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A Potted History of Calculus ...

A Potted History of Calculus. … a useful tool. A Potted History of Calculus. sequences. series. differential calculus. integral calculus. the link. the future. Babylonian Sequences (-2000 ish).

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A Potted History of Calculus ...

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  1. A Potted History of Calculus ... • … a useful tool.

  2. A Potted History of Calculus ... sequences series differential calculus integral calculus the link the future

  3. Babylonian Sequences(-2000 ish) • 100 shekels are to be divided among 10 brothers so that the shares form an arithmetic progression, and so that the eighth brother gets 6 shekels.

  4. Babylonian Series • 1 + 21 + 22 + … + 29 = ? You should add and find 512, subtract 1 from it, giving 511, add 511 and 512, so it is 1023.

  5. Greek APs • 1 + 3 + 5 + … + (2n+1) = ?

  6. Greek GPs(Euclid Bk IX, -325) • a + aq + aq2 + … + aqn-1 = ?

  7. Infinite GPs • Archimedes (-287:-212) • If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.

  8. na n Integral Calculus • Bonaventura Cavalieri • Geometria indivisibilibus continuorum nova quadam ratione promota (1629) Area of a triangle - Generalise!

  9. Integral Calculus (cont) • Cavalieri (1629)- • Torricelli (1644) - • Fermat (1646) -

  10. Differential Calculus • Fermat (1629) • Let a be any unknown of the problem. Let us indicate the maximum or the minimum by a in terms which shall be of any degree. We shall now replace the original unknown a by a+e and we shall express thus the maximum or minimum quantity in terms of a and e involving any degree. We shall adequate the two expressions and we shall take out their common terms. We shall divide all terms by e so that e will be completely removed from at least one of the terms. We suppress then all of the terms in which e or one of its powers will still appear, and we shall equate the others ...

  11. The Fundamental Theorem of Calculus • Toricelli (c.1646): y=xn • Isaac Barrow (1630 to 1677): • Lectiones Geometricae (1670) - • general result, but geometric and cumbersome.

  12. Newton and Leibniz

  13. Leibniz • Gottfried Wilhelm Leibniz (1646 to 1716) • Acta Eruditorum (1684): • A new method for maxima and minima as well as tangents, which is neither impeded by functional nor irrational quantities, and a remarkable type of calculus for them

  14. Isaac Newton(1642 to 1727)Fluxions ∙∙If the moment of x be represented by the product of its celerity x∙ into an indefinitely small quantity o (that is x∙o ), the moment of y will be y∙o, since x∙o and y∙o are to each other as x∙ and y∙. Now since the moments or x∙o and y∙o are the indefinitely little accessions of the flowing quantities, x and y, by which these quantities are increased through the several indefinitely little intervals of time, it follows that these quantities, x and y, after any indefinitely small intervals of time, become x+x∙o and y+y∙o. And therefore the equation which at all times indifferently expresses the relation of the flowing quantities will as well express the relation between x+x∙o and y+y∙o as between x and y; so that x+x∙o and y+y∙o may be substituted for those quantities instead of x and y.

  15. A century of pragmatismbut ... Y = 1-1+1-1+1 … ? Euler (1755): so

  16. The Great Debate Bishop George Berkely (1685-1753) The Analyst - Or a Discourse addressed to an Infidel Mathematician (1734) “He who can digest a second or a third Fluxion, a second or third Difference, need not, methinks, be squeamish about any Point in Divinity.”

  17. The Resolution - Limits “prime” and “ultimate” ratios - Principia Mathematica, Newton, 1723 “...ghosts of departed quantities.” - Berkely “… the limits of the ratios of the finite differences of two variables included in the equation.” - d’Alembert, 1754

  18. Cauchy Convergence Augustin-Louis Cauchy (1789-1857) The limit of the sequence xi, 1 = 1, 2, 3, … as n tends to infinity is x iff, given e, there exists N s.t. for all i > N, |xi-x | < e.

  19. Future History • Technology - will it: • help us to do it (computer algebras) • or • remove the need for it (cooling coffee)?

  20. Conclusion Nature and Nature’s Laws lay hid in Night. God said,Let Newton be!And All was Light. Alexander Pope Let Newton Be!, John Fauvel et al (ed), Oxford A History of Mathematics, Carl B Boyer, Wiley Makers of Mathematics, Stuart Hollingdale, Penguin http://www.math.bme.hu/mathhist/HistTopics/The_rise_of_calculus.html#91

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