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THE SCIENCE OF DYNAMICS 61 

at the end of t units of time is c + gt units, for if the 
direction downwards (of g) is reckoned positive, the direction 
upwards (of c) must be reckoned negative. If \ve project a 
body horizontally with the velocity of a units, and neglect the 
resistance of the air, Galileo recognised that it would describe, 
in the horizontal direction, a distance of at units in t units of 

at 2 

time, while simultaneously it would fall a distance of ? 

units. The two motions are to be considered as going on in- 
dependently of each other. Thus also, oblique projection may 
be considered as compounded of a horizontal and a vertical 
projection. In all these cases the path of the projectile is a 
parabola ; in the case of the horizontal projection, its equation 
in x and y co-ordinates is got from the two equations x at and 

at z 
y = ^-, and is thus 

2 v== 0^ 

y 2a 2 

Now, suppose that the velocity is neither uniform nor 
increases uniformly, but is different and increases at a different 
rate at different points of time. Then in the curve of 
velocities, the line OB is no longer straight. In the former 
case, the number s was the number of square units in the area 
of the triangle AOB. In this case the figure AOB is not a 
triangle, though we shall find that its area is the s units we 
seek, although v does not increase uniformity from O to A. 

Notice again that if, on OA, we take points C and E very 
close together, the little arc DF is very nearly straight, and 
the figure DGF very nearly a rectilinear triangle. Note that 
we are only trying, in this, to get a first approximation to the 
value of s, and so that, instead of the continuously changing 
velocities we know or think we know from our daily ex- 
perience, we are considering a fictitious motion in which the 
velocity increases (or decreases) so as to be the same as that of 
the motion thought of at a large number of points at minute 
and equal distances, and between successive points increases 
(or decreases) uniformly. 

Note also that we are assuming (what usually happens with 
the curves with which we shall have to do) that the arc DF 
which corresponds to CE becomes as straight as we wish if we 
take C and E close enough together.