REPORT ON CERTAIN BRANCHES OF ANALYSIS. 291 



The sines and cosines and the measures of angles defined 

 and detei'mined as above, are the only essential elements in a 

 system of trigonometry, and are sufficient for the deduction of 

 all the important formulge which are required either in algebra 



angle 6, would be perfectly indeterminate. It is the assumption of the measure 

 of an angle which is mentioned in the text which makes it necessary to re- 

 place A by 1. 



The knowledge of the exponential expressions for the sine and cosine would 

 furnish us immediately with all the other properties of these transcendents. 

 Thus, if the sines and cosines of two angles be given, we can find the sines 

 and cosines of their sum and difference ; and from hence, also, we can find 

 the sine and cosine of any multiple of an angle from the values of the sine 

 and cosine of the simple angle ; and also through the medium of the solution 

 of equations the sine and cosine of its submultiples. In fact, as far as the 

 symbolical properties of those transcendents are concerned, it is altogether 

 indifferent whether we consider them to be deduced primarily from the 

 assumed functional equations (1.), (2.), (3.), or from the primitive geome- 

 trical definitions of which those equations are the immediate symbolical con- 

 sequences. 



/*x dx /»« dv 



If we should denote the integrals / and / — (com- 



»/ V 1 — -r^ Jo Vl— y^ 

 raencing from respectively) by 6 and 6' respectively, then the integral of 

 the equation 



dx dy 



would furnish us with the fundamental equation 



sin {6 + 9') = sin 6 cos 6' -|- cos 6 sin 6', 03.) 



if we should replace x by sin 6, Vl — a? hj cos 6, y by sin 6', and Vl — y» 

 by cos i'. If the formulae of trigonometry were founded upon such a basis, 

 they would require no previous knowledge either of circular arcs considered 

 as the measures of angles, or of the geometrical definitions of the sines and 

 cosines, except so far as they may be ascertained from the examination of the 

 values and properties of the transcendents which enter into the equation (a.). 

 In a similar manner, if we should suppose 6 and 6' to represent the integrals 



/*•'* d X /* V d V 



of the transcendents / .,, , — -. and / — " then the integral 



Jo V(l+i-) Jo \/(l— r) ^ 



of the equation 



d X dy . , . 



would be expressed by the equation 



h sin {6 + d') ■=■ hsm 6 X h cos ^' + h cos 6 X h sin d', (8.) 



if we should make x z=z h sin 6 (the hyperbolic sine of 6), and -x/ (1 + •»!*) 

 = h cos 6 (the hyperbolic cosine of d),y=h sin 6', and Vl -j- ^2 _. ^ pq3 ^'^ 

 adopting the terms which Lambert introduced, and which have been noticed 

 in the note in p. 231 ; and it is evident that it would be possible from equa- 

 tion (S.), combined with the assumptions made in deducing it, to frame a 

 system of hyperbolic trigonometry (having reference to the sectors, and not 



u 2 



