230 Royal Society :— 
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 /aws 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 ofa 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 
the goniometrical cosines and sines of an angle, with which they are 
ultimately shown to have a close connexion, which can be rendered 
most evident by assuming as the unit-angle that subtended by a cir- 
cular are of the length of its radius. Studying these series quite in- 
dependently of these relations to angles, we discover that they bear to 
each other the same relations as the goniometrical cosines and sines, 
and that ifthe least tensor value of the variable for which the cosine 
series becomes null, is known, all its other values can be found by 
multiplying this by four times any scalar integer. This last product 
must be added to the least tensor value of the variable for which 
both the cosine or the sine series become equal to given scalars, in 
order to find all the solutions of such equations. Supposing the 
values of such series tabulated by the method of differences for all 
scalar values of the variable, so that such least tensor values can 
always be found, we are now able to assign the meaning of any 
power whose base and index are both clinants, and the logarithm of 
any clinant. This enables us to invert completely all the simple for- 
mations, sum, product, power with variable base and constant index, 
or constant base and variable index ; and hence to solve all equations 
