January 21, 1898.] 



SCIENCE. 



103 



Augustus De Morgan, who in his day waged ■ 

 such merry war with the circle squarers, got 

 half the delight of battle from the fact that he 

 had to meet his foes in single combat and 

 pepper them with small shot, a kind of warfare 

 from which he was sure to emerge with joyous 

 triumph and an appetite for more. To chase 

 his prey through a tangle of reasoning had to 

 his versatile mind the zest of a fox hunt. To 

 kill all the foxes at one discharge would have 

 spoiled his sport. Until very recently the 

 circle squarer had one safe retreat. Nobody 

 could logically dispose his general thesis. And, 

 beside, he had philosophic and scriptural au- 

 thority. A circle is a perfect figure. That 

 which is one span across is three spans around. 

 Even the august Legislature of Indiana was 

 lately beguiled by a savant from Hooppole 

 county into enacting that no circle should be 

 de rigeur in that State for which the ratio of 

 circumference to diameter was not exactly 

 three and two-tenths. But we have changed 

 all that. The circle squaring fraternity has 

 long had no standing in court, but now a per- 

 petual injunction is out against them. Not 

 only do we now possess a proof of the trans- 

 cendental nature of the number tt, but this 

 proof has been recently so simplified as to be 

 perfectly intelligible at a very early stage of 

 mathematical study. 



The mathematical pi is inedible without e. 

 The modern investigations begin with Hermite's 

 proof of the transcendence of the exponential 

 base in his paper ' Sur la fonction exponentielle,' 

 Comptes Bendus, 1873. Lindemann's celebrated 

 proof of the transcendence of jt appeared in the 

 Mathematische Annalen, 1882. The connect- 

 ing step is the establishment of the theorem 

 that in an equation Cg-\- Cie'' + c^e' + ■■■ = 0, 

 the exponents and the coefficients cannot all be 

 algebraic numbers. From Euler's equation 

 1 f e"' = 0, the transcendental character of n- 

 then follows at once. But it was first in 1893 

 that Hilbert, Hurwitz and Gordan did away 

 with the earlier formidable apparatus and re- 

 duced the proof to the present elementary 

 form. The results, together with the modern 

 disposition of the kindred problems of the du- 

 plication of the cube and the trisection of an 

 arbitrary angle, have since been made generally 



available by the publication in book form of 

 Klein's lectures on these subjects. These lec- 

 tures have already been translated into French 

 and Italian, and we have now, thanks to Pro- 

 fessors Beman and Smith, an excellent English 

 version. The present book is well edited and 

 well printed. Every teacher of even elemen- 

 tary mathematics will do well to obtain a copy, 

 not merely for his library, but to be actually 

 read. 



The Calculus for Engineers. By John Peeby, 

 Professor of Mechanics and Mathematics in 

 the Royal College of Science, London. Lon- 

 don and New York, Edward Arnold. 8vo. 

 Pp. viii + 378. 



From the title of this book one might natu- 

 rally expect to find in[,it a considerable deviation 

 from the prevalent stereotyped treatment of 

 what the author rather deprecatingly calls 

 ' academic ' calculus. On inspection, however, 

 the divergence turns out to be about as com- 

 plete as could well be imagined. The author's 

 aim is to make the methods of the calculus 

 available for the use of students who already 

 have a considerable knowledge of practical 

 physics and mechanics. A great deal can be 

 done in this direction by the aid of a few func- 

 tions and the simplest rules of difierentiation 

 and integration. In the present book the first 

 266 pages are divided into two chapters, one of 

 which deals, so far as the calculus proper is 

 concerned, with x", the other with e" and sin x. 

 These chapters are filled with an excellent col- 

 lection of examples drawn from every branch 

 of applied mathematics. To give an idea of 

 the diversity in this regard, I cannot do better 

 than to quote from the index, which is in itself 

 a commendable feature of the book. Under B, 

 which supplies one of the shortest lists in the 

 index, we have : Ballistic effects ; Basin, water 

 in ; Beams, fixed at ends, of uniform strength, 

 shear stress in, standard cases ; Beats in music f 

 Belt, slipping of ; Bending, in struts ; Bessels ;. 

 Bifilar suspension ; Binomial theorum ; Boiler, 

 heating surface of; Bramwell's valve gear ;. 

 Bridge, suspension. Very many of the exam- 

 ples are of a kind to be very appropriately in- 

 troduced into the ' academic ' books ; and con- 

 sidering how completely latter-day writers on. 



