expanding economy demanded a better
ability to calculate sums, percentages, for-
eign exchange, and rates of interest. The
new double-entry method of bookkeeping
allowed merchants to carefully track in-
come and expenses. Abacists were math
teachers who imparted the needed math-
ematical skills to the sons of traders, bank-
ers, and long-distance merchants. The in-
vention of printing in the middle of the
fifteenth century allowed mathematical
texts to circulate widely, beginning with
Theoricae Nova Planetarumof Georg von
Peuerbach in 1472 and a guide to arith-
metic, known as theTreviso Arithmetic,in
- TheElements, a study by the ancient
mathematician Euclid, first appeared in
printed form in 1482. The Arabic numer-
als, decimal places, and mathematical signs
and symbols borrowed from India came
into common use in Europe at about the
same time.
Sixteenth-century mathematicians be-
gan solving many thorny problems, such
as cubic and quartic equations. The Ger-
man philosopher Johann Müller, known as
Regiomontanus, wrote commentaries on
Ptolemy’sAlmagestand published his own
book of calculations, Detriangulus.An-
other German, Johann Widman, was the
first to use the plus and minus signs in a
published work. Other significant German
mathematicians were Adam Riese, Chris-
toph Rudolff (who pioneered the use of
root symbols), and Michael Stifel, who
wrote an algebra text, theArithmetica Inte-
gra, dealing with powers, radicals, and
negative numbers. In Italy, Geronimo Car-
dano wroteArs Magna, the first algebra
treatise written in Latin. Cardano’s follow-
ers included Niccolo Tartaglia, who drew
up the first “firing tables” for use by artil-
lery, and was the first to discover a for-
mula for solving cubic equations.
At the same time, astronomy was be-
coming a sophisticated mathematical
method of predicting planetary orbits, the
path of the stars, and the occurrence of
eclipses and other celestial phenomena.
The precise observation of the skies and
the measuring technique of trigonometry
were spurred by the demands of naviga-
tors, who needed accurate charts of newly
explored areas that lay thousands of miles
distant from familiar home shores. The
first textbook in this subject was theTrigo-
nometria, written by Bartholomaeus Pitis-
cus and published in 1595. The investiga-
tion of the heavens by telescope enabled
more precise astronomical calculations,
undertaken by Galileo Galilei, Tycho
Brahe, and Johannes Kepler, who devised a
systematic mathematical system for deter-
mining planetary orbits. The French phi-
losopher René Descartes developed a new
method of depicting calculations on charts
and a system of analytic geometry. The
culmination of Renaissance study of math-
ematics was the system of calculus, a
method of solving complex problems that
was first developed by Sir Isaac Newton
and the German scholar Gottfried Wil-
helm Leibniz.SEEALSO: Brahe, Tycho; Kepler, JohannesMaximilian I ...................................
(1459–1519)
Holy Roman Emperor who greatly ex-
panded the realm under the control of the
powerful Habsburg dynasty. He was born
in Wiener Neustadt, a suburb of Vienna,
the son of Emperor Frederick III and
Eleanor of Portugal. In 1477, he married
Mary of Burgundy, who brought the Low
Countries and Burgundy under Habsburg
control. In 1482, on the death of Mary,
Burgundy became a part of France while
the Netherlands, which had always resistedMaximilian I