Abraham de Moivre (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Abraham de Moivre" in English language version.

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Euler (1749). "Recherches sur les racines imaginaires des equations" [Investigations into the complex roots of equations]. Mémoires de l'académie des sciences de Berlin (in French). 5: 222–288. See pp. 260–261: "Theorem XIII. §. 70. De quelque puissance qu'on extraye la racine, ou d'une quantité réelle, ou d'une imaginaire de la forme M + N √-1, les racines seront toujours, ou réelles, ou imaginaires de la même forme M + N √-1." (Theorem XIII. §. 70. For any power, either a real quantity or a complex [one] of the form M + N √−1, from which one extracts the root, the roots will always be either real or complex of the same form M + N√−1.)
Euler (1749). "Recherches sur les racines imaginaires des equations" [Investigations into the complex roots of equations]. Mémoires de l'académie des sciences de Berlin (in French). 5: 222–288. See pp. 260–261: "Theorem XIII. §. 70. De quelque puissance qu'on extraye la racine, ou d'une quantité réelle, ou d'une imaginaire de la forme M + N √-1, les racines seront toujours, ou réelles, ou imaginaires de la même forme M + N √-1." (Theorem XIII. §. 70. For any power, either a real quantity or a complex [one] of the form M + N √−1, from which one extracts the root, the roots will always be either real or complex of the same form M + N√−1.)
Euler (1749). "Recherches sur les racines imaginaires des equations" [Investigations into the complex roots of equations]. Mémoires de l'académie des sciences de Berlin (in French). 5: 222–288. See pp. 260–261: "Theorem XIII. §. 70. De quelque puissance qu'on extraye la racine, ou d'une quantité réelle, ou d'une imaginaire de la forme M + N √-1, les racines seront toujours, ou réelles, ou imaginaires de la même forme M + N √-1." (Theorem XIII. §. 70. For any power, either a real quantity or a complex [one] of the form M + N √−1, from which one extracts the root, the roots will always be either real or complex of the same form M + N√−1.)
See:
- Stirling, James (1730). Methodus Differentialis … (in Latin). London: G. Strahan. p. 137. From p. 137: "Ceterum si velis summam quotcunque Logarithmorum numerorum naturalam 1, 2, 3, 4, 5, &c. pone z–n esse ultimum numerorum, existente n = ½ ; & tres vel quatuor Termini hujus Seriei $zl,z-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ [Note: l,z = log(z)] additi Logarithmo circumferentiae Circuli cujus Radius est Unitas, id est, huic 0.39908.99341.79 dabunt summam quaesitam, idque eo minore labore quo plures Logarithmi sunt summandi." (Furthermore, if you want the sum of however many logarithms of the natural numbers 1, 2, 3, 4, 5, etc., set z–n to be the last number, n being ½ ; and three or four terms of this series $z\log(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ added to [half of] the logarithm of the circumference of a circle whose radius is unity [i.e., ½ log(2 $π$ )] – that is, [added] to this: 0.39908.99341.79 – will give the sum [that's] sought, and the more logarithms [that] are to be added, the less work it [is].) Note: $a=0.434294481903252$

books.google.com

Jungnickel, Christa; McCormmach, Russell (1996). Cavendish. Memoirs of the American Philosophical Society. Vol. 220. American Philosophical Society. p. 52. ISBN 9780871692207. Well connected in mathematical circles and highly regarded for his work, he still could not get a good job. Even his conversion to the Church of England in 1705 could not alter the fact that he was an alien.
Tanton, James Stuart (2005). Encyclopedia of Mathematics. Infobase Publishing. p. 122. ISBN 9780816051243. He had hoped to receive a faculty position in mathematics but, as a foreigner, was never offered such an appointment.
Smith, David Eugene (1959), A Source Book in Mathematics, Volume 3, Courier Dover Publications, p. 444, ISBN 9780486646909
In 1738, de Moivre used trigonometry to determine the nth roots of a real or complex number. See: Moivre, A. de (1738). "De reductione radicalium ad simpliciores terminos, seu de extrahenda radice quacunque data ex binomio $a+{\sqrt {+b}}$ , vel $a+{\sqrt {-b}}$ . Epistola" [On the reduction of radicals to simpler terms, or on extracting any given root from a binomial, $a+{\sqrt {+b}}$ or $a+{\sqrt {-b}}$ . A letter.]. Philosophical Transactions of the Royal Society of London (in Latin). 40 (451): 463–478. doi:10.1098/rstl.1737.0081. S2CID 186210174. From p. 475: "Problema III. Sit extrahenda radix, cujus index est n, ex binomio impossibli $a+{\sqrt {-b}}$ . … illos autem negativos quorum arcus sunt quadrante majores." (Problem III. Let a root whose index [i.e., degree] is n be extracted from the complex binomial $a+{\sqrt {-b}}$ $a+{\sqrt {-b}}$ . Solution. Let its root be $x+{\sqrt {-y}}$ $x+{\sqrt {-y}}$ , then I define ${\sqrt[{n}]{aa+b}}=m$ ${\sqrt[{n}]{aa+b}}=m$ ; I also define ${\tfrac {n+1}{n}}=p$ ${\tfrac {n+1}{n}}=p$ [Note: should read: ${\tfrac {n+1}{2}}=p$ ${\tfrac {n+1}{2}}=p$ ], draw or imagine a circle, whose radius is ${\sqrt {m}}$ ${\sqrt {m}}$ , and assume in this [circle] some arc A whose cosine is ${\tfrac {a}{{m}^{p}}}$ ${\tfrac {a}{{m}^{p}}}$ ; let C be the entire circumference. Assume, [measured] at the same radius, the cosines of the arcs ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ , etc.
until the multitude [i.e., number] of them [i.e., the arcs] equals the number n; when this is done, stop there; then there will be as many cosines as values of the quantity $x$ $x$ , which is related to the quantity $y$ $y$ ; this [i.e., $y$ $y$ ] will always be $m-xx$ $m-xx$ .
It is not to be neglected, although it was mentioned previously, [that] those cosines whose arcs are less than a right angle must be regarded as positive but those whose arcs are greater than a right angle [must be regarded as] negative.)
See also:
- Braunmühl, A. von (1903). Vorlesungen über Geschichte der Trigonometrie [Lectures on the history of trigonometry] (in German). Vol. 2. Leipzig, Germany: B.G. Teubner. pp. 76–77.
- Moivre, A. de (1730). Miscellanea Analytica de Seriebus et Quadraturis [Analytical Miscellany of Series and Quadratures [i.e., Integrals]]. London, England: J. Tonson & J. Watts. pp. 103–104.
- From p. 102 of (de Moivre, 1730): "Problema III. Invenire Coefficientem Termini medii potestatis permagnae & paris, seu invenire rationem quam Coefficiens termini medii habeat ad summam omnium Coefficientium. … ad 1 proxime."
  (Problem 3. Find the coefficient of the middle term [of a binomial expansion] for a very large and even power [n], or find the ratio that the coefficient of the middle term has to the sum of all coefficients.
  Solution. Let n be the degree of the power to which the binomial a + b is raised, then, setting [both] a and b = 1, the ratio of the middle term to its power (a + b)ⁿ or 2ⁿ [Note: the sum of all the coefficients of the binomial expansion of (1 + 1)ⁿ is 2ⁿ.] will be nearly as ${\frac {2(n-1)^{n-{\frac {1}{2}}}}{{n}^{n}}}$ to 1.
  But when some series for an inquiry could be determined more accurately [but] had been neglected due to lack of time, I then calculate by re-integration [and] I recover for use the particular quantities [that] had previously been neglected; so it happened that I could finally conclude that the ratio [that's] sought is approximately ${\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ or ${\frac {2{\frac {21}{125}}{(1-{\frac {1}{n}})}^{n}}{\sqrt {n-1}}}$ to 1.)
  The approximation ${\frac {{2}{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{{n}^{n}}}$ is derived on pp. 124-128 of (de Moivre, 1730).

doi.org

Pearson, Karl (1924). "Historical note on the origin of the normal curve of errors". Biometrika. 16 (3–4): 402–404. doi:10.1093/biomet/16.3-4.402.
Stigler, Stephen M. (1982). "Poisson on the poisson distribution". Statistics & Probability Letters. 1: 33–35. doi:10.1016/0167-7152(82)90010-4.
Moivre, Ab. de (1707). "Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, & superiorum, ad infinitum usque pergendo, in termimis finitis, ad instar regularum pro cubicis quae vocantur Cardani, resolutio analytica" [Of certain equations of the third, fifth, seventh, ninth, & higher power, all the way to infinity, by proceeding, in finite terms, in the form of rules for cubics which are called by Cardano, resolution by analysis.]. Philosophical Transactions of the Royal Society of London (in Latin). 25 (309): 2368–2371. doi:10.1098/rstl.1706.0037. S2CID 186209627.
Moivre, A. de (1722). "De sectione anguli" [Concerning the section of an angle] (PDF). Philosophical Transactions of the Royal Society of London (in Latin). 32 (374): 228–230. doi:10.1098/rstl.1722.0039. S2CID 186210081. Retrieved 6 June 2020.
In 1738, de Moivre used trigonometry to determine the nth roots of a real or complex number. See: Moivre, A. de (1738). "De reductione radicalium ad simpliciores terminos, seu de extrahenda radice quacunque data ex binomio $a+{\sqrt {+b}}$ , vel $a+{\sqrt {-b}}$ . Epistola" [On the reduction of radicals to simpler terms, or on extracting any given root from a binomial, $a+{\sqrt {+b}}$ or $a+{\sqrt {-b}}$ . A letter.]. Philosophical Transactions of the Royal Society of London (in Latin). 40 (451): 463–478. doi:10.1098/rstl.1737.0081. S2CID 186210174. From p. 475: "Problema III. Sit extrahenda radix, cujus index est n, ex binomio impossibli $a+{\sqrt {-b}}$ . … illos autem negativos quorum arcus sunt quadrante majores." (Problem III. Let a root whose index [i.e., degree] is n be extracted from the complex binomial $a+{\sqrt {-b}}$ $a+{\sqrt {-b}}$ . Solution. Let its root be $x+{\sqrt {-y}}$ $x+{\sqrt {-y}}$ , then I define ${\sqrt[{n}]{aa+b}}=m$ ${\sqrt[{n}]{aa+b}}=m$ ; I also define ${\tfrac {n+1}{n}}=p$ ${\tfrac {n+1}{n}}=p$ [Note: should read: ${\tfrac {n+1}{2}}=p$ ${\tfrac {n+1}{2}}=p$ ], draw or imagine a circle, whose radius is ${\sqrt {m}}$ ${\sqrt {m}}$ , and assume in this [circle] some arc A whose cosine is ${\tfrac {a}{{m}^{p}}}$ ${\tfrac {a}{{m}^{p}}}$ ; let C be the entire circumference. Assume, [measured] at the same radius, the cosines of the arcs ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ , etc.
until the multitude [i.e., number] of them [i.e., the arcs] equals the number n; when this is done, stop there; then there will be as many cosines as values of the quantity $x$ $x$ , which is related to the quantity $y$ $y$ ; this [i.e., $y$ $y$ ] will always be $m-xx$ $m-xx$ .
It is not to be neglected, although it was mentioned previously, [that] those cosines whose arcs are less than a right angle must be regarded as positive but those whose arcs are greater than a right angle [must be regarded as] negative.)
See also:
- Braunmühl, A. von (1903). Vorlesungen über Geschichte der Trigonometrie [Lectures on the history of trigonometry] (in German). Vol. 2. Leipzig, Germany: B.G. Teubner. pp. 76–77.
- De Moivre determined the value of the constant $\textstyle 2{\frac {21}{125}}$ $\textstyle 2{\frac {21}{125}}$ by approximating the value of a series by using only its first four terms. De Moivre thought that the series converged, but the English mathematician Thomas Bayes (ca. 1701–1761) found that the series actually diverged. From pp. 127-128 of (de Moivre, 1730): "Cum vero perciperem has Series valde implicatas evadere, … conclusi factorem 2.168 seu ${\textstyle 2{\frac {21}{125}},\ldots }$ " (But when I conceived [how] to avoid these very complicated series — although all of them were perfectly summable — I think that [there was] nothing else to be done, than to transform them to the infinite case; thus set m to infinity, then the sum of the first rational series will be reduced to 1/12, the sum of the second [will be reduced] to 1/360; thus it happens that the sums of all the series are achieved. From this one series $\textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}$ $\textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}$ , etc., one will be able to discard as many terms as it will be one's pleasure; but I decided [to retain] four [terms] of this [series], because they sufficed [as] a sufficiently accurate approximation; now when this series be convergent, then its terms decrease with alternating positive and negative signs, [and] one may infer that the first term 1/12 is larger [than] the sum of the series, or the first term is larger [than] the difference that exists between all positive terms and all negative terms; but that term should be regarded as a hyperbolic [i.e., natural] logarithm; further, the number corresponding to this logarithm is nearly 1.0869 [i.e., ln (1.0869) ≈ 1/12], which if multiplied by 2, the product will be 2.1738, and so [in the case of a binomial being raised] to an infinite power, designated by n, the quantity $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ will be larger than the ratio that the middle term of the binomial has to the sum of all terms, and proceeding to the remaining terms, it will be discovered that the factor 2.1676 is just smaller [than the ratio of the middle term to the sum of all terms], and similarly that 2.1695 is greater, in turn that 2.1682 sinks a little bit below the true [value of the ratio]; considering which, I concluded that the factor [is] 2.168 or ${\textstyle 2{\frac {21}{125}},\ldots }$ ${\textstyle 2{\frac {21}{125}},\ldots }$ Note: The factor that de Moivre was seeking, was: ${\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots$ ${\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots$ (Lanier & Trotoux, 1998), p. 237.
  - Bayes, Thomas (31 December 1763). "A letter from the late Reverend Mr. Bayes, F.R.S. to John Canton, M.A. and F.R.S.". Philosophical Transactions of the Royal Society of London. 53: 269–271. doi:10.1098/rstl.1763.0044. S2CID 186214800.
  - See:
    - Archibald, R.C. (October 1926). "A rare pamphlet of Moivre and some of his discoveries". Isis (in English and Latin). 8 (4): 671–683. doi:10.1086/358439. S2CID 143827655.

jstor.org

Hald, Anders; de Moivre, Abraham; McClintock, Bruce (1984). "A. de Moivre:'De Mensura Sortis' or'On the Measurement of Chance'". International Statistical Review/Revue Internationale de Statistique. 1984 (3): 229–262. JSTOR 1403045.

semanticscholar.org

api.semanticscholar.org

Moivre, Ab. de (1707). "Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, & superiorum, ad infinitum usque pergendo, in termimis finitis, ad instar regularum pro cubicis quae vocantur Cardani, resolutio analytica" [Of certain equations of the third, fifth, seventh, ninth, & higher power, all the way to infinity, by proceeding, in finite terms, in the form of rules for cubics which are called by Cardano, resolution by analysis.]. Philosophical Transactions of the Royal Society of London (in Latin). 25 (309): 2368–2371. doi:10.1098/rstl.1706.0037. S2CID 186209627.
Moivre, A. de (1722). "De sectione anguli" [Concerning the section of an angle] (PDF). Philosophical Transactions of the Royal Society of London (in Latin). 32 (374): 228–230. doi:10.1098/rstl.1722.0039. S2CID 186210081. Retrieved 6 June 2020.
In 1738, de Moivre used trigonometry to determine the nth roots of a real or complex number. See: Moivre, A. de (1738). "De reductione radicalium ad simpliciores terminos, seu de extrahenda radice quacunque data ex binomio $a+{\sqrt {+b}}$ , vel $a+{\sqrt {-b}}$ . Epistola" [On the reduction of radicals to simpler terms, or on extracting any given root from a binomial, $a+{\sqrt {+b}}$ or $a+{\sqrt {-b}}$ . A letter.]. Philosophical Transactions of the Royal Society of London (in Latin). 40 (451): 463–478. doi:10.1098/rstl.1737.0081. S2CID 186210174. From p. 475: "Problema III. Sit extrahenda radix, cujus index est n, ex binomio impossibli $a+{\sqrt {-b}}$ . … illos autem negativos quorum arcus sunt quadrante majores." (Problem III. Let a root whose index [i.e., degree] is n be extracted from the complex binomial $a+{\sqrt {-b}}$ $a+{\sqrt {-b}}$ . Solution. Let its root be $x+{\sqrt {-y}}$ $x+{\sqrt {-y}}$ , then I define ${\sqrt[{n}]{aa+b}}=m$ ${\sqrt[{n}]{aa+b}}=m$ ; I also define ${\tfrac {n+1}{n}}=p$ ${\tfrac {n+1}{n}}=p$ [Note: should read: ${\tfrac {n+1}{2}}=p$ ${\tfrac {n+1}{2}}=p$ ], draw or imagine a circle, whose radius is ${\sqrt {m}}$ ${\sqrt {m}}$ , and assume in this [circle] some arc A whose cosine is ${\tfrac {a}{{m}^{p}}}$ ${\tfrac {a}{{m}^{p}}}$ ; let C be the entire circumference. Assume, [measured] at the same radius, the cosines of the arcs ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ , etc.
until the multitude [i.e., number] of them [i.e., the arcs] equals the number n; when this is done, stop there; then there will be as many cosines as values of the quantity $x$ $x$ , which is related to the quantity $y$ $y$ ; this [i.e., $y$ $y$ ] will always be $m-xx$ $m-xx$ .
It is not to be neglected, although it was mentioned previously, [that] those cosines whose arcs are less than a right angle must be regarded as positive but those whose arcs are greater than a right angle [must be regarded as] negative.)
See also:
- Braunmühl, A. von (1903). Vorlesungen über Geschichte der Trigonometrie [Lectures on the history of trigonometry] (in German). Vol. 2. Leipzig, Germany: B.G. Teubner. pp. 76–77.
- De Moivre determined the value of the constant $\textstyle 2{\frac {21}{125}}$ $\textstyle 2{\frac {21}{125}}$ by approximating the value of a series by using only its first four terms. De Moivre thought that the series converged, but the English mathematician Thomas Bayes (ca. 1701–1761) found that the series actually diverged. From pp. 127-128 of (de Moivre, 1730): "Cum vero perciperem has Series valde implicatas evadere, … conclusi factorem 2.168 seu ${\textstyle 2{\frac {21}{125}},\ldots }$ " (But when I conceived [how] to avoid these very complicated series — although all of them were perfectly summable — I think that [there was] nothing else to be done, than to transform them to the infinite case; thus set m to infinity, then the sum of the first rational series will be reduced to 1/12, the sum of the second [will be reduced] to 1/360; thus it happens that the sums of all the series are achieved. From this one series $\textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}$ $\textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}$ , etc., one will be able to discard as many terms as it will be one's pleasure; but I decided [to retain] four [terms] of this [series], because they sufficed [as] a sufficiently accurate approximation; now when this series be convergent, then its terms decrease with alternating positive and negative signs, [and] one may infer that the first term 1/12 is larger [than] the sum of the series, or the first term is larger [than] the difference that exists between all positive terms and all negative terms; but that term should be regarded as a hyperbolic [i.e., natural] logarithm; further, the number corresponding to this logarithm is nearly 1.0869 [i.e., ln (1.0869) ≈ 1/12], which if multiplied by 2, the product will be 2.1738, and so [in the case of a binomial being raised] to an infinite power, designated by n, the quantity $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ will be larger than the ratio that the middle term of the binomial has to the sum of all terms, and proceeding to the remaining terms, it will be discovered that the factor 2.1676 is just smaller [than the ratio of the middle term to the sum of all terms], and similarly that 2.1695 is greater, in turn that 2.1682 sinks a little bit below the true [value of the ratio]; considering which, I concluded that the factor [is] 2.168 or ${\textstyle 2{\frac {21}{125}},\ldots }$ ${\textstyle 2{\frac {21}{125}},\ldots }$ Note: The factor that de Moivre was seeking, was: ${\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots$ ${\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots$ (Lanier & Trotoux, 1998), p. 237.
  - Bayes, Thomas (31 December 1763). "A letter from the late Reverend Mr. Bayes, F.R.S. to John Canton, M.A. and F.R.S.". Philosophical Transactions of the Royal Society of London. 53: 269–271. doi:10.1098/rstl.1763.0044. S2CID 186214800.
  - See:
    - Archibald, R.C. (October 1926). "A rare pamphlet of Moivre and some of his discoveries". Isis (in English and Latin). 8 (4): 671–683. doi:10.1086/358439. S2CID 143827655.

st-andrews.ac.uk

mathshistory.st-andrews.ac.uk

O'Connor, John J.; Robertson, Edmund F., "Abraham de Moivre", MacTutor History of Mathematics Archive, University of St Andrews

stackexchange.com

hsm.stackexchange.com

zenodo.org

Moivre, A. de (1722). "De sectione anguli" [Concerning the section of an angle] (PDF). Philosophical Transactions of the Royal Society of London (in Latin). 32 (374): 228–230. doi:10.1098/rstl.1722.0039. S2CID 186210081. Retrieved 6 June 2020.