233 Cambiidge Philosophical Society. 



If a:b::A:B 



and b:c::B:C, 



then will a : c : : A : C.) 



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

 their hj^pothenuses, 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 ff, 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. li 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 a, 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 : — 



If pairs of tangents are drawn externally to each couple of three 

 unequal circles, the three intersections of the tangents of each pair 

 will be in one straight line. 



Also a paper was read by Professor De Morgan, " On a Proof 

 of the existence of a Root in every Algebraic Equation : with an 

 examination and extension cf Cauchy's Theorem of Imaginary 

 Roots ; and remarks on the proofs of the existence of Roots given 

 by Argand and by Mourey." 



The extension of Cauchy's theorem is very easily found, when the 

 proof is the first of those given by Sturm in Liouville's Journal. 

 The extended theorem is as follows : — 



Let <pz be any function of z, and let z^x + yV — 1. Let (x, y) 

 be a point on any circuit which does not cut itself. Let this point 

 describe the circuit in the positive direction of revolution ; and, 



P 



^(.r+y 'Z— 1) being P + Q. i/—\, let — • change sign k times as in 



+ — , and / times as in — 0-f. Let {x, y) be called a radical 

 point when (p(^x-\-y ^—\) =0, or=oo. Let there be m radical 

 points of the first kind within, and w' upon, the circuit ; let there 



