- NEWTON AND LEIBNIZ 469
Newton's first version of the calculus. Newton first developed the calculus in what
we would call parametric form. Time was the universal independent variable, and
the relative rates of change of other variables were computed as the ratios of their
absolute rates of change with respect to time. Newton thought of variables as
moving quantities and focused attention on their velocities. He used the letter
ï to represent a small time interval and ñ for the velocity of the variable x, so
that the change in ÷ over the time interval ï was op. Similarly, using q for the
velocity of y, if y and ÷ are related by yn = xm, then (y + oq)n = (x + op)m. Both
sides can be expanded by the binomial theorem. Then if the equal terms yn and
xm are subtracted, all the remaining terms are divisible by o. When ï is divided
out, one side is nqyn~l + ï A and the other is mpxm~l + oB. Ignoring the terms
containing o, since ï is small, one finds that the relative rate of change of the two
variables, q/p is given by q/p = (mxm~l) / (nyn~~l); and since y = xm^n, it follows
that q/p = {m/n)x^mln)-^1. Here at last was the concept of a derivative, expressed
as a relative rate of change.
Newton recognized that reversing the process of finding the relative rate of
change provides a solution of the area problem. He was able to find the area under
the curve y = axm^n by working backward.
Fluxions and fluents. Newton's "second draft" of the calculus was the concept of
fluents and fluxions. A fluent is a moving or flowing quantity; its fluxion is its rate
of flow, which we now call its velocity or derivative. In his Fluxions, written in
Latin in 1671 and published in 1742 (an English translation appeared in 1736), he
replaced the notation ñ for velocity by x, a notation still used in mechanics and in
the calculus of variations. Newton's notation for the opposite operation, finding a
fluent from the fluxion has been abandoned: Where we write j x(t) dt, he wrote x.
The first problem in the Fluxions is: The relation of the flowing quantities to
one another being given, to determine the relation of their fluxions. The rule given
for solving this problem is to arrange the equation that expresses the given relation
(assumed algebraic) in powers of one of the variables, say x, multiply its terms by
any arithmetic progression (that is, the first power is multiplied by c, the square by
2c, the cube by 3c, etc.), and then multiply by x/x. After this operation has been
performed for each of the variables, the sum of all the resulting terms is set equal
to zero.
Newton illustrated this operation with the relation x^3 — ax^1 4- axy — y^2 = 0,
for which the corresponding fluxion relation is 3x^2 x — 2axx + axy + axy — 2yy =
0, and by numerous examples of finding tangents to well-known curves such as
the spiral and the cycloid. Newton also found their curvatures and areas. The
combination of these techniques with infinite series was important, since fluents
often could not be found in finite terms. For example, Newton found that the
area under the curve æ = 1/(1 + ÷^2 ) was given by the Jyeshtadeva Gregory series
Later exposition of the calculus. Newton made an attempt to explain fluxions in
terms that would be more acceptable logically, calling it the "method of first and
last ratios," in his treatise on mechanics, the Philosophiae naturalis principia math-
ematica (Mathematical Principles of Natural Philosophy), where he said,
Quantities, and the ratios of quantities, which in any finite time
converge continually toward equality, and before the end of that