NATURE 



THURSDAY, MAY lo, 1877 



MATHEMATICS IN AMERICA 



Elements of the Differential and Integral Calculus, by a 

 new Method, founded on the True System of Sir Isaac 

 Newton, without the Use of Infinitesimals or Limits. 

 By C. P. Buckingham. (Chicago : S. C. Griggs and 

 Co., 1875. 343 pp.) 



Elements of the Infinitesimal Calculus, with Numerous 

 Examples and Applications to Analysis and Geometry. 

 By James G.Clark, A.M. (Ray Series. New York : 

 Wilson, Hinkle, and Co., 1875. 441 pp.) 



On a New Method of Obtainitig the Differentials of 

 Functions with Especial Reference to the Newtonian 

 Conception of Rates or Velocities. By J. Minot Rice 

 and W. Woolsey Johnson. (New York : D. van Nos- 

 brand, 1875. 32 pp.) 



AN American writer who had exceptional opportunities 

 of contrasting the methods of mathematical teach- 

 ing adopted in his own country with those which obtained 

 at Cambridge twenty-five years ago, strongly condemns 

 the Transatlantic system, and leads his readers to infer 

 that the attainments of the ordinary graduate in this par- 

 ticular branch of study vvere only on a par with those of 

 a fairly trained schoolboy here. It may be supposed, 

 then, that not many of the students ventured upon the 

 difficulties of the calculus. Indeed, he writes that " at 

 Yale where the course used to be thought a very difficult 

 and thorough one, the Differential was among the optional 

 studies at the end of the third year." (Bristed : " Five 

 Years at an English University," vol. ii , pp. 94, &c., 

 1S52.) 



We are not in a position to say that all this has 

 been changed in the interim, but among many evidences 

 of the increased interest taken in mathematical studies 

 \vc may surely refer to the three works now before us. 

 All three give evidence of careful study and honestly 

 grapple with the difficulties which beset the learner at the 

 very threshold of his inquiries. De Morgan long ago 

 wrote that " it is matter of common observation that any 

 one who commences the study, even with the best ele- 

 mentary works, finds himself in the dark as to the real 

 meaning of the processes which he learns, until, at a 

 certain stage of his progress, depending upon his capa- 

 city, some accidental combination of his own ideas 

 throws light upon the subject." The authors of the third 

 work under review refer to D'Alembert's precept, " AUez 

 en avant, et la foi vous viendra." 



iSIr. Buckingham takes as his fundamental idea of the 

 conditions under which quantity may exist to be that we 

 must not consider it only as capable of being increased 

 or diminished, but also as being actually in a state of 

 chaui^e. "It must (so to speak) be vitalised, so that it 

 shall be endowed with tendencies to change its value ; 

 and the rate and direction of these tendencies will be 

 found to constitute the groundwork of the whole system. 

 The differential calculus is the SCIENCE OF RATES, and 

 its peculiar subject is quantity in a state of 



CHANGE." 



Conceding to Leibnitz the honour of being the first to 

 Vol. XVI.— No. 393 



construct a system of rules for the analytical machinery 

 of the science, he will not allow that he ever got beyond 

 the ancient conception of the conditions of quantity. 

 " The only original birthplace of the fundamental idea 

 of quantity which forms the true germ of the calculus 

 was in the mind of the immortal Newton." 



An introduction of thirty-six pages discusses the method 

 of Descartes, the infinitesimal method (the results of 

 which are true, while the method is false — " true results 

 not because its principles are true, nor because its errors 

 are small, but because they are, whether great or small, 

 exactly equal, and exactly cancel and destroy each other. 

 . . . the system is but a mere artifice."), the method of 

 limits (here our author discusses Lemma I., Book I. of 

 the "Principia," considers Newton's defence of the Lemma, 

 and the opinions of Comte, Lagrange, and Berkeley, and 

 points out what he believes to be the fundamental errors 

 of this method and of the infinitesimal method). What 

 is called the true method of Newton is then treated of. 

 Referring to Newton's letter to J. Collins (December 10, 

 1672), he says that the theory on which Newton formed 

 his method of fluxions is contained in the second Lemma. 

 The lemma is given in full and discussed. " It is to be 

 remarked that the doctrine of limits is nowhere hinted 

 at, but the results are direct, positive, and substantial." 

 We cannot tarry longer over this matter, but in connec- 

 tion with this point refer to De Morgan's " On the Early 

 History of Infinitesimals in England" [Phil. Mag. Novem- 

 ber 1S52). Prof. Clifford, too, if our recollection of an oral 

 communication be correct, puts this lemma prominently 

 forward in his (.''unpublished) "Foundations of the Dif- 

 ferential Calculus and of Dynamics." In the work itself 

 we have the calculi (differential and integral) applied to 

 the subjects which usually find a place in similar treatises. 

 There is an appendix of thirteen pages on geometrical 

 fluxions. Many examples are worked out, but the merit 

 of the work does not lie at all, we think, in this direction, 

 but altogether in the numerous discussions which are to 

 be found in almost every chapter. 



Mr. Clark's work has been exceedingly well printed, 

 the type is very clear, and the paper good. This treatise, 

 too, is written with a view to remove " all grounds for 

 that feeling of uncertainty which often possesses the 

 student at the very outset, and from which he rarely finds 

 it possible to extricate himself." Much space is given to 

 an exposition of the Doctrine of Limits — the work being 

 founded mainly on that by Duhameh A large number of 

 examples have been taken from EngHsh treatises (Hall, 

 Walion, and Todhunter). Rather more ground is covered 

 in this treatise than in the former ; in neither, however, 

 have we any discussion of maxima and minima of func- 

 tions of more than two independent variables, nor of 

 methods of changing the variables in multiple integrals. 

 Here a few piges are devoted to definite integrals and 

 to differentiation and integration under the sign J . 

 Seven chapters are devoted to the elementary parts of 

 the theory of differential equations. The work, though 

 it does not reach the level of the like works by Messrs. 

 Todhunter and Williamson, is yet a compact and fair 

 elementary treatise. 



The third work on our list is a revised edition of a 

 paper read before the American Academy of Arts and 

 Sciences, January 14, 1873. It is the authors' intention 



