Mr. W.G. Horner an the Signs of the Trigonometrical Lines, 89 



cord witli the statements given in every treatise. But it is also 

 apparent from [a J, that the line A P and its continuation back- 

 'wards from A, complete not their revolution in less than eight 

 quadrants. The affections, in short, of the system of chords 

 are seen to be as under: 



In quadrant ... 



chord 



sec-chd 



1st co-chd ... 

 2nd ... 



&c. 



6. The same results may be obtained from the analytical 

 statements, chd A = 2 sin ^ A, sec-chd A = 2 cos | A, co- 

 chd A = 2 sin J f— + A], on exchanging A for % T A, 



2 -TT + A, 3 TT + A, 4 TT T A. It would therefore be better, 

 perhaps, to adhere to this mode of stating the value of the 

 chord, as Woodhouse does in the passing notice he gives it, 

 than by resting on the irrational form to leave the sign uncer- 

 tain. The plan of Mr. Hind's first chapter, it is true, re- 

 stricted him in this respect ; but his unguarded assertion, that 

 '' the chord is positive in every quadrant," has a direct ten- 

 dency to mislead, and should be revised. The versed sine 

 alone is always positive. 



In conclusion, it may be remarked, that whereas for every 

 other line or function {/) the formula of reduction to the 

 tables or diagrams is y (2 W7r+ A) =y(±A), for the chords 

 itis,chd(2w7r + A) = (-l)"-^chd(±A)= + (-l)"chdA. 



Mnemonic Hint, 

 Setting aside the versed sine and chord, whose aifection, as 

 well as value, is clearly expressed by 1 — cos A and 2 sin ^ A, 

 the affection of the principal lines is indicated by the order in 

 which it is most natural and usual to name them, viz. Sine, 

 Tangent, Secant. For, in the 1st quadrant all being + ; 

 in the 2nd all are — , except the Sink and its reciprocal the 

 cosecant. In the 3rd. all are — , except the Tangent and its 

 reciprocal the cotangent. In the 4th all are — , except the 

 Secant and its reciprocal the cosine. 



Note (relative to a formula in Lond. and Edinb. Phil. Mag., 

 vol. v. p. 191.) — Having still a space left, I take the oppor- 

 tunity of remarking, that the method of elimination by the 

 common measure, although tempting by its facility and ele- 

 mentary nature, is too defective to deserve recommendatioq. 



Tliird Series, Vol. 6. No. 32. Feb, 1835. N 



