Técnicas e instrumentos de recolección de datos

BIOGRAPHIES Legendre – Leibniz

deductions could be made in an algorithmic, computational manner (this idea would be taken up a century later by the English mathematician George Boole, the founder of symbolic logic). Leibniz was also the first to recognize the significance of the binary base—the number system that uses only two numerals, 0 and 1, the basis of modern computers; his interest in this base, however, was more philosophical than practical, seeing it as a gift from God to humanity. But by far his greatest contribution to mathematics was his invention of the calculus, which he developed during the decade 1666–75 independently of Newton. Whereas the reclusive Newton withheld

publication of his own results, Leibniz published his invention in 1684, precipitating one of the most ugly priority disputes in the history of science. The two great minds had arrived at the same results—in particular, they both discovered what is known today as the Fundamental Theorem of Calculus (the inverse relation between differentiation and integration), but their approach and notation were different, with Leibniz proposing the more efficient “d” notation (see DERIVATIVE).

Today we give Newton and Leibniz equal credit for the invention of calculus, an achievement that forever changed the course of mathematics.

Maclaurin, Colin (1698–1746) Scottish mathematician who was influential in disseminating the newly invented calculus throughout England. In his Treatise of Fluxions (1742) he attempted to give Newton’s differential calculus (“fluxion”

was Newton’s word for derivative) a geometric foundation.

This ran contrary to the trend that began to form in

continental Europe, where mathematicians were trying to put the calculus on firm, logical foundations, but it made the subject a lot more accessible to English scientists. Ironically, Maclaurin’s name is known today mainly for an infinite series that was actually discovered by his contemporary Brook Taylor (see MACLAURIN SERIES).

Napier, John (1550–1617) Scottish mathematician and the inventor of logarithms. His early life did not hint at any future

mathematical greatness. He was a practical man who invented a variety of mechanical devices to improve the crop on the farm on which he lived; these included a hydraulic screw to BIOGRAPHIES Maclaurin – Napier

BIOGRAPHIES Maclaurin – Napier

control the level of water. He also showed an interest in military hardware and drew plans for building a huge artillery piece and even a submarine. If that was not enough, Napier was also a religious activist who got himself embroiled in many controversies. It is not known what led him to the idea of logarithms, on which he worked for 20 years. His tables, published in 1614, were received with great enthusiasm by the scientific community, for they greatly reduced the labor of numerical computing (logarithms allow us to replace

multiplication and division by addition and subtraction). In his original logarithms he did not use a base in the modern sense, but with the help of the English mathematician Henry Briggs, who traveled to Scotland to meet Napier, they reworked the tables and made them into base 10 (“common”) logarithms. In this form they remained virtually unchanged, until the advent of the electronic handheld calculator in the 1970s made them obsolete. Napier also invented the Napier rods—a sort of mechanical calculator—and he devised a set of rules known as

“Napier analogies” for use in spherical trigonometry. And he advocated using the decimal point to separate the integral part of a number from its fractional part. But it is his invention of logarithms that made his name immortal.

Newton, Sir Isaac (1642–1727) English mathematician and physicist, by general consensus one of the three greatest scientists of all time (the others are Archimedes and Einstein). Newton’s early life was beset by misfortunes. His father died shortly before Isaac was born; his mother soon remarried, only to lose her second husband too. Young Newton was thus left in the custody of his grandmother. In 1661 he entered Trinity College (part of Cambridge University), where his mathematical genius flourished. He studied many of the classic works on

mathematics, including Euclid’s Elements and Descartes’s La Géometrie—none of which is easy reading even today. The fact that he studied these works on his own, with little help from the outside, set the stage for his future character—a reclusive man who was reluctant to share his thoughts with others. Indirectly it would contribute to his bitter priority dispute with Leibniz over the invention of the calculus.

In 1665 Cambridge University closed its doors due to the outbreak of the Great Plague. Newton returned to his family’s

BIOGRAPHIES Newton

BIOGRAPHIES Newton

farm, where he enjoyed two years of complete freedom to shape his scientific ideas; later he would refers to this period as his “prime years.” Newton’s first major discovery was the expansion of (a + b)ⁿinto powers of a and b when n is a negative integer or a fraction (the case when n is a positive integer had been known for a long time and involves the Pascal Triangle); the expansion in these cases is an infinite series. At about the same time Newton began to shape his thoughts on gravitation, reportedly triggered by seeing an apple fall from the tree (there is no evidence, however, that this actually happened). He also speculated on the nature of light and discovered the splitting of white light into its

spectrum of rainbow colors. And if these discoveries were not enough, he also worked out his “method of fluxions”—his differential and integral calculus. Unlike his rival Leibniz, Newton was always guided by physical intuition; he thought of a function as a relation between two variables, each of which “flows” continuously with time (hence the word fluxion). But his reluctance to publish his discoveries, while Leibniz published his own, precipitated a bitter priority dispute between the two, and the aftershocks lasted well after both men were dead. Newton’s work on gravitation was published in his great work, Philosophiae naturalis principia mathematica (Mathematical principles of natural philosophy, 1687). The Principia, as it is commonly known, has had an enormous influence on subsequent generations of scientists and was hailed as the greatest work in science since Euclid wrote his Elements around 300 B.^C.^E.; it marked the

beginning of the modern era in science. As for the calculus, a summary of it was not published until 1704 as an appendix to Newton’s other great work, Opticks, but a full account had to wait until 1736, nine years after his death. Newton died at the age of 85 and was given a state funeral; he was buried at Westminster Abbey in London, where an ornate tombstone marks the site.

Pascal, Blaise (1623–62) French mathematician, physicist, and philosopher who was educated by his father, himself a

mathematician. But the father, being a pedant, insisted that his son should first become acquainted with classical languages, so he forbade young Pascal to read any mathematics books.

BIOGRAPHIES Pascal

BIOGRAPHIES Pascal

Secretly, however, the 11-year-old Pascal read Euclid’s Elements, which he mastered all by himself. At the age of 16 he wrote an original paper on conic sections that amazed Descartes. Pascal pioneered (with Fermat) the theory of probability, made major discoveries in geometry, and built a calculating machine that could add and subtract numbers of up to eight digits. The famous Pascal Triangle (a triangular array of numbers whose nth row gives the coefficients in the expansion of (a + b)ⁿin powers of a and b) was not his invention, however; it had been known long before him. At 23 he turned to physics and discovered the law of hydrostatic pressure that bears his name. Then at 25 he suddenly lost interest in mathematics and science and spent his remaining years in religious penance. However, in one last flash of creativity, he found the area under the cycloid, thus anticipating the soon-to-be-discovered integral calculus.