A Potted History of Calculus ...

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A Potted History of Calculus ...

• a useful tool.

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

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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.
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Greek APs

• 1 3 5 (2n1) ?

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Greek GPs(Euclid Bk IX, -325)

• a aq aq2 aqn-1 ?

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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.

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Integral Calculus

• Bonaventura Cavalieri
• Geometria indivisibilibus continuorum nova quadam
  ratione promota (1629)

Area of a triangle -
Generalise!
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Integral Calculus (cont)

• Cavalieri (1629)-
• Torricelli (1644) -
• Fermat (1646) -

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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 ae 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 ...

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The Fundamental Theorem of Calculus

• Toricelli (c.1646) yxn
• Isaac Barrow (1630 to 1677)
• Lectiones Geometricae (1670) -
• general result,

but geometric and cumbersome.
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Newton and Leibniz
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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

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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 xo ), the moment of y
will be yo, since xo and yo are to each other
as x and y. Now since the moments or xo and
yo 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
xxo and yyo. And therefore the equation
which at all times indifferently expresses the
relation of the flowing quantities will as well
express the relation between xxo and yyo as
between x and y so that xxo and yyo may be
substituted for those quantities instead of x and
y.
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A century of pragmatismbut ...
Y 1-11-11 ?
Euler (1755)
so
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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.
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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. - dAlembert, 1754
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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 gt N, xi-x lt e.
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Future History

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

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Conclusion
Nature and Natures 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.html91