28 



NA TURE 



iNo 



1886 



fearful was he of looseness or slipshoddiness that he more 

 than once returns to this matter, and upon this very point 

 of an Association text-book writes as follows : — " There 

 are various considerations which seem to me to indicate 

 that if a change be made it will not be in the direction of 

 greater rigour^' (p. 172). He owns himself once to have 

 been in favour of hypothetical constructions, but that he 

 had subsequently seen reason to alter his opinions : in 

 many places in his essay he shows that he has not 

 renounced liypothetical statements. His idea of an 

 Associationist seems to have been that he is a being who 

 tries to evade the difficulty of passing a pupil in geometry 

 by asking for a less stringent text-book than that of 

 Euclid. 



It is vain to wish for the verdict of such able critics as 

 De Morgan and Todhunter on the work before us, but 

 we feel sure that the former would not have written con- 

 cerning it " Non est geometria," nor the latter have found 

 it wanting in Euclidian rigour. 



As to this matter of a different order from Euclid's 

 sequence we cite with cordial approval the following 

 remarks of a writer in our columns (vol. xxxiv. p. 50) : — 

 " We believe that those who have most carefully con- 

 sidered the question of a rival order of sequence of geo- 

 metrical propositions would agree that the best order in 

 a logical arrangement does not seriously conflict with 

 Euclid's order, except by simplifying it. Rather, by 

 bringing the proofs of each proposition nearer to the 

 fundamental axioms and definitions than Euclid does, it 

 renders less assumption of previous propositions neces- 

 sary for the proof of any given proposition. It stretches 

 the chain of argument straight instead of carrying it 

 round one or many unnecessary pegs." 



The influence which the Syllabus has had upon modern 

 editions of Euclid is patent to any reader of the works in 

 question. And now, little book, that the Association has 

 at the end of days sent forth on to (it may be) tempestuous 

 seas, we wish thee bon voyage! 



OUR BOOK SHELF 



American Journal of Mathematics. Vol. viii. No. 4. 



(Baltimore, August 18S6.) 

 The number opens with a memoir, by M. Poincard, " Sur 

 les Fonctions Abdliennes." The author gives here a 

 resume, with additional details, of a demonstration and 

 generalisation of twoof Weierstrass's theorems, which he 

 had previously published in the Proceedings of the Mathe- 

 matical Society of France (tome xii. p. 124). He then 

 extends a theorem of Abel's from plane curves to sur- 

 faces, and refers, for fuller details, to a crowned memoir 

 of M. Halphen's, " Sur les Courbes gauches algebriques." 

 He next discusses some properties of " fonctions inter- 

 mcdiaires," using the term in the sense given by MM. 

 Briot and Bouquet. This memoir occupies fifty-four 

 pages. The second paper, on '' A Generalised Theory of 

 the Combination of Observations so as to obtain the best 

 Result'' (24 pp.), is by the editor. Prof Newcomb. A very 

 valuable article, with important practical applications. 

 The final article (22 pp.), " On Symbolic Finite Solutions 

 and Solutions of Definite Integrals of the Equation — 

 d"y 



7- = "-'"'y^' 



ax 

 is by Mr. J. C. Fields. It discusses finite solu- 



tions analogous to the symbolic solutions of Riccati's 

 equation. 



A Sequel to the First Six Books of the Elements oj 

 Euclid ; containing an Easy Introduction to Modern 

 Geometry iunth numerous Examples). By John Casey, 

 LL.D., F.R.S. (Dublin : Hodges, 18S6.) 



This is the fourth edition of a book which has been 

 received with warm approval by English and Continental 

 geometers. The first eight sections present no notable 

 changes from the corresponding sections in the last edi- 

 tion. In our previous notice (NATURE, vol. xxix. p. 571) 

 we remarked that the author was " not so well up in the 

 literature of the modern circles as he might be." This 

 reproach is quite removed in the present edition. Indeed 

 in this direction the author has himself now done excel- 

 lent yeoman's service. The " supplementary chapter " of 

 fifty-eight pages gives an admirable account of this 

 modern branch in six sections. The first section states 

 and illustrates the theory of isogonal and isotomic 

 points, and of anti-parallel and symmedian lines. The 

 second discusses "two figures directly similar" in homo- 

 thetic figures. The third section is headed " Lemoine's 

 and Tucker's circles." The fourth discusses the " general 

 theory of a systein of three similar figures." The fifth gives 

 "special applications of the theory of figures directly 

 similar," more particularly with reference to Brocard's 

 circle and triangles. In the sixth section on the "theory 

 of harmonic polygons," the author, starting from Mr. 

 Tucker's extension of the Brocard properties to the 

 harmonic quadrilateral, and Prof Neuberg's continuation 

 of the same, gives his own beautiful generalisations to 

 the harmonic hexagon and other allied polygons. This 

 latter extension has been made the subject of a com- 

 munication by MM. Tarry and Neuberg to the French 

 Association meeting at Nancy in August of the present 

 year. The paper, which is not expected to be published 

 until April 18S7, contains a complete generalisation of 

 points of Lemoine and Brocard, and the modern circles 

 cited above for polygons and polyhedra. 



The success of the " Sequel " is due to the fact that the 

 author and the subject are exactly suited to each other : 

 the union is a most harmonious one, and the result is a 

 work indispensable to all lovers of geometry. 



Geometrical Drawijig for Army Candidates. By H. T. 

 Lilley, M.A. Pp. x., 54. (London : Cassell and Co., 

 1886.) 



In a short introduction to this little work the author 

 gives some useful advice to those beginning practical 

 geometry, and rightly lays stress on the proper method 

 of handling instruments, and on a good style of working. 



The book contains altogether 300 problems in plane 

 constructive geometry ; they are nearly all straight- 

 forward and easy, but iSo of them are specially indicated 

 as forming, according to the author's experience, a suit- 

 able first course for the majority of students. 



The problems are conveniently grouped together, and 

 hints are given in aid of the solution of typical ones, and 

 of those presenting extra difficulty. Beginning with the 

 construction of scales, we have the usual series on 

 polygons, proportionals, equivalent areas, and, in con- 

 clusion, several cases of circles touching other circles or 

 given lines. 



As a book of examples this collection seems likely to 

 prove useful in class-teaching. But in order to insure 

 sound instruction, much that is not contained herein will 

 have to be provided for the student. Thus in the notes 

 to the problems before us no reasons are given or indi- 

 cated for the various steps in the constructions, and there 

 is no distinction drawn between those methods of con- 

 struction which are exact, and those which do not admit 

 of proof 



