Epistola


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Monday, June 9

  1. page 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. page tablepractice (deleted) edited
    11:46 am
  2. page 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
    ...
    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
    (10$$6$e = lim_{2$x\rightarrow \inf}(1 + 1)^n\frac{1}{x})^x
    10^n + ... + 1
    {Picture_2.png}
    (view changes)
    11:46 am
  3. page 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} $
    <math>
    <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>
    </math>
    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. page tablepractice (deleted) edited
    7:45 am

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