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60 THE NATURE OF MATHEMATICS We have met this difficulty when considering the method of indivisibles, and will meet it again when considering the in- finitesimal calculus, and will only see how it is overcome when we have become familiar with the conception of a " limit." This new notion of velocity includes that of uniform velocity as a particular case. In fact, the rules of the infinitesimal calculus allow us to conclude, from the equation =a, where dt a is some constant, the equation 8 = at + b, where b is another constant. We must remember that all this was not expressly formulated until about fifty years after Galileo had published his investigations on the motion of falling. If we consider the curve of velocities, uniformly accelerated motion occupies in it exactly the same place as uniform velocity does in the curve of spaces. If we denote by v the numerical measure of the velocity at the end of t units of time, the acceleration, in the notation of the differential calculus, is measured by , and the equation = ft, where h is some dt dt constant, is the equation of uniformly accelerated motion. In Newtonian dynamics, we have to consider variably accelerated motions, and this is where the infinitesimal calculus or some practically equivalent calculus such as Newton's " method of fluxions " becomes so necessary in theoretical mechanics. We will now consider the curve of spaces for uniformly accelerated motion. On this diagram the arcs being t and s we will draw the curve where g denotes a constant. Of course, this is the same thing CJOC as drawing the curve y = y in a plane divided up by the m a;-axis and the y-axis of Descartes. This curve is a parabola passing through the origin. An interesting thing about this curve is that it is the curve that would be described by a body projected obliquely near the surface of the earth if the air did not resist, and is very nearly the path of such a projectile in the resisting atmosphere. A free body, according to Galileo's view, always falls towards the earth with a uniform vertical accelera- tion measured by the above number g. If we project a body vertically upwards with the initial velocity of c units, its velocity
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