History of Algebra: Leonardo of Pisa and His Successors

After ``Fibonacci Algebra Through History November 6, 2019

Recall from last time Leonardo of Pisa s Liber Abaci (1202 CE) was extremely influential, mostly because it effectively popularized the use of the Hindu-Arabic numerals, especially for practical calculations But it also contained a final chapter that gave an exposition of algebra -- techniques for solving linear and quadratic equations It followed Al-Khwarizmi s book on al-jabr w al muqabala very closely in that section (and also, earlier on, Al-Khwarizmi s discussion of the Hindu-Arabic numerals) Strong evidence that Leonardo knew some Arabic and drew on those or related sources

Successors There were a whole series of similar textbooks by other Italian authors (that came to be known as libri d abbaco in Italian) Used in ``algebra schools called botteghe d abbaco One of the first well-known books following Leonardo s book was called the Tractatus Algorismi by Jacopo of Florence, 1307 CE Covers essentially same ground but does not follow Al-Khwarizmi as closely -- evidence of an independent tradition starting But (ironically) the word algorism for the process of computation with the Hindu-Arabic numerals also came from Al-Khwarizmi s name!

Algorists vs. abacists Note: The Italian word abbaco does not usually refer to use of the abacus the people trained in these schools were mostly calculating by hand with the Hindu-Arabic numerals from a later book (1508 CE)

The strange case of Paolo Gerardi A Florentine, but he lived and worked at Montpellier in France Published a book in Italian called Libro de Raggione (Book of Problems) in 1328 CE In addition to linear and quadratic equations (as in Al-Khwarizmi, Fibonacci, Jacopo), he also ``branched out into cubic equations. Here s what he said about solving an equation ??3= ?? + ?: ? ? ?+ (? In modern notation, compute ? = 2?)2 2?+ But in fact this is not correct! He just followed the quadratic formula even though the equation is cubic and not quadratic

Paolo Gerardis misconceptions proliferate If he had checked an example, he would have seen that this cannot be correct. But he didn t(!) And this wasn t the only case he got wrong several others too! And, even more strangely, a number of later authors copied his mistaken solutions of cubic and quartic (fourth-degree) equations in books written over the next decades(!) Imagine this: Suppose you were a student learning ``algebra from one of these books, but what you were learning was just totally, hopelessly incorrect.

Master Dardi of Pisa Wrote a book called Aliabraa argibra in 1344 Gave correct solutions of some special forms of cubic and quartic equations; also made it clear that those solutions did not apply completely generally This was also the first book entirely about algebra to be written in Italian (not Latin) the unfortunate Gerardi had other topics in his book too. Dardi also popularized Italian terminology for terms in equations (still no consistent symbolic form for them):

Terminology for powers of an unknown The unknown itself cosa ( thing recall what Al-Khwarizmi did in the inheritance problem on Problem Set 3!) The square of the unknown censo The cube of the unknown cubo The fourth power of the unknown censo di censo We ll see versions of this over and over as we continue in this part of the story, extending to higher and higher powers(!)

Next steps People were gradually learning to solve more kinds of, and more difficult equations At the end of this phase, several large summaries of what had been learned were published One of the best known, and also the first printed book in Italian on algebra, was the Summa de arithmetica, geometria, proportioni, e proportionalita, 1494 by Luca Pacioli (c. 1447 1517) Pacioli came from Tuscany, knew the painter Piero della Francesca and the architect and general ``Renaissance man Leon Battista Alberti

Portrait of Luca Pacioli teaching

Luca Pacioli In the algebraic sections of the Summa, he used an abbreviated form of expressions that is different from, but somewhat reminiscent of, what Diophantos had done 1200 years previously: co. = cosa (the unknown), ce. = census (the square of the unknown) = square root p. = plus, m. = minus, v. = group terms (like our parentheses) For example: 1 co. m. v. 1 ce. m. 36 = ? (?2 36) Pacioli is also usually credited as being the inventor of double-entry bookkeeping, he s the ``unofficial patron saint of accountants(!)

The ideas spread to Northern Europe too Starting shortly after 1500, other schools of algebra also started to spring up in England, in Germany, etc. In England, Robert Recorde (1510 1558) introduced something like our modern = notation for equality in his book Whetstone of Witte In Germany, Christoph Rudolff (ca. 1500 ca. 1550) published a book called Coss (Germanized form of the Italian word cosa = unknown) A whole German school of cossists followed him, including Michael Stifel (published a second edition of Rudolff s book, apparently after Rudolff s death).

Rudolffs names for powers of the unknown Radix the first power Zensus the square Cubus the cube Zens de Zens the fourth power Sursolidum the fifth power Zensicubus the sixth power Bisursolidum the seventh power Zens Zens de Zens the eighth power Cubus de cubo the ninth power

A communication problem" One curious feature of this juncture in the history is that each new cossist algebra book that appeared put more and more of the expressions, operations, equations, etc. into symbolic forms But note I said forms and not form(!) The authors were all inventing their own notations and experimenting to find convenient and useful ways to write things Their symbols were frequently different and even inconsistent see discussion on pages 209 211 of Katz and Parshall if you are interested(!) It would take another 100 to 200 years before algebraic notation became completely standardized.