Newtons Principia First book Sections I Ii Iii With Notes And Illustrations

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46.
Join AC, and let pM) qN be the ordinates, and suppose the figure to revolve round AB, pM generates a circle whose area cc p M* cc A M* therefore this circle may be taken to represent the velocity at the time corresponding to AM, and the solid generated by pqNM represents the space described in time MN. The whole space is therefore represented by the cone generated by ABC) or ^AB. TrBC 2 ) which gives the same result as before.
(2) To find the space described from rest at any time by a parti
...cle under the action of a force ichose accelerating effect varies as the m^ power of the time.
This problem is more simply solved by applying directly the method of summation, since in order to find the area of the curve, constructed as in Lemma X. , we should eventually be obliged to have recourse to that method.
Let the time t be divided into n equal intervals, and let the acceleration by the force at the time t be pt m j hence, at the com- (rt\ m J, and, if the force be continued uniform during this interval, the velocity generated will be p ( j .


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