Editing History of mathematics (section)

== Renaissance ==
{{further|Mathematics and art}}
During the [[Renaissance]], the development of mathematics and of [[accounting]] were intertwined.<ref>Heeffer, Albrecht: ''On the curious historical coincidence of algebra and double-entry bookkeeping'', Foundations of the Formal Sciences, [[Ghent University]], November 2009, p. 7 [http://logica.ugent.be/albrecht/thesis/FOTFS2008-Heeffer.pdf]</ref> While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in [[Flanders]] and [[Germany]]) or [[abacus school]]s (known as ''abbaco'' in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing [[bookkeeping]] operations, but for complex bartering operations or the calculation of [[compound interest]], a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.

[[Piero della Francesca]] (c. 1415–1492) wrote books on [[solid geometry]] and [[Perspective (graphical)|linear perspective]], including ''[[De Prospectiva Pingendi]] (On Perspective for Painting)'', ''Trattato d’Abaco (Abacus Treatise)'', and ''[[De quinque corporibus regularibus]] (On the Five Regular Solids)''.<ref>della Francesca, Piero (1942). ''De Prospectiva Pingendi'', ed. G. Nicco Fasola, 2 vols., Florence.</ref><ref>della Francesca, Piero. ''Trattato d'Abaco'', ed. G. Arrighi, Pisa (1970).</ref><ref>della Francesca, Piero (1916). ''L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli'', ed. G. Mancini, Rome.</ref>

[[Image:Pacioli.jpg|thumb|left|''[[Portrait of Luca Pacioli]]'', a painting traditionally attributed to [[Jacopo de' Barbari]], 1495, ([[Museo di Capodimonte]]).]]
[[Luca Pacioli]]'s ''[[Summa de arithmetica|Summa de Arithmetica, Geometria, Proportioni et Proportionalità]]'' (Italian: "Review of [[Arithmetic]], [[Geometry]], [[Ratio]] and [[Proportionality (mathematics)|Proportion]]") was first printed and published in [[Venice]] in 1494. It included a 27-page treatise on bookkeeping, ''"Particularis de Computis et Scripturis"'' (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the [[mathematical puzzles]] it contained, and to aid the education of their sons.<ref>Sangster, Alan; Greg Stoner & Patricia McCarthy: [http://eprints.mdx.ac.uk/3201/1/final_final_proof_Market_paper_050308.pdf "The market for Luca Pacioli’s Summa Arithmetica"] {{Webarchive|url=https://web.archive.org/web/20180126012523/http://eprints.mdx.ac.uk/3201/1/final_final_proof_Market_paper_050308.pdf |date=2018-01-26 }} (Accounting, Business & Financial History Conference, Cardiff, September 2007) pp. 1–2.</ref> In ''Summa Arithmetica'', Pacioli introduced symbols for [[plus and minus]] for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. ''Summa Arithmetica'' was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized.

In Italy, during the first half of the 16th century, [[Scipione del Ferro]] and [[Niccolò Fontana Tartaglia]] discovered solutions for [[cubic equation]]s. [[Gerolamo Cardano]] published them in his 1545 book ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]'', together with a solution for the [[quartic equation]]s, discovered by his student [[Lodovico Ferrari]]. In 1572 [[Rafael Bombelli]] published his ''L'Algebra'' in which he showed how to deal with the [[imaginary number|imaginary quantities]] that could appear in Cardano's formula for solving cubic equations.

[[Simon Stevin]]'s ''[[De Thiende]]'' ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of [[decimal notation]] in Europe, which influenced all later work on the [[real number system]].<ref>[[Roshdi Rashed]] (1996) ''Encyclopedia of the History of Arabic Science'', chapter 10: Numeration and Arithmetic, page 315, [[Routledge]] {{doi|10.4324/9780203403600}}</ref><ref name="GS35">{{Cite journal |last=Sarton |first=George |date=1935 |title=The First Explanation of Decimal Fractions and Measures (1585). Together with a History of the Decimal Idea and a Facsimile (No. XVII) of Stevin's Disme |url=https://www.jstor.org/stable/225223 |journal=Isis |volume=23 |issue=1 |pages=153–244 |doi=10.1086/346940 |jstor=225223 |s2cid=143395001 |issn=0021-1753}}</ref>

Driven by the demands of navigation and the growing need for accurate maps of large areas, [[trigonometry]] grew to be a major branch of mathematics. [[Bartholomaeus Pitiscus]] was the first to use the word, publishing his ''Trigonometria'' in 1595. Regiomontanus's table of sines and cosines was published in 1533.<ref>{{cite book | last = Grattan-Guinness | first = Ivor | year = 1997 | title = The Rainbow of Mathematics: A History of the Mathematical Sciences | publisher = W.W. Norton | isbn = 978-0-393-32030-5}}</ref>

During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that were involved, were studied intensely.<ref name="Kline">
{{cite book
 | last = Kline
 | first = Morris
 | author-link =Morris Kline
 | title = Mathematics in Western Culture
 | publisher = Pelican
 | year = 1953
 | location=Great Britain
 | pages= 150–51}}</ref>