6o2 



NATURE 



[Oct. 27, 1887 



equal in importance to the discovery of the microscope 

 itself. And embryology, far more than any other depart- 

 ment of morphology, depends upon this art ; indeed, the 

 study may be almost said to date from the introduction 

 of this method of inqiiry. Hence there is a strong bias 

 in favour of representing structures in section ; and in 

 original papers and advanced treatises this custom is not 

 to be deplored, for thp reader knows exactly what is meant 

 ])y the figures. But even m sucn works I think that the 

 reader, and the author also, would benefit by the intro- 

 duction of a few additional illustrations representing the 

 organism, organ, or structure, as a solid object. But 

 there can hardly be two opinions on this subject in the 

 case of an introductory text-book. The beginner cannot 

 readily or correctly reconstruct in imagination the solid 

 structure from a representation of a section, and he must 

 infallibly lose considerably in time by the prevalent 

 custom of representing in only two dimensions objects 

 which really exist in three. Long descriptions might be 

 curtailed, and great additional clearness conferred by the 

 frequent illustration of solid objects, out of which a small 

 portion is represented as cut, on one side only, so as to 

 show the internal structure. But this necessity is not 

 fully recognized in any embryological text-book, although 

 some attempt is made to deal with it in this and in other 

 works. It is to be hoped that in future editions consider- 

 able attention may be paid to this mode of illustration, 

 which will be more than repaid by the advantage conferred 

 upon the young student. 



In conclusion, the author seems to have included every- 

 thing of importance in his subject up to the date at which 

 the book was written ; so that many important discoveries 

 or theories are described which are necessarily absent 

 from Balfour's work. When from the necessities of space 

 these are only briefly touched upon, the reference to thq, 

 original papers is to be found in Appendix B., containing 

 a bibliography of recently published works on embryology 

 Hence there is reason to hope that the volume may be 

 found useful to the student who is famiUar with Balfour's 

 work. E. B. P. 



SOME MATHEMATICAL BOOKS. 



The Conic Sections, with Solutions of Questions in 

 London University and other Examination Papers. 

 By G. Heppel, M.A. (London : Bailliere, Tindall, and 

 Cox, 1887.) 



A New Mode of Geometrical Demonstration, with Ex- 

 amples showing its Application to Lines and Angles, 

 Surfaces, and the Products of Three or more Straight 

 Lines, Sr^c. By D. Maver. (Aberdeen : A. Brown, 

 1887.) 



Easy Lessons in the Differential Calculus', indicating 

 from the Outset the Utility of the Processes called 

 Differentiation and Integration. By R. A. Proctor. 

 (London: Longmans, 1887.) 



First Steps in Geometry : a Series of Hints for the 

 Solution of Geometrical Problems, with Notes on 

 Euclid, useful working Propositions, and many Ex- 

 amples. By R. A. Proctor. (London : Longmans, 

 1887.) 



MR. HEPPEL'S little hand-book is not a complete 

 treatise on elementary analytical geometry as 

 usually presented to junior students, but it is a sequel to 



a previous small work in the same series (" Students' Aid 

 Series ''), " On the Geometry of the Straight Line and 

 Circle." The object aimed at in the two works is to 

 fully equip readers for the B.A. and B.Sc. examinations 

 of the London University and similar examinations. 

 Hence a limited portion only is discussed, viz. the 

 equations to the conies ; tangents, polars, normals, and 

 curvature ; sections of a cone, harmonic pencils, and 

 miscellaneous theorems. Though Mr. Heppel has treated 

 his subject concisely, he has not done his work in a per- 

 functory manner, for there is much originality exhibited 

 in his mode of treatment, and he has discussed the gene- 

 ral equation, not only for rectangular axes, but generally, 

 in a very clear manner. If we mistake not, this clear 

 exposition of a somewhat difficult part of the subject — 

 difficult, that is, to junior students — is the outcome of 

 some years' experience in tuition. In an appendix are 

 given " hints to students " founded on this experience, 

 and solutions to questions, illustrative of the text, which 

 have been taken from the London University papers. 

 There are a few errors in the printing, but they are not 

 of a character to seriously inconvenience the student. 

 We could have wished for a larger page, for then more 

 justice would have been done to the author in the pre- 

 sentment of some of the formulae. A student who care- 

 fully reads the text and transfers the formute for the 

 separate conies to larger pages, ought to require no other 

 text-book than this small one for the examinations named 

 above. 



Mr. Maver claims for his method the recommendation 

 that it is new. One can hardly expect such to be the 

 case, but we certainly do not remember to have come 

 across it as here applied. The nearest approximatior 

 we can lay hands upon is the method of parallel trans 

 formation, given in Petersen's " Methods and Theorie; 

 for the Solution of Problems of Geometrical Construe 

 tions" (pp. 46-47) ; but Mr. Maver has worked out th( 

 idea at considerable length and in an elegant manner 

 An illustration from the principles and from the body 

 the work will sufficiently explain the scope of the method 

 " Let AB and CD be two parallel straight lines, and AC 

 GK, and GM any lines whatever drawn from the line AI 

 to the line CD. If these lines, AC, GK, GM, move it 

 the direction of the parallels AB, CD, so that CF = Kli 

 = MD, then we have the space CE = KB = GE 

 (Euc. i. 36). Let the space CE be represented by s.\Q 

 which may be read space generated by AC, and so on 

 then ^AC = sGK = j'GM." Assuming results whici 

 readily flow from the above and which are given in th 

 "Principles," let us now take Example II. The side 

 AB, BC of a triangle are bisected by CE, AF, cutting i: 

 G, to prove CG = 2EG, AG = 2FG ; and if BG prcj 

 duced cuts AC in D, then AD = CD. " Let AF be thi 

 direction of motion, then sCG = sCY = .yBF = s'&A 

 = 2^EA = 2^EG ; . • . since .fCG = 2jEG, CG = 2ECI 

 In the same way AG = 2FG. Again, if DB be th| 

 direcrion of the motion we have jAD = .yAB = 2jEB = 

 2:rEG = .fCG = ^CD ; . • . since :f AD = jCD, AD = CD 

 There are five chapters, viz. one, containing the principles 

 two, applications to lines and angles ; three, to square^ 

 and rectangles ; four, the products of three or 

 straight lines ; five, to lines that are in the same stri 

 hne. In an appendix the author "proves Euc. i. 



