Review of the Principia of Newton. 33 
imvestigation of its parts, and demonstrates that these parts, 
when resolved, are identical with those’ a and conse- 
quently that the assumption or proposition is true 
It is demonstrated in this 39th pepadces , that if the space 
passed over and the force be assumed as co-ordinates, the 
eurve, which is the locus of these, will include an area, which 
will be as the square of the velocities in different places of the 
ne body. If the reciprocal of the velocities be made 
ordinates of another curve, corresponding to every place 
of the descending pil this curve will comprehend an area 
propor see to the tim 
I relation of pie ee velocity be given, or if the 
velacay be as any function of the time, and the con- 
sideration of the force generating the acceleration be ex- 
_ cluded, the relations of space, sine; and velocity, may be ex- 
pressed by different constructions, some of which, and those 
most common, are not truly mathematical. In these the time 
is — by an abscissa of a curve, the velocity by a co- 
ordinate, which is a given function of the time, or abscissa, 
pes the area included by the co-ordinates and curve, the lo- 
cus of the co-ordinates, will be the space passed over by the 
descending body. This mode of showing the relations of 
time, velocity and distance passed over, is, as before obsery- 
ed, unscientific, for the quantities compared are mathemati- 
cally incapable of a comparison, except by numbers whereof 
the units are nets ren eneons being the expressions of lines in 
one case, and areas or surfaces in another. But these rela- 
tions may be 5 Re by a locus, whereof all the peas 
may be homogeneous, or rights dass only. For this purpose 
we need only to considcr one oe pt Beceanboaes 
to represent the time uniform re pecan g commencing at 
the origin of the co-ordinates, or with any initial space, and 
the space passed over to be represented by a corresponding 
portion of the other axis, and the velocity will then be truly 
defined at any point by the trigonometrical tangent, which 
the tangent to the curve at that point makes with the axis axis rep- 
resenting the time ; for V the velocity is always as= The 
‘locus of a body acted on by a constant ae such as gravity, 
will be a parabola in this scheme. The first construction, 
however, has generally been adopted res "the elementary stu- 
dent, and with proper explanations, referring, as Newton al- 
ways does, to the lines which are the sides or boundaries of 
