invariable relation between scalars. By choosing three lines at right 

 angles to each other on which to project, we obtain three scalar re- 

 lations from every solid figure. If the figure is plane, then by pro- 

 jecting on a line and on a perpendicular to that line, we get two 

 scalar relations. 



Applying these results to transversals^ where a line parallel to one 

 unit radius cuts several other unit radii, produced either way if neces- 

 sary, we obtain, by considering two intersected radii, the results of 

 trigonometry, and by considering three or four intersected radii, 

 those of anharmonic ratios. 



As any line drawn from the centre of the unit-sphere may be con- 

 sidered as the appense of three lines drawn along or parallel to three 

 given unit radii, it may be expressed as the sum of the results of 

 three scalar operations performed on these radii respectively. By 

 properly varying these three scalars, the final point of the line may 

 be made to coincide with any point in space. But if there be a given 

 relation between the scalars, then the number of points will be 

 limited, and the whole number of the points constitutes the locus of 

 the original concrete equation referred to the accessory abstract equa- 

 tion. The consideration of this entirely new view of coordinate geo- 

 metry is reserved for a second memoir. 



Proceeding next to the laws of clinants, we readily demonstrate 

 that they are the same as the laws of scalars ; they introduce a new 

 conception, however, that of rotating through an angle not .necessarily 

 the same as a semi-revolution, that is, of a plane versor. By the con- 

 crete equation of coordinate geometry, it is immediately shown that 

 all clinants can be expressed as the sum of a scalar, and of the pro- 

 duct of a scalar by a fixed, but arbitrarily chosen versor. The 

 simplest versor to select is the quadrantal versor, which, under the 

 name of quadrantation, is now studied. The two addends of a clinant, 

 considered as a sum, are called its scalar and vector ; its two factors, 

 considered as a product, are its tensor and versor. The laws of these 

 parts are then studied. 



The statical algebra of clinants has for its object the reduction of 

 all combinations of clinants given in the standard form of the sum of 

 a scalar and vector, to a clinant of the same form. The application 

 of this to the series obtained for a general scalar power, leads to two 

 series, called cosines and sines of the variables, as distinguished from 



