16th-to-17th-Renaissance-mathematics.pptx

Ma. Julita B. Obiles
BICOL
UNIVERSITY
Legazpi City

16TH CENTURY MATHEMATICS
 Cultural, intellectual and artistic movement of the
Renaissance began in Italy around the 14th Century
 Gradually spread across most of Europe over the next two
centuries
 Science and art were still very much interconnected and
intermingled at this time, as exemplified by the work of
artist/scientists such as Leonardo da Vinci
 Just as in art, revolutionary work in the fields of philosophy and
science was took place

LATE 15TH, EARLY 16TH CENTURY
 Albrecht Dürer (German Artist)
- included an order-4 magic square in his engraving
“Melencolia I”
- super magic square” with many more lines of addition
symmetry than a regular 4 x 4 magic square (1514).
 Luca Pacioli (Italian Franciscan friar)
- published a book on arithmetic, geometry and book-keeping
at the end of the 15th Century

- became quite popular for the mathematical puzzles it
contained
- also introduced symbols for plus and minus for the first
time in a printed book (sometimes attributed to Giel Vander
Hoecke, Johannes Widmann and others)
- symbols that were to become standard notation

- also investigated the Golden Ratio of 1 : 1.618 in
his 1509 book “The Divine Proportion”
- concluded that the number was a message from
God and a source of secret knowledge about the
inner beauty of things.

16TH AND EARLY 17TH CENTURY
 The equals, multiplication, division,
radical (root), decimal and inequality
symbols were gradually introduced and
standardized

Simon Stevin Niccoloi
Tartaglia
Significant Names
Gerolamo
Cardano
Lodovico
Ferrari

Mathematical Contributions
Simon Stevin
 use of decimal fractions and decimal arithmetic
 enjoined that all types of numbers, whether
fractions, negatives, real numbers or surds (such
as √2) should be treated equally as numbers in
their own right.
Note: the decimal point notation was not popularized until
early in the 17th century

Mathematical Contributions and Short
Biography
Niccoloi Tartaglia
 Niccolò Fontana became known as Tartaglia (meaning
“the stammerer”) for a speech defect he suffered due to an
injury he received in a battle against the invading French
army
 a poor engineer known for designing fortifications, a
surveyor of topography (seeking the best means of defence
or offence in battles) and a bookkeeper in the Republic of
Venice.

 self-taught, but wildly ambitious, mathematician.
 distinguised himself by producing, among other
things, the first Italian translations of works
by Archimedes and Euclid from uncorrupted Greek
texts
Niccoloi Tartaglia

Niccoloi Tartaglia
 revealed to the world the formula for solving first one
type, and later all types, of cubic equations
(equations with terms including x3)
 in the Renaissance Italy of the early 16th Century,
Bologna University in one of its famed public
mathematics competitions
 stumped the best mathematicians of China, India and
the Islamic world.

 Encoded his solution in the form of a poem in an
attempt to make it more difficult for other
mathematicians to steal it
 His definitive method was leaked to Gerolamo
Cardano
 Engaged Cardano in a decade-long fight over the
publication of “Ars Magna” that included his cubic
solution.
Niccoloi Tartaglia

Niccoloi Tartaglia
 was thoroughly discredited and became effectively
unemployable when he decided not to show up when
challenged to a public debate (he initially accepted) by
Ferrari.
died penniless and unknown despite having produced
(in addition to his cubic equation solution) the
following:

 the first translation of Euclid’s “Elements” in a
modern European language
 formulated Tartaglia’s Formula for the volume of a
tetrahedron, devised a method to obtain binomial
coefficients called Tartaglia’s Triangle (an earlier
version of Pascal‘s Triangle)
Niccoloi Tartaglia

Niccoloi Tartaglia
 became the first to apply mathematics to the
investigation of the paths of cannonballs (work
which was later validated by Galileo’s studies on
falling bodies)
Even today, the solution to cubic equations is usually
known as Cardano’s Formula and not Tartgalia’s.

Gerolamo Cardano
a celebrated Italian Renaissance mathematician,
physician, astrologer and gambler.
 (or Cardan), a rather eccentric and confrontational
mathematician
doctor and Renaissance man, and author throughout
his lifetime of some 131 books.

Gerolamo Cardano
 Published himself in his 1545 book “Ars Magna”
(despite having promised Tartaglia that he would
not), Tartaglia’s cubic solution along with the work
of his own brilliant student Lodovico Ferrari
Even today, the solution to cubic equations is
usually known as Cardano’s Formula and not
Tartgalia’s.

Gerolamo Cardano
 an accomplished gambler and chess player, wrote a book
called “Liber de ludo aleae” (“Book on Games of
Chance“) when he was just 25 years old, which contains
perhaps the first systematic treatment of probability (as
well as a section on effective cheating methods).
 The book described the insight that, if a random event
has several equally likely outcomes, the chance of any
individual outcome is equal to the proportion of that
outcome to all possible outcomes.

Gerolamo Cardano
His “Liber de ludo aleae” remained unpublished
until 1663, nearly a century after his death.
It was the only serious work on probability
until Pascal‘s work in the 17th Century
 was also the first to describe hypocycloids, the
pointed plane curves generated by the trace of a
fixed point on a small circle that rolls within a larger
circle

Gerolamo Cardano
 The generating circles were later named Cardano
(or Cardanic) circles.
 colourful life remained notoriously short of money
thoughout his life, largely due to his gambling habits
 was accused of heresy in 1570 after publishing a
horoscope of Jesus (apparently, his own son
contributed to the prosecution, bribed by Tartaglia).

 an Italian mathematician famed for solving the
quartic equation.
 was born in 1522 in Bologna
 at the age of 14, became the servant of Gerolamo
Cardan.
Lodovico Ferrari

Lodovico Ferrari
 obtained a prestigious teaching post while still in his
teens after Cardano resigned from it and
recommended him, and was eventually able to retired
young and quite rich, despite having started out as
Cardano’s servant.
 Based on Tartaglia's formula, he and Cardan found
proofs for all cases of the cubic and, more impressively,
solved the quartic equation - this was reportedly
largely due to his work.

Lodovico Ferrari
 Won by default on a public debate with Tartaglia
when the later did not show up to the challenge
which he initially accepted
 After the win, fame soared and he was
inundated with offers of employment, including a
request from the emperor.

Lodovico Ferrari
 was appointed tax assessor to the governor of Milan
 after transferring to the service of the church,
retired as a young (aged 42) and rich man.
 He moved back to his home town of Bologna and in
with his widowed sister Maddalena.

Lodovico Ferrari
 Died in 1565 of white arsenic poisoning, most
likely administered by Maddalena.
 Maddalena did not grieve at his funeral and having
inherited his fortune, remarried two weeks later.
 Her new husband promptly left her with all her
fortune and she died in poverty.

 an unprecedented explosion of mathematical and
scientific ideas across Europe
 a period sometimes called the Age of Reason
 Hard on the heels of the “Copernican Revolution” of
Nicolaus Copernicus in the 16th Century, scientists
like Galileo Galilei, Tycho Brahe and Johannes Kepler
were making equally revolutionary discoveries in the
exploration of the Solar system, leading to Kepler’s
formulation of mathematical laws of planetary motion.

Significant Names
John Napier
Pierre de Fermat
René Descartes
Blaise Pascal
Isaac Newton
Gottfried Wilhelm
Leibniz

Significant Mathematical Contributions
 also spelled Neper, (born 1550, Merchiston Castle,
near Edinburgh, Scotland
 died April 4, 1617, Merchiston Castle)
 Scottish mathematician and theological writer who
originated the concept of logarithms as a
mathematical device to aid in calculations
John Napier

John Napier
His logarithm contributed to the advance of science,
astronomy and mathematics by making some
difficult calculations relatively easy
It was one of the most significant mathematical
developments of the age, and 17th Century
physicists like Kepler and Newton could never
have performed the complex calculatons needed for
their innovations without it

John Napier
 The logarithm of a number is the exponent when that
number is expressed as a power of 10 (or any other
base). It is effectively the inverse of exponentiation
 1622 William Oughted had produced a logarithmic
slide rule, an instrument which became indispensible
in technological innovation for the next 300 years.

John Napier
Improved Simon Stevin’s decimal notation and
popularized the use of the decimal point
and made lattice multiplication (originally developed
by the Persian mathematician Al-Khwarizmi and
introduced into Europe by Fibonacci) more
convenient with the introduction of “Napier’s Bones”,
a multiplication tool using a set of numbered rods.

RENé DESCARTES
 Sometimes considered the first of the modern school
of mathematics.
His development of analytic geometry and Cartesian
coordinates in the mid-17th Century soon allowed
the orbits of the planets to be plotted on a graph, as
well as laying the foundations for the later
development of calculus (and much later multi-
dimensional geometry).

RENé DESCARTES
 also credited with the first use of superscripts for powers
or exponents.
As a young man, he found employment for a time as a
soldier
After a series of dreams or visions, and after meeting
the Dutch philosopher and scientist Isaac Beeckman,
who sparked his interest in mathematics and the New
Physics, he concluded that his real path in life was the
pursuit of true wisdom and science.

RENé DESCARTES
 In France, as a young man he came to the conclusion
that the key to philosophy, with all its uncertainties and
ambiguity, was to build it on the indisputable facts of
mathematics.
 He moved from the restrictions of Catholic France to the
more liberal environment of the Netherlands, where he
spent most of his adult life, and where he worked on his
dream of merging algebra and geometry.

RENé DESCARTES
 In 1637, he published his ground-breaking
philosophical and mathematical treatise “Discours de
la méthode” (the “Discourse on Method”)
 one of its appendices in particular, “La Géométrie”, is
now considered a landmark in the history of
mathematics

RENé DESCARTES
introduced what has become known as the standard
algebraic notation, using lowercase a, b and c for
known quantities and x, y and z for unknown
quantities.
the first book to look like a modern mathematics
textbook, full of a‘s and b‘s, x2‘s, etc.

RENé DESCARTES
 In “La Géométrie” he first proposed that each point in two
dimensions can be described by two numbers on a plane,
one giving the point’s horizontal location and the other
the vertical location, which have come to be known as
Cartesian coordinates.
 He used perpendicular lines (or axes), crossing at a point
called the origin, to measure the horizontal (x) and
vertical (y) locations, both positive and negative, thus
effectively dividing the plane up into four quadrants.

RENé DESCARTES
Descartes’ ground-breaking work, usually referred
to as analytic geometry or Cartesian geometry, had
the effect of allowing the conversion of geometry
into algebra (and vice versa).
Thus, a pair of simultaneous equations could now be
solved either algebraically or graphically (at the
intersection of two lines).

RENé DESCARTES
 It allowed the development of Newton’s
and Leibniz’s subsequent discoveries of calculus.
 It also unlocked the possibility of navigating
geometries of higher dimensions, impossible to
physically visualize – a concept which was to become
central to modern technology and physics – thus
transforming mathematics forever.

RENé DESCARTES
 He also developed a “rule of signs” technique for
determining the number of positive or negative real roots
of a polynomial; “invented” (or at least popularized) the
superscript notation for showing powers or exponents
(e.g. 24 to show 2 x 2 x 2 x 2)
and re-discovered Thabit ibn Qurra’s general formula for
amicable numbers, as well as the amicable pair
9,363,584 and 9,437,056 (which had also been
discovered by another Islamic mathematician, Yazdi,
almost a century earlier).

RENé DESCARTES
He is perhaps best known today as a philosopher who
espoused rationalism and dualism.
His philosophy consisted of a method of doubting
everything, then rebuilding knowledge from the
ground, and he is particularly known for the often-
quoted statement “Cogito ergo sum”(“I think,
therefore I am”).

RENé DESCARTES
 Had an influential rôle in the development of modern
physics, a rôle which has been, until quite recently,
generally under-appreciated and under-investigated.
 Provided the first distinctly modern formulation of
laws of nature and a conservation principle of
motion, made numerous advances in optics and the
study of the reflection and refraction of light, and
constructed what would become the most popular
theory of planetary motion of the late 17th Century.

RENé DESCARTES
 His commitment to the scientific method was met
with strident opposition by the church officials of the
day.
 His revolutionary ideas made him a centre of
controversy in his day.
 Died in 1650 far from home in Stockholm, Sweden.
 13 years later, his works were placed on the
Catholic Church’s “Index of Prohibited Books”.

PIERRE DE FERMAT
born August 17, 1601, Beaumont-de-Lomagne, France
died January 12, 1665, Castres
French mathematician who is often called the founder
of the modern theory of numbers.
Together with Rene Descartes, he was one of the two
leading mathematicians of the first half of the 17th
century

PIERRE DE FERMAT
 Independently of Descartes, Fermat discovered the
fundamental principle of analytic geometry.
 His methods for finding tangents to curves and their
maximum and minimum points led him to be
regarded as the inventor of the differential calculus.
 Through his correspondence with Blaise Pascal he
was a co-founder of the theory of probability.

PIERRE DE FERMAT
 One example of his many theorems is the Two
Square Theorem, which shows that any prime
number which, when divided by 4, leaves a
remainder of 1 (i.e. can be written in the form 4n + 1),
can always be re-written as the sum of two square
numbers.

PIERRE DE FERMAT
 His so-called Little Theorem is often used in the testing of
large prime numbers, and is the basis of the codes which
protect our credit cards in Internet transactions today.
 In simple (sic) terms, it says that if we have two
numbers a and p, where p is a prime number and not a factor
of a, then a multiplied by itself p-1 times and then divided
by p, will always leave a remainder of 1. In mathematical
terms, this is written: ap-1 = 1(mod p). For example, if a = 7
and p = 3, then 72 ÷ 3 should leave a remainder of 1, and 49 ÷
3 does in fact leave a remainder of 1.

PIERRE DE FERMAT
He identified a subset of numbers, now known as Fermat
numbers, which are of the form of one less than 2 to the
power of a power of 2, or, written mathematically, 22n + 1. The
first five such numbers are: 21 + 1 = 3; 22 + 1 = 5; 24 + 1 = 17;
28 + 1 = 257; and 216 + 1 = 65,537.
Interestingly, these are all prime numbers (and are known as
Fermat primes), but all the higher Fermat numbers which
have been painstakingly identified over the years are NOT
prime numbers, which just goes to to show the value of
inductive proof in mathematics.

PIERRE DE FERMAT
 Fermat’s pièce de résistance, though, was his famous
Last Theorem, a conjecture left unproven at his death,
and which puzzled mathematicians for over 350 years
 It states that no three positive integers a, b and c can
satisfy the equation an + bn = cn for any integer value
of n greater than two (i.e. squared). This seemingly
simple conjecture has proved to be one of the world’s
hardest mathematical problems to prove.

PIERRE DE FERMAT
 Over the centuries, several mathematical and scientific
academies offered substantial prizes for a proof of the
theorem, and to some extent it single-handedly stimulated
the development of algebraic number theory in the 19th
and 20th Centuries.
 It was finally proved for ALL numbers only in 1995 (a
proof usually attributed to British mathematician Andrew
Wiles, although in reality it was a joint effort of several
steps involving many mathematicians over several years).

The final proof made use of complex modern
mathematics, such as the modularity theorem for
semi-stable elliptic curves, Galois representations and
Ribet’s epsilon theorem, all of which were unavailable
in Fermat’s time, so it seems clear that Fermat’s claim
to have solved his last theorem was almost certainly
an exaggeration (or at least a misunderstanding).
PIERRE DE FERMAT

PIERRE DE FERMAT
 In addition to his work in number theory, he
anticipated the development of calculus to some
extent, and his work in this field was invaluable later
to Newton and Leibniz.
 While investigating a technique for finding the centres
of gravity of various plane and solid figures, he
developed a method for determining maxima, minima
and tangents to various curves that was essentially
equivalent to differentiation.

 Also, using an ingenious trick, he was able to reduce
the integral of general power functions to the sums of
geometric series
 Fermat’s correspondence with his friend Pascal also
helped mathematicians grasp a very important
concept in basic probability which, although perhaps
intuitive to us now, was revolutionary in 1654, namely
the idea of equally probable outcomes and expected
values.
PIERRE DE FERMAT

BLAISE PASCAL
A Frenchman who was a prominent 17th Century
scientist, philosopher and mathematician.
a child prodigy and pursued many different avenues of
intellectual endeavour throughout his life.

BLAISE PASCAL
Much of his early work was in the area of natural and
applied sciences, and he has a physical law named after him
(that “pressure exerted anywhere in a confined liquid is
transmitted equally and undiminished in all directions
throughout the liquid”), as well as the international unit for
the meaurement of pressure.
In philosophy, Pascals’ Wager is his pragmatic approach to
believing in God on the grounds that is it is a better “bet”
than not to.

BLAISE PASCAL
a mathematician of the first order.
At the age of sixteen, wrote a significant treatise on the
subject of projective geometry, known as Pascal’s
Theorem, which states that, if a hexagon is inscribed in
a circle, then the three intersection points of opposite
sides lie on a single line, called the Pascal line.
 As a young man, he built a functional calculating
machine, able to perform additions and subtractions, to
help his father with his tax calculations

BLAISE PASCAL
 Best known for Pascal’s Triangle, a
convenient tabular presentation of binomial co-
efficients, where each number is the sum of the two
numbers directly above it.
 The co-efficients produced when a binomial is
expanded form a symmetrical triangle.

BLAISE PASCAL
 The Persian mathematician Al-Karaji had produced
something very similar as the Pascal’s Triangle as early as
the 10th Century, and the Triangle is called Yang Hui’s
Triangle in China after the 13th Century Chinese
mathematician, and Tartaglia’s Triangle in Italy after the
eponymous 16th Century Italian.
 But Pascal did contribute an elegant proof by defining the
numbers by recursion, and he also discovered many useful
and interesting patterns among the rows, columns and
diagonals of the array of numbers.

BLAISE PASCAL
For instance, looking at the diagonals alone, after the
outside “skin” of 1’s, the next diagonal (1, 2, 3, 4, 5,…)
is the natural numbers in order. The next diagonal
within that (1, 3, 6, 10, 15,…) is the triangular
numbers in order. The next (1, 4, 10, 20, 35,…) is the
pyramidal triangular numbers, etc, etc.
It is also possible to find prime numbers, Fibonacci
numbers, Catalan numbers, and many other series,
and even to find fractal patterns within it.

BLAISE PASCAL
 It fell to Pascal (with Fermat‘s help) to bring together
the separate threads of prior knowledge
(including Cardano‘s early work) and to introduce
entirely new mathematical techniques for the solution
of problems that had hitherto resisted solution.

 Two such intransigent problems which Pascal
and Fermat applied themselves to were the Gambler’s
Ruin (determining the chances of winning for each of two
men playing a particular dice game with very specific rules)
and the Problem of Points (determining how a game’s
winnings should be divided between two equally skilled
players if the game was ended prematurely).
 His work on the Problem of Points in particular, although
unpublished at the time, was highly influential in the
unfolding new field.
BLAISE PASCAL

 Later in life, he and his sister Jacqueline strongly
identified with the extreme Catholic religious movement
of Jansenism.
 Following the death of his father and a “mystical
experience” in late 1654, he had his “second conversion”
and abandoned his scientific work completely, devoting
himself to philosophy and theology.
BLAISE PASCAL

 His two most famous works, the “Lettres provinciales”
and the “Pensées“, date from this period, the latter left
incomplete at his death in 1662.
 They remain Pascal’s best known legacy
 usually remembered today as one of the most
important authors of the French Classical Period
 one of the greatest masters of French prose, much more
than for his contributions to mathematics.
BLAISE PASCAL

ISAAC NEWTON
 Born December 25, 1642 [January 4, 1643, New
Style], Woolsthorpe, Lincolnshire, England
 Died March 20 [March 31], 1727, London
 Physicist, mathematician, astronomer, natural
philosopher, alchemist and theologian

ISAAC NEWTON
 considered by many to be one of the most influential
men in human history.
 His 1687 publication, the “Philosophiae Naturalis
Principia Mathematica” (usually called simply the
“Principia”), is considered to be among the most
influential books in the history of science, and it
dominated the scientific view of the physical
universe for the next three centuries.

ISAAC NEWTON
 A giant in the minds of mathematicians everywhere (on a
par with the all-time greats like Archimedes and Gauss)
 greatly influenced the subsequent path of mathematical
development.
 Over two miraculous years, during the time of the Great
Plague of 1665-6, he developed a new theory of light,
discovered and quantified gravitation, and pioneered a
revolutionary new approach to mathematics: infinitesimal
calculus.

ISAAC NEWTON
His theory of calculus was built on earlier work by
his fellow Englishmen.
Calculus allowed mathematicians and engineers to
make sense of the motion and dynamic change in
the changing world around us, such as the orbits of
planets, the motion of fluids, etc

ISAAC NEWTON
Without going into too much complicated detail, he
(and his contemporary Gottfried
Leibniz independently) calculated a derivative
function f ‘(x) which gives the slope at any point of a
function f(x).
This process of calculating the slope or derivative of a
curve or function is called differential calculus or
differentiation (or, in Newton’s terminology, the
“method of fluxions”

ISAAC NEWTON
 instantaneous rate of change at a particular point on a
curve the “fluxion”
 the changing values of x and y the “fluents”.
His Fundamental Theorem of Calculus states that
differentiation and integration are inverse operations,
so that, if a function is first integrated and then
differentiated (or vice versa), the original function is
retrieved.

ISAAC NEWTON
Newton chose not to publish his revolutionary
mathematics straight away, worried about being
ridiculed for his unconventional ideas, and contented
himself with circulating his thoughts among friends.
in 1684, the German Leibniz published his own
independent version of the theory, whereas Newton
published nothing on the subject until 1693.

ISAAC NEWTON
 the Royal Society, after due deliberation, gave credit
for the first discovery to Newton (and credit for the
first publication to Leibniz)
 when it was made public that the Royal Society’s
subsequent accusation of plagiarism
against Leibniz was actually authored by none other
Newton himself, something of a scandal arose
causing an ongoing controversy which marred the
careers of both men.

ISAAC NEWTON
 credited with the generalized binomial theorem,
which describes the algebraic expansion of powers of
a binomial (an algebraic expression with two terms,
such as a2 – b2)
 made substantial contributions to the theory of finite
differences (mathematical expressions of the
form f(x + b) – f(x + a))

ISAAC NEWTON
 one of the first to use fractional exponents and
coordinate geometry to derive solutions to
Diophantine equations (algebraic equations with
integer-only variables)
 developed the so-called “Newton’s method” for
finding successively better approximations to the
zeroes or roots of a function; he was the first to use
infinite power series with any confidence; etc

ISAAC NEWTON
published his “Principia” or “The Mathematical
Principles of Natural Philosophy” in 1687
generally recognized as the greatest scientific book
ever written.
In it, he presented his theories of motion, gravity and
mechanics, explained the eccentric orbits of comets,
the tides and their variations, the precession of the
Earth’s axis and the motion of the Moon.

ISAAC NEWTON
 Later in life, he wrote a number of religious tracts
dealing with the literal interpretation of the Bible
 devoted a great deal of time to alchemy
 acted as Member of Parliament for some years
 became perhaps the best-known Master of the Royal
Mint in 1699, a position he held until his death in
1727.

ISAAC NEWTON
 In 1703, he was made President of the Royal Society
 in 1705, became the first scientist ever to be
knighted.
 Mercury poisoning from his alchemical pursuits
perhaps explained Newton’s eccentricity in later
life, and possibly also his eventual death.

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