NATURE 



573 



THURSDAY, OCTOBER lo, 1901. 



RATIONAL GEOMETRY. 



Plane and Solid Geometry. By Arthur Schultze, Ph.D., 

 and F. L. Sevenoak, A.M., M.D. Pp. ix4-37o. (New 

 York : The Macmillan Company, 1901. London : 

 Macrnillan and Co., Ltd.) Price bs. 



THIS is an excellent work for all young students who 

 wish to begin the study of geometry. In its order 

 of treatment it completely ignores Euclid, and thus 

 saves the young pupil from a long and wearisome waste 

 of time, giving him clearly and rapidly a knowledge of 

 the subject and an insight into its nature and purpose. 

 We wish that the English schoolboy could congratulate 

 himself on its appearance ; but this is forbidden by our 

 conservatism and the attachment of our public and pre- 

 paratory schools to mediasvalism. When an Educational 

 Reformation takes place in England — and there are 

 signs of its advent — such a work will be welcomed by all 

 of us who are interested in the scientific education of 

 the people. 



The book is divided into two parts — Plane Geometry 

 and Solid Geometry. The first part is divided into five 

 books (225 pages), and the second into three (93 pages). 

 The type is excellent, and the figures (especially those in 

 the second part) beautiful. 



To enter now into a i^\w matters of detail, we would 

 suggest to the authors that they should not have followed 

 the English plan of beginning with a catalogue of fifty 

 definitions before the pupil reaches the real work of the 

 subject : this makes for tediousness. The term straight 

 angle (adopted, apparently, from the A.I.G.T.) is, we 

 think, very objectionable, because the notion of straight- 

 ness should be kept quite distinct from that of an angle. 

 The first proposition in the book, "vertical angles are 

 equal," is Euclid's 15th ; prop. 2 is Euclid's 26th ; prop. 

 3 is Euclid's 4th ; prop. 4 is Euclid's i6th. Thus a 

 common-sense order of treatment is freely adopted. 

 Then comes the treatment of parallel lines in which all 

 of Euclid's results are given. The definitions of degrees, 

 minutes, and seconds are given in the preliminary defini- 

 tions (p. 5), but the protractor is not mentioned, so that 

 the actual way of reading the value of a given angle is 

 not exhibited. This omission of the protractor seems to 

 us to be a mistake. Some propositions are merely 

 enunciated, and, instead ot a formal proof, a " hint " to 

 the pupil in a few words is given. This is good, because 

 it e.xercises, without severity, the power of the young 

 thinker. The famous .Xsses' Bridge is given as prop. 14, 

 with the mere hint that it would be obvious if the bisector 

 of the angle at the vertex were drawn — as, of course, it 

 would be. Herein observe the contrast to Euclid, who 

 would not allow us to use this bisector unless he had 

 previously shown how to construct it — a perfectly useless 

 restriction which runs through the whole of Euclid. Of 

 course it is subsequently shown (p. 35) how to bisect an 

 angle and a line. The authors are generally precise in 

 their use of language, without adopting the grotesque 

 show of accuracy in our school Euclids. Nevertheless, 

 NO. 1667, VOL. 64] 



they occasionally make an absurd use of the word re- 

 spectively, which is so prominent in these works. Thus 

 (p. 30), " two triangles are equal if the three sides of the 

 one are respectively equal to the three sides of the 

 other " ; clearly no order of equality is necessary. See also 

 end of p. 74. 



In p. 33 and elsewhere the authors boldly define a 

 circle as an area, and distmguish it from its bounding 

 curve, which they call "a circumference." If we punch 

 a wad out of a sheet of cardboard, which area has the 

 right to be called the circle — the wad or the whole of the 

 outside area of the sheet ? 



In this respect, however, the authors are consistent, 

 while Euclid is not. Euclid's formal definition makes the 

 circle an area, while in his Book HI. he says that two 

 circles cannot have more than two points in common. 



The English barbarism involved in the proposition " if 

 two sides of a triangle are equal, the opposite angles 

 shall be equal," is consistently avoided, the simple word 

 " is " or " are" being always used instead of the compulsory 

 and ridiculous " shall be" of our school Euclids. 



The second book of Part i. treats of the circle, and 

 travels over the ground of Euclid's III. and a little more, 

 arithmetical examples being occasionally given — a great 

 desideratum in our English system. Here measurement 

 and ratio are introduced, as well as the notion of limits 

 — a great improvement. Euclid's first book problem, 

 "to construct a triangle when its three sides are given," 

 appears here as prop. 19 — a postponement of more than 

 doubtful value. 



The third book is on proportion and similar polygons, 

 and the propositions are illustrated and explained by 

 simple algebra and arithmetic ; thus the beginner can 

 learn the essence of the subject in a few minutes without 

 wasting a lifetime on Euclid's Book V. In this book the 

 authors give the propositions relating to the equality of 

 areas, of triangles, and parallelograms, while the pro- 

 position of Pythagoras now appears for the first time 

 (p. 147), founded on the similarity of the two triangles 

 into which a " right triangle " is divided by the perpen- 

 dicular from the vertex on the hypothenuse ; the old 

 proof and time-honoured figure are, however, given in 

 the next book (p. 178). Near the end of the third book 

 we have Euclid's well-known proposition whose trigono- 

 metrical form is c- = a''- + b- - lab cos C, the proof 

 being, of course, geometrical, but presented in algebraic 

 form. 



The fourth book treats of the areas of polygons, and 

 the proofs are presented in algebraic form. The exercises 

 all through are numerous and very appropriately placed. 

 In the part dealing with solid geometry and the funda- 

 mental properties of spherical triangles, the figures are, 

 as we have said, exceedingly good and realistic. 



In p. 264 we have the proposition "the sum of any 

 two face angles of a triedral angle is greater than the 

 third face angle" ; but the proof will have to be slightly 

 modified, as, in its present form, it is confusing for the 

 beginner. Thus, to the words " in the face AVC draw 

 VD equal to VB, making /_ DVA = Z. BVA," it may 

 fairly be objected that this is impossible if the points A 

 and C are already given. The line VD should first be 

 drawn, and then the lines ADC and BC. Again, in 



B li 



