202 



NATURE 



[May io, 191 7 



thing to which we are accustomed in this respect 

 receives masterly treatment here ; even the so- 

 called modern geometry of the triangle is handled 

 with a detail which to many readers will seem 

 excessive. The author's notation for angles and 

 segments is, however, in the reviewer's opinion, 

 needlessly tiresome. 



The sphere receives a similar elementary treat- 

 ment of less extent in chaps, v. and vi. (pp. 

 226-S2). Chap, iv., with the title, "On the Tetra- 

 cyclic Plane," introduces a method of present- 

 ment, followed also from chap. vii. to the end, 

 about which opinions may well differ. The book on 

 the whole has such value that criticism is a form 

 of praise, and we shall express our opinion freely. 

 We think the author might have introduced his 

 chapter on the tetracyclic plane, say, by a brief 

 account of Clifford's projection of the plane sections 

 of an ellipsoid from an umbilicus : when the plane 

 of projection is the tangent plane at the opposite 

 umbilicus, actual circles are obtained, and two of 

 these cut at right angles when the corresponding 

 plane sections of the ellipsoid are in conjugate 

 planes. If this were too elementary, he could, 

 even then without introducing the co-ordinates in 

 the first paragraph have defined quasi-circles as 

 projections on to an arbitrary plane of plane sec- 

 tions of a quadric taken from an arbitrary point of 

 the quadric. This would give at once the geo- 

 metrical meaning of the tetracyclic co-ordinates. 

 The projection of the intersection of the funda- 

 mental quadric with another quadric is then obvi- 

 ously a quartic curve with two nodes ; there seems 

 no great gain in calling such a curve a cyclic. The 

 tangent planes of the cones containing the curve 

 of intersection of the two quadrics intersect the 

 fundamental quadric in curves projecting into the 

 four systems of generating circles of the cyclic, 

 and it is easily seen that the centres of the circles 

 of one generation lie on a conic. All this seems 

 clearer without the co-ordinates. And it is curious 

 that a writer on non-Euclidean geometry should 

 not recognise that the admirable theorem quoted 

 from Jessop (p. 212), in regard to the angles be- 

 tween generating circles of different generations, 

 is a generalisation of an old friend, relating to the 

 difference of the distances of a variable point of 

 t>ie focal hyperbola of a system of confocal quad- 

 rics from two fixed points of the focal ellipse, the 

 absolute being one of the confocal quadrics. 



Similar remarks apply to chap, vii.., on 

 pentaspherical space, and chap. xiii. , on 

 circles in space ; we think the projective geometry, 

 of which these are translations, should be brought 

 forward first and made more fundamental. The 

 reader, after he has had the pleasure of turning the 

 author's theorems back into projective geometry, 

 will, we think, summarise them as such. The 

 author recognises that the famous pentacycle of 

 Stephanos is no more than a nearly obvious 

 theorem for lines in four dimensions ; he refers to 

 Segre's cubic variety in four dimensions (p. 506), 

 and there may be a real gain in calling it a cubic 

 complex of spheres. It certainly is very interesting 

 to have the theorems for spheres which he states; 

 NO. 2480, VOL. 99] 



but we think a greater insistence on the projective 

 geometry should have preceded his treatment^ 

 especially as a large part of the theory can be ob- 

 tained without the explicit use of co-ordinates. 



Some minor remarks may be added. There are 

 occasional misprints ; on p. 284 there are three 

 (lines I, 5, and 15 from the bottom.) The definitions 

 of the many technical terms are not given sufficient 

 prominence ; if one forgets the meaning of such 

 a term, it is in some cases a matter of hunting to 

 find it again. On p. 351 the author refers to. his 

 impression that Cayley spoke of the sign of the 

 radius of a circle ; one place where this is so at 

 least is in the " Collected Papers," vol. ii., p. 140. 

 The reference to Weitzenbock in the footnote 

 on p. 485 is obscure; the p. .2574 refers to the 

 Wiener Sitzungsberichte referred to in the foot- 

 note on p. 484. 



To use a phrase often employed by the author, 

 these minute criticisms, it is seen, are not trans- 

 finite in number. Copying again the concluding 

 phrase of what is a very gracious preface, which 

 all Englishmen will be glad to read, the reviewer 

 would like his last word, as was his first, to be 

 one of gratitude to the author. His book is a 

 noble addition to the geometrical literature in the 

 English language, and must have a great influence 

 on the prosecution of the study in all countries 

 where English is read. 



(2) This is a collection of elementary examples, 

 in which the details of calculation are given at 

 length, for the most part solved by Lagrange's 

 equations. Pp. 1-109 are occupied with com- 

 putations of centres of gravity, attractions and 

 potential, and moments of inertia. Then one page 

 is given to expounding the principle of virtual 

 work, and one to D'Alembert's principle; after 

 this the equations of Lagrange are briefly obtained. 

 Chap. v. (p. 121) begins with an example dis- 

 cussing the motion of a sphere on the surface 

 of a smooth cylinder (not in two dimensions), in 

 which it is assumed, without previous discussion, 

 that the motion of the centre is independent of 

 rotation. Pp. 262—323 are occupied with an 

 arithmetical discussion of the introductory formula? 

 of the theory of elliptic functions, but without any 

 application, to statical or dynamical problems. 

 Throughout the volume the computations are 

 arranged in a brief, businesslike way, and the dia- 

 grams are clear and numerous. As a workmg 

 class-book, in the hands of a competent teacher 

 concerned mainly in teaching the art of solving 

 concrete examples, the book might be of great use. 



A TEXT-BOOK OF GENETICS. 

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 of Biology and a Reference Book for Animal 

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SINCE the beginning of the present century,, 

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