44 



SAT URL 



Sep i i m bi 1: i<). 19 1 8 



tising ! 1 onscience. \\ e hope to see 



1 i these valuable essays in scientific 



d we would lake this oppoi tunit} "I 



wishing thi author success in his disinterested 



oi Scieniia, which is an indubitable 



factor towards true pacifism. 



MAI HEM 17 7C I/. BOOKS. 

 ■ ■■ \ 0) \ln \iniii iin,l Minima. By Prof. 

 Harris Hancock. Pp. xi\ 1 rgg. (Boston 

 (Mass.), London, etc.: (linn and Co., on,.) 

 Price COS. <nl. net. 

 (2) Analytic Geometry and Calculus. Bv Profs-. 

 F. S. Woods and P. II. Bailey. Pp. xi + 5 1 fi. 

 (Boston (Mass.), London, etc.: Ginn and Co., 

 11)17. I Price io.v. 6tt net. 

 (1) 'T'HE theory of maxima and minima con- 

 ■*■ tains pitfalls into which have fallen such 

 well-known mathematicians as Lagrange, Ber- 

 tram!. Serret, and Tbdhunter. A peculiar interest, 

 therefore, is attached to the subject, and the reader 

 will find Prof. Hancock's book well worth his 

 study. Except that there is no reference to cal- 

 culus of variations, the author has succeeded in 

 covering the ground fairly thoroughly, and that 

 without allowing the argument to be anywhere 

 tedious. He gives many references, and a leu 

 quite interesting examples. 



\!lci- a short investigation of maxima and 

 minima of functions of a single variable, he gives 

 in sonic detail ihe methods of Scheeffer and von 

 Dantscher, which introduced rigour into the dis- 

 cussion of functions of two or three variables-. 

 The theory here is intimately connected with the 

 theory of quad-ratio forms and singularities of 

 higher plane curves. The author seems not to 

 have read such books as Bromwich's "Quadratic 

 forms," Hilton's "Linear Substitutions," or 

 Ninth's " Elementartheilar," which would have en- 

 abled him in places to simplify his treatment of 

 quadratic forms. In tracing a plane curve near 

 a singularity, the author should have made use 

 of Newton's diagram. He should also have avoided 

 such a phrase as "cusps of the first and second 

 kind," which implies that the cusps in question 

 unparable, whereas the latter is a singularity 

 hei complexity than the former. 

 I lie chapter on relative maxima and minima is 

 -ting. The discussion usually 

 given in the text-books is very scanty, and the 

 fuller treatment here given is \ erv welcome. A 

 ible point is made in §§ 9&-107. The usual 

 proof thai the maximum triangle inscribed in a 

 given circle is equilateral runs as follows: "If 

 1 iot suppose Mil' to I..- ,!,, irreatesl triangle. If 

 AB + AC, let I) bis,, 1 tin- arc BAG. Then the 

 triangle BDC>BAC, etc. ' Is this argument ad- 

 tble? I he readei m ,■ 1 ompare the following 

 -1'iiing, th\c lo an Italian author: "I'nilv is 

 test integer. For, if not, supposes (+l) 

 vatest. Then >i'->)i, etc." li, proofs inn 

 parallel, but the tacit assumption 1 itesl tri- 



integer exists) is lawful in on< case and 

 not in the other. 



N'O. 2551, VOL. I02] 



(-•) This work is a revision and abridgment ot 

 the authors' two-volume "Course in Mathematics 

 for Students oi Engineering and Applied Science," 



and is intended 1 upj a Bwo-years' course for 



an average collegi class. Fhe book does noi 

 the impression oi being especiall) suited 1,, thi 

 needs of students of applied science, except for 

 thi- 1:11*1 that examples are included on finding 

 centre oi gravity, centre oi pressure, and so on. 

 In the main the book is apparently simply a 

 course on pure mat liema I ics designed tor the 



Amei i, an undergraduate. Vs such it may be 

 commended as quite clear and readable, and it i^ 

 furnished with some 2000 well-chosen examples. 



Naturally it is possible lo criticise certain portions 

 on the ground of absence of rigidity. But prob- 

 ably the authors have hit the happv mean between 

 a slovenliness which demoralises tli" beginner and 

 a precision which terrifies him. 



It is interesting to contrast the American and 

 English ideas of a suitable syllabus for the first 

 two years of a "pa^ss " mathematical course. 'Ihe 

 Americans include the co-ordinate geometrv of 

 straight line and plane; but the rest of the syllabus 

 consists almost entirely of the calculus and 

 elementary differential equations. Even the circle 

 and conic receive no more than a passing mention; 

 and very little algebra is inserted, such subjects 

 as determinants and the theory of equations being 

 del, ricd for subsequent study. Contrast this with 

 a certain English B.A. course, which demands no 

 calculus whatever, but requires the "simple pro- 

 perties of conic sections, including a discussion 

 of the general equation of the second degree and 

 the methods ol projection " I 'Ihe book under 

 review may give the student a somewhat false 

 idea of the importance of the conic (it is mentioned 

 casually along with the witch, the cissoid, and 

 the slrophoid), and he may find partial differentia- 

 tion studied by means of three-dimensional co- 

 ordinate geometry a little ton hard for him. Bui, 

 nevertheless, English teachers have very much to 

 learn from their allies. H. H. 



OUR BOOKSHELF. 



The Botany 0/ Iceland. Edited b} Dr. L. 



Kolderup Kosenvinge and Dr. Eug. Warming. 



Part ii. 3. Ernst 0strup : "Marine Diatoms from 



the Coasts ol Iceland." 4. Aug. Ilesselbo: 



"The Bryophyta of Iceland." Pp. 348 i>7.s. 



(Copenhagen : J. I-'rimodt ; London : J. W'hel- 



don and Co., 1918.) Price 11 .v. net. 



'lilts part completes vol. i. of "Ihe Botanv &i 



Iceland," the firsl part of which was issued in two 



sections, one on "'flu- Marine- Algal Vegetation, ' 

 by II- Jnnsson, in icJIZ, and a second on "'Ihe 

 Physical Geography of the Island," in mi|. 



'Ihe list ot marine diatoms from the coasts ol 

 Iceland comprises 205 species and varieties; seven 

 species and a number of varieties are hen 

 described as new. Mr. 0strup gives a tabular 

 list showing the universal distribution of the 

 forms, as well is their distribution on the different 



parts of the , -oasts ol Iceland from which it 



