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BO1 History of MathematicsLecture II

Analytic geometry and the beginnings of calculusPart 2: The appearance of symbolic notation

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Notation: compare Cardano (Ars magna, 1545)...

Having raised a third part of thenumber of things to a cube, towhich you add the square of halfthe number in the equation andtake the root of the total, considerthe square [root], which you willtake twice; and to one of them youadd half of the same, and you willhave the binome with its apotome,whence taking the cube root ofthe apotome from the cube rootof its binome, the difference thatcomes from this, is the value ofthe thing.

(Mathematics emerging, p. 327)

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... with Viete (c. 1590)...

Francois Viete(Francisci Vieta)Opera mathematica1646, p. 130

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... with Viete (c. 1590)...

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... and with Harriot (c. 1600)

British LibraryAdd MS 6784 f. 323available atThomas Harriot Online

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... and with Harriot (c. 1600)

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And here is Harriot’s own comparison

British Library Add MS 6782 f. 277; Thomas Harriot Online

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Elsewhere in the world

Seki Takakazu, Hatsubi Sanpo (1674), concerning thesolution of equations in several variables:

Equations written using the technique of boshoho(‘side-writing’; a.k.a. tenzan jutsu )

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Notation: Viete (Tours, c. 1590)

Francois Viete (1540–1603, France):

A, E, ... (i.e., vowels) for unknowns

B, C, D, ... (i.e., consonants) for knownor given quantities

symbols + , −

but otherwise verbal descriptions andconnections: quadratum (squared),cubus (cubed), aequatur (be equal), ...

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Notation: Harriot (London, c. 1600)

Thomas Harriot (1560–1621, England):

a, e, ... for unknowns

b, c, d, ... for known or given quantities

+, −

ab, aa, aaa

and many symbols: =, >, ...

(For another example of Harriot’s use ofnotation, see Mathematics emerging,§2.2.1.)

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Notation: Descartes (Netherlands, 1637)

Rene Descartes (1596–1650, France andHolland):

x, y, ... for unknowns

a, b, c, ... for known or given quantities

+, −

xx, x3, x4, ...

Descartes’ notation was widely adopted,although his ‘

∝

’ for equality eventuallygave way to ‘=’, and his ‘

pC ’ to ‘ 3

p’.

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Descartes’ notation

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Symbolism established in algebra

Frontispiece to: JohannesFaulhaber, Ingenieurs-Schul,Anderer Theil, Ulm, 1633(on fortification)

See: Volker Remmert,‘Antiquity, nobility, andutility: picturing the EarlyModern mathematicalsciences’, in The Oxfordhandbook of the history ofmathematics (Eleanor Robson& Jacqueline Stedall, eds.),OUP, 2009, pp. 537–563