M. C. Escher, The Horsman Great Minds
M. C. Escher, Stars
I try in my prints to testify that we live in a beautiful and orderly world, not in a chaos without norms, even though that is how it sometimes appears. My subjects are often playful; I cannot refrain from demonstrating the nonsensicalness of some of what we take for irrefutable certainties. It is, for example, a pleasure to deliberately mix together objects of two and of three dimensions, surface and spatial relationships, and to make fun of gravity.
Maurits Cornelious Escher (1898-1972)
3D surface 13D surface 33D surface 33D surface 23D surface
3D surface 33D surface 23D surface 13D surface 03D surface  1
3D surface 03D surface 13D surface 23D surface 33D surface 1
3D surface 03D surface3D surface 23D surface 33D surface 2
Suppose the world is bounded by a large sphere of radius R and the absolute temperature at a point within the sphere is R2 - r2, where r is the distance from the center of the sphere; suppose also that the dimensions of objects change from point to point, being proportional to the temperature at any given place. To inhabitants of such a world, the universe would appear to be infinite (a man walking away from the center reduces in size in a way that he would not be able to reach the edge of the sphere).
Henri Poincaré

He was younger than Plato, older than Archimedes, taught in Alexandria, and wrote the Elements . Euclid's Elements begins with a definition, "A point is that which has no part" and marches throughout 13 Books and 465 Propositions with no motivation. It is certainly one of the greatest and dullest books ever written; it ought to have been a student's and teacher's nightmare throughout the twenty-three centuries of its existence. One of the most fascinating aspects of Mathematics is that there exist statements that are both true and false. Perhaps the most famous of these is Euclid's controversial fifth postulate (two lines are parallel, it they are perpendicular to the same line).

Archimedes (Syracuse: Sicily, 287-212 BC) greatest contributions were in geometry. His methods anticipated the integral calculus 2,000 years before Newton and Leibniz. He calculated Pi to be 3.14163 by inscribing the inside and outside of a circle, polygons of many sides (the method was used until 19 centuries later). We don't know how much of the works of Archimedes were lost. He discovered fundamental theorems concerning the center of gravity of plane figures and solids. His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes' principal. Archimedes invented a device now known as Archimedes' screw. This is a pump, still used in many parts of the world.

Ptolemy (85-165) was a Greek mathematician born in Egypt and lived most of his life in Alexandria. One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory that prevailed for 1400 years. His two major works are the Almagest and the Geography . The Almagest is the earliest of his works and gives in detail the mathematical theory of the motions of the Sun, Moon, and planets. He made his most original contribution by presenting details for the motions of each of the planets. Ptolemy devised new geometrical proofs and theorems. Using chords of a circle he approximated pi as 3.14166.

Edward Fitzgerald's popular translation in 1859 of the Rubaiyat, made Omar Khayyam (1048-1122), an outstanding Persian mathematician and astronomer famous. His triangle is known to us as Pascal's triangle. Khayyam did not have any doubt as to the validity of Euclid's fifth postulate: "If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced infinitely, meet on that side on which are the angles less than the two right angles" it means "through a point not on a given line, there passes not more than one parallel to the line". In fact, he presented his own proof of the parallel postulate which he felt avoided.

Leonardo of Pisa (Italian 1180-1250) is better known as Fibonacci, or "son of Bonaccio", was an Italian merchant. The horizontal bar in fraction, was used regularly by Fibonacci (and was known before in Arabia). The slash was suggested in 1845 by De Morgan. His rabbit problem (see part I) gave rise to the "Fibonacci sequence" 1, 1, 2, 3, 5, 8, 13, . . . , Fn, . . . , , where Fn = Fn-1 + Fn-2 (each term after the first two is the sum of the two terms immediately preceding it). Today there is a quarterly journal devoted to studying mathematics related to this sequence.

Marin Mersenne (French, 1588-1648) was aware of Euclid's proof that if 2n - 1 is prime, then 2n-1(2n - 1) is perfect (Perfect numbers are those, like 6 = 1 + 2 + 3, which equal the sum of their proper divisors). He also knew that 2n - 1 cannot be prime if n is not, so this led him to the problem of finding those prime numbers p for which 2p - 1 is also prime. ... In science, Mersenne's work on sound was of such importance that he is sometimes called "the father of acoustics." He experimented with cords and brass wires of various lengths ( some more than 10 feet) and tensions. He discovered that sound travels 1038 ft/s which is quite close to the currently accepted figure of about 1087 ft/s in dry air at 32o F.

René Descartes (French, 1596 - 1650) was educated at the Jesuit academy of La Flèche in traditional Aristotelian philosophy. While in the school he was granted permission to remain in bed until 11 o'clock in the morning, a custom he maintained until the year of his death. He wrote a treatise on science; one appendix to this work, La Géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry. In 1649 Queen Christina of Sweden persuaded Descartes to go to Stockholm. However the Queen wanted to draw tangents at 5 a.m. and Descartes broke the habit of his lifetime of getting up at 11 o'clock. After 4 months in the cold northern climate, walking to the palace for 5 o'clock every morning, he died of pneumonia .

Pierre de Fermat (French, 1601-1665) was a lawyer and jurist who studied mathematics in his spare time. Fermat showed that there is no cube that is divisible into two cubes. Going further, he stated that there are no positive integer x, y, and z, such that xn + yn = zn (n>2). This theorem is called "Fermat's Last Theorem," was found written in the margin of his copy of Diophantus' "Arithmetica." He wrote that he had a proof of the theorem, but the proof was not found in his effects. Fermat was not in possession of the limit concept, but in a treatise entitled "Method of Finding Maxima and Minima", he explained a process now called differentiation which finds the slope of the tangent line at a point on a curve. It is appropriate to follow Laplace in acclaiming Fermat as the discoverer of the differential calculus.

Étienne Pascal (famed as the discoverer of the curve "Limacon of Pascal"), had unorthodox educational views and decided to teach his son Blaise (French, 1623 - 1662) himself. At the age of 14 Blaise started to attend Mersenne's meetings and presented a single piece of paper at the age of 16. It contained a number of projective geometry theorems, including Pascal's mystic hexagon. In correspondence with Fermat he laid the foundation for the theory of probability. His most famous work in philosophy is Pensées , a collection of personal thoughts on human suffering and faith in God. If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing . He died at the age of 39 in intense pain after a malignant growth in his stomach spread to the brain.

"Nature to him was an open book, whose letters he could read without effort" -- Albert Einstein.

Sir Isaac Newton (English 1643-1727) scientific genius emerged while Newton was still under 25 years old, he began revolutionary advances in mathematics, optics, physics, and astronomy. He laid the foundation for differential and integral calculus, several years before its independent discovery by Leibniz. Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions. Newton's novel idea of 1666 was to imagine that the Earth's gravity influenced the Moon, counter- balancing its centrifugal force.

Gottfried Wilhelm Leibniz (German, 1646-1716), better known as a philosopher than as a mathematician developed the present day notation for the differential and integral calculus. A law student at 15, he obtained his doctorate in law at 20. He once wrote: as soon as I began to learn logic, I was fascinated by its order and classification. My strongest pleasure lay in the categories, which seemed to me to call the roll of all the things in the world. At the age of 50, he proposed to a certain lady, but the lady asked for time to consider the matter, so Leibniz had a chance to think again, and withdrew his offer.

In the universe of Laplace, God did not play any role; the master was Leonhard Euler (1707-1783) of Switzerland with 13 children and a placid and uneventful personal life. There is considerable truth in the old saying that all calculus textbooks since 1748 are essentially copies of Euler or copies of copies of Euler. He was the first and greatest master of infinite series, infinite products, and continuous fractions. One of the most remarkable feature's of Euler's universal genius was its equal strength in both branches of mathematics, the continuous and the discrete. Euler showed that an imaginary power of an imaginary number can be real. He was the Shakespeare of mathematics. Universal, richly detailed, and inexhaustible.

The five volumes of Mécanique Céleste by Pierre Simon de Laplace (French, 1749-1827), which summed up the work on gravitation of several generations of scientists, omitted all references and left it to be inferred that the ideas were entirely his own. Napoleon tried to get rise out of him by protesting that he had written on the system of the world without once mentioning God as the author of the universe. He replied, "Sire, I had no need of that hypothesis." After the French Revolution, his political talent and greed for position came to full flower. They spoke ironically of his "suppleness" and "versatility" as a politician; smoothly adapted himself by changing his principles; he went back and forth from republican to royalist and each time he got a better job.

Joseph Fourier (French, 1768-1830), son of a tailor and orphan at eight, received his education through the Benedictine Order, in which he at a point intended to become a priest. The chief contribution of Fourier was that any function can be represented be a series (infinite sums) of sines and cosines. His work has ever been fundamental both in physics and mathematics. Functions no longer needed to be of the well-behaved form with which mathematicians had been familiar.

It is said that Goethe wrote and directed little plays for a puppet theater when he was 6 and Mozart composed his first childish minuets when he was 5, but Carl Friedrich Gauss (1777-1855), the son of a German gardener and Bricklayer, corrected an error in his father's payroll accounts at the age of 3. His doctoral dissertation was a milestone in the history of mathematics. He proved the fundamental theorem of algebra : The existence of real or complex root for any polynomial equation with real or complex coefficients. Apart from science, his main interests were history and world literature, international politics, and public finance. He owned a large library of about 6000 volumes in more than seven different languages.

An arrogant royalist, self-righteous and narrow-minded bigot, once a week appeared before the Academy to present a new paper. The Academy, largely on his account, was obliged to introduce a rule restricting the number of articles a member could request to be published in a year. This great French mathematician known as a smug hypocrite by his fellow scientists is known to us by the name of Augustine Louis Cauchy (1789-1857). His collected works fill 27 fat volume. His greater achievements lay in the field of analysis. Together with his contemporaries Gauss and Abel, he was a pioneer in the rigorous treatment of limits, continuous functions, derivative, integrals, and infinite series.

Niels Henrik Abel (Norwegian 1802-1829) solved a problem that had puzzled mathematicians for centuries: he showed that a general fifth degree polynomial equation cannot be solved algebraically, that is, in terms of radicals. Abel's contemporary, Evariste Galois (French, 1811-1832) , then proved that it was impossible to solve any general polynomial equation of degree greater than four in an algebraic manner. Every student of analysis encounters Abel's integral equation, Abel's convergence test and Abel's theorem on power series. A commutative group is called abelian group.

Gustave Lejeune Dirichlet (German, 1805-1859) was Gauss's student and his successor, at Götingen. Fluent in both German and French, Dirichlet served admirably as a liaison between the mathematics and mathematicians of the two countries. On July 16, 1849, exactly 50 years after the awarding to Gauss of his doctorate, Gauss enjoyed the celebration of his golden jubilee. As part of the show, Gauss tried to light his pipe with a piece of the original manuscript of his "Disquisitiones arithmeticae." Dirichlet who was present, was appalled at what seemed to him a sacrilege. He rescued the paper from Gauss's hand and treasured the memento the rest of his life; it was found by his editors among his papers after he died.

Charles Hermite (French, 1822-1901) never evoked a concrete image; he disliked geometry and was attracted to number theory. He discovered that the number e=2.7182... is transcendental. This discovery excited the mathematics society of the time, and was asked to show that pi also comes from the same family. He refused by saying " I shall risk nothing on an attempt to prove the transcendence of the number pi." Several of his purely mathematical discoveries were used in Physics. For example, The Hermitian forms and matrices were used for Heisenberg's 1925 formulation of quantum mechanics; his polynomials and functions are useful in solving Schrödinger's wave equation.

Eight years of poverty made Georg Friedrich Bernhard Riemann (German, 1826-1866) write his greatest works. He proved that it is possible to rearrange the terms of some (conditionally convergent) series and obtain an arbitrary sum. He clarified the notion of integral by defining what we now call the Riemann integral. He was an original thinker and a host of methods, theorems and concepts are named after him. In his trial lecture for an unpaid lectureship, Gauss made him lecture on the foundation of geometry, a subject on which he was unprepared. Once a full professor, his years of poverty were over, but his health was broken. He died of tuberculosis at the age of 39.

George Ferdinand Ludwig Philip Cantor was born of Danish parents in St. Petersburg in 1845, and moved with his parents to Frankfurt, in 1856. He exerted a considerable influence on much of the mathematics of present times. He founded set theory and introduced the concept of infinite numbers with his discovery of cardinal numbers. His early interests were in number theory and trigonometric series. He defined irrational numbers in terms of convergent sequences of rational numbers. In 1873 he proved the rational numbers are countable. Cantor's work was attacked by many mathematicians, the attack being led by Cantor's own teacher Kronecker. Today, Cantor's set theory has penetrated into almost every branch of mathematics. He died in a psychiatric clinic in Halle in 1918.

Henri Jules Poincaré (French, 1860-1912) can be said to have been the originator of algebraic topology and of the theory of analytic functions of several complex variables. In the field of celestial mechanics he studied the three-body-problem, and theories of light and electromagnetic waves. He is acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity. Like Laplace, Poincaré contributed noteworthy to the subject of probability. He had no ability whatever in drawing, and he earned a flat score of zero in the subject in school. At the end of the school year, his classmates jokingly organized a public exhibition of his artistic masterpieces. They labeled each item in Greek "This is a house", "This is a horse", and so on.

David Hilbert (German, 1862-1943) is considered one of the greatest mathematicians of recent times. His work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analyzed their significance. He contributed to many areas of mathematics including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.




Calculus with Anatytic Geometry, George F. Simmons; McGraw-Hill
A History of Mathematics, Carl B. Boyer; Princeton University Press
An Introduction to the History of Mathematics, Howard Eves; Holt, rinehart and Winston
MacTutor History of Mathematics archive

images: from these great minds and

M.C. Escher: The Work of M.C. Escher
Animated Ufo and the star knot:
Odd Knots
Saturn: Math Art Gallery
The Elastic Fractal: The Spanky Fractal Database
The Pavilion of Polyhedreality
Surface and Knot: The Geometry Center
The pi snake:
Pi Page
Ufo or the bright star
The Fractal Microscope
Fishes: Soap Films on Knots
The surface: Differential Geometry and Quantum Physics
Fractal trees:
Flying cigarets and the river: The CFM Home Page at Brown University
Black Hole: Black Hole Embedding Diagram Showing Space Warpage
The Cow: Mathematica
The Bone:
Fractal Analysis of Trabecular Bone
Viking ship:MATLAB Gallery
Tangrams: Massoud Malek
Note: Most images were altered.

Mathematics on the InternetHome