OPERATIVE ROOTS OF THE CIRCLE-FUNCTION 123 



We may call a and a' counter-chords (for a reason which will appear) ; and 

 — a and — a', the same. 



Let the angles made by a, a' with the positive tangent be called the 

 chord-angles of these chords ; and the same for any chords ; and denote the 

 corresponding chord-angles by Greek letters. By Euclid III, 32, all angles 

 subtended by a chord in the alternate arc (to its left) are equal to the chord- 

 angle ; but they may also be called arc-angles. The same will apply to — a 

 and — a' if we use the generaUsed idea of an angle. All angles are positive 

 when taken counter-clockwise. 



Chords drawn with the centre of the circle on the left may be called 

 right chords (a) ; and those drawn with the centre on the right, left chords 



(a'). In the positive circle, the chord- 

 angles of right chords are acute, those 

 of left chords obtuse ; and the sum of 

 the chord-angles of two counter-chords 

 is two right angles — that is, in the case 

 of a and a' the angles a + a = 2 in 

 Quadrantal Measure. 



From any point in the positive circle 

 a third chord may be drawn such that its 

 length is equal to the lengths both of 

 a and of a'. Its chord-angle (namely, 

 the angle which it makes with a positive 

 tangent drawn from its own point of 

 origin) must equal either a or a, but not 

 both. By rotating the circle containing 

 it, it can be brought to coincide either 

 with a or with a', but not with both. 

 We shall here assume therefore that 

 all chords of equal positive circles which 

 have equal chord-angles are themselves 

 equal. For example, if a chord be drawn 

 from the end of a equal in length to a 

 but in a reversed direction backwards 

 to the origin, then it equals, not a 

 nor — a, but a'. 



Any two chords which begin or end 

 at the same point may be coinitial, or 

 coterminous, or sequent at that point ; 

 and by rotating the circle containing 

 one of the chords it can be brought 

 into either of these three relations with the other chord. 



We assume (for brevity), regarding any chord a, that a ^^ r sin a. 

 The tangents t and — t may be considered to be counter-chords of zero 

 length, with chord-angles o and 2 respectively, and may be denoted by o 

 and — o, or o' respectively. 



With these definitions the i^-operations can now be employed for any 

 geometrical problem ; but I have space only to indicate the first steps, 



4.1. (i) To prove that r is its own counter-chord. For the chord-angle 

 of y is I, and of y' is 2 — i. Therefore it is immaterial in which direction r 

 be drawn, whether from the origin or toward it. So also for — r. 



(2) To prove that r = K^o and that o = K~ V. For 



KrO = cos 1.0 + sin i . ^/r'^ — o^ =» r ; 



K—^r = cos ( — i) , y + sin (— i) . Vr^ — y^ = o ; 



if we take the positive value of the radical (see 3 "04) 



K^a 



K^a 



'K^a' 



Fig. I. 



