VOL. LXVIXI.] PHILOSOPHICAL TRANSACTIONS. 359 



e*v'-' + c- V-' ti^grefoj-e sin. a X cos. b = 



2 

 L-j+ix-v/-'-c-''~*X\/-' + C'»-*XA/-'— c'-ax v/-i sin. (Z + 6 , sin, a — b 



5. Now it may be observed, tliat the imaginary value which has been found 

 for sin. a was obtained by bringing a fluxion, properly belonging to the circle, 

 under the form of one belonging to the hyperbola. It may therefore be worth 

 while to inquire, whether a similar expression may not be derived from the hyper- 

 bola itself. 



Let BAD be a rectangular hyperbola (fig. 3) of which the centre is c, and the 

 semi-transverse axis ac = 1 : let b be any point in the hyperbola, join bc, and 

 let BE be an ordinate to the transverse axis. Then, if the sector acb = ia, and 

 BE = z, it is known that a = -~ — jr; whence a = log. z + v'(l + z'^), and 

 ca =z z -\- v' (1 + z''). But if the sector be taken on the other side of the trans- 

 verse axis, a and z become negative, and c— « = — z -\- a/ {\ -{- z'). Hence 

 in like manner the absciss belonging to acb, that is ce — 



2 ' CO' 2 • 



These values of the ordinates and abscissae differ in nothing from those of the 

 sines and cosines already found, except in being free from impossible quantities; 

 for it is evident, that the quantity a is related in the same manner both to the cir- 

 cular and hyperbolic sectors. If now the ordinate a and abscissa b denote the 



ordinate and absciss belonging to the sectors ia, i^ respectively, then ord. a x 



c'-c-" c* + c-* (-(+i_f-a-* 4. c^-i — c*— " ord. a + 6 ord.a-b 



abs. b = —^- X —J— = = — y— + — — -. 



6. The conclusions in both the foregoing cases are perfectly coincident, and 

 the methods by which they have been obtained are similar; though with this 

 difference between them, that in the first all the steps are unintelligible, but in 

 the last significant. If then, notwithstanding a difference which might be ex- 

 pected so materially to affect their conclusions, they have been equally successful 

 in the discovery of truth, it can be ascribed only to the analogy which takes 

 place between the subjects of investigation; an analogy so close, that every pro- 

 perty belonging to the one may, with certain restrictions, be transferred to the 

 other. Accordingly, every imaginary expression, which has been found to be- 

 long to the circle in the preceding calculation, is by the substitution of real for 

 impossible quantities, or of -v/ 1 for \/ — 1, converted into a proposition which 

 holds of the hyperbola. The operations therefore performed with the imaginary 

 characters, though destitute of meaning themselves, are yet notes of reference 

 to others which are significant. They point out indirectly a method of demon- 

 strating a certain property of the hyperbola, and then leave us to conclude from 

 analogy that the same property belongs also to the circle. All that we are assured 

 of by the imaginary investigation is, that its conclusion may, with all the strict- 

 ness of mathematical reasoning, be proved of the hyperbola ; but if from thence 



