MATHEMATICS.

This article comprises a compact survey of the development of mathematics from ancient times until the early twentieth century. The treatment is broadly chronological, and most of it is concerned with Europe.

Unknown Origins

It seems unavoidable that mathematical thinking played a role in human theorizing from the start of the race, and in various ways. Arithmetic (as the later branch of mathematics became known) would have been one of them, motivated initially by forming integers in connection with counting. But other branches surely include geometry, linked to the appreciation of line, surface, and space; trigonometry, inspired by awareness of angles; mechanics, related to the motion of bodies large and small and the (in)stability of structures; part-whole theory, from consideration of collections of things; and probability, coming from judging and guessing about situations. In all cases the thinking would have started out as very intuitive, gradually becoming more explicit and less particular.

Some of the associated contexts would have been provided by study of the environment (such as days) and the heavens (such as new and full moons), which was a major concern of ancient cultures in all parts of the world; in those times mathematics and astronomy were linked closely. For example, the oldest recognized artifact is a bone from Africa, thought to be about thirty-seven thousand years old, upon which phases of the moon seem to have been recorded.

Among the various ancient cultures, the Babylonians have left the earliest extant evidence of their mathematical practice. They counted with tokens from the eighth millennium; and from the late fourth millennium they expressed numbers and properties of arithmetic in a numeral system to base 10 and handled fractions in expansions of powers of 1/60. Many surviving artifacts seem to relate to education—for example, exercises requiring calculations of unknown quantities, which correspond to the solution of equations but are not to be so identified. They also developed geometry, largely for terrestrial purposes. The Egyptians pursued similar studies, even also finding a formula (not the same one) for the volume of the rectangular base of a pyramid of given sides. They also took up the interesting mathematical problem of representing a fraction as the sum of reciprocals.

A major mathematical question for these cultures concerned the relationship between circles and spheres and rectilinear objects such as lines and cubes. They involve the quantity that we symbolize by, and ancient evidence survives of methods of approximating to its value. But it is not clear that these cultures knew that the same quantity occurs in all the relationships.

On Greek Mathematics

The refinement of mathematics was effected especially by the ancient Greeks, who flourished for about a millennium from the sixth century b.c.e. Pythagoras and his clan are credited with many things, starting with their later compatriots: the eternality of integers; the connection between ratios of integers and musical intervals; the theorem relating the sides of a right-angled triangle; and so on. Their contemporary Thales (c. 625–c. 547 b.c.e.) is said to have launched trigonometry with his appreciation of the angle. However, nothing survives directly from either man.

A much luckier figure concerning survival is Euclid (fl. c. 300 b.c.e.), especially with his Elements. While no explanatory preface survives, it appears that most of the mathematics presented was his rendition of predecessors' work, but that (some of) the systematic organization that won him so many later admirers might be his. He stated explicitly the axioms and assumptions that he noticed; one of them, the parallel axiom, lacked the intuitive clarity of the others, and so was to receive much attention in later cultures.

The Elements comprised thirteen Books: Books 7–9 dealt with arithmetic, and the others presented basic plane (Books 1–6) and solid (Books 11–13) geometry of rectilinear and circular figures. The extraordinary Book 10 explored properties of ratios of smaller to longer lines, akin to a theory of irrational numbers but again not to be so identified. A notable feature is that Euclid confined the role of arithmetic within geometry to multiples of lines (say, "twice this line is … "), to a role in stating ratios, and to using reciprocals (such as 1/5); he was not concerned with lengths—that is, lines measured arithmetically. Thus, he said nothing about the value of, for it relates to measurement.

The Greeks were aware of the limitations of straight line and circle. In particular, they found many properties and applications of the "conic sections": parabola, hyperbola, and ellipse. Hippocrates of Chios (fl. c. 600 b.c.e.) is credited with three "classical problems" (a later name) that his compatriots (rightly) suspected could not be solved by ruler and compass alone: (1) construct a square equal in area to a given circle; (2) divide any angle into three equal parts; and (3) construct a cube twice the volume of a given one. The solutions that they did find enlarged their repertoire of curves.

Among later Greeks, Archimedes (c. 287–212 b.c.e.) stands out for the range and depth of his work. His work on circular and spherical geometry shows that he knew all four roles for; but he also wrote extensively on mechanics, including floating bodies (the "eureka!" tale) and balancing the lever, and focusing parabolic mirrors. Other figures developed astronomy, partly as applied trigonometry, both planar and spherical; in particular, Ptolemy (late second century) "compiled" much knowledge in his Almgest, dealing with both the orbits and the distances of the heavenly bodies from the central and stationary Earth.

Traditions Elsewhere

Mathematics developed well from antiquity also in the Far East, with distinct traditions in India, China, Japan, Korea, and Vietnam. Arithmetic, geometry, and mechanics were again prominent; special features include a powerful Chinese method equivalent to solving a system of linear equations, a pretty theory of touching circles in Japanese "temple geometry," and pioneering work on number theory by the Indians. They also introduced the place-value system of numerals to base 10, of which we use a descendant that developed after several changes in adopted symbols.

This system of numerals was mediated into Europe by mathematicians working in medieval Islamic civilization, often though not always writing in Arabic. They became the dominant culture in mathematics from the ninth century and continued strongly until the fourteenth. They assimilated much Greek mathematics; indeed, they are our only source for some of it.

The first major author was al-Khwarizmi (fl. c. 800–847), who laid the foundations of algebra, especially the solution of equations. He and his followers launched the theory using words rather than special symbols to mark unknowns and operations. Other interests in geometry included attempts to prove Euclid's parallel axiom and applications to optics and trigonometry; an important case of the latter was determining the qibla (that is, the direction of Mecca) for any time and place at times of Muslim prayer. Their massive contributions to astronomy included theory and manufacture of astrolabes.

The Wakening Europe from the Twelfth Century

From the decline of the Roman Empire (including Greece) Euclid was quiescent mathematically, though the Carolingian kingdom inspired some work, at least in education. The revival dates from around the late twelfth century, when universities also began to be formed. The major source for mathematics was Latin translations of Greek and Arabic writings (and re-editions of Roman writers, especially Boethius). In addition, the Italian Leonardo Fibonacci (c. 1170–c. 1240) produced a lengthy Liber Abbaci in 1202 that reported in Latin many parts of Arabic arithmetic and algebra (including the Indian numerals); his book was influential, though perhaps less than is commonly thought. The Italian peninsula was then the most powerful region of Europe, and much commercial and "research" mathematics was produced there; the German states and the British Isles also came to boast some eminent figures. In addition, a somewhat distinct Hebrew tradition arose—for example, in probability theory.

A competition developed between two different methods of reckoning. The tradition was to represent numbers by placing pebbles (in Latin, calculi ) in determined positions on a flat surface (in Latin, abacus, with one b ), and to add and subtract by moving the pebbles according to given rules. However, with the new numerals came a rival procedure of calculating on paper, which gradually supervened; for, as well as also allowing multiplication and division, the practitioner could show and check his working, an important facility unavailable to movers of pebbles.

Mathematics rapidly profited from the invention of printing in the late fifteenth century; not only were there printed Euclids, but also many reckoning books. Trigonometry became a major branch in the fifteenth and sixteenth centuries, not only for astronomy but also, as European imperialism developed, for cartography, and the needs of navigation and astronomy made the spherical branch more significant than the planar. Geometry was applied also to art, with careful studies of perspective; Piero della Francesca (c. 1420–1492) and Albrecht Dürer (1471–1528) were known not only as great artists but also as significant mathematicians.

Numerical calculation benefited greatly from the development of logarithms in the early seventeenth century by John Napier (1550–1617) and others, for then multiplication and division could be reduced to addition and subtraction. Logarithms superseded a clumsier method called "prosthaphairesis" that used certain trigonometrical formulas.

In algebra the use of special symbols gradually increased, until in his Géométrie (1637), René Descartes (1596–1650) introduced (more or less) the notations that we still use, and also analytic geometry. His compatriot Pierre de Fermat (1601–1665) also worked in these areas and contributed some theorems and conjectures to number theory. In addition, he corresponded with Blaise Pascal (1623–1662) on games of chance, thereby promoting parts of probability theory.

In mechanics a notable school at Merton College, Oxford, had formed in the twelfth century to study various kinds of terrestrial and celestial motion. The main event in celestial mechanics was Nicolaus Copernicus's (1473–1543) De revolutionibus (1453; On the revolutions), where rest was transferred from the Earth to the sun (though otherwise the dynamics of circular and epicyclical motions was not greatly altered). In the early seventeenth century the next stages lay especially with Johannes Kepler's (1571–1630) abandonment of circular orbits for the planets and Galileo Galilei's (1564–1642) analysis of (locally) horizontal and vertical motions of bodies.

The Epoch of Newton and Leibniz

By the mid-seventeenth century, science had become professionalized enough for some national societies to be instituted, especially the Royal Society of London and the Paris Académie des Sciences. At that time two major mathematicians emerged: Isaac Newton (1642–1727) in Cambridge and Gottfried Wilhelm von Leibniz (1646–1716) in Hanover. Each man invented a version of the differential and integral calculus, Newton first in creation but Leibniz first in print. The use here of Leibniz's adjectives recognizes the superior development of his version. During the early 1700s Newton became so furious (or envious?) that he promoted a charge of plagiarism against Leibniz, complete with impartial committee at the Royal Society. It was a disaster for Britain: Newton's followers stuck with their master's theory of "fluxions" and "fluents," while the Continentals developed "differentials" and "integrals," with greater success. The accusation was also mathematically stupid, for conceptually the two calculi were quite different: Newton's was based upon (abstract) time and unclearly grounded upon the notion of limit, while Leibniz's used infinitesimal increments on variables, explicitly avoiding limits. So even if Leibniz had known of Newton's theory (of which the committee found no impartial evidence), he rethought it entirely.

Leibniz's initial guard was largely Swiss: brothers Jakob (1654–1705) and Johann Bernoulli (1667–1748) from the 1680s, then from the 1720s Johann's son Daniel (1700–1782) and their compatriot Leonhard Euler (1707–1783), who was to be the greatest of the lot. During the eighteenth century they and other mathematicians (especially in Paris) expanded calculus into a vast territory of ordinary and then partial differential equations and studied many related series and functions. The Newtonians kept up quite well until Colin Maclaurin (1698–1746) in the 1740s, but then faded badly.

The main motivation for this vast development came from applications, especially to mechanics. Here Newton and Leibniz differed again. In his Principia mathematica (1687) Newton announced the laws that came to carry his name: (1) a body stays in equilibrium or in uniform motion unless disturbed by a force; (2) the ratio of the magnitude of the force and the mass of the body determines its acceleration; and (3) to any force of action there is one of reaction, equal in measure and opposite in sense. In addition, for both celestial and terrestrial mechanics, which he novelly united, the force between two objects lies along the straight line joining them, and varies as the inverse square of its length.

With these principles Newton could cover a good range of mechanical phenomena. His prediction that the Earth was flattened at the poles, corroborated by an expedition undertaken in the 1740s, was a notable success. He also had a splendid idea about why the planets did not exactly follow the elliptical orbits around the sun that the inverse square law suggested: they were "perturbed" from them by interacting with each other. The study of perturbations became a prime topic in the eighteenth century, with Euler's work being particularly significant. Euler also showed that law 2 could be applied to any direction in a mechanical situation, thus greatly increasing its utility. He and others made important contributions to the mechanics of continuous media, especially fluid mechanics and elasticity theory, where Newton had been somewhat sketchy.

Mathematics in the Eighteenth Century: The Place of Lagrange

However, Newton's theory was not alone in mechanics. Leibniz and others developed an alternative approach, partly inspired by Descartes, in which the "living forces" (roughly, kinetic energy) of bodies were related to their positions. Gradually this became a theory of living forces converted into "work" (a later term), specified as (force x traversed distance). Engineers became keen on it for its utility in their concerns, especially when impact between bodies was involved; from the 1780s Lazare Carnot (1753–1823) urged it as a general approach for mechanics.

Carnot thereby challenged Newton's theory, but his main target was a recent new tradition partly launched by Jean d'Alembert (1718–1783) in midcentury and developed further by Joseph-Louis Lagrange (1736–1813). Suspicious of the notion of force, d'Alembert had proposed that it be defined by Newton's law 2, which he replaced by one stating how systems of bodies moved when disturbed from equilibrium. At that time Euler and others proposed a "principle of least action," which asserted that the action (a mechanical notion defined by an integral) of a mechanical system took its optimal value when equilibrium was achieved. Lagrange elaborated upon these principles to create Méchanique analitique (1788), in which he challenged the other two traditions; in particular, dynamics was reduced mathematically to statics. For him a large advantage of his principles was that they were formulated exclusively in algebraic terms; as he proclaimed in the preface of his book, there were no diagrams, and no need for them. A main achievement was a superb though inconclusive attempt to prove that the system of planets was stable; predecessors such as Newton and Euler had left that matter to God.

Lagrange formulated mechanics this way in order to make it (more) rigorous. Similarly, he algebraized the calculus by assuming that any mathematical function could be expressed in an infinite power series (the so-called Taylor series), and that the basic notions of derivative (his word) and integral could be determined solely by algebraic manipulations. He also greatly expanded the calculus of variations, a key notion in the principle of least action.

As in mechanics, Lagrange's calculus challenged the earlier ones, Newton's and Leibniz's, and as there, reaction was cautious. A good example for both contexts was Pierre-Simon Laplace (1749–1827), a major figure from 1770. While strongly influenced by Lagrange, he did not confine himself to the constraints of Lagrange's book when writing his own four-volume Traité de mécanique céleste (1799–1805; Treatise on celestial mechanics). His exposition of celestial and planetary mechanics used many differential equations, series, and functions.

The French Revolution and a New Professionalization

Laplace published his large book in a new professional and economic situation for science. After the Revolution of 1789 in France, higher education and its institutions there were reformed, with a special emphasis upon engineering. In particular, a new school was created, the École Polytechnique (1794), with leading figures as professors (such as Lagrange) and as examiners (Laplace), and with enrollment of students determined by talent, not birth. A new class of scientists and engineers emerged, with mathematics taught, learned, researched, and published on a scale hitherto unknown.

Of this mass of work only a few main cases can be summarized here. Joseph Fourier (1768–1830) is noteworthy for his mathematical analysis of heat diffusion, both the differential equation to represent it (the first important such equation found outside mechanics) and solutions by certain infinite series and by integrals that both now bear his name. From the 1820s they attracted much attention, not only for their use in heat theory but especially for the "pure" task of establishing conditions for their truth. New techniques for rigor had just become available, mainly from Augustin-Louis Cauchy (1789–1857), graduate of the École Polytechnique and now professor there. He taught a fourth approach to the calculus (and also function and series), based like Newton's upon limits but now fortified by a careful theory of them; although rather unintuitive, its mathematical merits gradually led worldwide to its preference over the other three approaches.

Ironically, Cauchy's own analysis of Fourier series failed, but a beautiful treatment following his approach came in 1829 from J. P. G. Dirichlet (1805–1859)—a French-sounding name of a young German who had studied with the masters in Paris. Dirichlet also exemplifies a novelty of that time: other countries producing major mathematicians. Another contemporary example lies in elliptic functions, which Carl Jacobi (1804–1851) and the young Norwegian Niels Henrik Abel (1802–1829) invented independently following much pioneering work on the inverse function by A. M. Legendre (1752–1833).

Jacobi and Abel drew upon a further major contribution to mathematics made by Cauchy when, by analogy with the calculus, he developed a theory of functions of the complex variable x + √ − 1y (x and y real), complete with an integral. His progress was fitful, from the 1810s to the 1840s; after that, however, his theory became recognized as a major branch of mathematics, with later steps taken especially by the Germans.

Between 1810 and 1830 the French initiated other parts of mathematical physics in addition to Fourier on heat: Siméon-Denis Poisson (1781–1840) on magnetism and electrostatics; André-Marie Ampère (1775–1836) on electrodynamics; and Augustin Jean Fresnel (1788–1827) on optics with his wave theory. Mathematics played major roles: many analogies were taken from mechanics, which itself developed massively, with Carnot's energy approach elaborated by engineers such as Gaspard-Gustave Coriolis (1792–1843), and continuum mechanics extended, especially by Cauchy.

Geometry was also taught and studied widely. Gaspard Monge (1746–1818) sought to develop "descriptive geometry" into a geniune branch of mathematics and gave it prominence in the first curriculum of the École Polytechnique; however, this useful theory of engineering drawing could not carry such importance, and Laplace had its teaching reduced. But former student Jean Victor Poncelet (1788–1867) was partly inspired by it to develop "the projective properties of figures" (Traité des propries projectives de figures, 1822), where he studied characteristics independent of measure, such as the order of points on a line.

The main mathematician outside France at this time was C. F. Gauss (1777–1855), director of the Göttingen University Observatory. Arguably he was the greatest of all, with major work published in number theory, celestial mechanics, and aspects of analysis and probability theory. But he was not socially active, and he left many key insights in his manuscripts (for example, on elliptic functions).

Other major contributors outside France include George Green (1793–1841), who, in An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), produced a wonderful theorem in potential (his word) theory that related the state of affairs inside an extended body to that on its surface. But he published his book very obscurely, and it became well-known only on the reprint during the 1850s initiated by William Thomson (later Lord Kelvin), who was making notable contributions of his own to the theory.

Midcentury Internationalism

By the 1840s Britain and the Italian and German states were producing quality mathematicians to complement and even rival the French, and new posts were available in universities and engineering colleges everywhere. Among the Germans, two figures stand out.

From around 1860 Karl Weierstrass (1815–1897) gave lecture courses on many aspects of real-and complex-variable analysis and parts of mechanics at Berlin University, attended by students from many countries who then went home and taught likewise. Meanwhile at Göttingen, Bernhard Riemann (1826–1866) rethought complex-variable analysis and revolutionized the understanding of both Fourier series and the foundations of geometry. Much of this work was published only after his early death in 1866, but it soon made a great impact. The work on Fourier series led Georg Cantor (1845–1918) to develop set theory from the 1870s. On geometry Riemann showed that the Euclidean was only one of many possible geometries, and that each of them could be defined independently of any embedding space. The possibility of non-Euclidian geometries, using alternatives to the parallel axiom, had been exhibited around 1830 in little-recognized work by Janos Bolyai (1802–1860) and Nicolai Lobachevsky (1793–1856) (and, in manuscript, Gauss); Riemann, however, went much further and brought us proper understanding of the plurality of geometries.

Weierstrass emulated and indeed enhanced Cauchy-style rigor, carefully formulating definitions and distinctions and presenting proofs in great detail. By contrast, Riemann worked intuitively, offering wonderful but often proof-free insights grounded upon some "geometric fantasy," as Weierstrass described it. A good example is their revisions of Cauchy's complex-variable analysis: Weierstrass relied solely on power series expansions of the functions, whereas Riemann invented surfaces now named after him that were slit in many remarkable ways. Among many consequences of the latter, the German Felix Klein (1849–1925) and the Frenchman Henri Poincaré (1854–1912) in the early 1880s found beautiful properties of functions defined on these surfaces, which they related to group theory as part of the rise of abstract algebras.

Another example of the gap between Riemann and Weierstrass is provided by potential theory. Riemann used a principle employed by his mentor Dirichlet (and also envisaged by Green) to solve problems in potential theory, but in 1870 Weierstrass exposed its fallibility by a counterexample, and so methods became far more complicated.

Better news for potential theory had come at midcentury with the "energetics" physics of Thomson, Hermann von Helmholtz (1821–1894), and others. The work expression of engineering mechanics was extended into the admission of potentials, which now covered all physical factors (such as heat) and not just the mechanical ones that had split Carnot from Lagrange. The latter's algebraic tradition in mechanics had been elaborated by Jacobi and by the Irishman William Rowan Hamilton (1805–1865), who also introduced his algebra of quaternions.

Among further related developments, the Scot James Clerk Maxwell (1831–1879) set out theories of electricity and magnetism (including, for him, optics) in his Treatise on Electricityand Magnetism (1873). Starting out from the electric and magnetic potentials as disturbances of the ether rather than Newton-like forces acting at a distance through it, he presented relationships between his basic notions as differential equations (expressed in quaternion form). A critical follower was the Englishman Oliver Heaviside, who also analyzed electrical networks by means of a remarkable but mysterious operator algebra. Other "Maxwellians" preferred to replace dependence upon fields with talk about "things," such as electrons and ions; the relationship between ether and matter (J. J. Larmor, Aether and Matter, 1900) was a major issue in mathematical physics at century's end.

The Early Twentieth Century

A new leader emerged: the German David Hilbert (1862–1943). Work on abstract algebras and the foundations of geometry led him to emphasize the importance of axiomatizing mathematical theories (including the axioms of Euclidean geometry that Euclid had not noticed) and to study their foundations metamathematically. But his mathematical knowledge was vast enough for him to propose twenty-three problems for the new century; while a personal choice, it exercised considerable influence upon the community. He presented it at the International Congress of Mathematicians, held in Paris in 1900 as the second of a series that manifested the growing sense of international collaboration in mathematics that still continues.

One of Hilbert's problems concerned the foundations of physics, which he was to study intensively. In physics Albert Einstein (1879–1955) proposed his special theory of relativity in 1905 and a general theory ten years later; according to both, the ether was not needed. Mathematically, the general theory both deployed and advanced tensor calculus, which had developed partly out of Riemann's interpretation of geometry.

Another main topic in physics was quantum mechanics, which drew upon partial differential equations and vector and matrix theory. One of its controversies concerned Werner Heisenberg's principle of the uncertainty of observation: should it be interpreted statistically or not? The occurrence of this debate, which started in the mid-1920s, was helped by the increasing presence of mathematical statistics. Although probability must have had an early origin in mathematical thinking, both it and mathematical statistics had developed very slowly in the nineteenth century—in strange contrast to the mania for collecting data of all kinds. Laplace and Gauss had made important contributions in the 1810s, for example, over the method of least-squares regression, and Pafnuty Chebyshev (1821–1894) was significant from the 1860s in Saint Petersburg (thus raising the status of Russian mathematics). But only from around 1900 did theorizing in statistics develop strongly, and the main figure was Karl Pearson (1857–1936) at University College, London, and his students and followers. Largely to them we owe the definition and theory of basic notions such as standard deviation and correlation coefficient, basic theorems concerning sampling and ranking, and tests of significance.

Elsewhere, Cantor's set theory and abstract algebras were applied to many parts of mathematics and other sciences in the new century. A major beneficiary was topology, the mathematics of location and place. A few cases had emerged in the nineteenth century, such as the "Möbius strip" with only one side and one edge, Riemann's fantastical surfaces, and above all a remarkable classification of deformable manifolds by Poincaré; most of the main developments, however, date from the 1920s. General theories were developed of covering, connecting, orientating, and deforming manifolds and surfaces, along with many other topics. A new theory of dimensions was also proposed because Cantor had refuted the traditional understanding by mapping one-one all the points in a square onto all the points on any of its sides. German mathematicians were prominent; so were Americans in a country that had risen rapidly in mathematical importance from the 1890s.

Some Reflections

The amount of mathematical activity has usually increased steadily or even exponentially, and the growth from the mid-twentieth century has been particularly great. For example, the German reviewing journal Zentralblatt Math published at the beginning of the twenty-first century a six-hundred-page quarto volume every two weeks, using a classification of mathematics into sixty-three numbered sections. To suggest the rate of increase, the other reviewing journal, the U.S. periodical Mathematical Reviews, published 3,800 octavo pages in 1980, 7,500 pages a decade later, and 9,800 pages in 2003. It would be impossible to summarize this mountain of work, even up to 1970; instead, some main points are noted relating to the previous sections and to the companion articles on algebras and on logic.

Not only has the amount increased; the variety of theories has also greatly expanded. All the topics and branches mentioned above continue to develop (and also many more that were not noted), and new topics emerge and fresh applications are found. For example, beginning with the 1940s mathematics became widely utilized in the life sciences and medicine and has expanded greatly in economics and other social sciences relative to previous practice.

Much of that work lies in statistics, which after its very slow arrival has developed a huge community of practitioners in its own right. Often it functions rather separately from mathematics, with its own departments in universities.

Another enormous change has been the advent of computing, again particularly since World War II and indeed much stimulated by war work as on cryptography and the calculation of parameters in large technological artifacts. Mathematics plays a role both in the design, function, and programming of computers themselves and in the formulation of many mathematical theories. An important case is in numerical mathematics, where approximations are required and efficient algorithms sought to effect them. This kind of mathematics has been practiced continuously from ancient times, especially in connection with all sorts of applications. Quite often algorithms were found to be too slow or mathematically cumbersome to be practicable; but now computer power makes many of them feasible in "number crunching" (to quote a popular oversimplification of such techniques).

A feature of many mathematical theories is linearity, in which equations or expressions of the form
(A) ax by cz … and so on finitely or even infinitely
make sense, in a very wide range of interpretations of the letters, not necessarily within an algebra itself (for example, Fourier series shows it). But a dilemma arises for many applications, for the world is not a linear place, and in recent decades nonlinear theories have gained higher status, partly again helped by computing. The much-publicized theory of fractals falls into this category.

From the Greeks onward, mathematicians have often been fascinated by major unsolved problems and by the means of solving them. In the late 1970s a proof was produced of the four-color theorem, stating that any map drawn upon a surface can be colored with four colors such that bounding regions do not share the same color. The proof was controversial, for a computer was used to check thousands of special cases, a task too large for people. Another example is "Fermat's last theorem," that the sum of the nth powers of two positive integers is never equal to the nth power of another integer when n 2. The name is a misnomer, in that Fermat only claimed a proof but did not reveal it; the modern version (1994) uses modern techniques far beyond his ken.

This article has focused upon the main world cultures, but every society has produced mathematics. The "fringe" developments are studied using approaches collectively known as ethnomathematics. While the cultures involved developed versions of arithmetic and geometry and also some other branches, several of them also followed their own concerns; some examples, among many, are intricate African drawings made in one unbroken line, Celtic knitting patterns, and sophisticated rows of knotted strings called quipus used in Mexico to maintain accounts.

A thread running from antiquity in all cultures, fringe or central, is recreational mathematics. Unfortunately, the variety is far too great even for summary here. Often it consists of exercises, perhaps posed for educational use, or perhaps just for fun; an early collection is attributed to Alcuin in the ninth century, for use in the Carolingian Empire. Solutions sometimes involve intuitive probability, or combinatorics to work out all options; with games such as chess and bridge, however, the analysis is much more sophisticated. Several puzzles appear in slightly variant forms in different cultures, suggesting transmission. Some are puzzles in logic or reasoning rather than mathematics as such, and it is striking that for some games the notion of decidability was recognized (that is, is there a strategy that guarantees victory?) long before it was studied metamathematically in the foundations of mainstream mathematics.

Lastly, since the early 1970s interest in the history of mathematics has increased considerably. There are now several journals in the field along with a variety of books and editions, collectively covering all main cultures and periods. One main motive for people to take up historical research was their dislike of the normal unmotivated way in which mathematics was (and is) taught and learned; thus, the links between history and mathematics education are strong. For, despite many appearances to the contrary, mathematics is a human activity.

See also Algebras ; Astronomy, Pre-Columbian and Latin American ; Calculation and Computation ; Geometry ; Logic ; Logic and Philosophy of Mathematics, Modern .

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Pier, Jean-Paul, ed. Development of Mathematics 1900–1950. Basel, Switzerland, and Boston: Birkhäuser, 1994. This and the follow-up volume focus mostly on pure mathematics.

——. Development of Mathematics 1950–2000. Basel, Switzerland, and Boston: Birkhäuser, 2000.

Roche, John J. The Mathematics of Measurement: A Critical History. London: Athlone Press, 1998.

Scriba, C. J., and P. Schreiber. 5000 Jahre Geometrie. Geschichte, Kulturen, Menschen. Berlin and New York: Springer, 2000.

I. Grattan-Guinness

Mathematics

Ancient mathematics

Classical analysis

Theory of numbers

Algebra

Vector spaces and matrix algebra

Topology and abstract spaces

Foundations

BIBLIOGRAPHY

The history of mathematics, and to some extent its content, can be thought of as involving three major phases. Ancient mathematics, covering the period from the earliest written records through the first few centuries A.D., culminated in Euclidean geometry, the elementary theory of numbers, and ordinary algebra. Equally important, this phase saw the evolution and partial clarification of axiomatic systems and deductive proofs. The next major phase, classical mathematics, began more than 1,000 years later, with the Cartesian fusion of geometry and algebra and the use of limiting processes in the calculus. From these evolved, during the eighteenth and nineteenth centuries, the several aspects of classical analysis. Other contributions of this phase include non-Euclidean geometries, the beginnings of probability theory, vector spaces and matrix theory, and a deeper development of the theory of numbers. About a hundred years ago the third and most abstract and demanding phase, known as modern mathematics, began to evolve and become separate from the classical period. This phase has been concerned with the isolation of several recurrent structures of analysis worthy of independent study—these include abstract algebraic systems (for example, groups, rings, and fields), topological spaces, symbolic logic, and functional analysis (Hilbert and Banach spaces, for example)—and various fusions of these systems (for example, algebraic geometry and topological groups). The rate of growth of mathematics has been so great that today most mathematicians are familiar in detail with the major developments of only a few branches of the subject.

Our purpose is to give some hint of these topics. The reader interested in a somewhat more detailed treatment will find the best single source to be Mathematics: Its Content, Methods, and Meaning, the translation of a Russian work (Akademiia Nauk S.S.S.R. 1956). Other general works are Courant and Robbins (1941), Friedman (1966), and Newman (1956). More specific references are given where appropriate. We do not here discuss probability, mathematical statistics, or computation, even though they are especially important mathematical disciplines for the social sciences, because they are covered in separate articles in the encyclopedia.

Ancient mathematics

The history of ancient mathematics divides naturally into three periods. In the first period, the pre-Hellenic age, the beginnings of systematic mathematics took place in ancient Egypt and in Mesopotamia. Contrary to much popular opinion, the mathematical developments in Mesopotamia were deeper and more substantial than those in Egypt. The Babylonians developed elementary arithmetic and algebra, particularly the computational aspects of algebra, to a surprising degree. For example, they were able to solve the general quadratic equation, ax² + bx + c = 0. An authoritative and readable account of Babylonian mathematics as well as of Greek mathematics is presented by Neugebauer (1951).

The second period of ancient mathematics was the early Greek, or Hellenic, age. The fundamentally new step taken by the Greeks was to introduce the concept of a mathematical proof. These developments began around 600 b.c. with Thales, Pythagoras, and others, and reached their high points a little more than a century later in the work of Eudoxus, who is responsible for the theory of proportions, which in antiquity held the place now held by the modern theory of real numbers.

The third period is the Hellenistic age, which extended from the third century b.c. to the sixth century a.d. The early part of this period, sometimes called the golden age of ancient mathematics, encompassed Euclid’s Elements (about 300 b.c.), which is the most important textbook ever written in mathematics, the work on conics by Apollonius (about 250 b.c.), and above all the extensive and profound work of Archimedes on metric geometry and mathematical physics (Archimedes died in 212 b.c.). The second most important systematic treatise of ancient mathematics, after Euclid’s Elements, is Ptolemy’s Almagest (about a.d. 150). Ptolemy systematized and extended Greek mathematical astronomy and its mathematical methods. The mathematical sophistication of Archimedes and the richness of applied mathematics evidenced by the Almagest were not equaled until the latter part of the seventeenth century.

Classical analysis

The intertwined and rapid growth of mathematics and physics during the seventeenth, eighteenth, and nineteenth centuries centered in a major way on what is now called classical analysis : the calculus of Newton and Leibniz, differential and integral equations and the special functions that are their solutions, infinite series and products, functions of a complex variable, extremum problems, and the theory of transforms. At the basis of all this are two major ideas, function and limit. The first evolved slowly, beginning with the correspondence, established in the Cartesian fusion of the two best-developed areas of ancient mathematics, between algebraic expressions and simple geometric curves and surfaces, until we now have the present, very simple definition of the term “function” A set ｆ of points in the plane (ordered pairs of numbers) of the form (x,y) is called a function if at most one y is associated with each x. If (x, y) is a member of ｆ, it is customary to write y = f(x); x is sometimes called the independent variable and y the dependent variable, but no causal meaning should be read into this terminology.

The notion and notation may be generalized to more than one independent variable; if g is a set of ordered triples (x, y, z) with at most one z associated with each pair (x,y), then z =g(x, y) is called a function of two arguments. Since the most general notion of function can relate any two sets of objects, not just sets of numbers, it is sometimes desirable to emphasize the numerical character of the function. Then ｆ is said to be a real-valued function of a real variable; here the term “real” refers to real numbers (in contrast to complex numbers, which will be discussed later).

Although a real-valued function has been defined as a set of ordered pairs of numbers, (x,y), where the domain of x is is an unspecified set of numbers, the subsequent discussion of functions is mostly confined to the familiar case in which the domain of x is an interval of numbers. Even when the discussion applies more generally, it is helpful to keep the interval case in mind.

A desire to understand limits was apparent in Greek mathematics, but a correct definition of the concept eluded the Greeks. A fully satisfactory definition, which was not evolved until the nineteenth century (by Augustin Louis Cauchy), is the following: b is the limit of ｆ at a if and only if for every positive number є there is a positive number δ such that, when the absolute value of x—a is less than є and greater than 0 (that is, 0 <│x - a│ < δ), the absolute value of f(x) – b is less than є (that is, │f(x) – b│ < є). In other words, b is the limit of ｆ at a if x can be chosen sufficiently close to a (but not equal to a) to force f(x) to be as close to b as de-sired. Symbolically, this is written lim_x→af(x) = b. The limit of ｆ at a may exist even though f(a) is not defined; moreover, when f(a) is defined, b may or may not equal f(a). If it does—that is, if f(x) is “near” f(a) whenever x is “near” a—then f is

said to be continuous at a. If ｆ is continuous at each a in an interval, ｆ is said to be continuous over that interval.

The calculus

The calculus defines two new concepts, the derivative and the integral, in terms of function and limit. They and their surprising relationship serve as the basis of the rest of mathematical analysis.

The derivative. The first definition arises as the answer to the question “Given a function ｆ, what is its slope (or, equivalently, its direction or rate of change) at any point x?” For example, suppose that y = ｆ(x) represents the distance, y, that a particle has moved in x units of time; then what is the rate of change of distance—the instantaneous velocity—at time x? If h is a short period of time, then an approximate answer is the distance traversed between x and x + h, that is, ｆ(x + h)— ｆ(x), divided by the time, h, taken to travel that distance (see Figure 1). The approximation is better the smaller the value of h, which suggests the definition of the rate of change of ｆ at x as the limit of this ratio as h approaches 0, that is,

This limit, if it exists, is denoted by ｆ(x) (or by dｆ(x)/dx or by dy/dx) and is called the derivative of ｆ at x. If f(x) exists, then ｆ can be shown to be continuous at x, but the converse is not true in general.

One of the earliest and most important applications in the social sciences of the concept of a derivative has been to the mathematics of marginal concepts in economics. For example, let x represent output, C(x) the cost of output x, and R(x) the revenue derived from output x; then C’(x) and R’(x) (or dC(x)/dx and dR(x)/dx) are the marginal cost and marginal revenue, respectively. Marginal utility, marginal rate of substitution, and other marginal concepts are defined in a similar fashion. Many of the fundamental assumptions of economic theory receive precise formulation in terms of these marginal concepts.

The integral. The second concept in the calculus arises as the answer to the question “What is the area between the graph of a function ｆ and the line y = 0 (the horizontal axis, or abscissa, of the coordinate system) over the interval from a to b?” (Regions below the abscissa are treated as negative areas to be subtracted from the positive ones above the abscissa; see Figure 2.) The solution, which will not be stated precisely, involves the following steps: the abscissa is partitioned into a finite number of intervals; using the height of the function at some value within each interval, the function is approximated by the resulting step function; the area under the step function is calculated as the sum of the areas of the rectangles of which it is composed; and, finally, the limit of this sum is calculated as the widths of the intervals approach zero (and, therefore, as their number approaches infinity). When this limit exists, it is called the Riemann integral of f from a to b and is symbolized as ∫^b_aｆ(x) dx. It can be shown that the Riemann integral exists if ｆ is continuous over the interval; it also exists for some discontinuous functions. For more advanced work, the concept of the length of an interval is generalized to the concept of the Lebesgue measure of a set, and the Riemann integral is generalized to the Lebesgue integral Roughly, the vertical columns used to approximate the area in the Riemann integral are replaced in the Lebesgue integral by horizontal slabs.

Although the interpretation of the integral as an extension of the elementary concept of area is

important, even more important is its relation (called the fundamental theorem of the calculus) to the derivative: Consider as a function of the upper limit, x, of the interval over which the integral is computed; it can then be proved that the derivative of this function, F’(x) exists and is equal to f(x). Put another way, the rate of change at x of the area generated by ｆ is equal to the value of ｆ at x; or put still another way, the operation of taking the derivative undoes the operation of integration. This fact plays a crucial role in the solution of many problems of classical applied mathematics that are formulated in terms of derivatives of functions.

Introductions to the calculus and elementary parts of analysis are Apostol (1961–1962) and Bartle (1964).

Implicit definitions of functions

An algebraic equation such as x² – 5 x – 3 = 0 implicitly defines two numbers (namely, the two values of x, 3 and –½) for which the equality holds. Other algebraic equations implicitly define sets of numbers for which they hold.

A functional equation is an equality stated in terms of an unknown function; it implicitly defines those functions (as in the algebraic case, there may be more than one) that render the equality true.

Ordinary differential equations. Suppose it is postulated that the amount of interest (that is, the rate of change of money at time t) is proportional to (that is, is a constant fraction, k of) the amount, ｆ(t), of money that has been saved. (This is the case of continuous compound interest.) Then ｆ satisfies the equation ｆ'( t ) = kｆ(t). This is a simple example of an ordinary differential equation, the solution of which is any function having the property that its derivative is k times the function. The solutions are f(t) = f(0) exp(kt), where f(0) denotes the initial amount of money at time t = Q. Another simple economic example is the differential equation that arises from the assumption that marginal cost always equals average cost (that is, dC (x)/dx = C(x)/x) which has the solution that average cost is constant, that is, that C(x) = kx for some constant, k.

Some laws of classical physics are formulated as second-order, linear, ordinary differential equations of the form

ｆ” (t) + P(t)ｆ’(t) + Q(t)ｆ(t) = R(t),

where ｆ” is the derivative of ｆ(ｆ” is called the second derivative of ｆ ) and P, Q, and R are given functions. If, for example, ｆ denotes distance, then this differential equation asserts that at each time t, a linear relation holds among distance, velocity, and acceleration.A vast literature is concerned with the solutions to this class of equations for different restrictions on P, Q, and R; most of the famous special functions used in physics—Bessel, hypergeometric, Hankel, gamma, and so on—are solutions to such differential equations (see Coddington 1961).

Partial differential equations Many physical problems require differential equations a good deal more complicated than those just mentioned. For example, suppose that there is a flow of heat along one dimension, x. Let ｆ(x,t ) denote the temperature at position x at time t. With t fixed, one can find the rate of change (the derivative) of temperature with changes in x; denote this by ∂ｆ(x,t)/∂x and its second derivative with respect to x by ∂² ｆ(x,t)/∂x². These are called partial derivatives. Similarly, holding x fixed, the derivative with respect to t is denoted by ∂ｆ(x,t)/∂t. According to classical physics, temperature changes due to conduction in a homogeneous one-dimensional medium satisfy the following partial differential equation:

where ｆ is the thermal conductivity, p the density, and σ thespecific heat of the medium. Problems involving two or more independent variables (usually, time and some or all of the three space coordinates)—fluid flow, heat dissipation, elasticity, electromagnetism, and so on—lead to partial differential equations. Their solution is often very complex and requires the specification of the unknown function along a boundary of the space. This requirement is called a boundary condition. (See Akademiia Nauk S.S.S.R. [1956] 1964, chapter 6.)

Integral equations. Some physical problems lead to integral equations. In one type, functions g and K of one and two variables, respectively, and a constant, λ are given, and the problem is to find those functions, f, for which

This equation is called Fredholm’s linear integral equation or the inhomogeneous linear integral equation. Basically, it asserts that the value of some quantity ｆ at a point x is equal to an impressed value, g(x), plus a weighted average of its value at all other points. Integral equations arise in empirical contexts for which it is postulated that the value of a function at a point depends on the behavior of the function over a large region of its domain. Thus, in the example just considered the value of ｆ at x depends on the integrand K(x,y)ｆ(y) integrated over the interval (a,b). There is a large body of literature dealing with the solution of various types of integral equations, especially those of interest in physics and probability theory.

Functional equations. Although both differential and integral equations (and mixtures of the two, called integrodifferential equations) are examples of functional equations, that term is often restricted to equations that involve only the unknown function, not its derivatives or integrals. A simple, well-known example is ｆ(xy) =ｆ(x) + ｆ(y), which implicitly defines those functions that trans-form multiplication into addition. If ｆ is required to be continuous, then the solutions are where K is a positive constant; this integral is called the natural logarithm. The choice of K is usually referred to as the selection of the base of the logarithm.

Difference equations are functional equations of special importance in the social sciences. They arise both in the study of discrete stochastic processes (in learning theory, for example) and as discrete analogues of differential equations. Here the unknown function is defined only on the integers (or, equivalently, on any equidistant set of points), not on all of the real numbers, and so the function is written ｆn = ｆ ( n ), where n is an integer. The equation states a relation among values of the unknown function for several successive integers. For example, the second-order, linear difference equation—the analogue of the second-order, linear, ordinary differential equation, described above—is of the form.

ｆ_n+2 + P_nｆ_n+1 + Q_nｆ_n = R_n.

In some probabilistic models of the learning process it is postulated (or derived from more primitive assumptions) that the probability of a particular response on trial n+ 1, denoted by p_n+1, is some function of p_n and of the actual events that occurred on trial n. The simplest such assumption is the linear one, that is, p_n+1 = αp_n + β where α and β are parameters that depend upon the events that actually occur. If there is a run of trials during which the same events occur, so that α and β are constant, then the solution to the above first-order, linear difference equation is

When different events occur on different trials, the equation to be solved becomes considerably more complex. An introduction to difference equations is Goldberg (1958).

Given a functional equation—in the most general sense—the answer to the question of whether a solution exists is not usually obvious. Exhibiting a solution, of course, answers the question affirmatively, but often the existence of a solution can be proved before one is found. Such a result is known as an existence theorem. If a solution exists, it is also not usually obvious whether it is unique and, if it is not unique, how two different solutions relate to one another. A statement of the nature of the nonuniqueness of the solutions is known, somewhat inappropriately, as a uniqueness theorem. Some rather general existence and uniqueness theorems are available for differential and integral equations, but in less well understood cases considerable care is needed to discover just how restrictive the equation is.

A general work on functional equations is Aczel (1966).

Three other areas of classical analysis

Three other branches of classical analysis will be briefly discussed.

Extremum problems For what values of its argument does a function assume its maximum or its minimum value? This type of problem arises in theoretical and applied physics and in the social sciences. In its simplest form, a real-valued function ｆ is defined over some interval of the real numbers, and the problem is to find those x_° for which ｆ(x_°) is a maximum or a minimum. If ｆ is differentiable and if x_° is not one of the end points of the interval, a necessary condition is that f'(x_°) = 0; moreover, x _° is a local maximum if ｆ(x_°) <0 and a local minimum ifｆ (x_°) >0. (These statements should be intuitively clear for graphs of simple functions.) From these results it is easy to find, for example, which rectangle has the maximum area when the perimeter is held constant: it is the square whose sides are each equal to a quarter of the perimeter.

A much more difficult and interesting problem—the subject of the calculus of variations—is to find which function (or functions) ｆ of a given family of functions causes a given function ｆ of ｆ (known as a functional) to assume its maximum or minimum value. For example, let ｆ be a continuous function that passes through two fixed points in the plane, and let F(ｆ) be the surface area of the body that is generated by rotating ｆ about the abscissa. A question that may be asked is “For which ｆ (or ｆ’s) is F(f) a minimum?” A major tool in the solution of this problem is a second-order, ordinary differential equation, known as Euler ’s equation, that ｆ must necessarily satisfy (just as the solution x₀ to the simpler problem necessarily satisfies ｆ '(x_O) = 0). (See Akademiia Nauk S.S.S.R. [1956] 1964, chapter 8.)

Within the past twenty years new classes of extremum problems have been posed and partially solved; they are mainly of concern in the social sciences, and they go under the names of linear, non-linear, and dynamic programming. An example of a linear programming problem is the following diet problem. Each of several foodstuffs, ｆ₁,ｆ₂,..., ｆ_k, contains known amounts of various nutritional components, such as vitamins and proteins. Let f_ij be the amount of component j in food f_i, j = 1, 2,..., n, and let a, be the minimum amount of component j acceptable in the diet. If X_i is the amount of food f_i in the diet, the diet will be acceptable only if the following n inequalities are fulfilled:

x₁ ｆ₁ + x₂ｆ_2j +...+ x_k ｆ_kj ≥ a_i, j = 1,2,...,n.

If P_i denotes the price of food ｆ_i, the problem is to choose the x_i so as to minimize the cost,

x₁p₁+ x₂p₂+...+ x_kp_k,

while fulfilling the above linear inequalities. [SeeProgramming.]

Functions of a complex variable One of the most beautiful subfields of analysis is the theory of functions of a complex variable, which was developed in the nineteenth century, starting with the work of Cauchy. It has been significant in the growth of several two-dimensional, continuous physical theories, including parts of electromagnetism, hydrodynamics, and acoustics, but so far its applications in the social sciences have been mainly restricted to mathematical statistics, as in the concept of the characteristic function of a probability distribution. A complex number, z, is of the form z = x+iy, where x and y are real numbers and i = Sums and products are defined in such a way that the resulting arithmetic reduces to that of the ordinary numbers when y = 0. Because a point (x,y) in the plane can be (usefully) identified with the complex number x + iy, functions from the plane into the plane can be interpreted as complex-valued functions of a complex variable. If the derivative of such a function exists at all points of a region, derivatives of all orders exist and the function can be expressed as a convergent power series of the form α₀ + α₁,z + α₂,z²+...+ for some circle of z’s within that region. It is clear from this result that the mere supposition that the derivative exists is a much stronger condition for complex-valued functions than for ordinary numerical functions. Such functions, which are called analytic, are very strongly constrained—among other things, specifying an analytic function over a small region determines it completely—and this fact has been effectively exploited to solve many two-dimensional problems of theoretical and practical interest. Interestingly, the theory cannot be neatly generalized beyond two dimensions. An introductory work on functions of a complex variable is Cartan (1961).

Integral transforms. Suppose that ｆ is any continuous, real-valued function defined over an interval from α to b and that K is a fixed, continuous, real-valued function of two variables, the first of which is also on the interval from α to b; then is called an integral transform of ｆ. If K satisfies certain restrictions, knowing I is equivalent to knowing ｆ. Nevertheless, if K is carefully chosen, I may have convenient properties not possessed by ｆ. For example, if α = 0, b=∞, and K(x,y) = e_-xy, then I, which is then known as the Laplace transform and which is closely related to the moment-generating function of statistics, has the property that it converts certain integrals (convolutions) of two functions into multiplications of their transforms. In statistics such a convolution represents the distribution of the sum of two independent random variables. Another well-known and important example is the Fourier transform, which is used widely in statistics, and to a lesser extent in probabilistic models of behavior, to obtain a probability distribution from its characteristic function.

Theory of numbers

Despite several intellectual crises that led mathematicians to introduce new types of numbers into mathematics, it was not until about a hundred years ago that numbers were treated as being something other than intuitively understood. The natural numbers, 1,2,3,..., and their ratios, the positive rationals, are ancient concepts. The Greeks first noted their incompleteness when they showed that they are inadequate to represent the length of the diagonal of a square whose side is of length 1. Certain irrational numbers had to be added, and later 0, negative numbers, and complex numbers were added so that certain classes of equations would all have solutions. To clarify this patchwork and to understand the uniqueness of the additions, nineteenth-century mathematicians undertook the axiomatization of various aspects of the number system. Perhaps the most subtle step was the definition of irrational numbers in terms of sets of rational numbers (roughly, the set of all rationals less than the irrational to be defined).

The axiomatization of numbers is not really the mainstream of the “theory of numbers.” When one sees a book or course with that title, it usually refers to the study of properties of the natural numbers, mainly the prime numbers. Recall that an integer is prime if it is divisible only by 1 and itself; the first few primes are 3, 5, 7, 11, and 13. In addition to the many results that can be proved directly (some of which were known to the ancients), such as that every integer can be represented uniquely as the product of powers of primes and that there are infinitely many primes, other results have depended upon the application of deep results from analysis. For example, parts of the theory of functions of a complex variable were used to show that the number of primes not larger than n divided by the number n/lnn, where Inn is the natural logarithm of n, that is, is a ratio that approaches 1 as n becomes large. Not only has this work greatly increased the depth of understanding of integers, but it has fed back into analysis and was one of the factors leading to the development of parts of contemporary abstract algebra.

Many applications of mathematics (for example, in statistics) involve counting the number of distinct events or objects that satisfy certain conditions; often these counting problems are quite difficult. Theorems providing explicit formulas or recursion schemes are called combinatorial theorems. One of the earliest important examples was the binomial theorem for the expansion of (a + b)ⁿ, which is now part of every elementary algebra course. [SeeProbability, article on Formal Probability.]

A general introduction to the theory of numbers is Ore (1948).

Algebra

Classically, algebra was the theory of solving equations expressed in terms of the four arithmetical operations—addition, subtraction, multiplication, and division. The linear and quadratic equations of elementary algebra are familiar examples. Historically, the expression of mathematical problems in the form of equations, using letters to stand for the unknown numbers, was a major step in clarifying and simplifying the mathematical nature of many kinds of problems. Perhaps the most important consequence of the introduction of letters and the use of equations was the extension of routine methods of calculation to quite complicated settings. The introduction of algebraic equations probably ranks in importance in the history of ideas with the earlier invention, probably first by the Babylonians, of the place-value system of notation for numbers; such a system was needed to develop simple algorithms for performing arithmetical computations.

The general theory of algebraic equations, the elementary parts of which are studied in high school, has a long and distinguished history in mathematics. The proof by Niels Henrik Abel in 1824 that solutions of an algebraic equation of degree five or greater, where the degree is the highest exponent of any term in the equation, cannot be expressed in terms of radicals (that is, expressions definable in terms of square roots) was one of the most important mathematical results of the first half of the nineteenth century. Another result of basic importance is the fundamental theorem of algebra, which was first proved in the eighteenth century but which was proved rigorously only in the last half of the nineteenth century. This theorem asserts that every algebraic equation always has at least one root that is a real or a complex number. Also of great significance were the proofs that not all numbers are roots of algebraic equations; numbers that are not such roots are called transcendental numbers. The most famous proofs of this sort are Charles Hermite’s (in 1873) that e is transcendental and F. Lindemann’s (in 1882) that π is transcendental.

Orderings

Much of the work in algebra during the present century has been devoted to generalized mathematical systems that are characterized not in terms of the four fundamental arithmetical operations but in terms of generalizations of these operations and of the familiar ordering relations of “less than” and “greater than.”

In a number of the social sciences the theory of binary relations has received extensive application. From an algebraic standpoint a binary relation structure may be characterized as consisting of a set A and a set R of ordered pairs (x,y), where x and y are both elements of A. Such an R is called a binary relation on A. A relation R is said to be a partial ordering of A when it is reflexive, antisymmetric, and transitive—that is, when it satisfies the following three properties: reflexive; for every x in A, xRx; antisymmetric: for every x and y in A, if xRy and yRx, then x – y; transitive: for every x, y, and z in A, if xRy and yRz, then xRz. If R is also connected in A (that is, if for any two elements x and y in A with x ≠ y, either xRy or yRx) then R is said to be a complete or simple ordering or, sometimes, a linear ordering of A. The concept of a complete ordering is a direct abstraction of the order properties of “≤” with respect to the real numbers. A familiar use of the concept of an ordering relation is in utility theory, particularly in the classical theory of demand in economics, in which it is assumed that each individual has an ordering relation over the set of commodity bundles or, more generally, over the set of alternatives with which he is presented. The general concept of ordering relations also has far-ranging applications in the theory of measurement within psychology and sociology, and more general binary relations have been extensively applied in anthropology in the study of kinship systems.

Partial orderings can be extended in another direction by imposing additional conditions to obtain lattices, which have also been used in the social sciences. In a different direction, but still within the framework of binary relations, is the theory of graphs, in which no restrictions are placed on the binary relation, R. Applications of graph theory have been made to social-psychological and sociological problems, especially to provide a mathematical method for representing various kinds of relationships between persons.

Groups, rings, and fields

Another direction of generalization of classical algebra has been to what are called groups, rings, and fields. A group is a set A together with a binary operation, o, satisfying the following axioms. First, the operation o is associative, that is, for x, y, and z in A, x o (y o z) =(x o y) o z. Second, there is an element e, called the identity, of the set A such that for every x in A, xoe = eo x = x. And, finally, for each element x of A there is an inverse element x^-1 such that x o x^-1 = e. It is obvious that if A is taken as the set of integers, o as the operation of addition, e as the number 0, and the inverse of x as the negative of x, then the set of integers is a group under the binary operation of addition. The theory of groups has had profound ramifications in other parts of mathematics and in the sciences, ranging from the theory of algebraic equations to geometry and physics. The reason for the fundamental importance of group theory is perhaps best summarized by stating that a group is the appropriate way to formulate the very important concept of symmetry. In the range of applications of group theory just mentioned, the underlying thread is the concept of symmetry, whether it is in the symmetry of the roots of an equation or the symmetry properties of the fundamental particles of physics. As a simple example, consider the finite group of rotations 90°, 180°, 270°, and 360°. A square does not change its apparent orientation under such a rotation about its center, but an equilateral triangle does. This group of rotations is the symmetry group of rotations for a square but not, of course, for an equi-lateral triangle. Although the methods and results of group theory have not yet had special applications of depth in the social sciences, they are important to many of the general mathematical results that have been applied.

The theories of rings and fields represent rather direct generalization of arithmetical properties of the number system. The theory of groups is fun damentally a generalization of the concept of a single binary operation, such as addition or multiplication, whereas rings and fields are algebraic systems that have two fundamental operations. The most familiar example of a field or of a ring is the set of rational numbers or of real numbers with respect to the operations of addition and multiplication.

Boolean algebras

Algebraic aspects of the theory of sets have been studied under the heading of Boolean algebras. The concept of an algebra of sets, that is, a collection of sets closed under union and complementation, is fundamental in the modern theory of probability, where events are interpreted as sets of possible outcomes and numerical probabilities are assigned to events. [SeeProbability, article on Formal Probability.]

Isomorphism and homomorphism

It should be mentioned that certain very general mathematical concepts find their most natural definition and application in modern algebra. One of the most important concepts is that of the isomorphism of two mathematical systems. An isomorphism is a oneto-one mapping of a system A onto a system B in which the operations and relations of A are preserved under the mapping and have the same structure as the operations and relations of system B. If the mapping is not one-to-one but the operations and relations are preserved, then it is called a homomorphism. A well-known application of the concept of isomorphism in the social sciences is in theories of fundamental measurement in which one shows that an appropriate algebra of empirical operations is isomorphic to some numerical algebra. It is this isomorphism that permits the direct application of computational methods to the results of measurement.

Introductory works on algebra, both for this and for the next section, are Birkhoff and MacLane (1941) and Mostow, Sampson, and Meyer (1963).

Vector spaces and matrix algebra

Linear algebra is one of the most important generalizations of classical elementary algebra. The objects to which the operations of addition and multiplication are applied are now matrices, vectors of an n-dimensional space, and linear transformations (an n × n matrix is a particular representation of a linear transformation in n-dimensional space). More particularly, linear algebra arises as a generalization of the linear equations so familiar in elementary algebra, and historically one of the most important tasks of linear algebra has been to find solutions of systems of linear equations. As many research workers in the social sciences know, the numerical solution of linear equations can be an extremely laborious and difficult affair when the number of equations is large. The set of coefficients of a system of linear equations gives rise to the concept of a rectangular array of numbers, which is precisely what a matrix is. An algebra of matrices in terms of addition and multiplication may be constructed; the distinguishing feature of this algebra, as compared with the algebra of the real numbers, is that multiplication is not commutative—that is, AB is not usually equal to BA, and the product of two nonzero matrices can be zero.

The intuitive geometric concept of a vector may be represented by a column or row of n numbers, and an algebra of vectors, which bears a close resemblance to the algebra of numbers, may be constructed. Simple (linear) transformations of vectors, such as rotations and stretches of the co-ordinate system in space, can be interpreted as multiplication by matrices. The interaction between the geometrical intuitions about n-dimensional space and the algebraic techniques of calculation provided by linear algebra and the theory of matrices have made them powerful tools in the application of mathematics to many parts of science. These applications have been particularly prominent in statistics (for example, in factor analysis), as well as in economics, where it is often useful to treat n-dimensional bundles of commodities as vectors.

Topology and abstract spaces

Intuitively, a topological transformation of a geometrical figure or object is a deformation that introduces neither breaks nor fusions in the object. Put more exactly, a topological transformation is one that is one-to-one, is continuous, and has a continuous inverse. If one starts with a circle—perhaps the best example of a simple closed curve—one can deform it topologically into an ellipse or into the shape of a crescent, but one cannot deform it topologically into a figure eight, for example, because then two distinct points of the circle are fused as the intersection point of the eight. Also, one cannot deform it into a straight line segment, because to do so would introduce a break in the circle. Many familiar qualitative geo-metrical properties are topological invariants in the sense that they are not altered (are invariant) under topological transformations. Examples are the property of being inside or outside a closed figure in the plane; the property of a surface being closed, such as the surface of a sphere or an ellipsoid; or the property of the dimension of an object. For example, the surface of a sphere cannot be topologically transformed into a one-dimensional curve or a three-dimensional sphere. We shall not attempt here to give an exact definition of continuity as it is used in topology; we simply remark that it is a reasonable generalization of the concept of continuity used in analysis.

Topological methods and results have far-reaching applications in many branches of mathematics, but as yet the methods themselves have not been directly applied in those parts of the social sciences concerned extensively with empirical data. The most direct applications have been in economics, where topological fixed-point theorems have been of great importance in investigating the conditions guaranteeing the existence of a stable equilibrium in a competitive economy. The classical example of a fixed-point theorem—first proved by L. E. J. Brouwer, at the beginning of this century—states that for every topological mapping of an n-dimensional sphere into itself there is always at least one point that maps into itself, that is, remains fixed. Familiar examples of such mappings are rotations in two or three dimensions for which the center of the rotation is the fixed point of the transformation.

Topological space

As a typical example of abstraction in modern mathematics, the intial concept of a topological transformation of familiar geometrical figures has led to the general abstract notion of a topological space. Roughly speaking, a topological space consists of a set, X , and a family, ℱ of subsets of X , called open sets, for which the following four conditions are satisfied: the empty set is inℱ X is in ℱ the union of arbitrarily many sets each of which is in ℱ is also in ℱ and the intersection of any finite number of sets from ℱ is also in ℱ. The concept of an open set is a generalization of the notion of an open interval of real numbers (an interval that does not include its end points). For example, the natural topology of the real line is the family of open intervals together with the sets that are formed from arbitrary unions and finite intersections of open intervals. Generally speaking, the notion of open set is used to express the idea of continuity. The important thing about a continuous function is that it does not jumble neighboring points too much, and this requirement may be expressed by requiring of a topological transformation that open sets be mapped into open sets and that the inverse of an open set be an open set.

Metric space Other kinds of abstract spaces have come into prominence in the development of topology. Perhaps the most important is the concept of a metric space. A set, X , together with a distance function, d, that maps pairs of points into real numbers is called a metric space if d satisfies the following conditions: d(x,y) = 0 if and only if x = y, that is, the distance between x and y is 0 if and only if x and y are the same point; d(x,y) ≥ 0, which asserts that distance is a non-negative real number; d(x,y) = d(y , x), that is, distance is symmetric; and, finally, d(x, y) + d(y, z) ≥ d(x, z), which is known as the triangle inequality. The concept of a metric space has had important applications in many parts of mathematics and is a fundamental concept in modern mathematics. It has been applied in recent work in scaling theory in psychology and sociology, particularly to the problems of multidimensional scaling, and also in certain areas of mathematical economics [see Scaling]. It is clear that the notion of a metric space generalizes, in a very natural way, the concept of distance in Euclidean space.

A typical metric problem raised in the social sciences is this: Given data in the form of “distances” among a finite set of points, what is the smallest dimensional Euclidean space within which the points can be embedded so that these distances equal the Euclidean or some other preassigned metric of that space? Recently this problem has been effectively generalized by permitting certain transformations of the “distances” that preserve their metric property. Little has yet been done about embeddings in non-Euclidean spaces.

An introductory work on topology is Hocking and Young (1961).

Foundations

As was remarked above, the concept of a rigorous mathematical proof originated in ancient Greek mathematics. The modern formal axiomatic method, characteristic of twentieth-century mathematical research and one of the most important topics to be clarified in modern research on foundations of mathematics, is conceptually very close to the approach followed in Euclid ’s Elements. The main difference is that the primitive concepts of the theory are now treated as undefined or meaningless. All that is assumed about them must be formally expressed in the axioms. In contrast, in the Elements primitive concepts such as those of point and line are given an interpretation or meaning from the very beginning. This modern conception originated with David Hilbert, who provided the first complete, modern axiomatization of geometry in 1889. It is customary to say that the concepts of the theory are implicitly defined by the axioms. What is not recognized often enough is that the collection of axioms together explicitly de fines the theory embodied in the concepts. Thus, in slightly more exact phrasing, the axioms of Euclidean geometry define the theory of Euclidean geometry by defining the phrase “is a model of Euclidean geometry.” In the same fashion, the axioms of group theory define the theory of groups by specifying what kinds of objects are called groups or, in other words, what kinds of objects are models of the theory of groups (here we are using the term “model” in the logical or mathematical sense).

A more particular aim of foundational research has been to provide a set of axioms that would serve as a basis for the main body of mathematics. At least three major positions on the foundations of mathematics have been enunciated in the twentieth century; they differ in their conception of the nature of mathematical objects.

Intuitionism

Intuitionism holds that in the most fundamental sense mathematical objects are themselves thoughts or ideas. The intuitionist holds that one can never be certain that he has correctly expressed the mathematics when it is formalized as a mathematical theory. As part of this thesis, the classical logic of Aristotle, in particular the law of excluded middle, has been challenged by Brouwer and other intuitionists because it permits the derivation of purely existential, nonconstructive statements about mathematical objects. In particular the validity of classical reductio ad absurdum proofs depends upon this logical law. Although intuitionists express themselves in a way which suggests a psychological analysis of mathematics, it should be emphasized that their conception of mathematical objects as thoughts has not been seriously explored by any intuitionists from the standpoint of scientific psychology.

Platonism

A second view of mathematics, the Platonistic one, is that mathematical objects are abstract objects that exist independently of human thought or activity. Those who hold that set theory or logic itself provides an appropriate foundation for mathematics (adherents of logicism) usually adopt some form of Platonism in their basic attitude. From the standpoint of working mathematics, set theory—and thus Platonism—has been the most influential conception of mathematics in this century. Set theory itself originated in the late nineteenth century with the revolutionary work of Georg Cantor. Its foundations were called into question by Bertrand Russell’s discovery of a simple paradox which arises in considering the set of all objects that are not members of themselves. If it is supposed that to every property there corresponds the set of objects having this property, then a contradiction within classical logic may easily be derived by considering the set whose members are those and only those sets that are not members of themselves. An apparently satisfactory foundation for set theory, which avoids this and related paradoxes, was formulated in 1908 by Ernst Zermelo, and with suitable technical extensions it provides a satisfactory basis for most of the mathematics published in this century.

Formalism

The third influential position on the foundation of mathematics, called formalism, was developed by Hilbert and others. This view is that the primary mathematical objects are the symbols in which mathematics is written. This carries to the extreme the development of the axiomatic method begun by the Greeks. Under the formalist account the interpretation and use of mathematics must then be given from outside pure mathematics. From a psychological or behavioral standpoint, there is much that is appealing about formalism, but again little effort has yet been made to relate the detailed results and methods of formalism to theoretical or experimental work in scientific psychology.

Relevance of research on foundations

In view of the high degree of agreement about the validity of most published pieces of mathematics, the skeptical social scientist may question the real relevance of these varying views about the foundations of mathematics to working mathematics itself. There is a highly invariant content of mathematics recognized by almost all mathematicians, including those concerned with the foundations of mathematics, and this invariant content is essentially untouched by radically different philosophical views about the nature of mathematical objects. A reasonable conjecture is that future research in the foundations of mathematics will attempt to capture this invariant content by concentrating on the character of mathematical thinking rather than on the nature of mathematical objects.

One other important aspect of foundational research in the twentieth century is the fundamental work on mathematical logic, in particular the attempt by Gottlob Frege, A. N. Whitehead, Bertrand Russell, and others to reduce all of mathematics to purely logical assumptions. These efforts have led to great clarification of the nature of mathematics itself and to vastly increased standards of precision in talking about mathematical proofs and the structure of mathematical systems. Of major importance were the deep results of Kurt Godel (1931) on the logical limitations of any formal system rich enough to express elementary number theory. His results show that any such formal system must be essentially incomplete in the sense that not all true sentences of the theory can be proved as theorems.

An introductory work on foundations is Knee-bone (1963).

Mathematics applied to social sciences

Applications of mathematics to specific social science problems are described, and detailed references are given, elsewhere in this encyclopedia. That material is not repeated here; several reason-ably general references are Allen (1938), Coleman (1964), Kemeny and Snell (1962), Luce (1964), Luce, Bush, and Galanter (1963-1965), Samuelson (1947). Suffice it to say that these applications involve only fragments of the whole of mathematics, and they have not been as successful as those in the physical sciences. The reasons are many, among them these: the effort so far expended is much less; the basic empirical concepts and variables have not been isolated and purified to the same degree; mathematics grew up with and was to some extent molded by the needs of physics, and so it may very well be less suited to social science problems if these problems are of a basically different character from those of physics; a typical social science problem appears to involve more variables than one is accustomed to handling in physics; and, finally, social scientists are generally not extensively trained in mathematics.

A social scientist who attempts to formulate and solve a scientific problem in mathematical terms is often disappointed with the mathematics he can find. This may happen simply because a mathematical system appropriate to his problem does not seem to have been invented, or, as is more common, the definite and often quite complex mathematical system that he happens to want to under-stand in depth has not been investigated in any detail. In this century especially, mathematicians have tended to focus on very general classes of systems, and the theorems concern properties that are true of all or of large subclasses of them; however, these results do not usually provide much detailed information about any particular member of the class.

As an example, the axioms of group theory are not categorical—that is, two groups need not be isomorphic. Therefore, theorems about groups in general tell one little about the specific properties of a particular group. But this is what is of interest when a particular group is used to represent an empirical structure, as in modern particle physics.

When this happens, it is necessary for the applied mathematician to carry out considerable mathematical analysis to achieve the understanding he needs to answer scientifically interesting questions.

We have already discussed two parts of mathematics in which highly specific systems have been explored in depth: classical analysis and matrix algebra. A primary motivation for this detailed work was the needs of physical science. In fortunate instances, a problem may be formulated in terms of one of these systems, in which case specific results can sometimes be extracted from the existing literature. Examples where this has been done are in the application of matrix algebra to factor analysis and of Markov chains (a part of probability theory) to several areas, including learning, social interaction, and social structure [see Factor Analysis; Markov Chains].

Theory as detailed as this, however, is not typical of contemporary mathematics. We have in mind such active areas as associative and non-associative algebras, homological algebra, group theory, topological groups, algebraic topology, rings, manifolds, and functional analysis.

The generality of contemporary mathematics can be seductive in that it invites sophistic treatments of scientific problems. It is often not difficult to find some general branch of mathematics within which to cast a specific social or behavioral problem without, however, actually capturing in detail the various constraints of the problem. Without these constraints few explicit results and predictions can be proved. Nevertheless, the real emptiness of such endeavors can be shrouded for the unwary in the impressive symbolism and ringing terms of whatever mathematics it is that is not being seriously used.

If the growth of the social sciences parallels at all that of the physical sciences, they will study in detail various systems, which, although of peripheral mathematical interest, are of substantive interest. Indeed, some examples already exist, including these: (1) Just as classes of maximum and minimum problems have been formulated and solved in the physical sciences, other classes have arisen in the social sciences, such as linear, nonlinear, and dynamic programming, game theory, and statistical decision theory. (2) Various mathematical structures that may correspond to (parts of) empirical structures have been investigated, for example, aspects of the theory of relations and the closely related theory of graphs, matrix algebra, and concatenation algebras, which arose in the study of grammar and syntax. (3) Underlying the success of much physical theory is the fact that many variables can be represented numerically. The theories that account for this in physics are not suitable for the social sciences, but alternative possibilities are under active development, particularly in terms of theories of fundamental and derived measurement. The mathematics is reasonably involved, although for the most part the proofs are self-contained. (4) Although the theory of stochastic processes is a well-developed part of probability theory, a number of the processes that have found applications in the social sciences had not previously been studied by probabilists; their properties have been partially worked out in the social science literature. Among the most prominent examples are the nonstationary processes that have arisen in learning theory. Some of these postulate that on each trial one of several operators Q_i transforms a response probability into the corresponding probability on the next trial. Two special cases have been most adequately studied. One assumes that the Q_i are linear operators and the other assumes that the operators commute with one another—that is, Q_i,Q_j; = Q_j,Q_i,.[See Learning.]

As increasing use is made of mathematics in the social sciences, one may anticipate the investigation of very specific mathematical systems and, ultimately, the isolation of interesting abstract properties from these systems for further study and generalization as pure mathematics.

R. Duncan Luce and Patrick Suppes

BIBLIOGRAPHY

AczÉl, J. 1966 Lectures on Functional Equations and Their Applications. New York: Academic Press.

Akademiia Nauk S.S.S.R., Matematicheskii Institut (1956) 1964 Mathematics: Its Content, Methods, and Meaning. Edited by A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Laurent’ev. 3 vols. Cambridge, Mass.: M.I.T. Press. → First published in Russian.

Allen, R. G. D. (1938) 1962 Mathematical Analysis for Economists. London: Macmillan.

Apostol, Tom M. 1961-1962 Calculus. 2 vols. New York: Blaisdell.

Bartle, Robert G. 1964 The Elements of Real Analysis. New York: Wiley.

Birkhoff, Garrett; and Maclane, Saunders (1941) 1965 A Survey of Modern Algebra. 3d ed. New York: Macmillan.

Cartan, Henri (1961) 1963 Elementary Theory of Analytic Functions of One or Several Complex Variables. Reading, Mass.: Addison-Wesley. → First published in French.

Coddington, Earl A. (1961) 1964 An Introduction to Ordinary Differential Equations. Englewood Cliffs,N.J.: Prentice-Hall.

Coleman, James S. 1964 Introduction to Mathematical Sociology. New York: Free Press.

Courant, Richard; and Robbins, Herbert (1941) 1961 What Is Mathematics? An Elementary Approach to Ideas and Methods. Oxford Univ. Press.

Friedman, Bernard 1966 What Are Mathematicians Doing? Science 154:357–362.

GÖdel, Kurt (1931) 1965 On Formally Undecidable Propositions of the Principia mathematica and Related Systems. I. Pages 4-38 in Martin Davis (editor), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Hewlett, N.Y.: Raven. → First published in German in Volume 38 of the Monatshefte fur Mathematik und Physik.

Goldberg, Samuel 1958 Introduction to Difference Equations: With Illustrative Examples From Economics, Psychology, and Sociology. New York: Wiley. → A paperback edition was published in 1961.

Hocking, John G.; and YOUNG, GAIL S. 1961 Topology. Reading, Mass.: Addison-Wesley.

Kemeny, John G.; and Snell, J. Laurie 1962 Mathematical Models in the Social Sciences. Boston: Ginn.

Kneebone, G. T. 1963 Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. New York: Van Nostrand.

Luce, R. Duncan 1964 The Mathematics Used in Mathematical Psychology. American Mathematical Monthly 71:364–378.

Luce, R. Duncan; Bush, Robert R.; and Galanter Eugene (editors) 1963–1965 Handbook of Mathematical Psychology. 3 vols. New York: Wiley.

Mostow, George; Sampson, Joseph H.; and Meyer, Jean-pierre 1963 Fundamental Structures of Algebra. New York: McGraw-Hill.

Neugebauer, Otto (1951) 1957 The Exact Sciences in Antiquity. 2d ed. Providence, R.I.: Brown Univ. Press.

Newman, James R. (editor) 1956 The World of Mathematics. 4 vols. New York: Simon & Schuster.

Ore, Øystein 1948 Number Theory and Its History. New York: McGraw-Hill.

Samuelson, Paul A. (1947) 1958 Foundations of Economic Analysis. Harvard Economic Studies, Vol. 80. Cambridge, Mass.: Harvard Univ. Press.

MATHEMATICS

Bible

The Bible does not deal directly with proper mathematical subjects; however there are some parts that do relate indirectly to different mathematical topics. These are widely discussed by the various commentators on the Bible and Talmud: the ratio of 300:50:30 between the three dimensions of Noah's ark (in the past, a basic ratio in shipbuilding), the mathematical model of a rainbow, the number of 220 sheep and goats sent by Jacob to Esau (220 as the first number of the smallest pair of amicable numbers), the calculations of the visibility of the crescent of the new moon, the total amount and volume of the daily *manna, Moses' financial report on the donations for the building of the Tabernacle (mishkan), the commandment of keeping exact measures and balances, chance and probability in relation to the fair division of the land of Israel, lot-drawing to insure the fair division of holy duties and privileges, the curve of "projectile motion" in relation to the unintentional killing of a man by throwing a stone, the surprising distribution of the 12 tribes into two equal groups of six on Mt. Gerizim and Mt. Ebal, and more.

The members of the tribe of Issachar were known as "Marei de-Ḥushbena" – the masters of calculations – as their elders specialized in astronomical and calendar calculations. Christian scholars have dealt extensively with Bible mathematics. Among others, an early 18^th-century scholar, J.J. Schmidt published an interesting tractate called Biblicus Mathematicus (Zuellichau, 1732) in which many biblical-mathematical subjects are discussed, often based on Jewish sources. In another tractate there is a report on a request to the rabbinical court of Frankfurt to elaborate upon the issue of the geometry of the "Sea of Solomon" in the holy Temple. Many examples of biblical mathematics can be found in Be'er Hetev (Vilna, 1866), a commentary on Leviticus by R. Aryeh Huminer.

Sefer Yeẓirah

According to ancient Jewish tradition, Sefer Yeẓirah is ascribed to Abraham, as stated at the end of Sefer Yeẓirah. Others ascribe the authorship of the current version of Sefer Yeẓirah to R. Akiba (second century). At its very beginning, "sefar" (arithmetic = the wisdom of mensuration and numbers) is mentioned as one of three dimensions in which the world was created (*Kuzari and others). Sefer Yeẓirah deals extensively with permutations and combinations of the 22 letters of the alphabet. The end of Chapter 4 concludes with the statement that the number of permutations of all the 22 letters of the alphabet – [22! = 1,124,000,727,777,607,680,000] – is a number "which the mouth cannot speak and the ear cannot hear." (It would take 3,564,182,926 people and more than 10,000 years to speak out this number, even at a rate of one number per second.) R. Joseph ben Kalonymos (the elder; mid-13^th century), in his commentary to Chapter 2 – erroneously ascribed to R. Abraham ben David (1120–1198) – was one of the earliest to use the decimal system to express large numbers using Hebrew letters. This was noticed by R. *Elijah of Vilna (1720–1797), who indicated that the Hebrew word "ג״אוח" which appears in the calculations is actually the result of 144 × 22 = 3,168, written in Hebrew numerals.

Mishnah and Talmud

The Mishnah and Talmud, dealing with all aspects of daily life, discuss many mathematical subjects. Yet the main reason for dealing with mathematics was mostly either to bolster the study of the Bible and its commandments or to clarify everyday applications of mathematical methods. As is clear from tractate Avot, the study and transmission of pure mathematical knowledge per se were matters of secondary importance. Nevertheless, for many practical halakhic issues, a considerable body of basic mathematical knowledge is required. The following are a few examples only: the basics of plane geometry in connection with the measuring of the Sabbath *eruv boundary (2,000 cubits) over hills and ditches (methods of leveling); elements of knot theory in relation to the Sabbath laws; the layout of family graves; the mathematics of calculating an optimal seeding area without violating the biblical laws of prohibition of sowing with mixed seeds; the number of grapes that are within a circle with a given radius (i.e., Gauss' circle lattice point problem); the mathematics of inheritance leading to geometric progressions and exponential equations; the calculation of square roots leading to irrational numbers; the mensuration of circular and polygonal geometric forms or the proper division of assets among people with different kinds of claims. Another well-known topic is found in tractate Kinnim, the last in the Mishnah order Kodashim. Among other things, it deals with the laws concerning the "confusion of birds," e.g., birds assigned as sin offerings mixed up with those assigned as burnt offerings. Especially the last chapter requires advanced algebra and logic. Among the more recent mathematical commentaries are those by Moshe Koppel of Bar-Ilan University (1998) and B. Engelman of the Nahal Sorek Nuclear Research Center (1992). Knowledge of basic trigonometry and astronomy was needed for the fixing of the new month by reckoning and calculating. The exact methods used by the rabbinical court (bet din) were not made public. It was Maimonides who described appropriate calculations in his "Laws of Sanctification of the New Month" in his MishnehTorah. Many of these topics were discussed and elaborated upon by commentators throughout the ages. It has to be remembered that most of these scholars had acquired their knowledge autodidactically and lacked any formal education. From a comment by R. Hai Gaon on Mishnah Kelim (16:1; 24:7) it seems that the Pythagorean writing table (abacus) and "Indian arithmetic" (i.e., numeration system and numerals) were known and in use at this time (Smith 2, 177). Although the Mishnah and Talmud use an approximation of 3 for the better value of 3.1415… for π, it is clear from various discussions in the Talmud, that the *amoraim must have been aware of much better values for both π and √2.

The Mishnah and Talmud mention a few individuals as having outstanding mathematical knowledge. Rabban *Gamaliel, who used a Heron-type of dioptra to reckon distances; R. *Eleazar Ḥisma and R. *Johanan ben Gudgada, whose vast mathematical knowledge was described as enabling them to estimate "the number of drops in the sea"; R. *Zadok, who revealed to the Romans an advanced system of finger-calculation as well as the underlying mathematics of what later came to be known as the "Roman Statyra" (steelyard). R. *Joshua b. Hananiah – called the Escolasticus – was versed in astronomy and mathematics, R. *Abbahu is mentioned as having calculated the length of the cycle of service of the different tribes in the Holy Temple. The amora*Samuel Yarḥina'ah was versed in calendar calculations. This enabled him to calculate the calendar for the Diaspora for more than 60 years in advance. Nevertheless this knowledge was called "simple calculation," as it did not show a deeper understanding of Jewish law proper. In the name of the amora*Adda a more accurate estimate of the duration of the seasons is reported. The Talmud also mentions a "Kippah shel Ḥesbonot," i.e., a covered place outside Jerusalem serving people visiting the holy city in arranging their financial calculations and transactions.

mishnat ha-middot

Bible commentators of the Middle Ages mention the existence of a treatise called the Baraita of 49 Rules (Middot). This treatise from the tannaitic era was said to contain geometrical formulas and calculations. It was Abraham ben Solomon, the son of R. Elijah of Vilna (the Gaon of Vilna) in his Rav Pe'alim (Warsaw, 1894), who first drew attention to this treatise, though no existing copy of it was known. In 1864 Moritz Steinschneider found a Hebrew mathematical manuscript, identified it as Baraitat ha-Middot and published it in 1864. A critical edition including the geometrical drawings omitted by Steinschneider was published by the mathematician Hermann Schapira in 1880 as Mishnat ha-Middot. Among others things, Mishnat ha-Middot contains the Pythagorean formula allowing the calculation of the square root of 2 and uses a value of 31/7 for π. Haim Horovitz of Frankfurt tried to prove that this Mishnat ha-Middot was actually part of a Tosefta to the Mishnah tractate Middot (describing the measurements of the Holy Temple). This suggestion was later supported by the discovery of additional fragments and accepted by Solomon *Gandz in his critical edition of the Mishnat ha-Middot. This suggests that the Mishnat ha-Middot is one of the oldest known Hebrew mathematical works.

The Era of the Geonim

R. *Nahshon bar Zadok, who headed the yeshivah of *Sura from 874 to 882, stated that the order of the weekdays on which any particular festival occurs in successive years repeats itself after a cycle of 247 years. Thus he was able to arrange these years and their characteristic dates in 14 tables. This system is known as "Iggul de-Rav Nahshon" (R. Nahshon's cycle). R. Abraham Azulai (1570–1643) explains in Nahshon Gaon's name the concept of "amicable" (or "friendly") number pairs. He also mentions the common belief in the peacemaking powers of these pairs of numbers. This was known to Jacob and explains the number of 220 sheep and goats that Jacob sent to his brother. The suggestion has been made that R. Nahshon Gaon received this information from his contemporary Thabit ibn Qurra (836–901), a Sabbean mathematician in Baghdad famous for his work in amicable numbers. R. *Hai ben Sherira (939–1038), head of the yeshivah of *Pumbedita, in one of his responsa, explains the use of the Heron-type dioptra used by Rabban Gamaliel. His mathematical description of the various methods is practically identical to the way Heron himself described it in his book, including the accompanying diagram. This gives rise to the conjecture that R. Hai Gaon was familiar with the original source of Heron himself. The gaon who dealt most extensively with mathematical subjects was R. *Saadiah ben Joseph (892–942), head of the yeshivah of Sura. Examples are his commentary on Sefer Yeẓirah and his Sefer ha-Yerushot ("The Book of Inheritances"). The latter is an extensive mathematical-halakhic text showing how to divide an inheritance according to Jewish law. Other figures from this era who commented on mathematical subjects were Rabbenu *Hananel ben Ḥushiel, head of the yeshivah of Kairouan (980–1050), in his commentary to the Talmud and R. *Shabbetai Donnolo (913–c. 982), an Italian physician and writer on medicine, in his Taḥkemoni, a commentary on Sefer Yeẓirah.

11^th and 12^th Centuries

france

The *tosafists concentrated their literary efforts on the elucidation of the Bible and the Talmud and did not hand down much mathematical work per se. Although they had no formal education in mathematics, and in most cases had no possibility of learning from Greek or Latin sources, they did acquire some basic knowledge in an autodidactic way. Thus they commented on various talmudic discussions involving arithmetic and simple geometry. On the one hand their comments show a basic knowledge of arithmetic, e.g., in proposing several ways to calculate the connection between the basic halakhic unit of a ¼ of a log (revi'it) and the required amount of 40 se'ah of pure water for the ritual bath – the *mikveh – yet it seems that some of the tosafists were unaware of the Pythagorean law and did not have a good approximation of π or of irrational numbers like √2 or √5. But evidently they were aware that surveyors used a better value than 3. They also used a method of applying and proving the Archimedean formula for the calculation of the area of a circle based on the radius and the circumference. It is highly probable that they adopted the method of their Spanish contemporary R. *Abraham bar Ḥiyya. His geometrical treatise Ḥibbur ha-Meshiḥah ve-ha-Tishboret ("Treatise on Mensuration and Calculation") was written for the rabbis of southern France. *Rashi (1140–1205), the forerunner of the tosafists, himself used a very interesting geometric method for calculating the square root, a method which had its sources in the Jerusalem Talmud. One of the tosafists, R. *Asher ben Jehiel (called the "Rosh"; c. 1250–1327), raised the following question: Why does the Talmud, as a book of law, discuss an inaccurate approximation of 3 for the value of π rather than using a better value, which had been known for a long time? The place for such an excursus would be a geometry text and not the Talmud. In reply, he showed that the famous Mishnah in question (Er. 1:6) was not intended to teach a geometrical principle but rather to introduce the halakhic rule that in certain instances this approximate value of 3 should be used. After his escape from Germany to Spain, R. Asher ben Jehiel asked one of his disciples – the astronomer R. Isaac ben Joseph *Israeli – to elaborate on the Pythagorean law and other geometric principles. It was in response to this request by R. Asher ben Jehiel that he wrote his famous trigonometric-astronomic treatise Yesod Olam.

spain and portugal

In Spain and Portugal Jewish intellectual and scientific growth continued to flourish. There was a lively exchange of mathematical knowledge between Jews and Arabs, and many Arabic mathematical ideas are reflected in Jewish literature. Likewise Jews contributed much to the Arabic corpus of scientific knowledge. Thus one finds much more elaborated mathematical ideas in both talmudic literature proper and original mathematical works.

Among the best-known mathematical figures of the 11^th and 12^th centuries was the above-mentioned Abraham bar Ḥiyya. Until the publication of Ḥibbur ha-Meshiḥah veha-Tishboret in 1910, this mathematical magnum opus was known only in manuscript. This text is probably the earliest post-talmudic mathematical text per se. The author states that the reason he compiled the discourse was the lack of knowledge of geometry among the Jews of southern France. As he writes in his introduction, the text was meant to serve as a textbook for judges who had to deal with legal issues concerning the surveying and measuring of fields. Towards the end Bar Ḥiyya provides an interesting demonstration – using a model built from concentric circles of thin rope – of the Archimedean formula for the calculation of the circle's area based on its radius (r) and its circumference (C) [½∙c∙r], without actually using the value of π. It is this very demonstration that is used by the tosafists in their Talmud commentary. Bar Ḥiyya gives the fair approximation of 1.4143 (1⅖ + 1/10) for the square root of 2 and 3.141593 for π. He was also among the first to introduce to Europe the complete solution of quadratic equations. This fine textbook was translated by Plato of Tivoli in 1145, just a few years after the death of Bar Ḥiyya, under the title Liber Embadorum.

Other works by Bar Ḥiyya dealing with mathematics are Ẓurat ha-Areẓ ("Form of the Earth"), a basic introduction to spherical trigonometry and astronomy, and Sefer ha-Ibbur ("Book of Intercalation"). Based on a statement by Maimonides, it seems that he knew this text. Another work of his is Yesodot ha-Tevunah u-Migdal ha-Emunah ("The Foundations of Understanding and the Tower of Faith"), an encyclopedia on arithmetic, geometry, optics, astronomy, and music. Parts of it were translated into Latin by the Hebraists Sebastian Münster (1488–1552) and his disciple Erasmus Oswald Schreckenfuchs (1511–1575).

abraham b. meir ibn ezra

*Ibn Ezra (1092–1167), a tosafist, was one of the most prolific commentators on the Bible, who at the same time also wrote extensively on mathematics. The best known of his mathematical contributions are Sefer ha-Shem ("Book of the Holy Name"), Sefer ha-Eḥad ("Book of the Number 1"), Sefer Keli ha-Neḥoshet ("Book of the Copper Instrument [i.e., the astrolabe]"), and Sefer ha-Mispar ("Book of Numbers"). (Some 100 years later, Jacob b. Machir ibn Tibbon, known as Profatius Judaeus, invented an improved version of the astrolabe, known as the quadrant.) Ibn Ezra also translated into Hebrew the commentary of al-Biruni on al-Khwarizmi's tables. In the introduction to this work he gives an historical account on the involvement of Jews in the introduction of Indian mathematics to the Arabic world. Ibn Ezra is one of the earliest Hebrew writers to introduce the "0," which he called galgal (wheel). In his Sefer ha-Mispar Ibn Ezra presents many exercises, which also appeared more than 250 years later in the Sefer ha-Mispar of Elijah *Mizraḥi. In addition, Ibn Ezra deals with the mathematics of inheritance and with the history of pi and mentions perfect numbers. He is also famous for his "prisoner problem," a mathematical puzzle first presented by *Josephus Flavius in his Jewish War and which is often used in introductory courses in mathematical programming. Many scientific articles have been written on Ibn Ezra's mathematical writings.

maimonides

Aside from a short tractate on calendar calculations, the works of Maimonides (1135–1204) are primarily nonmathematical. Yet in his Mishnah commentary Maimonides mentions that the ratio between the circumference of a circle and its diameter (π) cannot be expressed as a ratio of two natural numbers, and that this fact is not due to a lack of our understanding but is in the very nature of this number. He further states that there is no possibility to know the exact value of this ratio, that mathematicians have written various treatises on this subject, and that the approximation used by scientists is 22/7. Yet in his halakhic Mishneh Torah Maimonides requires the use of the old talmudic (and Babylonian) approximation of 3. A similar statement by Maimonides relates to two other irrational numbers, namely √2 and √5000. It is interesting to note that in the Western world it was Lambert (1728–1777) who first proved the irrationality of π in 1761. Much has been written in rabbinic literature about Maimonides' geometrical explanation of the mishnayyot in Kilayim 3:1 and 5:5. In his Moreh Nevukhim (Guide of the Perplexed), Maimonides mentions the difficulty in imagining the concept of the hyperbola and its asymptote, i.e., a curved line and a straight one, constantly approaching one another ad infinitum, without ever meeting. This subject – including the detailed explanations of the Jewish writers – was later elaborated by S. Motot, a 14^th-century Jewish mathematician, and by Francesco Barozzi in his Admirandum Illud Geometricum Problema, Tredicim Modis Demonstratum (Venice, 1585).

13^th and 14^th Centuries

levi ben gershom

*Levi ben Gershom (Ralbag/Gersonides; 1288–ca. 1344) was probably the most advanced Hebrew mathematician of his generation. Widely known for his biblical commentaries, he dealt with all the three branches of Arabic mathematics: arithmetic, geometry, and trigonometry. An extensive corpus of research about Gersonides has come into being (spearheaded by Bernhard Goldstein of Yale University). A comprehensive bibliography on Gersonides has been published by Menachem Kellner of Haifa University (1992). Gersonides mentions both Abraham Ibn Ezra and Abraham bar Ḥiyya as sources from which he derived some of his knowledge. His arithmetical works – Ma'aseh Ḥoshev ("The Practice of Arithmetic") and "De numeris harmonicis" have been studied since the publications of Ma'aseh Ḥoshev by R. Joseph Carlebach of Hamburg, and some years later, at the beginning of the 20^th century, by Gerson Lange. A hitherto missing part of problems of Ma'aseh Ḥoshev has been published by S. Simonson of Stonehill College. In this treatise Gersonides deals with arithmetic, algebra, and combinatorics. The short tractate on harmonic numbers was written as a response to an inquiry by Philip of Vitry, the bishop of Maux, shortly before Gersonides passed away. Another mathematical work is a commentary on Euclid's Elements. Gersonides' text on trigonometry, De sinibus, chordis et arcubus (originally written in Hebrew but immediately translated into Latin) is a commentary on the relevant chapters of Ptolemy's Almagest and was originally part of Gersonides' major work Milḥamot Adonai ("The Book of the Wars of the Lord"), part v, ch. 1. It was omitted in the printed Venice edition of 1560. In it, Gersonides presents a proof of the theorem of sines. He had arranged for a translation into Latin which is still extant. Carlebach showed that Gersonides was the inventor of the "cross-staff" (sometimes called "bacculus" or "Jacob's staff"), a simple yet powerful surveying device which allowed nautical and astronomical measurements. Carlebach even reconstructed a model of this instrument following Gersonides' description. The Jacob's staff was in use until the 17^th century.

immanuel ben jacob bonfils

Another Jewish mathematician of this era is Immanuel ben Jacob *Bonfils of Tarascon (1300–1377), a contemporary of R. Levi b. Gershom. He is known as Ba'al Kenafayim after his astronomical tables published under the name of Shesh Kenafayim ("Six Wings"). Bonfils taught astronomy and mathematics at Orange for some time. He also was one of the forerunners of exponential calculus, some 150 years before its adoption in Europe, as is evident from his Derekh Ḥilluk. A great number of his many mathematical and astronomical works are still in ms. A special volume on the history of science in the Middle Ages by G. Sarton was called Six Wings after Bonfils' astronomical tables.

Important findings relating to Jewish mathematics in the 12–14^th centuries have been contributed by various researchers, among them G. Freudenthal, G. Safatti, T. Levy, and D. Zeilberger.

15^th Century

moses ben abaraham provencal

Moses ben Abaraham *Provencal (1503–1575) was considered one of the greatest talmudists and most illustrious scholars of Italian Jewry in the Renaissance period. For many decades he was rabbi of the Italian community of Mantua, which therefore became a center of talmudic study. His mathematical knowledge is evident from his Be'ur Inyan Shenei Kavvim. In his Guide of the Perplexed Maimonides mentions the concept of the asymptote, a straight line constantly approaching a curved line without ever touching it, referring to the Conics of Appolonius. Provencal wrote a four-page Hebrew explanation of this subject, which was added to the Sabionetta (1553) edition of the Guide of the Perplexed. This kuntres (pamphlet), which became famous, was translated into Italian by Joseph Shalit (Mantua, 1550) and was included in the well-known volume on the concept of the asymptote by Franceso Barocius (Venice, 1586). The latter also contains geometric explanations by other Jewish commentators on the Guide of the Perplexed. The subject itself became a major topic of rabbinical mathematics and was discussed in rabbinical literature from the 14^th to the 19^th centuries. It has been suggested that Provencal was familiar with Simon Motot's book on algebra. In his pamphlet, Provencal includes an explanation of the Greek concept of the "mean and extreme proportion" (the "golden section") and proofs related to the connection between the lengths of the sides of a regular hexagon and a regular decagon, both inscribed to the same circle. This concept, described by Euclid and mentioned by Joseph *Albo in his Sefer ha-Ikkarim, inspired many discussions in the rabbinical literature, mainly because of the lack of knowledge of the proper definition of the Greek concept of "mean and extreme proportion." Provencal specifically refers the reader to the source in Euclid's Elements.

mordecai comtino

Mordecai Comtino (1420–d. before 1487) was the teacher of R. Elijah ben Abraham *Mizraḥi. He was on friendly terms with the Karaites and was the teacher of two of their leaders, Caleb *Afendopolo and Elijah *Bashyazi. His literary output includes Sefer ha-Ḥeshbon ve-ha-Middot on arithmetic and geometry and commentaries on Abraham Ibn Ezra's Sefer ha-Eḥad, Yesod Mora, and Sefer ha-Shem, in which various mathematical subjects are discussed. His Sefer ha-Ḥeshbon ve-ha-Middot was known only in manuscript until a careful analysis and partial translation was published by Moritz Silberberg of Schrimm in 1905. The plan of this treatise follows a statement of the Greek Nichomachus of Gerasa regarding the logical order of basic mathematical subjects. Following Ibn Ezra, Comtino introduces the full decimal numeration, including the "0," which again he called "galgal" (the wheel). After introducing standard subjects in the first part, on arithmetic, Comtino deals with the measurement and division of plane figures and then proceeds to calculation of volumes of geometrical bodies and their parts. He also provides a detailed vocabulary of the different scientific terms. A special addition is the collection of problems part of which are borrowed from Ibn Ezra. Mizraḥi in his Sefer ha-Mispar (see below) drew upon some of the problems presented by Comtino. Besides this, Comtino also dealt with the construction of astronomical instruments.

elijah ben abraham mizraḤi

Mizraḥi (Re'em; c. 1450–1526) is known primarily from his famous supercommentary to Rashi's Bible commentary. Mizraḥi's mathematical works are Sefer ha-Mispar (Constantinople, 1534), on arithmetic, and a commentary on Ptolemy's Almagest (no longer extant). The former book became a standard text for the study of arithmetic. It deals with whole numbers, fractions, and mixed numbers, with the extraction of the square and cube roots, proportions, and arithmetical and geometrical problems. In his lengthy introduction he describes the importance of the study of mathematics as a bridge between the different sciences. Sefer ha-Mispar is based on Ibn Ezra's Sefer ha-Mispar and the mathematical work of his teacher Mordecai Comtino (see above). A Latin abridgment by Sebastian Muenster was published by his disciple Schreckenfuchs (Basel, 1546). At the end of the 19^th century an in-depth description of Sefer ha-Mispar (Die Arithmetik des Elija Misrachi) was published by Gustav Wertheim (Frankfurt, 1893). An excerpt and analysis of those of his mathematical problems related to physics (Ueber physikalische Aufgaben by Elia Misrachi) was prepared by E. Wiedeman in 1910.

mordecai b. abraham finzi

*Finzi (c. 1407–1476), a banker and mathematician, was known mainly for his mathematical and astronomical works, which included Luḥot, tables on the length of days (published by Abraham Conat, Mantua, c. 1479), and an astronomical work entitled Netiv Ḥokhmah (unpublished). He translated into Hebrew the Algebra of the Arab mathematician Abu Kamil Soga (c. 850–930). In 1934, a young Jewish mathematician by the name of Joseph Weinberg, submitted a critical edition of Finzi's translation as a thesis to the University of Munich. (Weinberg later was murdered by the Nazis.) An English translation of Finzi's commentary on Abu Kamil's Algebra was published in 1966 by Martin Levey. Finzi also translated into Hebrew various works on astronomy and geometry, wrote commentaries on some of them, described and explained recently invented astronomical instruments, and wrote treatises on grammar and mnemonics.

abraham ben samuel zacuto

*Zacuto (1452–c. 1515) was known as a talmudic scholar, historian, mathematician, and astronomer. He was appointed professor of astronomy and mathematics at the University of Salamanca. His famous almanacs and astronomical tables became a principal base for Portuguese navigators. Thus, Columbus, whom he met in Salamanca, was able to garner important information before his famous expedition. Using the tables of Zacuto, Columbus was able to predict an eclipse of the moon and so save the lives of his men by demonstrating to hostile natives that he could shut out the light of the sun and moon. R. *Levi ben Habib, a contemporary of Zacuto, remarks in one of his responsa that Abraham Zacuto wrote a commentary on the Talmud, from which he quotes a small geometrical explanation. Zacuto's well-known Sefer Yuḥasin contains several references to mathematics.

16^th and 17^th Centuries

david b. solomon *gans

The astronomer, mathematician, and historian David b. Solomon *Gans (1541–1613) was raised and educated in the home of R. Moses *Isserles, the Rema. He also belonged to the circle of *Judah Loew ben Bezalel (the Maharal). Gans was in close contact with Tycho Brahe and Johann Kepler. Besides his history, Ẓemaḥ David, Gans compiled an astronomical-mathematical textbook called Neḥmad ve-Na'im (Jesnitz, 1743), an extract of which appeared in Prague in 1612 under the name of Magen David. In the preface to Neḥmad ve-Na'im he presents an abridged history of the transmission of mathematics and astronomy among the Jews, based on Jewish sources. The main part is devoted to pre-Copernican celestial mechanics, whereas toward the end he introduces basic instrumentation as well as trigonometry and its applications in daily life, enriching the text with many contemporary examples.

menahem zion porto (rafa)

Porto was an Italian rabbi born in Trieste toward the end of the 16^th century; he died in Padua around 1660. He was an excellent mathematician and astronomer. His works were highly praised by Andrea Argoli and extolled in Italian sonnets by Tomaso Ercaloni and Benedetto Luzzatto. In 1641 Gaspard Scüppius, editor of the Mercurius Quadralinguis, recommended Porto, in terms that were very complimentary to the rabbi, to Johannes *Buxtorf (the younger), with whom Porto later carried on an active correspondence. Among other works, Porto published a "Hand-book for the Merchant" (Over la-Soḥer, Venice, 1627), a compendium of basic arithmetic and many examples of business calculations for merchants. He also published a two-volume treatise of close to 400 pages – םימעה יניעל םכתניבו םתמכח – Porto Astronomico – dealing with trigonometry and astronomy.

joseph solomon rofe delmedigo

*Delmedigo (YaSHaR; 1591–1655) studied mathematics, mechanics, and astronomy under Galileo at Padua. His major work, Elim (Amsterdam, 1629), is a classic compendium of 16^th-century mathematics, physics and astronomy, and scientific instruments. He describes his use of Galileo's telescope to observe the planet Mars. Delmedigo displays a profound knowledge of the Greek, Renaissance, and contemporary literature dealing with mathematics and physics. The first part of Elim contains mathematical discourses dealing with both classical problems of Greek mathematics like the solution of the famous Alexander problem leading to Diophantine equations, the trisection of angles using the Conchoid of Nichomedes, the calculation of the octagon, and post-Renaissance mathematics like the solution of the cubic equations or the "squaring" of a circle, and the 355/113 and Ludolphine approximations to π. In Mayan Ganim Delmedigo deals mainly with spherical trigonometry, with the proof and application of the law of sines, the history of trigonometry, the prosthaphaeresis (trigonometric formulas for the conversion of a product of functions into a sum or a difference), and the law of tangents of the sine function. Gevurat Adonai deals mainly with astronomy but has some mathematics as well. Another part, Ma'yan Ḥatum, is devoted mainly to the discussion of physical problems and paradoxes – mostly from Aristotle's Mechanical Problems – as well as to topics from early 17^th-century classical physics.

jair Ḥayyim bacharach

*Bacharach (1638–1702), rabbi of Worms, had a keen interest in mathematics. His first volume of responsa, Ḥut ha-Shani ("Scarlet Thread," Frankfurt, 1679) contains a lengthy responsum (§98) dealing first with talmudic metrology and then, in the second part, with many mathematical subjects. His main source is Gevurot Adonai by Joseph Solomon Delmedigo (see above). He mentions Hero's formula for extracting the square root, the (Roman) system of finger calculation, the approximation of a circle's circumference by polygons, mathematical problems from the Sefer ha-Mispar of R. Elijah Mizraḥi (see above), the famous Alexander problem from R. Joseph Solomon Delmedigo, and others. In his second volume of responsa Ḥavvat Ya'ir, Bacharach deals at length with the Euclidean concept of "mean and extreme proportion" (the golden section) mentioned by R. Joseph Albo in his Sefer ha-Ikkarim (Responsa §111).

moses hefetz gentili

In both his Ḥanukkat ha-Bayit (Venice, 1696), on the architecture of the Temple, and Melekhet Maḥshevet, a commentary on the Torah (Venice, 1710), Gentili (1663–1711) presents material that reflects mathematical thinking. Referring to the weekly portion of Noah, he describes Descartes' mathematical model of the rainbow and in his commentary to the weekly Torah portion Ma'asei, he describes Tartaglia's model for the motion of a projectile. In Ḥanukkat ha-Bayit he makes extensive use of the "Pythagorean theorem" in discussing the structure of the altar.

elijah ben solomon zalman of vilna

Already in his very early life *Elijah ben Solomon Zalman (the Vilna Gaon; 1685–1779) showed great interest in the study of mathematics and astronomy as an aid to furthering and deepening the study of Jewish law. In his halakhic commentary on the Shulḥan Arukh he added a great number of notes which disclose his profound mathematical and astronomical knowledge. He encouraged his students to translate basic mathematical texts into Hebrew, and even wrote a small and very concise tractate on arithmetic, geometry, and trigonometry with an introduction to basic astronomy (Ayil Meshulash, 1834). An analysis and description of Ayil Meshulash was published by Elias Fink (Eliah Wilna und sein elementar-geometrisches Compendium, Frankfurt, 1903). Fresh interest in this compendium was aroused with the publication of a new edition with a modern Hebrew commentary. One of his students, Baruch *Schick, a rabbi and physician, published various Hebrew texts on astronomy and medicine as well as a Hebrew translation of the first six books of Euclid's Elements.

raphael levi of hanover

Raphael Levi *Hannover (1685–1779) showed his mathematical talent already as a child, when studying at the Jewish orphanage. One day, upon returning from his studies, observing the construction of the new royal stables in Hanover, he was able to prevent a serious engineering mistake. This drew the attention of Leibniz, the famous mathematician, who after meeting the young Raphael Levi offered to pay part of his tuition and to tutor him privately in mathematics and astronomy. In the last paragraph of his Tekhunat ha-Shamayim (Amsterdam, 1756) on geometry, trigonometry, astronomy, and calendar-making, Levi describes and strongly supports the Copernican system. In an addendum, Raphael Levi's disciple Moses Titkin elaborates a few difficult talmudic passages connected to mathematics. In addition to the Hebrew works, Raphael Levi also invented a system of using logarithms in currency conversions and wrote two mathematical compendia in German: (1) Wechsel-Tabellen Tractaetgen (Hanover, 1746), tables for currency conversion, and (2) Vorbericht vom Gebrauch der neuerfundenen Logarithmischen Wechsel-Tabellen (Hanover, 1747), a preliminary report on the use of the newly invented logarithmic currency conversion method. This latter text contains a good deal of advanced exercises and numerical examples. He also left a Hebrew manuscript of the first part of an introduction to algebra.

Among many other individuals who dealt with mathematical subjects were R. Yom Tov Lipmann *Heller Wallerstein (1579–1654), a disciple of the Maharal, who used his extensive knowledge of mathematics and astronomy throughout the whole of his commentary Tosefot Yom Tov to the Mishnah. R. David *Nieto of London (1654–1728), in his Kuzari Sheni, devoted a whole chapter to the explanation of some of the geometrical issues discussed in the Talmud and the commentary of the tosafists and many others.

18^th Century

The expansion of general knowledge caused a shift in the content of rabbinical mathematics, from basic arithmetic and geometry only to a much broader scope of talmudic subjects related to mathematics. This was made possible by the availability of the Hebrew texts of Abraham Ibn Ezra, Elijah Mizraḥi, Joseph Solomon Delmedigo, and others mentioned above. Topics like cubic equations, arithmetic and geometric progressions, logarithms, spherical trigonometry (for calculations not connected directly to astronomy), the use of trigonometric tables, logarithms, and methods similar to calculus – like analysis of functions for maxima and minima – were used. This brought forth a great number of texts containing mathematical excursus.

At the beginning of the 18^th century R. Samuel Schotten mentions in his Kos ha-Yeshu'ot (Frankfurt, 1711) his plan to publish a collection of talmudic-mathematical essays. His basic knowledge in mathematics enabled him to give several approbations to Hebrew astronomical texts. Some years later *Jonathan of Ruzhany published his Yeshu'ah be-Yisrael, a commentary on the laws of kiddush ha-ḥodesh (concerning the blessing of the New Moon) in Maimonides' Mishneh Torah (Frankfurt, 1720). This tractate has an interesting appendix on some halakhot requiring basic mathematical knowledge. In the same year, Jonathan published a compendium of three astronomical works which naturally contain various mathematical elaborations. Some years later R. Emanuel Hai Rikki (1680–1744) published his Ḥoshev Maḥashavot (Amsterdam, 1727), an interesting halakhic-mathematical discourse of 70 short chapters based on an inquiry concerning the measurements of a mikveh (ritual bath). Within these discussions he elaborates the subject of measuring the circumference of a circle, the calculation of √2, and the concepts of asymptotes, and shows how to calculate and to graph two curved functions, each approaching the other without ever touching it. R. Jonah *Landsofer's Me'il Ẓedakah (Prague, 1757) contains a responsum (§28) – an answer to an inquiry from a "learned man well versed in geometry" – elaborating the geometrical aspects of the altar in the holy Temple as discussed in the Talmud. This subject is often discussed in talmudic literature. In his answer, Landsofer shows his profound knowledge of the relevant texts and also provides a proof of the Pythagorean Theorem. This responsum follows the previous one (§27) dealing with the area calculations of various shapes, all of the size of a "lense." In 1794, David Pivani published Zikhron Yosef, a textbook on arithmetic, geometry, and plane and spherical trigonometry. In the introduction he explains some of the geometrical aspects underlying Maimonides' commentary to the Mishnah Kila'im 5:5. Three years later, David Friesenhausen published his Kelil ha-Ḥeshbon (Berlin, 1797), in collaboration with the Jewish Freeschool in Berlin. This text, an introduction to algebra, contains a variety of challenging examples and exercises. Among the more interesting topics are arithmetical and geometrical series, cubic roots, and Lambert's law concerning the brightness of an illuminated surface.

Towards the end of the 18^th century and in connection with the social emancipation and the resulting assimilation of the Jews, Jewish mathematics began to develop into two main streams: the traditional Jewish talmud scholar who used mathematical knowledge mainly for the purpose of expounding religious subjects and the new type of a modern mathematician of Jewish origin who pursues mathematics as an academic profession.

19^th Century

The study of mathematics in the 19^th century is widely characterized by the efforts of the *Haskalah movement to introduce general secular education among the Jews. It was one of the goals of the Haskalah to show that the Jewish people too have a basic mathematical tradition. Therefore one finds regular contributions on mathematics in periodicals like Sulamith in German, Ha-Me'assef, Ha-Ẓefirah, and Ha-Carmel in Hebrew – dealing with the study of mathematics in the framework of talmudic studies or dealing with mathematical problems as such. One of the first texts of the 19^th century was Beirurei ha-Middot by Tovia Segal of Horoshitz (Prague, 1807), on the geometry of Sabbath distances and diameters of levitical cities. This text contains an introduction to geometry and trigonometry using logarithms. At the beginning of the 19^th century Meyer Hirsch published a textbook/collection of exercises on geometry, Sammlung Geometrischer Aufgaben (Berlin, 1809). This collection of problems was used in Germany for almost a century and contains a formula proposed by the 15^th-century Simeon ben Ẓemaḥ *Duran. A little later (1828–31) Michael *Creizenach, a teacher at the Frankfurt Philantrophin Reform school, published a series of textbooks on descriptive geometry, algebra, and technical geometry. In the first quarter of this century the first Hebrew article on "binary numbers" (a basic concept in computer engineering and digital electronics) was written by R. Zechariah *Jolles. (The author passed away two years before George Boole published his paper taking up again the concept of "binary numbers" in 1854.) Jolles' paper is basically a translation and elaboration of Leibniz's famous paper on the same subject. It is included, among other mathematical writings, in his Ha-Torah ve-ha-Ḥokhmah (Vilna, 1913). In 1834 Ḥayyim Selig *Slonimsky published his Mosdei Ḥokhmah, an introduction to mathematics. A second edition of David Friesensohn's Kelil ha-Ḥeshbon was reprinted in Zolkiev in 1835. In 1845 the famous Leopold *Kronecker (baptized 1863) began his brilliant career as mathematician after receiving his Ph.D. In 1856 the great historian of mathematics Moritz Cantor published his fundamental paper "Ueber die Einfuehrung unserer gegenwaertigen Ziffern in Europa" ("On the Introduction of Our Present Numerals in Europe"). Later on he published his monumental Vorlesungen ueber Geschichte der Mathematik ("Lectures on the History of Mathematics"), which is considered as marking the beginning of the modern history of mathematics. In keeping with the spirit of the Haskalah, the works of Joseph Solomon Delmedigo were reprinted (Odessa, 1865) as was Shevilei de-Raki'a by Eliah Hochheim (Warsaw, 1863). The editor of this tractate, Baruch Lowenstein, added a monograph of his own, Bikkurei ha-Limmudiot ("Firstlings of Mathematics"), discourses on various historical topics in Jewish mathematics. About the same time Ẓevi ha-Cohen *Rabinowitz published his fine series of Hebrew texts on popular experimental physics, YesodeiḤokhmat ha-Teva (Warsaw, 1865), including special parts on mathematics and a short bibliography of Hebrew mathematical works. In those years Yom Tov Lipman *Lipkin – a son of the famous R. Israel *Lipkin Salanter, the founder of the *Musar movement – invented the "Lipkin linkage," a mechanical device to transform circular motion into linear motion, and became a famous mathematician who contributed mathematical problems to the Ha-Ẓefirah periodical. In the last quarter of the 19^th century the first modern systematic texts on Jewish mathematics appeared: Baruch *Zuckermann's Das Mathematische im Talmud (Breslau, 1878; see J. Szechtman, "Notes on Dr. Zuckerman's 'Introduction to his Mathematical Concepts in the Talmud,'" in: Scripta Mathematica, vol. 25 (1960), pp 49–62). The most important work in this field is Moritz *Steinschneider's series of articles on Jewish mathematics, published between 1893 and 1898 in Ennestroem's Bibliotheca Mathematica. Part of these articles (covering the 9^th–16^th centuries) were reprinted as Mathematik bei den Juden in 1964. Additional relevant information can be found in Steinschneider's contributions to the famous Realencyclopaedie by Ersch and Gruber. This was made possible with the opening of German and Italian libraries, allowing the study of ancient Hebrew mathematical manuscripts and books. In 1879 Hermann *Schapira, at that time still a student in Heidelberg, edited and published a German translation of the Mishnat ha-Middot, discovered by Steinschneider in 1864. Towards the end of the 19^th century Gustav Wertheim published Elemente der Zahlentheorie (Leipzig, 1887) and some years later an interesting monograph on the mathematics of Elijah Mizraḥi (Die Mathematik des Elia Misrachi, Frankfurt, 1893). A year later Israel Michel Rabbinowits of Paris published a Hebrew introduction to the Talmud containing an interesting appendix on the extraction of square roots, based on Heron's algorithm, as well as a discussion of negative and irrational numbers (Mavole-Talmud, Vilna, 1894).

[Shimon Bollag (2^nd ed.)]

20^th Century

Jewish mathematicians continued to make major contributions throughout the 20^th century and into the 21^st, as is evidenced by their extremely high representation among the winners of major awards: 27% for the Fields Medal (the "Nobel Prize of Mathematics") and 40% for the Wolf Prize. Of those still active around the outset of the 20^th century mention may be made of Rudolf Otto Sigismund *Lipschitz (1832–1903), whose contributions to mathematics and physical mathematics were mostly in the theory of numbers, the computation of variations, progressive series, and the theory of potential and analytic mechanics. With the French mathematician Augustin Louis Cauchy (1789–1857), he proved the theorem of prime importance in differential calculus and equations concerning the existing solutions to the equation dy/dx = f (x, y). Herman *Minkowski (1864–1909) is entitled to nearly all the credit for creating the geometry of numbers. He was one of the earliest mathematicians to realize the significance of *Cantor's theory of sets at a time when this theory was not appreciated by most mathematicians. The later work of Minkowski was inspired by *Einstein's special theory of relativity which was first published in 1905. He produced the four-dimensional formulation of relativity which has given rise to the term "Minkowski space." James Joseph *Sylvester (1814–1897) dominated the development of the theories of algebraic and differential invariants, and many of the technical terms now in use were coined by him.

In Italy Vito *Volterra (1860–1940) wrote numerous papers on partial differential equations, integral equations, calculus of variations, elasticity, and topology, and initiated the subjects of functionals and mathematical biology. Tullio *Levi-Civita (1873–1942) developed the absolute differential calculus, which was the essential mathematical tool required by Einstein for his development (in 1916) of the general theory of relativity. Levi-Civita's most important contribution in this field was the theory of "parallel displacement." He also produced significant papers on relativity, analytical dynamics, hydrodynamics, and systems of partial differential equations.

Two outstanding French mathematicians were Jacques Salomon *Hadamard (1865–1963), who produced important work in analysis, number theory, differential geometry, calculus of variations, functional analysis, partial differential equations, and hydrodynamics, and inspired research among successive generations of mathematicians, and Laurent *Schwartz (1915–2002), whose work broadened the scope of calculus and brought Paul Dirac's ideas of "delta functions" in quantum mechanics within the scope of rigorous mathematics. For this work he was awarded the Fields Medal in 1950.

Another winner of the Fields Medal was Paul Joseph *Cohen (1934– ), for his fundamental work on the foundations of set theory. Cohen used a technique called "forcing" to prove the independence in set theory of the axiom of choice and of the generalized continuum hypothesis. Felix *Hausdorff (1868–1942) was also an authority on set theory and its applications to sets of points and real analysis. His textbook Mengenlehre (Leipzig, 1935) is recognized as one of the great classics of set theory. The depth and simplicity of his research into fundamental problems was a source of inspiration in the rapid development of modern mathematics.

Johann Ludwig von *Neumann (1903–1957) sought to develop the subject of quantum mechanics as a mathematical discipline, which led him to research in Hilbert space and the initiation of continuous geometry. In addition, Von Neumann made important contributions to measure theory, ergodic theory, continuous groups, topology, classical mechanics, hydrodynamic turbulence, and shock wave, and was a pioneer of game theory. Issai *Schur (1875–1941) specialized in the theory of numbers, particularly with regard to finite groups and their representations. He is widely known as the author of "Schur's lemma," which states that the only operators that commute with a unitary irreducible representation are the scalar multiples of the identity operator. Schur is also credited with extending the finite group theory to compact groups, and is noted for his work in the representation theory of the rotation group. André *Weil (1906–1998) contributed widely to many branches of mathematics, including the theory of numbers, algebraic geometry, and group theory. Norbert *Wiener (1894–1964) invented the science of cybernetics. As a mathematician, Wiener's main innovation was to develop a mathematics based upon imprecise terms reflecting the irregularities of the physical world. He sought to reduce these random movements to a minimum in order to bring them into harmony. During World War ii, he applied his concepts to work connected with antiaircraft defense, and this led to advances in radar, high-speed electric computation, the automatic factory, and a new science he created called cybernetics, a word he coined from the Greek word for "steersman," meaning the study of control. This followed his attempt as a mathematician to find the basis of the communication of information, and of the control of a system based on such communication. Wiener suggested the use of cybernetics in diagnostic procedures and indicated the similarity between certain types of nervous pathology and servomechanism (goal-directed machines such as guns which correct their own fixing malfunctioning). See also Benoit *Mandelbrot and Robert *Aumann.

bibliography:

R. Aumann and M. Mashler, "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud," in: J. Economic Theory, 36 (1985), 195–213; W. Feldman, Rabbinical Mathematics … Astronomy (1931; repr. 1978); S. Gandz, Studies in Hebrew Astronomy and Mathematics (1971); M. Littman, Approaching Infinity, Selected Mathematical writings of R. Shlomo of Chelme (1989); G.B. Sarfatti, Mathematical Terminology in Hebrew Scientific Literature of the Middle Ages (Heb., 1968); M. Steinschneider, Die Mathematik bei den Juden (1964); N.E. Rabinovitch, Probability and Statistical Inference in Ancient and Medieval Jewish Literature (1973); J. Rosenberg, "Some Examples of Mathematical Analysis Applied to Talmud Study," in: Mathematical Analysis Applied to Talmud Study, at: www.math.umd.edu/~jmr/MathTalmud.html; B. Tsaban and D. Graber, Mathematics in Jewish Sources, at: http://www.cs.biu.ac.il/~tsaban/hebrew.html. websites: www.jinfo.org; http://imu.org.il (for mathematics in Israel); http://www5.in.tum.de/lehre/seminare/math_nszeit/SS03/vortraege/verfolgt/#gliederung (for mathematicians persecuted by the Nazis).

MATHEMATICS

MATHEMATICS. In his "Mathematical Praeface" to the Elements of Euclid of 1570, Elizabethan polymath John Dee (1527–1608) expounded on the importance and utility of mathematics to all fields of human endeavor. Field after field, he argued, from those we would find obvious (like navigation) to those we would find arcane (astrology) or outlandish (thaumaturgike), would benefit from the systematic application of mathematics. Although Dee was promoting a role for mathematics that was just taking shape during his lifetime, his vision did indeed prove prophetic. Undoubtedly, one of the most striking features of intellectual life in the early modern period is the startling expansion in the scholarly and practical domains covered by mathematics.

MATHEMATICS AND ITS CRITICS

Prior to the sixteenth century, mathematics in the West was a well-defined and circumscribed field consisting of two main branches: arithmetic, which had obvious practical applications in commerce and banking, and Euclidean geometry, which had few practical uses apart from astrology and, occasionally, optics. While mathematics was generally admired for the certainty and universality of its claims, the world as a whole, in keeping with Aristotelian tradition, was distinctly unmathematical, being governed by qualitative rather than quantitative rules. By the eighteenth century this view had been turned on its head: not only was an ever increasing number of fields being subjected to mathematical analysis, but the world itself had come to be understood as fundamentally mathematical in nature.

These developments were by no means a foregone conclusion in the sixteenth century; if anything, they seemed highly unlikely. For mathematics, far from being universally acknowledged as central to the intellectual and technological life of the age, was at the time being challenged as never before from various quarters.

Conservative critics, defending the established order of knowledge, challenged the truth claims of mathematics as incompatible with prevailing Aristotelian standards. Prominent among them were Italian philosopher Alessandro Piccolomini (1508–1578) and the Jesuit Benito Pereira (c. 1535–1610), who challenged the explanatory value of mathematical proofs. Proper scientific explanations, they argued with perfect Aristotelian orthodoxy, were causal arguments, proceeding from the true essence of objects to their properties. Mathematics, however, had no proper subject matter at all, and it could say nothing about the essential nature of physical objects. All mathematics could do was point to logical relations between hypothetical propositions, and thus it was a fundamentally inferior type of knowledge.

Mathematics did not fare much better among the new generation of reformers, who sought to uproot the Aristotelian framework and replace it with new conceptions of knowledge. In breaking the hold of Aristotelian standards on contemporary natural philosophy, many reformers found little use for mathematics. Its rigid procedures and unchanging truths seemed an unpromising basis for a radical reform of knowledge. The study of nature, many argued, should proceed through unmediated experience and systematic trial and error. The rigorous deductive reasoning characteristic of mathematics could only lead to predetermined and unvarying results. The maverick Italian philosopher Giordano Bruno (1548–1600), for example, argued that mathematics could only describe the external appearance of phenomena, but never penetrate their hidden secrets. Similarly, in England, Francis Bacon (1561–1626) in the Novum Organum insisted that mathematics "should only give limits to natural philosophy, not generate or beget it."

Mathematics did, of course, have many prominent defenders, ranging from the Jesuit Christopher Clavius (1537–1612) to Galileo Galilei (1564–1642) and René Descartes (1596–1650), each insisting in his way on the essential role of mathematics in any meaningful scheme of knowledge. But the very range of suggestions these and other natural philosophers offered for the role of mathematics in the general scheme of knowledge makes it clear that the fundamental questions raised by the challenges to mathematics did not go away. The critiques raised the fundamental questions that would guide the development of mathematics throughout the early modern period: what is mathematics, and how is it related to the natural world? The history of mathematics in this period is the story of the various answers that were given to these questions.

THE WORLD AS MIRROR OF MATHEMATICS

The fundamental answer to the critiques of mathematics was given by Galileo in his Assayer of 1623, when he wrote that the universe "is written in the language of mathematics." Galileo was expressing the widely held notion among practitioners that mathematics, far from being devoid of all subject matter as claimed by its critics, had the entire natural world as its object. But while most agreed that mathematics was closely integrated with the physical world, the precise nature of their relationship remained a matter of intense dispute.

One leading approach accepted the classical view of mathematics as a rigorous deductive science of number and magnitude. The universal laws of mathematics, in this view, were the fundamental laws that governed material reality. Thus when one is investigating mathematical and geometrical relationships, one is in fact investigating the basic structure of matter.

The chief promoter of this approach was René Descartes, who viewed mathematics as a fundamental rational law laid down by God for his creation. Once God, the divine architect, had set in motion his perfectly rational universe, it would henceforth operate forever in accordance with mathematical principles. Mathematical investigations are accordingly studies of the divine plan for the natural world, and the world is the direct expression of abstract mathematical principles.

Descartes's scientific work directly reflects this fundamental understanding. In his Meditations and the Discourse on Method, Descartes insisted that by following strict rational rules one could, in principle, follow God and "create" the world step by step. Rigorous rational deduction was therefore the key to knowledge of the natural world, and Descartes proceeded to demonstrate the effectiveness of this principle in short treatises on optics and the colors of the rainbow, which were attached to the early editions of the Discourse.

Descartes's most important contribution to mathematics was also a reflection of his religious and philosophical views. The Geometry was the founding text of analytic geometry and, like Descartes's other scientific treatises, was published as an appendix to the Discourse. In essence, the new field demonstrated the fundamentally mathematical nature of the physical world. Abstract algebraic relationships (that is, y =3Dax +b ) were shown to have actual physical manifestations (in this case, a straight line). In pointing out these hidden relationships Descartes was unveiling the divine mathematical laws that governed the world. Mathematics, in this view, was a perfectly rational and logical web of relationships that determined the nature of physical reality.

MATHEMATICS AS THE MIRROR OF THE WORLD

While Descartes was honing his analytical geometry, a very different mathematical approach, based on a very different understanding of the relationship of mathematics to the world, was being developed elsewhere in Europe. The use of infinitesimals, or "indivisibles" as they were most commonly called, in calculating lengths, areas, and volumes of geometrical figures was the most dramatic and important development in seventeenth-century mathematics. Fundamentally, the procedure involved reducing geometrical objects into an infinite number of their component parts: lines were viewed as an infinite collection of points, surfaces as made up of an infinite number of lines, and solids of surfaces. The length, area, or volume of the figure as a whole would then be calculated as the infinite sum of its elementary components.

The fundamental assumptions underlying this procedure were highly questionable and seemed to fly in the face of paradoxes that had been well known since antiquity. Descartes, who was much concerned with the perfect rational structure of mathematics, rejected infinitesimals and excluded them from the bounds of mathematics. Nevertheless, the effectiveness of this approach in reaching correct and previously unknown results was undeniable, and it was embraced enthusiastically by mathematicians across Europe. Thomas Hariot (1560–1621) and John Wallis (1616–1703) in England, Galileo and his disciples Bonaventura Cavalieri (c. 1598–1647) and Evangelista Torricelli (1608–1647) in Italy, Johannes Kepler (1571–1630) in Germany, and Blaise Pascal (1623–1662) in France were but a few of the most prominent practitioners of the new methods.

The infinitesimalist mathematicians' view of the relationship between mathematics and the world was, in many ways, the reverse of Descartes's approach. Whereas Descartes assumed that pure mathematical relationships governed the structure of matter, the infinitesimalists modeled mathematics on an intuition of the physical world. Geometrical bodies could be broken down into their indivisible components because, by analogy, physical bodies could be divided in the same way. As Cavalieri, whose Geometria Indivisibilibus was the most influential book about the theory and practice of indivisibles, wrote in his introduction, "plane figures should be conceived by us in the same manner as cloths are made up of parallel threads, and solids are in fact like books, composed of parallel pages."

The infinitesimalists' approach to mathematics drew much of its inspiration from the empiricist experimental philosophy that was gaining ground throughout Europe at this time. Much as the experimentalists sought to penetrate through external appearances and bring to light the inner structure of the material world, the new mathematicians sought to uncover the "inner structure" of geometrical figures, which in their view was the true cause of all geometrical relationships. Both groups, furthermore, adopted the imagery of geographical exploration as their guiding metaphor, presenting themselves as adventurous explorers on the hazardous seas of mathematics and natural philosophy.

Like their experimentalist colleagues, the infinitesimalists made the discovery of new and correct results the true test of their success, and like them they often adopted a methodology of trial and error in searching for the correct answers. This "experimental" approach to mathematics accounts for the infinitesimalists' relative disregard for the niceties of mathematical rigor and consistency. In their view, if a method produces true results, it must be fundamentally correct, and there was no point in spending too much time on clarifying the finer logical points. The most outspoken and unapologetic proponent of this approach was probably John Wallis, who advocated applying the experimentalists' "method of induction" to mathematics, in preference to traditional rigorous mathematical deduction.

While the new infinitesimalist approaches were in wide use in the seventeenth century, they were also seriously challenged in certain influential quarters. The issues at stake were not purely mathematical in nature, but involved wide-ranging philosophical, religious, and even political considerations. For one thing, the new approaches carried the taint of atomism—the ancient view that all material objects could be reduced to indivisible particles called "atoms" (from the Greek atomos, 'uncuttable'). Indeed there was no denying that the fundamental insights of the new mathematics and even its name strongly hinted that infinitesimalist mathematics was nothing but an expansion of atomism into mathematics.

This in turn led to a deeper difficulty: the suspicion that the new mathematics was based not just on atomism, but on materialism, which is the notion that the world was composed of nothing but matter, leaving no room for a providential spiritual realm. Geometry, after all, was often taken to be the very model of pure and abstract reasoning that governs the natural world. The notion that geometry itself, far from governing physical reality, is in fact a generalization of it, seemed to turn the proper hierarchy of mind and matter on its head, and challenge those who insisted that the world was ruled by a higher intelligence.

Finally, there was the question of the certainty of knowledge. Infinitesimalist mathematics seemed to be based on nothing more than a loose analogy with the physical world, trial and error, and a willful disregard for logical paradox. If even mathematics, that paragon of certain and unchanging knowledge, turned out to be so unsound, what hope could other, less rigorous fields have of attaining true knowledge?

In an age that still considered science, philosophy, and theology to be part of a single unified worldview, these criticisms cut deep. Descartes, concerned about the rational certainty of his method, excluded infinitesimal methods from proper mathematics. Even more significant was the reaction of the Society of Jesus, the most prominent religious order in Europe and the guardian of Catholic orthodoxy. Despite having among their members some of the most important and creative mathematicians in Europe, the Jesuits banned the teaching of infinitesimals from their educational institutions.

THE CALCULUS AND BEYOND

The invention of the calculus by Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) in the late seventeenth century was the most important development of early modern mathematics, and it quickly transformed the landscape of the field. The calculus took as its starting point the many practical techniquesand results achievedbythe infinitesimalist mathematicians, both in the determination of surfaces and volumes of geometrical figures, and in the calculation of tangents of curves. The fundamental insight of the calculus was that these two operations, calculating tangents (differentiation) and calculating surfaces and volumes (integration), are in fact the inverse of one another.

The importance of this discovery becomes clear when curves and geometrical figures are presented not as independent geometrical figures, but as expressions of algebraic formulations in the manner of analytic geometry. When presented in this manner, differentiation no longer deals with geometrical properties of particular geometrical objects, but becomes an abstract and general relationship between algebraic expressions. For example, one can say that the parabola expressed as y =3Dx 2 describes the area under the line y =3D 2x, and that y =3D 2x expresses the tangent of the parabola y =3Dx 2 at any point. But the relationship between the two algebraic expressions is no longer dependent on their particular geometrical representation: y=3D 2x is simply the differential of y =3Dx 2 and y =3Dx 2is the integral of y =3D 2x. The inverse relationship is a fundamental relationship between abstract algebraic expressions (or functions, as they came to be called later in the eighteenth century) independent of any particular geometric representation. Both Newton and Leibniz were quick to reduce the transformations back and forth between differentials and integrals (or "fluents" and "fluxions" as Newton called them) into systematic and reliable algorithms.

In the calculus, the two competing traditions of seventeenth-century mathematics were brought together. Although it clearly grew out of the techniques developed by infinitesimalist mathematicians, the calculus was equally dependent on the algebraic formulations of analytic geometry. Furthermore, the calculus detached the infinitesimalist methods from their dependence on an intuition of physical reality. If the older approaches could be viewed as growing out of an atomistic intuition of material reality, the calculus restored the primacy of abstract logical relationship to mathematics. Particular geometric figures could be seen as examples of these abstract algebraic relations, but these relations themselves were no longer dependent on any particular physical or geometrical instances.

MATHEMATICS IN THE ENLIGHTENMENT

The calculus, which positioned mathematics as both an abstract system of algebraic relationships and as intimately connected to the physical world, set the tone for eighteenth-century views of the field. The most eloquent formulation of attitudes toward mathematics in the Enlightenment was given by Jean Le Rond d'Alembert (1717–1783), in his "Preliminary Discourse" to the Encyclopédie, published in 1751. Whereas seventeenth-century practitioners viewed mathematics as either a generalization of material intuitions or as a universal law governing nature, for d'Alembert mathematics was necessarily both. On the one hand, he insisted, mathematics is clearly an abstraction from nature: it is nothing but the fundamental relationships among natural objects that are arrived at when the material features such as texture and color are stripped away. On the other hand, d'Alembert argued, the laws of nature are simply elaborations of mathematical relationships, arrived at by restoring matter's physical attributes to abstract disembodied mathematics. The world, then, according to d'Alembert, is fundamentally mathematical: mathematics is derived from the physical world, while the physical world is an extension of mathematical principles.

This view of an essentially mathematical universe manifested itself in the inclusion of an evergrowing number of scholarly fields that were brought under the sway of mathematics in this period. Years before, Galileo had already introduced mathematics into the study of falling bodies and statics, and he and his followers extended his work to the field of ballistics. Cartographic work was thoroughly mathematized in the seventeenth century, and Kepler and Newton transformed the ancient science of astronomy by extending the reach of mathematics from merely describing the motions of the heavens into the realms of celestial mechanics. In optics, Descartes's ingenious application of his "method" enabled him to explain such phenomena such as the formation of the rainbow with mathematical precision.

In the eighteenth century, a new generation of mathematicians, including the Bernoullis, Leonhard Euler (1707–1783), d'Alembert, Joseph-Louis Lagrange (1736–1813), and Pierre-Simon Laplace (1749–1827), among others, added increasingly precise theories of mechanics and argued famously about proper mathematical representations of abstract concepts such as vis viva, and concrete problems like the vibrations of strings and hydromechanics. Other fields that were seemingly less malleable for quantitative analysis, like doctrines of chance, or probability, and also the "moral" sciences, known today as social sciences, were also brought under the sway of mathematics, particularly in the work of the marquis de Condorcet (1743–1794). Institutionally, the eighteenth century saw mathematics gain a quickly growing foothold in newly established engineering and military colleges.

EPILOGUE

Unfortunately for d'Alembert and other promoters of the mathematical universe, rigorous mathematical analysis could not be easily derived from physical reality. Inconsistencies and paradoxes seemed to crop up repeatedly when mathematics was modeled on perceptions of the physical world, as critics of infinitesimal methods and the calculus, such as George Berkeley, were quick to point out. At the same time, the physical world proved to be far more varied and surprising than could ever be derived from bare mathematical principles.

Early in the nineteenth century the interdependence of mathematics and the physical world, so eloquently presented by d'Alembert, came to an end. In their work on the foundations of the calculus, mathematicians Bernhard Bolzano (1781–1848) and Augustin-Louis Cauchy (1789–1857) reformulated mathematical analysis as rigorous and logically self-consistent, a goal that had eluded their Enlightenment predecessors. They did so, however, at a price that would have seemed too heavy for d'Alembert and his colleagues: pure mathematics, in their scheme, was finally divorced from physical reality, existing in its self-enclosed Platonic realm.

The course and development of mathematics in the early modern period had come full circle. Criticized in the sixteenth century for being irrelevant to the developing sciences, mathematicians at the time had responded by forming a closer bond than ever before between their field and the physical world. Two and a half centuries later, in an attempt to save the identity and coherence of their field, mathematicians chose to sever those same conceptual ties, and establish mathematics in its own separate and insular domain.

See also Alembert, Jean Le Rond d' ; Aristotelianism ; Astronomy ; Bacon, Francis ; Cartesianism ; Descartes, René ; Empiricism ; Euler, Leonhard ; Lagrange, Joseph-Louis ; Leibniz, Gottfried Wilhelm ; Logic ; Newton, Isaac ; Scientific Method .

BIBLIOGRAPHY

Alexander, Amir. Geometrical Landscapes: The Voyages of Discovery and the Transformation of Mathematical Practice. Stanford, 2002.

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York, 1959.

Daston, Lorraine J. Classical Probability in the Enlightenment. Princeton, 1988.

Dear, Peter. Discipline and Experience: The Mathematical Way in the Scientific Revolution. Chicago, 1995.

Hankins, Thomas L. Science and the Enlightenment. Cambridge, U.K., and New York, 1985.

Amir Alexander

Mathematics

Leonardo’s admiration for mathematics was unconditional, and found expression in his writings in such statements as “No certainty exists where none of the mathematical sciences can be applied” (MS G, fol. 96v). It is therefore useful to consider the development of his mathematical thought, drawing upon the whole of his manuscripts in order to reconstruct its principal stages; such a study will also serve to illustrate the sources for much of his work.

Leonardo particularly valued the rigorous logic implicit in mathematics, whereby the mathematician could hope to attain truth with the same certitude as might a physicist dealing with experimental data. (Students of the moral and metaphysical sciences, on the other hand, had no such expectation, since they were forced to proceed from unascertainable and infinitely arguable hypotheses.) His predilection did not, however, presuppose a broad mathematical culture or any real talent for calculation. Certainly, his education was that of an artisan rather than a mathematician, consisting of reading, writing, and the practical basics of calculus and geometry, together with the considerable body of practical rules that had been accumulated by generations of craftsmen and artists.

The effects of his early education may be seen in Leonardo’s literary style (he preferred aphorisms and definitions to any prolonged organic development of ideas) and in his mathematics, which contains grave oversights not entirely due to haste. An example of the latter is his embarrassment when confronted with square and cube roots. In Codex Arundel (fol. 200r) he proposed, as an original discover), a simple method for finding all roots “both irrational and rational,” whereby the “root” is defined as a fraction of which the numerator alone is multiplied by itself one or two times to find the square or cubic figure. Thus the square root of 2 is obtained by multiplying 2/2 by 2/2 to find 4/2, or simply 2 ; while the cube root of 3 is reached by taking 3/9 x 3/9 x 3/9 = 27/9, or 3. He applied this erroneous method several times in his later work; however absurd they might be, Leonardo seems to have been prouder of his discoveries in mathematics than in any other field.

When he was in his late thirties, Leonardo began to try to fill the gaps in his education and took up the serious study of Latin and geometry, among other subjects. At the same time he began to write one or more treatises. He wished to write for scientists, although scholars of the period were not prepared to admit the knowledge possessed by an artist as science. (An illustration of this occurs in Alberti’s definition of the “principles” of geometry, in which he stated that he could not use truly scientific terms because he was addressing himself to painters.) It was only after Leonardo entered the Sforza service in Milan and became associated with philosophers and men of letters on a basis of mutual esteem that he took up his theory of the supremacy of painting. This theory was in part incorporated into the Treatise on Painting compiled by Leonardo’s disciple Francesco Melzi, who, however, omitted Leonardo’s definitions of the geometric “principles” of point, line, and area—definitions that Leonardo had drawn up with great care, since he wished them to be read by mathematicians rather than artists.

As a painter, Leonardo was, of course, concerned with proportion and also with “vividness of the actions” (this is how he translated Ficino’s definition of beauty as “actus vivacitas"). In his pursuit of the latter, Leonardo inclined toward physics, seeking motions that could be studied experimentally. Perspective, too, had an important place in the treatise on painting, but Leonardo was more concerned with aerial perspective (chiaroscuro) than with linear. MS C (1490) is rich in geometrical drawings, although limited to the depiction of the projection of rays of light; it contains no organic system of theorems.

Aside from those in MS C, the pages dedicated to geometry in other early manuscripts are few; there are nine in MS B (ca. 1489) and nineteen in MS A (1492). The prevalent matter in all of these is the division of the circumference of the circle into equal parts, a prerequisite for the construction of polygons and for various other well-known procedures. The notes set down are not connected with each other and represent elementary precepts that were common knowledge to all “engineers,” Because of these limitations, it is impossible to accept Caversazzi’s thesis concerning MS B, folios 27v and 40r, wherein Leonardo explained how to divide a circumference into equal parts by merely “opening the compass” ; Caversazzi sees this as an “exquisite geometric discovery,” one that may be applied to solve Euclidean problems of the first and second degree. But since at this time Leonardo had not yet begun to study Euclid, it is probable that he was referring to precepts known to all draftsmen.

Seven pages are devoted to geometry in Codex Forster 111 (1493-1495) and, again, these contain only elementary formulas, expressed in language more imaginative than scientific (the volume of a sphere, for example, is described as “the air enclosed within a spherical body” ). Only in Codex Forster II₁ written between 1495 and 1497, are there the first signs of a concentrated interest in geometrical problems, particularly those that were deeply to concern Leonardo in later years—lunes and the equivalence of rectilinear and curvilinear surfaces. An additional seventeen pages of this manuscript are devoted to the theory of proportion.

The two manuscripts that mark the close of Leonardo’s first sojourn in Milan-MS M and MS I-are of far greater importance. The first thirty-six pages of MS M contain translations of Euclid (“Petitioni” and “Conceptioni”) as well as derivations of propositions 1-42 (with a few omissions) of book I of the Elements, together with a group of propositions from the tenth book. Propositions 43-46 of book I appear in the first sixteen folios of MS I (which must for that reason be considered as being of a later date than MS M), as do propositions 1-4 and 6-10 of book 11, selections from book 111, and occasional references to book X. These evidences of Leonardo’s systematic study of Euclid may be related to his friendship and collaboration with Luca Pacioli.

Leonardo must have read the Elements and acquired some deeper knowledge of geometry before undertaking the splendid drawings of solid bodies with which he illustrated the first book of Pacioli’s Dirina proportione. His interest in this work is probably reflected in the pages of Codex Forster II₁ on proportion; it would also explain his study of the tenth book of the Elements, the book least read because of its difficulty and its practical limitation to the construction of regular polygons. Pacioli was thus responsible for arousing Leonardo’s enthusiasm for geometry and for introducing him to Euclid’s work; indeed, he may have helped him read Euclid, since the text would have been extremely difficult for a man as relatively unlettered as Leonardo. (It is interesting to note that MS I also contains some first principles of Latin, copied from Perotti’s grammar.) Pacioli’s influence on Leonardo was probably also indirect, through his Summa arithtietica and perhaps through his translation of the Elements.

That Leonardo had at hand the Latin text of the Elements of 1482 or 1491 can be seen through the identity of some of his drawings with those texts as well as by certain verbal correspondences. At the same time, his method was not to transcribe sentences from the text he was studying but to attempt to present geometrical ideas graphically. each page of his geometrical notes represents an aid to memorizing Euclid’s text, rather than a compendium; hence it is not always easy to trace the specific passages studied. For example, Leonardo would often begin with a sketch, a number, or occasionally a word which he must have used initially to impress upon his mind some of the intricacies of Euclid’s theses and later to recall the whole content. In MS M, folio 29v, three figures appear: a point and a line; the same point and line with an additional transversal line; and two parallel lines joined by the transversal, the original point having disappeared. The correspondence with the Euclidean thesis, together with the coincidence of similar notes on the preceding and following pages to propositions 27, 28, 29, 30, and 32, respectively, prove this to be Euclid’s book 1, proposition 31.

Notes on Euclid’s first books also appear in certain folios of the Codex Atlauticus, where they are set out in better order and in a more complete form. On folio 169r-b, for instance, the “Petitioni” and “Conceptioni” are presented symbolically, while on folio 177v-a proposition 1.7 is transformed into a series of thirteen drawings.

Leonardo continued to work with Pacioli after they both left Milan in 1499. In the Codex Atlanticus he stated that he would learn “the multiplication of roots” from his friend “master Luca” (fol. 120r-d). and he had in fact transcribed all the rules for operations with fractions from the Summa arithmetica (fol. 69, a-b). Pietro da Novellara recorded in 1503 that Leonardo was neglecting painting in favor of geometry; this activity is reflected in MS K, Codex Madrid, 11, and Codex Forster_I,as well as in many folios of the Codex Atlanticus.

Two-thirds of MS K₁(1504) are devoted to Euclid. From folio 15v to folio 48v Leonardo copied, in reverse order, almost all the marginal figures of books V and VI of the Elements. He transcribed none of the text, although the drawings are accoumpanied by unmistakable signs of his contemplation of the theory of proportion. MS K₂ contains notes similar to those in MS M and MS 1; here they refer to the whole of the second book of the Elements, to a few propositions of books I, II and III, and to nine of the first sixteen definitions of book V.

Leonardo’s interest in the theory of proportion is further evident in Codex Madrid, II, folios 46v–50, in which he summarizes and describes the treatise “De proportionibus et proportionalitatibus” that is part of Pacioli’s Summa arithmetica. A drawing on folio 78 of the same manuscript, graphically illustrating all the various kinds of proportions and proportionalities, is also taken from Pacioli’s book, while Euclid reappears in the last five pages of the manuscript proper. In the latter portion Leonardo transcribed, in an elegant handwriting, the first pages of an Italian version of the Elements, the author of which is not known (the list of Leonardo’s books cites only “Euclid, in Italian, that is, the first three books” ). Folio 85r contains the ingenious illustration of Euclid’s so-called algorithm (VII.2, X.2-3) that is repeated in more detail in Codex Atlanticus, folio 207r-b.

It is thus clear from manuscript sources that from the time that Leonardo began to collaborate with Pacioli on the Divina proportione, he concentrated on books I and II of Euclid—an indispensable base—and on books V and VI, the theory of proportion that had also been treated by Pacioli in the Summa arithmetica. The few references to book X deal with the ratios of incommensurate quantities; therefore there can be no doubt that the subject of proportions and proportionalities remained a constant center of Leonardo’s interest.

There is no evidence of a similar study of the last books of Euclid, but it must not be forgotten that about two-thirds of Leonardo’s writings have been lost. There is, moreover, no lack of practical applications of the propositions of the last books in Leonardo’s writings. But Codex Madrid, II, is of particular importance because it demonstrates that Leonardo, after conducting a modest study of Euclid from approximately 1496 to 1504, began to conduct ambitious personal research. Having copied out the first pages of the Elements as, perhaps, a model, Leonardo wrote on folios IIIr and 112r two titles indicative of the plan of his work, “The Science of Equiparation” and “On the Equality of Unequal Areas.” For this new science he wrote on folio 107v a group of “Petitioni” and “Conceptioni” based upon Euclid’s. Here Leonardo’s chief concern lay in the squaring of curvilinear surfaces, which he generally divided into “falcates” (triangles with one, two, or three curved sides) and “portions” (circular segments formed by an are and a chord, which could also be the side of a polygon inscribed in a circle). He carried out the transformation of these figures into their rectilinear equivalents by various means, some of which (such as superimposition and motion, or rotation) were mechanical.

The use of mechanical solutions, rarely accepted by Euclid, may perhaps be attributable to Leonardo’s engineering background; he would almost seem to have recognized the unorthodoxy of his procedures in applying to himself Giordano’s words: “This method is not simply geometrical but is subordinated to, and participates in, both philosophy and geometry, because the proof is obtained by means of motion, although in the end all mathematical sciences are philosophical speculations” (Codex Madrid, II,fol. 107r). Leonardo noted, however, that the squaring of curvilinear surfaces could be accomplished by more orthodox geometrical methods, provided the curved sides of the figure form parts of circles proportional among themselves (Codex Atlanticus, fol. 139v-a).

It is a short step from squaring falcates to squaring the circle, and Leonardo proposed several solutions to the latter problem. Again, some of them were mechanical and were suggested to him by studying Vitruvius. These consist in, for example, measuring the rectilinear track left by a wheel of which the width is one-quarter of the wheel’s diameter (MS G, fol. 61r; also MS E, fol. 25v, where the width of the wheel that makes “a complete revolution” is mistakenly given as being equal to the radius). The circumference is obtained by winding a thread around the wheel, then withdrawing and measuring it (MS K, fol. 80r).

Leonardo was also well acquainted with the method for the “quadratura circuli per lunulas,” which he knew from the De expetendis et fugiendis rebus of Giorgio Valla. A drawing corresponding to this method appears in MS K, folio 61r. (The identity of “Zenophont” or “Zenophonte,” a mathematician criticized by Leonardo on the previous page of the same manuscript, is probably resolvable as Antiphon,

whose method of quadrature was also criticized by Valla.)

Leonardo of course knew Archimedes’ solution to squaring the circle; Clagett has pointed out that he both praised and criticized it without ever having understood it completely. He accepted Archimedes’ first proposition, which establishes the equivalence of a circle and a rectangular triangle of which the shorter sides are equal to the radius and the circumference of the circle, respectively; he remained unsatisfied with the third proposition, which fixes the approximate ratio between the circumference (taken as an element separating two contiguous classes of polygons) and the diameter as 22:7. Leonardo tried to take this approximation beyond the ninety-six-sided polygon; it is in this attempt at extension ad infinitum that the value of his effort lies.

Valla provided Leonardo with a starting point when, in describing the procedure of squaring “by lunes,” he recommended “trapezium dissolvatur in triangula. “Leonardo copied Valla’s drawing, dividing the trapezium inscribed in a semicircle into three triangles; he extended the concept in MS K , folio 80r, in which he split the circle into sixteen triangles, rectified the circumference, then fitted the triangles together like gear teeth, eight into eight, to attain a rectangle equivalent to the circle. He took a further step in Codex Madrid, II, folio 105v, when departing from Euclid 11.2 (circles are to each other as the squares of their diameters), he took two circles, having diameters of 1 and 14, respectively, and divided the larger into 196 sectors, each equivalent to the whole of the smaller circle.

In Codex Atlanticus, folio 118v-a, Leonardo dealt with circles of the ratio 1:1,000 and found that when the larger was split into 1,000 sectors, the “portion” (the difference between the sector and the triangle) was an “imperceptible quantity similar to the mathematical point.” By logical extension, if each sector were “a millionth of its circle, it would be a straight portion” or “almost plane, and thus we would have carried out a squaring nearer the truth than Archimedes”’ (Codex Madrid, II, fol. 105v). In Codex Arundel, folio 137v, Leonardo squared the two diameters, writing “one by one is one” and “one million by one million is one million”; his disappointment in seeing that the squares did not increase suggested the variant “four million by four million is sixteen million” (that he corrected his error in Codex Atlanticus, fol. 118r-a, writing “a million millions,” reveals a poor grasp of calculation).

Leonardo was clearly proud of his discovery of the quadrature, and in Codex Madrid, II, folio 112r he recorded the exact moment that it came to him—on the night of St. Andrew’s day (30 November), 1504, as hour, light, and page drew to a close. In actual fact, however, Leonardo’s discovery amounted only to affirming the equivalence of a given circle to an infinitesimal part of another without calculating any measure; to have made the necessary calculations, he would have had to use Archimedes’ formula 22:7, which he rejected. Leonardo characterized Archimedes’ method of squaring as “ben detta e male data” (Codex Atlanticus, fol. 85r-a), that is, “well said and badly given.” It would seem necessary to describe his own solution simply as “detta,” not “data.”

The presumed discovery on St. Andrew’s night did, however, encourage Leonardo to pursue his geometrical studies in the hope of making other new breakthroughs. MS K₃, MS F, and a large number of pages in the Codex Atlauticus demonstrate his continuous and intense work in transforming sectors, “falcates,” and “portions” into rectilinear figures. Taking as given that curved lines must have equal or proportional radii, Leonardo practiced constructing a series of squares doubled in succession, in which he inscribed circles proportioned in the same way; to obtain quadruple circles, he doubled the radii. To obtain submultiple circles, he divided a square constructed on the diameter of the circle into the required number of rectangles, transforming one into a square and inscribing the submultiple circle therein. In order to double the circle, however, instead of using the diagonal of the square constructed on the diameter, he turned to arithmetic calculus to increase proportionally the measure of the radius, adopting a ratio of 1 :3/2, which is somewhat greater than the true one of 1:1/2. To obtain a series of circles doubling one another, he divided the greatest radius into equal parts, forgetting that arithmetic progression is not the same as geometric (Codex Madrid, II, fols. 117v, 132).

Alternating with the many pages in the Codex Atlanticus devoted to the solution of specific problems are others giving the fundamental rules of Leonardo’s new science of comparison. Within the same sphere falls a group of pages in Codex Madrid, II, which are concerned with transforming rectilinear figures into other equivalent figures or solids into other solids. These sections of the two collections, presumably together with pages that are now lost, are preparatory to the short treatise on stereometry contained in Codex Forster I₁ (1505) and entitled “Book on Transformation, That Is, of One Body Into Another Without Decrease or Increase of Substance.” In his study of this treatise, Marcolongo judges it to be suitable “for draftsmen more skilled in handling rule, square, and compass than in making numerical calculations” —Leonardo’s viewpoint was often that of the engineer.

One of the problems to which Leonardo chose to apply his new science was that of how to transform a parallelepiped into a cube, or how to double the volume of a cube and then insert two proportional means between two segments. In the Summa arithmetica, Pacioli had solved the problem and had described how to find the cube root of 8 by geometrical methods, although he provided no geometrical demonstrations. Leonardo, in Codex Atlanticus, folio 58r, stated that if the edge of a cube is 4, the edge of the same cube doubled will be 5 plus a fraction that is “inexpressible and easier to make than to express.” By 1504 Leonardo also possessed Valla’s book, which gave the solution of the problem, with numerous demonstrations from the ancients. Leonardo copied various figures from Valla (in Codex Arundel, fols. 78r-79v, where the last one is, however, Pacioli’s) and transcribed a vernacular translation of the part of Valla’s book referring to the demonstrations of Philoponus and Parmenius. (That, having the Latin text at hand, Leonardo felt the need to translate or to have translated the two pages that interested him confirms his difficulty in reading Latin directly-Clagett and Marcolongo have both pointed out many mistakes in the translation of this passage which, if uncorrected, distort the text.)

Leonardo repeatedly expressed his dissatisfaction with the solution reported by Valla. He particularly objected to having to make the rule swing until the compass has fixed two points of intersection; this “negotiation,” he stated (Codex Atlanticus, fol. 218v-b), seemed to him “dubious and mechanical.” Since Valla attributed this procedure to Plato, Leonardo further stated, in MS F (1508), folio 59r, that “the proof given by Plato to the inhabitants of Delo is not geometrical.” He continued to look for other, more truly “geometrical,” means to double the cube or find the cube root.

In folios 50v-59v of MS F and in the similar folios 159r-a and b of the Codex Atlanticus, Leonardo tried, in opposition to the “ancient system,” to resolve the problem by decomposing and rebuilding a cube; he further attempted to apply Pythagoras’ theorem by substituting three cubes for three squares constructed on the sides of a right triangle. Since every square is equal to half the square constructed on its diagonal, he attempted to halve each cube with a diagonal cut, on the assumption that the face of a doubled cube could be obtained through manipulating the rectangular face formed by the cut. He arranged nine cubes in the form of a parallelepiped upon which he analyzed the diagonals that he supposed to represent the square and cube roots, respectively (Codex Atlanticus, fols. 159r-b, 303r-b; Windsor Collection, fol. 19128). He forgot, however, that the progression of squares and cubes does not correspond to the natural series of numbers. Although he studied the proportions between the areas that “cover” a cube and those of the same cube doubled, he realized that he could not apply “the science of cubes based on surfaces, but based on bodies [volumes]” (Codex Arundel, fol. 203r). He was thus aware that his formulation of the problem was inaccurate.

Leonardo finally arrived at a solution. He recorded in Codex Atlanticus, folios 218v-b, 231r-b, that in order to “avoid the difficulty of the mechanical system” taught by Plato and other ancients, he had

eliminated the compass and the imprecise movement of the rule in favor of placing two sides of the rectangle in the ratio b = 2a (thereby unknowingly reestablishing the ratio used in the Greek text). Step by step, then, his procedure was to join two faces of a cube to make a rectangle, connect the upper-left angle to the center of the right square, then carry the line obtained to the upper extension of side a; from the extreme point thus reached, he drew a line that, touching the upper-right angle of the rectangle, cut the extension of its base. On this line Leonardo was able to determine the measure of the edge of the doubled cube, a measure that is to be found four times in the figure thus constructed. His results are reasonably accurate; with a = 2, Leonardo obtained a cube root for 16 as little as 0.02967 in excess of its true value

Leonardo was unable to demonstrate the truth of his “new invention” save through criticism of the “mechanical test” of the compass, which after “laborious effort” helped the ancients to discover the proportional means—whereas now, Leonardo claimed, “without effort I use it to confirm that my experiment was confirmed by the ancients “—which, he added, “could not be done before our time.” What Leonardo had discovered was that the third proportional number, used to determine the second, coincides with a line that can be more accurately constructed inside the rectangle; he thereby simplified the classical procedure without in any way discussing its scientific demonstrations, which he took for granted. He must, however, have checked his results experimentally and judged them with his unerring eye.

Other pages of the Codex Atlanticus contain summaries of problems to be solved, as, for example, folio 139r-a, which is entitled “Curvilinear Geometrical Elements.” This, together with other titles- “On Transformation,” folio 128r-a; “Book on Equation,” folio 128r-a; and “Geometrical Play” ( “De ludo geometrico” ), folios 45v-a, 174v-b, 184v-c, and 259v-a —including the previously mentioned “Science of Equiparation” —might suggest that Leonardo had actually written systematic books and treatises that are now lost. But it is known that he cherished projects that he did not complete; and it is unlikely that he composed treatises using methods and forms different from those set out in, for example, Codex Forster I_2. The titles of treatises projected but unwritten do provide some record of the stages of his mathematical work, in addition to which Leonardo has recorded some specific dates for his successes.

Leonardo’s discovery on St. Andrew’a night has already been mentioned: in the same manuscript, Codex Madrid, II, on folio 118r, he further mentioned that a certain invention had been given to him “as a gift on Christmas morning 1504.” In Windsor Collection, folio 19145, he claimed to have discovered, after prolonged research, a way of squaring the angle of two curved sides on Sunday, 30 April 1509. (Folio 128r-a of the Codex Atlanticus, the beginning of the “Book of Equation,” is concerned with the squaring of a “portion” of a single curved side and promises a second book devoted to the new procedure.)

Although the last discovery is exactly dated, Leonardo’s explanation of it is not very clear. (This is often the case in Leonardo’s work, since the manuscripts are marked by incomplete and fragmentary discussions, geometrical figures unaccompanied by any explanation of the construction, and the bare statements of something proved elsewhere.) His accomplishment is, however, apparent—he had learned how to vary a “portion” in infinite figures, decomposing and rebuilding it in many different ways. The examples drawn on the Windsor Collectionpage cited reappear, with variations, in hundreds of illustrations scattered or collected in the Codex Atlanticus. The reader can only be dismayed by their complexity and monotony.

The mathematical pages of the Codex Atlanticus (to which all folio numbers hereafter cited refer) provide an interesting insight into the late developments of Leonardo’s method. On folio 139r-a, among other places, he confirmed his proposal to vary “to infinity” one or more surfaces while maintaining the same quantity. It is clear that he received from the elaboration of his geometric equations the same pleasure that a mathematician derives from the development of algebraic ones. If, at the beginning of his research on the measure of curvilinear surfaces, his interest would seem to have been that of an engineer, his later work would seem to be marked by a disinterested passion for his subject. Indeed, the title of the last book that he planned— “Geometrical Play,” or “De ludo geometrico” —which he began on folio 45v-a and worked on up to the last years of his life, indicates this.

For example, one of Leonardo’s basic exercises consists of inscribing a square within a circle, then joining the resultant four “portions” in twos to get

“bisangoli.” The “bisangles” can then be broken up, subdivided, and distributed within the circumference to create full and empty spaces which are to each other as the inscribed square is to the four “portions” of the circumscribed circle. An important development of this is Leonardo’s substitution of a hexagon for the square, the hexagon being described as the most perfect division of the circle (fols. 111v-s and b [Figs. 3. 4]). This leaves six “portions” for subsequent operations, and Leonardo recommended their subdivision in multiples of six. (On the splendid folio 110v-a [Fig. 5], he listed and calculated the first fifty multiples of six, but through a curious oversight the product of 34 × 6 is given as 104, rather than 204: and all the following calculations are thus incorrect.)

To obtain the exact dimensions of submultiple “portions,” Leonardo devised the “square of proportionality” (fol. 107v-a), which he constructed on the side of the hexagon inscribed within the circle and then subdivided into the required number of equal rectangles. He next transformed one such rectangle into the equivalent square, the side of which is the radius of the submultiplied circle, the side of the corresponding inscribed hexagon, and the origin or chord of the minor “portions.” These were then joined in pairs (“bisangoli.”), combined into rosettes or stars, and distributed inside the circumference of the circle in a fretwork of full and empty spaces, of which the area of all full spaces is always equal to that of the six greater “portions” and the area of all the empty spaces to the area of the inscribed hexagon. The result is of undoubted aesthetic value and may be taken as the culmination of the geometrical adventure that began for Leonardo with Pacioli’s Divina proportione.

Although Leonardo remained faithful throughout his life to the idea of proportion as the fundamental structure of reality, the insufficient and inconstant rigor of his scientific thesis precludes his being considered a mathematician in the true sense of the word. His work had no influence on the history of mathematics; it was organic to his reflections as an artist and “philosopher” (as he wished to call himself and as he was ultimately called by the king of France). In his geometrical research he drew upon Archimedes and the ancients’ doctrine of lunes to develop the most neglected parts of Euclid’s work—curvilinear angles, the squaring of curvilinear areas, and the infinite variation of forms of unaltered quantity. Thus, in Leonardo’s philosophy, does nature build and infinitely very her forms, from the simplest to the most complex; and even the most complex have a rational structure that defines their beauty. It was Leonardo’s profound intuition that “Necessity... bridle and eternal rule of Nature” must have a mathematical foundation and that the infinite forms found in nature must therefore be the infinite variations of a fundamental “equation.”

Leonardo thus expressed a Pythagorean or Platonic conception of reality, common to many artists of the Renaissance. It was at the same time a revolutionary view, inasmuch as it identified form as function and further integrated into the concept of function the medieval notion of substance. Leonardo did not formulate his conception of reality in the abstract terms of universal scientific law. Rather, as the purpose of his painting was always to render the most subtle designs of natural structures, his last geometric constructions were aimed at discovering the mathematical structure of nature.

Augusto Marinoni