248
III - Convergence: Continuous variables
v, y, z, [ ... J, will be expressed by vo, yo, io, [ ... J seeing that vo, xo, yo and
io are to one another as V, x, y and i.
Now, since the moments (say, xo and yo) of fluent quantities (x and y,
say) are the infinitely small additions by which those quantities increase
during each infinitely small interval of time, it follows that those quantities
x and y after any infinitely small interval of time will become x + xo and
y + yo.
And Newton explains that, if one has a relation between x and y, one can
substitute x + xo and y + yo and calculate algebraically. In the relation
for example, one obtains
(x 3 + 3xox 2 + 3X 2 0 2 x + x 3 0 3 ) - (ax 2 + 2axox + ax 2 0 2 ) +
+ (axy + axoy + ayox + axy02) - (y3 + 3yoy2 + 3y 2 02y + y303) = O.
But by hypothesis x 3 -ax 2 +axy- y3 = 0, and when these terms are cancelled
and the rest is divided by 0, one obtains a relation that the reader can easily
write, by calculating as crudely as can be, as did Newton himself. There will
be terms not containing 0 and terms which contain 0:
But further, since 0 is supposed to be infinitely small so that it be able to
express the moments of quantities, terms which have it as a factor will be
equivalent to nothing in respect of the others. I therefore cast them out
and there remains
3xx 2 - 2axx + axy + ayx - 3yy2 = o.
Thus Newton, starting from a relation between his fluents x and y, functions of time, obtained a relation between x, y and their derivatives x and
y with respect to time, that he called their fluxions, i.e., as he explained
earlier, their "speeds of flow in time". What Newton denoted by 0 and xo
would be written as dt and x'(t)dt by Leibniz [up to the fact that Leibniz
used the notation dx and not the modern notation x'(t)dt] and gave rise to
an identical calculus, up to notation. For a long time, maybe wrongly, these
heuristic but very suggestive calculations have been replaced by rules that
allow one to calculate the derivatives of any algebraic combination of functions, but the current method is not necessarily quicker from the moment (in
the naive sense, and not in the sense that Newton meant in the preceding
citation!) when one has understood these ideas of Newton's, or, equivalently,
those of Leibniz. All this is theoretically incorrect since (i) infinitely small
quantities "are never met in Nature", as Newton said much later to annoy
Leibniz, (ii) the increment x(t + h) - x(t) corresponding to a "small", but
not infinitely small, increment h of t is not rigorously proportional to h. But
nevertheless this all works perfectly so long as one takes a few precautions.
III - Convergence: Continuous variables
v, y, z, [ ... J, will be expressed by vo, yo, io, [ ... J seeing that vo, xo, yo and
io are to one another as V, x, y and i.
Now, since the moments (say, xo and yo) of fluent quantities (x and y,
say) are the infinitely small additions by which those quantities increase
during each infinitely small interval of time, it follows that those quantities
x and y after any infinitely small interval of time will become x + xo and
y + yo.
And Newton explains that, if one has a relation between x and y, one can
substitute x + xo and y + yo and calculate algebraically. In the relation
for example, one obtains
(x 3 + 3xox 2 + 3X 2 0 2 x + x 3 0 3 ) - (ax 2 + 2axox + ax 2 0 2 ) +
+ (axy + axoy + ayox + axy02) - (y3 + 3yoy2 + 3y 2 02y + y303) = O.
But by hypothesis x 3 -ax 2 +axy- y3 = 0, and when these terms are cancelled
and the rest is divided by 0, one obtains a relation that the reader can easily
write, by calculating as crudely as can be, as did Newton himself. There will
be terms not containing 0 and terms which contain 0:
But further, since 0 is supposed to be infinitely small so that it be able to
express the moments of quantities, terms which have it as a factor will be
equivalent to nothing in respect of the others. I therefore cast them out
and there remains
3xx 2 - 2axx + axy + ayx - 3yy2 = o.
Thus Newton, starting from a relation between his fluents x and y, functions of time, obtained a relation between x, y and their derivatives x and
y with respect to time, that he called their fluxions, i.e., as he explained
earlier, their "speeds of flow in time". What Newton denoted by 0 and xo
would be written as dt and x'(t)dt by Leibniz [up to the fact that Leibniz
used the notation dx and not the modern notation x'(t)dt] and gave rise to
an identical calculus, up to notation. For a long time, maybe wrongly, these
heuristic but very suggestive calculations have been replaced by rules that
allow one to calculate the derivatives of any algebraic combination of functions, but the current method is not necessarily quicker from the moment (in
the naive sense, and not in the sense that Newton meant in the preceding
citation!) when one has understood these ideas of Newton's, or, equivalently,
those of Leibniz. All this is theoretically incorrect since (i) infinitely small
quantities "are never met in Nature", as Newton said much later to annoy
Leibniz, (ii) the increment x(t + h) - x(t) corresponding to a "small", but
not infinitely small, increment h of t is not rigorously proportional to h. But
nevertheless this all works perfectly so long as one takes a few precautions.