364 PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



14. The forms in the Harmonia Mensurarum might also be brought to con ■ 

 firm this theory: but, without accumulating instances any further, it may be 

 sufficient to remark two consequences that follow from it: 1. That the only cases 

 in which imaginary expressions may be put to denote real quantities, are those in 

 which the measures of ratios or of angles are concerned. 2. That the property 

 of either of those measures, so investigated, might have been inferred from ana- 

 logy alone. Now both these conclusions are agreeable to experience. It does 

 not appear, that any instance has yet occurred where imaginary characters serve 

 to express real quantities, if circular arches or hyperbolic areas are not the subjects 

 of investigation; and if the conclusion obtained may not be transferred from the 

 one to the other, by a mere substitution of corresponding magnitudes; that is, 

 of sines for ordinates, cosines for abscisses, and circular arches for the doubles 

 of hyperbolic sectors. The affinity between the circle and hyperbola is not how- 

 ever so close, bnt that it is subject to certain limitations, from considering which, 

 the truth of what is here asserted will be rendered more evident. 



(1.) Any proposition demonstrated of hyperbolic sectors may be transferred to 

 circular arches by substitution alone, without any change in the signs, when 

 only abscissae and their products enter into the enunciation, and conversely. 

 Thus abs. a X abs. h = 



abs (a + b) , abs. (a — l>) , ^, , cos. (a + i) , cos. (a— i) ^r., 



''°^- ^ ^ '- -\ \: -; and cos. a X cos. b = ^- -^ ^r — -. Thesame 



2 ' 2 '2 2 



holds when the simple power of the ordinate is combmed with any power what- 

 ever of the absciss: so in the theorems of art. 5 and 4, ord. a X abs. b = 

 ord^-ti) ^ orMpi) . and sin. a X cos. b = ''^-^'^ + T^ *). 



(2.) When an expression containing any property of hyperbolic sectors, in- 

 volves in it the rectangle of two ordinates, the value of that rectangle must have 

 a contrary sign, when a transition is made to the circle. Thus ord. a X ord. b 

 _ abs^(«+^ _ sbs^a-b) ^.^ ^.^^ ^ ^ _ c_^^^ cos^,(«-«) 



difference which, according to this rule, is found between the powers of ordinates 

 and of sines, may be seen in the following examples. If -{x denote any hyper- 

 bolic sector, then, by involving ^-^^:^ — , and again substituting for the exponen- 

 tial quantities as in art. 5, we have, 

 (ord. xY = 



;;, ■'■'■ I (ord. .r) 

 ' • ' , , ,. abs. 4.1 — -l abs. 2.1 + 3 ' 



(ord. xy = g ; 



, ., ord. 5.r — 5 ord. 3i + lOord. .i , ,, .... 



(ord. xy = j^ ; and universally, i( n be 



any number; a the co-efficient of the 2d term of a binomial raised to the power 



2 abs. 2i' — 1 



— ^ ; 



3 ord. 3j; — 3 ord. x 



— 1 ' 



