366 



MEMOIRS OF THE AMERICAN ACADEMY. 



rich husbands and American wives of any class of persons are wholly contained unde 



that class, and can be described without any discrimination of 



In point of fact 



by (156), the 



wives of any 



In order, 



rich husbands and American wives of the rich husbands and American 

 class of persons, are the rich Americans of that class. 

 Lobatchewsky has shown that Euclid's axiom concerning parallels may be supposed 

 o be false without invalidating the propositions of spherical trigonometry. 

 lien, that corresponding propositions should hold good in logic, we need not resort 

 o elementary relatives, but need only take S and V in such senses that every relative 

 f the class considered should be capable of being regarded as a sum of a scalar and 

 vector, and that a scalar multiplied by a scalar should be a scalar, while the product 



of a scalar and 



Now, to fulfil these conditions we have only 



take S^ as " self-q of," and V^ as " alio-q of" (q of another, that other being 



), and 



q may be any relative whatever. For, "lover," for example, is divisible into self-lover 

 and alio-lover ; a self-lover of a self-benefactor of persons of any class is contained 

 under that class, and neither the self-lover of an alio-benefactor of any persons nor the 

 alio-lover of the self-benefactor of any persons are among those persons. Suppose, 

 then, we take the formula of spherical trigonometry, 



cos a 



cosb cose -f- cos A sinb sine 



In quaternion form, this is, 



(165.) 



S(pq) 



(Sp)(S q ) + S((Yp)(Yq)). 



Let p be " lover," and q be " benefactor." Then this reads, lovers of their own b 

 factors consist of self-lovers of self-benefactors together with alio-lovers of alio-b 

 factors of themselves. So the formula 



sin b cos p b' 



sin a cos c cos pa' — sin c cos a cos pc' -f- sin a sin c sin b cos pb , 



where A', B', C, are the positive poles of the sides a, b, c, is in quaternions 



(166.) 



Y(pq) 



(Yp)(Sq) + (Sp)(Vq) + Y((Yp)(Yq)) , 



of 



and the logical interpretation of this is : lovers of benefactors of others 



lovers of self-benefactors, together with self-lovers of alio-benefactors, together with 



alio-lovers of alio-benefactors of others 



Euclidean 



It is a little striking that j ust as in the 



ian or imaginary geometry of Lobatchewsky the axiom concerning parallels 

 holds good only with the ultimate elements of space, so its logical equivalent holds 



good 



for elementary relatives 



