286 Cambridge Course of Malhemuhcs. 



themselves, independently of their connections, while a 

 multitude of others, without doubt, remain still undiscover- 

 ed. The general principles with a view to which an ele- 

 mentary treatise of geometry ought to be composed, in the 

 present state of mathematical science, we think, are these: 

 1st. 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. 'Sd. The 

 demonstrations should unite, as far as possible, the ele- 

 gance and rigor of the ancient geometers, with a greater 

 degree of conciseness. 3d. The demonstrations should 

 be so constructed as to exclude, as far as possible, propo- 

 sitions merely subsidiary ; that is, propositions which are 

 of no practical importance in themselves, but only steps in 

 demonstrating others. 4th. Indirect demonstrations should 

 be avoided, as much as possible. In general, those proofs 

 are the best, which establish the truth proposed upon an 

 immoveable basis, and, at the same time, clearly shew its 

 connection with other truths already known, and render 

 sensible, the transition from a proposition to that which 

 follows it. 5th. The truths 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 

 preserve a uniformity in the style of the demonstrations, 

 and an analogy between the different parts of the treatise. 

 We shall be better understood on this point, by taking an 

 example. Parallelograms upon the same base, and of the 

 same height, are equal. Also, parallelopipeds upon the 

 same base, and of the same height, are equal. The form- 

 er of these propositions, has the same relation to areas, 

 that the latter has to volumes. On account of the simi- 

 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 the 

 memory. 



The question, why students in mathematics should use a 

 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 



