17th Century Mathematics

The Mechanical World???

“they understood nature better because it was now mechanical;

and mechanics better because it was now mathematical”

ACHIEVEMENTS

SCIENCE REVOLUTION

MODERN NOTATION

LOGARITHM

INDIVISIBLES

PROJECTIVE

GEOMETRY

CALCULUS

PROBABILITY

ANALYTIC GEOMETRY

Inventions 1608 – telescope by Hans Lippershey

(Denmark) 1609 – thermostat by Cornelius Drebbel

(Britain) 1631 – multiplication sign x is first used. 1644 – barometer by Evangelista

Torrecelli (Italy) 1650 – air pump by Otto von Guericke 1660 – Royal Society founded in London 1668 – reflecting telescope by Newton 1680 – clocks have minute hand

Rene Descartes (1596 – 1650) “cogito, ergo sum”

born March 31, 1596 in Touraine, France

Father of Modern PhilosophyFather of Analytic GeometryTIMELINE1604 – enrolled in the Jesuit

School1612 – left school and followed

the usual path of a young man of wealth going to Paris

to taste the pleasures of its social life

1617 – joined the army as gentleman volunteer in the Dutch Republic

1619 – his future career as a mathematician and philosopher was revealed to him

Mathematical Legacy>Descartes' theory provided the basis for the

calculus of Newton and Leibniz, by applying infinitesimal calculus to the tangent line problem

>He created analytic geometry>Descartes rule of signs which is commonly

used to determine the negative and positive roots of a polynomial.

>discovered an early form of the law of conservation of momentum

>invented the notation which uses superscript to show the powers or exponents

>introduced the algebraic notation still in used today.

>Theory of Vortices - the solar system was filled with vaguely defined primordial matter, the so-called plenum or ether. The motion of the stellar mass set up an immense whirlpool or vortex, in the middle of which was the sun.

He died in Denmark on February 11, 1650 at the age of 53

    Did you know . . .? The man who invented analytic geometry,

never got out of bed before 11 in the morning!

Descartes never married, but he lived with a Dutch woman for several years and they had an illegitimate child

named Francine who died at age of 5. Descartes wrote to a friend that her death brought him the deepest sorrow he had ever known.

SIR ISAAC NEWTON (1642 – 1727)

Born: December 25, 1642 at LincolnshireDied: March 31,1727 (aged 84) at Kensington.Residence: England Citizenship: English Nationality: English Fields: Physics, Mathematics, Astronomy, Natural Philosophy, Alchemy, TheologyInstitutions : University of Cambridge

Royal Society Royal Mint

Alma mater: Trinity College, Cambridge

MAJOR CONTRIBUTIONS

In MECHANICS-Conception of the Law of Universal

Gravitation -Laws of motion

-Principles of conservation of both momentum and angular momentum.

In OPTICS-he built the first practical reflecting

telescope

-developed a theory of color based on the observation that a prism decomposes white light into the many colors which form the visible spectrum

-He also formulated an empirical law of cooling and studied the speed of sound.

In MATHEMATICS

- Method of Fluxions and FluentsKNOWN Today as differenTIal and integral

calculus. Applications: determine maxima and minima, tangents to curves, curvature of curves, points of inflection, and convexity and concavity of curves- The Generalized Binomial Theorem

- He contributed to the study of power series.

Newton’s Apples???“If I have seen farther than others, it is because I have stood on the

shoulders of giants.”

Gottfried Wilhelm LeibnizBorn in Leipzig, Germany in

1646 and studied law graduating with a bachelor’s degree at age 17.

After earning a doctorate in law, Leibniz entered a diplomatic service and spent most of his life traveling to the capitals of Europe on political missions.

1672- begins his serious study in Math while he was on his diplomatic mission to Paris.

There he built a calculating machine that can add, subtract, multiply, divide and extract square roots.

1665 – 1667: Newton’s golden years1672 – 1675: Leibniz’ fruitful years

ACCOMPLISHMENTS-Realization that coefficients of the system of

linear equations could be arranged into an array now called matrix, later called Gaussian elimination.

-introduced the dx and dy as differences between successive values of these sequences

- Introduced the rules d(x+y)= dx + dy and d(xy)= xdy + ydx

-find the differential of a quotient z=x/y-interpreted the term d∫y as area-The rule n(xn) = nxn-1

Though Newton invented calculus first, almost all of the notation that we use for calculus today is Leibniz's.

Leibniz also chose many of the terms that we use for graphing. He was the first person to use the word "coordinates," and he used the term "axes of coordinates." Leibniz also invented the terms "function" and "variable."

Guilty of plagiarism??? During the 18th

century the prevalent opinion was against Leibniz guilty of plagiarism but today the majority of writers incline to think that the inventions were independent.

Mathematics may not teach us to breathe oxygen and exhale carbon dioxide, or to love a friend and to forgive an enemy; but it gives us every reason to hope that every problem has a solution…<gudnight.//>

Galileo Galilei (1564 – 1642) – his name is associated with events of profound significance:

- - with the birth of modern science

- - with the Copernican revolution- (heliocentric theory)

- - with the dethronement of Aristotle as the supreme authority in schools

- - with the struggle against the external restrictions on scientific inquiry

William Oughtred introduced the symbol x for multiplication.Leibniz used a period/dot

> or < by Thomas Harriot

the use of the letters x, y, z for unknown quantities, and a, b, c for known &

successive powers of x denoted by exponents {Descartes}

3 Stages in the Development of Algebraic Notation:Rhetorical – statements and equations were written out in

ordinary language.Syncopated – familiar terms were abbreviated.Symbolic – every part of expression was characterized by

an ad hoc symbol.

Invention of Logarithm

-”the invention of logarithms saved astronomers a lot of troubles and doubled their lives” {Laplace}

-coined by John Napier - scientist and mathematicians used logarithms until the invention of calculator and computer

Napier’s Bones

Method of Indivisibles –area is made up of segments which are indivisibles and solid consists of areas which are indivisibles too.

-became the base of conception of definite integral.

Projective geometry- a kind of geometry which mathematicians study unchanging property to be drawn by projection.

Ex. A circle is change by projection into conic sections.

But the projection geometry was so difficult and different nature from former geometrical conceptions.  Thus; it wasn't acknowledged for 150 years.   

Beginning of Probability-started with the correspondence of Pascal and

Fermat in the problem of gambling.-Mathematicians were interested in distributing

gambling money.

Appearance of Analytic Geometry-while Desargues and Pascal were opening the new

field of projective geometry, Descartes and Fermat were conceiving ideas of modern analytic geometry.

-The projective geometry a ‘branch’ of geometry whereas the analytic geometry a ‘method’ of geometry.

-it combines algebra with geometry by introducing coordinates.

DESCARTES’ RULE OF SIGNS1. the number of positive roots of an

equation is either equal to the number of variations of sign or it is less than that number by an even integer.

2. The number of negative roots is either equal to the number of variations of sign in f(-x) or it is less than thE number by an even integer.

THUS; the largest number of + roots added to the largest possible number of – roots gives a sum equal or less than the degree of equation.

Descartes was prepared to recognized the existence of imaginary roots to fill out the number.