFIC MEMOIRS 
 
 BDJTED BT 
 
 J. S. AMES, Ph.D. 
 
 PHOMBBBOR OF PHYSICS IN JOHNS HOPKINS DNIVBRSITT 
 
 IX. 
 
 THE LAWis OF GRAVITATION 
 

 THE 
 
 LAWS OF GRAVITATION 
 
 MEMOIRS BY NEWTON, BOUGUER 
 AND CAVENDISH 
 
 TOOETUER WITH ABSTRACTS OF OTHER 
 IMPORTANT MEMOIRS 
 
 TRANSLATED AND EDITED BY 
 
 A. STANLEY MACKENZIE, Pn.D. 
 
 PROKKSSOR OF PHYSICS IN BRYN MAWR COLLKOJC 
 
 NEW YORK •:• CINCINNATI •:• CHICAGO 
 
 AMERICAN BOOK COMPANY 
 
V C- \-i s- M \ 
 
 >/ 
 
 1/ 
 
 I I 
 
 
 Copyright. 1900. by Ahkrican Book Cokpant. 
 
 W. p. I 
 
 ^ ' 
 
 0- '\-4-\\ 
 
 ■jr^iSfa^r*. .. 
 

 196101 
 
^^KNKRAL CO.VTKNT.S 
 
 I I 
 
 n '.Smplno,.. ekeiel, of B„„g, e 3" 
 
 TI»Bc.rtierc„,„r„vo«j.. * ,^ 
 
 Hislonciil account of h.« . 
 
 '^CX''"''''''^-'''--''---:-::-::- '" 
 
 Index... '43 
 
 H/i 
 
 157 
 
PREFACE 
 
 P'fi . 
 
 en- 
 
 PA(iK 
 V 
 
 I 
 
 
 19 
 
 2;} 
 
 44 
 47 
 58 
 50 
 107 
 
 111 
 143 
 145 
 157 
 
 In propariiij; this volume, the ninth in the Scientific Me- 
 moirs series, the editor has had in mind the fact that the most 
 important of the memoirs liere (hnilt with, that of Cavendish, 
 is frequently <jiven for detailed study to young physicists in 
 order to train tiicrn in the art of raading for themselves period- 
 ical scientific literature. Certainly no better piece of work 
 could be used for the purpose, whether one considers the 
 intrinsic importance of the subject-matter, the keenness of 
 argument and the logical presentation in detail, or the use and 
 design of apparatus and the treatment of sources of error. 
 The main objections to Cavendish's work are those he himself 
 pointed out, and it is important to notice that, notwithstand- 
 ing all the advance in the refinement and manipulation of 
 apparatus which has been made during the century that has 
 elapsed since the date of Cavendish's experiment, his value for 
 the mean specific gravity of the earth, 5.448, must still be con- 
 sidered one of the most reliable, being not far from the latest 
 results of Poynting, Konig and Richarz and Krigar-Menzel, 
 Boys and Braun. 
 
 Believing that we in America devote insufficient time, if 
 any, to a study of Newton's great work, the editor has thought 
 it well to incorporate with the memoirs on the experimental 
 investigation of gravitational attraction the statements of New- 
 ton himself concerning that subject. 
 
 The laws of gravitation are embodied in the formula, 
 
 mm' 
 
 which says that the attraction between two particles of matter 
 is directly proportional to the product of their masses, inverse- 
 ly proportional to the square of the distance between them, 
 and independent of the kind of matter and of the intervening 
 
'c«l value of o r " '. r^"'' ^''^^le usTo Z .. *['•"• ^"'='' 
 "•action beueon ? "" P"'''"'^ >ve mwt 1^^ ""^ """•«•- 
 
 Mperimcts vZ f" "' '" "^O'''- Aa the ,„al!'"'™"=""' ""at 
 
 """•'h's crusr «n , ""'S^ mountain ma^! „ !" '"""'s of an 
 been obseved tm '' ",' ""«k»e«8, a^da^ ^, '"?''""'<" "-e 
 ""«' 'he Su;;'; «;f different inr„mentr » ".r"'"" I"" 
 »nce and the bo '"k ,"" ^''''"' "valance tZ ^'"' P'"""" 
 always abo^^t tlT '"'"'™' "■"' ^et the 1, u P""*!"'"™ bal- 
 eonstitutin:, ftM"""' "^ <'''" wT '"'^ '''''■'e of G is 
 h-s aocordfn^,'" '7' P'-oo'of Ne /ton's la ^! "Periments as 
 
 «"™ing the efrfh f , ^"'"*«<' '" "sine the ,'.r"''."'« «<"'»•• 
 *'th the vX^/" "« a sphere, the vflne o/"'.^"^''- As- 
 "^ ".e eqnat!:: "' '"^ •"«"" 'Pacific gravity^'o^'.V::;;,^;'^ 
 
 Where n fg thfi o«« i ^^ti^ 
 
 the earth a„V ^^'^'•^^'«» d»e to ^raviti. ^ t> 
 
 «'"^ o the aboVr'^'^"^''^' ^'^ '•« quite usull^ "^ J^^ ^*^^'°« «' 
 
 the earth. '^"'^ ^^Penmeuts is I VdZ L""''. '^^^ *^« 
 
 ^^he work on ^^'^''J' ^^ 
 
 - -'Strr «»Serand'deTrd" ~ "^ *"« 
 
 P>-esen..d. It Z,u' '"""""" of the Hroifc t! , ''^^«''»M to 
 «f the method, wi ^k" T'" """ "'ey weTe th. n "''^'''« ''«'•« 
 g'-avitational attlf'' ""'"« ''een used for the '^ ""'"" '" ""> 
 feet instr,,ment''rT" ' '''■'<' "'tl.ongh 1 '"""'"'•^^ent of 
 merical results !"" ""'"'onrable ifcal o„n^v""' "' ''npef- 
 
 vi ^^® '"^"^^''•s as seemed 
 
 ^. *ifei . „ 
 
itation Con- 
 l»e law, than 
 en from the 
 ^al formula, 
 •fiirect, for, 
 show that 
 dies. Such 
 'he nnmer- 
 '"0 the at- 
 t a known 
 •acter that 
 3d in such 
 'ths of an 
 loIl of the 
 action has 
 »e plumb- 
 Jlum bal- 
 ;ie of G is 
 i^nents as 
 iie editor 
 en. As- 
 onnected 
 sarth. A, 
 
 PUEFACK 
 
 necessary to prevent the reader from wasting time over obscure 
 and iuaci^urate passages, and to suggest material for collateral 
 reading. 
 
 An effort luis been made to present along with the memoirs 
 I brief historical account of the various modes of experiment 
 used fur finding the mean specific gravity of the earth, and a 
 table of results is added. As the literature on the subject 
 before the present century is not always easily obtainable, the 
 treatment of the matter for that period is given in compara- 
 tively greater detail. Believing that a bibliography contain- 
 ing every important reference to the subject is an essential 
 feature of a work of this kind, the editor has endeavoured to 
 make himself familiar with the whole of the very extensive 
 literature relating to it, and accordingly is fairly confident 
 that no important memoir has escaped his observation. From 
 the mass of material ihus collected the bibliography given at 
 the end of the volume has been compiled. In order to keep 
 within the limits of space assigned, some references had to 
 be omitted, but they relate mainly to recent work, and it h 
 believed that they contain nothing of importance. 
 
 No effort has been made to deal with the mathematical side 
 of the subject; accordingly the memoirs of Laplace, Legendre, 
 Ivory, etc., which deal with the finding of the mean specific 
 gravity of the earth by means of analytical methods are not 
 referred to; but it is hoped that all the more important ex- 
 perimental investigations have been touched upon. 
 
 Bhtn Mawr, Ocicbfr, 1899. 
 
 A. Stanley Mackenzie. 
 
 ?tt 
 
 py the 
 eru is 
 Ives to 
 here 
 n two 
 nt of 
 nper- 
 nu- 
 and 
 lear- 
 med 
 
I If 
 
 1^ 
 
HISTORY OF THE SUBJECT 
 
 BEFORE THE 
 
 APPEARANCE OF NEWTON'S ''PRINCIPIA" 
 
 Dk. (cILHEUt's contributions to the spocultitions on i^ruvita- 
 tion jiro lunon.i!; the most iniportjint of the etuiy writinjj^s on tliat 
 Kuhject, ultlioiisjh to Ke}>k'r also must eredit he ^iven for a 
 deep insight into its nature; tiie hitter announces in his intro- 
 duction to tli^ Astronomid Soiui, published in 1(!0!>, his belief 
 iu the i)erfect recdpnxMty of the action of gravitation, and in its 
 application to the whole material universe. (Jilbert was led by 
 his researches on magnetism to the conclusion that the force of 
 gravity was due to the nnignetic properties of the earth; and 
 in IGOO announce<l [1*, I, 21] iiis opinion that bodies when re- 
 moved to a great distance from the earth would gradually lose 
 their motion downwards. The earliest proposals we find for 
 investigating whether such changes occur in ihe force of gravity 
 are in the works of Francis Bacon Y'i, Nov. On). II, 3G, and 
 Itist. Ndt. I, 153]. lie maintained that this force decreased 
 l)oth inwards and outwards from the surface of the earth, and 
 suggested experiments to test his views. He would take two 
 clocks, one actuated by weights and the other by the compres- 
 sion of an iron spring, and regulate them so that they would 
 run at the same rate. The clock actuated bv weights was then 
 to be placed at the top of some high steej)le, and at the bottom 
 of a mine, and its rate at each place compared with that of the 
 other, which remained at the surface. There is no record of 
 any trial of the experiment at that time. 
 
 After the founding of the Royal Society of London a stimu- 
 lus was given to experimenting upon this as upon many otiier 
 
 * The inimbers in brackets refer to the Bibliography. 
 
 1 
 
i» 
 
 I /( 
 
 ^^^M 01 its 0]v 
 wore uutTl ^'''""^ ^^^'^^''t and '^ ^^^^''raneous 
 
 - «;-..i to the ««.;:. ;L,:; ii^ix ""'"•■'r' "^ •-»- 
 
 ofthpm«H 1 ^ ^ Js worth repri.'tintr " ^."^ ^^e8tmlnster 
 . , '^ method employed in .„p), ^^' "^^ ^'^ing some idp;* 
 
 of ^^"owledge upon ti c f "t'tt'T?" '"'^' ^^"^ ^^ theVt^L 
 measured fho ^....fi xr ' "' ^'card havino- ;.. n -^ ^ 
 
THE L A W S OF ( J It A \' I T A T 1 O N 
 
 'V Dr. Power 
 ubternuieous 
 8 of tlireud 
 )oised. The 
 by means of 
 nouth of the 
 ce.* Tliree 
 'port to the 
 iV^estminster 
 g some idea 
 of the state 
 Newton first 
 n 1GG5 that 
 nagine that 
 all changes 
 md be the 
 irsuing his 
 le sun and 
 ws, it was 
 lie inverse 
 this law of 
 
 in deduc- 
 
 estimate, 
 nstead of 
 ion of the 
 hereupon 
 upon the 
 
 ect by a 
 neantime 
 
 ect data 
 
 t of the 
 Prom 
 
 fonnda- 
 
 It: 
 
 oncern- 
 ved be- 
 iy such 
 •y their 
 
 nearer or farther removal from that siirfaoe upwards. To this 
 end I took a [)air of exact scales and weights, and we!it to a 
 convenient place upon Westminster Abbey, where was a per- 
 pendicular height above the leads of a subjacent l)uildiiig. 
 which by measure I found tiireescore and eleven foot. Here 
 counterpoising a piece of iron (wliicb weighed about 15 ouiu;es 
 troy) aiul packthread enough to reach from the top to the bot- 
 tom, 1 found the counterpoise to be of troy- weight seventoon 
 ounces and thirty grains. Then letting down the iron by the 
 thread, till it almost touched the subjacent leads, I tried what 
 alteration there had happened to its weight, and fouiul, that 
 the iron preponderated the former counterpoise somewhat more 
 than ten grains. Then drawing up the iron and thread with 
 all the diligence possibly I could, that it might neither get nor 
 lose any thing by touching the perpend icniar wall, 1 found by 
 putting the iron and packthread again into its scale, that it 
 kept its last equilibrium; and therefore concluded, that it had 
 not received any sensible difference of weight from its nearness 
 to or distance from the earth. I repeated the trial in the sunie 
 place, but found, that it had not altered its e(|uilibrium (as in 
 the first trial) neither at the bottom, nor after I had drawn it 
 up again ; which made me guess, that the first preponderating 
 of the scale was fro!n the moisture of the air, or the like, that 
 had stuck to the string, and so made it heavier. In pursuance 
 of this experiment, I removed to another place of the Abbey, 
 that was just the same distance from the ground, that the for- 
 mer was from the leads; and upon repeating the trial there 
 with the former diligence, I found not any sensible alteration 
 of the equilibrium, either before or after I had drawn it up; 
 which farther confirmed me, that the first alteration proceeded 
 from some other accident, and not from the differing gravity of 
 the same body. 
 
 "I think therefore it were very desirable, from the determi- 
 nation of Dr. Power's trials, wherein he found such difference 
 of weight, that it were examined by such as have opportunity, 
 first, what difference there is in the density and pressure of 
 the air, and what of that condensation of gravity may be as- 
 cribed to the differing degrees of heat and cold at the top 
 and bottom, which may be easily tried with a common weather- 
 glass and a sealed-up thermometer ; for the thermometer will 
 shew what of the change is to be ascribed to heat and cold, 
 
 8 
 
I It 
 
 ! ' 
 
 i I 
 
 1 I 
 
 Ml 
 
 il, 
 
 f 
 
 ; ! 
 
 , !l 
 
 MK MO I IIS ON 
 
 Mini tlic wciUlicr j^Miiss will show tlic (lincriTi*,' condciiKHtion. 
 Next, for t,li(! knowing;, wliotlior this ultcrutioii of «j;riivity pro- 
 cchmI from tho doMsity mid ^'nivity of tho iiiiihicnt siir. it would 
 l»o requisite to niuke use of some vei-y li^lit ixxly, exteiidiMl 
 into hiv^ii dimensions, sueli jis u liirjje ;j;lol)e of j^diiss carefully 
 stopt, that no uir may get in or out; for if the alteration 
 proceeded from the magnetical attraction of the parts of the 
 earth, the hail will lose l)ut a sixteenth j)art of its weight (suj)- 
 posing a lunjp of glass held tlie same pro))ortion, that l>r. 
 Power found in hrass) ; hut if it jiroceed from the (lensity of 
 the air, it may lose half, or perhaps more. Further, it wen; 
 very desireahh^, that th(i current of the air in that })la(!e were 
 ohscrved, as Sir Rohert Moray intimated the last day. Fourth- 
 ly, I think it were worth trial to counterpoise a light and heavy 
 body one against anotlier above, and to carry down the scales 
 and them to the bottom, and observe what happens. Fifthly, 
 it were dcsireablc, that trials were n»ade, by the letting down of 
 other both heavier and lighter bodies, as lead, quicksilver, gold, 
 stones, wood, liquors, animal substances, and the like. Sixthly, 
 it were to be wished, that trial were made how that gravitation 
 docs decrease with the descent of the body — that is, by making 
 trial, how much the body grows lighter at every ten or twenty 
 foot distance. These trials, if accurately made, would afford 
 a great help to guess at the cause of this strange phaenom- 
 enon." 
 
 Dr. Power's experiment was repeated by Dr. Cotton, and an 
 account of his trials was given to the Society on June 1, 1004 
 [10, vol. 1, p. 433]. The weight was ^ lb., and the length of the 
 string 3(J yards. A loss in weight of ^ oz. was found. 
 
 On September 1, 16G4 [G, vol. 5, p. 307], we find a reference 
 to some experiments made at St. I*aul's Cathedral by a com- 
 mittee of the Koyal Society consisting of Sir R. Moray, Dr. 
 Wilkins, Dr. Goddard, Mr. Palmer, Mr. Hill and Mr. llooke. 
 The results of these experiments were given to the Society on 
 September 14, 1004 [10, vol. 1, p. 4GG] ; the weight was 15 lbs. 
 troy, the string about 200 ft. long, and the loss of weight 1 
 drachm. In a letter to Mr. Boyle [G, vol. 5, p. 53G], dated 
 September 15th, Mr. Ilooke gives more details, and remarks 
 that the balance was sensitive enough to be turned by a few 
 grains. He suggests the variation of the density of the air as 
 the cause of the loss in weight. Boyle [10, vol. 1, p. 470] pro- 
 
 Li! 
 
 ■K 
 
TIIK LAWS OK (iKAVITATIOX 
 
 posod tliiit II()()kt''s snijixostioii be tested by miikin<; tlio hus- 
 pended \vei<;lit of u l;iri;e <;liiss hull loaded with menuiry. 
 
 At a iiieetiii<if of the lioyal Society on Maieh 14, Hid."). Ilooke 
 rejxnted (10, vol. 'i, ]>. <»(l, and i>, vol. .">. p. ")44| that ho had 
 tried Dr. Power's (vxperimeni at some wells near Mpsom and 
 had found no loss in wei<jflit. Similar experirjients wen; made 
 hv Ilooke at Hanstead Downs, in Surrey, and reporteil on 
 Mar(d) '-il, Hidfl (10, vol. '■!, j). <»'.>, and (J, vol. a, pp. ;{">ri and 
 r)4<!l. The strini,' was X\() ft. lon^. and the halanee sensitive 
 to a i^raiji, yet a pound shewed no <dianj,'e in weii.dit when sus- 
 jteiided at the bottom of the well. He concludes that the 
 j)ower of «;ravity cannot bo niaj^netical, as (lilbort had sup- 
 posed, lie says: " Hut in truth upon the c(msidon:tion of the 
 nature of the theory, we may find, that supposing:; it true, that 
 all tlie constituent parts of the earth had a mMi,MU'tical power, 
 the decrease of <]^ravity would be almost a hundred tinu's less 
 than a jj^rain to a })ound, at as fjreat a depth as lifty fathom : 
 for if we consider the ])roportion of the parts of tlu^ earth 
 placed uj)on one side beneath the stone, with the parts on tho 
 other side ai)ove it, we may fiiul the disproportion greater. 
 Unless we sup|)oso the majjjnetism of the i)arts to act but at a 
 very little distaiuH'. which I think the experiments made in tho 
 Abbey and St. Paul's will iu)t allow of. If therefore there be 
 any su(di inequality of j]fr;ivity, we must have* some ways of 
 trial much more accuirato than this of scales, of whi(di I shall 
 propound two sorts," etc. It is interestiujr? to notice that the 
 considerati(ms upon which ho makes his (;omputations are prac- 
 tically those used by Airy in his Harton Colliery exj)eriment. 
 
 On December 7. 1081 [10, vol. 4, p. 110], Ilooke i)roduced 
 before tho society two pendulum-clocks adjusted to run at tho 
 .same rate, lie proj)()sod to put one at tho toj) and the other 
 at tho bottom of tho monument on Fish Str(;('t Hill, and ob- 
 serve wliether they would keep to<^ether. No notitui of his 
 having tried the experiment has been fouiul. Thh is the 
 method proposed by Ha(!on and used by Bouguer and many 
 others. 
 
 In 108:2, Ilooke read before the Royal Society *' A Discourse 
 of the Nature of Comets" [4, pp. I4H-11I1], in whicli he gives 
 his ideas on the subject of gravity (})articularly on pages 170- 
 18;}). Ho considers gravity to be a universal principle, in 
 herent in all matter, pro})agated by the same medium as that 
 
H 
 
 i ; 
 
 f t 
 
 MKMOIHS ON TIIK LAWS OK (JKAVITATION 
 
 l)y nietuiH of wlii(;li li^Hit is conveyed, witli unimaginable celer- 
 ity, to indefinitely gri^at distances, and with a power varying 
 witjj the distan(re. He sums up his conceptions on gravitation 
 in nine i)roj)ositions, wliich are of great interest, in that they 
 inchnhi many of the conceptions of Newton on tliis subject, 
 and yet were published four years before the Prinripia ap- 
 peared. 
 
 6 
 
ITATION 
 
 rinable celer- 
 )\ver varying 
 1 gravitation 
 in tliat they 
 this subject, 
 rincipia ap- 
 
 PHILOSOPHIAE Ni^TURALLS PRINCIPIA 
 
 MATHEMATICA 
 
 1*^ Edition, Iymdo,i, 1687. 2d Kditiou, ('a,„t,nd,,f, 1713 (CoteH' Edition) 
 ddMUton, London, 1736 {Pe>n/>ertoii'fi Addition) 
 
 AND 
 
 DE MUNDl SYSTEMATE 
 
 Ijmdon, 1727 
 
 BY 
 
 SIR ISAAC NEWTON 
 
 {Extracts taken from Dmrn's Edition of Motte's translation 
 3 volumes, J^/ulon, 1803) 
 
CONTENTS 
 
 PAdK 
 
 On the attraction of spfnirs \) 
 
 Lain of the diKtance 9 
 
 Jmw of the inaHHi'H 13 
 
 Variation of (j rarity on the earth' a mirfaee 14 
 
 All attraction in mutual l.") 
 
 Methods of xhowing the attraction between terrextrial Itodien 17 
 
 Proof of itH ej'intcnce 17 
 
 Similar discuxaion for the cane of celestial Itodien IH 
 
 Final statements concerning tfie laws of gravitation 19 
 
 8 
 
 i! 
 
^ 
 
 THE MATHEMATICAL PUIN(.MPLES OF 
 NATURAL PHJLOSOFHY 
 
 AND 
 
 SYSTEM OE THE WORLD 
 
 HY 
 
 SIR ISAA(J NKWTON 
 
 |i(K>K I. I*K(HM)SITI()N LXXIV. TllKollKM XXXIV. 
 
 Tin' same f/tinf/s .stf/i/tnscd (if to tho sovonil points of :i j^iven 
 8|)li(!r(i tluM'e toiul c(|iiJil cciitripotal forces (lecroasiriij in u dii- 
 plicjito ratio of tlio distiiiici'S from tiio points), / sat/, llnd a ntr- 
 pHsrle silnafe without tlic s/i/iere is ((ttntrtrd witli <( /nrm rcri/}- 
 rocftllj/ proportional to tlw sf/uarc of its distance from the 
 centre. 
 
 Book I. Puoposition^ LXXV. Tiikokkm XXXV. 
 
 //■ to the several points of a t/iren sphere there tend ef/aal cen- 
 tripetal forces decreasing in a dnplic((te ratio (f the dist(tnres 
 from tlie points ; I sat/, that another similar sphere will f/e altrart- 
 ed 1)11 it with a force recijrrocallf/ projwrtional to the square of the 
 distance of the centres. 
 
 For the attraction of every particle is reciprocally as tlie 
 square of its distance from the centre of the attracting spiiere 
 (by prop. 74), and is therefore the same as if that whole at- 
 tracting force issued from one single corpuscle placed in the 
 centre of this sphere. But this attraction is as great as on the 
 other hand the attraction of the same corpuscle would be, if 
 that were itself attracted by the several particles of the attract- 
 ed sphere with the same force with which they are attracted by 
 
 9 
 
MKNKHUS (jN 
 
 It. Rut timt iittnu'tio!! of tlio corpiisclo would he (hy prop. T4) 
 n'ci|>r(t("illy iiroporiioiuil to llir s(|iiar(' of its (listiiii(«( fiotn tlio 
 (■('Mtic of the splicrc ; I licrrforc llic jittnictioii of tlu^ spluTc, 
 iMjiiiil llicirto, is also in tlic siiiiic ratio. (^. K. D. 
 
 Cor. I. The atliiU'lions of s|»li('i«'s ttnvards otiior li(nnot;oii('- 
 oiiH spheres are as tlie attnietiiiiT sphcMcs applied to tin? squares 
 fd" the distatiees <d' their centres from the centres of those 
 \vlii(di they attract. 
 
 Cor. '^. 'IMie cas(f is tlio same wiien the attracted sphere does 
 also attract. Kor the several points of the one attract the sev- 
 «'ral points of the other with the sainc; force with whi<di they 
 themselves are attracted hy the (tthers a;;ain ; and therefore 
 since in all attractions (hy law 11) the attracted and attractin.i^ 
 point are both c(|indly acte(l on, tlu^ force will he doubled by 
 their mntnal attractions, the proportions retnainini;. 
 
 [ I'ro/m.si/ioH /y.\'.\' r/. /wnrcs I he sttniP Ihiiuj fur sfthprvs tHftrlr 
 up of honnif/t'iu'oHs ronrrnfrir liti/cr.s. | 
 
 Rook III. Rkoi'osition \'. Tiikokkm V. Sciioijim. 
 
 The force which retains the celestial bodies in their orbits 
 has been hitherto called centripetal force; but it being now 
 made plain that it can bo no other than a j;ravitatin<( force, 
 we shall hereafter call it gravity. For tlie cause of that cen- 
 tripetal force whi<di retains the moon in its orbit will extend 
 itself to all the planets. 
 
 ' 
 
 Rook III. Proposition VI. TiiKoi{K>r VI. 
 
 T/inf all Imlivx f/niri/dfc fofrnrds ct^'cri/ phiucf ; and that the 
 wcighfs of bodies tomirds (un/ the same planet, at er/fial distances 
 from the centre of the planet, are proportional to the quantities 
 of matter irhich Ihej/ severnllji contain. 
 
 It has been, now of a long time, observed by others, tbat all 
 sorts of beavy bodies (allowance being made for the inequality 
 of retardation wbicb they sutler from a small power of resist- 
 ance in tbc air) descend to the eartb from equal heifjhts in 
 equal times ; and tbat equality of times we may distinguish to 
 a great accuracy, by the help of pendulums. I tried the thing 
 in gold, silver, lead, glass, sand, common salt, wood, water, 
 and wheat. I provided two wooden boxes, round and equal ; 
 
 10 
 
5 
 
 •<» (l>y prop. 74) 
 iitici! fi-oni tlio 
 >f the splu'ic, 
 
 (^ K. [). 
 
 I*^'!' llOI||()(r(.||0. 
 to tll(! .Sfj||lirp.S 
 
 itrt'S of those 
 
 <l sphere does 
 tract tho Hov- 
 li which they 
 m(l therefore 
 ml JiMnietiriir 
 
 I' <loiihle(l hy 
 \<^ 
 
 sp/ierr.s tnade 
 
 tlieir orbits 
 t l)eiii^' now 
 
 iitiii;.r force, 
 of that (H'ji- 
 
 will exteiiil 
 
 VI. 
 
 nd that the 
 I flis/anrcft 
 qKanlities 
 
 ••s, that Jill 
 inequality 
 
 * of res is t- 
 hcifjhts in 
 n^uish to 
 the thing 
 
 5d, water, 
 
 id equal ; 
 
 TIIK LAWS OF (Mt.WITATloN 
 
 I filled the one with wood, juid suspended ;in e(|n;d weiirlit of 
 ^'old (as exactly as I coidd) in the I'cntre of oscilhUion of the 
 other. 'V\w boxes hMnirini; bv equal threads of 1 I feet nnide a 
 couple of pendidunis perfectly e(|ual in weii^Hit and lii^ure, and 
 erpudly receivin*? the resistance of the air. And. placiuij the 
 on»? by the other, I observed them to play tnijether fcu'wards 
 atid ba( kwards, for a loui; time, with eqind vil)rations. . . . 
 and the like happened in tht! other bodies. lU' these experi- 
 ments, in boilii's of the satne weiirlit, I could immifestly hav(» 
 discovere(l a dilTerence of nuitter less than tin' thousandth ]>art 
 of the whole, had any smdi been. Itiit, without all doubt, tho 
 luiture of gravity towards the planets is the sanu' as towards 
 the earth. . . . iMoreover, since the satellites of .lupiter per- 
 form their revolutions in times which observe the sesfpiiplicate 
 proportion of their distances from .lujtiter's centre, their acc»d- 
 erativc gravities towards .lupiter will be reciprocally as tin? 
 s(|uares of their distaiu.'es from .lupiter's centre — that is, e(pial 
 at equal distances. Ami, thertd'ore, these satellites, if sup- 
 posed to fall fofiutrds .In/tilrr from e(|ual luugbts, would describe 
 equal spaces in eipuil times, in lik(> manner as heavy bodies do 
 on our earth. . . . If, at e(|ual distanc^es from the sun. any sat- 
 ellite, in proportion to the <|uantity of its nuitter, did gravitate 
 towards the sun with a force greater than .lupiter in propor- 
 tion to his, according to any given proportion, suppose of d to 
 (' ; then the distance between the centres of the sun and of the 
 satellite's orbit would be always greater than the distance be- 
 tween the centres of the sun and (»f .lupiter nearly in the sid»- 
 duplieate of that proportion ; as by some computations I have 
 found. And if the satellite did gravitate towards the sun 
 with a force, lesser in the ])ro])ortion of e to (L the distance of 
 Llie centre of the satellite's orbit from the sun would be less 
 than the distance of the centre of .lupiter from the sun in the 
 subduplicate of the same proportion. Therefore if, at equal 
 distances from the sun, the accelerative gravity of any satell- 
 ite towards the sun were greater or less than the acu-elerative 
 gravity of .Jupiter towards the sun but by oiu' ^^}^^^^ part of the 
 whole gravity, the distance of the centre of the satellite's orbit' 
 from the sun would be trreater or less than th(» distar 
 
 o 
 
 u- 
 
 f J 
 
 piter from the sun by one ^^j,„ part of the whole distance — 
 that is, by a fifth part of the distance of the utmost satellite 
 from the centre of Jupiter ; an ; . lentricity of the orbit which 
 
 11 
 
 .Avxy.ij:'!^. 
 
i !!r 
 
 
 'ii: 
 
 '1 
 
 Ml 
 
 Ml 
 
 « 
 
 1 
 
 I' : 
 
 'Ii 
 i i| 
 
 mi: Mo I Its ON 
 
 would hn vory sfMHihlc hiil llir «>il)it« of tlic Hiitdlitoa ari^ 
 <'<)ti('(Mi( I'ic tn Jiipilri', iiiid t licrcfort; tlu; iKMUiici'iil ivc ^raviiics 
 of .hipitcr, iiml of .ill it.s Hiitt'lliti^s towiinU the sum, an; iMpiul 
 luiKni^ tliciiisclvt's. . . . 
 
 Hut fiirtlicr; tlio wcij^lits of nil tlu^ parts of every planot 
 to\var(ls any other planet art! on(» to another as the malter in 
 the several parts; for if sonie parts did ^M'avitate more, others 
 less, th;in for tln^ (pnintity of their matter, then tln^ whoh* 
 planet, aecordinij to tlu^ sort (d' parts with whi<di it most 
 aitonnds, would ;^M'avitato mori^ or less than in proportion to 
 the «(uantity of matter in the whol(\ Nor is it of any moment 
 whether these parts are extermil (U- internal : for if. for exam- 
 ple, W(^ should ima((ine th(^ terrestrial hodies with ns to ho 
 raised up to the (U'l) of the moon, to he lh(>r(f compared with 
 its hody; if the wcULjhts of suidi hodies were to the wei<fhts of 
 the external parts of the moon as the (|iuintities of malter in 
 the one and in tlui otluM* respectively ; but to the weij^hts of 
 the inttuMial parts in a <,'r(!at(M' or less proportion, then lik(!wiso 
 the wei.ujhts of those hodi(»s would he to the wei^dit of the 
 whole moon in a j^reater or less proportion ; uj^ainst what we 
 have slunved ahove. 
 
 Cor. 1. IFence the weiifhia of hodies do not de|)en<l upon 
 their forms and texturcis ; for if the weiufhts coidd ho altered 
 with the forms, tlu^y would he j:jreat(!r or less, a(M'ordinsf to the 
 variety of forins, ine(|ual matter; altogether a^ifainst experience. 
 
 Cor. )i. Universally, all hodies ahout the earth i^ravitate 
 towards the earth ; ami the weijj^hts of all, at (Mpuil distances 
 from the earth's centre, are as the (juantities of matter whicdi 
 they severally contain. This is tlie (|uality of all bodies within 
 the reach of our (ixp(M-iments ; and therefore (by rule '.i) to bo 
 artirmed of all bodies whatsoever. . . . 
 
 Cor. T). The power of ujravity is of a different nature from the 
 power of magnetism ; for the mairnetic attraction is not as the 
 matter attracted. Some bodies are attracted more by the 
 majjjnet ; others less ; most bodies not at all. The power of 
 maijfiu'tism in one and the same body may bo increased and 
 diminisluwl ; and is sometimes far stroni^er, for the quantity of 
 matter, than the power of Ljravity ; aiul in receding from the 
 magnet decreases iu)t in the duplicate but almost in the tri- 
 plicate proportion of the distance, as nearly as I could judge 
 from some rutle observations. 
 
 12 
 
Til K LAWS ()|« <i K \V n ^\TlnN 
 
 sllt('llit^^^ aro 
 iv(^ ^ruviiit's 
 III, uro l'(|iihI 
 
 fvory planol 
 \o iiijith'i- ill 
 Miorc. oiln'is 
 I llio wliolo 
 i<*li it iiiost 
 roportioii to 
 iiiiy iiiornt'iit 
 r. for oxiiin- 
 
 tll IIM t«» l)l> 
 
 n pared with 
 <' wci^'lits of 
 >f Mijitlcr ill 
 > \v('i«^r|its of 
 UMi likewise' 
 'i^'iit of tlio 
 ii.sL wliat we 
 
 I'pOIld llJ)OM 
 
 I 1)0 jiitei'i'd 
 
 ding to tiio 
 
 ''xpcrit'iioe. 
 
 I ijfi'iivitiite 
 
 I distiiiices 
 
 ttiT wiiicli 
 
 lies Willi in 
 
 lo ;}) to bo 
 
 from the 
 Inot us the 
 re by tlie 
 
 l)ower of 
 hiisod jiml 
 Quantity of 
 
 I from the 
 li tile tri- 
 
 bld judge 
 
 liooK III. I'liui'osnioN VII. 'riiroitiiM \'ll. 
 
 y/nt/ f/irrr is n fintnr nf f/rttrifif fr/it/iiti/ In if// Intiliis, inn- 
 fHiffinmt/ In lliv si'iri'ol iiiiiinl il iis nf' nuillvi' ivlnrh llirif i ntiliNH. 
 
 'riiiit all llic plaiu'ls iiiiitiially ;;i'avitat«' one towards aiiotlicr, 
 we have prove*! before; as well as that the fon-e ol' ;;iavity 
 towards every ""e of IIh'Mi. coiisidenMl apart, is reeiproeally as 
 the S(piar(! <d' (he distanet^ of plact's froiii the centre of the 
 plam't. And thence (by prop, till, book I, and its cor(dlaries) 
 it I'olhtws, that tli»! gravity lending towards all the planets is 
 proportional to the matter wliiidi tlu>y contain. 
 
 >loreovcr, since all the parts of any planet A gra\itate to- 
 wards any other planet H; and tlio gravity of every part is to 
 the gravity of the whole as tiie matter of the part to tlu^ matter 
 of the whole ; and (by law :{) to every ai^tion corresponds an 
 e(pial reaction ; theiud'oro tlu! planet B will, on the other hand, 
 gravitate towards all the parts of the planet A ; and its gravity 
 towards any one jiart will be to the gravity towanis the wlioh^ 
 as the matter of the part to the matter of tlu^ whole, (l. K. I). 
 
 Cor. 1. Therid'ore the force of gravity towards any whole 
 planet arises from, and is (jomponiided of, the fonies of gravity 
 towards all its parts. Magnetic and ele(!tri(! attractions alTord 
 us examples of this ; for all attraction towards the wlioh; arises 
 from the attractions towards the several parts. The thing may 
 be easily understood in gravity, if we consider a greater planet 
 us formed of a number of lesser i)lanets meeting together in 
 OIK! globe ; for ln'tirc it wotdd appear llial the force of the whole 
 must arise from the lorces of the component parts. If it is 
 objected that, according to this law, all bodies with us must 
 mutually gravitate one towards another, I answer, that since 
 the gravitation towards these bodies is to the gravitation to- 
 wards the whole earth as these ])odie8 are to the whole earth, 
 the gravitation towards them must be far less than to fall under 
 the observation of our senses. 
 
 Cor. )i,. The force of gravity towards the several equal par- 
 ticles of any body is reciprocally as the sepiare of the distance 
 of j)lace8 from the particles ; as appears from cor. 3, prop. 74, 
 book I. 
 
 [ Under proposil ion A'' occurs the fol/owiiuj important passage:] 
 However the planets have been formed while they were yet 
 in fluid masses, all the heavier matter subsided to the centre. 
 
 13 
 
!:,Vl 
 
 •ir^ 
 
 ; !(! 
 
 H 
 
 lii 
 
 M KM O I IIS ON 
 
 Since, tlioreforo, the oomrium matter of our earth on the 8nr- 
 fjice thenfof is ubout twUre hh licavy us water, and a little lower, 
 in niinos, is found about throe, or four, or even fivo times more 
 heavy, it is j)r()bablo that tlie quantity of the whole matter of 
 the cartii may be live or six times irreater tlian if it consisted 
 all of water.* 
 
 [l^ndi'r propnsifioits XVI 1 1, and XIX., Newton proves that 
 the axes of' the planets are less th((n the diamrters drawn perpen- 
 diridar to the axes. He shows how centrifugal force acts in 
 deterniininf/ the fornt of the earth, and dificnsses the nieasurenients 
 of terrestrial arcs k'nown. at that time; he deduces therefrom that 
 fjraritt/ will tte la. rned at the et/?(ator hj/ ggj (f itself and that 
 the earth will be higher at tlie erjuator titan at the poles bg 17.1 
 miles. ^ 
 
 Book III. Proposition XX. Pkobij:m IV. 
 
 To fnd and compare toget/ier the weights of bodies in the dif- 
 ferent regions of our earth. 
 
 Because the weights of the unequal legs of the eaiuil of water 
 ACQy/rrt are equal ; and the weights of the parts proportional 
 
 to the whole legs, and alike situated 
 in them, are one to another as the 
 weights of the wholes, and therefore 
 equal betwixt themselves; theweights 
 of equal parts, and alike situated in 
 Q the legs, will be reciprocally as the 
 legs — that is, reciprocally as 230 to 
 220. And the case is the same in 
 all homogeneous equal bodies alike 
 situated in the legs of the canal. 
 Their weights are reciprocally as the 
 legs — that is, reciprocally as the dis- 
 tances of the bodies 'rom the centre of the earth. Therefoie, 
 if the bodies are situated in the uppermost parts of the canals, 
 or on the surface of the earth, their weights will be one to an- 
 other reciprocally as their distances from the centre. And, by 
 the same argument, the weights in all other places round the 
 whole surface of the earth are reciprocally as the distances of 
 
 *[ Tliu Wiia (I wonderfnlli/ good f/ueits on Newton' a p(irt,unc£ tlie best of tlie 
 later deter mi nations give about 5.5 for tlie nican specific gravity of tlie eartli.^ 
 
 14 
 
 » I 
 
 4111 
 
rth on the sur- 
 1 a little lower, 
 fivu times more 
 ■'liole matter of 
 if it consisted 
 
 hn proves that 
 drawn perpen- 
 
 force acta in 
 "meusuremcnls 
 
 thcrefnm that 
 tse/f, and that 
 <e poles by 17.1 
 
 M IV. 
 
 iies in the dif- 
 
 oanal of water 
 
 } proportional 
 
 alike situated 
 
 notlier as the 
 
 and therefore 
 
 !s; the weights 
 
 :e situated in 
 
 •ocaliy as the 
 
 ily as 230 to 
 
 tlie same in 
 
 hodies alike 
 
 the canal. 
 
 ocaliy as the 
 
 y as the dis- 
 
 Therefoie, 
 
 the canals, 
 
 e one to an- 
 
 And, by 
 
 s round the 
 
 distances of 
 
 tlie best of the 
 of the earth.} 
 
 THE LAWS OF (J K A V I T A T 1 () N 
 
 the places from the centre; and, thoroforo, in the hypothesis 
 of the earth's being a spheroid, are given in projjortion. 
 
 [Xewton then states that ''the lengths of prndtfluins viln-atinrj 
 in equal times are as the forces of (jrartti/"; he ennnierates the 
 r.rper intents on the periods of pendnhuns made at differeut parts 
 'if the eart/t's surfare, and tests his roNtlusions. 
 
 The following re>narks appear on pp. 'ii)-l7} of .]fofte's transla- 
 fion of the 'Ule Mnndi Systemate,'' wherein Newton, after a 
 reference to his pendulum experiments, yiven on p. li of this 
 volume, says ;] 
 
 Since the action of the centripetal force upon the bodies at- 
 tracted is, at equal distances, proportional to the quantities of 
 matter in those bodies, reason requires that it should be also 
 proportional to the quantity of matter in the body attracting. 
 
 For all action is mutual, and (by the third law of motion) 
 makes the bodies tnutually to approach one to the other, and 
 therefore must be the same in both bodies. It is true that we 
 may consider one body as attracting, another as attracted; but 
 this distinction is more mathetnatical than natural. The at- 
 traction is really common of either to other, and therefore of 
 the same kind in both. 
 
 And hence it is that the attractive force is found in both. 
 The sun attracts Jupiter and the other {)lanets ; Jupiter at- 
 tracts its satellites ; and, for the same ri^ason, the satellites act 
 as well one upon another as upon Jupi er, and all the planets 
 mutually one upon another. 
 
 And though the mutual actions of two planets may be dis- 
 tinguished and considered as two, by which each attracts the 
 other, yet, as those actions are intermediate, they do not make 
 but one operation between two terms. Two bodies may be 
 mutually attracted each to the other by the contraction of a 
 cord interposed. There is a double cause of action, to wit, the 
 disposition of both bodies, as well as a double action in so far 
 as the action is considered as upon two bodies ; but as betwixt 
 two bodies it is but one single one. It is not one action by 
 which the sun attracts Jupiter, and another by whi(;h Jupiter 
 attracts the sun ; but it is one action by which the sun and 
 Jupiter mutually endeavour to approach each the other. By 
 the action with which the sun attracts Jupiter, Jupiter and 
 the sun endeavour to come nearer together (by the third law of 
 motion); and by the action with which Jii[»iter attracts the 
 
 15 
 
I! 
 
 |ill!. 
 
 if 
 
 
 I f 
 
 ■:'ii| 
 
 
 i::i 
 
 ' !i ' 
 
 Uli'i 
 
 !;'! 
 
 I I 
 
 I i 
 
 il 
 
 ! i 
 
 iiili 
 
 
 t; i 
 
 ! II. 
 
 H 
 
 MKMOIUS UN 
 
 sun, likewise Jii])itor and tlie sun endoavour to oomo nearer to- 
 gether. Hut the sun is not attracted towards Jupiter by a 
 twofohl action, nor Jui)iter hy a twofohl acliion towards tlie 
 sun ; but it is one siugk; intermediate action, by whicjj both 
 a})proacli nearer togetlier. 
 
 Tiuis iron draws tlio loadstone as well as the loadstone 
 draws the iron ; for all iron in tiie neighbourhood of tlie load- 
 stone draws otluM* iron, liut the action betwixt the loadstone 
 and iron is single, and is considered as single by tlie philoso- 
 ])liers. The action of iron upon the loadstone is, indeed, the 
 action of the loadstone betwixt itself and the iron, by which 
 both endeavour to come nearer together; and so it manifestly 
 appears, for if you remove the loadstone the whole force of the 
 iron almost ceases. 
 
 In this sense it is that we are to conceive one single action 
 to be exerted betwixt two planets, arising from the conspiring 
 natures of both; and this action standing in the same relation 
 to both, if it is proiiortional to the (juantity of matter in the 
 one, it will be also proportional to the quantity of matter in 
 the other. 
 
 Perhaps it may be objected that, according to this phil- 
 osophy (prop. 74, book 1), all bodies should mutually attract 
 one another, contrary to rlie evidence of experiments in ter- 
 restrial bodies; but I answer that the experiments in terres- 
 trial bodies come to no account ; for the attraction of homo- 
 geneous spheres near their surfaces are (by prop. 72, book 
 i) as their diameters. Whence a sphere of one foot in diam- 
 eter, and of a like nature to the earth, would attract a small 
 body placed near its surface with a force 20,000,000 * times less 
 than the earth would do if placed near its surface ; but so 
 small a force could produce no sensible effect. If two such 
 spheres were distant but by one-quarter of an inch, they would 
 not, even in spaces void of resistance, come together by the 
 force of their mutual attraction in less than a month's time;f 
 
 *\Tf the sphere in one foot in diameter, this number should be 40,000,000, 
 since the dinmeter of the earth is about 40.000,000/<. But perhaps Newton 
 intended to say a sphere of one foot in radius.] 
 
 f [The time is very much lefi. On the assumption that each of the spheres is 
 one foot in diameter, Poi/ntinf/ (185, p. 10) finds tite time to Ite about 320 sec- 
 onds. If, however, we take one foot as the radius of each spliere, Todhunter 
 (J 40, vol. 1, p. 401) >ih(noH that the time is less than 250 seconds.] 
 
 16 
 
 II 
 
THE LAWS OK GRAVITATION 
 
 ind loss splieres will come together ut a rate yet slower, viz., 
 Ill the proportion of their diameters. Nay, whole mountains 
 |ivill not be suttlcient to produce any sensible effect. A moun- 
 tain of an hemispherical figure, three miles high and six broad, 
 
 kvill not, by its attraction, draw the pendulum two min- 
 
 ^ites * out of the true perpendicular; and it is only in the 
 
 freat bodies of the planets that these fo;*ces are to be per- 
 
 •eived, unless we may reason about suuiller bodies in manner 
 
 followiug.f 
 
 Let A BCD represent the globe of 
 the earth cut by any plane, AC, into 
 
 [two parts, ACH and ACD. The part 
 
 IaCB bearing upon the part ACD 
 
 Ipresses it witli its whole weight; nor B| 
 can the part ACD sustain this press- 
 ure, and continue unmoved, if it is 
 
 {not opposed by an equal contrary 
 
 I pressure. And therefore the parts 
 
 |e({ually press each other by their 
 weights — that is, equally attract each 
 
 other, according to the third law of motion; and, if separated 
 and let go, would fall towards each other with velocities re- 
 ciprocally as the bodies. All which we may try and see in the 
 loadstone, whose attracted part does not propel the part at- 
 tracting, but is only stopped and sustained thereby. 
 
 Suppose now that ACB represents some small body on the 
 earth's surface ; then, because the mutual attractions of this 
 particle, and of the remaining part ACD of the earth towards 
 each other, are equal, but the attraction of the particle towards 
 the earth (or its weight) is as the matter of the particle (as we 
 have proved by the experiment of the pendulums), the at- 
 traction of the earth towards the particle will likewise be as 
 the matter of the particle; and therefore the attractive forces of 
 all terrestrial bodies will be as their several quantities of matter. 
 The forces (prop. 71, book I), which are as the matter in 
 
 Fig. b 
 
 * [Maskelyne {^\)mys with reference to this : "It will appear, Iry a very easy 
 calculation, i/iat such a mountain would attract Vie plumb-line 1' IH" from the 
 perpendicular."] 
 
 f [ TViis jxiragraph is of great imjyortance, becavse in it Newton indicates 
 the metliods of all tlie experiments yet made in order to measure gravitational 
 attraction in terrestrial bodies.] 
 
 B 17 
 
MEMOIRS ON 
 
 I i.ii 
 
 n: 
 
 ' i. 
 
 i I 
 
 f-l 
 
 m 
 
 r.i 
 
 terrestrial bodies of all forms, and therefore are not mutable 
 witii tiie forms, must be found in all sorts of bodies whatsoever, 
 celestial as well as terrestrial, and be in all proportional to their 
 quantities of matter, because among all there is no dilteretice 
 of substance, but of modes and forms only. But in celestial 
 bodies the same thi?ig is likewise proved thus. We have shewn 
 that the action of the circumsolar force upon all the planets 
 (reduced to equal distances) is as the matter of the planets; 
 that the action of the circumjovial force upon the satellites of 
 Jupiter observes the same law; and the same thing is to be said 
 of all the planets towards every planet; but thence it follows 
 (by prop. 09, book I) that their attractive forces are as their 
 several quantities of matter. 
 
 As the parts of the earth mutually attract one another, so 
 do those of all the planets. If Jupiter and its satellites were 
 brought together, and formed into one globe, without doubt 
 they would continue mutually to attract one another as before. 
 And, on the other hand, if the body of Jupiter was broken into 
 more globes, to be sure, these would no less attract one another 
 than they do the satellites liow. From these attractions it is 
 that the bodies of the earth and all the planets effect a spheri- 
 cal figure, and their parts cohere, and are not dispersed through 
 the aether. But we have before proved that these forces arise 
 from the universal nature of matter (prop. 72, book I), and 
 that, therefore, the force of any whole globe is made up of the 
 several forces of all its parts. And from thence it follows (by 
 cor. 3, prop. 74) that the force of every particle decreases in 
 the duplicate proportion of the distance from that particle; 
 and (by prop. 73 and 75, book I) that the force of an entire 
 globe, reckoning from the surface outwards, decreases in the 
 duplicate, but, reckoning inwards, in the simple proportion of 
 the distances from the centres, if the matter of the globe be 
 uniform. And though the matter of the globe, reckoning from 
 the centre towards the surface, is not uniform (prop. 73, book 
 I), yet the decrease in the duplicate proportion of the distance 
 outwards would (by prop. 76, book I) take place, provided that 
 difformity is similar in places round about at equal distances 
 from the centre. And two such globes wiP (by the same prop- 
 osition) attract one the other with a force decreasing in the 
 duplicate proportion of the distar'^e between their centres. 
 
 Wherefore the absolute force of every globe is as the qnan- 
 
 18 
 
TIIK LAWS OF GRAVITATION 
 
 not. mutable 
 ios whatsoever, 
 rtional to their 
 
 1 no diirerence 
 ut in celestial 
 V^e have shewn 
 ill the phinets 
 I' the jihmets; 
 he satellites of 
 ig is to be said 
 nee it follows 
 
 i are as their 
 
 le another, so 
 satellites were 
 without doubt 
 her as before. 
 8 broken into 
 tone another 
 raotions it is 
 feet a spheri- 
 rsed through 
 J forces arise 
 book I), and 
 de up of the 
 
 follows (by 
 decreases in 
 lat particle; 
 of an entire 
 eases in the 
 roportion of 
 he globe be 
 zoning from 
 V- 73, book 
 he distance 
 ovided that 
 1 distances 
 same prop- 
 ping in the 
 ientres. 
 
 the quan- 
 
 lityof matter which the globe contains; but the motive force 
 hv which every globe is attracted towards another, and which, 
 in terrestrial bodies, we commonly call their weight, is as the 
 coiitont under the quantities of matter in both globes applied 
 to the square of the distance between their centres (by cor. 4, 
 prop. ?(>, book I), .J which force the quantity of motion, by 
 which each globe in a give!i time will be carried towards the 
 other, is proportional. And the accelerative force, by which 
 every glol)e according to its quantity of matter is attracted 
 towards another, is as the quantity of matter in that other globe 
 applied to the square of the distance between the centres of 
 the twi (by cor. 2, prop. 76, book I), to which force the ve- 
 locity by which the attracted globe will, in a given time, be 
 carried towards the other is proportional. And from these 
 principles well understood, it will be now easy to determine 
 the motions of the celestial bodies among themselves. 
 
 Sir Isaac Newton was born at Woolsthorpe, near Grant- 
 ham, in Lincolnshire, in 1645J. He was educated at the Grant- 
 ham grammar-school, entered Trinjty College, Cambridge, in 
 1661, and received his degree four years later. He at once 
 began to make those magnificent discoveries in mathematics 
 and physics which have made his name immortal. In 1665 he 
 committed to writing his first discovery on fluxions, and shortly 
 afterward made the unsuccessful attempt, to which we have 
 already referred, to explain lunar and planetary motions. He 
 next turned his attention to the subject of optics ; his work in 
 that field includes the discovery of the unequal refrangibility 
 of differently coloured lights, the compositeness of white light 
 and chromatic aberration. Having erroneously concluded that 
 this aberration could not be rectified by a combination of lenses, 
 he turned his attention to reflectors for telescopes and made a 
 great advance in that direction. His name is also closely iden- 
 tified with the colours due to thin plates. From 1669 to 1701 
 he was Lucasian professor of mathematics at Cambridge. He 
 was elected to membership in the Royal Society in 1671, and 
 from 1703 until his death was its president ; he became a mem- 
 ber of the Paris Academy in 1609. The publication of his work 
 on Optics had caused some controversy, and such a lover of 
 peace was Newton, and so little did he care for the praise of 
 
 19 
 
^rmmmm!^^ 
 
 ME MO I Its ON THE LAWS OK (i It A V IT A T I O N 
 
 ' 
 
 i 
 
 '■'■\ 
 
 : ,,t 
 
 p!!l 
 Mi:; 
 
 •ij'i 
 
 : r 
 
 I t ! 
 
 ! it 
 
 the world, tluit it was only iit tlio onniost solicututioii of Ilalley 
 tiiat ho was williiii^ to give to tlio public tlio rosults of liis won- 
 derful resoarohes on coiitral orbits, and universal ^gravitation ; 
 these included an explanation of the lunar inequalities, the 
 figure of the earth, the precession of the equinoxiis and the 
 tides, and a method of comparing the masses of the heavenly 
 bodies. In lOOl) he became a member of Parliament, in 109(5 
 Warden of the Mint, and from lOU'J until his death was Master 
 of the Mint, lie gave much valuable aid in the rccoimige of 
 the money and in questions of finance at this period, lie was 
 knighted in 1705. During the latter years of his life much of 
 his time was devoted to his public duties. Ho died in 1737, 
 and was buried in Westminster Al>bey. 
 
 If 
 
LA FIGURE DE LA TERRE 
 
 netermint'e par les Obscrvivtioiis do Messieurs Boiiguer, et 
 (le la Coudarniiio, do rAeadeinie Uoyalc des S<;ieiHU's, eiivoyes 
 |)iir ordre du Roy au Peroii, pour observer aux environs do 
 l'f^(juatenr. 
 
 Avec uno Uelatiou abregee de ee Voyage, qui eontiont la 
 description du l*ays dans lequel les optM-ations ont etc faites. 
 
 Pau M. BOUGUER 
 X Paris, 1749 
 
 m 
 
 MM 
 
 Secliou 7. pp. 327-394 
 
 THE FIGURE OF THE EARTH 
 
 Determined by the observations of MM. Bouguer and de la 
 Condaniine, of the Royal Academy of Sciences, sent to Peru 
 by order of the King to make observations near the equator. 
 
 With a brief account of their travels and a description of the 
 country in which the investigations were made. 
 
 By PIERRE BOUGUER 
 
 Paris. 1749 
 
 id 
 
 w 
 
 fyl5 
 
 Section 7 pp. 327-394 
 
 21 
 
 3' 
 
1 
 
 I 
 
 
 k i 
 
 iif ! 
 
 ,1 ' 
 
 >' itfl! 
 
 Ji 
 
 CONTENTS OF SECTION VII 
 
 PAOR 
 
 Jntroditction 23 
 
 Chap. I, — Exjteriments Made in Order to find the Length of the Seconds- 
 Pendulum 24 
 
 Description of Pendulum 24 
 
 Method of Observation ( Omitted). 
 
 Observed Ijenyths of Seconds- Pendulum at Various Places 25 
 
 Corrections to be Made in the Observed Lengths 25 
 
 Corrected Lengths <f Seconds-Pendulum at Various Places. ... 27 
 Chap. 11. — Comjuirison of Attraction and Centrifugal Force (Omitted). 
 Chap. III. — liemarks on the Diminution in Attraction at Different 
 
 Heights above Sea-level 27 
 
 Calculation of the Attraction Due to a Plateau 29 
 
 Deduction of the Mean Density of the Earth from Pend- 
 ulum E.vperiments 32 
 
 Chap. IV. — On the Deflection of the Plumb line by a Mountain 33 
 
 Description of Mount Chimborazo 34 
 
 Its Deflection of the Plumb-line Calculated from the Theory. . 34 
 
 Various Ways Suggested for Showing the Deflection 35 
 
 Description of the Method Employed 38 
 
 Examination of tfie Attraction of Chimborazo 39 
 
 Meridian Altitudes at the First Station 40 
 
 Measurements Made toflnd the Itelative Positions of the two 
 
 Stations 41 
 
 Meridian Altitudes at the Second Station {Omitted). 
 
 Collected Meridian Altitudes at the Second Station 42 
 
 Calculations for the Observed Deflection of the Plumb-line. . . 42 
 
 Its Poor Agreement with that Calculated from Theory 43 
 
 Appendix {Omitted). 
 
 22 
 
 iiii 
 
PAOK 
 
 . . • • 
 
 23 
 
 )ndH- 
 
 
 • • • • 
 
 24 
 
 .... 
 
 24 
 
 
 25 
 
 
 35 
 
 s. . . . 
 
 27 
 
 ted). 
 
 
 f'erent 
 
 
 
 27 
 
 
 29 
 
 Pend- 
 
 
 
 32 
 
 . ... 
 
 33 
 
 
 34 
 
 ovy . . 
 
 34 
 
 
 35 
 
 
 88 
 
 
 39 
 
 
 40 
 
 e iwo 
 
 
 
 41 
 
 
 42 
 
 ne. . 
 
 42 
 
 
 . 43 
 
 SECTION VII. OF B0UGUER^8 FIGURE 
 
 OF THE EARTH 
 
 ACCOUNT OF THE KX PKIU MENTS OR OBSERVATIONS ON GRAV- 
 ITATION, WITH REMARKS ON THE CACSES OF 
 THE FIGURE OF THE EARTH 
 
 1. Having discussed evorytliitig that bears on the earth con- 
 sidered as a geometrical body, it remains for us, before terminat- 
 ing this wori<, to verify the facts whicli give us some slight 
 knowledge of the interior conformation of this great mass con- 
 sidered as a physical body. . . . 
 
 2. The first question which presents itself on this matter is 
 ji consideration of the part played in the flattening of tlie earth 
 by the attraction which compresses it from all sides, urging all 
 masses towards certain points. We know, since M. Richer first 
 remarked it (in Vill'Z in Cayenne), that this force is not every- 
 where the same. It is greater towards the poles, and less to- 
 wards the equator. This agrees perfectly with the figure of the 
 earth, which appears to have yielded a little to the great press- 
 ure at the poles, and to be slightly elevated, on the contrary, 
 at the equator, where the compressing force was more feeble. 
 But does the eifect correspond exactly to the cause upon which 
 we desire it to depend? Is the difference in attraction so great 
 that we can attribute to it all the inequality which exists, as 
 we have seen, between the two diameters of our globe ? To 
 answer this question it is necessary to determine, by exact ex- 
 periment, how much the attraction actually differs in different 
 })arts of the earth. . . . We have two methods for observing 
 tlie change in attraction as we pass from one region to another ; 
 we have only to examine how much more quickly or more 
 slowly a pendulum of given length oscillates ; or else to find 
 the length of the pendulum whose time of vibration is exactly 
 
 23 
 
MKMolliS ON 
 
 II spcoiul ; the (lifTorcnros whic^h wo shall tiiid in the? h'fi^th nf 
 this pciiiiiilniri will (Ictcrfniiio the chuii^eH uf the uttructioii us 
 we go from one region to another. 
 
 I 
 
 
 Ii! 
 ' i 
 
 1 
 
 1 ? 
 
 1' i 
 i 1 
 
 1 ' 
 
 1 1 
 
 
 N' 
 
 j ' 
 
 
 ! i 
 
 ! 
 
 ; 
 
 1 
 i 
 
 1 
 
 i 
 
 1 
 
 1 
 
 i' 1 
 
 \ i 
 
 i 
 
 i; i 
 
 f- 1 
 f •' 
 
 j 
 
 1 
 
 ' '' i 
 
 t ' 
 
 ■ 
 
 t: 
 
 1 
 
 ti 
 
 
 
 
 ACCOl'NT OK THK KXI'KIMMKNTS MADK FOR THK PUU|M)SK OF 
 I>KTKKMININ(} TIIK I.KNOTM OF THK HK(;ON DS-I'FN r)r LTM 
 
 .'J. My lirst experiments witli tiie pendulum were niiulo iit 
 I*etit-(ioiivo in the ishmd of St. Dominguc!. 'I'iiey iire reported 
 in the memoirs of the Academy for 1TIJ5 and 17JJ<». . . . 
 
 4. 'ri)e instrument whieli I almost always used, and which I 
 still use, is extremely i-imple. I make the pendulum always 
 exactly of the same len^t!».. and I compare its osirillations with 
 those of a clock which I regulaie hy (laily observations. Jt is 
 not, properly speaking, by the diiferent lengths of the ])end- 
 nlum that 1 judge of the intensity of gravitation at different 
 places ; I judge of it only by the greater or less nipidity of the 
 oscillations, or by the number of oscillations made by the pend- 
 ulum in 24 hours. ... It appears to me to be much easier to 
 count the number of oscillations than to measure directly dif- 
 ferences of a few hundredths of u line* in the length of the 
 pendulum. 
 
 [Then follows an account of his pciKhdum. The boh was of 
 copper, composed of two equal truncated cones joined at their 
 greater bases. The thread was a fibre of aloe, which is not af- 
 fected by the weather. The length was maintained constant by 
 having it always so that an iron ride just fitted in between the 
 clamp and the bob. The length of the equivalent simple pendulum 
 was 3(1 pouces, 7.015 lines. 
 
 Botiguer gives a description of a scale fixed behind the pend- 
 ulum, by means of which he could observe the decrement and the 
 time required by the pendulum to gain an oscillation on the 
 clock. ] 
 
 10. It is time to relate the experiments. ... I shall choose 
 one of those which I made on the rocky summit of Pichincha 
 [2434 toises above sea-level], in the month of August, 1737. The 
 
 * [73 pouces = 1 torn = 1.949 metres = 6.3945 ft. 13 lines = 1 pouce ] 
 
 34 
 
Til K LAWS (H- (',\l\\ I r.\ rioN 
 
 t'orco of attnictioii was fi'd))*?, not only btM-ausc we wcrr lu-aily 
 ovor tlio r(|iiatoi' at tliis place, liiil also ln'caiisc wo were ut u 
 very ;;;n'al. Ii«'i.nlit ahove the siirlace ul' the 8eu. . . . 
 
 I Dihtils of c.r/irriiHrnt. | 
 
 I",'. . . . We liinl in thin way that the pendulum wliieli heat.s 
 secomls at tiie equator, and in the hi^^Miest aecu'SsihU^ plaei; on 
 the eartli, is IJll polices (LOW lines in leiij^th. I inatle other ex- 
 periments at the same place which a;,'r»'e'i as exactly as possihie 
 with this result. \()ih' mmlr hij Ihni Aiitoxiitdv t'l/an i/air 'M\ 
 pnnrvs, (I.T15 lines. \Vc may lake us llie mean 'M\ jtouci's, O.io 
 lint's. I 
 
 i;{. I have found by the same proeeedini^s ami with the aid 
 of the same instriiimints, the len;^4h of the s(H3onds-pendiiliim 
 at t^iiito I 14<l(l laisrs ul/oiw sm-lcirl], to be 'M\ ponces, (t.H'i or 
 (l.s;{ lines. 1 have verified it at ditterent times and in all sea- 
 s«)ns of the year: at times of aphelion and perihelion, at the 
 e(|iiino\es, and when the sun was at intermediati^ points; the 
 extreme results were 'M\ pouees, O.T'.J lines and ().8r> lines, with 
 no dilTerciKies whioh could not bo attributed to the inevitable 
 errors of observation. . . . 
 
 I '/'//<' (jHCstion of n possible f/cnrlt/ cJuDtf/e is disrnssed. 
 
 HjjU'vinwnts were nuale irilh the same ((jtjxtrahis, in 1T4(>, (tt 
 I' hie tie r /nra. 14' or i^i' froni the et/aafor, ((nd searcehi 40 toises 
 above sea-lerel. /ioaf/aer regards this determination as that of 
 the true e(/ainoctial pendalnm.] 
 
 ir). 
 
 Place 
 
 I.OIl),'!!! 
 
 foiiiMl liy oxporlmonl 
 
 I 2484 toises absolute lieiglit. 
 Under the equator ul •] 1460 " 
 
 / Sen-level 
 
 36 pfHices, 
 
 6.70 liiH's. 
 
 
 
 683 " 
 
 
 
 7 07 " 
 
 At Poitnliell ., 9" 34' N. latitude 
 
 
 
 7.16 •• 
 
 At l»eiii-0(»iive, 18°37' " " 
 
 * * 
 
 
 7.33 " 
 
 At. Puiis 
 
 
 
 8.58 " 
 
 
 CORRECTIONS WHICH MUST BE API'LIKF) TO TlIK LHNGTIl OF 
 
 TIIE PENDULUM AS DETKUMIXKI) DIRECTLY FROM 
 
 THE EXI'KRIMKNTS. 
 
 16. [novf/uer remarks that these eorrertions arise from chanc/cs 
 in tenwerature and in the const it at ion of t lie atmosphere.] The 
 
 85 
 
1 
 
 i 
 
 
 1 
 
 ,1 
 
 ■ 
 t 
 
 1 
 
 ■ ! 
 ' 1 
 
 1 
 
 , 1 
 
 ', t 
 
 j 
 i 
 
 r| 
 
 1 
 
 IH.H 
 
 fini 
 
 I'!!'! 
 
 M KMOI KS ON 
 
 firHt niiiHc (loos not roftlly rlmtipc tlio length, it only iniikoH it 
 uppciir <litT<'r<Mit lUH^onliii/^ Jis tlic niciisureH wo iiho un* <lilTor- 
 cntly iilt<u'«M| hy licjit or ('(jid; hiit the otiior ciiusf brink's in ii 
 roul iniM|iiiility, simu! it prodiiccK nmrly the Huinu etToct uh if 
 tho weight wvrv. ^rnitcr or HtimlliT. . . . 
 
 IT. . . . Sinco tlu) t('in|K'riitnr(' of Quito (loos not (litTor from 
 thiit of {'aris in the rnitldio (tf sprin;,', wo huvo only to refer all 
 our results to it. 'i'hiit is, without altori.ig the lengths of the 
 pendulum found in these two cities, we huvo only to (M)rre(!t nil 
 the others hy in(!reasin<i or diminishing them, aoeording as the 
 metal rulos we used wore e,xpund(M| hy the hea^ or contraotod 
 hy the cohl. ( /A' ronrluilvs finni his r.rperimcnfs llutt a r/ianf/(', 
 of tenffffi of pcH(/it/inu of .0^ litirs rorrcspiunis fo <t <'linn(jp of 
 ffimpcrahnr of 'X' li. Uvncv hi had to mid .OT.") linos lo I ho lfinf/lh 
 found at sfff-lcrcl, (Hitl stihh'orl .0") lines from Ihal found at 
 /'irhinrhd. | 
 
 18. Thoro is little more diflieulty in finding? the altoratioii in 
 the hfUfifth of the p(M'.duluni eaused by the nuMlium in which 
 the experiments are made. This medium, whether rare or dense, 
 has a certain weight, atid that of the small nuiss of copper, of 
 which the hob of tho pendulum is formed, is a little lessened 
 by it. The small mass tends to fall to the earth with only the 
 excess of its weight above that ot the air which surroutuls it. 
 MMius our pendulums are acted on by a forte a little less than if 
 we had jierformcd tho oxperitnents in vnruo : and the lenjijth 
 of the seconds-pendulum, wliicdi wo found directly from experi- 
 ment, is a little too short in tlie same proportion. 
 
 10. The use of the barometer emibles us to find the ratio be- 
 tweeti the weight of mere. try and of air in all tho parts of the 
 atmosphere which are accessible. We observe how many feet 
 it is necessary to ascend or descend in order to change the 
 height of the mercury by a lino. ... I have found in this way 
 that it was only necessary to express the first (tho weight of 
 air) by unity, at (h j summit of Pichincha, if one expressed that 
 of copper by IK'OO. ... So I always found the seconds-pend- 
 ulum too small by jT^^ivT^th part. To correct for this error 
 we must add .04 lines [at Pichincha ; .05 at Quito; .06 at sea- 
 level.^ . . . This is the first time that any one has taken ac- 
 count of this small correction which enters into tho experi- 
 ments, but we catniot neglect it if wo wish to attain tho greatest 
 
 accuracy. ... 
 
 26 
 
Til i: LAWS (H' <i liA Vn ATluN 
 
 I /linii/Hcr f/ifft proves Hial Ihv limv of rihnilinn is unf n/tpn'ri- 
 tihljl nlf'nfctf hif till' nsishnirv of thr tiir. |* 
 
 •,'•.'. ('orrcftcMJ l»'ii;,'tlis nf jlic s( nds-jR-iitliiliim, or siicli \\n 
 
 tlicy would l)u if tlio oHcillutions wwv iiiailc /// rtniio. 
 
 I'liiro 
 
 t 2484 tolfli'H iili!4M|iiie height. 
 I ikIjt tli« i-qimtor Hi \ 146(1 
 
 f Sni-lcv«'l 
 
 At Portohcllo. 9' W N. lalltiid.; 
 A I IVtil O..UVI'. 18^27 " " . 
 At PariM 
 
 :)6 pollers. 0.01) lines 
 7'Jl " 
 
 7.;u) " 
 
 " '* 7 47 ' ' 
 
 " " H.07 " 
 
 
 i 
 
 II 
 
 COMPAHISON or ATTRACTION AND TIIK CKNTUIFlf} A I, TOKCK 
 
 WIIK 11 H>I)!KS ACt/riKK l«Y TIIK MOTION O I' TIIK KAKTII 
 
 A HO IT ITS AXIS, WITH K KM AUKS ON TIIK KFI'KCTS 
 
 Ol" THKSK TWO I'OKCKS. 
 
 I lioHi/urr ^'ti«is lh(tt I he prinn'/in- (iffntcfiott {fluff (s, f/tr affnir- 
 fioN flic mrfli irotifd Ihnw if if tri'ir af rrsf) is fit fhv cvufvifuijai 
 force (IS •/iSSJil : 1. /fc i/iirs n fitfjfr slioirimj flw ifirrcasr in flie 
 tvnijfli of flw siroiiifs-pfHilnhnti iif various lafifuilcs, due fo flic 
 vcnfrifiKjnl forvp. The folio iriuij hvadiiKis will (fiiw an idea of 
 llie Hiaffcr ronfirned in flie resf of fliis rlmpfcr. \ 
 
 The centrifugal force produced by the riiotioti of tlie earth 
 about its axis is not sufficient to produce the observed differ- 
 ences in weight. 
 
 The primitive attraction does not tend towards a common 
 point as centre. 
 
 t >a 
 
 III 
 
 REMARKS ON THE DIMINUTION IN THK ATTRACTION AT DIF- 
 FKRENT HEIGHTS ABOVE THE LEVEL OF THK SEA. 
 
 40. The experiments witli the pendulum which we have 
 made at Quito and on the summit of Fichincha teach us that 
 
 * [ike note at) puije 06] 
 27 
 
MKMOlllS ON 
 
 Hi 
 
 tlio iittnictioii oliaiiiifos with tlio distimco from tlio centre of the 
 ojirtli. 'riiis force goes on tlirniiiishiiig sih we Jisceiitl ; I have 
 found thi! pendulum iif Quit(» to he shorter than at sea-level by 
 .'V'l lines, or the iVsyth part: and in mount'iig to the summit of 
 Pichincha the pendulum is shortened again ijy ,11) lines, and is 
 (^lsth part shorter than at sea-level.* One cannot attribute 
 these differences to the centrifugal force, which, being greater 
 the higher we ascend, ought to diminish a little further the 
 primitive attraction. 'IMie centrifugal force is increased by the 
 lieight of the mountain l)y the T-jVit*''' P''-''^ <^>'»ly> smd J^s it is 
 itself but the jjj>th part of, the weight, it is clear that its new 
 incroasi! (!orresponds to .001 lines only in the length of the 
 pendulum, and oo does not sensibly coi^tribute to the dimi- 
 nution of the other force. 
 
 41. If we compare the shortening which the pendulum re- 
 ceives with the heigiit at whi(di the experiment was made, we 
 see that the forces do not decrease in the simple inverse ratio 
 of the distances from the centre of the earth, but that they 
 follow rather the proportion of the square. Quito is 14()(J 
 toises above sea-level, or ^^Vt^^' "' ^^^*^ radius of the eartli ; but 
 it has been found that the attraction is less by a fraction much 
 more considerable — mimely, by a itj^t^I^ part, which is nearly 
 double : this is not verv far from the inverse ratio of the 
 square of the distance. . . . We have a seconi'. example in the 
 experiment made on Pichincha. The absolute height of this 
 mountain, which is 2434 toises above sea-level, is j-jVs^'^ ^^ the 
 radius of the earth. The diminution of the length of the pend- 
 ulum, or of the attraction, ought then to be the jj^^jth part, 
 if it is to be in the inverse ratio of the square of the distance ; 
 but it was by no means so great — in fact, only the g^th part. 
 
 42. This diminution in attraction, as we go above sea-level, is 
 quite in conformity with what we otherwise know. We can 
 compare with the attraction here experimented upon that 
 which keeps the moon in its orbit, or wnich obliges it con- 
 tinually to perform a circle about us. These two forces are 
 exactly in the inverse ratio of the squares of the distances 
 from the centre of the earth. We can make the same ex- 
 
 * [Pendulum obsterratiom were nvule ni these and other places in Peru hy df 
 la Condnmine also (H, pp. 70, 144, 162-1G9). For a. complete hihliography of 
 pendnlitm e.rperiments, see that published by La Societe Franfaise de Physique 
 (178, ml. 4) 1 
 
 28 
 
TIIK LAWS OI' <;UAVITATI(>N 
 
 iiiiiiiiJitioii with rospoft to tlio principal phiiiots wiiicii liavo 
 suvt'i'iil satolliti's, or with respect to the siiii, to\var<ls wliich 
 ill! tlie })rincipal phiiiets are attracted, ami we shall always 
 liiid the law of the square. Why, then, do our experiments 
 constantly give a law not entirely in agreement with this? 
 Is it necessary to attrihnte the diirerence to some error on 
 our j)art ; or can it be that in the m'ighborhood of great masses 
 like the earth the law under consideration is oi)served in an 
 imperfect manner only ? 
 
 4IJ. We shall find ourselves in x position to solve this dilli- 
 culty, perhaps, by remarking that the Cordilleras, on which we 
 were placed, form a kind of [)lateau, or, what in certain ways 
 amounts to the sajne thing, the surface of the earth is there 
 carried to ii greater height or to a greater distance from the 
 centre. Tiiere is reason for believing that in this second 
 case the attraction would be a little greater ; for it is natural 
 to think that it depends upon the size of the attracting mass. 
 There are then two things to be considered in the cas(( of 
 the experiments on the pendulum which 1 have reported. 
 'IMiese experiments were made 
 at a great height above the av- 
 erage surface of the earth, and 
 therefore the attraction ought 
 to be found a little less. But, 
 on the other hand, the group / 
 of mountains on which Quito is j 
 
 placed and on which Pichincha 
 rises, and ail the other sum- 
 mits to which it acts as a 
 plinth, ouglit to produce nearly 
 the same elfect as if the earth 
 at this place were larger or had 
 a greater radius. The attrac- 
 tion on this account ought to increase. Thus it depends on 
 a kiml of chance, or, to speak niore ])hiloso])hically, it de- 
 pends on circumstances which we do not yet know, whether 
 the attraction at Quito will be equal to that at sea-level, or be 
 smaller or larger. 
 
 44. Suppose that the circle ADD represents tlie circum- 
 ference of the earth, of which C is the centre, and that Ar^ 
 is the amount by which Quito, situated at a, is elevated 
 
 89 
 
 ;i - 
 
M E M O I R S O X 
 
 i; 
 
 above spa -level. Inia^nne a now splierieal >«bell of terrestrial 
 matter, occiipyin^ nil the interval between the two concen- 
 tric surfaces ADD and ridd ; or, which conies to the same 
 thing, imagine that the eartii increases in radius, and that 
 Quito, without changing its position, remains at the level 
 of the sea, now supposed much higher. There is every reason 
 to think that the attraction at Quito would, as a consequence, 
 be found greater than it actually is at A or at D, in the 
 ratio of CA to Ca. It is necessary for that, however, to sup- 
 pose that the layer of earth enclosed between the two con- 
 centric surfaces is of the same density as all the rest ; for if 
 the density were different the increase would no longer be in 
 the same ratio. 
 
 45. Call /• the radius, and A the density of the earth. Then 
 rA is the attraction at all the points A, D, etc., suppo^-^ing 
 that the earth ends there. Call h the height Aa, which is 
 very small compared with r. Then the attraction at a is 
 less than at A, in the ratio of r' : (r + //)', or its diminution 
 will be as 2h :r; that is, if the attraction is rA at A, it is 
 (r — 27a)A at a, and this supposes that the earth has CA only 
 for effective radius. But all this will be subj>'C! \ ? change 
 if we add to our globe the layer Ar/D, whose density is h. 
 This new spherical laye/, if it had the same density as the 
 rest, would augment the attraction at the surface in the same 
 ratio as the radius of the earth became greater. The increase 
 would be in the ratio of r : r -^h. 
 
 46. Thus the added layer would not only make up for the 
 decrease which the attraction actually suffers when we go 
 away from the earth, in rising by the height Ka — hy but would 
 add a new amount to it, equal to half the diminution, since 
 it would make this attraction, which is actually r — 2h at the 
 point a, become r-{-h. It follows that the attraction which 
 the spherical layer can produce at its exterior surface ai i-^ 
 is expressed by dh, or three times its thickness; but \V'3 mv.A 
 multiply by the density S, because we suppose that the den 
 sity of the layer and that of the earth as a wholo are not 
 equal. 
 
 47. To recapitulate: When the earth has its radius, CA = r, 
 the attraction at A is ?-A, and at the height h is {r — 2h)A. 
 But when we add to the earth the spherical layer AdD, the 
 attraction at a becomes (r — 2/i)A+3//3. 
 
 30 
 
THE LA w s ( ) K ( ; II A \ 1 r a r i o n 
 
 48. All that remains now to he remarked is that the Cordill- 
 eras of Peru, however j^reat they may he, ought not to produce 
 tlie same effect as the spherical shell which we have assumed. 
 If the base EE of the Cordilleras were exactly double its 
 height, and this mass had the shape of the roof of a house of 
 indefinite length, then the Cordilleras would produce at a only 
 \ the effect of the entire spherical shell, as can he easily proved, 
 but there are further additions to be made in order to give a 
 more accurate idea of the Cordilleras of Peru. The base EE is 
 80 or 100 times greater than the height \((, which augments 
 the effect in precisely the same ratio as the angle at <( is 
 greater. This angle is oidy 90° when we find the effect \ of 
 that which the whole spherical layer would produce, but on 
 account of the great width of the base of the Cordilleras the 
 angle is nearer 170°, which doubles the effect. Moreover, the 
 Cordilleras do not terminate at the height of Quito in a single 
 summit like the ridge of a house ; it is, on the contrary, quite 
 10 or I'i leagues broad there. One can suppose then, without 
 fear of mistake, that the effect is the greatest which can be 
 produced by a chain of mountains. It is the ^ of that which a 
 spherical layer would produce, or |//^, and if we add to it the 
 attraction (r— 2/()A, which the globe ADD produces at a, we 
 shall have ( 7'-2h)A 4- pic* as the expression for the attraction 
 at Quito, when rA expresses that at sea-level. 
 
 49. The differciice between the two is 2//A — |//2, which 
 furnishes the subject of divers quite curious remarks. If the 
 matter of the Cordilleras were more compact than that of the 
 average of the whole earth, and their densities were as 4 : 3, 
 the difference 2/iA — |//^ would become zero, and the attrac- 
 tion at Quito would be the same as at sea-level. If the density 
 h were still greater, our expression In- the diminution would 
 change sign and become an increase, so. that the pendulum 
 would be longer at Quito than at sea-level. But it is evident 
 
 * [This formula is independently foundby D' Aleinhert {V3, vol.' 6, pp.85-9^i). 
 by Young (51 and 95, vol. 2, p. 27), and by Poixaon (65. vol. 1, pp. 492-6). 
 
 Under tJie form g*= ffo {^~— + oxJ *' *** known as "Dr. Youitg'-^ 
 
 Rule," ichere ,9' is the value of gravity at height h, and g^ is the valuj at the 
 sea-level. Faye (147) contends that the last term of the equation should be left 
 out ; and if Airy's "flotation theory" (94), oi' Faye's compensation theory 
 (146 J), be true, there is no doubt that this term reqvires correction.] 
 
 31 
 
 fftia 
 
 1;; 
 
 I' 
 
 
 ■I H 
 
 ■1* ^ji 
 
 - I 
 
MEMO I Its ON 
 
 it 
 
 il.i 
 
 tliiit tliiii[,'s}iro not so. Tlie difTorenoe in the huijiftli of the pend- 
 ulum is siiflicicMitly grent to lot us see that tlio density of tlie 
 matter of wliich tlio Cordilleras is formed is much smaller than 
 that of the rest of the ^lobe. 
 
 50. We have found by experiment a diminution of a -Y^jth 
 part in the length of the pendulum, or in the attraction, as we 
 go from the sea-level to Quito. So jj'jy corresponds to 2AA — 
 |/t^, as compared with rA, which expnssses the attraction at 
 
 sea-level ; that is, we have — r.— = ? — If we put - = 
 
 l.'ilil 
 
 r A 
 
 ,..,,, y which is the ratio of the height of Quito to the radius 
 
 of the earth, we shall have ~ = ^i^ x '^-^-fc. Whence 
 
 850 
 
 i3;u '^'^:i7 
 
 we deduce 5 = .TTwTi ^' which tells us that the Cordilleras of 
 
 Peru, in spite of all the minerals they contain, have less than ^ 
 the density of the interior of the earth.* 
 
 51. We admit that this determination may contain a few er- 
 rors on u<icount of the large number of elements we had to em- 
 ploy in order to arrive at it. Nevertheless, if we once admit 
 that the attraction, when the other circumstances are the same, 
 follows exactly the direct ratio of the masses, we cannot doubt 
 that the Cordilleras of Peru have a density considerably less 
 than that of the rest of the globe. If we suppose A and 5 equal, 
 our expression for the difference of the attractions at Quito and 
 
 at sea-level would become -— A ; which would make the differ- 
 
 2;- 
 
 ence between the lengths of the pendulum 4 times too small, or 
 the attractions as the square roots, instead of the squares, of 
 the distances from the centre of the earth. The attraction at 
 Quito would be less than at sea-level by only the ^^-Vf^^^ part, 
 and the pendulum would be really shorter by only 9 or 10 hun- 
 dredths of a line, and in appearance by 2 or 3, an account of the 
 different constitution and temperature of the air. The differ- 
 ence of the lengths of the pendulum is certainly greater. Thus 
 it is necessary to admit that the earth is much more compact 
 
 * r^'' .9^^^ Bouffuer's result more accurately, the density of the earth is 4.7 
 tinws that of the Cordilleras. Saif/ey (74, p. 149) ?ias made a recalculation of 
 tliese resultti, with the vro)y;r reduction to vacuo, and finds 4.25. He has done 
 the same for de la Condamine's pendulum experiments, with a result 4.50. 
 For Addendum^ see p. 160.] 
 
 82 
 
11 
 
 THE LAWS OK (UiAVlTATluN 
 
 below than above, and in tlie iiitorior than at tlie surface. For 
 the soil of Quito is like that of all other eountries ; it is a mixt- 
 ure of earth and stones, with some metallic constituents. . . . 
 'i'hose physicists who imagined a great void in the middle of 
 the earth, and who would have us walk on a kiml of very thin 
 crust, can think so no longer. We can make nearly the same 
 objection to Woodward's theory oi great masses of water in the 
 interior. But let us continue to limit Oiirselves to the facts, or 
 to the only immediate deductions which we can draw from them. 
 These deductions are confirmed by the observations described 
 in the next chapter, which is in the form 1 gave it in Peru be- 
 fore forwarding it to France. 
 
 IV 
 
 MKMOIR ON ATTRACTION AND ON TIIP] MANNER OF ORSERV- 
 
 INO WHETHER MOUNTAINS EXIIIMIT IT (rEAI) AT THE 
 
 ACADEMIE DES SCIENCES, IN OCTOBER, Vi'.V.i) 
 
 52. It is very difficult not to accept attraction as a principle 
 of fact or of experience. The most rigid Cartesians, like all 
 other philosophers, cannot dispense with it in this sense. All 
 they can do is to reserve to themselves the right of explaining 
 it. . . . Since all the planets circle about the sun, there must 
 necessarily be a force, I shall not say shoving them or drawing 
 them, but rather transporting them at each moment towards 
 this star. . . . Nothing prevents us from giving to this force 
 the name ** attraction," and from trying to assign to it a phys- 
 ical cause. . . . 
 
 [BongueraffinHs fhat in esfahh',sJtinf/ a nem principle, it is not 
 onlif necessary to prove the insufficiency of all others, bnt their 
 impossibility also. ] 
 
 54. While waiting for all this to happen, it will contribuce 
 to the perfection of physics if we examine more carefully into 
 attraction as a fact taught by experience. ... It appeared 
 to me that if all bodies act "at a distance." in proportion 
 to their mass, and according to the other laws which we know, 
 such enormous masses [the mountains of Peru] should pro- 
 duce a marked effect. I am well aware that they are very 
 small compared with the whole earth ; but one can approach 
 1000 or 2000 times nearer their centre, and if it is true that 
 c 88 
 
 ,i 
 
 i| 
 
MEMOIRS UN 
 
 the tittruotioiis increase not only simply in tlic Siimc ratio as 
 the (listunoes (liniinisii, but in the inverse ratio of their squares, 
 one ouglit to have a kind of eojupensation. 
 
 55. I sliall content mysi^lf with justifying tliia in the case 
 of a singhi mountain caMcd Chimborazo, the base of wliich 
 one is oiiligcd to j)aHS in going from tlie sea-side at fruaya- 
 fjuil to the more inhal)ite<i part of tlie provim^e of Quito, 
 which is enclosed between the two chains of mountains liere 
 formed by the Cordilleras, whose distance aj)art is 8 or 9 
 leagues. Chimborazo must be .'UOO or 3200 toises above the 
 sea-level [he aflcrmrrfls fntoid if to he 3:^17 toises], and 1700 
 or 1800 above the level of the plat(^au. We know exa(!tly the 
 relative heights of all the mountains we have seen, but not 
 having yet been able to compare any one witli sea-level, we 
 are ignorant of their absolute heights. Chimborazo has roots 
 which extend very far and become merged in those of the 
 other mountains, so that it is very difficult to determine the 
 true extent of its base. It must be more than 10,000 or 12,000 
 toises in diameter. But when we mount as high as possible, 
 to where the snow begins, which is 850 toises from the top 
 and renders the higher parts inaccessible, the mountain is still 
 more than 3500 toises in diameter. The top, instead of ter- 
 minating in a point, is rounded and blunt, and appears from 
 below to have a width of 300 or 400 toises. From these di- 
 mensions one can estimate its huge mass. In the present 
 investigation we need to know its height above ground only, 
 not above sea-level. Even so it must be 20,000,000,000 cubic 
 toises in volume. This is about the T.ToTn^oo.ooo^'^ P^^*- ^"^7 
 of the globe, and the effect of the attraction would be absol- 
 utely insensible, if one considered the quantity of matter 
 only. But as we can place ourselves at 1700 or 1800 toises 
 from the centre of gravity of the mountain, or 1900 times 
 nearer to it than to the centre of the earth, this proximity 
 ought to increase the effect about 3,000,000 times, and so 
 make it about 2000 times less than that which gravitation 
 produces, or the attraction caused by the whole mass of the 
 earth. This we get by employing only a rough calculation 
 and the lowest estimates. Calling the action of the moun- 
 tain 1, and that of the earth 2000, the direction of attraction 
 should be deflected from the vertical by about 1' 43". A 
 plumb-line which would be directed exactly to the centre of 
 
 84 
 
rill!: LAWS <M' (illA V ITATloN 
 
 the earth, if its jiiiias \vor«( ex|)(>se(l to the earth's attruftioii 
 alone, oiij^ht tlioii, o!i aoconnt of the action of tiie mountain, 
 to ho incliiUMl l)y tiiis sjiine fjuantity, whioii is, as we see, (juite 
 c!onsi<]orahh». 
 
 50. lUit how ran wo recognize tiii« inclination; for all 
 gravitating? hodies must ho ('(jually suhject to it, and wo 
 seem to lack a term of comparison? It would he useless to 
 have recourse to the level surfa(!es of the lieaviest lirjuids, 
 since the attraction heinjjf equally altered with respect to 
 them, their surfaces, instead of heing jjorfectly horizontal, 
 must sutler the siimo incliiuition. We see plainly, then, that, 
 in order to judge of the amount of this alteration, it will he 
 of no use to look just about us, we must seek another ver- 
 tical line far oil' which is subject to no action from the 
 mountain. Hut again, how are we to compare one vertical 
 with another; or measure the an<de which they make in 
 meeting towards the centre of the earth, and that with suf- 
 Hcient accuracy? If while on the mountain, we observe with 
 the quadrant the height of a point far (^tf, and then go to 
 that point and measure the height of the former place, it is 
 true that by the dilference of these two heights we can judge 
 of the relative positions of the two vertical lines. Hut be- 
 sides that we must know the exact distance from one to the 
 other, it will be necessary also to suppose that the visual ray 
 is a straight line ; and it is not oidy certain that this is not 
 true, we know that it is subject by refraction to a very ir- 
 regular curvature. We cai:not determine this (curvature with 
 sufficient exactness to enable us to find the effect of *Jie at- 
 traction. It seems to mc, therefore, that we must seek in 
 the heavens a term of comparison. By this means, however, 
 we shall easily overcome every difficulty; and what a moment 
 ago seemed an impossibility becomes at once very simple. 
 
 57. We have but to station ourselves to the north or to 
 the south of a mountain, and as near as possible to its centre 
 of gravity, and observe the latitude. This observation can bt^ 
 made with the greatest accuracy oidy by using a quadrant 
 or other equivalent instrument whose plumb-line will be de- 
 flected toward the mountain; this is the same as saying that 
 the zenith will recede from the mountain. Then we must 
 go east or west of this station to such a distance that the at- 
 traction is negligible; and if we observe the latitude in this 
 
 35 
 
 ■. 
 
 ' 
 
 I 
 
 I- 
 
 
mi: MO IKS ON 
 
 il I ,'' I 
 
 HPcohd \)\inv. with tlu? saiiio Ciiro siiid with thn same iiioans 
 UH ill tlic first, it is ovidciit tiiiit all tlic (lifTcroiicc which wo 
 Hhall ohsiirvo will In; due to attraction. In onit-r to have 
 tiiiH second .station precisely east or west of the first, we must 
 ohserve the azimuth of the sun at its rising; or setting, hy 
 findiji^ its position with reference to some easily distinguished 
 point on the horizon ; in doing so we must often suppose the 
 latituile known; hut the error we nuiy make on this supposi- 
 tion will he of no consefjuence, and it will always he easy to 
 find two stations on tlie same parallel of latitude to v/ithin 
 ',] or 4 sixtieths of a second. The latitude will he found pre- 
 cisely the same in the two places, if the vertical line has not 
 heen altered in the first. Suppose, however, that without 
 seeking the latitude, we ohserve simply the meridian alti- 
 tudes of a star at the two stations ; the dilTerence of these 
 two altitudes will indicate equally well the deflection of the 
 vertical line. It is evident that all the stars which pass the 
 meridian on the side of the apparent vei'tical line next to the 
 mountain will appear lower at the first station than at the 
 second ; for as the plumh-line approaches the mountain the 
 apparent zenith recedes from it and from these stars. It will 
 be quite the reverse with those stars which pass the meridian 
 on the other side of the apparent vertical line : they will ap- 
 pear higher at the first station.* 
 
 58. Instead of taking the stations both to the north or both 
 to the south, we could take them one to the north and the 
 other to the south, and exactly on the same meridian ; then 
 the effect of the attraction would be doubled, roughly speak- 
 ing, and we should find the sum of the contrary attractions. 
 The vertical line would be inclined in opposite directions at 
 the two stations ; and the altitudes of stars which would be 
 increased in the one would be decreased in the other. The 
 physical effect being doubled would be more sensible, and 
 more susceptible of observation. If the two points were 
 equally distant from the centre of gravity of the mountain, 
 the action would be equal at both, and in order to get each 
 
 *{_I7m metJiod of doubling the deflection caused by tJte mountain, hy observ- 
 ing not one star, but at least two, one north and one south of the statiotuj, 
 is due to de la Condamine. See his account of tJte expedition (8, p. 68), Zach 
 (49) and Poynting (185, p. 14). This is the method actually employed by 
 BouguerJ] 
 
 m 
 
THK LAWS OK (', li A V ITATIoN 
 
 ^T^ 
 
 of tliem wo should hsive merely to tiike hjilf of tlie (jUiintity 
 furnished hy the coiiiparison of the ohserviitions. In other 
 ciiscH the division wonhl hv ii little more ditlienlt; neverthe- 
 less it would he sunieient, Jis we sluill shew later, to divi<ie 
 tljo sum of the eontniry attnietions pronortionaliy to the pro- 
 ducts of the f|uantity hy which each station is more north or 
 more south, respectively, than the centre of ^'ravity of the 
 mountain and the cu])e of the distance of the other station, 
 respectively, from the sumo centre. Thus W(( are untler the 
 necessity of knowin^j the situation of each station with refer- 
 ence to the mountain ; hut w(( must know tin* distance from 
 one station Jji) the other also, in order to determines j^'eometric- 
 ally the difference of latitude hetween them. It is evident 
 that this difference' must itself j)roduce a change in the alti- 
 tude of each star, and we must know it hefore we can tell what 
 is the douhle effect of the attracticMi. To ohtain the difference 
 in latitude of the two places, it would suffice ordinarily to 
 measure to the east or to the west of the mountain a hase 
 directed nearly north and south, and to form on this hase two 
 triangles which end at the two stations. 
 
 50. This way of makinij two ohservations from different 
 sides of the same mountain in order to render the effect of the 
 attraction more sensihle, seems to me tiie more useful method, 
 as it depends less on the peculiarities of the places. We can 
 sotnetimes double the effect also by making the first observa- 
 tion at the north of one mountain and the second at the 
 south of another. If the two stations are not exactly on the 
 same east and west line, we have only to determine geometric- 
 ally their difference of latitude, and take account of it in the 
 comparison of the altitudes of the stars. 
 
 GO. Finally, it is not only by observations made at the north 
 or at the south that we can discover whether mountains are 
 capable of acting " at a distance" ; it can be done also by ob- 
 servations made at the east or the west; but with this differ- 
 ence, that it will be no longer a question of observing latitude, 
 or of taking the meridian altitudes of stars; it will be only a 
 (piestion of determining time exactly. It appears to me that 
 this last method would be often preferable to the preceding 
 ones, except that it requires two observers. Suppose that the 
 first of these is on the east side of a mountain, and the second 
 on the west side of another, or of the same, mountain. If each 
 
 37 
 
 ill 
 
 1 ^1 
 
 ; 
 
 t 
 
 i 
 
 ill 
 
f 
 
 M i:.M(>i ks ON 
 
 of ihom ro<^uliit('s ciircfiilly u clironornch'v hy (foncspoiulin^ al- 
 titudes, it is t'vidciit tliiit ull tlu'Ho iiltihidtts bLMii«;; altered by 
 tlio iittni(;ti()U \vlii(di dcilcM'ts tlie pliinil)-line, oacli (diroiioino- 
 tcr will l)(! n><,'iil!ittMl us if the nieridiim were not exactly vertic- 
 al, but in(diMed below toward the mountain, and above uwuy 
 from it. Let us sup|)ose that the attraction amounts to a min- 
 ut(t of arc. and that the two mountains uro un the equator ; 
 th(i lirst (dironometer will denote midday 4 seconds of time 
 too soon, and the otluM' 4 seconds too late. TIjus, nej;leet- 
 iiii,' tin; dilTerenco of lon«j^itude, which we could easily find by 
 measuriui:; tri^onometrically the distance of the two observ- 
 ers apart and reducing this distance to decrees and min- 
 utes, there would be a ditTerence of 8 seconds of time be- 
 tween the two chronometers. If the two mountains instead 
 of being on the equator were at latitude (10°, each minute of 
 imdijuition whicdi the attraction produced in a plumb-line 
 would produce 8 seconds of dilTerenco in the time of mid- 
 day, and tluM'efore HJ seconds difference in the chronometers. 
 Finally, to judju^e of the iittraction we need only know the exac^t 
 difference between the (dironometers; and to find this, it would 
 always b(^ sufficient to agree upon a signal, })y fire or other- 
 wise ; and to observe at both stations the minute and second 
 of the instantaneous ajjpearance of this signal. 
 
 (11. I return to tlu; first method because it appears to me to 
 be the simplest ; that is, suppose we station ourselves always 
 to the nortli or to the south of the mountain and confine our- 
 selves to observations of the latitude. It is evident that if we 
 take at each station the meridian altitude of one star only, 
 we must know to the last degree of nicety the condition of the 
 (juadrunt we are using. There is no lack of methods for veri- 
 fying this instrument, but there is one which is extremely 
 valuable in the present instance, because, at the same time as 
 we work at verifying the quadrant, we are making the observ- 
 ations which decide the question at issue ; and in thus abridg- 
 ing the operations we avoid opportunities for errors. This 
 method is to take the meridian altitudes of an equal number 
 of stars toward the north and toward the south, and, provided 
 that the state of the instrument does not vary from one observ- 
 ation to another, it does not matter if it does change from 
 day to d;iy. If it makes the altitudes of the stars on one side 
 the zenith too great, it will produce the same effect with re- 
 
 88 
 
Til K LAWS OK iillAVIT ATION 
 
 i« 
 
 Hpect to tlioso on tho otlior si«l('. Tlin?* tlio cliimjjo will inflii- 
 niro only tlic sum of the jiltitrnh's or llic coniplcnn'iits of the 
 jiltitucU'S, anil will not alter tlir tlilTcri'iici' of t he alt itndcs taken 
 on the (lilTerent sides. The jit tract ion, on the ('(uitrary, will n(»t 
 alter the sum, hut will ('han;^'i' the dilTerenee; heeanse at the 
 name time that it makes the stars on one sidt^ too hi^h, it makes 
 those ofj the other side too low. It will always ho easy to sep- 
 arate these two eaiises. and we shall not attrihnto to the one 
 that whieh arises from tln^other. To ohtain at one stroke the 
 efTect of attraction without heini,' ohli;r(.d to know the state of 
 tho quadrant or tlu^ decdinations (d' the stars, we need only ex- 
 amine whetlnu" the ditTereiices of the meridian altilndes taken 
 towards the north and towards tlu; south ar(^ the same at the 
 two stations, or whether they an^ suhject to a seeotul difT(!r- 
 ence. lint it is necessary to remark that the altitudes hein^' 
 incM'eased on tho one side while they are diminisluMl on tin; 
 other, it is the half of this S(!cond dilTc^'once whi. h diMiotes 
 the physical etToct of tho attraction, hoth when this elTect is 
 single and when it is douhle. In this lattor case, it will he 
 necessary to divide the total etTcct in tluf ratio wITudi the sep- 
 arate effe(3ts ought to have. 
 
 [/ioHf/nrr then proirs tliix rafio in he thai \nvnt\uned nlnire [p. 
 W'i) He adnnts that some nntuiitaiiis niiiftit shein tesx dttrartion 
 than that rcf/aired fit/ .\Vv/7o//'.s' tfttn {ar eren none), due to the 
 existence of great rarities in the tntiss. He discusses the dijf'erent 
 nnn(nt(fins in the neit/hbtHirhood of Quito, and for various reasons 
 decides upon Chiniborazo as the one most suit alAe for tlie experi- 
 ment.^ 
 
 Examination of the Attraction of Ciiimborazo 
 
 65. I did not ascend this mountain alone as \ did the pre- 
 ceding one. I had some time hefon^ communicated my design 
 and all my views to M. de la Condaminc, and when on the 
 point of carrying them out I mentioned tlu'm to M. de Ulloa, 
 one of tho two naval lieutenants who had assisted in the oh- 
 servations both of myself and of i\r. de la Condamine ever since 
 our arrival In the domains of His Catholic Majesty. These 
 gentlemen obligingly offered to accom})any me, not oidy in the 
 preparatory examination, but also during the stay it was necess- 
 ary to make on the mountain side ; and as I knew it would be 
 to the advantage of the observations, I hastened to accept the 
 
 80 
 
MKMOIKS ON 
 
 P 
 
 ' ill 
 
 ofT**!'. I IukI alivudy tlioii^^lit tlitit ('liiFiiboruzo fiiHillcil p- 
 proxitniiti'ly tlu; lu't'osHitry conilitioris : I knew tlitit it wii .y 
 niwy of iKMM^ss ; it (M)iiI(I be Htutn from (^iiih), or rather from 
 IMcliiiicliii, from wliicii it wuh Tr>,<MM) t(»is(>8 ilirttuiit ; and I liud 
 
 ulrciidy mi'iiHiirtMl its lu'i;,dit On DiMutmbcr 4tli wo ('stub- 
 
 lisht'd oiirHclvt.'K on the Hotitli Hidu of the mountain, at the bot- 
 tom of tbo Hnow line, S'V.i toi.ses lutlow tlu; summit, but about 
 •J40() abovo sni-levi'l, ami exacitly IMO toises above the pbuso tit 
 Quito wliero I have always made my observations, and IJ44 
 toises al)ove that part of IMeliiiudia wliero tlioro i8 a (M'okh 
 wliicdi (;an be stieii from all [uirts »»•■* the eity, and where I passed 
 some days in Mareli, 1137, in order to observer the astronomitial 
 refraction. I shall not sptMik of the(U)ld and the other diseuni- 
 forts we ha<l to put up with ; snow covered our tent and all 
 the j^round around as far as SOO or IMKI toises below us, and we 
 lived ill fear of beinj; buried undtu' its wei«;ht. It needecl con- 
 tinual vi<(ilance in onler to avoid it. [M. dc intotf fell ill, and 
 had lo di'sreml the tnininhihi on Dcmnhcr \'yfh. ) 
 
 [ Li'/l alone, /ioNt/Hcr and dv la ('Onditniinv ohsrrird the alii- 
 fades of 10 sfars, 4 on the sindh side and (5 on the north. The 
 fidlotrinf/ are the altitudes as ajf'erted /»// the error of the instrn- 
 nieat and hi/ refraction; ttiej/ arc the means of t lie rvadinijs of the 
 two ot)ser vers. \ 
 
 On tlio norlli side : 
 
 Ciipellii 
 
 First head of Gemini. 
 
 Second heud of Geiniii 
 
 Second horn of Aries. 
 
 First horn of Aries. . . 
 
 Aldehariiii 
 
 On the soutli side : 
 
 Acarnar 
 
 Canopus 
 
 Tail of Cetus 
 
 Siriiis 
 
 Moridiiui HllitnJoH 
 
 ul the flrnt Htation 
 
 On 14tli 
 
 •w. 
 
 On 15tli Dnc. 
 
 O 
 
 9 
 
 ■ 1 
 
 o ' 
 
 ** 
 
 43 
 
 50 
 
 43J 
 
 43 50 
 56 6 
 
 30 
 5 
 
 59 
 
 55$ 
 
 55 
 
 59 53 
 
 57i 
 
 66 
 
 18 
 
 15 
 
 66 18 
 
 50 
 
 69 
 
 
 
 30 
 
 69 
 
 17i 
 57f 
 
 73 
 
 33 
 
 43 i 
 
 Ta 33 
 
 ;{3 
 
 58 
 
 10 
 
 33 58 
 
 25 
 
 :{8 
 
 58 
 
 53* 
 
 38 58 
 
 55 
 
 73 
 
 6 
 
 36^ 
 
 73 6 
 
 35 
 
 75 
 
 9 
 
 13^ 
 
 75 9 
 
 40 
 
 68. Wo observed the meridian altitude of the sun three 
 times. De la Condamino found it at the lower edge on De- 
 cember 15th to be 07° 54' ^(5'^ This altitude, wliich is cor- 
 rected for the error of the instrument, but not for refraction 
 
 40 
 
 'FBI 
 
Til K LAWS OK rnt A VITAT ION 
 
 ii!ul punillux, ^ivos T 'Vy M** Koiitli for tlio liititmlo of the 
 l»lin'(5 wlicrt! we wwr. I ulist-rvrd it on tlio ."»tli uikI l\itli ; on 
 tlir .Mil \vt! hail not yrt rc^^MilattMl tlu' rhronofnctor nor tnirtMJ 
 tlie meridian, and I t'onnd 1 ' iio' 10". On tliu I'^tli I oIihoi'vimI 
 llic apparent attitiidi^ of tlio lower udgo of the 8iin to be 
 IIS" :)':J4"; whieh ^ivert I" 'M)' 0". 
 
 iWK As soon as wo wore estal)lislio(l on ('hirnhorazo, \ lui<I sent 
 a tent al)ont a league and a half to tlio west to a place (billed 
 rAi'viml to servo as the seeonti station, [/iotnfm'r then dcscriht's 
 the nwnsHirnii'itls inadr tn Jind Iki' cxdrl position of the scrond 
 station : it nuts '.i'ui) ,'oisrs distant from t ft e first, 1T4 toiscs fotrrr, 
 and sonivwhitt snutli nf trrsf of it. Tlifif tnyan tttrir otiscrrtttions 
 from tin' strond stdlion on Dvc. IVtth. Ilcrv tln'if snffcrvd niorr 
 front tlic wind and cold tlian at tlie more derated station, as tlieif 
 were more exposed to the prevailint/ east wind. It filled their 
 eifes with dust and rontinaal/tf threatened to overturn the tents.* 
 The screws of the t/aadrant ronld not tte tamed at ni(jht withant 
 applffint/ heat to them.. Then follows a talde of the altitades as 
 of/serred. Sinre the second station was 505 toises, or W^", south 
 of the first, we mast increase hij Wl" all the altitudes of stars ob- 
 served toward the north, and diminish the others by tlie same 
 amount. Moreover since the second station was lower than the 
 first by 174 toises, the altitudes observed at the second station 
 must be diminished, to reduce them to the level of the first, on ac- 
 count of the cxces.s of astronomical refraction. The folio wimj 
 are the corrected altitudes, and the differences for each star of 
 the mean determinations at the two stations .•] 
 
 I*, 
 
 ^1 
 
 ^ i\ 
 
 * [For de la Condnmine'H account of (tie experiments, see his Journal (8, 
 /). 69 and 8h pt. 2, p. 146).] 
 
 41 
 
 II il 
 
 
 \ 
 
 \. 
 
IT 
 
 MEMOIRS ON 
 
 CORRBCTED ALTITI'DKH AT TIIK SkCOND STATION 
 
 Oil the norih side: 
 
 (Jnpella 
 
 First liead of Gemini . . . 
 
 Second head of Omiiiii. 
 
 Second horn of Aries.. . 
 
 First horn of Aries 
 
 Aldelmraii 
 
 On the soulii side 
 
 Acarnar 
 
 Canopiis 
 
 Tail of Cetus 
 
 Sirius 
 
 On 2lBt Doc. 
 
 o 
 
 » 
 
 n 
 
 43 
 
 40 
 
 10 
 
 66 
 
 16 
 
 59 
 
 68 
 
 59 
 
 8 
 
 72 
 
 33 
 
 3U 
 
 33 
 
 56 
 
 53 
 
 38 
 
 57 
 
 16 
 
 73 
 
 4 
 
 47* 
 
 75 
 
 8 
 
 
 
 ON 
 
 
 
 On 
 
 'iM 
 
 Doc. 
 
 o 
 
 t 
 
 n 
 
 43 
 
 49 
 
 15 
 
 59 
 
 53 
 
 23* 
 
 66 
 
 17 
 
 16 
 
 68 
 
 59 
 
 11 
 
 73 
 
 33 
 
 36i 
 
 33 
 
 56 
 
 38 
 
 38 
 
 57 
 
 36 
 
 73 
 
 4 
 
 50 
 
 75 
 
 8 
 
 n 
 
 F.XCCHH (>r ItltitlKtOH 
 
 nt Iho llrHt HlHlioii 
 ovor thoHC at iIh 
 BGcoiiil, iifior lilt' 
 liitler hiivo bec" 
 corroded 
 
 1 24 
 
 1 25 
 
 10 
 6* 
 
 1 37i 
 1 28 
 1 48 
 
 1 33* 
 
 III 
 
 ':J ' 
 
 t ! 
 
 [Bovgtier considers the observations on the tail of Vetns and 
 the first horn of Aries as the best, but thinks it most legitimate 
 to take the mean of ttie altitudes of each star at each station and 
 to give equal tveight to their differences. Tttese differences are 
 given in the last column of the preceding table, and are, lie main- 
 tains, too large and too uniform to be due to any defect in the ob- 
 servations. Tlie averages of the excess of all the stars on the 
 north side and all those on the south side are now to be taken.] 
 
 74. . . . Tliey Jjive about 1' 19" us the mean excess for the 
 north stars, and 1 34" for the south. The second difference is 
 15". I leave it to my readers to say whether such a quantity 
 is sufficiently established by the means employed. My quad- 
 rant was 2.5 ft. in radius, and it must be remarked that any 
 errors which may exist in its graduation are of no importance 
 here ; since we have to do not with the altitudes tliemselves, 
 but with their differences. Suppose we admit the 15", it will 
 ^\\Q 7".5 for the effect of attraction; it would be much greater 
 if we compared the tail of Cetus with the first horn of Aries. 
 However this is not the complete and absolute effect ; for if 
 J^ttraction really takes place, the mountain must have some 
 effect at the second station, which was about 4572 toises from 
 the centre of the mountain, and 61°. 5 to the west of south. 
 At the first station we were nearly 16° west of south, and 1753 
 
 * [T/iis is evidently a misjyrintfor 1' 16".] 
 43 
 
THE LAWS OF GHAVITATION 
 
 toises distant. From tliese data we find that tlio effect at the 
 iioarer station is to the e.*^?ect we ouglit to tind at tiie otiier as 
 lIJoS : 100, '31* as 13 j\ : 1 nearly, lint since our observations 
 ^nve only the difference of the two effects, we must increases 
 T".5 by a 13tli or 14th part of itself in order to have the total 
 effect [which makes if <S"J. 
 
 75. We must admit that this effect is very different from 
 what we had expected. But we know so little about the 
 earth's density, and on the other hand that of the mountain 
 may be so different from that which we have assumed it to be, 
 that there is no reason to be surprised at anything.* [// is a 
 tradition among the natives that Chiinborazo is an extinct vol- 
 cano, and, if so, its density would 1)e vtry hard to estimate, 
 lionyuer thinks it miyht be better to experiment on smaller and 
 denser monntains.] It is very probal)le that we shall find in 
 France or in p]ngland some hill of sufficient size, especially if 
 we double the effect ; and 1 shall be delighted if 1 find on my 
 return that the experiments that shall have been made either 
 confirm mine or throw new light on the matter. [At Kiobamba 
 in Peru, December 30, 1738.] 
 
 [In an appendix Bouyuer states that after a more thoronyh 
 snrvey of the Cordilleras he failed to find a more satisfactory 
 place at which to repeat his experiment. He snygests ttiat the 
 converse effect be experimented upon : viz. , the decrease in grav- 
 ity due to some deep canon amony mou.Jains. Assuminy such 
 yreat cavities in Chimborazo as would make its real only half 
 its apparent vohwie, he finds his results would make it G or 
 7 times \ less dense than the earth; this he thinks not unreason- 
 
 * [De la Condnmine also laid little stress on the numerical result of tJie plumb- 
 I'ne ex})eriment ; foi' tie says {S, p. 69), " if we can deduce from it nothing 
 decisively in favour of t/ie Newtonian attraction, at least we find not/iing con- 
 trary to tJtat theory. "] 
 
 f [In a critical analysis of all the experiments made todeto'mine the density 
 of the earth vp to t/iat time, Saigey (74, p. 151 and in his ''Petite P/iysique du 
 Globe," Paris, 1842, pt. 2, p. 151), in 1842, stated that liouguer's calcnlations 
 were erroneous, because tie confused llie centre of attraction with the centre of 
 gravity of tJie mountain; he refers to a metJiod, by means of whicJi, using 
 Bouguer's own mean result, lie deduced the density of tlie earth to be 4.62 
 times ttiat of Uie mountain ; but adds tliat if he had used only the results from 
 observations on tlie tail of Cetus and tJie first horn of Aries, the result would 
 tiave been 1.83, which is almost exactly tJie same as ttoit found by MasJcelyne by 
 tlie same metliodfor the Iiill S^heliallien.] 
 
 43 
 
if I 
 
 M 
 
 t'. 
 
 i i ; 
 
 -1 
 I: 
 
 ,4 5 
 
 ii 
 
 II 
 
 MEMOIRS ON THE LAWS OF GRAVITATION 
 
 able. Bonguer further remarks that the distribution of density/ 
 in the earth may be such that the maximum attraction of the 
 earth is not at its surface but at some J' stance beneath «7.] 
 
 [In connection with this work of Bouguer shoiild be read a 
 paper (9) presented by him to the Academy on April 28, 1750, 
 on the possibility of detect irig the deflection of a pluuib-line due 
 to the ebb and flow of the tide. 
 
 Valuable accounts and discussions of Bonguer' s work in Peru 
 are given by von Zach (43, 44, and 40), Schmidt (04, vol. 2, 
 p. 475), Todhunter (140. chap. 12), Za7iotti- Bianco (\4S^, pt. 
 2, pp. 122-25), and Poynting (185, pp. 10-14.] 
 
 Pierre Bouguer was born in 1098 at Oroisic, Bretagne, and 
 was educated at the Jesuit College at Vannes. He succeeded 
 his father as professor of hydrography at Oroisic in 1713, and 
 in 1730 accepted a similar position at Havre. His investiga- 
 tions concerning the intensity of light, embodied in a work, 
 Essai d'optiqne sur la gradation de la lumiere, published in 
 1729, led to his being elected a member of the Academy of 
 Sciences in 1731 ; he was promoted to the ottico of pensioned 
 astronomer in 1735. Along with two other members of the 
 Academy, MM. de la Condamine and Godin, he was sent to 
 Peru in 1735 to measure the length of an arc of the meridian 
 near the equator. Their labors there lasted ten years, and the 
 results of their observations were published in 1749 in La 
 Figure de la Terre, from which we have given the preceding ex- 
 tracts. For several years afterwards Bouguer was engaged in 
 a bitter controversy with de la Condamine concerning their re- 
 spective shares in the Peruvian researches. In addition to his 
 works on photometry, ho published several valuable treatises 
 on navigation, and various papers on atmospheric refraction 
 and other optical problems, and on mechanics. He died at 
 Paris in 1758. 
 
 44 
 
45 
 
 i| 
 
 
 Ml 
 
 
 THE BERTIER CONTROVERSY 
 
 
THE BERTIER CONTROVERSY 
 
 1 
 
 In June, 1709, there appeared in the Journal den Sciences ef 
 lies Beaux Arts a letter (11) by a M. (Joultaud, wlio signed 
 liimself Former Professor of Physics at Turin. In it he de- 
 scribed some pendulum experiments madc^ in the Alps of Savoy. 
 Ho claimed to have found that at a height of 1085 toises above 
 the base of the mountain the pendulum gained 28' in 2 months ; 
 20' 22" in 'A months at a height of 514 toises; an.l 15' 4" in 175 
 days at a height of 210 toises. So that it appeared as if the 
 attraction of gravitation increased with the distance from the 
 earth's centre, instead of behaving according to the Newtonian 
 law. A full account of the apparatus and observations was 
 added, and there seemed no reason why credence should not be 
 given to the results. The advocates of the Newtonian theory 
 felt called upon to account for this phenomenon consistent- 
 ly with their doctrine. D'Alembert (12 and 13, vol. 6, pp. 
 85-92) attacKed the problem and found that the Newtonian 
 theory was adequate to explain the fact, provided the mean 
 density of the earth were about three eighths of that of the 
 mountain.* An abstract of Coultaud's alleged observations is 
 given by David (14). 
 
 In Dec, 1771, another letter appeared in the same journal 
 (15) signed by one Mercier, and addressed to Gessner, Pro- 
 fessor of Physics in the Univ. of Geneva. It described experi- 
 ments made in Valois similar to those of Coultaud and with 
 similar results, namely, that the attraction of gravitation is 
 directly proportional to the square of the distance. D'Alem- 
 bert then discussed the question again (13, vol. G, pp. 93-98). 
 Further explanations on the Newtonian theory were forth- 
 
 ^i 
 
 f..^:t 
 
 :U1 
 
 'IIM 
 
 * Compare the conclusion of Bout^uer on p. 31 of ibis volume. 
 
 47' 
 
MKMUlltS UN 
 
 II 
 
 n 
 
 ff 
 
 corning from Le Sago (10) and Laliindo (17). RoitTe also dis- 
 cussed the experirnonts (18). 
 
 Tliese results of Coultaud and Mercier seem, however, to 
 have been a cause of great exultation to a certain number of 
 scientists, especially ecclesiastics, who contended that the New- 
 tonians wished to take from them their Father, their God, in 
 asserting that bodies attract and move of themselves without 
 any Prime Mover. It is hard to believe that this feeling 
 existed so late as a century ago. One of the most active of 
 the opponents of the theory of Newton was Father Bertier, 
 de rOratoire, who founded his 4th volume of Les Prinriprs 
 Physiques on the above experiments. For several years a warm 
 discussion raged among French physicists over the question. 
 
 Le Sage, having had his suspicions aroused by some passage 
 in Mercier's letter, began a careful investigation into the gen- 
 uineness of the experiments both of Coultaud and Mercier. 
 lie found them to be fabrications from beginning to end (I'J). 
 Le Sage does not mention whom he supposes to be the j)er- 
 petrators or instigators of the fraud. 
 
 A new impetus was given to the discussion by the publica- 
 tion of 2 letters (20) from Father Bertier describing experi- 
 ments with the balanoe similar to those performed by members 
 of the Royal Society of London a century before ; but Bertier 
 writes as if the idea were entirely a new one. The length of 
 the string used to suspend the weight from one arm of the 
 balance, after it had been counterpoised in the pan above, was 
 74 ft. In one case weights of 25 lbs. were used, and when one 
 of them was suspended at the end of the string it lost in weight 
 1 ounce 3.5 drachms. Bertier concluded, much to his satisfac- 
 tion, that bodies weigh more the farther they are from the 
 centre of the earth. Roiflfe followed with a paper (21) discuss- 
 ing the experiments made thus far and remarking that Bertier 
 had not taken account of the difference in the density of the 
 air at the two levels. Le Sage also criticised Bertier very 
 harshly (22). Repetitions of Bertier's experiment were made 
 by M. David, and Fathers Cotte and Bertier (23), the one with 
 a string of 170 ft. and weights of 1220 lbs., the others with 
 a string of 45 ft. and weights of 150 lbs. The one reported a 
 loss in weight of 1 oz., the others of 2 lbs., in the same direc- 
 tion as indicated by Bertier's iSrst experiment. An article by 
 David (24) in answer to Le Sage contains some scornful strict- 
 
 48 
 
TlIK LAWS OK GRAVITATION 
 
 iiros on Nowton anil liis principles which form amusing read- 
 ing at this late date. Uozier ('^5) criticised all the experi- 
 ments made on the laws of gravitation ; he refers to some 
 more made by Bertier (:iG) frotn which the latter concluded 
 that the loss in weight was proportional to the length of stri?ig 
 and to the weight. Uozier then announced the details of some 
 experiments of a similar kind made by himself, which gav(f 
 quite discordant results. David wrote another letter ('^7) with 
 more details of his experiments, but adding nothing of value. 
 Bertier followed with a siinilar letter ('iS). A committee of 
 thfc Academy of Dijon repeated the experiments with a sensit- 
 ive balance, and found (2!J) no change in weight except that 
 due to the different densities of the air at the higher and 
 lower levels. Some experiments on this same subject were 
 made by Achard (IJ3) ; he found by using first a string aiid 
 then a brass chain with which to suspend the masses, that the 
 changes in weight of the suspended nuiss could be ascribed to 
 variations in the temperature and dampness of the air. Dolo- 
 mieu (30) made some experiments with a weight sus])ended in 
 a mine, similar to those of Dr. Power (page 2) ; his results 
 permitted no definite conclusions as to change in weight. In 
 connection with this work we might notice a valuable article 
 by Le Sage (34) on the history of the theory of gravitation and 
 the experiments made concerning it. lie gives a brief account 
 of the views of Gilbert, 15acon, Kepler, Beaugrand, Fermat, 
 Pascal, Roberval, Descartes, and Gassendi ; finally he dis- 
 cusses Dolomieu's experiments and the possible variation in 
 density beneath the surface of the earth. 
 
 The controversy closes with the appearance of a letter (30) 
 from Bertier making an humble retraction of his statements 
 regarding the deductions to be drawn from his experiment ; he 
 admits that such experiments do not prove that bodies weigh 
 more as they are farther from the earth, but he declines to give 
 up his belief that such is the case, lie again inveighs against 
 those who "by means of 101 different laws, which they make 
 God create to cover their ignorance, explain everything with a 
 facility i-hat is truly delightful." 
 D 49 
 
 
 m 
 
 > i 
 
 
 i liFi 
 
51 
 
 I i, 
 
 /■, ii 
 
 
 
 THE SCHEHALLIEN EXPERIMENT 
 
 .1 
 
 J! r 
 
 i 
 
THE SCHEHALLTEN EXPERIMENT 
 
 In 1772, \fjiskolyno, the Kii<(lisli AstroiionuM- Uoyjil, pro- 
 posed to tlio Uoyiil Sooioty {'M) tluit i\\v cxporimciit of lioii- 
 ^'uer on the attriictioii of a inoimt.'iiii ho repojitcd in (Jroat 
 Britain, as Hoiit^iior InrnHclf had sii«;<ijest(Ml 'M) years Ix'foro. 
 Maskelyne liad heoii informed of two phiccs which iui.i,dit Im' 
 convenient for the purpose. One was near the i'onliiies of 
 Yorksliire and Lancashire, on the hill Whernside ; the otiier 
 in (Jund)erland, on the iiiil Ilelvellyn. The proposal was 
 favoural»iy received hy the Society, and Mr. ('iiarU^s Mason was 
 sent to examine various hills in Kn<riand and Scotland, and to 
 select tiie most suitahle (-i'-i). Mason found that the two hills 
 referred to by Maskelyne were not suitable; and fixed upon 
 Schehallien in Perthshire as offeriiii^ the best situation. At 
 the earnest solicitation of the Royal Society, Maskelyne him- 
 self undertook to make the necessarv observations. He had at 
 his disposal a lO-foot zenith sector, and all his other instru- 
 ments were the best of their kind at the time. The work was 
 begrun in the summer of 1774. The niethod of findin<j; the de- 
 tiection of the plumb-line due to the hill was exactly the second 
 of the methods described by liouguer (page JJO) ; he took read- 
 ings of the zenith-distances of certain stars at two stations, one 
 north and one south of the hill, aiul by this means doubled the 
 deflection of the plumb-line. Between Juno.'JOth and Septem- 
 ber 2'^d he took 100 star observations from the south station, 
 and 108 from the north station; in all IV.17 observations on 43 
 stars. At the same time a very elaborate survey by triangida- 
 tion was made of the dimensions and form of the hill. This 
 was considered as made up of a very large number of prisms, 
 sufficient data for the determination of each of which were col- 
 lected during the survey. 
 
 53 
 
 
 1 
 
 
 i I 
 
 f:f^ 
 
 \ ll 
 
 <• f J^i 
 
MKMOIIIS ()N 
 
 lii^ 
 
 III lii.H \ni\)or {'.Vi) di'scrilHii^ tho operations, iMuskelyiio cal- 
 (Milutcs, from 4(> only out of tlx* XVi ohKcrvations, that tho up- 
 piirciit (lilTrri'iicc of latitiule hctwcoii tlio two HtatioiiH \h r)4".(».* 
 Tho triio (lilTrrciuM) of latitiulu is 4;{", Icaviii^j 11". (} duo to tho 
 • oiitrary attractions of the hill. 
 
 From a rouj^h cahMilatioii, assuming tho density of tlu; 
 mountain to ho tho samo as tho nu>an density of tho oarth, 
 and that tho law of attracrtion is that of tho invorso scjuaro 
 of the distan(M\ Maskolyint found that tho attra(;tion should bo 
 twioo that found hy ohservatiojj. lionco tho moan density of 
 tho earth is twiet; that of tho hill. A more exact calcidation 
 was promised for tho future. Maskelyno draws two imiin con- 
 clusions: (1), tiiat Schohallion has an attraction, and so, there- 
 fore, has every mountain ; {'i), that tho inverse scjuare law of 
 till' distance is confirmed ; for if the force wore only a little 
 atT(fctiM| by the distan(!o, the attraction of tho hill would be 
 wholly insensible. 
 
 The survey of the hill and its environs was made during the 
 years 1774, 1775 and ITTO. The calculation of the attraction 
 of tho hill from thosii measurements was undertaken by Mut- 
 ton,! who employed several new and interostiiiit^ methods. A 
 full act'ount will bo found in his paper (l{7 and 47, vol. "i, 
 pp. l-(JS). Assumin*^ that the density of the hill is the same 
 as the mean density of the earth, Ilutton found that the attrac- 
 tion of the earth is to tho sum of the contrary attractions of 
 the hill as iWV,] : 1. Now Maskelyno had found the deflection 
 due to the contrary attractions of the hill to be IT'.G; whence 
 the attraction of the oarth is to the sum of the attractions of 
 the hill as 1 : tan. ll".tJ, or as 17781 : 1 ; or, allowing for the 
 
 m 
 
 if 
 
 • f 
 
 J -I 
 
 ,.k. 
 
 * Von Zaeli (49, App., pp. 686-693) Una calculated the results from all 
 of the H87 observjilifins, and finds for the iippureut difference of latitude 
 54". 651, and for the deflection due to the contrary attractions of tlie lull 
 11". 632 ; whi(!h is in i'litire accord with Maakelyne's calculations. Saigey 
 (74, p 153) also subjected the result to a test which was satisfactory. 
 Zanolti-Bianco states (148A, pt. 3, p. 134) that Saigey maintained that 
 Maskelyne did not choose his station at the most favourable part of the 
 hill-side, and that if he had done so he would have found the deflection 
 14" inst<'ad of 11".6. 
 
 f For Ilutton 's own estimate of his share in the work, and for his con- 
 tempt for Cavendish's experinieiu, see bibl. No. 45. For a good account 
 of Button's method of calculation, see Zanotti-Bianco (148^, pt. 3, pp. 126- 
 32); see also Helmert (148, vol. 3, pp. 368-80). 
 
 54 
 
Til K LAWS OK i. KAVI'IArinN 
 
 rontrifupjil forco, uh ITH()4 : I iHMirly. I!<mht thp iiicuii donsity 
 of the eurtli in to tlu» donHity of the liill us 1TS(H : W.\:\, or iih 
 1> : ft noarly. Assuming the Hp(M'itl(' jjruvity of tlio hill to ho 
 about ^.5, lliitton ivriiiirks that, this would ^ivc 4.."» an tho 
 mean HpcM'ilic gravity of the earth. lliifto?i revised this result 
 ill his •' Tracts " (IT, vol. 'i, p. •14); he takes the sperilic 
 jjravity of the hill as .'I and henco the wpeeilic ^'ravity of the 
 earth would be 5.4 !iearly. 
 
 Play fair, with the aid of liord Webb Seymour, made a care- 
 ful litholo«;iral Hurvoy of S<diehallieu, and published his re- 
 sultH in ISII (4<> and 4S). Ife found that the hill was made 
 up of two (tIasHOS of rock, (|uartz of specific ^raviiy v*.<!.*{!»Si(;, 
 and micaceous rock, including calcareous, of specific jjravity 
 '^.H10.*J!>. Krom two suppositions as to the distribution of these 
 two components in the interior of the hill, usin^ iliitton's data 
 for the attraction, IMayfuir calculated the mean density of the 
 earth to bo 4.5r)SH»; and 4.H«H;!l'.>7 respectively. IMayfair con- 
 sidered the experiment on Suliehallien so exact that he took the 
 mean of the above results, 4.713, as the best determination of 
 the mean density of tho earth. 
 
 Ilutton prefers to take '>i.77 as the mean of IMayfair's deter- 
 minations for tho density of the hill, and the density of the 
 earth as ^ of ;i.77, or 5 nearly. In a paper j)ublished in 18:;il 
 (52, 53 and 54), Ilutton complains that his share in the S(!he- 
 hallien experiment has always been underestimated ; he ^ives 
 a brief account of tho observations, cahuilations and results, 
 and considers 5 as the most probable value of the mean density 
 of the earth. Ho shews that the Schehallien experiment could 
 not bo made to give tho same result as that of Cavendish, 5.48, 
 unless the deflection 11". be diminished to about 10". 5 or 
 10". 4. which is manifestly too great an error to have been 
 committed by Maskclyne, considering the accuracy of the 
 observer and of the instruments, and the large number of ob- 
 servations made. Ilutton suggests the repetition of the ex- 
 periment at one of the pyramids in Kgypt. Some years later 
 Peters (80^) made a calculation of the attraction of the Great 
 Pyramid. 
 
 For brief accounts of the Schehallien experiment and criti- 
 cisms upon it, reference should be made to Ilutton (38 and 47, 
 vol. 2, pp. G9-77), von Zach (43, 44 and V.)), Muncke (01, vol. 
 3, pp. 944-70), Schmidt (64, vol. 5J, pp. 474-9), Meiiabrea (71), 
 
 55 
 
 \'\ 
 
 \ 
 
 ill 
 
 fii 
 
 i 
 
 U' 
 
 I'M 
 
i: 
 
 l:i 
 
 m 
 
 mi 
 
 
 !i 
 
 I 
 
 MKMOIliS UN THE LAWS OF GRAVITATION 
 
 Schell (135), Todliiinter (140, vol. 1, pp. 459-60). Ztinotti- 
 Hiaiico (I4Hi, pt. 2, pp. 125-35) hikI Fiesdorf (ISfl^, pp. 5-7). 
 
 dipt. J(u;ob lias remarked (118 and 15il)tliat by this method 
 we may measure the attraction of the mass of the mountain 
 above the surface, yet we do not know how much ouglit to be 
 added or subtracted due to that delow it. 
 
 Von Zach makes mention of several early astronomers who 
 assign anomalies in theirgeodetic measurements to the influence 
 of mountains on the plumb-lines of their instruments; the 
 reader is referred to von Zach, Humboldt (8!:i, vol. 1, notes, 
 pp. 45-7) and Helmert (148, vol. 2, chap. 4), and to the ac- 
 count in this volume (p. 123) of the work of James and 
 Clarke. Von Zach himself niude a very careful determination 
 in 1810, after the method used by Bouguer, of the attraction 
 of mount Mimet, near Murseilh^s. He found a deflection of 
 the plumb-line amounting to 2". lie did not calculate the 
 density of the earth. His observations were published in book 
 form in 1814 (41)). 
 
 For this work Maskelyne was presented by the Iloyal Society 
 with the Copley medal. At the presentation the President, Sir 
 John Pringle, delivered an address (35) on the attraction of 
 gravitation, giving a critical account of the state of the subject 
 before the time of Newton, as woU as of hi later developments. 
 
 56 
 
 ' IW 
 
 ') I. ■:■ 
 
 .4-: 
 
 m 
 
EXPERIMENTS TO DETERMINE THE 
 DENSITY OF THE EARTH 
 
 BY 
 
 HENRY OAVENDISir, Esq., F.R.S. and A.S. 
 
 Read June 21, 1798 
 
 (From t/w PhilosophiMl Tmnsaetiom of the Ronnl S^meU/ of London for the 
 
 year 1798, Part II., pp. 469-536) 
 
 57 
 
 («f t ' 
 
 .'*? 
 
.'^] 
 
 If 
 
 i » 
 
 
 CONTENTS 
 
 PACK 
 
 Introduction 59 
 
 Description of the appanituH 61 
 
 Method of observing the deflection 64 
 
 " time of vibration 64 
 
 Effect of the resistance of the air 65 
 
 Account of the experiments 67 
 
 Testing for magnetic effects 68 
 
 Testing tfie elastic properties of the wire 72 
 
 Further tests for magnetic effects 75 
 
 Testing the effect of variation of temperature about the box 76 
 
 Final observations 80 
 
 On the tfieoi'g of the erperiment 88 
 
 Corrections to be made in the tlieory as first given 91 
 
 Effect of the variable position of the arm on the equations 97 
 
 When and how to apply the corrections 98 
 
 Table of results 99 
 
 Conclusion 99 
 
 Appendix : to find the attraction of the mahogany case on the balls 102 
 
 58 
 
u 
 
 EXPERIMENTS TO DETERMINE THE 
 DENSITY OF THE EARTH 
 
 BY 
 
 HENRY CAV^ENDISII, Esq., F.H.S. and A.S. 
 
 !:'■! 
 
 
 i. 3 
 
 Many years ago, the late Rev. John Micliell, of this society, 
 nontrived a metliod of determiiiinf:^ tiie density of the eai i,h, hy 
 rendering sensible the attraction of small (|uantities of matter ; 
 but, as he was engage<l in other pursuits, he did not (iompiete 
 the apparatus till a short time before his death, aiul did .not 
 live to make any experiments with it. After his death, the 
 apparatus came to the Rev. Francis John Hyde Wollaston, 
 Jacksonian Professor at Cambridge, Avho, not having conven- 
 iences for making experiments with it, in the manner he could 
 wish, was so good as to give it to me. 
 
 The apparatus is very simple ; it consists of a wooden arm, (j 
 feet long, made so as to unite great strength with little weight. 
 This arm is suspended in an horizont' 1 jmsition, by a slender 
 wire 40 inches long, and to each extremity is hung a leaden 
 ball, about 'I inches in diameter ; and the whole is inclosed in 
 a narrow wooden case, to defend it from the wind. 
 
 As no more force is required to make this arm turn round on 
 it centre, than what is necessary to twist the suspending wire, 
 it i-^ plain, that if the wire is sufficiently slender, the most min- 
 ute force, such as the attraction of a leaden weight a few inches 
 in diameter, will be sufficient to draw the arm sensibly aside. 
 The weights which Mr. Michell intended to use were 8 inches 
 diameter. One of these was to be placed on one side the case, 
 opposite to one of the balls, and as near it as could conveniently 
 be done, and the other on the other side, opposite to the other 
 
 59 
 
 ''Mi 
 
If 
 
 1 1 
 
 iVIKMOIUS ON 
 
 itiil 
 
 pit J 
 
 
 ball, so tliJit tlio attraction of both tlieso weiglits would con- 
 spire in (Iniwiiig the arm aside ; and, when its position, as af- 
 fected by these weights, was ascertained, the weights were to 
 bo removed to the other side of the case, so as to draw the arm 
 the contrary way, and the position of the arm was to be again 
 determined ; and, copseqiiently, half the difference of these po- 
 sitions would shew how much the arm was drawn aside by the 
 attraction of the weights. 
 
 In order Lo determine from hence the density of tlie earth, 
 it is necessary to ascertain what force is required to draw the 
 arm aside through a given s})ace. This Mr. Michell intended to 
 do, by putting the arm in motion, and observing the time of its 
 vibrations, from which it may easily be computed.* 
 
 Mr. Michell had prepared two wooden stands, on which the 
 leaden weights were to be su})ported, and j)ushed forwards, till 
 they came almost in contact with the case; but he seems to 
 have intended to move them by hand. 
 
 As the force with which the balls are attracted by these 
 weights is excessively minute, not more than 6„, 00*0,000 ^^ 
 their weight, it is plain, that a very minute disturbing force 
 will be sufficient to destroy the success of the experiment; and, 
 from the following experiments it will appear, that the disturb- 
 ing force most ditticult to guard against, is that arising from 
 the variations of heat and cold ; for, if one side of the case is 
 warmer than the other, the air in contact with it will be rare- 
 fied, and, in consequence, will ascend, while that on the other 
 side will descend, and produce a current wliich will draw the 
 arm sensibly aside. f 
 
 * Mr. Coulomb lias, in a variety of cases, used a contrivance of this kind 
 for trying small attractions ; but Mr. Micliell informed me of liis intention 
 of making lliis experiment, and of tbe meiiiod lie intended to use, before 
 tbe publication of any of Mr. Coiilomb'.s experiments. 
 
 f M. Cassini, in observing tbe vnriation compass placed by him in tbe 
 observatory (wbieli was constructed so as M^iivke very minute changes of 
 position visilde. nnd in wliicli tlie needle was suspended by a silk thread), 
 found that standing near tbe box. in order to observe, drew tbe needle 
 sensibly aside ; which I have no doubt was caused by this current of air 
 It must be observed, that his compass box was of metal, which transmits 
 heat faster than wood, and also was many inches deep ; both which causes 
 served to increase the current of air. To diminish the effect of this cur- 
 rent, it is by all means advisable to make the box, in which the needle 
 plays, not much deeper than is necessary to prevent the needle from strik- 
 ing against the lop and bottom. 
 
 60 
 
T UK I. A \v s o I' (1 li A \' 1 r A r i o n 
 
 As I was convinced of the necessity of jruunling aj^ainst this 
 source of error, 1 resolved to place the a{)paratii8 in a room 
 wiiicli should remain constantly slnit, aiul to observe the mo- 
 tion of the arm from without, by means of a telescope ; and to 
 suspend the leaden weij^fhts in such manner, that I could move 
 them without entering into the room. This difference in the 
 manner of observing, rendered it necessary to make some al- 
 teration in Mr. Michell's apparatus ; and, as there were some 
 I)arts of it which I thought not so convenient as could bo 
 wislied, I chose to make the greatest part of it afresh. 
 
 Fig. 1 is a longitudinal vertical section through the instru- 
 ment, and the building in which it is phuied. AI5(Ji)I)CB- 
 AEFFE is the case ; .^• and x are the two balls, whiidi are sus- 
 pended by the wires hx from the arm yhnih, which is itself 
 suspended by the slender wire (jl. This arm consists of a 
 slen<ler deal rod hinh, strengthcniMl by a silver wire hijh ; by 
 wiiich means it is made strong enough to support the balls, 
 though very light.* 
 
 The case is supported, and set horizontal, by four screws, 
 resting on posts fixed firmly into the ground ; two of them are 
 represented in the figure, by S and S; the two others are not 
 represented, to avoid confusion. (J(r and (id are the eiul 
 walls of the building. W and W are the leaden weights ; 
 which are suspended by the copper rods IJrPrR, and the 
 wooden bar rr, from the centre pin Vp. This pin passes 
 through a hole in the beam II II, perpendicularly over the cen- 
 tre of the instrument, and turns round in it, being prevented 
 from falling by the plate p. MM is a pulley, fastened to this 
 pin; and Mm, a cord wound round the pulley, and passing 
 through the end w.all ; by which the observer may turn it 
 round, and thereby move the weights from one situation to the 
 other. 
 
 Fig. 3 is a plan of the instrument. AAAA is the case. SSSS, 
 the four screws for supporting it. hh, the arm and balls. W 
 and W, the weights. MM, the pulley for moving them. When 
 
 
 *Mr. Michell's rod was entirely of wood, and was much stronger and 
 stiffer than this, thougli not much heavier; but, as it had warped when it 
 came to me, I chose to make another, and preferred this form, partly as 
 being easier to construct and meeting with less resistance from the air, 
 and partly because, from its being of a less complicated form, I could more 
 easily compute how much it was attracted by the weights. 
 
 61 
 
'U 
 
 MKiNJUlliS ON 
 
 lit 
 
rilK IwWVS OF (}UA VITATION 
 
 tlio weights are in this position, l)()th conspire in drawing the 
 arm in the direetion h\\ ; hut, when they are removed to the 
 situation tn and //', represented hy the dotted lines, hoth con- 
 spire in drawing the arm in the contrary direction /nr. These 
 weights are prevented from striking the instrument, in' pieces 
 of wood, wiiicii stop tiiem as soon as they come witiiin I of an 
 inch of the case. Tiie pieces of wood are fastened t) the wall 
 
 ,' -^^M 
 
 
 Fig- 2 
 
 of the building; and I find, that the weights may strike against 
 them with considerable force, without sensibly shaking the in- 
 strument. 
 
 In order to determine the situation of the arm, sli})s of ivory 
 are placed within the case, as near to each end of the arm as 
 can be done without danger of touching it, and are divided to 
 20ths of an inch. Another small slip of ivory is placed at each 
 and of the arm, serving as a vernier, and subdividing these 
 divisions into 5 parts ; so that the position of the arm may be 
 observed with ease to lOOths of an inch, and may be estimated 
 to less. These divisions are viewed, by means of the short 
 telescopes T and T (Fig. 1), tiirougli slits cut in the end of the 
 case, and stopped with glass; they are enlightened by the lamps 
 L and L, with convex glasses, placed so as to throw the light 
 on the divisions ; no other light being admitted into the room. 
 
 The divisions on the slips of ivory run in the direction Ww 
 (Fig. 2), so that, when the weights are placed in the positions 
 wand w, represented by the dotted circles, the arm is drawn 
 aside, in such direction as to make the index point to a higher 
 number on the slips of ivory ; for which reason, I call this the 
 positive position of the weights. 
 
 FK (Fig. 1) is a wooden rod, which, by means of an endless 
 screw, turns round the support to which the wire yl is fastened, 
 
 63 
 
 ■M 
 
 >* 
 
_im.-jjv.w:^-i 
 
 'U 
 
 i; 
 
 MKMOIKS UN 
 
 tl 
 
 
 and tlier('})y oniiblos tlie obsorvor to turn round tho wire, till 
 the arm settloH in tlio iniddht <tf tli(> cnao, without danger of 
 touching? oitlier side. Tiio win? /// is fastened to its support at 
 top, and to the centre of tlie arm at bottom, by brass clips, in 
 whicli it is pin(died by sorows. 
 
 In these two figures, the difTorent ])arta are drawn nearly in 
 the proper proportion to eaeli otiier, and on a scale of one to 
 tiiirteen. 
 
 Before I proceed to the account of the experiments, it will 
 be pro))er to say something of the manner of observing. Sup- 
 pose the arm to be at rest, and its position to be observed, let 
 the weights be then moved, tlie arm will not only be drawn 
 aside thereby, but it will be made to vibrate, and its vibrations 
 will continue a great while ; so that, in order to determine how 
 much the arm is drawn aside, it is necessary to observe the ex- 
 trenie points of the vibrations, and from thence to determine 
 the point which it would rest at if its motion was destroyed, or 
 the point of rest, as I shall call it. To do this, I observe three 
 successive extreme points of a vibration, and take the mean 
 between the first ami third of these points, as the extreme 
 point of vibration in one directioii, and then assume the mean 
 between this and the second extreme, as the point of rest ; for, 
 as the vibrations are continually diminishing, it is evident, that 
 the mean between two extreme points will not give the true 
 point of rest. 
 
 It may be thought more exact, to observe many extreme 
 points of vibration, so as to find the point of rest by different 
 sets of three extremes, and to take the mean result ; but it 
 must be observed, that notwithstanding the pains taken to pre- 
 vent any disturbing force, the arm will seldom remain perfect- 
 ly at rest for an hour together; for which reason, it is best to 
 determine the point of rest, from observations made as soon 
 after the motion of the weights as possible. 
 
 The next thing to be determined is the time of vibration, 
 which I find in this manner: I observe the two extreme points 
 of a vibration, and also the times at which the arm arrives at two 
 given divisions between these extremes, taking care, as well as I 
 can guess, that these divisions shall be on different sides of the 
 middle point and not very far from it. I then compute the 
 middle point of the vibration, and, by proportion, find the time 
 at which the arm conies to this middle point. I then, after a 
 
 64 
 
TIIK LAWS OK (JliA V ITATloN 
 
 number of vibrutioiis, ropoat this oj)omtion, and divide the in- 
 terval of time, between the cominjjf of the arm to these two 
 middle points, by tlio number of v'brations, whieh jjives the 
 time of one vibration. The following example will explain 
 wiiat is here said more clearly : 
 
 Fxtreiiu! 
 poiiitH 
 
 DiviHioua 
 
 Tinio 
 
 I'oiht of 
 
 Timi> (irmltlillc 
 (if vihrttlKiK 
 
 27.2 
 
 
 
 
 25 
 24 
 
 10" 2:r 4 ) 
 
 57 \ 
 
 — 
 
 10'' 2:j 23 ' 
 
 22 1 
 
 — 
 
 
 24 <l 
 
 
 27. 
 
 — 
 
 — 
 
 24 7 
 
 
 22.6 
 
 — 
 
 — 
 
 24 75 
 
 
 2tJ.H 
 
 — 
 
 — 
 
 24.8 
 
 
 2:}. 
 
 — 
 
 — 
 
 24.85 
 
 
 26.6 
 
 — 
 
 — 
 
 24.9 
 
 
 
 25 
 24 
 
 11 5 22 ) 
 
 6 48 
 
 — 
 
 11 5 22 
 
 23.4 
 
 
 
 
 
 The first column contains the extreme points of the vibra- 
 tions. The second, the intermediate divisions. The third, 
 the time at whinh the arm came to these divisions ; and the 
 fourth, the point of rest, which is thus found : the mean be- 
 tween the first and third extreme points is ^7.1, and the mean 
 between this and the second extreme point is ;^4.(), which is 
 the point of rest, as found by the three first extremes. In like 
 manner, the point of rest found by the second, third, and 
 fourth extremes, is 24.7, and so on. The fifth column is the 
 time at which the arm came to the middle point of the vibra- 
 tion, which is thus found-: the mean between 27.2 and 22.1 is 
 24.05, and is the middle point of the first vibration ; and, as 
 the arm came to 25 at 10'' 23' 4", and to 24 at 10'' 23' 57", we 
 find, by proportion, that it came to 24. G5 at 10'' 23' 23". In 
 like manner, the arm came to the middle of the seventh vibra- 
 tion at 11'' 5' 22"; and, therefore, six vibrations were performed 
 in 41' 59", or one vibration in 7' 0". 
 
 To judge of the propriety of this method, we must consider 
 in what manner the vibration is affected by the resistance of the 
 air, and by the motion of the point of rest. 
 
 Let the arm, during the first vibration, move from D to B 
 (Fig. 3), and, during the second, from B to ^/ ; Bd being less 
 than DB, on account of the resistance. Bisect DB in M, and 
 B 65 
 
f 
 It 
 
 H 
 
 '{\ 
 
 M KM OIKS UN 
 
 IW in m, luul bisect M/// in n, and let .r ho nny ])oint in the 
 vihnition ; tlion, if the rcsistiiiico is pioportionul to the HCjuaro 
 of the veU)(!ity, the whole time of a vihnition iu very little al- 
 tered ; hut, if T '^} taken to the time of one vihration, aa tiio 
 diameter of a circle to its Hemi-circuimference, the time of 
 
 moving; from \\ to n ex(!ced8 \ a vihration, hy .. nearly ; 
 
 and the time of moving from B to m falls short of I a vihration, 
 
 M 
 
 -I — f- 
 
 fi 
 
 n m 
 
 *m 
 
 by as much ; and the time of moving from H to .r, in the sec- 
 ond vihnition, exceeds that of moving from ;/• to H. in the first, 
 
 TxD^/xHr' 
 by , " supposing \)d to be bisected in h ; so that, 
 
 if a mean is taken, between the time of the first arrival of 
 the arm at x and its returning hack to the same point, this 
 moan will be earlier* than the true time of its coming to B, by 
 TxPr/xBa;" 
 
 The effect of motion in the point of rest is, that when the 
 arm is moving in the same direction as the point of rest, the 
 time of moving from one extreme point of vibration to the 
 other is increased, and it is diminished when they are moving 
 in contrary directions ; but, if the point of rest moves uni- 
 formly, the time of moving from one extreme to the middle 
 point of the vibration, will be equal to that of moving from 
 the middle point to the other extreme, and, moreover, the time 
 of two successive vibrations will be very little altered ; and, 
 therefore, the time of moving from the middle point of one 
 vibration to the middle point of the next, will also be very 
 little altered. 
 
 * [ThiH word should be "later" as is observed by TodJiunter (140. vol. 2, p. 
 KJ;")). For an elementary discxission of this kind of motion see Williamson and 
 Tarletoii's " Treatise of Dynamics" e.v. 13, ii? 117 Poisson (65, vol. 1, pp. 
 353-36i) and Menabrea (71) have given very elaftorate analyses of the problem. 
 Cornu and Bailie (137, 141, 142, 143. and 157) jn-oved in 1878 tfiat the re- 
 sistance in tJi€ case under consideration is proportional to tlie first power of 
 tJis velocity.^ 
 
 66 
 
 
TIIK LAWS OK (JKA V ITATIoN 
 
 
 It appotirs, tliereforc, that on account of the resistance of 
 the air, the time at whi(;h the arm comes to the middle point 
 of the vihration, is not exactly the mean hetween th«' times of 
 its coming to thi; extreme points, wliii'h «'aiises some inaccur- 
 acy in my method of lindin^ the time of a vihration. It must 
 he ohservcd, however, that as the time of (Mjinin^' to the miihlh- 
 point is hoforo tiie middle of the vihration, hoth in tint lirst 
 and last vihration, and in ^^'ueral is nearly e(|ually so, the error 
 produced from this cause must he inconsiderahhs and, on the 
 whole, I see no method of finding the tinn^ of a vihration whi(di 
 is liahle to less ohjection. 
 
 'I'he time of a vihration tmiy he determined, either hy previous 
 trials, or it may he done at each experiment, hy ascertaining the 
 time of the vihrations which the arm is actually put into by 
 the motion of the weights ; hut there is one advantage in the 
 latt(>r method, luimely, that if there should bo any accidental 
 attra(!tion, such as electricity, in the glass })late8 through which 
 the motion of the arm is seen, which should increase the forcse 
 necessary to draw the arm aside, it wouhl also diminish tlie 
 time of vibration ; and, consequently, the error in the result 
 would be much loss, when the force required to draw the ami 
 aside was deduced from experiments made at the time, than 
 when it was taken from previous experiments. 
 
 If 
 
 a 
 
 ACCOFNT OF TIIK EXPERIMENTS 
 
 re- 
 of 
 
 In my first experiments, the wire by which the arm was sus- 
 pended was 3()| inches long, and was of copper silvered, one 
 foot of whicii weighed 2^ grains ; its stiffness was such as to 
 make the arm perform a vibration in about 15 minutes. I im- 
 mediately found, indeed, that it was not stiff enough, as tlie 
 attraction of the weights drew the balls so much aside, as to 
 make them touch the sides of the case ; I, however, chose to 
 make some experiments with it, before I changed it. 
 
 In this trial, the rods by which the leaden weights were sus- 
 pended were of iron ; for, as I had taken care that there should 
 be nothing magnetical in the arm, it seemed of no signification 
 whether the rods were magnetical or not ; but, for greater se- 
 curity, I took off the leaden weights, and tried what effect the 
 rods would have by themselves. Now I find, by computation, 
 that the attraction of gravity of these rods on the balls, is to 
 
 67 
 
MKMOlltS ON 
 
 Hi 
 
 I'i! 
 
 II 
 
 tliiit of tlio W(i|,'fjtH, iHMirly us 17 to '<ir)00 ; ho that, hh the ut- 
 tni(;tioii of the wri^lits iippeiircMl, l»y tlio fon';^'oiM;; triiil, to he 
 Hiiniciciit to ciniw tin; urrii asido l»y about 15 tlivi.sioiiH, tliu at- 
 trai^tioii of \\h' rotl.H jiloiic kIioiiM draw it asidi; about ^^ of a 
 division ; and, tiKM'oforo, tbo iiiotioii of tbo rotls from one iiuar 
 position to till! otb(!r, should move it about ^ of a division. 
 
 'I'ho result of tho oxpcrinu-nt was, that for the llrst ir» niin- 
 ntus after tin; rods were rutiiovod from om; nuur position to thu 
 othor, very little motion was produced in the arm, and hardly 
 mor(! than ou^ht to be produc^ed by the action of gravity ; but 
 the motion tluMi increased, so that, in about a (juarter or luilf 
 an hour more, it was found to have moved ^ or [^ division, in 
 tlio same ilirection that it ought to have done by tho action of 
 gravity. On returning the irons ba(!k to their former position, 
 tho arm moved backward, in the same manner that it before 
 moved forward. 
 
 It must be observed, that tho motion of tho arm, in tlioso ex- 
 periments, was hardly more than would sometimes take place 
 without any apparent cause ; but yet, as in tlirec experiments 
 which wore made with those rods, tho motion was constantly of 
 the same kind, though differing in quantity from ^ to 1^ divis- 
 ion, tiiero seems great reason to think that it was produced by 
 tlio rods. 
 
 As this effect seemed to mo to be owing to magnetism, though 
 it was not such as I should have expected from that cause, I 
 changed the iron rods for copper, and tried them as before ; 
 tho result was, that there still sconiod to bo some effect of the 
 same kind, but more irregular, so that I attributed it to some 
 accidental cause, and therefore hung on tho leaden weights, and 
 proceeded with tho experiments. 
 
 It must bo observed, that tho effect which seemed to be pro- 
 duced by moving tho iron rods from one near i)08ition to tho 
 other, was, at a medium, not more than one division ; whereas 
 the effect produced by moving tho weight from the midway to 
 tho near position, was about 15 divisions ; so that, if I had con- 
 tinued to use the iron rods, the error in tho result caused there- 
 by, could hardly have exceeded -^^ of the whole. 
 
 68 
 
Til K LAWS OF <Jli.\ VITATION 
 
 ExpKuixfKMT I. Aro. 5 
 
 Wi iifht» in Viiihnifi iumtion 
 
 KxlDMIIA 
 |Nllll(M 
 
 PIvImIoiih 
 
 Timt 
 
 ri.4 
 
 11.5 
 11.5 
 
 li'' 42 
 
 55 
 
 U) 5 
 
 I'ollil ol 
 
 rcHi 
 
 11.0 
 
 Tlino of mill of 
 vlhrallou 
 
 Diflvrenc* 
 
 23.4 
 27.0 
 24.7 
 27 :i 
 25.1 
 
 At 10'' 5 , weiyhtH mured to ponUive jHmtion 
 
 25. H2 
 2«l.()7 
 20.1 
 
 Atn 
 
 18.2 
 
 GO 
 
 10.8 
 
 7.7 
 
 11 
 12 
 
 12 
 11 
 
 11 
 12 
 
 12 
 11 
 
 •• 0', ireif/htH returned Ixick to midwdy jtosition 
 
 0" 1 13 
 
 4S i 
 1 30 S 
 
 10 20 i 
 
 17 20 S 
 
 30 24 ) 
 
 31 11 ) 
 
 45 58 / 
 
 47 4 )" 
 
 12 
 11.72 
 
 10 9 
 30 45 
 
 45 58 
 
 14 56 ' 
 
 14 30 
 
 15 13 
 
 Motioei on moving froni midway to pos. = 14.32 
 ' pos. to midway = 14.1 
 
 Time of ono vilnation = 14' 55" 
 
 It must be observed, tlmt in this experiment, the attnictioii 
 of the weiglits drew tlie arm from 11.5 to )ir>.H, so that, if no 
 contrivance had been used to prevent it, the momentum ac- 
 quired thereby would have carried it to near 40, and would, 
 therefore, liave made the balls to strike against the case. To 
 prevent this, after the arm had moved near l."» divisions, I re- 
 turned the weights to the midway position, and let them re- 
 main there, till the arm came nearly to the extent of its vibra- 
 tion, and then again moved them to the positive position, 
 whereby the vibrations were so much diniinishiMl, that the balls 
 did not touch the sides; and it was this which ])revented my 
 observing the first extremity of the vibration. A like method 
 was used, when the weights were returned to the midway posi- 
 tion, and in the two following experiments. 
 
 69 
 
 fill I 
 
 11 
 
 -!! 
 
rrv 
 
 MKMOIRS ON 
 
 The vibrations, in moving the weights from the midway to 
 the positive position, were so small, that it was thought not 
 worth while to observe the time of the vibration. When the 
 weights were returned to the midway position, I determined 
 the time of the arm's coming to the middle pcint of each vibra- 
 tion, in order to see how nesirlv the times of the different 
 vibrations jigreed together. In great part of the following ex- 
 ])eriments, 1 contented myself with observing the time of its 
 coming to the middle point of only the first and last vibration. 
 
 EXFKUIMENT II. AUO. G 
 Weigh'" in midway position 
 
 Fxtrerne 
 poiiitH 
 
 DiviHiijiiH 
 11 
 
 Time 
 
 I'oilil ot 
 rest 
 
 Time of mid ol 
 vibration 
 
 DifTeronce 
 
 
 lOh 4' 0' 
 
 
 
 
 11 
 
 11 
 
 
 
 
 
 11 
 
 17 
 
 
 
 
 
 11 
 
 25 
 
 11. 
 
 
 
 
 
 Weights moved 
 
 to positive fMsition 
 
 
 29.3 
 
 
 
 
 
 
 24.1 
 
 — 
 
 — 
 
 26.87 
 
 
 
 30. 
 
 .. 
 
 — 
 
 27.57 
 
 
 
 r.2 
 
 — 
 
 — 
 
 28.02 
 
 
 
 V -.7 
 
 — 
 
 — 
 
 28.12 
 
 
 
 2(5. !» 
 
 — 
 
 — 
 
 28.05 
 
 
 
 28.7 
 
 
 — 
 
 27.85 
 
 
 
 27.1 
 
 
 — 
 
 27.82 
 
 
 
 28.4 
 
 
 
 
 
 
 
 
 Weights returned 
 
 '' to midway position 
 
 
 6. 
 
 
 
 
 
 
 
 12 
 
 1 3 50 \ 
 
 
 1" 4' 1" 
 
 
 
 13 
 
 4 34 f 
 
 
 
 18.5 
 
 — 
 
 — 
 
 12.37 
 
 — 
 
 > 14' 52' 
 
 
 13 
 
 18 29 } 
 
 
 18 53 
 
 
 
 12 
 
 19 18 f 
 
 
 
 6 5 
 
 — 
 
 — 
 
 11.67 
 
 — 
 
 14 46 
 
 
 11 
 12 
 
 33 48 \ 
 
 34 51 S 
 
 — 
 
 33 39 
 
 
 15.2 
 
 — 
 
 — 
 
 11. 
 
 — 
 
 13 46 
 
 
 13 
 12 
 
 45 8 ,) 
 
 46 22 S 
 
 — 
 
 47 25 
 
 
 7.1 
 
 — 
 
 — 
 
 10.75 
 
 — 
 
 15 25 
 
 
 11 
 
 12 
 
 3 3 48 / 
 5 18 S 
 
 — 
 
 2 2 50 
 
 
 13.6 
 
 
 
 
 
 
 Motion of arm on movins? weights from midway to po.s. = 15 87 
 
 pos. to midway — 15.45 
 Time of one vibration = 14' 42 ' 
 
 70 
 
 It- '!■ 
 
T UK L A W S ( ) F < ; K A \' I T A 1' I O N 
 
 125 
 
 ExpKiiiMKNT III. Arc;. 7 
 7'he weifjhfn ha'nf/ in the ponitiee ixisitioii, and the arm. a little in motion 
 
 Kxin'iiie 
 
 ni 
 
 visions 
 
 |MIIIIIS 
 
 
 
 Hl.T) 
 
 
 
 29 
 
 
 — 
 
 »1 
 
 
 — _ 
 
 29.1 
 
 
 
 Timo 
 
 I'oint of 
 
 Tiinn of mill, of 
 vibration 
 
 9. 
 20.5 
 
 9.2 
 
 17.4 
 
 10.1 
 
 15.6 
 32. 
 
 23.7 
 31.8 
 25.8 
 
 31.1 
 
 14 
 15 
 
 15 
 14 
 
 14 
 15 
 
 It 
 13 
 
 13 
 14 
 
 28 
 
 27 
 
 27 
 28 
 
 30.12 
 30.02 
 
 Weights moved to midway jjosition 
 
 10" 34' 55 " 
 
 DiflTerenco 
 
 10" 34' 18" ) 
 35 6 f 
 
 49 31 / 
 
 50 27 f 
 
 11 5 7 } 
 
 6 18 <i 
 
 18 46 / 
 
 19 58 \ 
 
 33 46 / 
 
 35 26 S 
 
 14.8 
 14.07 
 13.52 
 13.3 
 
 49 39 
 
 11 4 17 
 
 19 4 
 
 33 31 
 
 14' 44 ' 
 14 38 
 14 47 
 14 37 
 
 Weights moved lo positive position 
 
 2 59 
 
 47 40 
 
 2 48 / 
 
 
 3 56 S 
 
 
 — 
 
 27.8 
 
 — 
 
 28.27 
 
 — 
 
 28.62 
 
 44 58 1 
 
 
 46 50 S 
 
 
 Motion of the arm on moving weights from pos. to mifi.= 15.22 
 
 mid. tf) pcKS. = 14.5 
 Time of one vibration, wlicn in mid. position — 14 39" 
 
 pos. position = 14' 54 " 
 
 Tliese experiments are sufficient to shew, tiiat tlie attraction 
 of tlie weights on the balls is very sensible, and are also suf- 
 ficiently regular to determine the quantity of this attraction 
 pretty nearly, as the extreme results do not differ from each 
 other by more than ^ part. But there is a circumstance in 
 them, the reason of which does not readily appear, namely, 
 that the effect of the attraction seems to increase, for half an 
 
 71 
 
 t^ 
 
I '. 
 
 i.i'i 
 
 MEMOIRS ON 
 
 1 
 
 1^ M-'fi ! 
 
 honr, or an hour, after the motion of the weights ; as it may 
 be observed, that in all three experiments, the mean position 
 kept increasing for that time, after moving the weigiits to the 
 positive position ; and kept decreasing, after moving them 
 from the positive to tlie midway position. 
 
 Tiie first cause wliicii occurred to me was, that possibly there 
 might be a want of elasticity, either in the suspending wire, or 
 something it was fastened to, which might make it yield more 
 to a given pressure, after a long continuance of that pressure, 
 than it did at first. 
 
 To put this to the trial, I moved the index so much, that the 
 arm, if not prevented by the sides of the case, would have stood 
 at about 50 ('"visions, so that, as it could not move farther than 
 to 35 divisiouo, it was kept in a position 15 divisions distant 
 from that which it would naturally have assumed from the 
 stiffness of the wire ; or, in other words, the wire was twiste<l 
 15 divisions. After having remained two or three hours in this 
 position, the index was moved back, so as to leave the arm at 
 liberty to assume its natural position. 
 
 It must be observed, that if a wire is twisted only a little 
 more than its elasticity admits of, then, instead of setting, as 
 it is called, or acquiring a permanent twist all at once, it sets 
 gradually, and, when it is left at liberty, it gradually loses part 
 of that set which it acquired ; so that if, in this experiment, 
 the wire, by having been kept twisted for two or three hours, 
 lia<l gradually yielded to this pressure, or had begun to set, it 
 would gradually restore itself, when left at liberty, aiul the 
 point of rest would gradually move backwards ; but, though 
 the experiment was twice repeated, I could not ))crceive any 
 such efff'ct. 
 
 The arm was next suspended by a stiffer wire. 
 
 1 ^ 
 
 Experiment IV. Aug. 12 
 Weights in midway position 
 
 Kxtrcnie 
 poiDts 
 
 Divisions 
 
 21.6 
 21.5 
 21.5 
 
 Time 
 
 9'' 30' 0" 
 52 
 10 13 
 
 I'oint of 
 rest 
 
 21.5 
 
 Time of mid. of 
 vibration 
 
 Difference 
 
 7» 
 
 i«f i 
 
' 
 
 TIIK LAWS OK (IIIAVITATION 
 
 Weif/htH morcdfroin vtidiray to poKitive ponidon 
 
 it, 
 
 !VS, 
 
 it 
 
 ce 
 
 27.2 
 
 
 
 
 
 
 22.1 
 
 — 
 
 — 
 
 24.6 
 
 
 
 27. 
 
 — 
 
 — 
 
 24.(57 
 
 
 
 22 6 
 
 — 
 
 — 
 
 24.75 
 
 
 
 26 8 
 
 — 
 
 — 
 
 24 8 
 
 
 
 28 () 
 
 — 
 
 — 
 
 24.85 
 
 
 
 26.6 
 
 — 
 
 — 
 
 24 9 
 
 
 
 23.4 
 
 
 
 
 
 
 
 
 Wt'if/hts mo red to ntf/dlire ]if)iiitiofi 
 
 
 15. 
 
 
 
 
 
 
 
 17 
 19 
 
 19 25 } 
 
 20 41 f 
 
 — 
 
 10" 20' 31 ' 
 
 
 22.4 
 
 — 
 
 — 
 
 18.72 
 
 — 
 
 7' 0' 
 
 
 20 
 19 
 
 26 45 } 
 
 27 22 f 
 
 — 
 
 27 81 
 
 
 15.1 
 
 — 
 
 — 
 
 18.52 
 
 — 
 
 6 57 
 
 
 19 
 20 
 
 35 1 / 
 
 48 f 
 
 — 
 
 34 28 
 
 
 21.5 
 
 ~~~ 
 
 — 
 
 18.85 
 
 — 
 
 7 23 
 
 
 20 
 19 
 
 40 23 } 
 
 41 18 <i 
 
 — 
 
 41 51 
 
 
 15.8 
 
 — 
 
 — 
 
 18.22 
 
 — 
 
 6 48 
 
 
 18 
 19 
 
 48 36 } 
 
 49 24 f 
 
 — 
 
 48 89 
 
 
 20.8 
 
 — 
 
 — 
 
 18.1 
 
 — 
 
 6 58 
 
 
 19 
 
 18 
 
 54 45 / 
 
 55 45 S 
 
 — 
 
 55 37 
 
 
 15.5 
 
 
 
 
 
 
 
 
 Wcir/htH moved 
 
 to positive position 
 
 
 31.3 
 
 
 
 
 
 
 
 25 
 28 
 
 11 10 25 } 
 11 3 f 
 
 — 
 
 11 10 40 
 
 
 17.1 
 
 — 
 
 — 
 
 24.02 
 
 — 
 
 7 3 
 
 
 22 
 28 
 
 17 6 1 
 26 S 
 
 — 
 
 17 48 
 
 
 30.6 
 
 — 
 
 — 
 
 24.17 
 
 — 
 
 7 1 
 
 
 25 
 28 
 
 24 83 ) 
 
 25 17 f 
 
 — 
 
 24 44 
 
 
 18.4 
 
 — 
 
 — 
 
 24 32 
 
 — 
 
 7 5 
 
 
 23 
 25 
 
 31 21 } 
 
 32 9 S 
 
 — 
 
 81 49 
 
 
 29.9 
 
 — 
 
 — 
 
 244 
 
 — 
 
 6 59 
 
 
 25 
 28 
 
 88 89 ) 
 39 31 \ 
 
 — 
 
 38 48 
 
 
 19.4 
 
 — 
 
 — 
 
 24 5 
 
 — 
 
 7 6 
 
 
 23 
 25 
 
 45 16 / 
 
 46 12 \ 
 
 — 
 
 45 54 
 
 
 29.3 
 
 
 
 
 
 
 Moiion of arm on moving weights from midway to pos. — 3.1 
 
 pos. to nei?. = 6.18 
 iieg. lo pos. r- 5.92 
 
 Time of one vibration in neg. position —7' l" 
 
 pos. position 
 
 ; it 
 
f 
 
 Iff 
 
 '^TT'rrfV' | I IPP i^V ''"'•»!, ^.' I 
 
 : 
 
 MEMOIRS ON 
 
 Experimf<:nt V. Aru. 20 
 
 7'/<« weights being in the pfmtive poHitinn, the, urm wan made to vibrate, by 
 
 intiiHiig the iiitirx 
 
 Kxtromo 
 points 
 
 DiviHions 
 
 Time 
 
 h.int or 
 
 roHt 
 
 TiMio of mid. of 
 vibration 
 
 DifTerence 
 
 29.6 
 
 
 21.1 
 
 — 
 
 — 
 
 25.2 
 
 
 
 29. 
 
 — 
 
 — 
 
 25.17 
 
 
 
 21.6 
 
 
 
 
 
 
 
 
 Weightu iiiored to negative jwsition 
 
 
 22.6 
 
 
 
 
 
 
 
 20 
 19 
 
 10" 22' 47 " } 
 23 30 S 
 
 — 
 
 10" 23' 11 " 
 
 
 16.3 
 
 — 
 
 — 
 
 19.27 
 
 
 
 21.9 
 
 — 
 
 — 
 
 19.15 
 
 
 
 16.5 
 
 — 
 
 — 
 
 19.1 
 
 
 
 21.5 
 
 — 
 
 — 
 
 19.07 
 
 
 
 16.8 
 
 — 
 
 — 
 
 19.07 
 
 
 
 21.2 
 
 — 
 
 — 
 
 19.07 
 
 
 
 17.1 
 
 — 
 
 — 
 
 19.05 
 
 
 
 20.8 
 
 — 
 
 — 
 
 19.02 
 
 
 
 17.4 
 
 — 
 
 — 
 
 19.05 
 
 
 
 20.6 
 
 — 
 
 — 
 
 19.02 
 
 
 
 
 20 
 19 
 
 11 32 16 \ 
 33 58 S 
 
 ~ 
 
 11 33 53 
 
 
 17.5 
 
 — 
 
 — 
 
 18.97 
 
 — 
 
 7' 13" 
 
 
 19 
 
 41 16 / 
 
 
 41 6 
 
 
 
 20 
 
 43 f 
 
 
 
 20.3 
 
 
 
 
 
 
 
 
 Weight f< moved 
 
 to jioHitire jWHition 
 
 
 20.2 
 
 
 
 
 
 
 
 24 
 26 
 
 49 10 / 
 
 50 19 ( 
 
 — 
 
 49 37 
 
 
 29.4 
 
 — 
 
 — 
 
 24.95 
 
 
 7 7 
 
 
 26 
 25 
 
 56 15 / 
 47 S 
 
 — 
 
 56 44 
 
 
 20.8 
 
 _.. 
 
 
 24 92 
 
 
 
 28.7 
 
 — 
 
 — 
 
 24.87 
 
 
 
 21.3 
 
 — 
 
 — 
 
 24.85 
 
 
 
 28.1 
 
 — 
 
 — 
 
 24.75 
 
 
 
 21.5 
 
 — 
 
 — 
 
 24.67 
 
 
 
 27.6 
 
 — 
 
 — 
 
 24.67 
 
 
 
 22. 
 
 — 
 
 — 
 
 24.7 
 
 
 
 
 24 
 
 45 48 } 
 46 43 \ 
 
 — 
 
 46 21 
 
 
 27.2 
 
 
 
 — 
 
 24.7 
 
 — 
 
 7 1 
 
 
 25 
 24 
 
 53 11 ) 
 
 54 9 J' 
 
 — 
 
 53 22 
 
 
 22.4 
 
 
 
 
 
 
 Motion of arm on moving weights from pos. to neg.= 5.9 
 
 neg. to po8. = 5.98 
 Time of one vibration, when weights are in neg. position = 7' 5" 
 pos. position — T 5" 
 
 74 
 
TIIK LAWS OF (iUAVlTATlON 
 
 In the fourtli oxperiment, the effect of the weights seemed 
 to increase on stjinding, in all tliree motions of the weiglits, 
 conformably to wliat was observed with the former wire ; but 
 in tlie last experinjent tiie case was dilfereTit ; for though, on 
 moving the weiglits from positive to negative, the effect seemed 
 to increase on standing, yet, on moving them from negative to 
 positive, it diminislied. 
 
 My next trials were, to see whether this effect was owing to 
 magnetism. Now, as it happened, the case in which the arm 
 was inclosed, was placed nearly parallel to the magnetic; east, 
 and west, and therefore, if tiiere was anything magnetic in the 
 balls and weights, the balls would acquire polarity from the 
 earth ; and the weights also, after having remained some time, 
 either in the positive or negative position, would acquire po- 
 larity in the same direction, and would attract the balls; but, 
 when the weights were moved to the contrary position, that 
 pole which before pointed to the north, would point to the 
 sonth, and would repel the ball it was approached to ; but yet, 
 as repelling one hall towards the south has the same effect on 
 the arm as attracting the other towards the north, this would 
 have no effect on the position of the arm. After some time, 
 however, the poles of the weight would be reversed, and would 
 begin to attract the balls, and would therefore produce the 
 same kind of effect as was actually observed. 
 
 To try whether this was the case, I dettiched the weights 
 from the npper })art of the copper rods by which they were 
 suspended, but still retained the lower joint, namely, that 
 which passed through them ; I then fixed them in their posi- 
 tive position, m such manner, that they could turn round on 
 this joint, as a vertical axis. I also made an apparatus by 
 which I could turn them half way round, on these vertical axes, 
 without opening the door of the room. 
 
 Having suffered the apparatus to remain in this manner for a 
 day, I next morning observed the arm, and, having found it to 
 be stationary, turned the weights half way round on their axes, 
 but could not perceive any motion in the arm. Having suf- 
 fered the weights to remain in this position for about an hour, 
 I turned them back into their former position, but without its 
 having any effect on the arm. This experiment was repeated 
 on two other days, with the same result. 
 
 We may be sure, therefore, that the effect in question could 
 
 75 
 
MEMOIRS ON 
 
 . 1'! 
 
 not be produced by magnetism in tbe woiglits ; for, if it was, 
 turning them half round on tlieir jt.xe«, would immediately have 
 changed tlieir magJietic attraction into repulsion, and have 
 produced a motion in the arm. 
 
 As a further proof of this, I took off th( leaden weights, and 
 in tlieir room placed two 10-inch magnets , the apparatus for 
 turning them round being left as it was, and the magnets being 
 placed horizontal, and pointing to the balls, and with their 
 north poles turned to the north ; but I could not find that 
 any alteration was produced in the place of the arm, by turn- 
 ing them half round ; which not only confirms the deduction 
 drawn from the former experiment, but also seems to shew, that 
 in the experiments with the iron rods, the effect produced could 
 not be owing to magnetism. 
 
 The next thing which suggested itself to me was, that pos- 
 sibly the effect might be owing to a difference of tempera- 
 ture between the weights and the case ; for it is evident, 
 that if the weights were mnch warmer than the case, they 
 would warm that side which was next to them, and prodnce a 
 current of air, which would make the balls approach nearer to 
 the weights. Though I thought it not likely that there should 
 be sufficient difference, between the heat of the weights and 
 case, to have any sensible effect, and though it seemed im- 
 probable that, in all the foregoing experiments, the weights 
 should happen to be warmer than the case, 1 resolved to ex- 
 amine into it, and for this purpose removed the apparatus nsed 
 in the last experiments, and supported the weights by the cop- 
 per rods, as before ; and, having placed them in the midway 
 position, 1 put a lamp under each, and placed a thermometer 
 with- its ball close to the outside of the case, near that part 
 which one of the weights approached to in its positive position, 
 and in such numner that 1 conld distinguish the divisions by 
 the telescope. Having done this, I shut tlie door, and some 
 time after moved the weights to the positive position. At first, 
 the arm was drawn aside oidy in its usual manner ; but, in 
 half an hour, the effect was so much increased, that the arm 
 was drawn 14 divisions aside, instead of about three, as it 
 would otherwise have been, and the thermometer Avas raised 
 near l^.f) ; namely, from (51" to (>''^°.5. On opening the door, 
 the weights were found to be no more heated, than just to pre- 
 vent their feeling cool to my fingers. 
 
 76 
 
TIIK LAWS OK (JUAVlTA'iON 
 
 As tho oiToot of ji (litTcrcMicc of tcriiixM'iitiire jippi-aiTtl to hv 
 so grt'iit, I bored a siiiull liolo in oiiu of the wri^Mits, about 
 tlirec-quiirters of jui iiicb (loop, iind iiisortod tlio bull of a sriiull 
 tliormomotor, jiiid thou oovorocl up tlio opouiuj,' witii ooiuont. 
 Another suuill therniorueter was plaoed with its ball close to 
 tho case, aud as near to that part to which the woi<,dit was 
 approached as could be done with safety ; the theruioniotors 
 being so i)laced, that when the weights were in the negative 
 position, both could be seen through one of the telesco])es, by 
 means of light rellected from a concave mirror. 
 
 or 
 ,rt 
 
 >y 
 
 18 
 
 ^t, 
 
 in 
 im 
 
 lit 
 
 d 
 
 Exrr<:KFMKNT \'I. Skit. 
 
 Wt'if//ilx ill miihriOf jittsift'ori 
 
 
 
 
 
 Tliornioinetor 
 
 iTtI romp 
 
 Divi.sions 
 
 Time 
 
 I'oinl of 
 
 
 
 TjAI'I l-IIK' 
 
 
 
 points 
 
 
 
 rust 
 
 III iilr 
 55.5 
 
 ill \v{<i)jlit 
 
 
 18.9 
 
 9'' 43 
 
 
 
 18.85 
 
 10 3 
 
 18.85 
 
 
 
 
 Weighta moved to inyative pmtion 
 
 
 13.1 
 
 — 
 
 10 12 
 
 — 
 
 55.5 
 
 55.8 
 
 18.4 
 
 — 
 
 18 
 
 15.82 
 
 
 
 13.4 
 
 — 
 
 25 
 
 
 
 
 missed 
 
 
 
 
 
 
 13.6 
 
 — 
 
 39 
 
 — 
 
 55.5 
 
 55.8 
 
 17.6 
 
 — 
 
 46 
 
 15.65 
 
 
 
 13.8 
 
 — 
 
 53 
 
 15.65 
 
 
 
 17.4 
 
 — 
 
 11 
 
 15.65 
 
 
 
 14.0 
 
 — 
 
 7 
 
 15.65 
 
 
 
 17. 'i 
 
 — 
 
 14 
 
 — 
 
 55.5 
 
 
 
 Weif/hta moved to jiositive jwsition, 
 
 
 25.8 
 
 — 
 
 23 
 
 
 
 
 17.5 
 
 — 
 
 30 
 
 21.55 
 
 
 
 25.4 
 
 — 
 
 37 
 
 ','1 .6 
 
 
 
 18.1 
 
 — 
 
 44 
 
 21 (i.-) 
 
 
 
 25.0 
 
 — 
 
 51 
 
 
 
 
 niissiid 
 
 
 
 
 
 
 24.7 
 
 — 
 
 5 
 
 
 
 
 19. 
 
 — 
 
 12 
 
 21.77 
 
 
 
 24.4 
 
 — 
 
 19 
 
 
 
 
 M 
 
 otiou of arm on moving weights from midway to — = 
 
 3.03 
 
 
 - to + = 
 
 5.9 
 
 77 
 
■ rw '■■ ""T- 
 
 M KM OIKS UN 
 
 0; ii 
 
 1:51 1 
 
 ill 
 
 MM: 
 
 ^;!ii 
 
 i '!">'' 
 
 Extreme 
 pnliilH 
 
 13.6 
 
 18.8 
 13.8 
 
 16.9 
 14.5 
 16.6 
 
 26.4 
 17.2 
 26.1 
 
 19.3 
 25.1 
 19.7 
 
 EXFEUIMKNT VII. SkI'T. 18 
 Weifjfitu in viuhnay posit ion 
 
 DIVIHiOMK 
 
 19.4 
 19.4 
 
 Time 
 
 I'diiii or 
 
 rcHl 
 
 Thertnomoler 
 111 iiir in woiK>it 
 
 Sh 30 — 56.7 
 
 9 32 — 56.6 
 
 Weif/ht« mored to net/dtivc poHilion 
 
 40 - 
 
 47 16.25 
 
 54 
 
 Eiijht cjctveme jvUiUh niinHed 
 
 10 58 
 
 11 5 15.62 
 12 
 
 Weighta mored to positive position 
 
 57.2 
 
 20 
 
 28 
 35 
 
 21.72 
 
 56.5 
 
 Four extreme points missed 
 
 10 
 
 17 22.3 
 
 24 
 
 Motion of arm on moving weights from midway to — = 3.15 
 
 - to + = 6.1 
 
 ExPKRiMEivr VIII. Sept. 23 
 Weights in midway position 
 
 Extreme 
 points 
 
 13.5 
 18.6 
 13.6 
 
 17.4 
 14.1 
 
 17.2 
 
 Divisions 
 
 Time 
 
 9" 46' 
 10 45 
 
 Tolnl of 
 rest 
 
 Thermometer 
 
 in air 
 
 in weight 
 
 19.3 
 19.2 
 
 19.2 
 
 53.1 
 53.1 
 
 
 T 
 
 Veights mored to negative position 
 
 
 — 
 
 56 
 
 11 3 
 
 10 
 
 16.07 
 
 
 53.6 
 
 
 Four extreme points missed 
 
 
 — 
 
 44 
 51 
 
 58 
 
 15.7 
 
 
 53.6 
 
 78 
 
TlIK LAWS UK <J1:AV1TATIUN 
 
 Weiyhtn vifyred to jMmitive jumition 
 
 15.7 
 20.7 
 16.0 
 
 25.0 
 18.1 
 25.5 
 
 0'' 1 
 
 i; 
 
 21.42 
 
 53.15 
 
 Ttto extreine jiouitti ininHetl 
 
 :iO 
 48 
 
 50 
 
 21.9 
 
 Motion of arm on moving wiii^lit.s from niiilwny to — = 8.1!i 
 
 -to+=:5.72 
 
 In these tliree experiinonts, the ctTect of tlie weiglit appeared 
 to inerease from two to live tenths of a (livisioii, on stanilin*; 
 an liour ; and tlie tljerniometers shewed, that the weights were 
 three or five tenths of a degree wanner tlian tlu; air close to the 
 case. In the two hist experiments, 1 pnt a himp into the room, 
 over night, in liopesof making the air warmer than tlie weights, 
 but without effeet, as the lieat of the weights exceeded tliat of 
 tlie air more in tliese two experiments than in the former. 
 
 On the evening of Oc;tol)er 17, the weights being phieed in 
 tlie midway position, lamps were pnt under them, in order to 
 warm them ; the door was then shut, and the lamps suffered to 
 burn out. The next morning it was found, on moving the 
 weights to the negative position, that they were 7°. 5 warmer 
 than the air near the case. After they had continued an hour 
 in that position, they were found to have cooled l^.T), so as to 
 be only 0° warmer than the air. Tiiey were then moved to the 
 positive position ; and in botii positions the arm was drawn 
 aside about four divisions more, after the weights had remained 
 an hour in that position, than it was at first. 
 
 May 22, 17!J8. The experiment was repeated in the same 
 manner, except that the lamps were made so as to l)urn only a 
 short time, and only two hours were suffered to elapse before 
 the weights were moved. The weights were now found to be 
 scarcely 2° warmer than the case ; and tlie arm was drawn 
 aside about two divisions more, after the weights had remained 
 an hour in the position they were moved to, than it was at first. 
 
 On May 23, the experiment was tried in the same manner, 
 except that the weights were (tooled by laying ice on them; the 
 ice being confined in its place by tin plates, which, on moving 
 the weights, fell to the ground, so as not to be in the way. «)n 
 moving the weights to the negative position, they were found 
 
 79 
 
 -u 
 
^^ 
 
 MK Mollis ON 
 
 S<1 
 
 111 
 
 I'^'^M 
 
 to bo lUmiit S'' ooldcr tluui the air, and tlioir rfTont on the arm 
 BocniLMl now to (liininiHh on Hlandin;;, instcud of iniM'oasin^, as 
 it did boforo ; as tbc arm was drawn asido about *^J divisions 
 less, at tbo ond <>f an iionr al'tor tiio motion of the wnights, 
 than it was at first. 
 
 It Kooms snfliciontly provcMl, tbtM'cforo, that the efTect in ques- 
 tion is produced, as above (ixphiinod, l)y the dilTeren(!e of tem- 
 perature Ix'tween tiie wei«?lits and ease; for in tiie (Ith, Htl). and 
 IHh* experiments, in whieh the wei<;hts were not much warmer 
 than tile case, tiieir (dfec^t iiu^reased but little on standing; 
 whereas, it increased much, when they were much warmer than 
 the case, and decreased fiuHsh, when they were much cooler. 
 
 It must be observed, that in this a])))aratus, the box in which 
 the balls play is pretty deej), and the balls hang near the bot- 
 tom of it, whi(!h makes the efTect of the current of air more 
 sensible than it would otherwise be, and is a defect which 1 iu- 
 tend to rectify in some future experiments. 
 
 '11 
 
 '', 
 
 iiii: 
 
 ;iitl 
 
 i:|!il 
 
 ;ii'(i 
 
 •I ill 
 
 I 
 
 ExpKiiiMENT IX. April 29 
 
 Weightx in pimtlre ixmtion 
 
 Kxtromo 
 
 Divi.sioiiH 
 
 Time 
 
 Toiiit of 
 
 Time or middle of 
 
 poiiils 
 
 
 
 rt'sl 
 
 vliinilioii 
 
 34.7 
 
 
 
 
 85. 
 
 — 
 
 — 
 
 34.84 
 
 
 34.05 
 
 
 
 
 
 
 Weights moved to negntive p<mtio7i 
 
 
 23.8 
 
 
 
 
 
 
 28 
 29 
 
 11" 18' 39" \ 
 
 58 S 
 
 — 
 
 11" 18' 43" 
 
 33.2 
 
 — 
 
 — 
 
 28.52 
 
 
 
 29 
 
 28 
 
 35 27 ) 
 57 \ 
 
 — 
 
 25 40 
 
 33.9 
 
 — 
 
 
 28.25 
 
 
 33. 
 
 — 
 
 — 
 
 28.01 
 
 
 24.15 
 
 — 
 
 — 
 
 37.82 
 
 
 31. 
 
 — 
 
 — 
 
 27.(53 
 
 
 34.4 
 
 — 
 
 — 
 
 37.55 
 
 
 30.4 
 
 28 
 
 7 4 1 
 
 53 S 
 
 27.47 
 
 
 
 27 
 
 
 7 26 
 
 24.7 
 
 
 
 
 
 
 Motion of ivnn — 6.32 
 
 
 
 Time of vibnition =: 6' 58" 
 
 
 * [^Tku is evidently a misprint for Qth, 1th, and 8///.] 
 
 80 
 
f 
 
 T 11 !•; 1. A \V S ( ) K ( J K A \' I T A T 1 U N 
 
 I 
 
 Kxi'KUiMKNT \. May 5 
 
 Rxtmine 
 
 |Hllll(M 
 
 a4.5 
 
 ;m.5 
 
 84.4 
 
 22.3 
 
 88.2 
 
 22.0 
 
 Wi.Ti 
 
 2:i.2 
 
 31.45 
 
 23.5 
 
 30.7 
 
 23.95 
 
 80.25 
 
 IHvIhIuUH 
 
 2S 
 29 
 
 2S 
 27 
 
 27 
 
 28 
 
 28 
 27 
 
 27 
 
 28 
 
 Weif/htx in jxmiliir jutnition 
 
 Time 
 
 I'liinl or 
 rvMl 
 
 33.9^ 
 
 TIlID' 1)1' llilililli' III' 
 
 viliriilKiii 
 
 Weiyhts moral to myiitirf jioxitiou 
 
 10" 43 42 I 
 
 44 ( 
 
 r,{) 33 / 
 
 51 S 
 
 11 25 20 } 
 
 58 )■ 
 
 32 / 
 
 32 40 S 
 
 39 19 ) 
 
 40 2 ] 
 
 07 tJO 
 
 27.72 
 27.7 
 
 27.58 
 27.4 
 
 27.28 
 
 27.21 
 27.21 
 
 m 43 30 
 
 50 30 
 
 11 25 24 
 32 27 
 39 23 
 
 Motion of arm =0.15 
 Time of vibration = 59 " 
 
 l»in«'r<'iic( 
 
 7' 
 
 7 3 
 50 
 
 EXFKHIMENT XI. MaY 6 
 
 Wrights in jtom'tive jxmtion 
 
 Extreme 
 puiutH • 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibration 
 
 34.9 
 
 
 
 34.1 
 
 — 
 
 — 
 
 34.47 
 
 
 34.8 
 
 — 
 
 — 
 
 34.49 
 
 
 34.25 
 
 
 
 
 
 
 Weights maved to negative position 
 
 
 23.3 
 
 
 
 
 
 
 28 
 29 
 
 9'' 59' 59 } 
 10 27 S 
 
 — 
 
 lO" 0' 8' 
 
 33.3 
 
 — 
 
 — 
 
 28.42 
 
 
 
 29 
 27 
 
 6 52 ) 
 
 7 51 S 
 
 — 
 
 7 5 
 
 23.8 
 
 — 
 
 — 
 
 28.35 
 
 
 F 
 
 
 81 
 
 
 
^. 
 
 -f^^-,- 
 
 ^MAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 ^ 5.^t% 
 
 ^> M^ 
 
 m 
 
 y. 
 
 :/. 
 
 4C 
 
 % 
 
 1.0 
 
 1.1 
 
 |io "'^" mH 
 
 ■^ ^ ■2.2 
 2.0 
 
 us 
 
 ■It 
 
 IL25 111 1.4 
 
 I 
 
 M 
 1.6 
 
 ^ 
 
 V] 
 
 
 / 
 
 ^7 
 
 "^S 
 
 y 
 
 Hiotographic 
 
 Sciences 
 
 Corporation 
 
 23 WEST MAIN STREET 
 
 WEBSTER, N.Y. MS80 
 
 (716)l7'i-4S03 
 
 V 
 
 iV 
 
 H^ 
 
 ■\ 
 
 
 '<^ 
 
 

 1 
 
MEMOIUS ON 
 
 82.5 
 
 __ 
 
 
 — 
 
 28.3 
 
 
 24.4 
 
 
 
 
 
 
 missed 
 
 
 
 
 
 
 24.8 
 
 
 
 
 
 
 81.3 
 
 — 
 
 
 — 
 
 28.17 
 
 
 
 29 
 
 28 
 
 
 10" 48' 37" ) 
 49 21 \ 
 
 — 
 
 10" 49' 8" 
 
 25.3 
 
 — 
 
 
 — 
 
 28.2 
 
 
 
 28 
 29 
 
 
 56 8 I 
 56 \ 
 
 — 
 
 56 13 
 
 30.9 
 
 
 
 
 
 
 
 
 Motion of arm =6.07 
 
 
 
 
 Time of vibralion = 7' 1 " 
 
 
 In the three foregoing experiments, the index was purposely 
 moved so that, before the beginning of the experiment, the 
 balls rested as near the sides of the case as they could, without 
 danger of touching it ; for it must be observed, that when the 
 arm is at 35, they begin to touch. In the two following ex- 
 periments, the index was in its usual position. 
 
 Experiment XII. May 9 
 
 Weif/htK ill, negative position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibration 
 
 
 17.4 
 
 9" 45 " 
 
 
 
 
 17.4 
 
 58 
 
 
 
 
 17.4 
 
 10 8 
 
 
 
 
 17.4 
 
 10 
 
 17.4 
 
 
 
 Weights moved to positii 
 
 'e position 
 
 
 28.85 
 
 
 
 
 
 
 24 
 22 
 
 20 50 ) 
 
 21 46 ( 
 
 — 
 
 10" 20' 59" 
 
 18.4 
 
 — 
 
 — 
 
 23.49 
 
 
 28.3 
 
 — 
 
 
 23.57 
 
 • 
 
 19.3 
 
 — 
 
 — 
 
 23.67 
 
 • 
 
 27.8 
 
 — 
 
 — 
 
 23.72 
 
 
 20. 
 
 — 
 
 — 
 
 23.8 
 
 
 27.4 
 
 — 
 
 . — 
 
 23.83 
 
 
 
 24 
 28 
 
 11 3 13 ) 
 54 \ 
 
 — 
 
 11 3 14 
 
 20.55 
 
 — 
 
 — 
 
 23.87 
 
 
 
 28 
 24 
 
 9 45 I 
 
 10 28 f 
 
 — 
 
 10 18 
 
 87. 
 
 
 
 
 
 
 Motion of arm = 
 
 :6.09 
 
 
 
 Time of vihrulinn = 
 
 :7'3' 
 
 
 88 
 
8" 
 
 18 
 
 29.6 
 17.4 
 28.9 
 
 18.4 
 
 28.4 
 19.3 
 
 27.8 
 
 19.9 
 
 27.3 
 
 13.5 
 21.8 
 
 13.9 
 
 21.1 
 14.4 
 20.5 
 14.7 
 20. 
 
 15. 
 
 19.5 
 
 THE LAWS OF GRAVITATION 
 Experiment XIII. May :e5 
 
 WeifjhtH in tm/ntive poHition 
 
 25 
 ?4 
 
 23 
 24 
 
 24 
 23 
 
 23 
 24 
 
 24 
 23 
 
 17.2 
 
 Weights moved to positive position 
 lOh 22' 22" 
 
 22" ) 
 45 f 
 
 29 59 ; 
 
 30 23 f 
 
 36 58 1 
 
 37 24 / 
 
 44 
 
 i\ 
 
 11 
 
 23 
 24 
 
 5 26 
 6 
 
 u 
 
 12 12 
 
 23.32 
 
 23.4 
 
 23.52 
 
 23.63 
 
 23.7 
 
 23.7 
 
 23.72 
 
 18 
 17 
 
 17 
 18 
 
 50 } 
 
 Weights moved to negative position 
 
 17.75 
 
 37 34 ) 
 
 38 10 f 
 
 44 26 ) 
 
 4 f 
 
 45 
 
 18 
 17 
 
 17 
 18 
 
 19 57 ) 
 20 52 \ 
 
 27 15 
 
 28 15 
 
 \ 
 
 Motion of the arm on moving weights 
 Time of vibration at + 
 
 17.67 
 
 17.62 
 
 17.6 
 
 17.52 
 
 17.47 
 
 17.42 
 
 17.37 
 
 Time of middle of 
 vibration 
 
 10" 22' 56" 
 
 30 3 
 
 37 7 
 
 44 14 
 
 11 5 31 
 12 35 
 
 37 39 
 44 45 
 
 20 24 
 27 30 
 
 from- to + = 
 + to - = 
 
 83 
 
 6.12 
 5.97 
 
 :7'6" 
 
 :7'7" 
 
! 
 
 MEMO ins UN 
 
 • 
 
 
 i ExpKRiMKNT XIV. May 26 
 I Wcif/htM in nef/afire position 
 
 
 
 • Kxlromo 
 |> imintH 
 
 DIvisiuiiM 
 16.1 
 
 Time 
 
 9'' IH 0" 
 
 Point ol 
 rest 
 
 TImo of middle n( 
 vibriition 
 
 
 
 
 16.1 
 
 24 
 
 
 
 
 16.1 
 
 46 
 
 
 
 
 16.1 
 
 49 
 
 16.1 
 
 
 Wei(/htH moved to positive jnysition 
 
 
 27.7 
 
 
 
 
 
 i:^^ 
 
 23 
 22 
 
 10 46 ) 
 1 16 ^ 
 
 — 
 
 10" r 1" 
 
 ^-: 17.3 
 
 — 
 
 — 
 
 22.37 
 
 
 
 22 
 
 7 58 / 
 
 
 8 5 
 
 
 23 
 
 8 27 f 
 
 
 ,..;: 27.2 
 
 — 
 
 — 
 
 22.5 
 
 
 
 23 
 22 
 
 15 2 ( 
 32 i 
 
 — 
 
 15 9 
 
 18.3 
 
 — 
 
 — 
 
 22.65 
 
 
 26.8 
 
 — 
 
 — 
 
 22.75 
 
 
 19.1 
 
 — 
 
 — 
 
 22.85 
 
 
 26.4 
 
 — 
 
 — 
 
 22.97 
 
 
 -. 
 
 23 
 22 
 
 43 40 } 
 
 44 22 f 
 
 — 
 
 43 32 
 
 20. 
 
 — 
 
 — 
 
 23.15 
 
 
 
 22 
 23 
 
 49 53 ) 
 
 50 37 \ 
 
 — 
 
 50 41 
 
 26.2 
 
 
 
 
 
 Weiijhts moved to negative position 
 
 
 12.4 
 
 
 
 
 
 
 16 
 17 
 
 11 7 53 ) 
 8 27 S 
 
 — 
 
 11 8 25 
 
 21.5 
 
 — 
 
 — 
 
 17.02 
 
 
 
 17 
 16 
 
 15 30 I 
 
 16 3 \ 
 
 — 
 
 15 27 
 
 12.7 
 
 — 
 
 — 
 
 16.9 
 
 
 uiT 20.7 
 
 — 
 
 — 
 
 16.85 
 
 
 Ih! 13.3 
 
 
 — 
 
 16.82 
 
 
 '^'•'"^ 20. 
 
 — 
 
 — 
 
 16.72 
 
 
 13.6 
 
 — 
 
 — 
 
 16.67 
 
 
 
 16 
 17 
 
 50 33 ) 
 
 51 19 f 
 
 — 
 
 50 58 
 
 19.5 
 
 — 
 
 — 
 
 16.65 
 
 
 ■ 
 
 17 
 16 
 
 57 53 I 
 
 58 44 \ 
 
 — 
 
 58 6 
 
 14. 
 
 
 
 
 
 Motion of arm by moving weights from — to + : 
 
 = 6.27 
 
 
 
 + tO-: 
 
 = 6.13 
 
 j : Time of vibratioi 
 
 1 at -f 
 
 = 7' 6" 
 
 ■1.1' » """ 
 
 = 7' 6" 
 
 84 
 
 ■ J ■'•'-■■ 'I \ 
 
 1 1 ■ . 1 ." 
 ' ■ ',1 1 
 
THE LAWS OF (JUAVITATlUN 
 
 J5 
 J7 
 
 r 
 
 6 
 
 III the next oxperinieiit, the balls, before the motion of the 
 weights, were inude to rest as near as possible to the sides of 
 the case, but on the contrary side from what they did in the 
 i)th, loth, and 11th experiments. 
 
 ExfKiiiMKNT XV. May 27 
 
 WiUjhtx in negative jxtnilion 
 
 Extreme 
 puiiits 
 
 Divisions 
 
 Time 
 
 I'oint of 
 res I 
 
 Time of middle of 
 viliriitioii 
 
 8.9 
 
 
 
 
 
 
 8.85 
 
 — 
 
 
 — 
 
 8.61 
 
 
 3.85 
 
 — 
 
 
 ^— 
 
 8.61 
 
 
 3.4 
 
 
 
 
 
 
 
 
 Weights moved to poxiti 
 
 ve jHitiitioii 
 
 
 15.4 
 
 
 
 
 
 
 
 10 
 9 
 
 
 10" 5' 59" ) 
 
 6 27 \ 
 
 — 
 
 10" 5' 56" 
 
 4.8 
 
 — 
 
 
 — 
 
 9.95 
 
 
 
 9 
 10 
 
 
 12 43 I 
 18 n f 
 
 — 
 
 13 5 
 
 14.8 
 
 — 
 
 
 — 
 
 10.07 
 
 
 
 10 
 9 
 
 
 20 24 } 
 56 f 
 
 
 20 13 
 
 5.9 
 
 — 
 
 
 — — 
 
 10.23 
 
 
 14.35 
 
 — 
 
 
 — 
 
 10.85 
 
 
 6.8 
 
 — 
 
 
 — 
 
 10.46 
 
 
 18.9 
 
 — 
 
 
 — 
 
 10.52 
 
 
 
 11 
 10 
 
 
 48 80 1 
 
 49 11 \ 
 
 — 
 
 48 42 
 
 7.5 
 
 — 
 
 
 — 
 
 10.6 
 
 
 
 10 
 
 11 
 
 
 55 26 ) 
 
 56 10 \ 
 
 — 
 
 55 48 
 
 13.5 
 
 
 
 
 
 
 
 
 Motion of tlie arm — 
 
 6.34 
 
 
 
 
 Time of vibration = 
 
 7' 7" 
 
 
 The two following experiments were made by Mr. Gilpin, 
 who was so good as to assist me on the occasion. 
 
 Experiment XVI. May 28 
 
 Weights in negative j)osition 
 
 Extreme 
 points 
 
 UivisioDS 
 
 Time 
 
 Point or 
 
 rest 
 
 Time of middle of 
 vibration 
 
 22.55 
 8.4 
 
 21. 
 9.2 
 
 — 
 
 — 
 
 15.09 
 14.9 
 
 
 85 
 
MEMOIRS ON 
 
 
 Wei(/fits moved to jwintive position 
 
 26.6 
 
 
 
 
 
 
 22 
 
 ai 
 
 10" 22 53 ) 
 23 20 
 
 — 
 
 lOh 23' 15 ' 
 
 15.8 
 
 — 
 
 — 
 
 21. 
 
 
 
 20 
 21 
 
 36 
 
 — 
 
 30 30 
 
 25.8 
 
 — 
 
 — 
 
 21.05 
 
 
 
 32 
 21 
 
 37 23 I 
 55 \ 
 
 — 
 
 37 45 
 
 16.8 
 
 — 
 
 — 
 
 21.11 
 
 
 
 20 
 21 
 
 44 29 I 
 
 45 4 
 
 — 
 
 45 1 
 
 25.05 
 
 — 
 
 — 
 
 21.11 
 
 
 
 22 
 21 
 
 51 54 / 
 
 52 32 ] 
 
 — 
 
 52 20 
 
 17.57 
 
 — 
 
 — 
 
 21.2 
 
 
 
 21 
 22 
 
 59 31 / 
 11 13 f 
 
 — 
 
 59 34 
 
 24.6 
 
 — 
 
 — 
 
 21.28 
 
 
 
 22 
 21 
 
 6 24 1 
 
 7 9 f 
 
 — 
 
 11 6 49 
 
 18.3 
 
 
 
 
 
 
 Motion of the iirrn : 
 
 = 6.1 
 
 
 
 T 
 
 irne of vibration : 
 
 = 7' 16" 
 
 
 13 !M!: 
 
 ,, -if!' ?. 
 
 Experiment XVII. May 30 
 
 Weights in negative position 
 
 Extreme 
 poiutH 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle or 
 vibration 
 
 
 17.2 
 
 10" 19' 0" 
 
 
 
 
 17.1 
 
 25 
 
 
 
 
 17.07 
 
 29 
 
 
 
 
 17.15 
 
 40 
 
 
 
 
 17.45 
 
 49 
 
 
 
 
 17.42 
 
 51 
 
 
 
 
 17.42 
 
 11 1 
 
 17.42 
 
 
 
 Weights moved to positive position 
 
 
 28.8 
 
 
 
 
 
 
 24 
 23 
 
 11 11 23 ) 
 49 
 
 — 
 
 ll"" 11' 87" 
 
 18.1 
 
 — 
 
 — 
 
 23.2 
 
 
 
 22 
 23 
 
 18 13 / 
 43 
 
 — 
 
 18 42 
 
 27.8 
 
 — 
 
 — 
 
 23.12 
 
 
 
 24 
 23 
 
 25 19 ) 
 49 
 
 — 
 
 25 40 
 
 18.8 
 
 — 
 
 — 
 
 23.2 
 
 
 
 23 
 24 
 
 32 41 ) 
 
 33 13 y 
 
 86 
 
 — 
 
 32 43 
 
 .'.iv».J*!i.;,v.:--:=Vr 
 
 ..^■...i-:;.sf^.^>,.. 
 
 ^.Hs: 
 
27.38 
 
 19.7 
 
 37. 
 
 20.4 
 
 36.5 
 
 20.8 
 
 36.25 
 
 13.3 
 
 22.4 
 
 13.7 
 
 21.6 
 
 14. 
 
 20.8 
 
 14.3 
 
 20.1 
 
 14.6 
 
 TJIK LAWS OF (JKAVITATION 
 
 23.31 
 
 24 
 23 
 
 23 
 24 
 
 24 
 
 33 
 
 28 
 
 24 
 
 24 
 23 
 
 23 
 24 
 
 11" 30 28 ) 
 40 3 J 
 
 46 33 ) 
 
 47 11 f 
 
 53 36 ) 
 
 54 17 f 
 
 34 / 
 
 1 18 f 
 
 7 34 } 
 
 8 21 f 
 
 14 30 ) 
 
 15 24 f 
 
 23.44 
 
 23.53 
 
 33.57 
 
 23.55 
 
 23.59 
 
 Weights fmved to neffative position 
 
 17 
 
 18 
 
 18 
 17 
 
 17 
 
 18 
 
 18 
 17 
 
 17 
 
 18 
 
 18 
 17 
 
 17 
 
 18 
 
 18 
 17 
 
 32 19 / 
 
 
 48 f 
 
 — 
 
 39 46 ) 
 
 17.95 
 
 40 19 f 
 
 — 
 
 46 26 ) 
 
 17.85 
 
 .47 f 
 
 -^ 17.72 
 53 43 } 
 
 54 20 f — 
 
 . 0-89 , ''■' 
 
 1 30 f - 
 
 7 39 ) 
 
 17.47 
 
 H 31 f 
 
 : — 
 
 14 54 / 
 
 17.37 
 
 15 43 f 
 
 — 
 
 31 33 ) 
 
 17.27 
 
 33 33 )" 
 
 — 
 
 11" 39 44' 
 46 46 
 53 48 
 55 
 7 50 
 14 58 
 
 32 
 
 44 
 
 39 
 
 44 
 
 46 
 
 48 
 
 53 
 
 50 
 
 1 
 
 55 
 
 7 
 
 59 
 
 15 
 
 4 
 
 22 5 
 
 Motion of the ann on moving weigh.s fn.m - to + :. 5.73 
 
 Time of vibmtion at + + to - = 5 64 
 
 = 7' 2" 
 
 = r 3" 
 
MKMOlltS UN 
 On thk Mktiioh ok C'ompitino tiik Dknsity av Tirn Eauth 
 
 I" IK ) M '!• 1 1 i;s !•; K X !• K K I M K N IS 
 
 I sliiill first ooriipiito tluH, on tlio supposition that tlio arm 
 iuul copper rods luive no wciglit, and that the \voi«i:lits exert no 
 sensible attra(!tion, ex(X'pt on tlie nearest ball ; and shall then 
 examine what corrections are necessary, on account of the arm 
 and rods, and some other small ctiuses. 
 
 The first thing is, to find the force required to draw the arm 
 aside, which, as was before said, is to be determined by the time 
 of a vibration. 
 
 The distaiKse of the centres of the two balls from eacdi other 
 is 73. IJ inches, and tlierefore the distance of each from the 
 centre of motion is IJfi.Cr), and the length of a pemluluni vi- 
 brating seconds, in this climate, is 3!).l-t; therefore, if the 
 stiffness of the wire by which the arm is suspended is such, 
 that the force which must be applied to each ball, in order to 
 draw the arm aside by the aiigle A, is to the weight of that 
 ball as the arch of A to the radius, the arm will vibrate in the 
 same time as a pendulum whose length is 3G.G5 inches, that is, in 
 
 V ')n I 1 f^cconds ; and therefore, if the stiffness of the wire is 
 
 such as to make it vibrate in N seconds, the force which must 
 be applied to each ball, in order to draw it iiside by the angle 
 
 A, is to the weight of the ball as the arch of Ax^.,x'. ',' 
 *^ ^ ;V.J.14 
 
 to the radius. Hut the ivory scale at the end of the arm is 
 
 38.3 inche.". from the centre of motion, and each division is gij, 
 
 of an inch, and therefore subtends an angle at the centre, whose 
 
 arch is ^^^ ; and therefore the force which must be applied to 
 
 each ball, to draw the arm aside by one division, is to the weight 
 
 of the ball as 
 
 7G0N* 
 
 30.05 , , 1 i. 1 * 
 
 * [Or thitu: using the ordinary notation for the simple lendulum vibrating 
 through small arcs, if the force on each ball drawing the arm aside through an 
 
 arc subtending an angle of A° were mg x 
 
 arc 
 radius 
 
 , tlie arm would vibrate like a 
 
 jtendulum of the same length, and have a period 
 
 
 seconds, because the 
 
 period of a pendulum varies as tfie squarz root of its length. But the force varies 
 as r-T-j ; therefore tJie force required to draw the arm through A° with 
 
 {periodf 
 
 88 
 
Til K LAWS OK (illAV ITATloN 
 
 Tlio next tiling is, ^o tiixi tlio proportion wliicli tlie iittraction 
 of the wcij^ht on tlio hail hciirs to that of the carlli thi'roon, sup- 
 posini; the i)all to hv place*! in the middle of tlic case, that is, 
 to he not ncarcM* to one side than the other. When the \veij,'ht8 
 are approached to tlie hiills, their centres are S.S.") inches from 
 the middle line id" tlit? ejise ; hnt, thronjifh inadvertence, the 
 distance, from eacdi other, of the rods whi(di snpport these 
 weif^hts, was made ecpial to the distaiu^e of the centres of the 
 halls from each other, whereas it ou^ht to have heen somewhat 
 greater. In conse(juence of this, tin; centres of the wei<;hts are 
 not exactly opposite to those of the halls, when they are ap- 
 proached together; and the effect of the weights, in drawing 
 the arm aside, is less than it wonid otherwise have heen, in the 
 
 triplicate ratio of ' '' ' to the chord of the an<:le whose sine is 
 
 u t,r .)o.M,) 
 
 .T;r-'!r' or ill the trii)li( ate ratio of the cosine (d' i this angle to 
 ob.bo 
 
 the radius, or in the ratio of .977!) to 1.* 
 
 diosc 
 kI to 
 sight 
 
 \'ating 
 jhan 
 
 like a 
 
 se the 
 
 Uivies 
 
 with 
 
 . , „„ are of A" ^0.6.") 
 
 N*. And the force required to draw 
 
 the arm through ) .scale divmoti with period N" 
 
 36.65 1 
 
 = ^^^-36 65 ^39.14^^ 
 
 * [lA!t W be tile position of 
 the ''weight " of mans W, B 
 the position it was intend- 
 ed that it should hate, and m 
 that of the "ball" of mass m. 
 The distance mB, or WA, is 
 8.S5 inches, and OVV and Om 
 36.65 inches. Call WA and 
 Wm a and b resjiectively, and 
 G the gravitation constant. 
 Then it was intended that 
 tJie attraction to move the arm 
 
 should be ; — , but it is 
 
 //* 
 
 GWm a 
 
 was intended in the ratio of 
 
 36.65 
 
 39.14- '"^ "" H18N» 1 
 
 B W. 
 
 a' 
 
 ; and so is less than 
 
 a 
 
 6 
 
 ,3 to 1, or of Oos^ ^ to 1.] 
 
 Fig. d 
 

 !'' 'f\ 
 
 il!!i 
 
 V 
 
 |i-; 
 
 i| , 
 
 ; H 
 
 M I:M(H lis ON 
 
 Viiich of tlio uoijjlits woiijlis 'l,A:V.),(H)i) fjniins, and tlioroforo is 
 0(jiihI ill \v('i,i,Hit to 10.(14 splicricul l'i»ot of wiitor ;* jiiid tlu-nifore 
 its iitti'iKiinii on a piiiticlc pluccd at, tlit' ('ciitroof tlio ball, is to 
 tlie attraction of a spliorical l'(»ot of watcM- on an (Mjiial jiarticlo 
 
 piacod on its siirfa<'i% as 10.04 x .I'iT'.t X ( , ,. ) to 1. Tho 
 
 mean dianiotcr of tiic oartli is 41,S(K»,(MMI IVt't ;f and tliorefore, 
 it the mean density of the earth is to that of waiter as 1) to one, 
 the attraction of the leaden weii^hton the hall will he to that of 
 
 tlu! earth thereon, as 1().«>4 x.Oi^Dx 
 1 • 8,;;}!),(K)(U).t 
 
 \S.S,»/ 
 
 lo 41,80(),0(K) I) : 
 
 * [ T/iii/ is, t's (f/Kiil to tlic weifiht ofn xjifieir of water which ran he iiiHcrihed in 
 a I'lihc whose nil nine in 10.64 en. ft., or n^e ean erpretm the rolinne of the sphere 
 />!/ the unni/ter 10.04, when the nnit of rolnnie is that of a sphere of \ foot in. 
 
 diameter, that is, of — en. ft The radins of a spherical f<H>t of water is, aecf>rd- 
 
 inf/lt/, 6 inches. (\irendi.sh ecidtntli/ nses Kirwan's estimate (fiT^W.'^Tt f/rainH 
 to the CH. in. of wafer. 
 
 The ensnin;/ ca/cnfation can he stated thns: ('<dl o' and d' the densities of 
 water and (f the earth respect iceh/, m the mass ff the hall, and (» the r/ra cita- 
 tion constant. The coin me of the earth in spherical nnifs is (41 800 000)'. and 
 its radius Qx 41 800 000 inches. 
 
 Gx lOM xdxm 
 
 Aft met ion of iceifiht on hall at 8.85 inches (8.8r))- 
 
 Attraction of earth on ball 
 
 X 9779 
 
 Gx(41 80(MM)0)'xr/'x m. 
 (6x41800 000)' 
 
 .9779 x 10.64 xf-^^_Y 
 \H.8.v 
 
 But we have already fov nd {par/e 89) 
 Forc^ required to draw the arm fhrout/h 1 div. 
 
 Weight of hall 
 
 Diridinfi equation (1) by (2) we have 
 Attraction tf Wtif/ht on hall 
 
 41 800 000 ~ 
 a 
 
 
 1 
 
 '8 7;jy 000 D 
 
 . . . . 
 
 (1) 
 
 1 
 
 818N« • • 
 
 . . . 
 
 (3) 
 
 818 N2 
 
 N2 
 
 
 8 739 GOOD ~ 
 
 " 10 6831) 
 
 
 Force required to draw the arm throuf/h 1 die. 
 
 = no. of die. throvgh which th^ arm i,i dnurn~li div.] 
 
 f III strictii(!Ks, we oiigiit, inslcad of tlie nieiiti diameter of the earth, to 
 take the (liiiineter of that sphen; whose aitiiiction is equal to the force of 
 gravity in this climnte; but the difference is not worth regarding. 
 
 X [JIutton has pointed out (54) that this numl)er should he 8,740,000; but it 
 will not make any appreciable change in the value of D.] 
 
 90 
 
 .')li 
 
Til K LAWS (>!• (i ItA V IT ATION 
 
 «)' 
 
 0) 
 
 (2) 
 
 1, to 
 
 be of 
 
 \ut it 
 
 It is slunvii, tlicn'foro. that tlic force uliich miiHt Ix; applied 
 to oju'li ball, in order to dniw tjie arm oiu division out of its 
 
 natural position, is ..^ of the weiijlit of the hall ; and. if the 
 
 mean density of tlie earth is to that of water as I) to I, I ho at- 
 
 tra(!tion of the wjMi;ht on the hall is — ^; MMi> u **' ''"' ^V('i«^'lit (»f 
 
 that hall ; and therefore the attraction will he ahle to draw the 
 
 arm out of its jiatnral position by ^, ' ' ..or,- , ^ divis- 
 
 ions ; and therefore, if on movin<( the weii^hts from tlm mid- 
 way to a near position the arm is found to move h divisions, or 
 if it moves 'Z B divisions on nn)ving the wei;,dits Irom one near 
 position to tho other, it follows that the density of the earth. 
 
 We must now consider the corrections! which Tuust be ap- 
 plied to this result ; Hrst, for the ert'ect which the resistance of 
 the arm to motion has on the time of the vibration : t>d, for the 
 attraction of the weights on the arm : )M, for their atirac^tion 
 on the farther ball : 4th, for the attraction of the copper rods 
 on the balls and arm : r)th, for the attracttion of the case on the 
 bulls and arm : and 0th, for the alteration of the attraction of 
 the weights on the balls, according to the position of the arm, 
 and the effect which that has on the time of vibration. None 
 of these corrections, indeed, except the last, are of much sig- 
 nification, but they ought not entirely to be neglected. 
 
 As to the first, it must be co!isidered, that during the vibra- 
 tions of the arm and balls, part of the force is s])ent in acceler- 
 ating the arm ; and therefore, in order to find the force re- 
 quired to draw them out of their natural position, we must 
 find the proportion which the forces spent in accelerating the 
 arm and balls bear to each other. 
 
 Let EDC«/6' (Fig. 4) be the arm. B and <'j the balls. C.s- 
 the suspending wire. The arm consists of 4 parts ; first, a deal 
 rod l)cd, 73.3 inches long ; :^d, the silver wire \)Cd, weighing ITO 
 grains ; 3d, the end pieces DE and ed, to which the ivory 
 
 * [This nvmber should be 10,085. See lust note.] 
 
 f [For a discussion of these corrections, similar to that of Cavendish, but 
 with modern mathematical treatment, see Keirh (07).] 
 
 91 
 
mi: Mollis ON 
 
 : a^^l 
 
 Wm 
 
 I^B' ' 
 
 
 E 
 
 
 c 
 
 A 
 
 D 
 
 
 a 
 
 
 
 
 \ 
 
 B 
 
 
 
 I 
 
 voniior ih fastoiiod, oiicli of wliinh wei^jlis 45 p;rjiiiiH ; luul 4tli, 
 Homo bnisH work iU', at tiio et'utro. The doiil rod, whoii dry, 
 w('i«^lis )l'.\'i() grains, but wlion very diimp, sis it ooiimioidy wais 
 during tiie exporirneiits, woiglis '^400 ; tlio trjuis\orse seotioii is 
 
 of tlio slisipi! roproscMited in Fig. 5 ; the tlii(^k- 
 noss liA, and the dimensions of tlie ])art 
 1)1W, Itt'ing till' same in all j)arts ; but the 
 breadth \M/ diminishes gradually, from the 
 middle to the ends. The area of this see- 
 tion is .133 of a, square inch at the middle, and .140 at the 
 end; and therefore, if any point x (Fig. 4) is taken in cd, 
 
 and — , IS called .r, this rod weighs --— per inch at 
 
 the muUUe; ^^.,^-^, at the end, mul j^x -;^^^ = 
 at :r; and therefore, as the weight of the wire is 
 
 Fig. 5 
 
 7:j.;j 
 
 170 
 73.3 
 
 per inch, the deal rod and wire together may be con- 
 
 .- , , , . ,, , 341)0-1848 a; . . 
 sidered as a rod whose weight at x= ~,y^n P^^' i»ch. 
 
 But the force required to accelerate any quantity of matter 
 placed at x, is proportional to x^ ; that is, it is to the force re- 
 quired to accelerate the same quantity of matter placed at d as 
 a;' to 1 ; and therefore, if cd is called /, and x is supposed to 
 flow, the fluxion of the force required to accelerate the deal 
 
 92 
 
 fili; 
 
 ■liiUii 
 
Til K l,A\VS (M- (IKA V I TAT ION 
 
 -|C 
 
 at 
 
 X 
 
 ire 18 
 
 con- 
 
 itter 
 
 re- 
 
 \d us 
 
 to 
 
 Ideal 
 
 rod iUid wire is proportioiiiil to 
 
 
 tlie IliK'tit 
 
 of wliicli, iit'iicratt'il wliil«' ./• flows from r to d. 
 
 I 
 
 r.VVM) [.^4S\ 
 
 {:)0 
 
 so tliiit the force required to luieelenite eucli lialf of the deal 
 rod and wire, is the sjiiiie ais is retjuired to u(!eelerute ;$.'»(► i^nuiihs 
 phieed at d. 
 
 'I'he resistauee to tn«)tion of each of the pieces di\ is e(|iial to 
 tliat of 4S j^M-ains phiced at ^/ ; as the distance of their centres of 
 
 jrravitv from (' is ;{S inche.- 
 
 Th(( resistance! of tln^ l»rass work 
 
 at the centre nniy l)e disref!;ardc(| ; ant! tlierefore the whoh' Uwvv 
 refpiired to acceh;rate the arm, is tiie same as that re(|nired to 
 accelerate :)tlS grains phiced at cAvh of the points I) and d. 
 
 Kach of tlie balls weighs 11. "^O^ ;,n'ains, and they art^ phuM'd 
 at the same distance! from the (sentre as I) ami d; ami there- 
 fore, the force required to accelerate the halls ami arm to- 
 f^cther, is the same as if each hall vveijifhed II, <!»;(>, and the arm 
 had no weight; and therefore, supposing:: the time; of a vibra- 
 tion to be given, the force required to draw the arm aside, is 
 greater than if the arm had no weight, in the i)roportion of 
 ii,(;(;o to \\,WZ, or of I.o;ja3 to 1. 
 
 To find the attraction of the w 'ights on the arm, through d 
 draw the vertical plane dwb perpendicular to D^/, and let tn 
 be the centre of the weight, which, though not accurately in 
 this plane, nuiy, without sensible error, be considered as placed 
 therein, and let b be the centre of the ball; then wh is hori- 
 zontal and =8.85, and ^/6 is vertical and =5. 5; let wd=a, v)b 
 
 dx 
 z=.b, and let ~, or \—x, = z; then the attraction of the weight 
 
 on a particle of matter at x, in the direction bw, is to its at- 
 traction on the same particle placed at h:\b^ : {a^-^-z^ry, or is 
 
 proportional to ;,, and the force of that attraction to 
 
 h''x{l—z) 
 move the arm, is proportional to ^^ -, and the weight of 
 
 {(t^-^-z'TY 
 
 the deal rod and wire at the point x, was before said to be 
 
 3490-1848 .r 1G42+I848z . , , .. * -^ ^ 
 
 — — = -~- per inch; and therefore, if dx 
 
 7o.u io.o 
 
 flows, the fluxion of the power to move the arm 
 
 93 
 
ni^ 
 
 on 
 
 r He 
 
 I 'i': 
 
 
 gurg^j 
 
 1 
 
 
 
 
 \ 
 
 1 
 
 * 
 
 ( 
 1 
 
 
 i 
 
 liJ 
 
 M KM OIKS ON 
 
 = hx -^ ~ X ~=::ZX (821+024 z) X ^^ 1 
 
 /rzxh'il-hU):\zV^y^j 
 
 h^zx(H'n■^^^)'^z-9Uz') 
 
 924//2x('j' + ^') 
 
 (a'-^-rz'V^ 
 
 a' 
 
 ; wliicli, as -^^ = .08, 
 
 {a-'-\-Pzy 
 J>'-'i X (80') + 1 (>;{;<;) 0:>4 h'z_ 
 
 805/// io:w>' low/ 024 /»' 
 
 The fluent of this 
 
 Fa 
 
 P 
 
 log 
 
 a 
 
 and the force with wliioh the attraction of the weight, on the 
 nearest half of the deal rod and wire, tends to move the arm, 
 is i)roportional to this fluent generated while z flows from 
 to 1, that is, to 128 grains. 
 
 The force with wliich the attraction of the weight on the end 
 
 piece de tends to move the arm, is proportional to 47 x -3 [npproxi- 
 
 mairlt/], or 20 grains; and therefore the whole power of the 
 weight to move the arm, by means of its attraction on the near- 
 est part thereof, is equal to its attraction on 157 grains placed 
 
 157 
 at^, which i^' ' . , or .0130 of its attraction on the ball.* 
 
 It must be observed, that tiie effect of the attraction of the 
 weight on the whole arm is rather less than this, as its attrac- 
 tion on tlie farther half draws it the contrary way; but, as the 
 attractiou on this is snuiU, in comparison of its attraction on 
 the nearer half, it may be disregarded. 
 
 The attraction of the weight on the furthest ball, in the 
 direction biv, is to its attraction on the nearest ball :: wb^ : tvh'j; 
 ::.0017;1; and therefore the effect of the attractioi. of the 
 weigiit on both balls, is to that of its attraction on the nearest 
 ball::. 9083 : 1. 
 
 * [A few minor misprints in the last two paragraphs in the oi'iginal paper 
 have been, corrected. A recalculation seems to give 142.5 instead of 128, and 
 28 instead of 29, grains; (his would change the valve of J} hy 1 jmrt in 1000.] 
 
 f [^This is erroneously printed in the oi'iginal as icd^ : icW.] 
 
 94 
 
TIIK LAWS OK (JRAVITATION 
 
 the 
 
 the 
 
 the 
 Lrest 
 
 To find tlie attraction of the copper rod on the nourost hall, 
 let b and w (Fig. (J) he the centres of the hall and weight, and 
 ea the perpend icnlar part of the copper rod, which consists of 
 two parts, ad and dc. ad weighs :^2,0(/() grains, and is l(j inches 
 long, and is nearly bisected by w. de weighs 41, (MM), and is 4(1 
 inches long, wh is 8.85 inches, and is perpendicular to ew. 
 Now, the attraction of a line eWy 
 of uniform thickness, on b, in the 
 direction bw, is to that of the same 
 quantity of matter placed at w :: bw 
 '.eb; and therefore the attraction 
 of the part da equals that of 
 
 r, , or 1G,-J00, placed at w ; 
 
 db 
 
 and the jittraction of de equals 
 
 that of 41,000 x''"'xv^- 41,000 
 
 I'd be 
 
 X — rX, ,> or 2500, placed nt the 
 ed bd 
 
 same point ; so that the attraction 
 
 of the perpendicular part of the w\ \h 
 
 copper rod on b, is to that of the 
 
 weight thereon, as 18,800 : 2,439,- 
 
 000, or as .00771 to 1. As for the 
 
 attraction of the inclined part of 
 
 the rod and wooden bar, marked ng. c 
 
 Pr and rr in Fig. 1, it may sa^'ely 
 
 be neglected, and so may the attraction of the whole rod on the 
 
 arm and farthest ball ; and therefore the attraction of the weight 
 
 and copper rod, on the arm and both halls together, exceeds the 
 
 attraction of the weight on the nearest ball, in the proportion 
 
 of . 9983 -}-.0139 + . 0077 to one, or of 1.0199 to 1. 
 
 The next thing to be considered, is the attraction of the- ma- 
 hogany case. Now it is evident, that when the arm stands at 
 the middle division, the attractions of the opposite sides of the 
 case balance each other, and liave no power to draw the arm 
 either way. When the arm is removed from this division, it is 
 attracted a little towards the nearest side, so that the force re- 
 quired to draw the arm aside is rather less than it would other- 
 wise be ; uut yet, if this force is proportional to the distance of 
 the arm from the middle division, it makes no error in the re- 
 
 95 
 
\l 
 
 il 
 , '1? 
 
 n 
 
 "I 
 
 1 
 
 'HI k 
 
 MEMUIUS ON 
 
 suit ; for, thougli tlio attraction will draw tho arm aside more 
 than it would otherwise do, yet, as the accelerating force by 
 which the arm is made to vibrate is diminished in the same 
 proportion, the square of the time of a vibration will be in- 
 creased ill the same })roportion as the space by which the arm 
 is drawn aside, and tiiereforo the result will be the same as if 
 the case exerted no attraction ; but, if the attraction of the 
 case is not proportional to tiie distance of tiie arm from the 
 middle point, the ratio in which the accelerating force is di- 
 minished is different in diiferent parts of the vibration, and 
 the square of the time of a vibration will not be increased in 
 the same proportion as the quantity by which the arm is drawn 
 aside, and therefore the result will be altered thereby. 
 
 On computation, 1 find that the force by which the attrac- 
 tion draws the arm from the centre is far from being propor- 
 tional to the distance, but the whole force is so small as not to 
 be worth regarding; for, in no position of t!ie arm does the 
 attraction of the case on the balls exceed that of ^th of r« spheric 
 inch of water, placed at the distance of one inch from the cen- 
 tre of the balls ; and the attraction of the leaden weight equals 
 that of 10. G spheric feet of water placed at 8.85 inches, or of 
 234 spheric inches placed at 1 inch distance ; so that the at- 
 traction of the case on the balls can in no position of the arm 
 exceed y^\^ of that of the weight. The computation is given 
 in the Appendix. 
 
 It has been shown, therefore, that the force required to draw 
 the arm aside one division, is greater than it would be if the 
 arm had no weight, in the ratio of 1.0353 to 1, and therefore 
 
 — ' ' ^ ..j of the weight of the ball ; and moreover, the attraction 
 
 of the weight and copper rod on the arm and both balls to- 
 gether, exceeds the attraction of the weight on the nearest ball. 
 
 in the ratio of 1.0199 to 1, and therefore = 
 
 1.0199 
 
 8,739,000D 
 
 of the 
 
 818N' 
 
 weight of the ball ; consequently D is really equal to - 
 
 1.0199 N* N** l.Odo.J 
 
 X '.. ' .. , or ^ ,. ^, instead of „ , as by the for- 
 
 , as by the for- 
 
 8,739,()(K)B' 10,844B*' 10,083B 
 
 mer computation. It remains to be considered how much this 
 is affected by the position of the arm. 
 
 * [T/iis should be 10,846 ; see note on pp. 90 and 91.] 
 
 96 
 
THE LAWS UK (iUAVlTATlON 
 
 Suppose the weights to be approached to tlio balls ; let W 
 (Fig. 7) be the centre of one of the weights; let M be the cen- 
 tre of the nearest ball at its mean position, as when the arm is 
 at 20 divisions ; let B be the point which it actually rests at ; 
 and let A be the point which it would rest at, if the weight was 
 removed ; consequently, AI5 is the spa(!e by which it is drawn 
 aside by means of the attraction ; and let M/3 be tiie space by 
 which it would be drawn aside, if the attraction on it was 
 
 Us to- 
 t ball, 
 
 .0353 
 |e for- 
 
 h this 
 
 W 
 
 B 
 
 M 
 
 t 
 
 to 
 
 Pig. 
 
 the same as whcr. it is at M. But the attraction at B is greater 
 than at M, in the proportion of \VM'^ : WB"; and therefore, 
 
 AB = M/3 X Y^ j^,= M/3 X ( 1 + ^^h very nearly. 
 
 Let now the weights be moved to the contrary near position, 
 and let w be now the centre of the nearest weight, and h the point 
 
 of rest of the centre of the ball ; then Ai = M/?xM-f w.,r)> 
 
 and Bi = M/3x(2 + j^,^+ ,...r l = 2M/3xM + jr|..;^ so that the 
 
 whole motion BZ» is greater than it would be if the attraction 
 on the ball was the same in all places as it is at M, in the ratio 
 
 of l + TfTu to one ; and, therefore, does not depend sensibly 
 
 on the place of the arm, in either position of the weights, but 
 only on the quantity of its motion, by moving them. 
 
 This variation in the attraction of the weight, affects also 
 the time of vibration ; for, suppose the weights to be ap- 
 proached to the balls, let W be the centre of the nearest 
 weight ; let B and A represent the same things as before ; and 
 let X be the centre of the ball, at any point of its vibration ; let 
 AB represent the force with which the ball, when placed at B, 
 is drawn towards A by the stiffness of the wire ; then, as B is 
 the point of rest, the attraction of the weight thereon will also 
 equal AB ; and, when the ball is at ^\ the force with which it 
 is drawn towards A, by tlie stiffness of the wire, = Aa;, and that 
 with which it is drawn in the contrary direction, by the attrac- 
 
 WB' 
 tion,=ABx 
 
 W;c^ 
 
 .^ , so that the actual force by which it is drawn 
 97 
 
towards A = Aa:— 
 'ZBxxAB 
 
 MK.MUIUS UN 
 ABxWlV . ,, . „ ,„ /, 2B2:\ ,, 
 
 VVB 
 
 , very neurly. So that the actual force with which 
 
 the ball is drawn towards the middle point of the vibration, is 
 less than it would be if the weights were removed, in the ratio 
 
 of l — TiTTT to one, and the square of the time of a vibration is 
 
 VVB 
 
 3AB 
 
 increased in the ratio of 1 to 1— Trrrr ; which differs very little 
 
 from that of l + Trwr to 1, which is the ratio in which the 
 
 motion of the arm, by moving the weights from one near posi- 
 tion to the other, is increased. 
 
 The motion of the ball answering to one division of the arm 
 
 30.05* 
 
 ' ' ' — ; and, if MBf is the motion of the ball answering 
 
 MB 30.05^/ d 
 
 to d divisions on the arm, 
 
 and 
 
 WM :eOx 38.3x8.85 185' 
 
 therefore, the time of vibration, and motion of the arm, must 
 
 be corrected as follows : 
 
 If the time of vibration is determined by an experiment in 
 
 which the weights are in the near position, and the motion of 
 
 the arm, by moving the weights from the near to the midway 
 
 position, is d divisions, the observed time must be diminished 
 
 2d 
 in the subduplicate ratio of 1 — 7777 ^^ 1> ^^^^ ^s> i^ ^^^ ratio 
 
 185 
 
 d 
 
 of 1 — TTjr to 1; but, when it is determined by an experiment 
 
 in which the weights are in the midway position, no correction 
 must be applied. 
 
 To correct the motion of the arm caused by moving the 
 weights from a near to tlie midway position, or the reverse, 
 observe how much the position of the arm differs from 20 
 divisions, when the weights are in the near position ; let this 
 be n divisions, then, if the arm at that time is on the same side 
 of the division of 30 as the weight, the observed motion must be 
 
 * [This number, 36.65, here, and again in tlie next line, is eiToneously 
 printed in Cavendis/i's memoir as 36.35.] 
 f [In tlie original this is erroneously pnnted as wB.] 
 
 98 
 
which 
 
 ion, is 
 3 ratio 
 
 tioii is 
 
 f little 
 
 3h the 
 ,r posi- 
 
 lie arm 
 waring 
 
 - ; and 
 1, must 
 
 nent in 
 tion of 
 id way 
 inishcd 
 
 |e ratio 
 
 iriment 
 reciion 
 
 [ig the 
 averse, 
 )m 20 
 let this 
 kie side 
 lust be 
 
 \neou8ly 
 
 T H K I. A W S i)V < ; li A V I I' A 'V \ ( ) N 
 
 "Zn 
 diminished by tlie part of tlie wiiole ; but, otherwise, it 
 
 must be as mucli increased. 
 
 If the weights are moved from one near position to the otlier, 
 
 and the motion of tlie arm is 'Zd divisions, the observed motion 
 
 Z<1 
 must be diminished by the -— part of the whole. 
 
 loo 
 
 If the weights are moved from one near position to the other, 
 
 and the time of vibration is determined while the weights are 
 
 in one of those positions, tiicre is no need of correcting either 
 
 the motion of the arm, or the time of vibration.* 
 
 Conclusion 
 
 The following table contains the result of the experiments 
 
 Expor. 
 
 Mot. weig..t 
 
 Mot. arm 
 
 Do. corr. 
 
 Time vib. 
 
 Do. corr. 
 
 Density 
 
 M 
 
 m. to + 
 
 14.82 
 
 13.42 
 
 
 
 
 5.5 
 
 + to m. 
 
 14.1 
 
 13.17 
 
 14' 55' 
 
 — 
 
 5. SI 
 
 ^1 
 
 m. to + 
 
 15.87 
 
 14.69 
 
 — 
 
 — 
 
 4.88 
 
 + to m. 
 
 15.45 
 
 14.14 
 
 14 42 
 
 — 
 
 5.07 
 
 «l 
 
 + to m. 
 
 15.22 
 
 13.56 
 
 14 39 
 
 — 
 
 5.26 
 
 m. to 4- 
 
 14.5 
 
 13.28 
 
 14 54 
 
 — 
 
 5.55 
 
 ^1 
 
 111. to + 
 
 3.1 
 
 2.95 
 
 
 6' 54' 
 
 5.36 
 
 + to - 
 
 6.18 
 
 — 
 
 7 1 
 
 — 
 
 5.29 
 
 - to + 
 
 5.92 
 
 — 
 
 7 3 
 
 — 
 
 5.58 
 
 H 
 
 + to - 
 
 5.9 
 
 — 
 
 7 5 
 
 — 
 
 5.65 
 
 -. to + 
 m. to — 
 
 5.98 
 3.03 
 
 2.9 ^ 
 
 7 5 
 
 — 
 
 5.57 
 
 e] 
 
 
 
 5.53 
 
 - to + 
 
 5.9 
 
 5.71 
 
 7 4 
 by 
 
 
 5.62 
 
 TJ 
 
 111. to — 
 
 - to + 
 
 3.15 
 6.1 
 
 3.03 
 5.9 ■ 
 
 6 57 
 
 5.29 
 5.44 
 
 8J 
 
 m. to — 
 
 3.13 
 
 3.00 
 
 mean 
 
 
 5.34 
 
 - to + 
 -1- to - 
 
 5.72 
 6.32 
 
 5.54^ 
 
 6 58 
 
 
 5.79 
 
 9 
 
 — 
 
 5.1 
 
 10 
 
 + to - 
 
 6.15 
 
 — 
 
 6 59 
 
 — 
 
 5.27 
 
 11 
 
 -f to - 
 
 6.07 
 
 — 
 
 7 1 
 
 — 
 
 5.39 
 
 12 
 
 - to + 
 
 6.09 
 
 — 
 
 7 3 
 
 — 
 
 5.42 
 
 X3| 
 
 - to + 
 
 6.12 
 
 — 
 
 7 6 
 
 — 
 
 5.47 
 
 + to - 
 
 5.97 
 
 — 
 
 7 7 
 
 — 
 
 5.63 
 
 u] 
 
 - to -f 
 
 6.27 
 
 — 
 
 7 6 
 
 — 
 
 5.34 
 
 + to - 
 
 6.13 
 
 — 
 
 7 6 
 
 — 
 
 5.46 
 
 15 
 
 - to + 
 
 6.34 
 
 — 
 
 7 7 
 
 — 
 
 5.3 
 
 16 
 
 - to + 
 
 6.1 
 
 — 
 
 7 16 
 
 — 
 
 5.75 
 
 n| 
 
 - to + 
 
 5.78 
 
 — 
 
 7 2 
 
 — 
 
 5.68 
 
 + to - 
 
 5.64 
 
 — 
 
 7 3 
 
 — 
 
 5.85 
 
 * [ The corrections neutralize each other, since they are the same for N* and 
 B, whose ratio enters into the expression for D.] 
 
 99 
 
MK MO I IIS ON 
 
 From til is tablo it jippeurs, tluit tiiougli tlie experiments 
 agree pretty well together, yet the difference between them, 
 both in the f|iijintity of motion of the arm and in the time of 
 vibration, is greater than ean proceed merely from the error of 
 observation. As to the difference in tiie motion of the arm, it 
 nniy very well be acconnted for, from the current of air pro- 
 duced by the difference of temperature ; but, whether tiiis can 
 account for the difference in the time of vibration, is doubtful. 
 If the current of air was regular, and of the same swiftness in 
 all parts of the vibration of the ball, 1 think it could not; but, 
 as there will most likely be nnudi irregularity in the current, it 
 may very likely be suf!icient to iiccount for the difference. 
 
 hy a mean of the experiinents made with the wire first used, 
 the density of the earth comes out 5.48* times greater than 
 that of water ; and by a mean of those made with the latter 
 wire, it comes out the same ; and the extreme difference of the 
 results of the ^3 observations made with this wire, is only .75 ; 
 so that the extreme results do not differ from the mean by more 
 than .38, or ^ of the whole, and therefore the density should 
 seem to be determined hereby, to great exactness. It, indeed, 
 may be objected, that as the result appears to be influenced by 
 the current of air, or some other cause, the laws of which we 
 are not well acquainted with, this cause may perhaps act al- 
 ways, or commonly, in the same direction, and thereby make a 
 considerable error in the result. But yet, as the experiments 
 were tried in various weathers, and with considerable variety 
 in the difference of temperature of the weights and air, and 
 with the arm resting at different distances from the sides of 
 the case, it seems very unlikely that this cause should act so 
 uniformly in the same way, as to make the error of the mean 
 result nearly equal to the difference between this and the ex- 
 
 f 
 
 * [This should be 5.31. Had thethird number in tJiecolumn of densities been 
 5.88, instead f>/4.88, the average would have been as Cavendish gave it. But 
 Baily (79, p. 90) rectilculated tJte densities from Cavendish's data, and found 
 4.88 to be correct. Curiously enough Cavendish made the same error in deduc- 
 ing the mean result of the whole number of experiments. It should be 5.448, not 
 6 ^ {which icould be had by putting 5.S8 inj)lace r>/"4.88), withaprobable error 
 <j/'.033. The mean result of the last 23 observations is 5.48. The greatest dif- 
 ference of the single results from one another is .97 ; and tJie extreme result 
 differs from the mean of all by .57, or ^.^ of tlie whole. For an account of 
 Iluttons reflections on Cavendish's ''pretty and amusing little experiment " 
 see his paper (45 and 54) ; they are referred to in this volume on page 105.] 
 
 100 
 
 ...->Li Wi* .-'I'ltiilfclil :>■. '. .^^ 
 
es been 
 
 But 
 
 I found 
 
 \deduc- 
 
 18, not 
 
 error 
 
 [at dif- 
 
 result 
 
 mt of 
 
 mt,"' 
 
 |>5.] 
 
 THE LAWS OK CiKAVITATlUN 
 
 treme ; and, thoreforo, it sooma very unlikely that tlio density 
 of the earth sliould dilTor I'roin r>.48 by so luueii as ^^^ of tlio 
 whole.* 
 
 Another objection, perhaps, may be made to these experi- 
 ments, namely, that it is uncertain whether, in these small 
 distances, the force of gravity follows exactly the same law as 
 in greater distances. There is no reason, howt^ver, to tiiiniv 
 that any irreguhirity of this kind takes place, until the bodies 
 come within the action of what is called the attraction of co- 
 hesion, and which seems to extend only to v(n*y minute dis- 
 tances. With a view to see whether tlie result could be affected 
 by this attraction, I made the 0th, 10th, lltli, and 15th exjjeri- 
 ments, in which the balls were made to rest as close to the sides 
 of the case as they (M>uld ; but there is no difference to be de- 
 pended on, between the results under that circumstance, and 
 when the balls are placed in any other part of t!ie case. 
 
 According to the experiments made by Dr. Maskelyne, on 
 the attraction of the hill Schehallien, the density of the earth 
 is 4|^ times that of water ; which differs rather more from the 
 preceding determination than I should have expected. lUit I 
 forbear entering into any consideration of which determination 
 is most to be depended on, till I have examined more carefully 
 how much the preceding determination is affected by irregular- 
 ities whose quantity I cannot measure. 
 
 * [See note on page 100.] 
 101 
 
i 
 
 APPENDIX 
 
 ON TITH; attraction of the MAMOdANY (!ASE ON THE BALLS 
 
 The lirst tiling is, to find tlio iittraction of tlie rectangular 
 
 ;, piano ckiM (Fig 8) on tiie 
 
 point a, placed in tjje line 
 ac perpendicular to this 
 plane. 
 /^ Let ac = rt, ck = b, ch — 
 
 ic, and let -5-- — ^ = «t>', and 
 a ■\- X 
 
 Fig- 8 
 
 -s — - — ;; = v", tlien the at- 
 a + ;' 
 
 traction of the line bfi on 
 a, in the direction ab, = 
 
 , —. — - — , ; and therefore, if 
 t^ ab X afi' 
 
 cb flows, the fluxion of the 
 
 attraction of the plane on the point a, in the direction cb, = 
 bx ^ _ ~ ^''^^ _ ~ ^^'^ 
 
 __ - — the variable part of the fluent of which = — losr (v 
 
 4- Vi-\-vY, and therefore the whole attraction = lou: ( — '^ — -x 
 
 nb \ \ ac 
 
 vTs si ; "^o that the attraction of the plane, in the direction 
 
 cb, is found readily by logarithms, but I know no way of finding 
 its attraction in the direction ac, except by an infinite series.* 
 
 * [Playfair has given an expression in finite terms for this attraction on pp. 
 225-8 of his pajyr in the Tnins. Rov. Soc Edin., vol. 6, 1812. pp. 187-243, 
 
 103 
 
M KM OIKS ON TIIK LAWS OK <; K A V I T A T I (> N 
 
 Tlio two most ronvcnioiit Horios I know, are tlio followini^ : 
 Fiirt St'i'ios. Let - — tt, unci lot A=un' wlioso tau^. i.s n-, li 
 
 = A— TT, (^' = H-|--— , D— (' — . , t'te. 'riuMi llm attniction ill tlio 
 
 direction ar 
 
 
 1 
 
 For the second series, let A = iirc'. whose tanj;. = , IJ = A — , 
 
 «■ IT 
 
 C=:lH.i-3» I) = C-;r-5, etc. Tlicii tlie attraction = uic.!K)" 
 
 IJtt 
 
 On- 
 
 It must 1)0 observed, that the first scries fails wiien tt is 
 greater tlian unity, and the second, when it is h;ss ; hut, il' h is 
 taken equal to the least of the two lines rk and vh, there is no 
 case in which one or the other of them may not be used con- 
 veniently. 
 
 By tlie iielp of these series, I computed the following table : 
 
 |log(v 
 ak 
 
 X 
 
 action 
 
 jmling 
 'ies.* 
 
 on pp. 
 37-243, 
 
 
 .lOfi'i 
 
 .3714 
 
 .6145 
 
 .6248 
 
 .7071 
 
 .7808 
 
 .8676 
 
 .9285 
 
 9H15 
 
 1. 
 
 .1962 
 
 .00001 
 
 
 
 
 
 
 .3714 
 
 .00039 
 
 .00148 
 
 
 
 
 
 
 
 
 
 .5145 
 
 .00074 
 
 .00277 
 
 .00521 
 
 
 
 
 
 
 
 
 .6248 
 
 .00110 
 
 .00406 
 
 .00778 
 
 .01183 
 
 
 
 
 
 
 
 .7071 
 
 .00140 
 
 .00522 
 
 01008 
 
 .01525 
 
 .02002 
 
 
 
 
 
 
 .7808 
 
 .00171 
 
 .00637 
 
 .01245 
 
 .01896.02405 
 
 03247 
 
 
 
 
 
 .8575 
 
 00207 
 
 00772 
 
 .01522 
 
 02339.03116 
 
 .03964 
 
 .05057 
 
 
 
 
 .9285 
 
 .00244 
 
 .00910 
 
 .01810 
 
 .02807.03778.04867 
 
 06319 
 
 .08119 
 
 
 
 .9815 
 
 .00271 
 
 .01019 
 
 .02084 
 
 .03193 
 
 .04368 .05639 
 
 07478 
 
 09931 
 
 .12849 
 
 
 1. 
 
 .00284 
 
 .01054 
 
 .02135 
 
 .03347 
 
 .04560.05975 
 
 .07978 
 
 .10789 
 
 .14632 
 
 .19612 
 
 ck 
 
 Find ill this table, with the argument ~, at top, and the ar- 
 
 (tic 
 
 gument -^ in the left-hand column, the corresponding logarithm ; 
 
 entitled " Of the Solids of Gredtent Attraction, or those which, among all the 
 Solids that have certain Properties, attract with the greatest Force in a given 
 Direction.'^'] 
 * [In the la fit term of the series Ute cotfjicicni D was omitted in the original.] 
 
 103 
 
 i 
 
M KM OIKS ON 
 
 .JKi' 
 
 m 
 
 1' w^^ 
 
 tlioji JuM toorotlior tliis lojjjiritlini, tlio lofjiiritlirii of , , suid tlio 
 
 rfj 
 
 nk' 
 
 i<)*;uritlim of . ; tlic sum is tlio loguritlirn of the uttniction. 
 
 To ('onipiite from licm'.u Llio sittnictiou of tlu; ciiso ofi tlio 
 ^ E ball, lot thobox l)(!BA(Fi^'. 
 
 1), in which th(! ball pluyK, bo 
 tiividod into two piirts, by a 
 vortical sootioii, porpendio- 
 ular to tho length of tho 
 caHo, and passinji^ through 
 the centre of the ball ; and, 
 in Kig. '.), Kit the parallel- 
 opij)e(l \\\\)Knh<lc bo one of 
 these [)arts, ABDK being tho 
 above-men tionod vertical sec- 
 tion ; let X bo tho centre of 
 tho ball, and draw t!ie par- 
 allelogram i^npnilx parallel to 
 IVvr/I), and xfirp parallel to 
 fiWbn, and bisect /5^ in c. 
 Now, the dimensions of the 
 box, on tho inside, arc Bd=:1.75 ; Hi) = ;J.(5 ; H/3=1.7r); and 
 /5A — 5 ; whem^c I find that, if xc and /3./' are taken as \\\ tho two 
 upper linos of tho following table, the attractions of the differ- 
 ent parts are as set down below. 
 
 B 
 
 1 \ 
 
 
 n 
 
 
 
 \ 
 
 1 A 
 
 X 
 
 c 
 
 i\ 
 
 I 
 
 1 
 
 
 K 
 
 p 
 
 
 \ 
 
 1 n 
 
 
 
 1 
 1 
 
 \ 
 
 \ 
 
 \ 
 
 o\^ 
 
 
 d' 
 
 m 
 
 Fig. 9 
 
 Excess of attract ion 
 
 Slim of these 
 Excess of attraction 
 
 xc 
 
 ftx 
 
 of Ddi'd above Bhrg 
 
 mdr p above nb r p 
 
 mesp al)ove nasp 
 
 of BA;j/3 al)Ove DdmS 
 Aanfi above EeviS 
 
 Whole attraction of the inside surface of the 
 half box. 
 
 .75 
 1.05 
 .2374 
 .2374 
 .3705 
 
 .8453 
 .5007 
 .4677 
 
 .1231 
 
 .5 
 1.3 
 .1614 
 .1614 
 .2516 
 
 "5744 
 .3271 
 .3079 
 
 .0606 
 
 .25 
 1.55 
 .0813 
 .0813 
 
 .2897 
 .1606 
 .1525 
 
 .0234 
 
 It appears, therefore, that tho attraction of tho box on x in- 
 creases faster than in proportion to the distance xc. 
 
 The specific gravity of tho wood used in this case is .61, and 
 its thickness is f of an inch ; and therefore, if the attraction 
 of the outside surface of the box was the same as that of the 
 
 104 
 
THK LAWS OK (iUAV 11 A T I ( ) N 
 
 inside, tho whole attniotioii of the hox on the ball, when rr 
 = .75, would he equal to 2 x.r>31 x.'il x I cnhic! inches, or .'H)\ 
 ephericj inches of water, placed at the distance of one inch from 
 the centre of the ball. In reality it vm\ never bo so ^reat as 
 this, as the attraction of the outside siirfacui is rather less than 
 that of the in-^ide ; ami, moreover, the <listan<H» of ./• from r can 
 never he quite so great as .To of an inch, as the greatest motion 
 of the arm is only l^ inch. 
 
 .25 
 .55 
 
 .0813 
 .0813 
 ,1271^ 
 
 :2897 
 ,1606 
 ■ 1 525 
 
 .0234 
 
 X in- 
 
 L and 
 
 Iction 
 If the 
 
 Much has been written concerning the Cavendish experi- 
 ment ; the following references may be consulted to advantage. 
 
 ( I ilbert (40), in 1 71M), translated the greater part of ( 'avendish's 
 paper into German for his Anmilcn, adding many explanatory 
 notes. A few years later Jirandes (4"^) gave a frc^sh mathemati- 
 cal analysis of the experiment, incduding the equations for the 
 time of swing of the torsion pendulum in the experiment pro- 
 posed by Muncke (see below). In 181."), the original j)aper of 
 Cavendish was translated entire into French by M. Chomprc 
 (50). 
 
 In 1821, Hutton (54) recalculated the results of the experi- 
 ment after Cavendish's own formuhe, ami found, as he thought, 
 a "copious list of errata, some of which are large or import- 
 ant." Tho mean of the first experiments so corrected is 5.19, 
 and of the other 23 is 5.43 ; the mean of these two means is 
 5.31, which Hutton takes as the correct result given by the 
 Cavendish experiment. Baily states, however (79, pp. 92-9()), 
 that Hutton himself had fallen into error, and that the com- 
 putations of Cavendish are correct except in the one detail re- 
 ferred to on page 100 of this volume. Haily gives a very care- 
 ful criticism of the experiment on pp. 88-91 of his memoir. He 
 remarks that *' Cavendish's object, in drawing up his memoir, 
 appears to have been more for the purpose of exhibiting a 
 specimen of what he considered to be an excellent method of 
 determining tills important inquiry, than of deducing a result 
 that should lay claim to the full confidence of the scientific 
 world." Baily points out that the time was not determined 
 with due accuracy ; that the experiments were not arranged in 
 groups, in order to eliminate the error arising from the march 
 of the resting point; and that the distance between tiie weight, 
 
 105 
 
MKMolltS ON 
 
 I 
 
 I 
 
 1111(1 tlic hull WHH jiHHumod «MniHtut)t. Wo hIiuII SCO liitor from 
 tlio u<;(;oiiiit.s of tlii> iiivoMti;;iit ions of JNtioli, Kiiily, ('oriiii itiul 
 hiiillc, imd hoyw how tho ornns in (.'iivculish's ox|)(U'iriioiit liiivo 
 hcM'M uvoiih'il. 
 
 MiiiKtko (<l|, vol. .*{, pp. 040-70) Ims ^ivoii iin jiooount of tlio 
 oxpcrinuMit iind an lulrninihlo critioisin of it, and cornparos the 
 result with that ohtaiiiiMi hy MaskclyiKt and Mutton. Il(! pro- 
 positi another nicthod of nsin^ the torsion halani^t^ to Iind tho 
 in(!an density of tlu^ earth ; he wonld Iind the tini(;of vihratioii 
 with th(^ masses lirst in the line of the ))allH and then in a 
 line at ri^lit anj^les to that direetion. 'I'lierc; would ho no de- 
 llection to ho !neaHiir«Ml. We Inivo seen ahove that Brandos 
 pive the theory of this experiment. Investigations of this 
 nature have heen made hy liei(di (81)), Kotvos (M.)'-i) and 
 Hraun (H)JJ); for accounts of whicdi see the latter part of this 
 volume. 
 
 A useful resume of Cavendish's paper was j^iven hy Sehmidt 
 ((14, vol. '4, pp. 4H1-7). lie formed anew equations from which 
 to derive the value of the density of tho earth, and found 5.52 
 usinj( Cavendish's data. 
 
 Anotluir mathematical investigation of the dynamical proh- 
 lem underlying the Caveiulish experiment was made hy Mena- 
 brea (71, 7'i and 713), in l.S4(>. It is a very elahorato analysis of 
 the whole prohlom. He examines the effect of the resistance 
 of the air on the time of vihration, and also shews how to find 
 the fuass of the earth supposing tliat it is composed of spheroidal 
 layers of variable density. In Baily's memoir (7!>) is another 
 elaborate analysis, hy Airy, of the matheunitical theory of the 
 investigation. It treats especially of Baily's modificatioii of 
 the Cavendish experiment (reproduced in Routh's Jliffid Dy- 
 namics 1882, pt. 1, pp. 350-304). 
 
 An elementary treatment of the problem involved is given by 
 Cosseiin (127), and from the formula he arrives at he derives 
 the value of the mean density of the earth as given by Caven- 
 dish's experiment, and gets 5.00. A similarly elementary treat- 
 ment by liabinet (132) gives 5.5. 
 
 An excellent account of Cavendish's work is given by Zanotti- 
 Bianco (148|) and by Poynting (185, pp. 40-8) ; in the latter is 
 to be found a diagram showing the closeness of Cavendish's 
 separate results to the mean. 
 
 106 
 
;n by 
 Irives 
 [von- 
 [•etit- 
 
 Tll K LAWS OK (JKAV I T A I' h > N 
 
 IIkNUY Cw TA'DlSIf, HOU of LfU'd ( 'llJlllt'S ( IjlViMulisIl uiid a 
 
 lu'plu'w of tlic third I)iikr of l>t'Voii«liir«'. wha horn ut Nico in 
 lT>n litiil (liotl ut Ijondoii in 1K|(». |[e stiidiod ut ('iiinl)rid<{(>, 
 and luuioniin;; poMsrHscd, l>y tho di'iith of an uncit', of >i hir^o 
 fortiint^ hu di'votcd his iifu unostcntutiously to privntr KvMcntilic 
 studies. Hcsid(s i\\v. inv(>sti.L,^ition on <^n'avitalion:il attraction 
 h(>r(> n>print(>(l, he is rrrnarkahh' for liis rcscandics in th«> fudd 
 of (duMnistry, and has hocn calitMl tlio '* Newton " of that suh- 
 je(;t. lie worked on the eonstitnents of the atmosphere ami 
 on hydro^^en ; he made tho first syntiu\sis of water, liy Inirnin^ 
 iiydroifen in air. and found thti density of liydro^en to lio ,', 
 (instead of ^4) of tliat of air. lie det(M*min(M| the ratio of de- 
 pido<fistieated to phloj^istieated air to he about as 1 : 4. Caven- 
 dish also made numy rc^searelies of «;reat importamio in th»! sub- 
 ject of (d(!etri(,'ity ; these iiave been coibHjtcd and edited by 
 Clerk-Maxwell, i'erhaps the most important of his electricMil 
 investigations is that whi(di proved that (dectrostatic attrac^tion 
 takes ])lace according to the law of the inverse s«|uare of the 
 distance, lie is also the author of sev(M'al paptu's on astromdui- 
 oal questions. Most of his writings are to be found in the 
 Philosophical Transactions of the period. 
 
 107 
 
 lotti- 
 
 jr is 
 
 lish's 
 
illl 
 
 
JSgjBr -^r«jv^T-,;:,^i< 
 
 HISTORICAL ACCOUNT OF THE EXPERT 
 
 MENTS MADE SINCE THE TIME 
 
 OF CAVENDISH 
 
fv|^ ■■^?^ ((«ii ii!-Ti«f ^pkivn^'ijwfljfj'ii"! ■ M i|uvm imm 
 
 m 
 
HISTORICAL ACCOUNT OF THE EXPERI- 
 MENTS MADE SINCE THE TIME 
 OF CAVENDISH 
 
 Carlini. Ill 1821, Cfirliiii, director of tlie Hrera ohservtitory 
 iit Milan, made a series of experiments at tlie Hospice on Mt. 
 Cenib in the Alps, to determine tlie length of the seconds-pend- 
 ulum (55). He was led to do so from considering that tiio 
 Alps offerad a favourable situation for a determination of the 
 mean density of the earth, and that no pendulum experiments 
 had been made there since the publication of the fictitious 
 ones of Coultaud and Mercier (see p. 47). Carl in i compared 
 the time of vibration of a simple pendulum, made after the 
 general style of Borda's, with that of a standard clock whose 
 rate was noted daily. The height of the observing station was 
 194:3 metres above sea-level, in latitude 45° 14' 10". The cor- 
 rected length of the seconds -pendulum reduced to sea -level 
 was found to be 993.708 mm. Matthieu and Biot had found 
 the length of the "decimal" seconds-pendulum at Bordeaux, 
 in lat. 44° 50' 25", to be 741.G151 mm. The calculated length 
 at Mt. Cenis would be 741.0421 mm. ; or, for the *' sexagesimal " 
 seconds -pendulum (100 000 decimal seconds = 80 400 sexages- 
 imal seconds) 993.498 mm. The difference between this length 
 and the observed length is .210 mm., which represents the at- 
 traction of the mountain on the pendulum. 
 
 The mountain Is composed of schist, marble and gypsum, of 
 specific gravities 2.81, 2. 80 and 2.32 respectively. Carlini took 
 the average of all three, 2.00, as the mean density of the hill. 
 Assuming tliat the hill was a segment of a sphere 1 geographical 
 mile in height and had a base of 11 miles in diameter, the at- 
 traction was calculated to be 5.0203, where B is the specific 
 gravity of the hill. With the same units the attraction of the 
 
 111 
 

 MEMOIRS ON 
 
 I 
 
 eartli is 14 IK)4A, where A is the mean density of the earth.* 
 wi ^-^^^^^^ -^10 , ^ , on 
 
 We cannot place very great confulence in this result, not only 
 on account of the fact that the oxtreme value of the length of 
 the seconds-pendulum varies from the mean by .032 mm. and 
 only 13 determinations were made ; but especially because the 
 size and density to be assigned to the mountain are largely a 
 matter of conjecture. 
 
 A resume of Carlini's paper was given by Saigey (56), and 
 by Schell (135). Excellent accounts of the experiment, with 
 criticisms of it, have been given by Zanotti-Bianco (148ii^, P^* '^f 
 pp. 130-45), Poynting (185, pp. 22-i) and Fresdorf (186^, pp. 
 8-11). 
 
 Sabine (58 and 82, notes, p. 47) remarks that Biot and Car- 
 lini had not properly reduced to vacuo the observed pendulum 
 lengths, and states that the corrected length of ^,he seconds- 
 pendulum on Mt. Cenis is 39.0992 in. From the observations 
 made in the Formentera-Dunkirk survey he finds by interpo- 
 lation a pendulum length of 39.1154 in. for the latitude of the 
 Hospice. The difference between the observed and calculated 
 lengths is .01G2 in. The difference calculated from the inverse- 
 square law is .0238 in. With Carlini's data and equations he 
 derives 4.77 for the value of A. 
 
 Schmidt (G4, vol. 2, p. 480) gives a concise account of the 
 theory of the experiment, and remarks that Carlini made an 
 error in determining the attraction of a spherical segment. 
 Making the necessary correction he finds A = 4.837, a result 
 not far from that of the Schehallien experiment. 
 
 In 1840, Giulio (70) also gave the true expression for the at- 
 traction of a spherical segment, and noted that several other 
 corrections must be made in Carlini's calculations ; the height 
 of the segment is 1.05 miles, instead of 1 mile ; the length of 
 the pendulum as determiued by Biot must be corrected for an 
 error in the rule used to find the length, and for the altitude of 
 Bordeaux. Moreover, the red uctions to vacuo had not been made 
 properly in either case. When all these corrections had been 
 applied to Carlini's results, Giulio found the value of A to be 4.95. 
 
 * This sigoificatioD of A will be retained throughout the rest of the 
 .volume. 
 
 112 
 
THE LAWS OF (J II A V I T A T I () N 
 
 Saigey (74, p. 155) niiikos tlio observed pendiiliini length 
 corrected to vacuo !)9;J.75r) mm., a!id tlio calculated length 
 903.017 mm. With thesr' uumbers the value of A becomes 0.15. 
 
 Zanotti- Bianco (148|^, pt. 2, p. 130) mentions that Knopf 
 (149J) has compared the value of gravity iis observed by C'ar- 
 lini on the top of the tnountaiu with the value calculated for 
 the same place from observations made on the same parallel of 
 latitude, and found for A, 5.08. 
 
 .f the 
 
 ie an 
 
 lent. 
 
 •esult 
 
 at- 
 )ther 
 jight 
 th of 
 )r an 
 le of 
 
 »ade 
 Ibeen 
 
 .95. 
 
 If the 
 
 Airy, Wiiewell and SnRRPSFiA>fKs at Dolooatii Mink. 
 In 1820, Drobisch, in an appendix to si piimphlet on the figure 
 of the moon (57), suggested that experiuients be made on tlu^ 
 change in the period of a pendulum when carried from the sur- 
 face of the earth to the bottom of a mine; he gave the theory 
 of the experiments and calculated the change resulting from 
 certain hypotheses. It is interesting to recall the fact that 
 Bacon proposed the same investigation two centuries earlier. 
 (See p. I.) 
 
 At the very same time, unknown to Drobisch, experiments 
 of this nature were being tried in England by Airy and Whew- 
 ell at the copper mine of Dolcoath in Cornwall. Their meth- 
 od was to swing one invariable pendulum at the mouth of 
 the pit and compare its rate, by Kater's method of coincidences, 
 with that of a standard clock, and at the same time perform 
 the same operation upon anoilier pendulum and another clock 
 at a depth of 1220 ft. in the mine. The pendulums were then 
 exchanged and the operations repeated. The greatest difficulty 
 experienced was that of comparing the rates of the two clocks. 
 The first series of experiments was abruptly stopped on account 
 of the damage received from fire by the lower pendulum. A 
 short account of the method was published in 1827 (00), and 
 Drobisch translated it for Poggendorlf's Annalen (59), wherein 
 he gives also a more complete account of the tiieory and an ap- 
 plication of hjs equations to Airy's observations. Assuming 
 the mean density of the surface layer of the earth to be 2.587, 
 the experiments gave about 20 for the value of A. Drobisch 
 contends that the surface density should be taken to be 1.52, 
 considering how large an amount of the surface layer is water. 
 
 Two years later Airy and Whewell, assisted by Mr. Sheep- 
 shanks and others, attempted to repeat the experiments ; but 
 after overcoming various anomalies in the motions of the pend- 
 H 113 
 
mi: MO I lis ON 
 
 uhunH, ilio observations \V(Mo stopped by a fall of rock in the 
 mine. The value of A foinul from this series was about 0. A 
 full account of the exi)eriments was print(Ml j)rivately ((>:i) in 
 IHJiH, and Drobisch translated the pamphlet for the Annalen 
 (03). 
 
 )■ 
 
 Reich's First Fxpruiment. In 1838, F. Tieieh. Professor 
 of Physi(!s ill the Herijfakademie at Freiberc^, ])iil)lishe(l in book 
 form (07) the account of a series of experiments carried on by 
 him since 1835 to find, after the method of Cavendish, the 
 mean density of the earth. The adoption of the mirror and 
 scale method of measuring deflections seemed to iiim to prom- 
 ise a !n(!Jins of overcoming many of the ditticulties against whicdi 
 ('avondish had contended. The final observations were made 
 in the year 1837. 
 
 In order to avoid the effects due to irregularities of tempera- 
 ture, the api)aratus was set up in a cellar room wiiich was care- 
 fully c1os(m1 up, and the observations made through a hole in 
 the door. The arm of the balance was '.i.Oll) m. long, and its 
 moment of inertia was found after the manner used by (Jauss 
 for a magnet. Tiie average weight of each of the balls was 
 484.213 gr., and their distance below the arm was 77 cm. They 
 were composod of an alloy of about 1)0 parts tin, 10 parts bis- 
 muth, and a little lead. The attracting masses were of lead 
 45 kg. in weight and about 20 cm. in diameter, and hence 
 much smaller than those used by Cavendish. They were sus- 
 pended from pulleys running on rails parallel to the arm of the 
 balance, and could be quickly moved from the null to the at- 
 tracting positions. Only one mass was in the attracting posi- 
 tion at a time, on account of the fsict that in every one of the 
 four attracting positions the distance from the mass to the ball 
 was slightly different ; whereas Cavendish used both masses at 
 once. The distance from mass to ball was measured at each 
 observation by means of a telescope moving along a horizontal 
 scale, and not once for all as was done by Cavendish. 
 
 After the suspended system was set up, Reich found a con- 
 tinual changing of the zero-point, which often lasted for G 
 months. In his final observations this was not noticeable be- 
 cause of the length of time, 1^ years, which intervened between 
 the initial and final experiments. Accordingly the second 
 means of the elongations were found by him to be more con- 
 
 114 
 
Til K LAWS (H- <i i;.\ V IT.\Th»N 
 
 stunt tlian Ciivcndisli found (lu'in. 'I'lu' Tollowinj^' tuhlu will 
 show this, iind Jilso illustnilo Kuicli's nictlioil of tiniliii^ tlii' 
 timo of vibisition : 
 
 hence 
 
 pos\- 
 of tiro 
 Hi ball 
 3ses at 
 
 each 
 Izontal 
 
 con- 
 
 for G 
 
 )le be- 
 
 jtween 
 
 lecond 
 
 con- 
 
 
 
 
 Tiinoiir 
 
 IHlHSilKO 
 
 
 Kxlrcmos 
 
 IhI meiiii 
 
 '2i\ iiicaii 
 
 
 
 
 
 
 
 III 74 5 
 
 Hi ' 
 
 5.5 
 
 05.3 . 
 
 
 
 
 
 
 c • ■ 
 
 7(5.00 
 
 
 iv 37 yo H 
 
 IV' 37 
 
 yO'.H 
 
 50.8 
 
 
 
 
 
 
 ( 
 
 7a. H5 
 
 74.035 
 
 ^4 31 3 
 
 34 
 
 30 .3 
 
 00.9 
 
 
 
 
 
 
 f 
 
 75.00 
 
 74.K75 
 
 41 33 .4 
 
 41 
 
 13 .4 
 
 60.0 
 
 
 
 
 
 
 87.6^ '" 
 
 74.35 
 
 75.075 
 
 47 59 .3 
 
 48 
 
 10 .8 
 
 
 
 
 
 
 Av(M-ii,i;(! or M inciin 74.0583 
 Time of vibniiioii (it'U'niiiiicil froni |)ii.s.«iip' of 74.5 
 
 IV" 41' 33".4- IV" 37 ;'{G.8-i;} 'ir; .i\ ) ,.,, ....^ 
 
 47 50 .3- U 31 .3=13 38 .0 j "^'''- ^^ *^ '^ 
 Time of vibration (letcrmincd from piissa^c of 75.5 
 
 IV" 41' 13".4-IV" 37' 30'.H^13' i\ A\ } ,„,... ,.., ,.„ .. 
 
 48 10.8- 34 30.3 = 13 41 .0 [ *'^'''- ^"^ '*^ " 
 
 Wy interpolation the time of a double vibration across 74.!)r).s;j 
 is II)' 41". 70S. This differs somewhat from Cavendish's method, 
 as will be seen by a reference to page (15. lieich considered it 
 a more accurate method tinui that of ('avendish, and remarks 
 that when he applied the hitter's method to the above observa- 
 tions he got results not very ditrerent, but on ap[)lying his own 
 method to Cavendish's observations lie got somewluit dit!erent 
 results. Heidi's method was adopted by Baily (70, pp. 44 and 
 47) in his experiments. 
 
 When in the above way at least three passages across the two 
 median points bad been observed, lieich waited until the arm 
 was in its next extreme position and seemingly at rest and then 
 rapidly moved the attracting mass to its new position. It was 
 always so movoil as to increase the swing. He assumed that 
 the motion was instantaneous, and used the last extreme of the 
 one series as the first of the next. Baily (79, p. 4G) followed 
 him in this departure from the procedure of Cavendish. Cornu 
 arul Bailie (14:i) have pointed out that this method leads to 
 error, and we shall refer to the matter again when describing 
 the results obtained by Baily (page 11(5). From each such set 
 of 4 extremes the restiug-point and the time of vibration arc 
 
 115 
 

 MEMOIRS ON 
 
 found. From )l sucli sets tlio dcviution could bo found, uiid 
 the mciui density of tlie eurtli culeu'.ated. iii>i(di did not, how- 
 cvor, procood in tluit way, but deduced one value of A from all 
 the observations of each day ; that is, he took the average of 
 all the deviations of that dav for the final mean deviation, and 
 the average of all the times of vibration for the llnal nu'an time 
 of vibration, Jind from these deduced one value for the mean 
 density of the earth. 
 
 I na})|)lying corrections to the equations derived from a simpli- 
 fied form of the theory of the experiment, Keich followed Cav- 
 endish exactly. 57 observations were made, from which 14 de- 
 terminations of the value of A were deduced. The mean of all, 
 when corrected for the centrifugal force, was 5.44 ±.0233, a re- 
 sult almost coinciding with Cavendisli's. Reich admits at the 
 end of his paper that there were certain anomalies in the 
 motioft of the beam which he could not account for. 
 
 A second series of observations with iron masses 30 kg. in 
 iveight and 20 cm. in diameter gave for A, 5.4522, which proves 
 that no disturbance could have arisen from magnetic action. 
 
 Valuable concise accounts of Reich's experiment are given 
 by Beaumont (00), Baily (08, 09 and 70, pp. 00-8), Schell 
 (135), Poynting (185, pp. 48-50) and Fresdorf (180^, pp. 20-2). 
 
 Baily. While Reich was making the investigations just re- 
 ferred to, a very comprcliensive and elaborate series of experi- 
 ments upon almost the same plan was being carried on by the 
 English astronomer F. Baily. These experiments were under- 
 taken at the instance of the Royal Astronomical Society, and 
 in aid of them a grant of £500 was made by the British govern- 
 ment. The results were published in 1843 (79). They were 
 carried on in one of the rooms of Baily's residence, a one-story 
 house standing detached in a large garden. The apparatus was 
 almost the counterpart of that of Cavendish, except that the 
 balls were not suspended from the balance arm, but were 
 screwed directly on to its ends. The balance and its mahogany 
 case were, moreover, suspended from the ceiling, and the at- 
 tracting masses rested on the ends of a plank movable on a 
 pillar rising from the floor. As a protection against changes 
 of temperature this apparatus was then surrounded by a wooden 
 enclosure. The masses were of lead rather more that 12 in. in 
 diameter, weighing 380.409 lbs. each. Torsion rods of deal and 
 
 116 
 
TJIE LAWS OK (JliAVITATluN 
 
 on a 
 |anges 
 
 )oden 
 liii. in 
 111 and 
 
 of brass, each about 77 in. long, wore omployed, and their 
 motion was observed by the mirror and seaU> nuitliod. lialKs of 
 (HtTerent materials and of various diameters weic experimented 
 upon : viz., 1.5 in. platinum, "i in. lead, 2 in. /iiie, 2 in. glass, 
 2 in. ivory, 2.5 in. lead, and '^.h in. hollow brass. 'IMie mode 
 of suspension was varied greatly, both single and double sus- 
 pension wires being used, and the material and distance apart 
 of tho bifilar wires being frerjuently changed. The lengtli of 
 the suspending wires was ordinarily al)out 00 in., and tho time 
 of vibration varied from about 100 to 580 seconds. 
 
 The experiments were begun in Oct. IS;J8, and carried on 
 for 18 months, until about 1300 observations had been made ; 
 when, on account of tho great discordance of the results, a 
 stop was nuide. Prof. Forbes suggested that these anomalies 
 might arise from radiation of heat, and advised the use of gilt 
 balls and a gilt case. These changes were made, and the tor- 
 sion box also lined with thick flannel. They ttirned out to be 
 decided improvements, although sotne anomalies still existed, 
 and it is evident that the choice of a place for setting up the 
 apparjitus was not a good one. 
 
 Baily adopted the method of Ueich for reducing the time re- 
 quired to make the number of turning-points recjuisite for cal- 
 culating the deviation and period ; that is, the masses- were 
 moved quickly from one near position to the other, and the last 
 turning-point of one series served for the tirst of the next. 
 Three new turning-points were observed at each position of the 
 masses, and each group of 4 was called an ** experiment." 
 2153 such experiments were made during the years 1841-2. 
 The time of vibration was found for each experiment after the 
 method adopted by Reich. In deducing the mean density of 
 the earth from the observations Baily proceeded quite differ- 
 ently from Reich. There was always a slow motion of the zero- 
 point, and Baily, in order to fake account of this, combined 
 the deflections and periods in threes. The difference between 
 the deflection^'of the 2d experiment and the average of the 1st 
 and 3d is twice the mean deviation. The average of the period 
 of the 2d experiment with the average of the 1st and 3d is the 
 mean period. From the mean deviation and mean period so 
 found a value of A is deduced. Another was then found from 
 comparing the 3d experiment with the 2d and 4th, and so on. 
 The mean of all the experiments gave forA, 5.G747zt.0038. Some 
 
 117 
 
T 
 
 ■!» ^am* «.i*i»if. f'^t'i 
 
 I 
 
 M KM o I lis ON 
 
 of tli(» oxpcrinuMits woro miulc with the brass rod uloiio, without 
 any halls, \\h<! mean result for \vlii(;li was .'>.(i(i(l(;d=.<K>:jS. 
 
 Till' Miatlicmatical analysis of th(» prolijcni was ;;ivt'ii by 
 Airy, aiul is iii('ori)orati'(l in Haily's paper (ID, pp. !i'.>-lll); it 
 is also to bo found iu lioutli's Uiijid hytntniirs, lS8:i, pt. 1, 
 ]>p. :jr»!)-(l4. 
 
 liaily published a oondens(Ml a(!(!ount, of his work in several 
 journals (?'), 7'», 77, 7.S iind HO). A cari^ful diseussiou of it is 
 given by ScludI (i:{"») ami by l*oyntin<; (is:», pp. r>ti-7). 
 
 In lS4v*, Sai<,'ey (M) wrote a full aeeonnt of all the experi- 
 in(Mits nuide befori! that date ; \w ju^ives his reasons for consider- 
 ing th(^ pcMidnluin nu'tluul of finding A the htast aecurati;, the 
 mountain method somewhat better, and the torsion method thu 
 best, lie limls great fault witli the work of liaily, aiul eon- 
 Hidora that iiis results are not so wor^hv of (M)nlide!H;e as thoso 
 of Cavendish. Saigi^y eontiMuls that the anomalicss observed 
 by ('avendish, IJeitdi, aiul i^tily cannot be accounted for by 
 radiation of heat, as Korbes suggested, because the bahmco 
 swings in an enclosure all points of which are at the same tem- 
 peratur(f (thus begging the (juestioii); he (lonlldently renuirks 
 that these anomalies are caus(Ml by the passage of air into or 
 out of the case astluf barometric; pressure (diang(\s. The values 
 of A found by Haily increased from T).!)! to 5.77 as the density 
 of the balls used changed from 'Z\A) to 1.'.) respectividy; Saigey 
 thinks that this must arise from an error in calculating the 
 monuuit of inertia of the balance arm. II(^ devises a graphical 
 method of making proper allowance for this supposed error, 
 and deduces as the linal im^an of all the experiments of Baily a 
 value 5.52, the extremes being 5.4l> and 5.55- 
 
 Saigey made a new determination of A (74, vol. VZ, p. 377), 
 from the ditTerence (»".8<l of the astropomical and geodetieal 
 latitudes of Evaux as calculated by Puissant. Applying the 
 method used by Ilutton for Schehallien, and hiter by James 
 and (Jlarko for Arthur's Seat, he found the ratio of A to the 
 surface density of France to be 1.7. Assuming the latter den- 
 sity to be 2.5, the former becomes 4.25. 
 
 In 1847, liearn tried to account for the anomalies in Baily's 
 results by assuming a magnetic action. lie worked out the 
 theory (81) of sindi action, and found that it must bo of a very 
 fluctuating nature and may be either positive or negative, and 
 even greater in magnitude than the force of gravitation. That 
 
 118 
 
Til !•: LAWS ()|- (; li.\ VriATK^N 
 
 377), 
 
 lotical 
 
 llamcs 
 \o the 
 den- 
 
 aily's 
 It the 
 very 
 », and 
 'That 
 
 Hiudi a majUMit'lic net ion docs not icallv rxist is to \h>. dcdncrd 
 I'roni licith's results willi iron masses (see paije lit;). 
 
 Monli<;ny olTere«| to the h'oyal A<'adeinv of Ueliiiiini, in lS,"r.\ 
 a memoir in vvliieh lu> att rihnted the peeuliarit iits in tlii^ he- 
 liaviour of the torsion pendiilnm in the experiments of Caven- 
 dish and of Huily to the rotation of the earth. Sehaar (S.*)), to 
 whom the nn'moir was referred hy llu' Society, proved that the 
 rotation (d' the earth eonld not produce Huch elTects, and tin* 
 memoir was not piihlished. 
 
 It was Cornii and liaille who lirst pointiMl out (11'.*), in ISTS, 
 the main error in liailv's method. It lies in his tal\in<' the Ith 
 i'eadin<; of the tiirnin^f-point of «»ne series of experinn-nts as t he 
 Ist of the next, as already explainc^d. They shewed that the 
 rotation of the plank Inddini; the massijs coidd not he per- 
 formed rapidly enoii<;h to ;^^'t the masses into the new position 
 hefore the arm had l)e<^iin its return journey. They theriffore 
 rejcK'ted the 1st of ea<di series of 4 readinii^s, ami cahudatetl A 
 from the other '.\ in II) (;ases taken at random t'rom .-tonu^ of 
 liaily's most diveri^ent values, and fouiul r).ljir> instead of r».7i;{. 
 iteducinj^' liaily's llnal value in the same proportion they ^ot 
 
 ').r)5. 
 
 A curious ridation hetween density and temptu'aturo us pre- 
 8ente(l in Haily's deteriiiinations was pointed out hy llicks 
 (1(1(1), in 18S(I. The nu'an density seems to faP with rise of 
 temperature. The most |)rohahle explamition of this is j^ivini 
 by I'oyntinj; (LS."), p. 5(1), who renuirks that the exi)eriim'nts 
 with the light balls happened to be made in winter, and those 
 with the heavy balls in summer, llicks also refers to several 
 slight corrections to l)e made in Airy's discussion of the thciu-y 
 — viz., for the air displaced by the attracting nuisses, for the 
 inertia of the air in which the balls move, and for expansion 
 with change of temperature. 
 
 Reich's Skcoxd Exi'KUIMKNT. Ten years after the appear- 
 ance of Baily^s memoir, Ueich published (s:}) an ac(!ouut of 
 some further experiments with his apparatus. In the begin- 
 ning of his paper he pointed out that Baily's method of com- 
 bining the results of the separate experiments was better than 
 that used by himself. He pro(!eeded to calculate the results 
 of his first experiments by IJaily's method and found for A the 
 value r).49±.();i0. 
 
 tl9 
 
1 
 
 w 
 
 mi: Mollis ON 
 
 ij 
 
 r< :! 
 
 :j 
 
 h«'iii;j imjH'csst'd with llut luioiiialics in Huily'H «»l)s<'rviition«, 
 iiiid cspcriiilly witli i lie Viiiiatioii of tlif liiiul rcHiiits with tho 
 ilciisity of \\\v hulls, Ucich (JcterniiiitMl to repent his (experi- 
 ments. Mis iippiiratiis was set up this time in u see(Mi<l-story 
 rnnm, and haily's «h'vi<M's were! empioyet! in order t<» avoid ihv 
 elT('<;ts of tiMnperatiire (duin^(>s. The only important change in 
 th(! arran^cnnMit of the apparatns was in tin; piacin^' of the lar^o 
 inaHS. it was now set in om; of fonr depressions tM) ' apart in a 
 eir(;nlar tahle ntvolvin;^ nnder the halanco ahout a vertical axiu 
 passinj^ tiirou^'h the centre of ontf of the halls; thns no corroc- 
 tion was !iec(!ssary for the attraction of the tahle and its sup- 
 ports upon the hall. The halls and masses were those nsed in 
 the first expt;riment. Three series of experiments wcirc; inado 
 durin<;f the years 1817-50, one with a suspending wire of thin 
 copper, one with thi(d\ coppitr, and ont; with a hililar iron sus- 
 pension. The final mean density of the eartli was found to ho 
 
 r».r)H:{-^dr.oi4'.>. 
 
 In order to make a test of I learn's explanation (sec page IIH) 
 of the pecMiliarities in Haily's results, IJeicdi made sonu! further 
 experiments, lie ke[)t tlu^ North pole of a stronj; nuignct near 
 tlio attra(!ting lead nniss for a whole day, and then suddenly 
 rotated tho mass throu<j^h ISO" ahout a vertical axis; hut no 
 elteet was evident. Hence variations in the result arc not duo 
 to tho nnignctizing of the nnisses l)y the earth, or similar 
 cauvscs. lie then took off tlu^ tin hulls and suhstitutcd success- 
 ively halls of hismuth and of iron. The values of d wore ro- 
 spectivoly 5.5^1};^ and 5.088? ; the largeness of the latter denotes 
 possihly u diamagnetic action rf tho lead mass; hut it shews 
 that under tho original circumstances no measurahlc effect 
 could have arisen from rmignetic action. 
 
 Prof. Forhes had suggested * to Reiidi that A could bo found 
 from tho period of the halanco only, hy noting tho variation of 
 tho time of vibration witU the position of the attracting masses. 
 IJeich nuido some experiments of this nature by placing two 
 lead masses diametrically opposite to each other, first so tha^ 
 tho lino joining tnem was perpendicular to tho vertical plane 
 through the torsion arm, and next was in tho plane. This 
 caused no deviation, but only a change in the time of swing of 
 
 * We have seen (pajio 106) that this method was suggested earlier, in 
 Gehler's Physikalische Worterbucfi, and the equations given by Brandes (42), 
 
 120 
 
Til K LA WS <>K (iU.\ VIT ATImN 
 
 lound 
 m of 
 
 ISSCS. 
 
 two 
 
 tllilu 
 
 |)1(inc 
 JTlns 
 [got 
 
 iv, in 
 
 (42). 
 
 -V-' 
 
 tiie Ituiiuire. 'Vhci viiliio of A foiiiKl in this wity wuh ('i.'s','), htit 
 tlir tippiiriit us Wiis not Wfll ilrviscd for \\\v work. 
 
 Scvcnil ul)sliiirt8 of Uuii'li's |iaiK'r nrv. to be found (84, hO, 
 ST un.i 18.*., in». :»0-',»). 
 
 AlUV's IJAUTON ('OM.IKIIY Kx I'l: Kl M KNT. W" llUVO lllllMldy 
 
 icffrrt'd t«) Ally's cxpuriuiunts in t.ho Dcdcoiitli luinu in l8*^»>-8. 
 In IS.M, lio ii^iiin undertook to c;uri-y out invcsti^itions ( KM)) 
 along tlio Kunic lines, the introduction of tin; tidei^ruph having 
 nuido eiisy the eoinpurison of the eloeks iit the top and bottom 
 of the mine, lie stdected the llarton ('olliery, near South 
 Shields, for the experiments, which were carried out by six ex- 
 perienced assistants o\' whom Mr. Dunkin was the chief. The 
 two stations were vertically above each other a»nl l*ir»»! ft. apart. 
 The apparatus was the best obtainable, and special precautions 
 were taken in order that the p(!nduliim supports miglit be rigid. 
 
 Simultaneous (diservations of the two petidulunis were kej)t 
 up night and day for a week ; then the pendulums were ex- 
 changed and observations taken for another week. Two more 
 exchanges were nuide, but the observations for them both were 
 made in one week. Each pendulum bad six swings of nearly 
 4 hours each on every day of observation, aiul between success- 
 ive swings the clock rates were compared by telegraphic siginils 
 given every 15 seconds by a journcynum clock. 
 
 The corrections and reductions wore carried out by Airy in a 
 verv elaborate manner. The results of the 1st and Ikl series 
 agree very closely, as do those of the 'M and 4th, sbowing that 
 the pendulums Inid undergone no sensible change. Hy com- 
 paring the mean of the 1st and 3d series with the mean of the 
 2d and 4th, the ratio of the pendulum rates at the upper and 
 lower stations is obtained independently of the pendulums em- 
 
 Sed. The final result gave gravity at the lower station 
 Be than gravity at the upper by jT^ireth part, with an un- 
 
 V certainty of ^\o^h part of the increase ; or the acceleration of 
 the seconds-i>vHdulum below is 2".24 per day, with an uncer- 
 tainty of less than 0".01. 
 
 In order to calculate what this difference should be, suppose 
 the earth to be a sphere of radius r and mean density A, sur- 
 rounded by a spherical shell of thickness h and density I, then 
 
 , 1 . , ^, , gravity below ^ 2h 3//2 , 
 
 a simple analysis shews that t-^ — r — = 1 H r (com- 
 
 '■ *' gravity above • -^ \ 
 
 131 
 
 rA 
 
Iff 
 
 MEMOIRS ON 
 
 ! i: 
 
 4\ 
 
 pure p. 'M). Airy gives n discussion of tlio effect of ^urfuce ir- 
 regularities ; it is sliewii that, supposing tiie surfa(;e of tiie eartii 
 near the mine bo liave no irregularitie.s, the effect of tiioseat dis- 
 tant parts of tiie earth nniy be neglected. lie also assumes that 
 there is no sudden change of density just under the mine, lie 
 proves that the effect of a plane of '3 miles in radius and of the 
 ;,liickness of the sIijII is 'i'j of that of the uhole shell, so that 
 only the neighbouring country need be surveyed. Since the 
 ui)per station is oidy 74 ft. above high water, it will be sutllcient 
 to assume that any excess or defect of matter exists actually on 
 the surface. A careful survey of the environs of the mine was 
 made, and allowance made for each elevation and depression. 
 The general result is that the attraction of the regular shell of 
 
 ^ . , , T • • 1 1 I 1 *. 1 .1 . gi'Ji-vity below 
 matter is to be diminished by about wJ-yth part ; •/ , 
 
 •*»" 1 gravity above 
 
 =.-1.00012O;J3-.OOO179.S4x-. Now from the pendulum ex- 
 
 periments Airy found 
 
 gravity below 
 
 = 1 . 00005 1 S5 ± . ()( KK »00 1 ; 
 
 gravity above 
 hence | = 2.0'^(;(>±.OO73. Prof. W. II. Miller found the aver- 
 
 
 
 age density of the rocks in the mine to be 2.50; hence A = C.50G 
 ±.0l8:i. 
 
 Airy had intended that the temperatures at the two stations 
 should be the same, but the temperature of the lower station 
 was 7°. 13 F. higher than that of theupper. In a supplement- 
 ary paper (101) Airy makes a correction for this temperature 
 difference in two distinct ways, giving for the corrected A, G.800 
 and Ci.ij'Zd respectively. In this paper Stokes (102) investigates 
 tlie effect of the earth's rotation and ellipticity in modifying 
 the results of the Ilarton experiments. It was found to be 
 small, changing A from G.50G to G,5G5. *.* 
 
 Airy published several preliminary notices of his work (88, 89 
 and 122), abstracts of which appeared in several journals (1)0, 
 91, 92, 98 and 111). Valuable re^jumes of the main paper are 
 also to bo found (105, 107, 109, 112 and 119). 
 
 Haughton (lOG, 110, 113 and IIG) gave a rough but simple 
 method of deducing A from Airy's figures, and arrived at 5.48 
 as the value of A. Knopf (149^) has severely criticized this 
 calculation. Another simple formula for the same purpose 
 was given by an anonymous writer (114). On the effect of 
 
 122 
 
m 
 
 T U E L A W S O K (i It A V I T A T I ( ) N 
 
 great clijuigcs in density ])elo\v tlic inider station one should read 
 the paper by Jacob (118 and 1*21) already referred to. Schef- 
 fler {l'^^) pnl»'ihhed in IHO."), though it is dated 18r)(>, the pro- 
 posal of an experiment similar to Airy's, hut made no rd'erenco 
 to any earlier j)roj)osal8 of the same kind. Folie (KM) calcul- 
 ated, in 1^72, tiie attraction at the two stations in a nianncr 
 diirercnt from Airy's, by considering the shell as made up of "i 
 parts. Using Airy's data he arrived at 0.4JJ0 as the value of A. 
 Valuable summaries and criticisms of Airy's work are given 
 by Schcll (i:;*)), Zanotti-Hianco (USJ, pt. '^, pp. UO-00), I'oyiit- 
 ing (185, pj). '^4-!>) and Fresdorf (180^, p}). 13-7). 
 
 ex- 
 
 Jamks and (!lauke. As a result of the calculations mad(^ 
 from the observations taken for the Ordnance Siirvey of (ireat 
 Britain and Ireland (104, 117, 120, 124, 125 and 120) by Lt. 
 (Jol. James, it was found that the plumb-line was considerably 
 deflected at several of the principal trigonometrical stations. 
 It was evident from the nature of the ground at the places 
 under consi<leration, that this deflection was due to irregulari- 
 ties of the surface. In order to study this action more care- 
 fully James decided to have the Scheliallien experiment re- 
 peated at Arthur's Seat, near Edinburgh (lo;} and 125, pp. 572- 
 024). The observations were made during Sept. and Oct., 
 1855, with Airy's zenith -sector, on the summit of Arthur's 
 Seat (A), and at points near the meridian on the north (N) 
 and south (S) of that mountain, at about one-third of its al- 
 titude above the surrounding country. After corrections had 
 been applied, the results were as follows : 
 
 'ing 
 be 
 
 k89 
 
 ;yo, 
 
 are 
 
 iple 
 
 .48 
 
 :his 
 
 ^ose 
 
 of 
 
 Station 
 
 Astiononiical lat.=A 
 
 Geodetical lat. ~G 
 
 A-G 
 
 s 
 
 55° 50' 20". 09 
 
 55° 50' 24". 25 
 
 2". 44 
 
 A 
 
 50 415 .00 
 
 50 38 .44 
 
 5 .25 
 
 N 
 
 57 9 .22 
 
 57 2 .71 
 
 C .51 
 
 It will be noticed that even on the summit of the hill there 
 is an attraction of more than 5" toward the south, which can 
 not be due to the hill. Similarly, to the south of the hill the 
 attraction is not toward the north jis we might expect. It is 
 evident that there is present so!n(; other attracting force, be- 
 sides that of Arthur's Seat, which aj)pears to produce a general 
 deflection of 5" toward the south, 
 
 123 
 
IT 
 
 1 ' < 
 1 I 
 
 l>l 
 
 ii 
 
 'A~' 
 
 
 hi 
 
 U i 
 
 MEMOIRS ON 
 
 Capt. Clarke, who made all the calculations, in order to find 
 the attraction according to Newton's law, used a modification of 
 the method of Hutton. He took account of all the surface 
 irregularities within a radius of about 24000 ft. Tiie resulting 
 value for the ratio of the density of tlie rock composing the hill 
 to that of the whole earth was .517JJ±.0053. James investi- 
 gated the density of the rocks of Arthur's Seat and found it 
 to be on the average 2.75. This gives for A the value 5.31G 
 =h.054. 
 
 In order to see whether the general deflection of 5" could be 
 accounted for by the presence of the hollow of the River Forth 
 to the north and the high land of the Pentland Hills to the 
 south, Clarke extended the calculated attraction to the borders 
 of Edinburghshire, some 13 miles away. He was able in this 
 way to account for a general deflection of 2". 52, and he thought 
 that by carrying the calculations to Peeblesshire the whole 5" 
 might be accounted for. 
 
 Several abstracts of the original paper have been puMis! '^ 
 (108, 115 and 123). Poynting (1,S5, pp. 19-22) has given a 
 valuable criticism of the work. 
 
 In connection with this investigation might be mentioned 
 the various writings on the subject of local attraoLions. Any 
 one wishing to become acquainted with this subject should read 
 Airy's account of his ** flotation theory " (94 and 97), Faye's 
 account of his '^compensation theory" (130, 14G| and 147), 
 Pratt's papers (93, 90 and 99), Saigey (74), Struve (129), Pech- 
 mann (131), the treatises of Pratt (133), Clarke (149) and Hel- 
 mert (148, vol. 2). Many other references to papers by these 
 men as well as by Schubert, Peters, Keller, Bauernfeind and 
 others are to be found in the Roy. Soc. Cat. of Scientific papers 
 and in Gore's "A Bibliography of Geodesy" (174). See also 
 note on page 31 and remarks on page 56. We might here re- 
 call the determination of A by Saigey from local attraction (se& 
 page 118). Pechmann (131) in the same way found in the Tyrol, 
 in 1864, two different values for A, 6.131 l=h. 1557 and 6.352 dz 
 .726, having assumed the density of the earth's crust to be 
 2.75. We shall refer later on to the determinations of Men- 
 denhail and Berget. 
 
 CoRNU AND Baille. In 1873, Cornu and Bailie published 
 a short paper (137) stating that they had undertaken to repeat 
 
 124 
 
 
THE LAWS OF GRAY ITATION 
 
 ipers 
 also 
 
 re re- 
 (seo 
 
 lyi'ol, 
 
 be 
 Len- 
 
 
 the Cavendish experiment under conditions as different as pos- 
 sible from those previously employed. They began by making 
 a thorough study of the torsion-balance in order to learn under 
 what conditions it would have the greatest precision and sensi- 
 tiveness. They found among other things that the resistance of 
 the air was proportional to the velocity (141, 14:^, 143 and 157). 
 
 The apparatus was set up in the cellar of the ^ficole Polytech- 
 nique. The arm of the balance waa a small aluminium tube 
 50 cm. long, carrying on each end a copper ball 109 gr. in 
 weight. The suspension wire was of annealed silver 4.15 m. 
 long, and the time of vibration of the system (V 38". The at- 
 tracting mass was mercury which could be aspirated from one 
 spherical iron vessel on one side of one of the copper balls to 
 another vessel similarly situated on the other side of the ball. 
 This method got rid of the disturbances arising from the move- 
 ment of the lead masses in the Cavendish form of the experi- 
 ment. The iron vessel was 12 cm. in diameter and the mer- 
 cury weighed 12 kg. Another great improvement was the 
 reduction of the dimensions of the apparatus to ^ of that used 
 by Cavendish, Reich and Baily, the time of oscillation and the 
 sensitiveness remaining the same. The motion of the arm was 
 registered electrically. 
 
 Two series of observations were made ; one in the summer 
 of 1872 gave A = 5. 56, and the other in the following winter 
 5.50. The difference was explained by a flexure of the torsion- 
 rod, and the former result was considered the better. 
 
 In a later report (142) they refer to some changes made in 
 their apparatus ; they increased the force ol attraction by using 
 4 iron receivers, 2 on each side of each copper ball, and they 
 reduced the distance between the attracting bodies in the ratio 
 of V'2 to 1. The time of vibration, 408", remained the same 
 within a few tenths of a second for more than a year. The new 
 value of A was 5.56. 
 
 We have already referred (page 119) to the fact that Cornu 
 and Bailie foulul out the error in the Baily experiments. 
 
 A final account of these experiments has not yet been pub- 
 lished. Abstracts of the papers cited are given by Poynting 
 (185, pp. 57-8) and by several journals (138 and 139). 
 
 Ished 
 peat 
 
 Jolly. In 1878, von Jolly of Munich published an account 
 (144 and 145) of the results of his study of the beam balance 
 
 125 
 
1 
 
 M H: M O I R S ON 
 
 V '• 
 
 iij 
 
 iiu an instninieiit for measuring gravitational attractions. Tie 
 disciisaod the sources of error in the balance readings and 
 methods of eliminating them. IMie variations due to tempera- 
 tures etfects are very ditliciilt to avoid, but by working in the 
 mornings only, and by covering the balance case with another 
 lined inside and out with silver paper, it was found to bo pos- 
 sible to get ([uite concordant results. 
 
 Jolly applied the balanee to test the Newtonian law of the 
 distance. Two extra scale pans were suspended by wires from 
 the ordiiKiry scale pans of the balance ami 5.^!) tn. below them. 
 The wires and lower scale pans were enclosed to prevent oscilla- 
 tions from air currents. Two kilogramme masses of polished 
 nickel-plated brass were balanced against each other, first both 
 in the upper scale pans, and then one in the upper and the 
 other in the lower pan, in each case double weighings being made 
 after the manner of (jJauss. The motion of the beam was noted 
 bv the mirror and scale method, the mirror being fixed at the 
 middle of the beam and perpendicular to its length. If r is the 
 radius of the earth at sea-level, and h a height above it, then a 
 
 riniss Qi at sea-level weighs Q2 at h, where Q2=Qi (l— " ) ap- 
 
 Q2 1 000 000- 1. :»()!)<) 
 
 1 000 000 
 The differ- 
 
 proximately. Jolly found by experiment— = 
 
 Qo 1 000 000-1. GG2 
 whereas the equation g^ves^^^ 1 OOO 000 
 
 ence, .153 mg., Jolly thought, was due to local attractions. He 
 proposed to repeat the experiment at the top of a high tower, 
 and at the same time to find the mass 01 the earth by noting 
 the change in weight of one of the masses in the balance when 
 a large lead ball was brought beneath it. 
 
 The results of these experiments (153 and 154) were published 
 in 1881. The distance between the scale pans was now 21.005 
 m. The arm of the balance was 60 cm. long, and the maxi- 
 mum load 5 kg. Four hollow glass spheres of the same size 
 were made and in each of two 5 kg. of mercury were put, 
 and all were sealed up. Each scale pan had always one sphere 
 in it, and thus Jiir corrections were avoided. An observation 
 was made as follows : first the mercury-filled spheres were bal- 
 anced in the upper pans, and then one in the upper pan was 
 balanced against the other in the lower. The change in weight 
 
 126 
 
THE LAWS OF (JitA VITATION 
 
 TTc 
 
 observed was 31.G8G mg.; whereas the olian;;e as calciihited 
 from the formula sliould have been 33.05!)* ing. The differ- 
 ence is in tlie same direction as in the earlier experiment. 
 
 A sphere of radius .4975 m. and weight 5775. !;i kg. was tlion 
 built up out of lead bars under tlie lower scale pan which 
 received the mercury-filled globe. The distance from the 
 centre of this sphere to that of the globe was then .508G m. 
 The attraction of the sphere for the mc>rcury-filled globe when 
 in the upper pan was neglected. 
 
 Observations were made exactlv as before, and the chanfje in 
 weight was 3'2.^75 mg. The increase in weigiit due to the 
 presence of the lead is therefore .5(S() mg. Knowing tlie den- 
 sity of the lead to be 11.18(1, a simple calculation gives for the 
 mean density of the earth 5.(>!)"^±.(K;8. 
 
 An account of these experiments is given by Ilelmert (14S, 
 vol. 2, pp. 380-2), Zunotti- Bianco (U8|, vol. i, pp. 175-8->i), 
 Wallentin (154^), Keller (107), Poynting (185, pp. 01-4) and 
 Fresdorf (180|,pp. 23-5). 
 
 ap- 
 
 l)!)<) 
 
 Mendeniiall. In 1880, Prof. T. C. Mendenhall described 
 (150) a method of finding the period of a pendulum such that 
 a determination required 20 or 30 minutes only. At the begin- 
 ning and end of this time the pendulum throws a light trip- 
 hammer of wire which breaks a circuit and makes a, record on 
 a chronograph on which a break-circuit clock is also marking. 
 The advantage of such an arrangement, in addition to the 
 short time required, is that the arc of vibration may be small 
 and will change very little. Mendenhall expressed a deter- 
 mination to find the variation of the acceleration due to gravity 
 on going from Tokio to the top of Mount Fujiyama. 
 
 A year later the results of these experiments were published 
 (151), having been made in Aug., 1880. An invariable pend- 
 ulum was used, made from a Kater's pendulum by removing 
 one ball and knife-edge. Its period at Tokio (barometer 30 in. 
 and temperatiire 23°. 5 C.) was .999834 sec. On the top of 
 Fujiyama the barometer stood nearly stationary at 19.5 in. 
 during the observations, and the thermometer at 8°. 5. After 
 approximate corrections were made for buoyancy, the time, 
 reduced to Tokio conditions, was 1.000330 sec. Assuming g at 
 
 * According to Helmert this should be 33.108 und according to Zanotti- 
 Bianco 33.053. 
 
 127 
 
T 
 
 ™fWfp"Yi,;« I'.jm'jm' 
 
 I ■ li«i |iimwiii(|ppii*f II q v^^i ,' « 
 
 I 
 
 I 
 
 i 
 
 MEMOIRS ON 
 
 Tokio to be Sk7084, as he had found in the previous year, it fol- 
 lows that at the summit of Fujiyama it is 0.78HC. 
 
 No exact triangulation of the region had been made, but 
 Mendenhall assumed Fujiyama to be a cone 2.35 miles high 
 standing on a plain of considerable extent. The angle of the 
 cone was measured from photograjdis and found to be lo8°. 
 Fujiyama is an extinct volcano, said to have been made in a 
 single night, and hence its composition ought to be homogene- 
 Oiis. Its average density was taken as 2.VZ, but no great re- 
 liance can be placed on this number. Corrected for the differ- 
 ence in latitude, 19', between Tokio and Fujiyama, the time at 
 its base, supposing the hill taken away, would be .9!)9847 sec. 
 The density of the earth, calculated from these data after the 
 manner of Oarlini, was found to be 5.77. 
 
 Fresdorf (18C|, pp. 11-13) describes fully the experiments 
 and points out an error in Mendenhall's calculations ; the cor- 
 rected value for A is 5.CG7. Poynting (185, pp. 39-40) gives 
 an abstract of the papers referred to. 
 
 Stern KCK. Major von Sterneck has made several investiga- 
 tions of the variation of gravity beneath the earth's surface. 
 The earliest experiments (155) were made, in 1882, in the 
 Adalbert shaft of the silver mine at Pribram in Bohemia. The 
 method employ-jd was to carry an invariable half-second pend- 
 ulum and a comparison clock from one station to another, and 
 find the period by the method of coincidences, the clock being 
 compared with a standard clock by carrying a pocket chron- 
 ometer from one to the other. The pendulum, of brass, was 
 a rod 24 cm. in length carrying a lens-shaped bob weighing 1 
 kg. The knife was of steel whose edge was so cut away that 
 it rested on a glass plate on two points only. The apparatus 
 was always enclosed in a glass case to prevent air currents. 
 The 3 stations !at which observations wore made were at the 
 surface, 516.0 in. and 972.5 m. below the surface respectively. 
 The respective periods at these stations were .5008550, .5008410 
 and .5008415 seconds, and the resulting values of A, found 
 from Airy's formula, were G.28 and 5.01, the density of the 
 surface layer being taken as 2.75. It will be noticed that the 
 values of ^ at the two underground stations are practically the 
 same, and the results are unsatisfactory. 
 
 A year later (156) von Sterneck repeated his experiments at 
 ♦ 128 
 
 * 
 
TUK LAWS OF UiiAVlTATlON 
 
 )eing 
 
 was 
 ing 1 
 that 
 ratus 
 lents. 
 the 
 Ivel y. 
 18410 
 )uiul 
 tlie 
 the 
 the 
 
 bs at 
 
 the same stations and at two additional ones. In order that 
 liis observations might ho independent of the rates of the clocks 
 nsed in finding tiie periods, Sterneck introduced an important 
 modification of tlie metiiod adopted by Airy and by himself in 
 his earlier investigations. lie made another penduiiim simihir 
 to the one described above ; one of these was always at the 
 surface station and the other at one of the underground sta- 
 tions, and their rehitive periods were compared by means of 
 electric signals sent simultaneously from a single clock. This 
 clock kept a circuit closed for half a second every other half 
 second and operated a relay with a strong current at each sta- 
 tion. The passage of the *' tail " of the pendulum in front of 
 a scale was observed by means of a telescope, in the focal plane 
 of which was a shutter moved by the relay current every half 
 second, and at those instants only was the picture of the tail 
 of the pendulum allowed to pass to the eye through the tele- 
 scope. The time of a coincidence was when at one of these 
 flashes the tail appeared exactly at the middle of the scale ; the 
 time between two successive coincidences determines the period 
 of the pendulum. The observer at each of the two stations is 
 thus finding the period of his pendulum ii *;erms of exactly the 
 same nnit of time. When the observations were corrected, it 
 was found that the period at the highest underground station 
 was less than that at the next lower station, and the determin- 
 ation at the former station was consequently not used. The 
 values of A as determined from observations at the other sta- 
 tions were 5.71, 5.81 and 5.80, with a mean of 5.77. Helmert 
 (148, vol. 2, p. 499) has made a recalculation and finds that 
 these numbers should be 5.54, 5.71, 5.80 and 5.71 respectively. 
 Von Sterneck used his results at the surface and at these 
 underground stations to express y as a function of the depth. 
 Calling the value of g at the surface unity, and measuring r 
 from the centre of the earth and calling it equal lo unity at 
 the surface, he deduced the following expression for the value 
 of g at any depth : 
 
 ^=2.6950 r-1.8087 r'-f-.1182 r\ 
 This would make g a maximum, l.OG, at r = .78. The density 
 would be expressed by the formula (/=:15.1;)G — 12.513 r, giving 
 15.136 for its value at the centre of the earth, and 2.624 at the 
 surface. These relations are at least suggestive if not con- 
 vincing. 
 
 I . 129 
 
r 
 
 i 
 
 
 W ' 
 
 
 m ' 
 
 f 
 ' 
 
 l[ ^: 
 
 1, 
 
 i ' 
 
 ME MO I lis OX 
 
 During tlio year 1883 von Sterneck used tlio saino metljoC 
 and apparatus to doterniino tlio variation in gravity for 13 sta- 
 tions above tlie earth's surface at Kronstadt. lie found (l')H) 
 gravity greater at a higiier point (Schlossborg) tlian at a lower 
 (Zwinger), and proved that neither the formula of Young (see 
 page 31) nor that of Faye and Ferrei for the reduction to sea- 
 level gave satisfactory results. 
 
 Twice in this year Sterneck made investigations at Krusna 
 hora in Boliemia. Here there was a mine witii a horizontal 
 gallery 1000 m. long, and he wished to find the effect of the 
 overlying sheet of earth upon the value of gravity at various 
 points in the gallery. The same apparatus was used after some 
 im[)rovements had been made. Observations were taken at the 
 mine mouth and at points 31)0 and 780 m. from the mouth, and 
 G2 and 100 m. respectively below the surface of the ground. 
 The results shewed that gravity in the plateau increased with 
 the depth of the super-incumbent layer by the half of the 
 amount by which it would have changed in free space when 
 the distance from the centre of the earth was changed by the 
 same amount. Observations were made at 4 stations above 
 ground also at different elevations, and it was found that the 
 Faye-Ferrel rule accounted for the differences between them 
 much better than did the Bouguer- Young rule. 
 
 Further experiments (164) were made, in 1884, at Saghegy 
 in Hungary, and elsewhere, with results similar to those de- 
 scribed above. An important improvement was made in the 
 method of observing the coinciilences. They were now ob- 
 served by the reflections of an electric spark from two mirrors, 
 one fixed on the pendulum stand, and the other attached to 
 the pendulum and when at rest parallel to the first. The spark 
 was made by the relay circuit every half second. 
 
 In 1885, Sterneck made a series of observations (165) at the 
 mouth and at 4 underground stations in the Himmelfahrt- 
 Fundgrube silver mine at Freiberg in Saxony. He was led to 
 do so by the publication of the results of some pendulum meas- 
 urements made there, in 1871, by Dr. C. Bruhns, who had found 
 that gravity decreased with the depth. Using Airy's formula, 
 von Sterneck found the following values for zl at the 4 under- 
 ground stations in the order of their depth : 5.66, 6.66, 7.15 
 and 7.60, the density of the mine strata being 2.69. These re- 
 sults indicate an abnormal increase of gravity with depth. Von 
 
 130 
 
 f ' 
 
TilK LAWS OK <;KAVlTATiUN 
 
 3thO(I 
 
 'i sta- 
 (158) 
 lower 
 ? (sec 
 
 sea- 
 
 rusna 
 zontal 
 ot the 
 arious 
 1* some 
 at the 
 ,h, and 
 round, 
 d with 
 of the 
 e when 
 by tlie 
 above 
 bat the 
 
 1 them 
 
 aghegy 
 ose de- 
 in the 
 3\v ob- 
 lirrors, 
 hed to 
 B spark 
 
 at the 
 
 Ifahrt- 
 
 led to 
 
 meas- 
 
 found 
 
 rmula, 
 
 nnder- 
 7.15 
 
 lese re- 
 Von 
 
 Sterneck tiotioed tluit in tlieso experiments, as well as in those 
 made at I'rihrum, the increase in gravity is nearly proportional 
 to the increase in temperature. \U\i altliougii i[i('i\s (1(1(5), as 
 we have seen (page 11!)), disoovcred a connection between the 
 values of A and the tiimperaturcs in Ha-'y's cxiu'riinents, and 
 Cornu and Bailie (lagt* \'i^)) got a larger rcstilt for A in snmnicr 
 than in winter, we have no reason for looking ui)on tlu* varia- 
 tions in temperature as an explanation of the* afiomalics under 
 consideration. An interesting criticism of von Sterncck's work 
 is given by Poynting (185, pp. yU-3!>). Short a(!counts of it 
 are given by Fresdorf (18(jJ, pp. 17-9) and Giinther (lOGJ, vol. 
 1, p. 189). 
 
 WiLsiNO. In 1887, .1. AVilsing (170) made at Potsdam a de- 
 termination of the mean density of the earth by means of jiii 
 instrument which is called the pendulum balance, and is the 
 common beam balance turned through 90°. It is practically a 
 pendulum made of a rod with balls at each end and a knife- 
 edge placed just above the centre of gravity. The instrument 
 used by Wilsing consisted of a drawn brass tube 1 m. long, 
 4.15 cm. in diameter and .10 cm. thick, strengthened near the 
 middle wliere the knife-edge is atlixed. The knife-edge and 
 the bed on which it rested were of agate, and (5 cm. long. To 
 the ends were screwed the balls of brass weighing 540 gr. each, 
 and on the upper ball was a pin carrying discs which were used 
 for finding the moment of inertia and the position of the cen- 
 tre of gravity of the pendulum. Its motion was observed by 
 the telescope and scale method, a mirror being attaclied to the 
 side of the pendulum parallel to the kjiife-edge. The pend- 
 ulum was mounted on a massive pier in the basement of the 
 Astrophysical Observatory in Potsdam, and was protected from 
 air currents by a cloth-lined wooden covering. 
 
 The attracting masses were cast-iron cylinders each weighing 
 325 kg. They^vere so arranged on a continuous string passing 
 over pulleys that when one was opposite the lower brass ball on 
 one side of the pendulum the other was opposite the upper ball 
 on the other side. Their relative positions could be quickly 
 changed from without the room, so that the former mass came 
 opposite the upper ball and the latter mass opposite the lower ; 
 the deflection was now in the opposite direction from what it 
 
 was in the first case. 
 
 131 
 
If 
 
 MKMolliS ON 
 
 Tlio (loublo (loflontioii duo to tho oliiin^o in poHJtion of the 
 lUiisHos, and tlui tiino of vihnitioii art; tlui (|inuititic's rcfjiiirtMl 
 for tlui (lotcM'iniMatiou of A. 'V\\v. roiidini^s for tlinso (|iiiWititio.s 
 woro iimdo by the mcitlnxl of Uuily. which has hcon iilniudy do- 
 s(M'ilK'(l. 'VUo, linio of vihraLioii was dctiUMiiiiUMl (irst witii the 
 dis(!s on top of tho ii[)|nM' ball, tluui with one rorriovcMl and tinMi 
 with still anotlxu* rcriiovcd. In this way tlu> moment of inertia 
 was obtaiiu'd. Tho theory of tho instrumcMit is eomplioated, 
 and for it roforeiK^o must be inadi! to tho original paper. Tho 
 residt obtained for A was r».r>I)4 t.(>:i*i. 
 
 In ISHI), WilsiuLj pubiisluMJ (W'l) an acoount of some further 
 ol)sorvations with tho same apparatus, some slight changes 
 baving been \uiu\v in it in tho meantime. Kxtra pro(Miutions 
 wore taken in order to avoid tho elTe(!ts of variations of tom- 
 poraturo. K.\p(M'iments svere made with the old balls, with new 
 load balls, and with the pendulum rod alone. Tho mean re- 
 sult from thes«; was r)..')SS±.()i;j ; and tho final average of all 
 his «lotorminationH r>.r)i!)dr.OI^. 
 
 A preliminary })aper (l(J'3) was read by Wilsing before tiio 
 Berlin Academy, ami also an extract (H>1>) of his first paper. 
 A condensed translation of both papers was made by Prof. J. 
 II. (Joro (ITl) for the Smithsonian Report for 1888, and a short 
 account of the work is given by Poynting (185, pp. G5-9) and 
 by Fresdorf (KSOJ, p. 28). 
 
 
 I 
 
 r 
 
 PoYNTrxo. Prof. J. II. Poynting published in 1878 the 
 results (140) of a study of tho beam balance. He found tliat 
 tho sources of error were temperature changes producing con- 
 vection currents and unequal expansion of the arms, and tlie 
 necessity of frequently raising the knife-edges from the planes. 
 Ho tried to overcome tlie former difficulty by taking the same 
 precautions as those employed by users of the torsion balance ; 
 and he did away altogether with the raising of the beam be- 
 tween weighings, and when the weights had to be exchanged 
 held the pan tixed in a clamp. 
 
 The paper gives a description of his balance and illustrates 
 how it can be used, (1) to compare two weights, and (2) to find 
 the mean density of the earth. The motion of the beam was 
 observed by means of a telescope and scale, the mirror being 
 fixed at the centre of the beam. The deflection of the ray 
 could be multiplied by repeated reflections between this mirror 
 
 132 
 
TIIK r.AWS (»F <;KAV ITATIoN 
 
 and unotlicr whicli wiw flxod jiiid nearly paralli'l to tlic fortnor. 
 'I'lio ceiitro of oscillation was (U'tciiniiitd after tlio tnetliod of 
 hiiily with tlio torsion l)alan<*<'. As a result of II observations 
 I'rof. I'oyntini; found tln^ mean density o1 tint earth to ho 
 A.OOdi.lo. ll(} felt jiistilied, therefore, in proceeding,' to havi^ a 
 Miort5 Huitahlo halancis constructed in order to niake a more 
 careful determination of this <|uantity. 
 
 The inv(!stigation (Mnitinued through many years, and the 
 results (I8<)autl IH."», j)[». TI-IT)*;) were not puhlished until IH'.M. 
 Many unforeseen diniculti(\s arose during,' the pro^^ress of the 
 work, hut by patience and skill I*oyntin<^ was al)le to overcome 
 these difli(!ulties and to be<jfin to take observations in 18!M). 
 Tiie balance was of the larj^'e bullion type, Vrl'.i cm. long, and 
 made with extra rigidity by Oertling. It was set up in a base- 
 ment room at Mason (/(dlege, liirmingham. The prin(Mple 
 upon whieh the experiment is based is as follows : two balls 
 of about the satno mass are suspended from the two arms of 
 the balance. Beneath the balance is a turn-table carrying a 
 heavy spherical mass vertically under one of the balls. The 
 position of the beam is observed and the turn-table moved un- 
 til he mass is nnder the other ball and the position of the beam 
 again observed. The deflection measures twice the attraction 
 of the mass for the ball. The attra(!tion of the mass for the 
 beam and wires, etc., is then eliminated by repeating these ob- 
 servations with the balls suspended at a different distance be- 
 low the arm, for then the attraction of the mass on the balance 
 remains the same, and we find the change in the attraction of 
 the mass for the ball with change oi" distance. The calculation 
 was complicated by the presence on the turn-table of another 
 mass as a counterpoise to the former one ; it was snuiller than 
 that one and at a correspondingly greater distance from the 
 centre. It was used because certain anomalies could be ac- 
 counted for only on the supposition that the floor tilted when 
 the turn-table was rotated with the large mass only upon it. 
 
 Instead of the ordinary mirror fixed on the beam, Poynting 
 used the double-suspension mirror (see Darwin B. A. Rep., 
 1881). The riders were manipulated by mechanism from with- 
 out, and the observer was stationed in the room above, whence 
 he could make all changes and observations without opening 
 the balance room. The attracted and attracting masses were 
 made of an alloy of lead and antimony. The balls were gilded 
 
 183 
 
mr^-t^ II < ^ I 
 
 If i. 
 
 MKMolltS ON 
 
 ftnd woi^liod ovor '^1 (HK> ^r. cucli. 'I'lin lurgo masH wei^. „vt 
 l5(MMK)^ri'., ami til,. (!omit('i|)(>is»' jilxmt Imlf iis niiicli. A MrHt 
 Hot of ()l)Ht'rvat ions pivo A = .*>.")"i. 'I'lio iittnuititi^ ImmUoh wore 
 tlu'ii all iiiviM'tcd ill onicr to (>liiiiiimt(t tlio t'tTtu^ts of wunt of 
 Hyntiiietry in tlu^ ponitioii of tlio tiini-tablts and of lioino^e- 
 nt'ity ill tlio masses. A now sc^t of ol)si'rvatioiiH ^av(^ ^^5.40. 
 The ililTercnco bc'tweeii tlie results of tlie two sot8 must havo 
 been caustMl by a cavity or irre<;ular distrihiitioii of density in 
 tlie iar^M^ mass, and l»y other experiments l*rof. I'oynting found 
 that its (U'litro of gravity was not .»i> its centre of ligiiro, but 
 was nearly at tli(» place at which his gravitational experiments 
 would have suggested it to be. The mean result for A is taken 
 to lu^ r».4pJU, and for \\w gravitation constant, (i, 0.('>!)H4x 10-^ 
 
 I'oynting remarks that the effects of (!onvection (lurrents are 
 greater in the beam balance than in the torsion balance, since 
 the motion of the former is in a vertical plane. lie thinks 
 that a balaiKH! of greatly reduced dimensions would have been 
 preferable. The admirable way in wliich Prof. Poynting has 
 utilized tlie common balance for absolute measurements of 
 force caused the University of Cambridge to award him the 
 Adams l*rize in ISW.i. 
 
 For a short account of this work see Wallentin '^54J). 
 
 It. : !-' ■ 
 
 Reikjkt. In 1750, Hougujr read before the ^icademy of 
 Sciences the results (0) of some experiments made by him to 
 determine whether the plumb-line was affected by the tidal 
 motion of the ocean. He was not able to detect any such 
 effoct. Towards the middle of last century Boscovitch pro- 
 posed (140, vol. 1, pp. 314 and 327) to phme a long ])endulum 
 in a very high tower by the edge of the sea, where the height 
 of the tide is very great, and to observe the deviation due to 
 the rise of the water, and thence to calculate the mean density 
 of the earth. Von Zach suggested (40, vol. 1, p. 17) a modification 
 of the experiment. Boscovitch also proposed the use of a reser- 
 voir after the manner about to be described, used by Berget. 
 In 1804, Robison, in his ''Mechanical Philosophy'' vol. 1, page 
 339, points out that a very sensible effect on the value of grav- 
 ity might be observed at Annapolis, Nova Scotia, due to the 
 very high tides there. The theory of this local influence is 
 given in Thomson and Tait's "Natural Philosophy " pt. II., 
 page 389. Struve (129) proposed to find A from observations 
 
 134 
 
I 
 
 TllK LAWS OF GKAVITATIUN 
 
 «»ri plumb-lines plucod on ouch aide of the liristol Clunmel, uful 
 Keller (UIH) culculiited tlie <lefleetion of the pliirnh-liiie due 
 to the dniiiiiii^ of Luke Fuciiio. In IH'.KJ, M. lierj^et iitilizecl 
 thin principle in order to find (IHI) the denHity of the oiirth. 
 
 lie hud the nne of u hike of '.\'i he(;tures in un>u in the Com- 
 niiino of Iluhuy-lu-ncuve in ikd^ian Luxeinhour^. The levtd of 
 the hike eoiild he lowered 1 in. in u few hourw, iind hh ({uickly 
 regained. lie could thus introduce under his iiiHtrunient a 
 I)ra(^ticiilly infinite phino of nuitter whose iittriiotion couhl he 
 ctilcu lilted and observed. The apparatus used to rneaHuro the 
 attraction was the hydrogen graviineter such as Boussingault 
 and Mascart (('onip. Uend., vol. !)r), j)p. I'^d-S) used to find the 
 diurnal variation of gravity. The variation of the column of 
 mercury was observed by the interference fringes in vacuo be- 
 tween the surface of the mercury and tlie bottom of the tube, 
 \^'hich was worked o])tically plane. A first series of observa- 
 tions was made when the lake was lowered 50 cm., and another 
 when it was lowered 1 n). A change of 1 m. caused a displace- 
 ment of the mercury column of l.ijOx 10-« cm. The value of 
 the gravitation constant found was O.80xl()~^ of A, 5.41 and 
 of the mass of the e;i' th 5.85 x 10" gr. 
 
 M. (louy remarks (182) that such a result would iniply that 
 the temperature remained constant during hours to y ttfJ (TTnr 
 of a degree, which is impossible. Pavilion, with the greatest 
 care, was able to reach xirVinr of a degree only. So that the 
 result given by lierget can not be so accurate as he supposed. 
 
 For )i short account of the experiment see Fresdorf (186^, 
 pp. 29-30). 
 
 Boys. Prof. C. V. Boys read before the Royal Society, in 1880, 
 an important paper (175 and 176) on the best proportions and 
 design for the torsion balance as an instrument for finding the 
 gravitation constant. He shewed that the sensibility of the 
 apparatus, if the period of oscillation is always the same, is in- 
 dependent ofs the linear dimensions of the apparatus ; and re- 
 marked that the statements of Cornu on this point (page 125) 
 are not correct. There are great advantages to be gained by 
 reducing the dimensions of the apparatus of Cavendish 50 or 
 100 times ; the main one is that the possibility of variation of 
 temperature in the apparatus is enormously minimized. Then, 
 too, the case can be made cylindrical and corrections for its at- 
 
 185 
 
iMKMOiU.S (IN 
 
 ;i.i 
 
 h-t- 
 
 traction avoided. Until quartz fibres existed it would have 
 been impossible to have made this reduction in the dinuMisions 
 of the apparatus and retained the period of 5 to 10 minutes. 
 The introduction of tiiis invaluable new moans of suspension 
 is also due to I^rofessor Boys. Another improvement in the 
 form of the apparatus devised by him is the suspending of the 
 small bails at diiferent distances below the arm (the masses 
 must bo at corresponding levels), so that each mass acts prac- 
 tically on one ball only. 
 
 Boys showed to tho Society a balance of this design ; it l»ad 
 an arm of only 13 mm. in length, was 18.7 times as sensitive as 
 that of Cavendish and behaved very satisfactorily. He pro- 
 posed to prepare a balance of this kind especially suitable for 
 absolute determinations and capable of determining the gravi- 
 tation constant to 1 part in 10 000. 
 
 An account of his completed work (187 and 181)) was read in 
 1804. For the details of this beautiful experiment and the in- 
 genious way in which the apparatus was designed the original 
 paper must be consulted. The general design was that of his 
 earlier apparatus, but very great attention was given to the 
 minutest details, and especially to the arrangements for meas- 
 uring the dimensions. Some idea of the accuracy aimed at 
 may be got from considering that in order to obtain a result 
 correct to 1 in 10 000 it was necessary to measure the large 
 masses to 1 in 10 000, the times to 1 in 20 000, some lengths to 
 1 in 20 000 and angles to 1 in 10 000. The dimensions finally 
 used were, diameter of masses 2.25 and 4.25 in.; distance be- 
 tween masses in plan 4 and 6 in. ; distance between balls in 
 plan 1 in. ; diameter of balls .2 smd .25 in. ; diflference of level 
 between upper and lower balls « in. The masses were of lead 
 formed under great pressure, and the balls of gold. 
 
 The moment of inertia of the beam was determined by find- 
 ing the period when the balls were suspended from it, and 
 when they were taken away and a cylindrical body of silver, 
 equal in weight to the balls with their attachments, suspended 
 from the middle of the beam. The apparatus was enclosed by 
 a series of metallic screens to prevent temperature changes, 
 and outside of all was a double -walled wooden box with the 
 space between the walls filled with cotton-wool. The final result 
 for the gravitation constant was 6.6576 x 10~^ and for ^, 5.5270. 
 The last figure in each ca&e has no significance, but Boys con- 
 
 136 
 
 I 
 
TllK LAWS OF GRAVITATION 
 
 Ind- 
 land 
 ver, 
 Ided 
 
 by 
 
 jes, 
 the 
 uilt 
 570. 
 ton- 
 
 sidored that the next to tlie hist couhl not he more tlian 2 in 
 error at the outside, lie is still ronviiiced that I part in 
 10 000 can be reached, but would increase the length of tin; 
 beam to 5 cm., since the disturbing moments due to convection 
 are proportioual to the r)th power of the linear dimensions, not 
 to the 7tli as he liad originally supposed. An excellent resume 
 of the experiment is to be found in the lecture delivered by 
 Boys before the Royal Institution (188). 
 
 EoETVOES. A series of investigations upon g»"ivitation is now 
 under way by Prof. R. von Eotvcis of Budapest. He has pub- 
 lished a preliminary account (H):i) only of his experiments, but 
 they promise to be very elaborate :i;id exhaustive. His paper 
 begins with a mathematical discussion of the space -variation 
 of gravity as deduced from the potential function. He investi- 
 gates the equipotential surface and the measurements necessary 
 to determine tlie principal radii of curvature, the variation of 
 gravity along the surface, and the variation perpendicular to the 
 surface. The latter has already been measured with the pend- 
 ulum, and by Jolly (144 and 145), Keller (15'^), Thiesen (171)) 
 and others with the common balance. For the measurement 
 of the other quantities von Eutvos uses the torsion balance. 
 This he makes in two forms : the first is of the same general 
 type as that of Baily and is called the K'7'ummfut(/sr((ri<miefcr, 
 since it is used to measure the difference of the reciprocals of 
 the principal radii of curvature ; the second is like that of 
 Boys in that one ball is on one end of the rod and the other 
 suspended 100 cm. below the other end by means of a wire, 
 and is called the Horizontalvarionieter. The peculiarity of 
 these instruments is in the method devised for getting rid of 
 convection currents ; von Eotvos makes the case with double 
 walls of thin metal with an air-space of from 5 to 10 mm. So 
 steady is the motion that the balance can be used i?i any room 
 in the laboratory, and even in the free air at night. The period 
 is usually froiA 10 to 20 minutes, and the suspension wire is of 
 platinum of 100 to 150 cm. length. The rod swings in a flat 
 cylindrical box 40 cm. in diameter and 2 cm. deep. 
 
 Some investigations have been made of the variation of 
 gravity in the neighbourhood of the hill S^ghberg, where von 
 Sterneck found great peculiarities. Some preliminary deter- 
 minations have been made of the constant of gravitation also, 
 
 137 
 
f 
 
 I i- 
 
 
 [' i 
 
 
 MEMOIRS ON 
 
 with a result 6.05xl0~^ Von Eotvos speaks of tlie method 
 cr,ploye(] as jin entirely new one, but it is only a variation of 
 that already einj)loyed by Reich, and later by Dr. Braun, the 
 oscillation method. The instrumeTit (the Kriimmungsvariom- 
 eter) is set up between two piUars of lead, and the time of vib- 
 ration observed both when the torsion rod is in the line joining 
 the pillars and when it is perpendicular to this line. The 
 paper is characterized by an almost total disregard of the work 
 already done in the field of gravitation. 
 
 Eotvos gives a description of two new instruments for use 
 in the study of gravitatioil. One he calls the Graviintioncom- 
 penmtor ; in design it is similar to the others, but tde arm 
 swings in a narrow tube. The tube is surrounded at each end 
 by the compensating masses having the balls at their centres ; 
 tliese masses are of the shape of a disc with two almost com- 
 plete quadrants taken away, just enough being left to hold the 
 remaining two quadrants together. By orienting these masses 
 any amount of compensating attraction required can be pro- 
 duced. The other instrument is called the Gravitationmulti- 
 plicator ; underneath the torsion balance is a turn-table with 
 the attracting mass ; when the ball has reached its maximum 
 elongation in the direction of the mass, the latter is suddenly 
 moved to the opposite side, and so on. From the difference 
 of two successive elongations, and a knowledge of the damp- 
 ing, the amount of the attraction can be determined. This 
 is rather similar to a piece of apparatus proposed by Joly (177), 
 in 1890. 
 
 Braun. One of the latest and most elaborate determinations 
 of the mean density of the earth is that made by Dr. Carl 
 Braun, S.J., at Mariaschein ^n Bohemia (193). In its gen- 
 eral form the apparatus is like that employed by Reich in his 
 later experiments. Like Reich, too, he uses two distinct 
 methods of finding his results, the deflection and the oscilla- 
 tion methods. The experiments of Dr. Braun differ however 
 in several very important respects from those of Reich ; the 
 dimensions of the apparatus are much reduced, the masses are 
 suspended from wires and the deflection is determined differ- 
 ently ; but the respect in which it differs from all previous de- 
 terminations is in the fact that the torsion rod swings in a 
 partial vacuum of about 4 mm. of mercury. This was sug- 
 
 138 
 
THE LAWS OF GRAVITATION 
 
 gested by Faye (130) anrl by Boys, but no investigation of the 
 kind had ever been made. 
 
 The experiments were begun in 1887. In a corner of a liv- 
 ing-room a heavy stone slab was set into the stone wall ; on 
 this was a glass plate from wliieh arose a brass tripod to carry 
 the suspension wire, 1 m. long, and the torsion-rod ; and on 
 the plate fitted airtight a conical glass cover within which a 
 vacuum could be made. The apparatus was so tight that the 
 pressure inside did not change in 4 years. Suspended from a 
 movable ring encircling the glass cover were the two masses, 
 about 42 cm. apart ; the masses used were two sets of spheres, 
 one of brass weighing 5 kg. each, and the other of iron, 11'^ mm. 
 in diameter, filled with mercury, and weighing about 0.15 kg. 
 each. The torsion rod was a triangle of coj)per wires and the 
 balls were suspended from its ends and lay in the same hori- 
 zontal plane 24.6 cm. apart. Each ball was of gilded brass and 
 weighed about 54 gr. In order to provide against temperature 
 changes, the whole apparatus was surrounded by metal screens, 
 cloth hangings and wooden enclosures. 
 
 The deflection method of observation is practically that of 
 Cavendish, but the position of the centre was found rather 
 differently. Dr. Braun observed the time of the passage in 
 each direction across the wire of the telescope of several scale 
 divisions near the centre, and took as the centre that point 
 with reference to which the time of oscillation was the same in 
 both directions. He found for the final corrected mean result 
 A = 5-. 52962. 
 
 In the oscillation method the period was determined when 
 the masses were in the line joining the balls, and also when they 
 were in a line at right angles to that direction. The final cor- 
 rected mean result gave A = 5. 52020. The mean of all is 
 5.52945±.0017. The extremes were 5.5004 and 5.5511. The 
 mean of all the results found in 1892 was 5.52770, and in 1804 
 was 5.53048. 
 
 The final result for the gravitation constant was 6.655213 
 X 10-8. 
 
 In an appendix is given a further discussion of the correc- 
 tions to be made on account of damping. Dr. Braun found his 
 former estimate to be in error, and after lamination gave as 
 the most probable final results, A = 5.52700±:.0014, and G 
 =(6. 65816 ±.00168) x lO-^. 
 
 139 
 
 H 
 
 I ! 
 
 ii! 
 
 il 
 
i 
 
 I*; 
 
 
 > I 
 
 1; 
 
 
 It. ' 
 
 E« » 
 
 r 1 
 
 
 
 M K M () I II S () N 
 A concise Jiccoinit of the work la given in Nature (107). 
 
 KoKNio, Ru'HAiiz AND KuiUAU - Mknzki.. Ill 1884, Pro- 
 fessors A. Konig sind V. Uidiiirz proposed (150 und UiO) to 
 doterniine tlie gnivitiition constjint by a nietliod wliicli is ii 
 niodificution of tlisit used by Jolly (page Vlij). In the hitter 
 experiment the h)\ver set of scale pans was 'i\ in. beneatli the 
 upper and ditTerences of temperature were unavoidable. Tiie 
 improvement proposed was to have the sets of scale pans much 
 closer together, to measure the cliange of weight with height 
 after the manner of Jolly and then to insert between the upper 
 and lower pans a huge block of lead with holes in it for the 
 passage of the wires supporting the lower pans. A weighing 
 was made with two nearly equal masses, one in the right upper, 
 the other in the left lower pan ; then the former in the right 
 lower is balanced against tlie latter in the left upper pan. From 
 these weighings, taking account of the result of similar ob- 
 servations without the block, the value of 4 times the attrac- 
 tion of the block is determined, and from a comparison of this 
 result with the calculated attraction, the gravitation constant 
 can be determined. Professors Konig and Richarz seem to 
 have hit upon the same idea independently of each other. In 
 1881, Keller proposed (1H7) a somewhat similar modification of 
 the Jolly experiment. Professor A. M. Mayer suggested (101) 
 the use of mercury instead of lead for the attracting mass, but 
 Konig and Richarz replied (1<)'^) that Mayer had misunder- 
 stood the forin of the experijuent, and gave a lucid and simple 
 explanation of their method. 
 
 In 1803, appeared a report (183 and 184) on the observations 
 made to find the decrease of gravity with increase of height. 
 A description is given of the balance and of the improvements 
 introduced into it in order to overcome the liability to varia- 
 tion in its readings. The masses weighed against each other 
 were 1 kg. each, and the balance had a sensitiveness of 1 part 
 in 1 000 000. All exchange of weights was made au^^omatically 
 without opetiing the covers. The apparatus was carefully sur- 
 rounded with metal screens to ward off temperature changes. 
 It was set up in a bastion of the citadel at Spandau, and in 
 consequence of the departure of Konig to accept a professor- 
 ship at Berlin, Dr. Krigar-Menzel assisted in the carrying on 
 of the research. The pans were '^.20 m. apart vertically. The 
 
 140 
 
 3"'! 
 
THK LAWS OF GRAVITATION 
 
 m. 
 
 whereas the 
 
 01) 
 
 n\t 
 
 er- 
 
 iple 
 
 iilly 
 
 in 
 
 for- 
 
 011 
 
 ^he 
 
 change in ffnivitv observed was .0000005:^3 , 
 
 cjilculjited value was .O0OOOG07. The difToronce is ascribed to 
 tlie local attraction of walls, etc. 
 
 The paper [V.^S) embodying the final results was presented 
 to the Berlin Academy in Dec, 1897. A most exhanstive ex- 
 amination had been made of the jiossible sources of error, and 
 the devices for overcoming these difficnlties were most ingen- 
 ious and elaborate. In the cases where the sources of error 
 could not be eliminated, as in the variations of temperature with 
 time and place, the otfect is carefully considered and allowed 
 for. Observations were made continuously from Sept., 18!M), to 
 Feb., 181M), and from the elegance of the method and the time 
 and care devoted to the working out of the result, this deter- 
 mination of the gravitation constant and the mean density of 
 the earth must be taken as one of the very best. 
 
 The block of lead weighed 100 000 kg., was '^00 cm. high and 
 JilO cm. square and was built up out of bars of lead 10 x 10x30 
 cm. on the top of a massive pier. The amount of the settling 
 of the pier was measured and found to be not important, and 
 the shape of the block was not distorted by its own pressure. 
 The final value for U was (0. 085 ±.011) 10-s, and for A. 5.505 
 ±.009. 
 
 Professors Richarz and Krigar-Menzel published (191 and 
 199), in 1890, f condensed account of their work. Other ab- 
 stracts are also to be found (185, pp. 04-5, 180^, pp. 20-7, and 
 194). 
 
 Minor Notices. In 1889, Dr. W. Laska of Prague proposed 
 (173) a method of finding the density of the earth. At the toj) 
 of a rod projecting above a pendulum is a lens which is so close 
 to a fixed plate of glass that Newton's rings are visible. A 
 hollow ball near the bob of the pendulum is then filled with 
 mercury and attracts the bob, bringing the lens nearer the 
 plate ; an observation of the movement of the Newton's rings 
 will measure the deflection of the bob. No further report has 
 been published. An account of his method is given by Gttnther 
 (19G|, vol. 1, p. 197). 
 
 About the same time Professor Joly of Dublin suggested 
 (177) a resonance method for the same purpose. A pendulum in 
 a vacuous vessel has the same ])eriod as two massive ones kept 
 
 141 
 
M KM 1 IIS ON 
 
 
 going outsiile tlio vessel. The amplitude of tlie motion of the 
 inner pendulum duo to a given number of swings of the outer 
 ones would give a measure of the constant of gravitation. 
 
 In 1895, Professor A. S. Mackenzie of Bryn Mawr College 
 publislied an account (100) of some experiments with the Boys' 
 form of torsion balance to determine whether the gravitational 
 properties of crystals vary with direction. No such variation 
 was found in the case of calc-spar, the crystal under investiga- 
 tion. He shewed further that the inverse square law holds 
 good in the neighbourhood of a crystal to one-fifth per cent. 
 
 Two years later ai)peared an account (190) of an investigation 
 by Professors Austin and Thwing of the University of Wiscon- 
 sin to determine whether gravitational attraction is indepen- 
 dent of the intervening medium, that is, whether there is a 
 gravitational permeability. No effect was found due to the 
 medium within the limits of error of the method. 
 
 At a meeting of the '* Deutcher Natnrforscher und Aerzte" 
 in Brunswick, in 1807, Professor Drude read a paper (195) on 
 action at a distance, wiiicli contains a very valuable account of 
 the theory of gravitation, and should be consulted by any one 
 wishing to find a brief resume of that subject, and especially 
 for a discussion of the velocity of propagation of gravitation. 
 
 The latest work on the laws of gravitation is that of Profess- 
 ors Poynting and Gray (200) on the search for a directive 
 action of one quartz crystal on another. A small crystal was 
 suspended and its time of rotative vibration noted ; a large 
 crystal in the same horizontal plane was then rotated about 
 a vertical axis through its centre with a period either equal to, 
 or twice, that of the smaller crystal. If there were any directive 
 action the small crystal should be set in vibration by forced 
 oscillations ; no such effect was found. 
 
 ^; 
 
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 142 
 
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MEMOIUS UN 
 
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 II; 
 
THE LAWS UF GKAVITATIUN 
 
 BIBLIOORAPITY 
 
 ^■2 
 
 SS 
 -H-H 
 S| 
 
 u: o 
 II II 
 ->? 
 
 a 
 
 io 
 
 s 
 
 c 
 u 
 
 a 
 o 
 
 a 
 
 •o 
 
 S 
 
 p 
 
 3 
 
 to 
 
 3 
 
 es 
 
 c 
 S 
 
 a 
 
 3 
 
 2 
 
 CQ 
 
 1 
 
 1600 
 
 W. Gilbert. 
 
 2 
 3 
 
 1665 
 1687 
 
 P. Bftcon. 
 I. Newton. 
 
 4 
 5 
 6 
 
 7 
 8 
 
 1705 
 1727 
 1744 
 1749 
 1751 
 
 R. Hooke. 
 I. Newton. 
 R. Boyle. 
 P. Bongner 
 Ch. M. (\c 1 
 
 77i« numbers marked with an asterisk have not been cmwilled by the nlUnr. 
 
 De magnete magneticiscinc corporibus et de 
 
 nmgno mngnete tellnre pliysiologia nova. Lon- 
 
 flon. 4*°. 
 
 Opera omnia— Frankfurt "/m. Folio. 
 
 Philosophiae loiturulis principiii miithematica. 
 
 London. 4'". 
 
 Posthumons works. London. Folio. 
 
 De mundi sysninute. London. 8*". 
 
 Works, edited hy Birch. 5 vol. Loudon. Folio. 
 
 La figure dc la terre. Paris. 4'". 
 Ch. M. dc la Conduniinc. Journal du voyage. Paris. 4'°. 
 8i* 1752-4 Ch. M. de la Condamine. Supplement au journal historique. 
 
 2 parts. Paris. 4'°. 
 
 Sur la direction qu' afifectent les fils4plomb. 
 
 Hist, de I' Acad Roy. den Sc. avec les Mem. de 
 
 Math, et de Phys. ,1-10 and 150-168. 
 
 The history of the Royal Society of London. 
 
 4 vol. London. 4*°. 
 
 (Letter of 30 pp.). Journ. des 8c. et des Beaux 
 
 Arts. June. 
 J. d' Alembert. (Letter). Journ. des Sc. et des Beaux Arts. July. 
 i7bl-80 J. d'Alembert. Opuscules mathematiques. 8 vol. Paris. 4'°. 
 1771 J. P. David. Dissertation sur la figure de la terre. La Haye. 
 
 8'«>. 
 
 (Letter of 27 pp.). Journ. des Sc. et des Beaux 
 
 Arts. Dec. 
 
 Solutions des doutes. Journ. des Sc. et des Beaux 
 
 Arts. April. 
 17* 1772 J. J. ^a Laude. Remarques sur de nouvelles experiences de 
 
 pesantcur. Le Journal des Sfavans. Aug 
 de la P. de Roiffe. Experience du pendule de le McrciCk mx 
 
 AlpesduValais. Journal Encyclopedique, Xi^O, 
 
 250. 
 G. L. Lesage. Lettre sur la faussete de deux suites d*experi- 
 
 ences. [Bozier^ Journ de Phys., 1 , 249-260. 
 Father Bertier. (Letters). {Rozier'\ Journ. de phys., 3, 251-2 
 
 and 275. 
 
 145 
 
 9 1754 P. Bouguer. 
 
 10 1756 T. Birch. 
 11* 1769 J. Coultaud. 
 
 12* 
 
 13 
 
 14* 
 
 1769 
 
 15* 1771 Mercier. 
 
 16* 1772 G. L. Lesage. 
 
 18* 1772 
 
 19 1773 
 
 20 1773 
 
I 
 
 21 1773 
 
 22 1773 
 
 23 1774 
 
 24 1774 
 
 25 1774 
 20* 1774 
 
 27 1775 
 
 28 1775 
 
 29 1775 
 
 30 1775 
 
 31 1775 
 
 32 1775 
 
 33 1776 
 
 34 1776 
 
 35 1776 
 
 36 1777 
 
 .1. P. Diwld. 
 Abbu Uozier. 
 
 MEiNK^lKS ON 
 
 (l«!liiP. <1(! UoifTe. OhHorvations HiirlVxperlcnccMlc |H*r('IJ«!rli(r. 
 \Iiozi,r] Jonrn. de P/i!/n., 2, 374-H. 
 
 G. L. Lefliige. KutlcxioiiH 8tir uiic noiivclle ('Xporicnce dii Ku- 
 vtircnd Pi^re Ucrtier. [Hnzier] Journ. de Phyn., 
 2, 378-81. 
 
 J. P Diivhl iuul Fiilhers VoiW, nnd Ik-ilicM*. (Notice of their ex- 
 p(Miin«;iits). {Uozitr] .hmni. dr Phyn., 4, 338. 
 |{epoi)geuiix rOrtoxioiiH tic M. Leaiige. [Roziei\ 
 JoHvn. de P/iyx., 4. 431-41. 
 Observiilions hut l.i Ictirt! de Pere Bertier. 
 [liozier] Journ. de I*/iys., 4, 454-61. 
 
 Father Bertier. (Account of experiments). Jonrn. de Verdun, 
 148-185. 
 
 .1. P. David. Sur la pesanteur des corps. {Rozier'\ Journ. de 
 Pkyit., 5, 120-139. 
 
 Fatlier Bertier. (Letter). {R>zin] Jonrn. ile Pfiyn., 5, 305-18. 
 — (Account of cxp'lH. tniMle by coin, of Acad, of 
 
 Dijon). [liozier] Journ. de P/iyn., 5, 314-26. 
 
 Chev. de Doloniieii. Experiences siir la pesantenr des corps t\ 
 differentes distances dn centre de la tene. 
 [Rozier] Journ. de P/iyn., l\, 1-5. 
 
 N. Maslielyne. A proposal for measuring llie attraction of some 
 bill in this king<lom by astronomical observa- 
 tions. Phil. Trann. Jjond., 495-9. 
 
 N. Maskelyne. An account of observations made on the moun- 
 tain Schehallien for finding its attraction. Phil. 
 Trann. Loud., 500-42. 
 
 F. K. Achard. Bemerkung<!n Uber die von Herrn Bertier an- 
 geslelllen Versuclie. lieschiift. der Berl. Oes. 
 Naturf. Freunde, 2, 1-11. 
 Experiences et vues sur I'intensite de la pes- 
 anteur dans I'inlerieur de la terre. [Rozier] 
 Journ. de Phyn., 7, 1-12. 
 
 Discours sur I'altraction des montagnes ; traduit 
 par M. le Hoy. [Rozier] Journ. de Phys., 7, 
 418-34. 
 
 Retraction du P^re Bertier de I'Oratoire, sur la 
 consequence qu'il a tire de son experience d'un 
 corps, pesant plus dans un lieu haut que dans 
 un bas. [Rozier] Journ. de Phys., 9, 460-6. 
 An account of the calculations made from the 
 survey and measures taken at Schehallien in 
 order to ascertain the mean density of the earth. 
 Phil. Trans. Lond., 68, 689-788. 
 Calculations to determine at what point in the 
 side of a hill its attraction will be the greatest. 
 Phil. Trans. Lond., 1-14. 
 
 Experiments to determine the density of the 
 earth. Phil. Trans. Land., 88, 469-526. 
 146 
 
 Q. L. Lesage. 
 
 J. Pringle. 
 
 Father Bertier. 
 
 37 1779 C. Hutton 
 
 38 1780 C. Hutton. 
 
 39 1798 H. Cavendish. 
 
T II K LAWS ( ) K ( J U A \' I T A T I ( ) N 
 
 89i 1799 
 40 1799 
 
 L. W. Gilbert. 
 
 an- 
 
 Oes. 
 
 the 
 I in 
 
 mb. 
 
 the 
 
 test. 
 
 the 
 
 41 1808 A. Motte 
 
 43 1806 
 
 43 1809 
 
 44 1810 
 
 45 1811 
 
 F 
 
 P. X. von Znch. 
 
 C. Hiittou. 
 
 46 1811 J. Play fair 
 
 47 1812 
 
 48 1818 
 
 49 1814 
 
 50 1815 
 
 51 1819 T. Young 
 
 52 1820 
 
 53 1821 
 
 54 1821 
 
 (U.'vi«'w of 80). imi. lint., 11. 288-41. 
 
 VtMHUclic, iiin (lie Dichti^ktit /.u Ixstinitnfti, 
 von llenrv Ciivnidlsli, Ksq. [GUlHi't] Ann. (Ur 
 /»/<//«., 2/1-02. 
 
 The niutlii tnatieal principles of natural |)lii 
 losopby by Sir I. Nuwton, traiiHlateil into Knglisli 
 Ity Andrew IMotte, to whicli are added Newton's 
 syHteni of the world, etc. VV. Davis' e<l". 8 
 vol. London. 8'". 
 
 H. W. Rranch'.s. Theoretisclie I'nter.sueliungen nbi-rdieOseiila- 
 tionen d<'r Drehwaage bei Oavendish's Ver- 
 8U(!hen Uber die Attraction kleiner MaNsen. 
 Mag. fiu' den Nt'iienten ZKntand tltr Aalurkiindc, 
 12, 800-810. 
 X. von Zacli. lJel»er dit; IM(\i;llchkeit die Oestalt der Krde 
 atiH Gradniessiingen /ai bestiniwen. Momill. 
 Corre>ip.,*li\\\A). 
 
 Ueber Densiidt der Krde und deren EintUis.s 
 anf geographisclie Ortsbestininuingen MomiU, 
 6V/v.<*/>, 21,298-810. 
 
 On the ealenlations for ascertaining the meiin 
 den.sity of the earth. [T:il<>cfi] Phil. Mny.,l\H, 
 112-6. 
 
 Account of a lithological Hurvey of Schehallien. 
 made in order to determine the .specitic gravity 
 of the rocks which compose that mountain. 
 Phil. TmuH. Lond./Ml-n. 
 Tracts on mathematieid and philosophicul Nub- 
 jects. 3 vol. London. 8"°. 
 Bericlit von einer lithologischon Aufnahme des 
 Schehallien, um das specifiscrhe Oewicht der 
 Gebirg.sarten dessejhen, und daraus die mittlere 
 Diehtigkeit der Erde zu be^tiinmen, von J. Play- 
 fair. Esq. Pof/f/ Atin., 43. 62-75. 
 
 F. X. von Zach. L'attraction des montagnes, et ses effets sur les 
 tils il plomb ou sur les niveaux des instruments 
 d'ustroMomie. 2 vol. Avignon. 4*°. 
 
 N. M. Chompre. E.xperiences pour determiner la denslte de la 
 terre ; par Henry Cavendish. Traduil de I'An- 
 glais. Jonrn. de Vfjc Hoy. Polytechnique. Cfthier 
 ^ 17. 10, 268-320. 
 
 Remarks on the prol)ahilities of error in physical 
 observations, and on the density of the earth. 
 Phil. Trans. Lond., 70-95 ; Misc.'Workfi, 2,8-28. 
 (Letter to liaplace). [Blainville] Journ. de Phys., 
 90, 307-12. 
 
 On the mean density of the earth. ITilloch] 
 PJiil. Mag., 68. 3-13. 
 
 (Same title as 53). Phil. Tranx. Lond. , 276-292. 
 147 
 
 C. lIultoD. 
 L. W. Gilbert 
 
 C. Hutton. 
 
 C. Hutton. 
 
 C. Hutton. 
 
MKMUIUS ON 
 
 5.") IH'.M F. Curlinl. 
 
 50 1825 8. 
 
 57 1H20 M 
 
 5H 18'^7 K 
 
 Ml IH'J? M 
 
 60 1827 — 
 01 1825-45 
 02* 1828 — 
 
 68 1828 M. 
 
 04 1829-30 
 
 65 185}:} S, 
 00 185J7 K. 
 
 07 1838 F. Heicb. 
 
 08 1838 — 
 
 69 1839 F 
 
 70 1840 C. 
 
 71 1840 L. 
 
 72 1840 A. 
 
 73 1841 L. 
 
 74 1842 J. 
 
 OfiservHzlonI drllii Iiini?liP//,ii drl pon<l<ilo sem- 
 plicr fiittt! 111!' hII(>/./ii (li iiiilli; Ichc niiI livolldtiel 
 miiro. Kff. Axtr. Ui }filtiii(>, h\\\). 28-40. 
 (Notice of 55). [FeruHnar] Hull, di-n Se. Math., 
 a. 298-301. 
 
 VV. Drobiflcii. Do vein lunne fl^^iiru. liipsiiie. 12""'. 
 
 S(iiliiiie). All tu;cotiiit of Pruf. Ciuliiii'H expcriinciits on 
 Moni (;<'iiiH. Quart. Joiirn. ofSe., 24, 153-9. 
 
 W. DrobiHch. I'cbei'dii^ in dcii Mincii von Dolcoatli in Ooiii- 
 wuli ncucrlidi Hiiv;(^stellt(!n l'cndcll)eobiU!htiiiiff- 
 en. /%//. A nn . . I O. 444 450. 
 (Notice of Dolcouth cxpi). IViil. May., [2|, 1, 
 385-0. 
 
 .I.S. T. Oeblei. Pliysiliiilisclies WOiterbuch. 22 vol. Leip- 
 zig. H"". 
 
 Account uf e.\pci'iraent.s inudc ui Dolcoatli mine 
 in C'ornwull, in 1826 and 1828, for tlie pur|)osc 
 of detei'inining ilic density of tlie eurth. Cum 
 bri«lge. 8*". Printed privuiely. 
 
 \y. Drobiscli. AiisfUhrlielier Ucrlcht llbcr inelirerc in den 
 Jahren 1820 iind 1828 in den Minen von Dolcotitii 
 io Cornvvull ziir Bestiramung der niittleren Dicli- 
 tlgkeit der Knle ungeHtellte I'endelversucbe. 
 Pogff. Ann., 14,409-27. 
 
 J. C E. Scliinidt. Jidirbiicli der mutliemiitisclien und pliy- 
 sisclien Ocograpliic. 2 vol. GOtiiiigen. 8*'". 
 
 I). PoissoD. Traitede iiu'c,iiii(pU'. 2'' ed". 2 vol. F^aris. 8'". 
 
 d(; Beaumont. Fxtruitd un incinolrc de M. Ueicli Hiir la dcii- 
 site dc la terrc Cotitp. Uend.. 5, 097-700. 
 Versuclie ttbcr die iniitlcie Diclitigkeit der Erde 
 mittelsl der Drehwage. Freiberg. 8*". 
 On the repetition of the Cavendish experiment, 
 for determining the mean density of the earth. 
 Phil. Mag.,[^, 12, 283-4. 
 (Same title aa 08). Man. Nut. Roy. Aatr. iSoc, 
 4, 90-7. 
 
 Sur la determination de In densite moyoime de 
 la terre, deduite de I'observation du peminle faite 
 a I'Hospice du Mont-Cenis par M. Caiiini en 
 Septcmbre, 1821. Mem. Accad. Torino, [2], 2, 
 379-84. 
 
 F. Menubrea. Calcul de la densite de la terre. Mem. Accad. 
 
 Torino. [2], 2, 305-08. 
 
 G. Calcul de la densite de la terre, par L. F. Mena- 
 brea. Bibl. Univ. de Geneve, [nouv.], 27. 108-75. 
 
 F. Menabiea. On (Cavendish's experiment. Phil }faf/., [3], 
 
 19, 02-3. 
 F. Saigey. Densite du globe. Rev. Scient. etind., [Quesne- 
 
 lille]. 11. 149-60 and 242-53. and 12. 373-88. 
 148 
 
 Baily. 
 I. Giulio. 
 
Til K LAWS OK <i KA VII'ATIoN 
 
 [3], 
 
 'fine- 
 88. 
 
 
 75 IH43 F. Haily. 
 
 76 IH42 F. FtuHy. 
 
 77 1842 
 
 78 1H4;} 
 
 79 1843 
 
 80 184;{ 
 80i 1845 
 
 F. Ilaily. 
 
 F. Bully. 
 F. Bully. 
 
 A. G. 
 
 ('. A F. Pouts, 
 
 81 1847 G. W, Ilearn. 
 
 82 1849 E. Sabine. 
 
 88 185a F. Reich. 
 
 84 1853 — 
 
 85 1853 Scliaar. 
 
 86 1853 — 
 
 87 1853 
 
 88 1855 
 
 89 1855 
 
 90 1855 
 
 F. R^ich. 
 
 G. B. Airy. 
 G. B. Airy. 
 
 An atM-otiMt of Honx- <x|H>riini'iiiM with tlm tor 
 hIoii rod, for ilctcriiiinlii^ tin- mean ilniMiiy of 
 the •Mirlh. /»/*//. .\/,i;/ , |;{|, 21, il I •,»!. 
 Bi'Sllllats lie (|l|)-|<|ll)'H c.XlttM iriirf> lulliH iivcc lit 
 
 hulann' ilc iiiisiiiii, |i(iiir tU'-irniiini'r In dciiHiiC' 
 niovcMiinlr III Irrro. Ann. ilr Chiin. tttlr l*/ii/tt., 
 
 |:<|! rt. :{;iH-5:{. 
 
 licriclit von cinl^rii Vcrsiichcti mil dcr Dich- 
 
 wa^e /iir ItiNtimmimgdtr mlltlcnn Dichligkelt 
 
 dcr Krde. /•»/////. Ann., 57. 4:.:{ 67. 
 
 (Siiint! litic as 75). .\t<>ii. .\i<f Uoi/. Axtr. .*v>r. , 
 
 rt, IHH II nd 197-'.»()6. 
 
 KxpcriiiuMits with the torsion tod ft)r dctcrndn- 
 
 iuu; the mean diMisily of the earth. Mmi li'ii/. 
 
 Astr. .*^/r , 14. 1-130 and i.-eexlviii. 
 
 (Same liile as 76). liihl. Unii\ ik (ienhr, [none], 
 
 4: J, 177-81. 
 
 Von <ien kleinen Ahlenkiin;;* n dcr FiOthlinic 
 (iiid dcs Niveaus, welelic diireli die An/iehuiip;- 
 er> tier Sonne, d«-s Mondes, initl eiiiiu;rr lern'S- 
 trischt'ii Grgmsiilndc jn'ivorgeltniclil wcKh-n. 
 Autt: X,irfi.,*2*2, 33-43. 
 
 On tlu! cause of the discrepancies observed by 
 Mr. Baily with the (.'avendi^h apparatus tor de 
 teiinining the mean density of the earth. Phil. 
 TinnH. Land., 317-39 
 
 Cosmos, by Alexander von llumitoldt, trans- 
 lated under the superintendence of Lieut. Col. 
 Edward Sabine. 6"' Ed". 1 . liondon. H*". 
 Neue Versuche nni der Di'ehwaage. Jj<ip. Abh. 
 math. phy. cL, 1. 383-430. 
 
 Neue Ver.HUche rd)er die ndlllere Dielititrkeit der 
 Erde. von F. Reich. /V/.V- Ami-, 8*>. lHO-98. 
 Rapport de M. Schaar sur un nienutire de M. 
 Montigny relalif aux experiences pour deter- 
 miner la densite tie la terre. hull. Acad. Ron, 
 Belg., 19. pt. 3, 476-81. 
 
 Nonvelles experiences sur la densite moyenne 
 de la lerre. Ann. de Chim. et de P/iyn., [3], HH, 
 383-3. 
 
 New experiments on the mean density of the 
 earth. P/iil. Mof/., [4], 5, 1.54-9. 
 (Report on lliirton expts.). Mon. Not. Roy., 
 Astr. Soc., 15,35-6. 
 
 Noterespeciing the recent experiments in the 
 Harinn Colliery. Mon. Not. Roy. Antr. Soc, 
 15, 46. 
 
 (Report on Harton expts.). Mon. Not. Roy, 
 Astr. Soc., 16, 135-6. 
 149 
 
MKMOIIIS ON 
 
 
 Ft> 
 
 
 91 1855 — 
 
 93 1855 — 
 
 93 1855 J. H. Pratt. 
 
 94 1855 G. B. Airy. 
 
 95 1855 
 
 96 1856 
 
 97 1856 
 
 98 1856 
 
 99 ,856 
 
 T. Young. 
 J. H. Pnitt. 
 G B. Airy. 
 
 J. H Pratt. 
 
 100 1856 G. B. Airy. 
 
 101 1856 G. B. Airy. 
 
 103 1856 G. G. Stokes. 
 
 103 1856 H. James and 
 
 Nolo sur Irs olwervations dii pendule executet-s 
 dans Ics inincs d«! Marlon |)our deierminer la 
 dcMisiie m<»y(!nn('d(!la terre : par M. Airy. Ann. 
 de C/iim. H dc PIiiih., [3], 4,*$, 381-3. 
 Exlrait (III rapp« rt preseiiit' tl la 35"'« seance 
 annivcr-saire de laSocieti' H^yale Astrononiiqiie 
 <Je Londies par \v. conseil (Ic ccile societ''; le 9 
 Fevricr, 1855. Arch, dex Sc. Phj/n. et Nat., liJ>, 
 188-191. 
 
 On the atinietion of the Himalaya mountains, 
 and of the elevaled rcgioris beyond them, upon 
 the plumb-line in India. Phil. Trans. Ijond., 
 145,53-100. 
 
 On the eDUiputation of the effect of the attrac- 
 tion of njounlain-masscs. a.s disturbini? the ap- 
 parent astronondcal latitude of siations in 
 ffeodetic surveys. Phil. Trans. Loud., 14l>, 
 101-4. 
 
 Miscellaneous works and lift^ by Peacock and 
 Leitch. 4 vol. London. 8'''. 
 (Same iltle as 93). Man. Not. lioy. Astr. Soc, 
 Hi, 36-41 and 104-5. 
 
 (Sa'ue title as 94). Mon. Not. Roy. Astr. Soc, 
 lO. 42-43. 
 
 (Report on Ha''ton expts.). Mon. Not. Hoy. 
 Astr. Hoc, 16, i04. 
 
 On the effect of local attraction upon the plumb- 
 line r-l stations on the Euirli^h arc of the merid- 
 ian, between Dunnose and Burleigh Moor ; and 
 a meihi'd of computing its amount. Phil. Trans. 
 jMnd., 14«. 31-5'>. 
 
 Account of p(!ndi;lum cxperimenis umlertaken 
 in th(! Ilarton Colliery, for the purpose of de- 
 termining the mean density of theearih. Phil. 
 Trans. Lond., 146, 997-342. 
 Supplement to the "account of pendulum ex- 
 perinients undertaken ia the Harlon Colliery" ; 
 lieing an account of experiments undertaken 
 to determine the correclioii for the temperatiu'c 
 of the pendulum. Phil. Trans. Lond., 146, 
 343-55. 
 
 (Addendum to 101 ; on the effect of the earth's 
 rotation and ellipticity in modifying the numer- 
 ical results of th<; Harton experiment). Phil. 
 Trans. Lond., 146, 353-5. 
 A. R. Clarke. On the deflection of the plumb- 
 line at Arthur's Seal, and tl.j mean specitie 
 (gravity oi the earth. Phil. Trans. Lond., 146, 
 591-6CG. 
 
 150 
 
THE LAWS OF GRAVITATION 
 
 
 104 1850 II. James. 
 
 105 1856 
 
 106 1856 S. Haughton. 
 
 107 1856 G. B. Airy. 
 
 108 1856 H. Jumes. 
 
 109 1856 G. B. Airy. 
 
 110 1856 — 
 
 111 1856 G. B. Airy. 
 
 113 1857 E. R. 
 
 113 1857 — 
 
 114 1857 — 
 
 115 1857 — 
 
 116 1857 — 
 
 117 1857 H. James. 
 
 118 1857 W. ^. Jacob. 
 
 119 1857 G. B. Airy. 
 
 120 1857 H.James. 
 
 On the flgur,", dimension.s and mean specific 
 gravity of llie earlli, as derived from the ord- 
 nance trigoiKiinetrical survey of Great Britain 
 and Ireiaiui. Pliil. Traus. Land., I4«, 607-26. 
 Ucher dii: in der Iv«)ideiigrnl)e von llarton /,nr 
 Bestiniiniiiig der tnittleren Di(;lile der £rde nn- 
 ternonimenen Pendelbeohaciilungen ; von G. B. 
 Ai ry . Pogg. Ann.,\)7, 599-605. 
 On tlie density of tlie jiirili, deduced fronfl tlie 
 experiments of the Astronomer Iloyal, in the 
 Harton coal-pit. Phil. Mag., [4], 12, 50-1. 
 (Same title as 100). Phil. May., [4J, 111. 226-31. 
 Account of the observations and computiU^ons 
 made for the purpose of ascertaining the amount 
 of the deflection of the plumb-line at Arthur's 
 Seat, and the mean specific gravity of the earth. 
 Phil. Mag., [4], 12,314-6. 
 (Same title as 101). Phil. Mag., [4]. 12, 467-8. 
 Ueber die Dichtigkeit der Erde, hergeleitet aus 
 den Verauchen des Kttnigl. Astronoraen (Hrn. 
 Airy) in der Kohlengrube Harton ; von Sr. Ehr- 
 wtlrd. Samuel Haughton, Fellow des Trinity 
 College in Dublin. Pogg. Ann., »9, 332-4. 
 On the pendulum experiments lately made in 
 the Harton Colliery, for ascertaining the mean 
 densitv of the earth. Am. Journ. Sc., [2], 21, 
 359-64. 
 
 Memoire sur lea experiences enterprises dans la 
 mine de Harton pour determiner la densite 
 moyenne de la terro, par G. B. Airy. Arch, des 
 Sc.'Phi/.'i. et Nat., 35, 15-29. 
 Ueber die Diciitigkeit der Erdc, hergeleitet aus 
 den Pendelbeobachtungen des Herrn Airy in 
 der Kohlengrube Harton von Herrn S. Haugh- 
 ton, Fellow am Trinity-Collfge in Dublin. Zeit. 
 filr Math. u. Phya., 2. 68-70. 
 Ueber die Bestimmungder nnttleren Dichtigkeit 
 der Erde. Zeit. filr Math. u. Phys.. 2, 128-30. 
 (Same title as 103). Proc. Boy. Soc. Edin., ii, 
 364-6. 
 
 (Notice of 106). Am. Journ. Sc., [2], 24, 158. 
 (Same title as 104). Phil. Mag. , [4], 1 3, 129-32. 
 On the causes of the great variation among the 
 different measures of the earth's mean density. 
 Phil. Mag., [4], 13. 525-8. 
 (Same title as 101). Proc. Roy. Soc. Land., 8, 
 58-9. 
 
 (Same title as 104). Proc. Boy. Soc. Lond., 8, 
 111-6. 
 
 151 
 
MEMOIRS ON 
 
 
 131 
 
 1857 
 
 W. 8. Jacob. 
 
 122 
 123 
 
 1858 
 1858 
 
 G. B. Airy. 
 
 124 
 
 1858 
 
 — 
 
 125 
 
 1858 
 
 H. James and 
 
 Proe. Roy. Soc. Lnrm., 8, 
 
 Proc. Hoy. Ind.,2, 17-22. 
 Man. Not. Roy. Astr. Sac, 
 
 Moil. Not. Roy. Adr. Soc, 
 
 r 
 
 126 
 127 
 
 128* 
 
 1859 
 1859 
 1859- 
 
 P. F. 
 
 60 
 
 129 1861 O. Struve 
 
 130 
 
 131 
 
 1863 
 1864 
 
 137 1873 
 
 (Same title as 118). 
 
 295-9. 
 
 (Sume title as 111). 
 
 (aame title as 103). 
 
 18,220. 
 
 (Siimc litlc as 104). 
 
 18, 220-2. 
 A. It. Clarke. Ordnance trigonometrical Survey 
 
 of Great Britain and Ireland, Account of llie 
 
 observiitions and calcniations of the principal 
 
 triangul.-ition ; and of the figure, dimensions and 
 
 mean specific gravity of the earth as derived 
 
 therefrom. 2 vol. London. 4'°. 
 
 (Same title as 125). Mon. Not. Roy. Astr. Soc, 
 
 1», 194-9. 
 J. Gosselin. Nouvel examen sur la densite moyenne de la 
 
 terre. Mem. Ac<ul. Imp. de Metz, [2], 7, 469-85. 
 E. Sergent. Sidla densita della materia nell' intorno del 
 
 globo, e suila polenza della crosta terrestre. Atti 
 
 della Soc. lUil. di Se. Nat. Milano. 2, 169-175. 
 
 Ueber eiiien von General Schubert an die Aka- 
 
 demie gerichteten Antrag betreflFend die Rus- 
 
 sisch • Scandinavische Meridian - Gradmessung. 
 
 Bull. Acad. St. Petersb. phys. math, cl., 3, 395- 
 
 424. 
 H. A. E, A. Faye. Sur les instruments geodesiques el sur la 
 
 densite moyenne de la terre. Comp. Rend., 56, 
 
 557-66. 
 E. Peclimann. Die Abweichung der Lothlinie bei astrono- 
 
 mischen Beobachtungsstationeii und ilire Be- 
 
 rechnung als Erforderniss einer Gradmessung. 
 
 Denkftchr. Acad. Wm. Wien. math.-naturw. cl., 
 
 22, 41-88. 
 
 Note surle calcul de I'experiencede Cavendish, 
 
 relative A la masse el fi, la densite moyenne de la 
 
 terre. Cosmos, 24, 543-5. 
 
 A treatise on attractions, Laplace's functions, 
 
 and the figure of the earth. '6^. Ed". Cambridge 
 
 and London. 8*". 
 
 Ueber die nnltlere Dichtigkeit der Erde. Zeit. 
 
 fur Math. n. Phys., lO, 224-7. 
 
 Ueber die Hestimniungder millleren Dichtigkeit 
 
 der Erde. G5ttingen. 4'°. 
 
 Sur le calcul de la densite moyenne de la terre, 
 
 d'apr^s les observations d'Airy. Ball. Acad. Roy. 
 
 Belg., [2], 33, 369-372 and 389-409. 
 A. Cornu et J, B. Bailie. Determination nouvelle de la con- 
 
 stante de I'attraction et de la densitfe moyenne de 
 
 la terre. Comjt. Rend., 7<», 954-8. 
 152 
 
 132 
 
 1864 
 
 J. Babinet. 
 
 133 
 
 1865 
 
 J. H. Pratt. 
 
 134 
 
 1865 
 
 H. Scheffler. 
 
 135 
 
 1869 
 
 A. Scbell. 
 
 136 
 
 1872 
 
 F. Folic. 
 
T II E L A W S O K R A \' I T A T I () N 
 
 la 
 
 .'it 
 
 138*1873 — (Notice of 137). Bull. Ilebd. de l\Umt\ Scient. (fr 
 
 France, [IJ. 12, 70. 
 
 139 1873 A. Coinu and J. H. Hiiille. Miitniii (Ictcnniniiiion of \\w cnn 
 
 stiuit of iiUriictioii, and of the mcim density ol 
 tlic eaiMi. Chemifdl Nrwx, 27, 211. 
 
 140 1873 I. Todliunler. A history of Wm iimiliemfUicil theories of at 
 
 tractirtn iitid the fiijiire of the earth, from the 
 time of Newton tothatof Laplace. 2 vol. Lon- 
 don. 8*". 
 
 141 1878 A. Cornii et J. B. Bailie, ^^tnde de la renistance de I'air dans 
 
 la balance de torsion. C'omp. lieitd., 8($, 571-4. 
 
 142 1878 A. Cornii et.I. li. Bailie. Sur la mesiire de la densite moyenne 
 
 de la terre. Cnnp. Jiaid., 8«, 61)9-702. 
 148 1878 A. Cornu et J. B. Bailie. Iidliieiice des terrnes proportionels an 
 
 cane des ecarts, dans U' nioiivemt-nl oscillatoin; 
 do la balance de torsion. Comp. liend., 80, 
 1001-4. 
 
 144 1878 Ph. von Jolly. Die Anwendiini; der Waagc auf Probleme der 
 
 Gravitation. Part 1. Abh. Bay. Aknd. Wisn.cl. 
 2. 1», Ahth. 1. 157-176. 
 
 145 1878 Ph. von Jolly. (Same title as 144). 117^^. ^mh.., 5, 112-34. 
 
 146 1879 J. IL Poynting. On a method of using the Indance with gnat 
 
 delicacy, and on its employment to determine 
 the mean density of the earth. Pr<>c. Rotj. JSoc. 
 Land., 28, 2-35. 
 H. A. E. A. Faye. Sur les variations seculaires de la figure 
 mathematique de la terre. Comp. Rend., OO, 
 1185-91. 
 Faye. Sur la reduction des observations du pen- 
 dule au niveau de la mer. Comp. Rend., 90, 
 1443-6. 
 F. R. Helmert. Die mathen^atischen tind physikalischen 
 Theorieen der hOlierei? jreodasie. 2 vol. Leip- 
 zig. 8*°. 
 O. Zanolti-Bianco. U problema meccanico della figuradella 
 terra. 2 parts. Firenze Torino-Roma. 8'°, 
 A. R. Clarke. Geodesy. Oxford. 8^". 
 O. Knopf. Ueber die Methoden zur Bestimmung der mitlleren 
 
 Dichtigkeit der Erde. Jena. 
 T. C. Mendenhall. Determination of the acceleration due to the 
 force of gravity, at Tokio, Japan. Am. Jonni. 
 i, Se., [3], 20, 124-32. 
 
 T. C. Mendenhall. On a determination of the force of gravity 
 atlhe summit of Fujiyama, Japan. Am. Journ. 
 Sc., [3J. 21,99-103. 
 153 1881 F. Keller. Sulla diminuzione della gravita coiraltezza. 
 
 Atti Accnd. TAncei. Mem. d. sc, [3]. J), 103-17. 
 153 1881 Ph. von Jolly. (Same title as 144). Part 2. Ahh. Bay. Akad. 
 
 WisHcl. 2. 14. Ahth. 2. 3-26. 
 153 
 
 146i 1880 
 
 147 1880 H. A. E. A 
 
 148 1880-4 
 
 148i 1880-5 
 
 149 1880 
 149^* 1880 
 
 150 H80 
 
 151 1881 
 
MEMOmS ON 
 
 M: 
 
 •I 
 
 1 
 
 
 154 1881 
 1544 188a 
 
 155 1883 
 
 156 1883 
 
 157 1883 
 
 158 1884 
 
 159 1884 
 
 160 1885 
 
 161 1885 
 
 162 1885 
 
 163 1885 
 
 164 1885 
 
 165 1886 
 
 166 1886 
 
 167 1886 F.Keller. 
 
 168 1887 F. Keller. 
 
 169 1887 J. Wilsing. 
 
 170 1887 J. Wilsing 
 
 Ph. von Jolly. (Siinie title as 153). Wied. Ann., 14. 331-55. 
 
 J. G. Walleiilin. UelxT <lie Metluxlen zur Bt'stinunung ilcr 
 initlleren Diclilc dcr Erde iiiid eine neue dies- 
 hezttglicliu Anvveiidung der Wage. Humboldt, 
 1, 312-7. 
 
 R. voij Steineck. Unterauclmngeri ttber die Schwere itn In- 
 nern der Erde. Mitth. Mil.-Oeog. Inst. Wieiii, 2, 
 77-130. 
 
 R. von Sterneck. Wiederlioliing der Untersucliungen Uber die 
 Schwere im Iniiern tier Erde. Mitth. Mil.-Qeog. 
 fust. Wien, ',i, 59-94. 
 
 J. B. Bailie. Sur la resistance de I'air dans les niouvements 
 oscillatoires trt's Icnts. Coinp. Rend. , OO, 1493-5. 
 
 R. von Sterneck, Unlersnchniigen liber die Schwere aiif der 
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THE LAWS OF (JUAVITATION 
 
 171 1888 J. H. Gore. 
 
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 173 
 
 1889 J. VVilsing. 
 
 173 
 
 1889 VV. Laska. 
 
 174 
 
 1889 J. H. Gore. 
 
 175 
 
 1889 C. V. Boys. 
 
 176 
 
 1889-90 (;. V. Br 
 
 177 
 
 1889-90 J. Joly. 
 
 178 1889-91 
 
 179 1890 Tliieseii. 
 
 180 1891 J 
 
 181 1893 A. Berget. 
 
 183 
 183 
 
 185 
 186 
 
 1893 
 1893 
 
 184 1894 
 
 1894 
 1894 
 
 187 
 188 
 
 1894 
 1894 
 
 Deterininaiioii of the mean density of the earlli 
 
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 Ueher eincn neuen Appiirat zur Be.slimmung 
 
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 A bibliogniphy of geodesy. Wasliingtnu. 4'". 
 
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 1887. 
 
 On the (.'avendisli Experiment. Proc. Hot/. Soc. 
 
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 Collection de memoires relatifs i\ la physique, 
 
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 Determination experimcntale de la constantede 
 
 I'atlraction universelle, ainsi que de la masse et 
 
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 Sur la realisation des temperatures constanles. 
 
 Comp. Rend., 117, 96-7. 
 F. Richarz und O. KriuaiMeuzel. Die Abnahnie der Schwere 
 
 niit d«r I lolie l»<si immt dureli Wilgungen. Sitz- 
 
 unffsh. A/>(i»f. Wixtt. Berlin, 163-83. 
 F. Richarz und O. Krigar-Menzel. (Same title as 183). Wied. 
 
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 J. H. Poynting. The mean density of the earth. London. 8'". 
 J. H. Poynting. A hisiciy of the methods of weighing tln^ 
 
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 Die Methoden zur Bcstimmung der mittleren 
 
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 On the Newtonian consiantof gravitation. Proc. 
 
 Roy. Soc. Lond., 56, 131-2. 
 
 (Same title as 187). Nature, 50, 330-4, 366-8, 
 
 417-9 and 571. 
 155 
 
 Gouy. 
 
 186i 1894 G. Fresdorf 
 
 C. V. Boys. 
 C. V. Boys. 
 
MKMOIKS ON TIIK LAWS OF GRAVITATION 
 
 II 
 
 19:5 1896 C. Brauu. 
 
 194 1896-7 — 
 
 195 1897 P. Dnidc. 
 
 189 1895 C. V. Bovs. (Same title as 187). Phil. TmuH. Lond., [A], 
 
 18«, 1-72. 
 
 190 1895 A. S. Mackenzie. On the attractionsof crystalline and isotropic 
 
 masses at small distances. /*////. Jicv., *2, 331- 
 48. 
 
 191 1896 F. Kicharz und (). Kii^ar-Menzel. Gravitationsconstante «nd 
 
 mittlere Dichtigkeit der P^nle, hestitnml durcli 
 Wilj^iingcn. Sitzin>f/nh. A/cad. Wixx. Berlin, 
 1305-18. 
 I!)2 1896 R. von E5tv5s. Unlersuclmnircn [Wwv Gravitation und Erd- 
 
 ma/L^nielisniiis. \Vi<'</. Ann., 51>, 354-400. 
 Di(! Graviiiitiotisconstanle, die Ma.ssc und mitt- 
 lere Diclite d(!r Enle nacli ciner neuen experi- 
 meniellen Hcstimmung. Denkxehr. Ahid. Wi»H. 
 Wien. niath.-nalnvw. d., «4, 187-258c. 
 The gravitation constant and the mean density 
 of tlu! eartii. Nature, 55, 296. 
 Uebcr Fernowirkungcn. Wied. Ann., 62, i.- 
 xli.x. 
 
 196 1897 L. W. Austin and ('. H. Thwing. An experimentiil research on 
 
 gravitational permeability. IVn/. Rev., 5, 294- 
 300. 
 196i 1897-9 S. Gllnther. llandbucli der Geophysik. 2 vol. Stuttgart. 
 
 197 1897 J. H. P(oynling). A new determination of the gravitation 
 
 constant, and the mean density of the earth. 
 Nature, 56, 127-8. 
 
 198 1898 F. Richarz und O. Krigar-Menzcl. Ucstimmung der Gravita- 
 
 tionsconslantc! und mittlercn Diclitigkeit der 
 Erde dunth Wilunngen. Anhanr/ Ahh. Aknd. 
 WiHH. Berlin. 1-196. 
 
 199 1898 F. Richarz und O. Krigar-Mcnzd. (Same title as 191). Wied. 
 
 ^l//«., 66, 177-193. 
 
 200 1899 J. H. Foynting and P. L. Gray. An experiment in sean h of 
 
 a directive action of one quartz crystal on an- 
 other. Phil. Trans. Lond., [A], 102, 245-56. 
 
 156 
 
INDEX 
 
 Acimrd, 49. 
 
 Airy. 5, IOC). li:{, 118. 119. 121-124. 
 128-130; Thcoiy of Cuvcndish Ex 
 perinieiit, 100, 118; Dolcoiith Ex- 
 periments. 113; lliirton Expcri 
 niciits, 121. 
 
 Arthur's Bent. 118, 123. 124. 
 
 Altraclion, Newton's Theorems on. 
 9 ; Newton's Error in Ciilculutioii 
 of, 10, 17 ; Primitive, 27 ; Of a 
 Pliiteuu. 29-32 ; Of a Spliericul 
 Segment, ('iilciiliiled by Newton, 
 17 ; l)y (.'iirliiii, Sclimidt and Giii- 
 lio. 111. 112 ; Shown by I)etle(;lion 
 of Pliiinl)-line. 33-43 ; Of Ciiiml)0- 
 razo. 34. 39 ; Of Soiiehallien, 43. 
 53-50 ; Due to Tides, 44. 134 ; Of 
 uiiy Hill. (;aiculated by Iluiton, 
 54; Of the Great Pyramid. 55; 
 Local. 50, 122-124. 120, 134, 135, 
 141 ; Of Mount Mimet, 50 ; Of 
 Mass Benealli Earth's Surface, 50, 
 
 122. 123; Of Arthur's Seat, 118. 
 
 123, 124 ; Of t\M\x, 118 ; Of a 
 Cone, 128 ; Of an Infinite Plane, 
 135. 
 
 Austin and Thwing, 142. 
 
 B 
 
 Babiiiet, 100. 
 
 Bacon, 1, 2. 5. 49, 113 
 
 Baily, 100, 105, 100. 115-120, 125, 
 131-133. 137 ; Cavendish Experi- 
 ment Criticized by, 105 ; Error of, 
 Pointed out 6y Cornu and Bailie, 
 119 ; x\nomalies in Results of, and 
 their Explanations, 118, 119. 
 
 Balance, Experiments with Beam. 
 2-5, 48. 49. 125. 132. 140 ; Experi- 
 ments with Torsion. 59-105. 114- 
 121, 124, 135, 137-139. 142; Mich- 
 ell Devised Torsion. 60; Experi- 
 ments with Pendulum, 131, 132. 
 
 Bauernfeind. 124. 
 
 lieaiimont. 110. 
 
 Beruet, 124, 134, 135. 
 
 Bertier. 47-49. 
 
 Boscoviteh, 134. 
 
 Bougiier. 5, 21 , 23-25, 27, 32. 33, 36. 
 39-44, 47. 53, 50. 130, 134; On 
 Tides, 44, 134 ; Life of, 44 ; First 
 to Take Account of Buoyancy of 
 Air. 26. 
 
 Boyle. 4. 
 
 Boys, 100. 135-137. 139, 142. 
 
 Brandes, 105. 100, 120; Theory of 
 Cavemlisli Experiment. 106; The- 
 ory of Oscillation Method, 105, 
 100, 120. 
 
 BrauM, 100. 138. 139. 
 
 C 
 
 Carlini. 111-113, 128. 
 
 Cavendish. 54, 55. 59. 90. 91. 98. 100, 
 
 105-107. 114-110, 118, 119, 125. 135. 
 
 130, 139; Error in Calculation of, 
 
 100, 105 ; Life of. 107. 
 Chimborazo, 22, 34. 39-41. 43. 
 Clarke, 56. 118. 123, 124. 
 Condamiiie, de la. 21. 28, 32. 36. 39- 
 
 41. 43. 44 ; Pendulum Experiments 
 
 of, 28 ; Method of, for Doubling 
 
 Deflection of Plumb-line, 36. 
 Cornu and Bailie. 66, 106. 115. 119. 
 
 124. 125, 131. 135. 
 Cotte, 48. 
 Cotton, 4. 
 Coulomb Balance, First Proposed by 
 
 Michell, 60. 
 Coultaud, 47, 48, 111. 
 
 D 
 
 D'Alembert. 31. 47. 
 Damping. Method of Finding A, 
 138, 141, 142 ; Effect of, 139. 
 
 157 
 
INDKX 
 
 
 !»•:, 
 
 Ocvid. 47-49 
 
 Dftleclion. of Arm of Torsion H:il 
 uiice, How McHsiircd, by (Javcn- 
 (lish. 04. 98; by U.icli. il6. 119; 
 l)y Biiilv, 117. 119. i:{2. 133; by 
 liriiun. 138; AlTittt.s ihc Period, 
 97; Error in iJaily's Method of 
 ()bs<'rviiiir. 119, l'i5 ; Mulliplied 
 by Poyntin^r. 133. 133. 
 
 DcHcjirtes. 2. 49 ; Sufjgestj'd Method 
 of McHsuring Griiviiy, 2 
 
 Diiiu'iisionH of Torsion Bidimce, Ef- 
 fVcts of, 125. 135, 137, 138. 
 
 Doleoath, 113, 121. 
 
 Doloinien. 49. 
 
 Drobisch. 113, 114. 
 
 Dnidc, 142. 
 
 E 
 
 EOtviis. 106, 137, 138. 
 
 F 
 
 F)iye*31. 124, 130. 189; Compensa- 
 lion Theory of. ('oriection of " Dr. 
 Young's Rule," 31, 124. 
 
 Ferrel, 130. 
 
 Flotation Theory. 31. 124. 
 
 Folic, 123 
 
 F(.rl.es, 117, 118, 120. 
 
 Forced Vibrations, 138, 141, 142. 
 
 Fr(?sdorf. 56. 11 3, 116. 123, 127. 128, 
 131, 132, 135. 
 
 Fujivuma, 127, 128. 
 
 G 
 
 Gilbert, Dr.. 1, 5.49. 
 
 Gilbert, L. W., 105. 
 
 Giulio. 112. 
 
 Gore, 124, 132. 
 
 Gossflin, 106. 
 
 Gouy. 135. 
 
 Gnivinieter, 135. 
 
 Gravitation. Early Conceptions of, 
 1, 49. 56 ; Early Experiments on, 
 by Menibers of Royal Society, 2-5 ; 
 As Explanation of Planetary Mo- 
 tion, bv Newton, 2, 10-19 : Msig- 
 net.io Theory of. 1, 4. 5, 12 ; liooke's 
 Ideas Concerning, 5, 6; Compen- 
 sator. 138 ; Mnltiplicator, 138 ; Per- 
 meability, 142 ; Velocity of Prop- 
 agation i>f. 142. 
 
 Gravity, Proposed Experiment on, 
 by Bacon, 1 ; by Descartes, 2 ; 
 
 Decrease of. with Height. 27-83, 
 47-49. 111-113, 126-128, 180, 187. 
 140 ; Law of Increase of, with 
 Deptli, 129, 130 ; Increase of, with 
 Temperature, 131 ; Mathematical 
 DLscussHni of, frt>m Potential, 137. 
 
 Gray, 142. 
 
 GUuther, 181. 141. 
 
 Harton Colliery. 5, 121,122 
 Hanghton, 122. 
 Hearri, 118, 120. 
 Hebnert, 54, 56, 124, 127, 129. 
 Hicks. 119, 131. 
 Hooke, 2, 4, 5. 
 Horizontal variometer, 137. 
 Humboldt, 56. 
 
 Hutton. 54, 55,90, 100, 105, 106, 118, 
 124. 
 
 J 
 Jacob, 56, 123. 
 
 .binie.s and Clarke, 56, 118, 128, 124. 
 .Icily, 125, 126, 187, 140. 
 Joly, 138, 141. 
 
 K 
 
 Keller, 124. 127, 135, 137, 140. 
 Kepler, 1. 2. 49. 
 Kn«»pf. 113, 122. 
 KOnig. 140. 
 
 Krigar-Menzel. 140. 141. 
 KrUmmungsvariometer, 137, 138. 
 
 Lalande, 48. 
 
 Laska. 141. 
 
 Law. of the Distance, 2. 9. 29, 47. 
 
 101. 126. 142 ; Of the Masses, 18. 
 
 32 ; Of the Material, 12, 142 ; Of 
 
 the Medium, 142. 
 Lesage, 2, 48, 49. 
 
 M 
 
 Mackenzie, 142. 
 
 Maun(!tism. Gilbert's Explanation of 
 Gravitation by. 1. 4. 5 ; Contrasted 
 with Gravitation, 12; Testa for 
 Eflfecis of. by (Cavendish, 67. 68, 
 75. 76 ; by Reich. 116, 120 ; Sug- 
 gested by Hearn to Account for 
 Anomalies in Baily's Results, 118- 
 120. 
 
 158 
 
INDEX 
 
 MaHkelyne. 17, 48, 53-50, 101, 100. 
 
 Miiyer, 140. 
 
 Meimbreii, 55. 00, 100. 
 
 MriMiunliiill, 124, 127, 128. 
 
 Mcrcier, 47,48, 111. 
 
 Micliell, 59, GO, 01. 
 
 Minu Experiments, 1, 2, 4, 5, 40, 
 
 113. 121, 128, I'Jl. 
 iMontigny, 119. 
 Miiiicke, 55, 105, 100. 
 
 N 
 
 Nc'wion, 2, 0. 7. 9. 14-17. 19. 39. 43. 
 47-49, 50. 107, 110, 124, 120. 141 ; 
 ExpliiiDition of PliiiK-taiy .Mmioiis 
 by, 2, 10; Pciidiiliim Kxpcriiiujiit.s 
 of, 11. 15 ; Guess as lo V^ahie of 
 A by, 14 ; Errors in ('iileiiliiiioiis 
 of, 10. 17 ; Indicates Metlioils of 
 Finding A, 17; Calculates Attnic 
 tion oi u Mouniain, 17 ; Lite of, 
 19; Attempts to Upset Tlieorv of, 
 47. 
 
 P 
 
 Peclnnann, 124. 
 
 Pendulum, Experiment with. Pro- 
 posed by Bacon, 1 ; by Hooke, 5 ; 
 Experiments wi ii, by Newion. 
 10, 11, 15; by liouguer, 24-33; 
 by Coidlaud and Mercier. 47 ; by 
 Curlini, 111 ; by Airy, 113. 121 ; 
 by iMi'ndeidiall, 127 ; by Slerneck, 
 128-131 ; l.y Luslia, 141 ; Correc- 
 tion for Bunyjincy of Air on. First 
 Used, 20 ; (Jorrection for. Due to 
 Resisian(;e nf Air, 27, 00 ; Methods 
 of Comparing One with Another, 
 113, 121, 128-130; Balance, 131. 
 
 Peters, 55. 124. 
 
 Playfair, 55, 102. 
 
 Plumb-line, Deflection of, Observed 
 at Chimliorazo, 33-43 ; at Sche- 
 hallien, 43, 53-50; at Arthur's 
 Seal, 123; at Evaux. 118. 124; 
 in Tyrol. 124 ; Deflection of, Cal- 
 culated for Chimborazo, 34; how 
 to Observe. 35-39, 53 ; by Tides, 
 44, 134, 135. 
 
 Poisson, 31. 00 *• 
 
 Power, 2-5. 40. 
 
 Poynting, 10, 30, 44, 100. 112. 110, 118, 
 119. 123-125, 127, 128. 131-134. 142. 
 
 Pratt. 124. 
 
 Pringle, 50. 
 
 Puissant, 118. 
 
 Pyramid, Attraction of the Great, 55. 
 
 1 
 
 Uelc'.!, 91. 100, 114-121, 125, 138. 
 Kesislance of Air, Discussed by Bon- 
 
 guer. 27; by Cavendish, 05-07; 
 
 by I'oisson, Alenubrea, and Cornu 
 
 and Bailie, 00, 100, 125. 
 Itiehar/, 140, 141. 
 Itobi.soii, 134. 
 Koille, 48. 
 Uoyal Hoci 'y, Experimeutn by 
 
 Members ..f, 2-0, 48. 
 Uo/.ier, 49. 
 
 Sabine, 112. 
 
 8aigey, 32, 43. 54, 112. 113. 118. 124; 
 
 Corrt'ction of Peiiiviaii Pendulum 
 
 Experinients by. 32, 43. 
 Schaar, 119. 
 Schettler. 123. 
 Schehallien. 43, 53-55, 101, 112, 118, 
 
 123 
 SeheVl, 50. 112. 110, 118, 123. 
 Sclnnidl, 44. 5.). 100, 112. 
 Schubert. 124. 
 Sheepshanks, 113. 
 St. Paul's Cathedral, Experiments 
 
 at, 4, 5. 
 Sterneck, 128-131, 137. 
 Stokes, 122. 
 Struve, 124, 134. 
 
 Temperature, Effects of, on Torsion 
 Balance, Discussed by Cavendish, 
 00, 70-80; by Reich. 114; by 
 Baily. 110, 117; by HicUs and 
 Poynting, 119 ; by Boys. 135. 130 ; 
 by E«)tv(is. 137 ; by Braiin, 139 ; 
 Change of A with. 125, 131 ; 
 fjinni of Constancy of, 135. 
 
 Thiesen, 137. 
 
 Thomson arxl Tait, 134. 
 
 Tides, Action of, on Plumb-line, 44. 
 134. 
 
 Time of Vibration, How Found by 
 Cavendish. 04-07, 70 ; by Reich, 
 115, 119 ; by Baily, 117 ; by Men- 
 denhall. 127 ; As Affected by De- 
 flection, 97 ; As AlTecied i>y Con- 
 vection Currents, 80, 100, 134, 137 ; 
 A Found From, 105, 100, 120, 
 138, 139. 141. 
 
 Todhunter, 10, 44. 50, 00. 
 59 
 
u 
 
 Ullott. 25, 39, 40. 
 
 INJJEX 
 
 Wilsing, 131, 182. 
 
 Viiruum, Experiment Made in, 138, 
 141. 
 
 W 
 
 Wallenlin, 127. 134. 
 
 We.HtininHUr Ahhuy, Experiments 
 
 Miule ut, 2. 8, r». 
 W lie well, 113. 
 
 Young, Kuleof, 31, 180. 
 
 Z 
 
 Zach. 3«. 44, 54- 56, 134 ; Mnskelyne 
 
 Exporiinenl Culculaled by, 64 ; 
 
 Finds Atlruction of Mount Mi- 
 
 Miet, 50. 
 Zunotti Bitmco, 44, 54. 50. 100, 112. 
 
 113, 123, 127. 
 
 ADDENDUM 
 
 Page 32. [Fut/c (146J) fum calculated the diminution in the attrnction ac- 
 cording to hiii fornuda {see note on p. 31), and Jinda it to be the fi^i^fi P'lrt, 
 tehich in Hot far from that resulting from the e^perime?it. llin calculation can 
 aim be stated in the following way: taking no account of the attraction of the 
 plateau, t/ie observed pendulum, lengths reduced to sea-level by Siiigey are at 
 
 L'IsledeVInca 990.935 wj7m. 
 
 Quito 991.009 " 
 
 Difn'ence 074 " 
 
 which difference is of the orucr of the errors of the observations. ^See this vol- 
 ume, p. 130, Ilelmert (148. vol. 2, chap. 3), and Zanotti-Bianco (li8^,pt. 1, 
 ehap. 8, a7idpt. 2, p. 182).] 
 
 160 
 
 THE END