68 Mr. W. G. Horner on the Signs of the Trigonometrical Lines, 



Now (Euc, i. 32.), 90° + IAP= ACP + APC=2IAP 

 + A P C .-. 90° -I A P = A P C. But (Euc, i. 29), 90° 

 -I A P = C A P ; wherefore A P C = C A P. Consequently 

 C P = C A constantly; which is the defining property of the 

 circle. Q. e. d. 



3. This demonstration obviously applies to angles in every 

 quadrant. And if a is substituted for A, at a quadrant's 

 distance from it (fig. 2.), and i, t, for I, T, the same applies 

 verbatim to the complementary arcs or angles also. Now, T, 

 or /, being placed at the concurrence of C P and A T, or a t 

 and B P being moreover supposed to revolve in the same di- 

 rection /^^ about the point B, or 6, diametrically opposite 

 to A, or «, it is manifest that we have 



1^ of ACP. 



4. Now, in the first quadrant, to the conditions of which 

 all other quadrants are supposed to conform, the lines A T, 

 A P, C P, B P, are terminated at A, C, B, and extensible 

 only in the directions T and P. If that which is affirmed re- 

 specting any of these lines requires, in any specified quadrant, 

 to be accommodated to the continuation of such line in the 

 opposite direction, the variation will be indicated by a nega- 

 tive sign, as is well known. In my diagrams, the original and 

 accidental or positive and negative lines are distinguished, the 

 former by being drawn with a full stroke, and the latter by 

 being dotted. And to render the whole as completely eluci- 

 datory as possible, I have annexed, rather than suffixed, to 

 each P, I, and T, the number of the quadrant to which it ap- 

 pertains, following in this respect the convenient notation of 

 Mr. Hind. A dotted arrow is also introduced, to show the 

 negative character of such arcs or angles as imply a revolu- 

 tion contrary to the direction originally assumed. 



If these minutiae answer the purpose intended, of making 

 these diagrams an intelligible substitute for dry tables of 

 + and — , I am sure those who have no need of either will 

 have the good nature to excuse them, for the sake of those 

 who will find them useful. 



5. From the diagrams, then, it is apparent that the varia- 

 tions of affection of all the lines, except the chords, are com- 

 pleted within the course of four quadrants, and that they ac- 



