286 Cambridge Course of Mathematics. 
themselves, independently of their connections, while a 
multitude of others, without doubt, remain still undiscover- 
ed. The general principles with a view to whichan ele- 
mentary treatise of geometry ought to be composed, in the 
present state of mathematical science, we think, are these: 
ist. All those truths should be selected, which admit of 
extensive applications as well to ordinary practical purpo~ 
ses, as in the higher branches of mathematics. 2d. The 
connection with other truths already known, and render 
sensible, the transition from a proposition to that which 
follows:it. 5th. The traths demonstrated should be ar- 
ranged in the most natural order, and well connected with 
each other. 6th. The synthetic method of demonstration 
should be employed, as being peculiarly appropriate to el- 
ementary geometry. 7th. Great care ought to be used to 
larity of the propositions, their demonstrations ought to be 
similar. ‘The preservation of this analogy not only gives 
elegance to the demonstrations, but much assistance to 
memory. . , as? 
The question, why students in mathematics should use @ 
modern treatise of geometry, in preference to the Elements 
of Euclid, is of far more difficult discussion than the prece- 
ding. Euclid’s Geometry has come down to us clothed 
with the authority of the high antiquity of two thousand 
