Download Leon Battista Alberti: Della pittura (On painting), 1435 Johannes

MATH 300 History of Mathematics
Figures in Renaissance Mathematics
Fifteenth Century
Leon Battista Alberti: Della pittura (On painting), 1435
Laid out the geometric principles of linear perspective (vanishing point, station point, convergence of parallel lines in the picture plane) for artists and painters
Johannes Müller (Regiomontanus): De triangulis omnimodis (On triangles of every kind), 1463, pub. 1533
First systematic study of plane and spherical trigonometry in Europe, conceived as a preparation for reading Ptolemy’s Almagest
Mentioned that sines could depend equally on their central angles as on the circular arcs
Constructed an extensive sine/cosine/versine table; proved all propositions
Luca Pacioli: Summa de arithmetica, geometria, proportioni et proportionalità (A Compendium of Arithmetic, Geometry, Proportion and
Proportionality), 1494
A Franciscan friar, Pacioli, was a maestro d’abbaco, or abacist, a professional mathematician
responding to the mathematical needs of the merchant class
Pacioli wrote this encyclopedia of mathematics in Italian (not Latin) to help the sons of Tuscan
merchants learn commercial arithmetic, practical geometry, and algebra (proportions, linear
and quadratic equations)
Used standard abbreviations for algebraic terms: no = numero (number), .c. = cosa (thing),
.ce. = censo (square), .cu.= cubo (cube), “ = radice (root), p = più (plus), m = meno (minus)
Claimed that the general cubic equation is as unsolvable as the quadrature of the circle
First occurrence of double-entry bookkeeping for accounting purposes
Sixteenth Century
Solving the Cubic: Before 1515, Scipione del Ferro, professor at the University of Bologna,
discovered a method to solve the cubic equation x3 + cx = d, and disclosed this technique to his
student, Antonio Maria del Fiore, and successor at the university, Annibale della Nave
Christoff Rudolff: Coss, 1521
First algebra in German
Formulated the law of “exponents” in terms of powers of 2 (but with abbreviated symbols, not
numerical exponents!)
√
Introduced symbols + for addition, − for subtraction, for root
Albrecht Dürer: Underweysung der Messung (Treatise on Mensuration),
1525
Accomplished artist, well known today, especially for his fine woodcuts
First study of geometry written in German, essentially a technical manual for artists
Developed new terminology for geometric terms: “crescent” = lune, “fork-line” = hyperbola,
“egg-line” = ellipse
Focused on geometrical symmetry and methods for constructing regular polygons, conic sections, and perspective drawings of three-dimensional figures
Solving the Cubic: Niccolo (Fontana) Tartaglia of Brescia boasted that he had also solved
cubic equations; in 1535 he was challenged by della Nave to a contest consisting of 30 equations
to solve in 50 days’ time, and was able to solve all della Nave’s cubic equations while della Nave
could not solve a single one of Tartaglia’s!
Nicolas Copernicus: De revolutionibus orbium coelestium (On the revolutions of the heavenly spheres), 1543
Turned to a heliocentric theory for the solar system, supported by thirty years of observations
and awareness that they did not match the Ptolemaic model
The manuscript was completed by the early 1530s, but Copernicus delayed publication, finally
convinced to publish
Michael Stiffel: Arithmetica Integra (A Complete Arithmetic), 1544
Student at the University of Wittenberg and associate of Martin Luther
Introduced concatenation of symbols to represent multiplication, whole number exponents
Compressed the three forms of quadratic equation (x2 = bx + c, x2 + bx = c, x2 + c = bx into one
by allowing negative coefficients
Developed a procedure for computing nth roots using “Pascal’s triangle”
Gerolamo Cardano: Ars Magna (The Great Art), 1545
Cardano was professor at the University of Milan
After learning of Tartaglia’s success with cubic equations, Cardano entreated him to teach
him the method, agreeing not to publish, but when he learned that del Ferro had known the
solution years before, he considered himself free to go ahead with publication
Also included the solution to the general quartic equation devised by Cardano’s student, Ludovico Ferrari (who in 1548 bests Tartaglia in another mathematical contest)
Robert Recorde: The Whetstone of Witte, 1557
First popular algebra in English
Introduced equal sign: “a pair of parallels, . . . thus ===, because no 2 things can be more equal”
Rafael Bombelli: Algebra, 1560
2
3
c
, he worked with “another
In describing the solution to the cubic x3 = cx + d in which d4 < 27
sort of cube root,” neither positive nor√negative, but of numbers which are “plus of minus” or
“minus of minus”, i.e., of the form a ± b, where b is negative
Bombelli showed that such (complex) numbers conform to many of the rules of normal arithmetic
Thomas Finck: Geometriae rotundi libri XIIII (On the geometry of the
circle, in fourteen books), 1583
First introduced the terms “tangent” and “secant” with sine for the trigonometric quantities;
referred to sine complement, tangent complement and secant complement for the remaining
values
Simon Stevin: De Thiende (The Art of Tenths), 1585
Dutch engineer, Stevin developed use of decimal numeration for fractional numbers:
782
“875 0 7 1 8 2 2 3 should be 875
1000 ”
Set out rationale for and rules of arithmetic with decimal system
Argued that Euclidean distinction between (discrete) number and (continuous) magnitude was
no longer valid once arbitrarily fine division of numbers was possible with decimal system
François Viète: In artem analyticem isagoge, (Introduction to the analytic art), 1591
Member of the Privy Council of Henry III and Henry IV of France, successful code breaker
with the kings’ Spanish enemies
Reviving the tradition of Greek analysis, Viète treated algebra as the subject of analysis: assume the solution is known, then work through a series of operations until the unknown is
expressed in term of knowns
Divided analysis into three parts: zetetics, finding the proper equation to represent the problem; poristics, performing appropriate symbolic manipulation; and exegetics, applying the rules
of algebra to solve the problem
Used vowels to represent unknowns, consonants for given quantities; this use of symbols enabled him to treat general examples rather than specific ones, and study the structure of algebraic equations and their solutions
Solved the cubic equation by using the triple angle formula cos3 α − 34 cos α = 14 cos 3α
Realizing that bx − x2 = c can have two positve roots x1 and x2 , he showed that b = x1 + x2
and c = x1 x2 , and a similar result linking the coefficients of the cubic, quartic, and quintic
polynomial equations with the two, three, four or five positive roots of the polynomial
Seventeenth Century
Johannes Kepler
Lutheran astronomer-mathematician, taught at the University in Graz (modern Austria)
• Mysterium cosmographicum (The Secret of the Universe), 1596
Explained the structure of the solar system by a model in which the sizes of the
orbits were determined by nesting the five Platonic solids between the six planets
• Ad Vitellionem paralipomena, quibus astronomiae pars optica tra-
ditur (Paralipomena to Witelo and the optical part of astronomy),
1604
Unified presentation of the three conic section curves, in which the parabola is
viewed as the boundary curve between the family of hyperbolas and the family
of ellipses (as one focus of a hyperbola/ellipse retreats from the other, the curve
approaches a parabola)
• Astronomia nova (The New Astronomy), 1609
Using extensive observations of Mars and the other planets, Kepler formulated his
First Law (planets orbit the sun in ellipses with the sun at one focus) and Second
Law (planets sweep out equal orbital areas in equal times)
• Harmonice mundi (The Harmonies of the World), 1618
Kepler stated his Third Law of planetary motion (the ratio “between the periodic
times of any two planets is precisely the ratio of the 32 th power of the mean distances” of the planets to the sun)
John Napier
Scottish landowner and amateur astronomer
• Mirifici logarithmorum canonis descriptio (A Description of the
Wonderful Table of Logarithms), 1614
Initial appearance of modern decimal notation, with the use of a decimal point
Attempting to improve the speed of arithmetical calculation involving trigonometric formulas for astronomical ephemeris table construction, Napier invented and
tabulated “logarithms”
For angles from 0◦ 00 to 45◦ 00 , he tabulated sines and the “logarithms” of these sines
“From the logarithms of four numbers in proportion, the sum of the middle numbers. . . is equal to the sum of the extremes”
• Mirifici logarithmorum canonis constructio (The Construction of
the Wonderful Table of Logarithms), 1619
Napier explained how he devised his logarithms: setting two number lines side by
side, one with values increasing arithmetically the other with values decreasing
geometrically, the former numbers are “logarithms” of the latter
Henry Briggs: Arithmetica Logarithmica (Logarithmical Arithmetic),
1624/1628
Held the chair of geometry at Gresham College, London
After meeting with Napier in 1615, he simplified Napier’s logarithms by arranging that, rather
than having the logarithm of 10,000,000 equal to 0, he set log 1 = 0 and log 10 = 1, creating
“common” logarithms
Tabulated fourteen-place logarithms for the numbers 1-20,000 and 90,001-100,000 in 1624;
four years later, Belgian printer Adrian Vlacq extended the tables for arguments 20,001-90,000
Galileo Galilei
Mathematician and astronomer who taught at the universities in Pisa and Padua; widely
known as the “father of modern science” by virtue of formulating and testing theories with
his own empirical observations (e.g., he discovered the moons of Jupiter by telescope)
• Dialogo sopra i due massimi sistemi del mondo (Dialogue Concern-
ing the Two Chief World Systems), 1632
Despite warnings from the Inquisition in 1616 not to publicly support the Copernican heliocentric model, Galileo asserted this very theory, for which he was officially
censured by the Catholic Church through Jesuit Cardinal Robert Bellarmine
• Discorsi e dimostrazioni matematiche, intorno a due nuove scienze
(Discourses and Mathematical Demonstrations Concerning Two New
Sciences), 1638
Written in the form of a dialogue among three characters, Galileo described his
theories of objects in free fall (that they are uniformly accelerated, and that the
distances they fall are proportional to the squares of their times) and projectile
motion (projectiles move in parabolic orbits)
Incorrectly surmised that a the path of an object that falls from one point to another
(not directly below it) in minimum time, known as the brachistochrone problem, is
that of a semicircular arc