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Enlightening SymbolsA Short History of Mathematical Notation and Its Hidden Powers$

Joseph Mazur

Print publication date: 2016

Print ISBN-13: 9780691173375

Published to Princeton Scholarship Online: January 2018

DOI: 10.23943/princeton/9780691173375.001.0001

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(p.81) Part 2 Algebra

(p.81) Part 2 Algebra

Enlightening Symbols

Joseph Mazur

Princeton University Press

Going back in time once again, before Indian numerals were brought to Europe.

Significant Initiators

Many of these initiators were either the first or best known for putting symbols into print:

DIOPHANTUS (205 15–290 15). Alexandrian Greek. Mathematician.

Wrote the Arithmetica in ca. 250 AD. First to use symbol for minus Part 2 Algebraand unknown Part 2 Algebra

HYPATIA (ca. 370–415). Greek. Mathematician.

First notable woman mathematician, and commentator of the Arithmetica.

ARYABHATTA (476–550). Indian. Mathematician-astronomer.

Used letters to represent unknowns.

BRAHMAGUPTA (598–668). Indian. Mathematician-astronomer.

Possibly the first writer to use zero (a small black dot) as a number (621 AD). Wrote the Brahmasphutasiddhanta (628), which used abbreviations for squares and square roots and for each of several unknowns occurring in special problems. Introduced rules for manipulating negative and positive numbers.

(p.82) AL-KHWĀRIZMĪ (ca. 780–ca. 850). Arab. Mathematician-astronomer-geographer.

Scholar in the House of Wisdom. Wrote the Compendious Book on Calculation by Completion and Balancing (Algebra) (830 AD). Organized rhetorical algebraic expressions according to the various species of forms.

MAESTRO DARDI DI PISA (Jacopo). Italian. Mathematician.

Unpublished manuscript dated 1344; the Aliabraa argibra earliest manuscript written in the Italian vernacular that exclusively treats algebra.

FRA LUCA BARTOLOMEO DE PACIOLI (1446/7–1517). Italian. Mathematician.

His treatise on algebra was the first to be printed; gave it the Arabic name Alghebra e Almucabala (Restitution and Comparison, or Opposition and Comparison, or Resolution and Equation) (1478).

NICOLAS CHUQUET (1455–1488). French. Mathematician.

Triparty en la Science des Nombres (Three-part Book on the Science of Numbers) (ca. 1484). Labeled species of powers as Part 2 Algebra, …, and square root as Part 2 Algebra

JOHANNES WIDMANN (1460–1498). German. Mathematician.

In his 1489 work, Behende und hubsche Rechenung auff allen Kauffmanschafft (Nimble and Neat Calculation in All Trades), he introduced + as a symbol for plus.

MICHAEL STIFEL (or Stefleius) (1487–1567). German. Mathematician.

Published an edition of Die Coss in 1553. Used the letters “M” and “D” for multiplication and division, respectively. So 3 ②D secM ter ②would indicate 3 x 2 z 2 y , where sec and ter stand for second and third unknown.

CHRISTOFF RUDOLFF (1499–1545). German. Textbook author.

Die Coss (1525). Incorporated the symbols Part 2 Algebraand Part 2 Algebrafor square, cube, and fourth roots, respectively.

GEROLAMO CARDANO(1501–1576). Italian. Physician, mathematician, astrologer.

Mathematician who in 1545 wrote the Ars Magna, solving cubic and quartic equations. Recognized the value of imaginary and complex solutions.

(p.83) ROBERT RECORDE (ca. 1512–1558). Welsh. Physician and mathematician.

Whetstone of Witte (1557) was widely read, so it introduced the equal symbol (=) to northern European countries.

RAFAEL BOMBELLI (1526–1572). Italian. Mathematician.

Involved with solutions of cubic and quartic equations (1572). Used Part 2 Algebrato represent the unknown, its square, its cube, and so on.

GUILIELMUS XYLANDER (also Wilhelm Holzmann) (1532–1576). German. Scholar.

A classical scholar. Translator of Euclid’s Elements and Diophantus’s Arithmetica into Latin.

FRANÇOIS VIÈTE (1540–1603). French. Mathematician.

Used letters to represent numbers as general objects, and subjected them to the same algebraic reasoning and rules as numbers.

SIMON STEVIN (1548–1620). Flemish. Mathematician and engineer.

In his L’Arithmetique (1585), he used the so-called Index Plan for writing exponents—that is, x2 − 3x +2 would be written as Part 2 Algebra

THOMAS HARRIOT (1560–1621). English. Astronomer, mathematician, ethnographer.

Set polynomials equal to zero, and thereby saw that if a were a root to the polynomial equation degree less than five, then xa is a factor of the polynomial.

WILLIAM OUGHTRED (1574–1660). English. Mathematician.

Clavis mathematicae (1631). Invented more than one hundred symbols, but less than a dozen survive the seventeenth century. Usedto indicate multiplication and the colon “:” to denote division.

PIERRE HÉRIGONE (1580–1643). French. Mathematician and astronomer.

Cursus mathematicus (1634). Wrote a six-volume algebra text almost entirely in symbols. Invented ⊥ (“is perpendicular to”) and ∠ (“angle”).

CLAUDE GASPARD BACHET(1581–1638). French. Mathematician, linguist, scholar.

First to translate Arithmetica from Greek to Latin (1621).

(p.84) RENÉ DESCARTES (1596–1650). French. Mathematician, philosopher.

La Géométrie (1637). Used numerical superscripts to mark positive integral exponents of a polynomial. Ranked individual powers numerically. Established the convention of reserving beginning letters of the alphabet for fixed known quantities and latter letters for variables or unknowns.

JOHN WALLIS (1616–1703). English. Mathematician.

Mathesis Universalis and Arithmetica Infinitorum (1655). Used negative exponents and indicated infinity by the symbol∞.

ISAAC NEWTON (1642–1727). English. Physicist-mathematician, alchemist.

Conceived of unknown variables as Fluents (what we call “dependent variables”), quantities flowing along a curve. Derivatives are denoted as singly dotted forms , , ż, so-called pricked letters.

GOTTFRIED LEIBNIZ (1646–1716). German. Mathematician, philosopher.

Understood the limits and conceptional powers of symbols. Made symbols a priority in his attempts at clear writing. Invented the proper symbols for the differential and integral calculus.

LEONARD EULER (1707–1783). Swiss. Physicist, mathematician.

Represented 1 as i in his Recueil des pieces qui ont remporte les pris de l’academie royale des sciences (1777).

WILLIAM JONES (1746–1794). Welsh. Philologist, ancient India scholar.

Introduced the Greek letter π‎.

GUSTAVE-PETER LEJEUNE DIRICHLET (1805–1859). German. Mathematician.

Introduced the modern function concept.

WILLIAM ROWAN HAMILTON (1805–1865). Irish. Physicist-mathematician.

Introduced the “quaternions,” a new number system in four dimensions that contained the complex numbers.