Gauss's dissertation gave the first proof of the fundamental theorem of algebra. At the age of 24 he published his theory of numbers, one of the most brilliant achievements in the history of mathematics.

A child prodigy, Gauss taught himself to read and to count by the age of three. Recognising Gauss's talent, the Duke of Brunswick in 1792 provided him with money to allow him to pursue his education. He attended Caroline College from 1792 to 1795 and at this time Gauss formulated the least-squares method and a conjecture on the distribution of primes. This conjecture was proved by Jacques Hadamard in 1896.

In 1795 Gauss went to Göttingen where he discovered the fundamental theorem of quadratic residues.

Gauss developed the concept of complex numbers and in 1799 the University of Helmstedt granted Gauss a Ph.D. for a dissertation that gave the first proof of the fundamental theorem of algebra. In his dissertation Gauss severly criticized Legendre, Laplace and other major mathematicians of the day for their lack of rigour.

At the age of 24 he published "Disquisitiones arithmeticae", his theory of numbers, one of the most brilliant achievements in the history of mathematics. The construction of regular polyhedra occur in this work as do integer congruences and the law of quadratic reciprocity.

He also calculated orbits for the minor planets Ceres and Pallas. The asteroid Ceres had been briefly observed in January 1801 but had then, after it had been tracked for 41 days, was lost in the brightness of the Sun. Gauss computed the orbit using his least squares method and correctly predicted where and when Ceres would reappear. After this he accepted a position as astronomer at the Göttingen Observatory.

In 1820 Gauss invented the heliotrope, an instrument with a movable mirror which reflected the Sun's rays. It is used in geodesy. During the late 1820s, in collaboration with the physicist Wilhelm Weber who he met while the guest of Alexander von Humboldt in Berlin, Gauss explored many areas of physics doing basic research in electricity and magnetism, mechanics, acoustics, and optics. In 1833 he constructed the first telegraph.

When in his 80th year a fellow mathematician met him and described him as follows:

... a venerable, fine old fellow, with a contented manly expression. There is an extraordinary aspect of power about him and his every word. He is about 80 years of age, but not a trace of superannuation is seen about him.

Gauss made a careful study of foreign papers in the reading room at Göttingen and in particular made a systematic study of the financial news. This stood him in very good stead since he was able to gain a considerable personal fortune through his dealings on the stock exchange. He died a very rich man.

Newton's life can be divided into three quite distinct periods. The first is his boyhood days from 1643 up to his graduation in 1669. The second period from 1669 to 1687 was the highly productive period in which he was Lucasian professor at Cambridge. The third period (nearly as long as the other two combined) saw Newton as a highly paid government official in London with little further interest in mathematics.

Isaac Newton was born in the manor house of Woolsthorpe, near Grantham in Lincolnshire. Although he was born on Christmas Day 1642, the date given on this card is the Gregorian calendar date. (The Gregorian calendar was not adopted in England until 1752.)Newton came from a family of farmers but never knew his father who died before he was born. His mother remarried, moved to a nearby village, and left him in the care of his grandmother. Upon the death of his stepfather in 1656, Newton's mother removed him from grammar school in Grantham where he had shown little promise in academic work. His school reports described him as 'idle' and 'inattentive'. An uncle decided that he should be prepared for the university, and he entered his uncle's old College, Trinity College, Cambridge, in June 1661.

Newton's aim at Cambridge was a law degree. Instruction at Cambridge was dominated by the philosophy of Aristotle but some freedom of study was allowed in the third year of study. Newton studied the philosophy of Descartes, Gassendi, and Boyle. The new algebra and analytical geometry of Viète, Descartes, and Wallis; and the mechanics of the Copernican astronomy of Galileo attracted him. Newton talent began to emerge on the arrival of Barrow to the Lucasian chair at Cambridge.

His scientific genius emerged suddenly when the plague closed the University in the summer of 1665 and he had to return to Lincolnshire. There, in a period of less than two years while Newton was still under 25 years old, he began revolutionary advances in mathematics, optics, physics, and astronomy.

While Newton remained at home he laid the foundation for differential and integral calculus several years before its independent discovery by Leibniz. The 'method of fluxions', as he termed it, was based on his crucial insight that the integration of a function is merely the inverse procedure to differentiating it. 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 their maxima and minima. Newton's "De Methodis Serierum et Fluxionum" was written in 1671 but Newton failed to get it published and it did not appear in print until John Colson produced an English translation in 1736.

Barrow resigned the Lucasian chair in 1669 recommending that Newton (still only 27 years old) be appointed in his place.

Newton's first work as Lucasian Professor was on optics. He had reached the conclusion during the two plague years that white light is not a simple entity. Every scientist since Aristotle had believed this but the chromatic aberration in a telescope lens convinced Newton otherwise. When he passed a thin beam of sunlight through a glass prism Newton noted the spectrum of colours that was formed.

Newton argued that white light is really a mixture of many different types of rays which are refracted at slightly different angles, and that each different type of ray produces a given spectral colour. Newton was led by this to the erroneous, conclusion that telescopes using refracting lenses would always suffer chromatic aberration. He therefore proposed and constructed a reflecting telescope. Newton was elected a fellow of the Royal Society in 1672 after donating a reflecting telescope.

Also in 1672 Newton published his first scientific paper on light and colour in the Philosophical Transactions of the Royal Society.

Newton's paper was well received but Hooke and Huygens objected to Newton's attempt to prove by experiment alone that light consists in the motion of small particles rather than waves. Perhaps because of Newton's already high reputation his corpuscular theory reigned until the wave theory was revived in the 19th C.

Newton's relations with Hooke deteriorated and he turned in on himself and away from the Royal Society. He delayed the publication of a full account of his optical researches until after the death of Hooke in 1703. Newton's "Opticks" appeared in 1704. It dealt with the theory of light and colour and with (i) investigations of the colours of thin sheets (ii) 'Newton's rings' and (iii) diffraction of light.

To explain some of his observations he had to use a wave theory of light in conjunction to his corpuscular theory.

Newton's greatest achievement was his work in physics and celestial mechanics, which culminated in the theory of universal gravitation. By 1666 Newton had early versions of his three laws of motion. He had also discovered the law giving the centrifugal force on a body moving uniformly in a circular path. However he did not have a correct understanding of the mechanics of circular motion.

Newton's novel idea of 1666 was to imagine that the Earth's gravity influenced the Moon, counter- balancing its centrifugal force. From his law of centrifugal force and Kepler's third law of planetary motion, Newton deduced the inverse- square law.

In 1679 Newton applied his mathematical skill to proving a conjecture of Hooke's, showing that if a body obeys Kepler's second law then the body is being acted upon by a centripetal force. This discovery showed the physical significance of Kepler's second law.

In 1684 Halley, tired of Hooke's boasting, asked Newton whether he could prove Hooke's conjecture and was told that Newton had solved the problem five years before but had now mislaid the paper. At Halley's urging Newton reproduced the proofs and expanded them into a paper on the laws of motion and problems of orbital mechanics.

Halley persuaded Newton to write a full treatment of his new physics and its application to astronomy. Over a year later (1687) Newton published the "Philosophiae naturalis principia mathematica" or "Principia" as it is always known.

The "Principia" is recognised as the greatest scientific book ever written. Newton analysed the motion of bodies in resisting and non resisting media under the action of centripetal forces. The results were applied to orbiting bodies, projectiles, pendulums, and free-fall near the Earth. He further demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalised that all heavenly bodies mutually attract one another.

Further generalisation led Newton to the law of universal gravitation:

all matter attracts all other matter with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

Newton explained a wide range of previously unrelated phenomena:- the eccentric orbits of comets; the tides and their variations; the precession of the Earth's axis; and motion of the Moon as perturbed by the gravity of the Sun.

After suffering a nervous breakdown in 1693, Newton retired from research to take up a government position in London becoming Warden of the Royal Mint (1696) and Master(1699).

In 1703 he was elected president of the Royal Society and was re-elected each year until his death. He was knighted in 1708 by Queen Anne, the first scientist to be so honoured for his work.

Leonhard Euler was one of top mathematicians of the eighteenth century and the greatest mathematician to come out of Switzerland. He made numerous contributions to almost every mathematics field and was the most prolific mathematics writer of all time. It was said that "Euler calculated without apparent effort, as men breathe...." He was dubbed "Analysis Incarnate" by his peers for his incredible ability.

Leonhard Euler was born in Basel, Switzerland, on April 15, 1707. His father, a Calvinist pastor and former mathematician, planned the life of a clergyman for his son and originally Leonhard followed that path. He graduated from the University of Basel in 1724 where he studied theology and Hebrew. During his time at the school, however, he was privately tutored in mathematics by Johann Bernoulli. Johann was so impressed by his pupil's ability that he convinced Euler's father to allow Leonhard to become a mathematician.

Euler took up a position at the Academy of Sciences in St. Petersburg, Russia, in 1727 and became the professor of mathematics six years later. During his stay, he was married and would over his lifetime have thirteen children, five of which would survive to adulthood. While in Russia, he lost sight in one eye after working day and night for three days to solve a problem. The question, which was a public contest, took all the other mathematicians involved months to figure out. He also discovered that the Czar's government was far from democratic as he was followed by secret police. He looked for a way out.

He found it in 1741, when he moved his family to Berlin to take over as director of mathematics at the Academy of Sciences under Frederick the Great. While in Prussia, his home was destroyed by invading Russian armies, but he was held in such high esteem by both countries that he was compensated for more than he lost. He also frustrated Frederick's mother to no end by refusing to engage in conversation. When she finally asked him for a reason, he responded: "Madam, it is because I have just come from a country where every person who speaks in hanged." He also could not handle the intrigues and feuds that plagued the Academy. When the previous president died, Euler should have been the obvious successor except for the fact that Frederick disliked him. The monarch asked D'Alembert, a French mathematician, to take the position. D'Alembert, who saw the injustice, refused on the basis that no one could be placed above Euler. However, it became clear it was time for Leonhard to find a new home.

Meanwhile, Russia had come under the rule of the more liberal Catherine the Great. In 1766, he returned to St. Petersburg and became the director of the Academy. Soon afterwards, he went completely blind but continued his mathematical work by dictating to a secretary. His house burned down in 1771 and his life was saved only by the heroic efforts of a servant to carry him out of the flames. He died of a stroke on September 7, 1783. Appropriately to this simple mathematician, his final words were simply "I die."

Euler was especially famous from his writings. Simply put, he produced more scholarly work on mathematics than anyone. It was said that he could produce an entire new mathematical paper in about thirty minutes and had huge piles of his works lying on his desk. Even more impressive, Euler contemplated new problems not in quiet privacy but in the presence of his young children. It was not uncommon to find "Analysis Incarnate" ruminating over a new subject with a child on his lap.

Though Euler is best remembered for his contributions to mathematics, he was involved in some extent in almost all fields. Especially close to his heart was philosophy. While in Berlin, he would constantly get involved in philosophical debates, especially with Voltaire. Unfortunately, Euler's philosophical ability was limited and he often blundered to the amusement of all involved. However, when he returned to Russia, he got his revenge. Catherine the Great had invited to her court the famous French philosopher Diderot, who to the chagrin of the czarina, attempted to convert her subjects to atheism. She asked Euler to quiet him. One day in the court, the French philosopher, who had no mathematical knowledge, was informed that someone had a mathematical proof of the existence of God. He asked to hear it. Euler then stepped forward and stated: "Sir,,hence God exists; reply!" Diderot had no idea what Euler was talking about. However, he did understand the chorus of laughter that followed and soon after returned to France.

Euler's contributions to every mathematical field that existed at the time. He standardized modern mathematics notation when he used symbols such as f(x), e, , i and in his textbooks. He was the first person to represent trigonometric values as ratios and prove that e is an irrational number. His invention of the calculus of variations led to the general method to solve max and min value problems. In physics, he developed the general equations for hydrodynamics and for motion. He was also one of the first people to recognize that infinite series had to be convergent to be used safely. Possibly his most impressive work was his approximation of the three-body problem of the sun, earth and moon, which he solved while completely blind and performing all the computations in his head. Among his other endeavors were proofs of Fermat's final theorem for cubes and quads, the use of calculus in mechanics and the computation of logs for negative and imaginary numbers.

Little is known of his life other than the fact that he taught at Alexandria, being associated with the school that grew up there in the late 4th cent. B.C.

He is famous for his Elements, a presentation in thirteen books of the geometry and other mathematics known in his day. The first six books cover elementary plane geometry and have served since as the basis for most beginning courses on this subject. The other books of the Elements treat the theory of numbers and certain problems in arithmetic (on a geometric basis) and solid geometry, including the five regular polyhedra, or Platonic solids.

The great contribution of Euclid was his use of a deductive system for the presentation of mathematics. Primary terms, such as point and line, are defined; unproved assumptions, or postulates, regarding these terms are stated; and a series of statements are then deduced logically from the definitions and postulates.

Although Euclid's system no longer satisfies modern requirements of logical rigor, its importance in influencing the direction and method of the development of mathematics is undisputed.

One consequence of the critical examination of Euclid's system was the discovery in the early 19th cent. that his fifth postulate, equivalent to the statement that one and only one line parallel to a given line can be drawn through a point external to the line, can not be proved from the other postulates; on the contrary, by substituting a different postulate for this parallel postulate two different self-consistent forms of non-Euclidean geometry were deduced, one by Nikolai I. Lobachevsky (1826) and independently by János Bolyai (1832) and another by Bernhard Riemann (1854).

A few modern historians have questioned Euclid's authorship of the Elements, but he is definitely known to have written other works, most notably the Optics.

René Descartes (March 31, 1596 – February 11, 1650), also known as Cartesius, was a noted French philosopher, mathematician, and scientist. Dubbed the "Founder of Modern Philosophy" and the "Father of Modern Mathematics," he ranks as one of the most important and influential thinkers of modern times. For good or bad, much of subsequent western philosophy is a reaction to his writings, which have been closely studied from his time down to the present day. Descartes was one of the key thinkers of the Scientific Revolution in the Western World. He is also honoured by having the Cartesian coordinate system used in plane geometry and algebra named after him.

Descartes frequently contrasted his views with those of his predecessors. In the opening section of the Passions of the Soul, he goes so far as to assert that he will write on his topic "as if no one had written on these matters before". Nevertheless many elements of his philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differs from the Schools on two major points: first, he rejects the analysis of corporeal substance into matter and form; second, he rejects any appeal to ends—divine or natural—in explaining natural phenomena. In his theology, he insists on the absolute freedom of God’s act of creation.

Descartes was a major figure in 17th century continental rationalism, later advocated by Baruch Spinoza and Gottfried Leibniz, and opposed by the empiricist school of thought, consisting of Hobbes, Locke, Berkeley, and Hume. Leibniz, Spinoza and Descartes were all versed in mathematics as well as philosophy, and Descartes and Leibniz contributed greatly to science as well. As the inventor of the Cartesian coordinate system, Descartes founded analytic geometry, that bridge between algebra and geometry crucial to the invention of the calculus and analysis. Descartes' reflections on mind and mechanism began the strain of western thought that much later, impelled by the invention of the electronic computer and by the possibility of machine intelligence, blossomed into, e.g., the Turing test. His most famous statement is Cogito ergo sum (French: Je pense, donc je suis or in English: I think, therefore I am), found in §7 of Principles of Philosophy (Latin) and part IV of Discourse on Method (French).