Mar. 26, 1874], 



NATURE 



399 



of adequate means for its pursuit. By sucli means will 

 our rulers show themselves to be the real well-wishers and 

 benefactors of their country. 



TODHUNTER'S ''MATHEMATICAL 



THEORIES OF ATTRACTION"* 



A His/01 y of the Matliematical Theories of Attraction. 



and the Figure of the Earth from, the time of Newton. 



to that of Laplace. By I. Todhunter, M.A., F.R.S. 



Two vols. (London: Macmillan, 1874.) 



II. 



OF the great Scotch mathematician, Maclaurin, we 

 read — " The importance of his investigations may 

 be seen by observing how great has been his influence on 

 succeeding writers. Clairaut, D'Alembert, Lagrange, Le- 

 gendre, Gauss, Ivory, and Chasles show by reference, ex- 

 plicit or implicit, their obligations to the creator of the 

 theor>' of the attraction of ellipsoids. 



Maclaurin well deserves the memorable association of 

 his name with that of the great master in the inscription 

 which records that he was appointed Professor of Mathe- 

 matics at Edinburgh, ^' ipso Newtono suadente." His main 

 contribution to the theory of the figure of the earth was 

 an exact demonstration of Newton's postulate, of which 

 only approximate solutions had previously been given. 

 We may note on § 260 that we_have seen the French ver- 

 sion of his "Treatise on Fluxions," "traduit de I'Anglois, 

 par le R. P. Pezcnas, Paris, 1749; 2 vols." The first 

 volume has li. pp. of Introduction ; v.-viii. Avertisse- 

 mcnt par le traducteur ix.-li.. Translation of Author's 

 Preface and Introduction with Table dcs Matieres ; then 

 344 pp. of text, plates, and 4 pp. of errata. The second 

 volume has viii. pp. of contents, 322 pp. of text, plates, 

 with 6 pp. at end (4 pp. of errata), for errata, approbation, 

 2i.x\d^ privilege. 



The next noteworthy name is again that of an English 

 writer, Thomas Simpson. His contributions are of 

 eminent importance. The analysis he employed, Mr. 

 Todhun'er observes, " would not have been unworthy of 

 Laplace himself.'' There is here an interesting biographi- 

 cal note of a kind which the writer so well knows how to 

 introduce, and \- hich adds a charm to themore general 

 details. In writing our notice we have especially dwelt 

 upon the English contributors to our subject ; on the 

 whole it can hardly be denied " that Newton's country- 

 men have left to foreigners the glory of continuing and 

 extending his empire." Singularly enough Mr. Tod- 

 hunter gives no account of Simpson's work, "A New 

 Treatise of Fluxions . . . with a variety of new and 

 curious problems." London, 1737. 8vo. This is six 

 years earlier than the " Mathematical Dissertations." 

 Problems XXI. to XXIII. (§§ 201-206) deal with 

 attractions of a circular plate on a point on the axis ; 

 of a cylinder on a point on its axis ; of a sphere 

 on a particle on its surface, or any distance above it, for 

 law varying as inverse of (distance)^ and for (distance)". 

 They correspond to Problems II., IV., V., VI. of 1S23 

 edition of the " Doctrine and Application of Fluxions." 



The great work of Clairaut, " Theorie de la figure de la 

 terre," &c., appeared in 1743. In this branch "no other per- 

 son has accomplished so much as Clairaut; and the subject 

 * Continued from p. 380. 



remains at present substantially as he left it, though the 

 form is different. The splendid analysis which Laplace 

 supplied adorned but did not really alter the theory 

 which started from the creative hands of Clairaut.'' 

 Laplace, too, places it " au rang des plus belles produc- 

 tions mathdmatiques." 



The expedition to Peru gave rise to much paper war- 

 fare, and Mr. Todhunter has collected together, in a useful 

 form, the titles of the original pamphlets. We think he 

 has overlooked the following, " Nouvcau projet d'une 

 mesure invariable proprc ^ devenir universelle, extrait 

 d'un memoire lu . . . le 24 avril, 1 748, par M. de la Con- 

 damine," viii. pp. A copy we have consulted of No. xx. 

 (p. 236) is dated " Plombieres, juin 30, 1754." (Consult 

 Lalande, p. 455.) 



D'Alembert need not long detain us. Laplace points 

 out that his writings want ''clarte." Mr. Todhunter says of 

 him, " The errors of D'Alembert are certainly surprising ; 

 they seem to me to indicate that he was little in the habit 

 of enlarging his own views by comparing them with those 

 of others. His criticisms of Clairaut prove that he had 

 not really mastered the greatest work which had been 

 written on the subject he was constantly studying. His 

 readiness to publish unsound demonstrations and abso- 

 lute errors is abundantly shown in the course of our 

 criticism. On the whole the blunders revealed in the 

 history of the ' Mathematical Theory of Probability,' and 

 in the present history, constitute an extraordinary shade 

 on a fame so tiright as that of D'Alembert." 



Here we must give an account of a work not men- 

 tioned in the History. The " Considerazioni sopra la 

 Figura della Terra " * of Tommaso Narducci • appeared 

 about the year 1747. It comprises two Lemmas (in 

 modern geometrical conies they would be for the ellipse 



(i), GN equal — , (2) radius of curvature equal 



PG^ CA' 

 CB'- 



CA' 

 ), and nine problems. The first problem is 



" Dati due gradi di meridiano e loro latidudine, trovare 

 la raglone degl' assi, e gl' assi stessi ;" the last is " Data 

 la ragione de' due assi, che sia di i ad ni, trovare nel 

 meridiano un grado, che sia eguale al grado dell' equa- 

 tore." It is an' interesting piece of geometrical work. 



In his §490 Mr. Todhunter considers it curious that the 

 (Cambridge) University library does not possess a com- 

 plete copy of the famous work of Stay and Boscovich. 

 His surprise will probably be increased when we state 

 that, if we are not mistaken, neither do the libraries of 

 the British Museum or that of the late Mr. Graves ; at any 

 rate, we do not remember to have met with Boscovich's 

 commentary on the poem. " These writings furnish ele- 

 mentary accounts of the most important results which 

 had been obtained up to their date, and reveal apparently 

 great knowledge and judgment in Natural Philosophy." 

 A copy of Boscovich's " Dissertatio de telluris figura, 

 habita in seminario romano Soc. Jesu nunc primum aucta 

 et illustrata abipsometauctore, P. R. J. Boscovich," forms 

 pp. l6l-2l8 of vol. ii. of Giuliani's memoirs (cited at the 

 foot, date 1744;. In p. 184 he speaks of Maclaurin's 

 " Fluxions " as " Newtono ipso dignissima ;" there is a 

 noteworthypassage, pp.217, 2i8,and also a notice on p.xii. 



* It occupies pp. 225-266 of vol. iii. of tiie " Rlemoria sopra la Fisicae 

 Istoria Naturaic di diversi valentuomini," edited by Carl Antonio Giuliani, 

 in 4 vols. (1743-1747)- -^ - 



