279 



If the sphere is then to become only indeterminate, and 

 not necessarily Infinite, we must suppose that the numerator 

 of the same expression (13) also vanishes; that is, we must 

 have in this case the condition 



V . ABCDA = 0. (15) 



In words, as the product of the five successive sides of an 

 uneven but rectilinear pentagon inscribed in a sphere, has 

 been seen to be purely a line, so we now see that the product 

 of the four successive sides of a quadrilateral inscribed in a 

 circle is (in this system) purely a number: whereas, for every 

 oMer rectilinear quadrilateral, t^'Aei/ier plane or gauche, the 

 grammarithm obtained as the product of four successive sides 

 involves a grammic part, which does not vanish. This 

 condition (15), for a quadrilateral inscribable in a circle, 

 could not be always satisfied, when d approached to a, 

 and tended to coincide with it, unless the following theorem 

 were also true, which can accordingly be otherwise proved : 

 the product abca, or ab x bc X ca, of three successive sides of 

 any triangle abc, is a pure vector, in the direction of the tan- 

 gent to the circumscribed circle, at the point a, where the 

 sides which are assumed as first and third factors of the pro- 

 duct meet each other. If a^ be the point upon this circum- 

 scribed circle which is diametrically opposite to a, we find for 

 the length and direction of the diameter aa^ in this notation, 

 that is, for the straight line^o a. from a^, the expression :* 



abca 



AA ■=. ; (Id) 



V . ABC ^ ' 



the denominator denoting a line which is in direction perpen- 



* With respect to the notation of division, in this theory, the author pro- 

 poses to distinguish between the two symbols 



q' 

 Q~' q' and — , 



