Department of Mathematics, Physics and Statistics
VISAYAS
STATE UNIVERSITY
Visca,
Baybay City, Leyte
WRITTEN REPORT
IN
HISTORY OF MATHEMATICS
(Math 105)
Submitted by:
Apas, Aluna Mae
Espinosa, Ian Mae
Racal, Joanna Crystel
Taneo, Marifie
Cagasan, Roberto
Submitted To:
Ms.
Irish S. Coco
Instructor
THE 17th CENTURY MATHEMATICS
In the wake of the Renaissance, the
17th Century saw an unprecedented explosion of mathematical and scientific
ideas across Europe, a period sometimes called the Age of Reason.
.The invention of the logarithm in
the early 17th Century by John Napier (and later improved by Napier and Henry
Briggs) 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.
PROJECTIVE GEOMETRY
What
is projection
Ø Development of Projective Geometry
Ø The study of the
"properties of figures which remain unaltered (invariant) in
projection."
Ø branch of mathematics that deals with the relationships between geometric
figures and the images, or mappings, that result from projecting them onto
another surface.
Ø Study of
geometry without measurement, just the study of how points align with each
other.
v The process of projecting an
image onto a screen or other surface for viewing.
v The image of a geometric
figure reproduced on a line, plane, or surface.
v Look at a checker board head
on. All of the lines are parallel. Turn that same board at an angle keeping
your perspective the same and what you see is quite different. The lines are no
longer parallel. From a geometrical standpoint, what you are seeing is a
projection of the lines of the checkerboard onto another plane.
CROSS-RATIO
v It is a numerical quantity
preserved by projection. It is the “ratio of ratios” of lengths.
v a fundamental quantity in projective geometry, and is comparable to the
notion of "distance" in traditional geometry, which is invariant
under projection.
Desargues (1591 - 1661)
v considered
by many to be the true founder of Projective Geometry.
v Desargues was an engineer and architect who became interested in the
concept of projection.
v His most notable work, however was the Brouillon projet which was
published in 1639. This essay dealt with conics.
DESARGUES’ THEOREM
Two
triangles are in perspective axially
if and only if they are in perspective centrally.
HISTORY
v Desargues
never published this theorem, but it appeared in an appendix entitled Universal
Method of M. Desargues for Using Perspective (Maniére universelle de M.
Desargues pour practiquer la perspective) of a practical book on the use of
perspective published in 1648 by his
friend and pupil Abraham Bosse (1602 – 1676).
Blaise pascal
v He focused on simplifying the properties of conic sections.
v Pascal produced an essay that unfortunately was lost but was read by
Leibniz who called it, "so brilliant that he could not believe it was
written by so young a man."
Pascal's Theorem
v If a hexagon is inscribed in a conic, the three points of intersection of
the pairs of opposite sides lie on one line."
Philippe de la hire
v was also heavily influenced by Desargues and strongly interested in
projective geometry. He is most noted for his work, Sectiones Conicae
("Conic Sections"). This work dealt entirely with projective
geometry. La Hire believed strongly that the methods of projection were
stronger by far than Appollonios' methods. He thus attempted to prove all 364
of Appolonios' theorems. He came very close to this goal, proving 300 of them.
Number theory
A branch
of pure mathematics devoted primarily to the study
of the integers.Study prime numbers as well as the properties of
objects made out of integers (e.g., rational numbers) or defined as generalizations of
the integers (e.g., algebraic integers).
v Arithmetic was the old term of number theory. By the 12th
century it was superseded by number theory.
v In particular, arithmetical is preferred as an adjective to
number-theoritic.
Pierre de Fermat
q A french mathematician, who is often called the “founder of modern
theory of numbers.”
q One of the leading mathematician
of the first half of 17th century.
Great Works
ž Given the credit for early development that led to modern calculus, and
the progress in probability theory.
ž Discovered the fundamental principle of analytic geometry.
ž Inventor of “Differential Calculus”.
ž Evaluated the integral power functions.
Theorems
Two Square Theorem
“states that any odd prime number will be the
sum of two square numbers if and only if it leaves a remainder of 1 if divided
by 4”.
• This dichotomy among primes ranks as one of the landmarks of number
theory.
Fermat’s Little Theorem
“states that if p is prime
and a is any whole number, then p divides
evenly into ap − a.”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.”
Fermat’s Last Theorem
Ø His famous last theorem.
Ø A conjecture left unproven at his death and which puzzled mathematicians
for over 350 years.
Fermat’s Little Theorem
Ø Until it was proved for ALL numbers only in 1995.
Ø (a proof usually attributed to British) mathematician, Andrew Wiles.
Ø Though, it was a joint effort, of several steps involving many
mathematician over several years.
Ø The final proof made use of complex
Analytic Geometry
René
Descartes has been dubbed the "Father of Modern Philosophy", but he
was also one of the key figures in the Scientific Revolution of the 17th
Century, and is sometimes considered the first of the modern school of
mathematics.
In
1637, he published his ground-breaking philosophical and mathematical treatise
"Discours de la méthode" (the “Discourse on Method”), and one of its
appendices in particular, "La Géométrie", is now considered a
landmark in the history of mathematics. Following on from early movements
towards the use of symbolic expressions in mathematics by Diophantus, Al-Khwarizmi and
François Viète, "La Géométrie" 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. It was perhaps the first book to look like a modern mathematics
textbook, full of a's and b's, x2's, etc.
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). 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.
Although
analytic geometry was far and away Descartes’ most important contribution to
mathematics, 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).
Elementary Probability
Galileo
Galilee
Ø Wrote down some ideas about dice
games that lead to the formation of the earlier parts of probability theory.
History
Ø Before the theory of probability was formed Gambling was popular.
Gamblers were crafty enough to figure simple laws of probability by witnessing
the events at first hand.
Pascal
and Fermat
Ø A gambler dispute in 1654 led to the creation of a Mathematical theory
of probability by these two mathematician.
Chevalier
de Méré
Ø A French nobleman interested in gaming and gambling questions.
Ø This problem posed by de Méré led to an exchange of letters between
Pascal and Fermat in which the fundamental principles of probability theory
were formulated for the first time.
Christian
Huygens
Ø A Dutch scientist and a teacher of
Leibniz
Ø Published the first book on
probability entitled De Ratiociniis in
Ludo Aleae.
Pierre
de Laplace
Ø Introduced a host new ideas and
Mathematical techniques in his book, Theorie
Analytiquedes Probabilites.
Applied
probabilistic ideas to many scientific and practical problems.
Ø The development of probability
theory has been stimulated by the variety of its application.
Ø Mathematical statistics is one
important branch of applied probability.
Ø The ideas have been refined somewhat
and probability theory is now part of a more general discipline known as
‘measure theory’.
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