290 Cambridge Course of Mathematics. 
ceeding and higher parts of mathematics. The seventh 
proposition, B. [. difficult for beginners, is given only for 
the sake of the eighth, and isof 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 tediousness and difficulty of their demonstrations. They 
are omitted by Hutton and other late English writers, as 
well as by Legendre and Lacroix. 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 extensive 
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- 
