During the Renaissance, the development of mathematics and
of accounting were intertwined. 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 schools (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.
Luca Pacioli's "Summa de Arithmetica, Geometria,
Proportioni et Proportionalità" (Italian: "Review of Arithmetic,
Geometry, Ratio and 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. 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. It is important to note that Pacioli
himself had borrowed much of the work of 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
equations. Gerolamo Cardano published them in his 1545 book Ars Magna, together
with a solution for the quartic equations, discovered by his student Lodovico
Ferrari. In 1572 Rafael Bombelli published his L'Algebra in which he showed how
to deal with the imaginary quantities that could appear in Cardano's formula
for solving cubic equations.
Simon Stevin's book De Thiende ('the art of tenths'), first
published in Dutch in 1585, contained the first systematic treatment of decimal
notation, which influenced all later work on the real number system.
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.
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 involved, were
studied intensely.