Page:Somerville Mechanism of the heavens.djvu/96

ma; hence at the end of that time the particle would be found in B, having described the diagonal mB. Were the particle now left to itself, it would move uniformly to C in the next equal interval of time; but the action of the second impulse of the attractive force would bring it equably to b in the same time. Thus at the end of the second interval it would be found in D, having described the diagonal BD, and so on. In this manner the particle would describe the polygon mBDE; but if the intervals between the successive impulses of the attractive force be indefinitely small, the diagonals mB, BD, DE, &c., will also be indefinitely small, and will coincide with the curve passing through the points m, B, D, E, &c.

64. In this hypothesis, no error can arise from assuming that the particle describes the sides of a pulygon with a uniform motion; for the pulygon, when the number of its sides is indefinitely multiplied, coincides entirely with the curve.

65. The lines mA, BC, &c., fig. 17, are tangents to the curve in the points, m, B, &c.; it therefore follows that when a particle is moving in a curved line in consequence of any continued force, if the force should cease to act at any instant, the particle would move on in the tangent with an equable motion, and with a velocity equal to what it had acquired when the force ceased to act.

66. The spaces ma, Bb, CD, fig. 18, &c., are the sagittæ of the indefinitely small arcs mB, BD, DE, &c. Hence the effect of the central force is measured by ma', the sagitta of the arc mB described in an indefinitely small given time, or by ${\frac {({\text{arc}}\,mB)^{2}}{2\,.\,om}}=ma,$ $om$ being the radius of the circle coinciding with the curve in $m$.

67. We shall consider the element or differential of time to be a constant quantity; the element of space to be the indefinitely small