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LVIII. Notices respecting New Books. 



The Elements of Plane Geometry for the use of Schools and Colleges. 

 By Richard P. W right, formerly Teacher of Geometrical Drawing 

 &c. hi Queenwood College, Hampshire. With a Preface by T. Archer 

 Hirst, F.R.S. 8?c, Professor of Mathematics in University College, 

 London. London: Longmans and Co. 1868 (pp. 211). 



TN noticing this work we shall, in the first place, speak of it simply 

 as a Treatise on Plane Geometry, and abstain from weighing 

 its merits against those of any other manual. Regarded from this 

 point of view, we may mention that it consists of four books ; of 

 which the first two are elementary, the third treats of ratio and of 

 similar figures, the fourth of areas. The first two books are in sub- 

 stance very nearly the same as props. 1-34 of the 1st book, props. 

 1-33 of the 3rd book, and a few propositions in the 4th book of 

 Euclid's ' Elements;' the differences, which at first sight look much 

 greater than they really are, consist mainly in arrangement and 

 in modes of statement and proof. For instance, the theorem "Through 

 any three points ABC, not in the same straight line, one, and only 

 bne, circumference can always be drawn," and the problem " To de- 

 scribe a circle which shall pass through three given non-collinear 

 points, P, Q, R," are neither more nor less than the theorem " One 

 circumference of a circle cannot cut another at more than two points," 

 and the problem " To describe a circle about a given triangle." The 

 arrangement, however, is very different ; e. g. the first chapter of 

 the first book is devoted to theorems equivalent to props. 13, 14, 15 

 of Euclid's 1st book. But perhaps the capital difference is that 

 Mr. Wright reserves the solution of problems to the end of the 

 second book. It might be thought that to assume a variety of 

 constructions in proving theorems without showing that previously 

 proved theorems give the means of making those constructions, 

 would lead to reasoning in a vicious circle ; and of course this might 

 easily happen. But we believe that Mr. Wright has avoided the 

 danger, and that the deduction of his various theorems is perfectly 

 rigorous. 



The first chapter of the third book is devoted to a very elaborate 

 discussion of the doctrine of ratio and proportion. The line taken 

 will be most easily understood by a citation of two definitions. 

 Ratio is defined thus : — " Between every two magnitudes A and B 

 of the same kind, there exists a relation answering to the inquiry how 

 often the first contains the second, which is called the ratio of A to 

 B." Proportionality is thus denned : — "When two associated and 

 variable magnitudes are so related that the ratio of any two values, 

 A v A 2 of A, is equal to the ratio of the corresponding values, B x , B a 

 of B, these magnitudes are said to be proportional." It is tolerably 

 plain, from the definition of ratio, that the discussion has to be pre- 

 ceded by an introductory discussion of commensurability and incom- 

 mensurability, which, again, introduces the conception of limits. 



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