Newton's Letters to Leibniz

Epistola Prior (1676)

"Though the modesty of Mr Leibniz, in the extracts from his letter which you have lately sent me, pays great tribute to our countrymen for a certain theory of infinite series, about which there now begins to be some talk, yet I have no doubt that he has discovered not only a method for reducing any quantities whatever to such series, as he asserts, but also various shortened forms, perhaps like our own, if not even better. Since, however, he very much wants to know what has been discovered in this subject by the English, and since I myself fell upon this theory some years ago, I have sent you some of those things which occurred to me in order to satisfy his wishes, at any rate in

Fractions are reduced to infinite series by division; and radical quantities by extraction of the roots, by carrying out those operations in the symbols just as they are commonly carried out in decimal numbers. These are the foundations of these reductions: but extractions of roots are much shortened by this theorem,

where P + PQ signifies the quantity whose root or even any power, or the root of a power, is to be found; P signifies the first term of that quantity, Q the remaining terms divided by the first, and m/n the numerical index of the power of P + PQ, whether that power is integral or (so to speak) fractional, whether positive or negative. For as analysts, instead of aa, aaa, etc., are accustomed to write a2, a3, etc., so instead of
I write aab(a3 + bbx)-3: in which last case, if
is supposed to be (P + PQ)"In in the Rule, then P will be equal to a3, Q to bbx/a3, m to -2, and n to 3. Finally, for the terms found in the quotient in the course of the working I employ A, B, C, D, etc., namely, A for the first term, Pml"; B for the second term, m/n AQ; and so on. For the rest, the use of the rule will appear from the examples.
Example 1

Example 2
as will be evident on substituting 1 for m, 5 for n, c5f or P and (c4x - x ) ~ /fco~r Q , in the rule quoted above. Also -x5 can be substituted for P and (c4x + c5)/(-x5) for Q. The result will then be
The first method is to be chosen if x is very small, the second if it is very large."

Epistola Posterior (1676)

Newton's second letter to Leibniz reveals the former's derision and outrage at Leibniz's appropriation of his work. The letter goes on to demonstrate Newton's mastery of the topics at hand, series, and hints at the problem of tangents, or differentiation.

In the opening of the letter, Newton condescendingly applauds Leibniz's "discovery" of a "method for obtaining convergent series" and then notes that he himself had understood and explored several such methods, the most famous of which is still referred to as Newton's Method for finding the roots of a function:

"I can hardly tell with what pleasure I have read the letters of those very distinguished men Leibniz and Tschirnhaus. Leibniz's method for obtaining convergent series is certainly very elegant, and it would have sufficiently revealed the genius of its author, even if he had written nothing else. But what he has scattered elsewhere throughout his letter is most worthy of his reputation-it leads us also to hope for very great things from him. The variety of ways by which the same goal is approachedshas given me the greater pleasure, because three methods of arriving at series of that kind had already become known to me, so that I could scarcely expect a new one to be communicated to us. One of mine I have described before; I now add another, namely, that by which I first.chanced on these series-for I chanced on them before I knew the divisions and extractions of roots which I now use. And an explanation of this will serve to lay bare, what Leibniz desires from me, the basis of the theorem set forth near the beginning of the former letter."


He notes that powers of 11 can be used to find the coefficients, because 11^n
$$6$e = lim_{2$x\rightarrow \inf}(1 + \frac{1}{x})^x

10^n + ... + 1

Using the factor of 11 he can figure out not only the coefficients of a binomial with an integer power, but also that for a fractional power. Whereas an integer power yields an alternating series which ends nicely, a fractional power just keeps going on forever. The way he did this is show below, using m=1/2 as an example.

Since the factorial of a fraction is not defined (as far as we know and Newton knew), Newton decided to approach the problem just using fractions to interpolate. With this he can find the area under the circle (1-x^2)^(1/2), which was his point in doing this in the first place. Essentially he's finding the integral and then using a basic derivative method to find the actual function.

Statement of the area of the "circular segment"

Again stating the series, that he will try to interpolate.

"But in that treatise infinite series played no great part. Not a few other things I
brought together, among them the method of drawing tangents which the very skilful
Sluse communicated to you two or three years ago, about which you wrote back [to
him] (on the suggestion of Collins) that the same method had been known to me also.
We happened on it by different reasoning: for, as I work it, the matter needs no proof.
Nobody, if he possessed my basis, could draw tangents any other way, unless he were
deliberately wandering from the straight path. Indeed we do not here stick at equations
in radicals involving one or each indefinite quantity, however complicated they may
be; but without any reduction of such equations (which would generally render the
work endless) the tangent is drawn directly. And the same is true in questions of
maxima and minima, and in some others too, of which I am not now speaking. The
foundation of these operations is evident enough, in fact; but because I cannot proceed
with the explanation of it now, I have preferred to conceal it thus:
6accdm13effli319n404qrr4~8t 12vx. On this foundation I have also tried to simplify the
theories which concern the squaring of curves, and I have arrived at certain general