Mr, A. J. Ellis on the Systematization of Mathematics. 229 
intermediate values of the variable ; that is, the algebra of differences. 
Considering the two operations of altering a formation by increasing 
the variable, and taking the difference between two different values 
of the formation (of which operations the first is necessarily unity 
added to the second), we regard them as formators, and immediately 
apply the results of that algebra, which furnishes all the necessary 
formule. For approximating to the roots of equations, we require 
to consider the case where the variable changes infinitesimally, thus 
founding the algebra of differentials, which is, in fact, a mere sim- 
plification of that of differences, owing to all the results being ulti- 
mately calculated for assignables only. Finally, to find the alteration 
in a formation of commutative formators, when the variable formator 
is increased by any other formator, we found the algebra of deri- 
vatives. 
In applying the results of scalar algebra to geometry, we-start with 
the fundamental propositions that the appense of the sides of an en- 
closed figure taken in order is a point, and that when the magnitude 
and direction of the diagonal of a parallelogram or parallelopipedon, 
and lines parallel the sides which have the same initial point as the 
diagonal, are given, the whole figures are completely determined. In 
order to introduce scalars, a unit-sphere is imagined, with its radii 
parallel to the lines in any figure, and in known directions. Any line 
ean then be represented as the result of performing a scalar operation 
on the corresponding radius. 
The first object is to reduce the consideration of angles to that of 
straight lines, by the introduction of cosines and sines, which are 
strictly defined as the scalars represented by the relation of the 
abscissa to the abscissal radius, and the ordinate to the ordinate 
radius respectively. These definitions immediately lead to the rela- 
tions between the cosines and sines of the sums of two angles, and 
those of the angles themselves, whatever be their magnitude or direc- 
tion, and thus found goniometry. 
Defining a projection of any figure on any plane to be that formed 
by joining the points on that plane corresponding according to any 
law with those of the figure, we have the fundamental relation that, 
if the first, and therefore the second figure is enclosed, the appense of 
the sides of the second in the order indicated by the sides of the first, 
is a point. The orthogonal projection of any figure, by means of 
planes drawn perpendicular to any line, being all in one line, each 
projection can be represented as the result of a scalar operation per- 
formed on the same unit radius, and hence this projection leads to one 
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 ¢ransversals, where a line parallel to one 
unit radius cuts several other unit radii, produced either way if neces- 
sary, we obtain, by considering ¢wo intersected radii, the results of 
Phil. Mag. 8. 4. Vol. 19, No, 126. Mar. 1860. R 
