If this way of interpreting the second law seems perverse, keep in mind that the geometric mathematics Newton used in the Principia — and others were using before him — had no way of representing acceleration as a quantity in its own right. Newton, of course, could have conceptualized acceleration as the second derivative of distance with respect to time within the framework of the symbolic calculus. This indeed is the form in which Jacob Hermann presented the second law in his Phoronomia of 1716 (and Euler in the 1740s). But the geometric mathematics used in the Principia offered no way of representing second derivatives. (Newton employed curvature — that is, the circle “touching a curve” — in place of the second derivative with respect to distance throughout the Principia). Hence, it was natural for Newton to stay with the established tradition of using a length as the measure of the change of motion produced by a force, even independently of the advantage this measure had of allowing the law to cover both discrete and continuously acting forces (with the given time taken in the limit in the continuous case). - plato.stanford.edu