290 Cambridge Course of Mathematics. 



ceeding and higher parts of mathematics. The seventh 

 proposition, B. 1. difficuh for beginners, is given only for 

 the sake of the eighth, and is of no further use whatever. 

 The sixteenth is evidently implied in the thirty-second, and 

 therefore is of no use, except as being subsidiary to the de- 

 monstration of others. Propositions forty-fourth and for- 

 ty fifth, are not of sufficient use to compensate for the space 

 which they occupy In B. II. the sixth, eighth, tenth 

 and eleventh propositions, with some others relating to the 

 properties of straight lines variously divided and produced, 

 are very unimportant and tend to discourage beginners by 

 the (ediousness and difficulty of their demonstrations. They 

 are omitted by Hutton and other late English writers, as 

 well as by Legendre and Laoroix. Many of the proposi- 

 tions in B. III. are, also, of small practical utility, and are 

 not used in subsequent parts of the science. The demon- 

 strations of many of them are indirect, of some of them, ar- 

 tificial ; and the construction of some of the figures, is un- 

 natural and difficult to be conceived. To one or another 

 of these objections, the following propositions are liable ; 

 fourth, fifth, sixth, tenth, eleventh, twelfth, thirteenth. The 

 fourth B. contains an incomplete view of that part of the 

 science which it embraces. It ought at least, to comprise 

 an investigation of the approximate ratio of the circumfer- 

 ence to the diameter of a circle. Of B. sixth, we have 

 to say only, that M. M. Legendre and Lacroix have demon- 

 strated the same truths in a more simple and equally rigor- 

 ous manner, that they have divested them of much techni- 

 cal language which rendered them difficult to be understood, 

 and that they have supplied many propositions of extensi).ie 

 use in the subsequent parts. 



Another particular, on account of which we must give the 

 preference to the Elements of Legendre and Lacroix, respects 

 the arrangements. It is by a different and more skillful ar- 

 rangement, that they have contrived to avoid, much more 

 than Euclid has done, subsidiary propositions,indirect demon- 

 strations, and unnatural constructions. Perhaps we may here 

 be expected to furnish the instances, in which their arrange- 

 ment is superior to that of Euclid. But by way of excusing 

 ourselves from this, we must beg leave to observe, that a 

 question of arrangement is of so extensive a nature, that we 

 could not do justice to our views of the subject, without en- 



