G. B. AIRY, ESQ., ON THE SUBSTITUTION OF METHODS, &c. 



167 



place before the Society the series of propositions which I suggest as sufficient for these pur- 

 poses, and (as an example) their application to the particular Theorem to which I have 

 alluded. I have omitted several merely formal steps in the demonstrations. It will be seen 

 that the demonstrations which I offer, though applying to the properties of lines only, require 

 the use of areas; but in this respect they are simpler than Euclid's, which, though applying 

 to lines only, require the use both of areas and of the process of equimultiples. 



Proposition (A). If the rectangle contained under the sides a, B, be equal to the rect- 

 angle contained under the sides 6, 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. 



Because the opposite vertical angles of the two rectangles are equal at the point of meeting, 

 A and B will be in the same straight line. Produce the external sides of the rectangles till 

 they meet in D, join DE ; and, as the sum of the angles GFD, EDF, is less than two right 

 angles, produce the lines FG, DE, till they meet in H; and draw ^/parallel to FD or GE. 

 If the rectangle under h and A is not terminated in the line HI, let it be terminated by the 

 line KL. Since KL is parallel to 6 or GE and therefore parallel to HI, it will be entirely 

 above or below HI. Now by Euclid, the complements FE, EI, are equal ; but, by hypothesis, 

 FE, EL, are equal ; therefore EL is equal to EI, which is impossible if the line KL is above 

 or below ///; therefore KL coincides with HI, and the rectangle 6, A, coincides with the 

 complement EI, and the two given rectangles therefore are the complements, &c. a.E.D. 



Peoposition (B). If the rectangle contained under the lines a, B, is equal to the rect- 

 angle contained under the lines A, h ; and if the rectangle contained under the lines 6, C, is 

 equal to the rectangle contained under the lines B, v ; then will the rectangle contained under 

 the lines a, C, be equal to the rectangle contained under the lines A, c. 



[This is equivalent to the ordinary ex cequali theorem. 



If a : b :: A : B, 



and b : c :: B : C, 



Then will a : c :: A : C] 



