November 14, 1901] 



NA TURE 



27 



another : so also have the sectors.'' For the proof of 

 these two propositions onh is the fifth definition usually 

 employed. All the remaining properties of proportion 

 required for use in the sixth book are either assumed or 

 proved (it would be inore correct to say they are supposed 

 to be proved) algebraically. Some teachers, probably a 

 small minority, use the syllabus of the Association for the 

 Improvement of Geometrical Teaching. This syllabus, 

 following Euclid's line of argument, uses the properties 

 of unequal ratios to prove properties of equal ratios, thus 

 making the proofs unnecessarily artificial and therefore 

 difficult ; so that, though it is quite possible for a clever 

 pupil to follow the reasoning in each separate proposi- 

 tion, it is very difficult for him to grasp the argument as 

 a whole. 



In the book under notice the fundamental definitions of 

 the fifth book of Euclid, viz., the test for equal ratios. No. 5, 

 the test for distinguishing between unequal ratios, No. 7, 

 and the definition of the compounding of ratios, with 

 which may be included the definition of duplicate ratio 

 find no place. And it may be conceded at once that, if 

 these definitions are not properly explained, it is much 

 better that they should not appear in an elementary 

 text-book, because the beginner, to whom the simple 

 ideas on which they rest are not carefully expounded, is 

 far more likely to make progress in geometry with the 

 aid of Prof Holgate's book than with an ordinary Euclid. 



The author, after defining the ratio of two incommen- 

 surable magnitudes as the limit of a rational fraction, 

 obtained by a definite process (Art. 229), proves a general 

 theorem on limits, viz., " that if there are two variable 

 quantities dependent on the same quantity in such a way 

 that they remain always equal while each approaches a 

 limit, then their limits are equal " (Art. 230). 



With the aid of this, the two propositions, Euc. VI., i 

 and 33, already referred to, are proved, and also the pro- 

 position " that a straight line parallel to one side of a 

 triangle divides the other two sides in the same ratio," 

 Euc. \"I., 2 (first part}. 



Taking the proof furnished of this last theorem as a 

 type, and leaving out of it the use made of the general 

 theorem on limits quoted above, it may be pointed out 

 that the proof is in effect a demonstration (though some 

 expansion would be necessary to make this clear) of the 

 proposition that if a certain definite process be applied to 

 the segments of each of the sides of the triangle the result 

 will be to determine the same irrational number' in each 

 case, so that this irrational number may be taken as the 

 measure of the ratio of the two segments of each side, 

 and that consequently these ratios are equal. With such 

 a change the treatment would accord with modern ideas 

 regarding irrational numbers as set forth by Dedekind 

 in his tract on continuity and irrational numbers - and 

 now generally accepted. 



But when such propositions as " that if two ratios are 

 equal, their reciprocal ratios are equal " are required 

 (see Arts. 234, 237-240), then the proofs supplied are 



ne which separates all rational 



)wer, such that 



less than every fraction in the 



^ An irrational number is definec 

 fractions into two classes, an upper a 



(i) Every fraction in the lower cl; 

 upper class. 



(2) The lower class contains no greatest fraction. 



(3) The upper class contains no least fraction. 



- Translated into English by Prof, lieman. (Chicago : The Open O 

 Publishing Co., 1901.) 



NO. 1672, VOL. 65] 



valid for commensurable ratios only. This change in 

 the mode of treatment should have been clearly indi- 

 cated in the text or preface. To have proved these 

 propositions on the lines on which the treatment of the 

 subject of ratio had been begun would soon have carried 

 the author beyond the comprehension of those for whom 

 his book was intended. To have proved them upon 

 Euclid's lines would have made it necessary to add a 

 large number of additional explanations. This, however, 

 was the only practicable logical alternative. 



The fourth chapter deals with the areas of plane 

 polygons and with the measurement of the circle. Archi- 

 medes proved that tt lies between 3i and 3*^ by a con- 

 sideration of the regular inscribed and circumscribed 

 polygons of ninety-six sides. The author obtains a much 

 closer approximation, but finds only a series of values 

 increasing up to tv by using the inscribed regular polygons. 

 The result would have been more impressive if a series 

 of values decreasing down to tt had been found as well. 



The sixth chapter deals with lines and planes in space. 

 It is a matter of opinion whether the modes of construct- 

 ing the perpendicular to a plane in §§ 403 and 408 are 

 or are not more difficult than Euclid's ; but these last 

 are so useful in Spherical Trigonometry that it seems a 

 pity they have not had more prominence given to them 

 than is furnished by § 406. For a similar purpose it 

 would have been useful to give some further account of 

 the angles between a line which meets a plane and the 

 lines in the plane than Prop. xx. in § 449, viz., "The 

 acute angle which a straight line makes with its own pro- 

 jection upon a plane is the least angle it makes with any 

 line of that plane." It is advantageous to know, not only 

 the least and the greatest angles between a fixed line 

 meeting a plane and lines in the plane, but also the 

 way in which the angle varies as the line in the plane 

 revolves. 



The statement of Prop, xxii., § 458, " that the sum of 

 any two face angles of a trihedral angle is greater than 

 the third angle," should be limited by inserting the word 

 " convex" before the word " trihedral." 



The seventh, eighth and ninth chapters deal with 

 prisms, pyramids, cylinders, cones and spheres. 



There are several features of interest in the book, such 

 as the use of the principle of continuity in §§ 204, 264, 

 323, and the directions given in § 540 for constructing 

 the regular polyhedra. The proofs of many of the pro- 

 positions seem to be new. The work is evidently that of 

 an experienced teacher, and is written on the lines of 

 good class teaching, in which the teacher suggests steps 

 in the argument to the pupil, and the demonstrations are 

 worked out by both together. The book is calculated to 

 arouse and stimulate those who have at heart the teaching 

 of their subject. 



M0S<2UIT0ES AND MALARIA IN MA URITIUS. 



Les Moustiques : Anatoiiiie et Biologic. By A. Daruty 

 de Grandpre and D. d'Emmerez de Charmoy. (Port 

 Louis, Mauritius : Planter^ and Commercial Gazette, 

 1900.) 



THIS work is a contribution to the study of the 

 Culicid;e, and principally of the genera Culex and 

 Anopheles, of their role in the propaga,tion of malaria 



