properties of addition hold good in like manner for quater- 

 nions : results which were indeed stated to the Academy in 

 November, 1843, as consequences from the algebraical defini- 

 tions of a quaternion, and of operations performed thereon, 

 but have now been mentioned again, as following from more 

 geometrical definitions also. 



Comparing the view here proposed with that which was 

 submitted to the Academy in November 1844, a quaternion 

 may be said to reduce itself to a scalar (or ordinary real num- 

 ber), when the two straight lines, of which it is the quotient, are 

 parallel; the scalar being positive when those lines are simi- 

 lar, but negative when they are opposite in direction. And, on 

 the other hand, the scalar part vanishes, and the quaternion 

 becomes a pure vector, when it is a quotient of two rectangu- 

 lar lines: and, in this last case, it may be conveniently con- 

 structed by a third line perpendicular to both of them, namely, 

 by one drawn in the direction of the positive axis of the quater- 

 nion, with a length which bears to an assumed unit of length 

 the ratio marked by the modulus. This third line, which thus 

 represents or constructs the quotient of two other lines per- 

 pendicular to it and to each other, may, by a suitable choice of 

 those two lines, receive any proposed length, and any pro- 

 posed direction ; and every straight line having length and 

 direction in space may, in this view, be regarded as a particu- 

 lar quaternion, namely, as one of the class above called vec- 

 tors. It is easy to prove that when lines are thus treated as 

 quotients, they have the same sums, differences, and quotients, 

 as those obtained by the processes or conceptions above de- 

 scribed or alluded to : and hence it would be natural to define^ 

 as we should be at liberty to do, that the product of two 

 lines is also in general a quaternion, obtained by multiplying 

 two vector factors together, according to the rules of multipli- 

 cation of quaternions. We should then be able to establish, 

 in this new way, all the rules, already communicated to the 

 Academy, for the multiplication of straight lines in space; 



