Mathematics, history of
From Encyclopædia
mathematics is as old as
civilization itself. By the
Neolithic Period, as life became settled and villages began to appear, writing and counting became increasingly useful, if not necessary. With counting, the history of
mathematics began. To count the passage of time, to weave intricate patterns in baskets or fabrics, and to apportion goods, crops, and livestock required a basic sense of arithmetic. Similarly, even in the most rudimentary cultures, the ability to decorate pottery with intricate designs, to distinguish constellations among the
stars, or to arrange stones, obelisks, and
tombs in ritualistic formations indicates a sense of space and geometry.EGYPTIAN, BABYLONIAN, AND GREEK MATHEMATICSThe earliest knowledge of
mathematics is preserved in Egyptian papyruses, Babylonian
cuneiform tablets, and Greek manuscripts. They indicate that the first mathematical concerns involved ARITHMETIC, ALGEBRA, GEOMETRY, and
trigonometry.Arithmetic and AlgebraAmong the earliest surviving mathematical texts are the famous Rhind papyrus (c.1750 BC) and the Golonishev papyrus. They reveal that the Egyptians used a DECIMAL system; the unit was represented by a single line, and tens, hundreds, and thousands by hieroglyphic symbols. Arithmetic for the Egyptians was essentially additive; repeated doubling was used for multiplication. Except for the FRACTION 2/3, for which there was a special hieroglyph, all fractions were expressed as unit fractions of the form 1/n; a relatively simple fraction like 2/59 was always handled in the more complex though equivalent form 1/36 + 1/236 + 1/531 = 2/59.Unit fractions were extremely cumbersome and would not have facilitated computation or the advance of arithmetic. Even so, Egyptian
mathematics was apparently suited for applications in commerce and agriculture. To deal with such problems as storing grain or apportioning loaves of bread, the Egyptians even applied a rudimentary algebra, although it did not advance beyond the simple LINEAR EQUATION in one unknown.In contrast, Babylonian arithmetic, which made use of a place-valued sexagesimal system, made certain computations, such as multiplication and division, considerably easier than the Egyptian method. The Babylonian base 60 is still used in measuring time (1 hour = 60 minutes, 1 minute = 60 seconds) and in measuring the degrees of a circle. The Babylonians also surpassed the Egyptians in their use of algebra.
cuneiform tablets from the Hammurabic period (about 1950 BC) reveal an ability to solve even quadratic and simple cubic equations.
cuneiform tablets from later periods (about 600 BC to AD 300) also reflect the algebraic-arithmetic strengths of the Babylonians and show the advances they made in applying their
mathematics to astronomy. To facilitate their complicated computations, tables for multiplication, reciprocals, and
square roots (see
square ROOT) were prepared, as well as tables for solving certain basic forms of equations.The first major discoveries in Greek
mathematics are ascribed to
Pythagoras of Samos and his followers. Pythagorean arithmetic regarded
numbers as sums of units or points and consequently has often been interpreted as an
abstract form of atomism. A group centered around ZENO OF ELEA (5th century BC) opposed this Pythagorean atomism and formulated ZENO'S PARADOXES. The ultimate effect of Zeno's arguments was to
stress the need to study the definitions and foundations of
mathematics more closely. The
Pythagoreans also provided the first general proof of the so-called PYTHAGOREAN THEOREM and discovered the existence of the IRRATIONAL
number, then known as an incommensurable magnitude.The discovery of incommensurable magnitudes was very troubling to Pythagorean
philosophy, which asserted that all magnitudes could be expressed in terms of integers or ratios of integers. The discovery made it clear that Pythagorean arithmetic was insufficient to express such geometric quantities as the diagonal of a
square. Some have called this the first great crisis in the history of
mathematics. Although EUDOXUS OF
Cnidus later solved the dilemma by working out a theory of proportion, after Pythagoras's time Greek
mathematics became essentially geometric rather than algebraic. This trend was reinforced by
Plato, the teacher of Eudoxus, who regarded geometry as the model of certain reasoning.Geometry and TrigonometryThe best-known mathematician of antiquity is EUCLID (fl. 3d century BC), whose Elements of Geometry provides a systematic treatment of geometry in the form of definitions, axioms, postulates, and theorems. The progression of Euclid's arguments was taken as a model of logical rigor in ancient times; since then axiomatization has represented the highest form of scientific argument.Because Euclid was a compiler and editor of existing ideas, the greatest mathematician of ancient times judged by the quality of his own original work was ARCHIMEDES (287-212 BC), who applied the method of exhaustion to determine rigorously the areas and volumes of numerous geometric figures. His younger contemporary, APOLLONIUS OF PERGA, introduced the terminology for the ELLIPSE, the H?PERBOLA, and the PARABOLA and determined the specific properties of each type of curve, all of which are CONIC SECTIONS. The circle remained the most important curve, because astronomy in ancient times was based upon the geometry of perfect circles and uniform circular motions. Eudoxus, ARISTARCHUS OF SAMOS, HIPPARCHUS OF NICAEA, and Claudius
Ptolemy made fundamental contributions in developing geometric models for planetary motions (see
ASTRONOMY, HISTORY OF).The last of the noteworthy geometers of ancient times was PAPPUS (fl. 3d century AD), whose Collection is a final synthesis of the work of his predecessors since Euclid.In the Almagest
Ptolemy introduced a kind of
trigonometry in terms of a table of chords that was equivalent to a
sine table. Not only did the Almagest contain formulas for the
sines and cosines of the sums and differences of angles, it also provided the rudiments of SPHERICAL
trigonometry, the study of triangles projected onto the surface of a sphere. Spherical
trigonometry was given considerable
attention by
Menelaus (about AD 100), who was responsible for showing the significance of the arcs of
great circles in dealing with such problems as spherical triangles.Although some work in algebra was carried out in this period, ancient mathematicians never managed to amalgamate geometry and algebra in the way that mathematicians in the 16th and 17th centuries did when they created ANALYTIC GEOMETRY. Instead, the Greeks confined most of their work to geometry, the only branch of
mathematics in which they could deal successfully with continuous magnitudes and with quantities that were incommensurable.ISLAMIC AND MEDIEVAL MATHEMATICSAfter the fall of the Roman Empire in the western Mediterranean, the Greco-Roman tradition was maintained and transmitted to the Latin West by the Byzantines in Constantinople and by scholars in intellectual centers such as Isfahan, Jundishapir, and
Baghdad. Islamic scholars also helped to spread mathematical discoveries made in India and China. Among the earliest examples of such activity was Al-Farazi's translation of the Hindu Siddhantas. Al-Khwarizmi's book on arithmetic contained a thorough exposition of the Hindu system of
numbers and brought the idea of place-valued decimal notation to the West.Because Arabic
science was greatly interested in astronomy, largely for reasons related to
astrology and the casting of horoscopes, considerable
attention was devoted to
Ptolemy's Almagest and to the advance of
trigonometry. The astronomer Al-BATTANI advanced the study of spherical
trigonometry and produced a table of cotangents for use in astronomical computations. The Persian poet
Omar Khayyam devoted considerable time to the study of algebra and geometry. Al-Haitham, also known as Alhazen, applied geometry to the study of
light, and Abul-Wafa advanced spherical
trigonometry.Like their Islamic counterparts, scholars in the Latin West first brought new life to their
mathematics by translating basic works from Greek or Arabic sources, especially after Toledo fell into Christian
hands in 1085. LEONARDO PISANO, better known as Fibonacci, wrote his Liber Abaci (c.1202) based on bits of arithmetic and algebra that he had accumulated during his travels. He brought both the Arabic place-valued decimal system and the use of Arabic numerals to the Latin West.The most striking advance in medieval
mathematics was its innovative application of
mathematics to
physics, particularly to the problems of uniform and accelerated motion. In this respect the French scholastic Nicole Oresme and a group of mathematicians, including Thomas Bradwardine, at Merton College,
Oxford, are noteworthy.With the fall of Constantinople (1453) many Eastern scholars left for western
Europe and brought knowledge of Greek manuscripts (and often the manuscripts themselves) with them. The astronomer Georg von PEUERBACH began a translation of
Ptolemy's Almagest that one of his students, REGIOMONTANUS, completed. In
Italy artists studied VITRUVIUS, applied geometry to the construction of great buildings? and pioneered in the mathematical study of
perspective. LEONARDO DA VINCI, Leon Battista ALBERTI, and
Piero della Francesca wrote treatises on the
mathematics of
perspective.
