Sir Issac Newton was born to a poor family in Woolsthorpe, England on January 4,1642. At the time of Newton's birth, England had not adopted the Gregorian calendar and therefore his date of birth was recorded as Christmas Day, December 25 1642. He attended Trinity College in Cambridge, England only after it became apparent that he would never be a successful farmer. While there, he took interest in mathematics, optics, physics, and astronomy.

While a student, Newton was forced to take a two year hiatus when plague closed Trinity college. At home, he continued to work with optics, using a prism to separate white light, and became the first person to argue that white light was a mixture of many types of rays, rather than a single entity. He continued working with light and color over the next few years, and published his findings in “Optics” in 1704.

Disturbed by the problems with telescopes at the time, he invented the reflecting telescope, grinding the mirror and building the tube himself. Relying on a mirror rather than lenses, the telescope presented a sharper image than refracting telescopes at the time. Modern techniques have reduced the problems caused by lenses, but large telescopes such as the use mirrors.

As a student, Newton studied the most advanced mathematical texts of his time. While on hiatus, he continued to study mathematics, laying the ground for differential and integral calculus. He united many techniques that had previously been considered separately, such as finding areas, tangents, and the lengths of curves. He wrote De Methodis Serierum et Fluxionum in 1671, but was unable to find a publisher.

Newton also established a cohesive scientific method, to be used across disciplines. Previous explorations of science varied depending on the field. Newton established a set format for experimentation still used today.

His most famous work came with the publication of his "Philosophiae Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), generally called Principia. In it, he determined the three laws of motion for the universe. The first describes how objects move at the same velocity unless an outside force acts upon it. His second law of motion provided a calculation for how forces interact. The force acting on an object is equal to the object's mass times the acceleration it undergoes. Newton's third law states that for every action in nature, there is an equal and opposite reaction.

After his graduation, he began to teach at the college, and was appointed as the second Lucasian Chair there. Today, the chair is considered the most renowned academic chair in the world. In 1689, Newton was elected as a member of parliament for the university. In 1703, he was elected as president of the Royal Society, a fellowship of scientists that still exists today. He was knighted by Queen Anne in 1705. He never married. Newton died in 1727, at the age of 84. After his death, his body was moved to a more prominent place in Westminster Abbey. During the exhumation, large amounts of mercury were found in the scientist's system, likely due to his work with alchemy.

Leibniz was born in Leipzig, Saxony (now Germany), on July 1, 1646, four years after the birth of Newton. His father, Friedrich Leibnütz, was a lawyer and professor of moral philosophy at the University of Leipzig; Gottfried's mother, Catherina Schmuck, was Friedrich's third wife. Both sides of the family enjoyed social standing and scholarly reputations. Leibniz (who changed the spelling of his name) had a half-brother, Johann Friedrich; a half-sister, Anna Rosina; and a sister, Anna Catherina, whose son, Friedrich Simon Löffler, became his sole heir. His father died when he was six, but young Gottfried had already begun to demonstrate a passion for knowledge and omnivorous reading. He studied his father's library of classic, philosophical and religious works, and his school syllabus included German literature and history, Latin, Greek, theology and logic. By age 12, Leibniz read Latin and was adept at writing Latin verse. He began to formulate his own ideas in logic, among them the ideas of an alphabet of human thought and a universal encyclopedia.

In 1661, at age 15, Leibniz entered the University of Leipzig, where he studied with philosophy professor Jakob Thomasius. He received his bachelor's degree in 1663, then spent the summer in Jena studying with the mathematician Erhard Weigel, who introduced him to elementary algebra and Euclidean geometry. After receiving his master's degree in 1664, Leibniz wrote a dissertation for the Doctor of Law degree, but because of his youth, the university refused to award it to him. Leibniz subsequently entered the University of Altdorf in Nuremburg, where his dissertation was accepted and his degree was awarded in 1666. The university then offered him a professorial position, but he declined.

France had been encroaching on the Rhineland, and in the winter of 1671-1672, Leibniz devised a plan to distract the French by encouraging them to conquer Egypt and build a canal across the isthmus of Suez. The plan eventually came to nothing, but it allowed Leibniz to accompany a diplomatic mission to Paris, meet prominent philosophers and scholars, and immerse himself in Parisian salons. This was the time of Leibniz' greatest advances in mathematics, and he formed a lifelong friendship with Christiaan Huygens, who was in the employ of the Académie Royale des Sciences. He was also able to obtain and copy unpublished manuscripts of Blaise Pascal and René Descartes, and he devised a calculating machine that could add, subtract, divide, multiply and extract roots.

As a young man, Leibniz had a reputation as a savant, a wit, and an elegant courtier. In his last years he was increasingly disregarded. On November 14, 1716, after a week in bed suffering from arthritis, gout and colic, Leibniz died in Hanover in the presence of his secretary, Johann Georg Eckhart. His funeral and burial took place on December 14 in Neustüdter Church. No one from the Court attended his funeral, and his grave went unmarked for 50 years. An elegy was read at the Académie Royale des Sciences a year after Leibniz' death, but neither the Royal Society of London nor the Berlin Academy published an obituary.

Leibniz was a versatile and prolific contributor to mathematics. In On the Secrets of Geometry and Analysis of Indivisible and Infinite Quantities(1686) Leibniz first used the integral sign. In a 1693 letter, Leibniz used multiple indices to state the result of three linear equations. He also worked on the problem of elimination in the general theory of equations and laid the foundations of the theory of determinants. In another letter, he suggested expanding cube roots into infinite series. He introduced terminology, including the term "function," borrowed by Bernoulli.

In 1702, Leibniz published A Justification of the Calculus of the Infinitely Small, in which he attempted to justify his algorithms for differentation and integration. Between the years 1702-1703 he integrated rational fractions in trigonometric and logarithmic functions, produced a theory of special curves, and stated equations important in navigation. Leibniz was one of the first to work out the properties of the binary number system; he anticipated the central concerns of modern computer science, bringing human reasoning under mathematical law.

Sir Isaac Newton and Gottfried Wilhelm Leibniz are considered the inventors of calculus (known in the past as "the calculus"). Calculus is mostly the study of infinity (large without bound) and the infinitesimal. And these two people certainly made great discoveries to advance this field of mathematics. But it must be said that they had many predecessors, like Archimedes and Euclid who dealt with the infinite and infinitesimal in very natural ways. Besides, many of the ideas of these two concepts were being discussed and studied in the years before Newton and Leibniz worked on them.

Pierre de Fermat and Isaac Barrow made important discoveries before Newton and Leibniz. Newton began creating calculus around 1664-1666, but did not publish. Some people were aware of what he was doing, through letters and papers which Newton showed to people. He did not publish until 1687 and later. Leibniz first studied calculus around 1672-1676, and published in 1684 and 1686. The ideas were similar, but the notation was different. And many of the lesser discoveries were different. Nowadays, our calculus notation is mostly that of Leibniz.

In the late 1690's British scientists began accusing Leibniz of having plagiarized Newton's great discovery. Nowadays, it seems unlikely that Leibniz knew very much about what Newton had discovered; and his discoveries would seem to be mostly original. Of course, neither Newton's nor Leibniz' discoveries were totally original. In 1711, Leibniz appealed to the Royal Society of London, of which he was a member and Newton was President, to clear up these accusations. In public, Newton pretended to have nothing to do with the scandal, remaining silent about it. But, it seems that he secretly was the motivating force behind the accusations. The Royal Society appointed a commission, and essentially found Leibniz guilty of plagiarism.

By taking sides against Leibniz, Newton and his followers refused to use Leibniz' superior notation. Newton founded astrodynamics, based largely on calculus, with which he mostly solved the motions of the planets. After Newton, British mathematics went into a decline, while German mathematics (and the mathematics of other countries) prospered. Today, Newton and Leibniz are considered the co-inventors of calculus. This is Newton's dot notation compared to Leibniz notation.