193 



since in the case of numbers, and this the author of this paper 

 attempts in some cases in which geometrical lines only are the 

 subject of consideration, by a new treatment of a theorem equivalent 

 to Euclid's simple ex aquali and of the doctrine of similar triangles, 

 referring to nothing more advanced than Euclid, Book II. 

 The author proves, — 



1. If the rectangle contained under the sides a, B be equal to 

 the rectangle contained under the sides b, A ; and if these rectangles 

 be so applied together that the sides a and b shall be in a straight 

 line and that the side B shall meet the side A, the two rectangles 

 will be the complements of the rectangles on the diameter of a 

 rectangle. 



2. If the rectangle contained under the lines a, B is equal to the 

 rectangle contained under the lines b, A ; and if the rectangle under 

 the lines b, C is equal to the rectangle contained under the lines 

 c, B ; then will the rectangle contained under the lines a, C be 

 equal to the rectangle contained under the lines c, A. 



(This is equivalent to the ordinary ex tequali theorem. 



If a : b : : A : B 



and b : c : : B : C, 



then will « : c :: A: C.) 



3. If two right-angled triangles are equiangular, and if a, A are 

 their hypothenuses, and b, B homonymous sides, the rectangle con- 

 tained under the lines a, B is equal to the rectangle contained under 

 the lines b, A. 



(The equivalent theorem in proportions is 



a:b::A:B.) 



4. If a, c and A, C are homonymous sides of equiangular triangles, 

 the rectangle contained under a, C will be equal to the rectangle 

 contained under c, A. 



5. If b, c and B, C are homonymous sides including the right 

 angles of two equiangular right-angled triangles, the rectangle con- 

 tained under b, C will be equal to the rectangle contained under 

 c, B. 



6. If the rectangle contained under the lines a, B is equal to the 

 rectangle contained under the lines b, A ; the parallelogram con- 

 tained under the lines «, B will be equal to the equiangular paral- 

 lelogram contained under the lines b, A. 



(This is equivalent to the proposition, 



If a : b : : A : B 



then a : b : : A cos a : B cos a.) 



These propositions will suffice for the treatment of the first 

 thirteen propositions of Euclid's sixth book (Prop. I. excepted), and 

 of all the theorems and problems apparently involving proportions 

 of straight lines (not of areas, &c.) which usually present themselves. 

 The author then proceeds, as an instance of their application, to 

 prove by means of them the following theorem : — 



