MM. ID- IIS -I HI IlCES. .'337 



change it> station while it keeps it- aspect, or it may alter its 

 aspect while it maintains its station, these two constituents of 

 position being entirely independent of each other. It is evi- 

 dent, moreover, that the station of a body depends upon three 

 arbitrary variables, the three coordinates of (> : whereas it> 

 aspect is a function of the nine angles which the three body- 

 axes make with the three axes in space. As the angles which 

 a straight line makes with axes to which it is referred, are 

 elements of very frequent use in geometrical and mechanical 

 speculations, I shall take the liberty, for the purpose of avoid- 

 ing tedious repetitions, to call them the axe-angles of the line. 

 distinguishing also between the space-axe angles and the body- 

 axi angles; thus u. «', a", /'. I>. I> . c, c. c (winch is the usual 

 notation) will denote the cosines of the space-axe angles of the 

 body-axes. Between these nine cosines there exist six equa- 

 tions of condition, so that, in ultimate analysis, the aspect of 

 a body will, as well as its station, depend upon the values of 

 three independent variables. The choice of these becomes 

 therefore a matter of importance. Euler. who must be re- 

 ded as the inventor of this interesting branch of analysis. 

 showed as early as the year 1771. in a paper published in (lie 

 fifteenth volume of the Now, Commentarii of the Academy of 

 St Petersburg, under the title of Problema algebraicum oh 

 affectiones prorsus singulares memorabile, how these nine quan- 

 tities might be expressed in terms of three independent angles, 

 namely, the inclination of one of the moveable to one of the 

 fixed planes, and the distances from their intersection to an 

 axis in i ach plane. The author begins l>\ considering the 

 question analytically; and this view of it gives rise to a prob- 

 lem altogether similar, with respect to the determination of 

 sixteen quantities connected l>\ ten analogous conditions, from 

 which he proceeds, with hi- characteristic habit of gradual 

 _ aeralization, to extend bis analysis to twenty-live quantities 

 with fifteen connecting relations, and so on. It is only the 

 first case ot' the problem that can have any application to geo- 

 metry, but the whole paper is deserving of attention as fur- 

 nishing one of the earliest specimens of the improved methods 

 of modern analysis. Of all the solutions of the first case ot 



