mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.
In many cultures—under the stimulus of the needs of practical pursuits, such as commerce and agriculture—mathematics has developed far beyond basic counting. All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms. Inquiries into the logical and philosophical basis of mathematics reduce to questions of whether the axioms of a given system ensure its completeness and its consistency. For full treatment of this aspect, see mathematics, foundations of.
This article offers a history of mathematics from ancient times to the present. As a consequence of the exponential growth of science, most mathematics has developed since the 15th century CE, and it is a historical fact that, from the 15th century to the late 20th century, new developments in mathematics were largely concentrated in Europe and North America. For these reasons, the bulk of this article is devoted to European developments since 1500.
This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe, it is necessary to know its history at least in ancient Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the 9th to the 15th century. The way in which these civilizations influenced one another and the important direct contributions Greece and Islam made to later developments are discussed in the first parts of this article.
India’s contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Islamic mathematics during its formative years. A separate article, South Asian mathematics, focuses on the early history of mathematics in the Indian subcontinent and the development there of the modern decimal place-value numeral system. The article East Asian mathematics covers the mostly independent development of mathematics in China, Japan, Korea, and Vietnam.
Ancient mathematical sources
It is important to be aware of the character of the sources for the study of the history of mathematics. The history of Mesopotamian and Egyptian mathematics is based on the extant original documents written by scribes. Although in the case of Egypt these documents are few, they are all of a type and leave little doubt that Egyptian mathematics was, on the whole, elementary and profoundly practical in its orientation. For Mesopotamian mathematics, on the other hand, there are a large number of clay tablets, which reveal mathematical achievements of a much higher order than those of the Egyptians. The tablets indicate that the Mesopotamians had a great deal of remarkable mathematical knowledge, although they offer no evidence that this knowledge was organized into a deductive system. Future research may reveal more about the early development of mathematics in Mesopotamia.
In modern times the invention of printing has largely solved the problem of obtaining secure texts and has allowed historians of mathematics to concentrate their editorial efforts on the correspondence or the unpublished works of mathematicians. However, the exponential growth of mathematics means that, for the period from the 19th century on, historians are able to treat only the major figures in any detail. In addition, there is, as the period gets nearer the present, the problem of perspective. Mathematics, like any other human activity, has its fashions, and the nearer one is to a given period, the more likely these fashions will look like the wave of the future. For this reason, the present article makes no attempt to assess the most recent developments in the subject.
Mathematics in ancient Mesopotamia
Until the 1920s it was commonly supposed that mathematics had its birth among the ancient Greeks. What was known of earlier traditions, such as the Egyptian as represented by the Rhind papyrus (edited for the first time only in 1877), offered at best a meagre precedent. This impression gave way to a very different view as historians succeeded in deciphering and interpreting the technical materials from ancient Mesopotamia.
Owing to the durability of the Mesopotamian scribes’ clay tablets, the surviving evidence of this culture is substantial. Existing specimens of mathematics represent all the major eras—the Sumerian kingdoms of the 3rd millennium BCE, the Akkadian and Babylonian regimes (2nd millennium), and the empires of the Assyrians (early 1st millennium), Persians (6th through 4th century BCE), and Greeks (3rd century BCE to 1st century CE). The level of competence was already high as early as the Old Babylonian dynasty, the time of the lawgiver-king Hammurabi (c. 18th century BCE), but after that there were few notable advances. The application of mathematics to astronomy, however, flourished during the Persian and Seleucid (Greek) periods.
The numeral system and arithmetic operations
Unlike the Egyptians, the mathematicians of the Old Babylonian period went far beyond the immediate challenges of their official accounting duties. For example, they introduced a versatile numeral system, which, like the modern system, exploited the notion of place value, and they developed computational methods that took advantage of this means of expressing numbers; they solved linear and quadratic problems by methods much like those now used in algebra; their success with the study of what are now called Pythagorean number triples was a remarkable feat in number theory. The scribes who made such discoveries must have believed mathematics to be worthy of study in its own right, not just as a practical tool.
Geometric and algebraic problems
In a Babylonian tablet now in Berlin, the diagonal of a rectangle of sides 40 and 10 is solved as 40 + 102/(2 × 40). Here a very effective approximating rule is being used (that the square root of the sum of a2 + b2 can be estimated as a + b2/2a), the same rule found frequently in later Greek geometric writings. Both these examples for roots illustrate the Babylonians’ arithmetic approach in geometry. They also show that the Babylonians were aware of the relation between the hypotenuse and the two legs of a right triangle (now commonly known as the Pythagorean theorem) more than a thousand years before the Greeks used it.
A type of problem that occurs frequently in the Babylonian tablets seeks the base and height of a rectangle, where their product and sum have specified values. From the given information the scribe worked out the difference, since (b − h)2 = (b + h)2 − 4bh. In the same way, if the product and difference were given, the sum could be found. And, once both the sum and difference were known, each side could be determined, for 2b = (b + h) + (b − h) and 2h = (b + h) − (b − h). This procedure is equivalent to a solution of the general quadratic in one unknown. In some places, however, the Babylonian scribes solved quadratic problems in terms of a single unknown, just as would now be done by means of the quadratic formula.
Although these Babylonian quadratic procedures have often been described as the earliest appearance of algebra, there are important distinctions. The scribes lacked an algebraic symbolism; although they must certainly have understood that their solution procedures were general, they always presented them in terms of particular cases, rather than as the working through of general formulas and identities. They thus lacked the means for presenting general derivations and proofs of their solution procedures. Their use of sequential procedures rather than formulas, however, is less likely to detract from an evaluation of their effort now that algorithmic methods much like theirs have become commonplace through the development of computers.
The sexagesimal method developed by the Babylonians has a far greater computational potential than what was actually needed for the older problem texts. With the development of mathematical astronomy in the Seleucid period, however, it became indispensable. Astronomers sought to predict future occurrences of important phenomena, such as lunar eclipses and critical points in planetary cycles (conjunctions, oppositions, stationary points, and first and last visibility). They devised a technique for computing these positions (expressed in terms of degrees of latitude and longitude, measured relative to the path of the Sun’s apparent annual motion) by successively adding appropriate terms in arithmetic progression. The results were then organized into a table listing positions as far ahead as the scribe chose. (Although the method is purely arithmetic, one can interpret it graphically: the tabulated values form a linear “zigzag” approximation to what is actually a sinusoidal variation.) While observations extending over centuries are required for finding the necessary parameters (e.g., periods, angular range between maximum and minimum values, and the like), only the computational apparatus at their disposal made the astronomers’ forecasting effort possible.
Mathematics in ancient Egypt
The introduction of writing in Egypt in the predynastic period (c. 3000 BCE) brought with it the formation of a special class of literate professionals, the scribes. By virtue of their writing skills, the scribes took on all the duties of a civil service: record keeping, tax accounting, the management of public works (building projects and the like), even the prosecution of war through overseeing military supplies and payrolls. Young men enrolled in scribal schools to learn the essentials of the trade, which included not only reading and writing but also the basics of mathematics.
One of the texts popular as a copy exercise in the schools of the New Kingdom (13th century BCE) was a satiric letter in which one scribe, Hori, taunts his rival, Amen-em-opet, for his incompetence as an adviser and manager. “You are the clever scribe at the head of the troops,” Hori chides at one point,
a ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments—it is 60 cubits high, 30 cubits in the middle…and the generals and the scribes turn to you and say, “You are a clever scribe, your name is famous. Is there anything you don’t know? Answer us, how many bricks are needed?” Let each compartment be 30 cubits by 7 cubits.
This problem, and three others like it in the same letter, cannot be solved without further data. But the point of the humour is clear, as Hori challenges his rival with these hard, but typical, tasks.
The geometric problems in the papyri seek measurements of figures, like rectangles and triangles of given base and height, by means of suitable arithmetic operations. In a more complicated problem, a rectangle is sought whose area is 12 and whose height is 1/2 + 1/4 times its base (Golenishchev papyrus, problem 6). To solve the problem, the ratio is inverted and multiplied by the area, yielding 16; the square root of the result (4) is the base of the rectangle, and 1/2 + 1/4 times 4, or 3, is the height. The entire process is analogous to the process of solving the algebraic equation for the problem (x × 3/4x = 12), though without the use of a letter for the unknown. An interesting procedure is used to find the area of the circle (Rhind papyrus, problem 50): 1/9 of the diameter is discarded, and the result is squared. For example, if the diameter is 9, the area is set equal to 64. The scribe recognized that the area of a circle is proportional to the square of the diameter and assumed for the constant of proportionality (that is, π/4) the value 64/81. This is a rather good estimate, being about 0.6 percent too large. (It is not as close, however, as the now common estimate of 31/7, first proposed by Archimedes.
Assessment of Egyptian mathematics
The papyri thus bear witness to a mathematical tradition closely tied to the practical accounting and surveying activities of the scribes. Occasionally, the scribes loosened up a bit: one problem (Rhind papyrus, problem 79), for example, seeks the total from seven houses, seven cats per house, seven mice per cat, seven ears of wheat per mouse, and seven hekat of grain per ear (result: 19,607). Certainly the scribe’s interest in progressions (for which he appears to have a rule) goes beyond practical considerations. Other than this, however, Egyptian mathematics falls firmly within the range of practice.