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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 

 formulae. 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 comnratative*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 parallelopipedori, 

 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 

 can 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 



