Translated and Annotated by Ian Bruce. BOOK I Section 1
ref: https://www.17centurymaths.com/contents/newtoncontents.html
The mathematical foundations of the work are set out here in a series of Scholia, which are of considerable interest, as they are given in terms of what Newton calls the first and last ratios of sums and ratios, being a geometric approach to the limiting processes involved in integration and differentiation.
Alla fine della Sectio I, we are made aware by Newton of
apart from the ponderous reducti ad absurdum type geometrical methods of the ancient Greeks, e.g. the approach of Archimedes to solving certain problems, and as used by Huygens in his Horologium in deriving the isochronous property of his cycloidal pendulum:
which appears to be none other than extracting a limit from first principles by seeking closer and closer upper and lower bounds indefinitely; and
which is akin to the modern calculus.
At present, for the sake of
economy and to avoid further controversy it would seem regarding vanishing
quantities, only the first of the two has been used and
shown in some detail.
Newton goes to great pains to construct a geometrical method which embodies the
ideas both of integration and differentiation, but which avoids directly the
forming of integrals and derivatives as we know them now; instead, the idea of a
limit is set out initially, which incidentally demolishes Zeno' s Paradox in a
sentence, and which in
words is more or less the present definition of such, the difference between the
limiting value and nearby values can be made as small as it pleases without end,
without actually being made equal. In the method of first and last sums and
ratios, a geometrical method is established for carrying out this process, and
so for treating integration and differentiation.
At first we look at slightly greater and slightly smaller rectangles enclosing a
curve, which approach each other and the segment of the area under the curve
closer and closer as their number is increased and the bases diminished, which
on the whole seems highly credible. Thus a formulation of integration is
obtained. In the second a parallelogram is presented in which a small arc of the
curve lies near the diagonal, crossing at the ends,
and one side of the parallelogram is a tangent at one vertex; as the
parallelogram is diminished in size, the diagonal, the curve, and the tangent
finally merge together closer and closer; on elaboration, we are to follow
points on lines in diminishing similar triangles approaching an ultimate point
at their intersection, where the two similar triangles vanish, but their finite
ratios of corresponding sides is maintained by two other trustworthy similar
triangles which remain finite all along, this method also seems intuitively ok,
as we are told that the final ratio is to be evaluated from the finite similar
triangles, at least in the case of the curve being a circle, which follows
readily from elementary geometry, while of course Newton rightly asserts that
the dreaded zero on
zero is never used.
Thus the objections of the doubters were allayed, or at
least they had less to argue about. This approach is and was not enough for the
pure mathematicans at the time, at least those on the continent; we must
remember that Newton was (in my view) essentially a physicist doing mathematics,
at which he was extraordinarily adept, as well as an experimentalist cum
alchemist cum theologian, rather than a pure mathematician. The method considers
points moving along lines in time, which we now would consider as a mere
parameter, and we are asked to retrace these positions into the past to arrive
at the starting point, or very close to it. Thus, Newton's calculus in the
Principia is about the rates of change of quantities with time.