epistolaSam

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### Monday, June 9

1. home edited ... theories which concern the squaring of curves, and I have arrived at certain general Theorems…
...
theories which concern the squaring of curves, and I have arrived at certain general
Theorems."

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### Thursday, June 5

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### Monday, April 28

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### Sunday, April 20

1. tablepractice (deleted) edited
11:46 am
2. home edited Newton's Letters to Leibniz Epistola Prior (1676) "Though the modesty of Mr Leibniz, in…

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
part.
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:
...
those very
distinguished
distinguished men Leibniz
...
for obtaining
convergent
convergent series is
...
revealed the
genius
genius of its
...
has scattered
elsewhere
elsewhere throughout his
...
also to
hope
hope for very
...
goal is
approachedshas
approachedshas given me
...
arriving at
series
series of that
...
expect a
new
new one to
...
another,
another, namely, that
...
on them
before
before I knew
...
And an
explanation
explanation of this
...
of the
theorem
theorem set forth
{Picture_1.png}
He notes that powers of 11 can be used to find the coefficients, because 11^n
(10$$6e = lim_{2x\rightarrow \inf}(1 + 1)^n\frac{1}{x})^x 10^n + ... + 1 {Picture_2.png} (view changes) 11:46 am 3. home edited Newton's Letters to Leibniz \mathop {\lim }\limits_{x \to \infty } \sqrt {b^2 - 4ac}  <… Newton's Letters to Leibniz \mathop {\lim }\limits_{x \to \infty } \sqrt {b^2 - 4ac}  $<semantics> <mrow> <munder> <mrow> <mi>lim</mi><mo>&#x2061;</mo> </mrow> <mrow> <mi>x</mi><mo>&#x2192;</mo><mi>&#x221E;</mi> </mrow> </munder> <msqrt> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>&#x2212;</mo><mn>4</mn><mi>a</mi><mi>c</mi> </mrow> </msqrt> </mrow> <annotation encoding='MathType-MTEF'> </annotation> </semantics>$ 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 part. 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: ... those very distinguished distinguished men Leibniz ... for obtaining convergent convergent series is ... revealed the genius genius of its ... has scattered elsewhere elsewhere throughout his ... also to hope hope for very ... goal is approachedshas approachedshas given me ... arriving at series series of that ... expect a new new one to ... now add another, another, namely, that ... on them before before I knew ... And an explanation explanation of this ... of the theorem theorem set forth {Picture_1.png} He notes that powers of 11 can be used to find the coefficients, because 11^n$$6$e = lim_{2$x\rightarrow \inf}(1(10 + \frac{1}{x})^x1)^n
10^n + ... + 1
{Picture_2.png}
(view changes)
11:44 am

### Thursday, January 3

1. tablepractice (deleted) edited
7:45 am