Mathematics, often referred to as the science of structure, order, and relationships, has evolved from basic practices of counting, measuring, and describing the shapes of objects. It involves logical reasoning and quantitative calculations and has progressively incorporated greater levels of idealization and abstraction in its subject matter. Since the 17th century, mathematics has become an essential companion to the physical sciences and technology. In more recent times, it has also played a crucial role in the quantitative aspects of the life sciences.
In various cultures, driven by the demands of practical endeavors like trade and agriculture, mathematics has developed well beyond simple counting. This expansion has been most prominent in societies with the complexity to sustain such pursuits and the luxury of time for contemplation and building upon the achievements of earlier mathematicians.
All mathematical systems, such as Euclidean geometry, are composed of sets of axioms and theorems that can be logically derived from these axioms. Questions about the logical and philosophical foundations of mathematics often revolve around whether a given system’s axioms ensure both completeness and consistency. A comprehensive examination of this aspect can be found in the field of mathematical foundations.
This article provides a historical account of mathematics, spanning from ancient times to the present day. Due to the rapid growth of science, the majority of mathematical developments have occurred since the 15th century CE. Notably, from the 15th century to the late 20th century, Europe and North America played central roles in these developments. Consequently, the primary focus of this article is on European mathematical advancements since 1500.
However, it is essential to recognize the significance of developments elsewhere. Understanding the history of mathematics in Europe necessitates an understanding of its roots in ancient Mesopotamia, Egypt, ancient Greece, and Islamic civilization from the 9th to the 15th century. The interactions and influences among these civilizations, as well as the direct contributions of Greece and Islam to subsequent developments, are explored in the initial sections of this article.
India’s contributions to contemporary mathematics are notable, primarily through the influence of Indian achievements on Islamic mathematics during its formative stages. Separate articles, such as “South Asian mathematics,” focus on the early history of mathematics in the Indian subcontinent and the development of the modern decimal place-value numeral system. Another article, “East Asian mathematics,” delves into the largely independent evolution of mathematics in China, Japan, Korea, and Vietnam.
The various branches of mathematics are covered in dedicated articles, encompassing algebra, analysis, arithmetic, combinatorics, game theory, geometry, number theory, numerical analysis, optimization, probability theory, set theory, statistics, and trigonometry.
Ancient mathematical sources
It is crucial to consider the nature of sources when studying the history of mathematics. The history of Mesopotamian and Egyptian mathematics relies on original documents authored by scribes. In the case of Egypt, these documents are scarce but consistently reflect the practical and elementary nature of Egyptian mathematics. Conversely, Mesopotamian mathematics benefits from a wealth of clay tablets that showcase more advanced mathematical achievements compared to Egypt. However, these tablets do not provide evidence of an organized deductive system within Mesopotamian mathematics. While future research may shed light on early Mesopotamian mathematics and its influence on Greek mathematics, the current understanding of Mesopotamian mathematics is likely to remain largely intact.
Before the era of Alexander the Great, there are no preserved Greek mathematical documents, except for fragmented paraphrases. Even for the subsequent period, it’s important to note that the oldest copies of Euclid’s Elements date from Byzantine manuscripts in the 10th century CE. This contrasts with the availability of Egyptian and Babylonian documents. While the general outline of Greek mathematics is secure, significant questions, such as the origins of the axiomatic method, pre-Euclidean theories of ratios, and the discovery of conic sections, remain subjects of debate among historians due to fragmented texts, quotations from non-mathematical sources, and conjecture.
Many important early Islamic mathematical treatises are lost or exist only in Latin translations, leaving unanswered questions about the connections between early Islamic mathematics and the mathematical traditions of Greece and India. Additionally, the vast amount of surviving material from later Islamic centuries has not been thoroughly studied, making it challenging to determine what later Islamic mathematics did not contain and, subsequently, to assess the original contributions of European mathematics from the 11th to the 15th century.
In modern times, the invention of printing has resolved the issue of obtaining reliable texts, allowing historians to focus on editing the correspondence or unpublished works of mathematicians. However, due to the exponential growth of mathematics, historians from the 19th century onwards can only delve into detailed accounts of major figures. Furthermore, as one approaches the present, the challenge of maintaining perspective arises. Mathematics, like any human endeavor, experiences trends, and the closer one is to a particular era, the more likely these trends may appear to be the future direction of the field. Hence, this article refrains from evaluating the most recent developments in mathematics.
Mathematics in ancient Mesopotamia
Until the 1920s, it was widely believed that the origins of mathematics could be traced back primarily to the ancient Greeks. Earlier mathematical traditions, such as the Egyptian knowledge represented by the Rhind papyrus (only edited for the first time in 1877), were seen as offering limited precedents. This perspective underwent a significant transformation as historians managed to decipher and understand technical materials from ancient Mesopotamia.
The durability of clay tablets used by Mesopotamian scribes has resulted in substantial surviving evidence from this civilization. These mathematical artifacts span various significant periods, including the Sumerian kingdoms of the 3rd millennium BCE, the Akkadian and Babylonian empires in the 2nd millennium BCE, the Assyrian empire in the early 1st millennium BCE, the Persian empire from the 6th to the 4th century BCE, and the Greek influence from the 3rd century BCE to the 1st century CE. Mathematical proficiency was already at a high level as early as the time of the Old Babylonian dynasty, during the era of King Hammurabi (around the 18th century BCE). However, significant mathematical advancements were relatively scarce after that period. Nevertheless, the application of mathematics to astronomy thrived during the Persian and Seleucid (Greek) periods.
The numeral system and arithmetic operations
In contrast to the Egyptians, mathematicians during the Old Babylonian period displayed a remarkable depth in their mathematical pursuits that extended beyond their routine accounting responsibilities. They introduced a versatile numeral system, reminiscent of the modern one, which capitalized on the concept of place value. Moreover, they developed computational methods that harnessed this numerical representation, solving linear and quadratic problems akin to contemporary algebra. Their accomplishments in exploring what we now call Pythagorean number triples marked a significant achievement in number theory. These early mathematicians must have recognized the intrinsic value of mathematics as a field of study, not merely as a practical tool.
The Sumerian numeral system that preceded the Old Babylonian era followed an additive decimal (base-10) principle, similar to the Egyptian system. However, the Old Babylonian system transformed this into a place-value system with a base of 60, known as sexagesimal. The reasons behind the choice of 60 remain unclear, but a possible mathematical advantage might have been the presence of numerous divisors (2, 3, 4, 5, and multiples) of this base, greatly simplifying division operations. For numbers from 1 to 59, they employed a simple additive combination of symbols (e.g., mathematics mathematics mathematics mathematics mathematics represented 32). To represent larger values, they applied the concept of place value. For instance, 60 was represented as mathematics, 70 as mathematics mathematics, 80 as mathematics mathematics mathematics, and so forth. In essence, mathematics could denote any power of 60, with the context determining the intended power. By the 3rd century BCE, Babylonians appeared to have developed a placeholder symbol resembling zero, though its exact meaning and usage remain uncertain. Furthermore, they lacked a symbol to distinguish integral from fractional parts (as in the modern decimal point). Consequently, the three-place numeral 3 7 30 could represent 31/8 (i.e., 3 + 7/60 + 30/602), 1871/2 (i.e., 3 × 60 + 7 + 30/60), 11,250 (i.e., 3 × 602 + 7 × 60 + 30), or a multiple of these numbers by any power of 60.
The four fundamental arithmetic operations were executed similarly to those in the modern decimal system, with carrying occurring when a sum reached 60 rather than 10. Multiplication was aided by reference tables; one such tablet listed multiples of numbers from 1 to 20, 30, 40, and 50. To multiply two multi-digit numbers, a scribe would break the problem down into several multiplications, each by a one-digit number. They would then look up the value of each product in the corresponding tables, finally obtaining the solution by adding up these intermediate results. These tables also facilitated division, as the values listed were all reciprocals of regular numbers.
Regular numbers were those whose prime factors were divisors of the base; consequently, reciprocals of regular numbers had finite representations (unlike reciprocals of non-regular numbers, which produced infinitely repeating decimals). For example, in base 10, only numbers divisible by 2 and 5 (e.g., 8 or 50) were regular, and their reciprocals (1/8 = 0.125, 1/50 = 0.02) had finite representations. The reciprocals of other numbers (such as 3 and 7) resulted in infinite repeating decimals (e.g., 1/3 = 0.333… and 1/7 = 0.142857…). In base 60, only numbers with factors of 2, 3, and 5 were regular. For instance, numbers like 6 and 54 were regular, making their reciprocals (10 and 1 6 40) finite. The entries in the multiplication table for 1 6 40 simultaneously represented multiples of its reciprocal 1/54. To divide a number by any regular number, one could consult the table of multiples for its reciprocal.
An intriguing tablet from Yale University features a square with its diagonals. One side of the tablet bears the number “30,” under one diagonal is “42 25 35,” and along the same diagonal appears “1 24 51 10” (equivalent to 1 + 24/60 + 51/602 + 10/603). This third number represents the accurate value of the square root of 2 to four sexagesimal places (equivalent to 1.414213… in the decimal system, with only a slight discrepancy in the seventh place). The second number is the product of the third number and the first, providing the length of the diagonal when the side is 30. This implies that the scribe employed a method akin to the lengthy process for finding square roots. Furthermore, by selecting 30 (equivalent to 1/2) as the side length, the scribe obtained the reciprocal of the square root of 2 (since 1/2 = 1/Square root of 2) as the diagonal length, a valuable result for division purposes.
Geometric and algebraic problems
A Babylonian tablet, now housed in Berlin, provides an intriguing example of geometric problem-solving. In this tablet, the diagonal of a rectangle with sides measuring 40 and 10 is determined as 40 + 102/(2 × 40). Here, an effective approximation rule is employed, similar to the later Greek geometric writings, where the square root of the sum of a^2 + b^2 can be estimated as a + b^2/2a. Both of these examples illustrate the Babylonians’ arithmetic approach to geometry. Importantly, they also reveal that the Babylonians were aware of the relationship between the hypotenuse and the two legs of a right triangle, now recognized as the Pythagorean theorem. This awareness predates the Greek utilization of the theorem by over a thousand years.
One common type of problem found in Babylonian tablets involves determining the base and height of a rectangle when given their product and sum. The scribes would deduce the difference since (b − h)^2 = (b + h)^2 − 4bh. Similarly, if provided with the product and difference, they could find the sum. Moreover, when both the sum and difference were known, each side could be ascertained through the equations 2b = (b + h) + (b − h) and 2h = (b + h) − (b − h). This procedure is equivalent to solving a general quadratic equation with a single unknown. Occasionally, the Babylonian scribes also resolved quadratic problems in terms of a single unknown, a method akin to modern usage of the quadratic formula.
However, it’s important to note that these Babylonian quadratic procedures differ from what is commonly referred to as algebra today. The scribes lacked algebraic symbolism, and while they certainly grasped the generality of their solution methods, they presented them as specific cases rather than as the application of general formulas and identities. Consequently, they did not have the means to convey generalized derivations and proofs of their solution methods. Nonetheless, their use of sequential procedures rather than formulas remains significant, especially considering the prevalence of algorithmic methods in the modern era due to computer development.
As mentioned earlier, Babylonian scribes were aware of the relationship between the base (b), height (h), and diagonal (d) of a rectangle, expressed as b^2 + h^2 = d^2. When values were chosen arbitrarily for two of these terms, the third term usually turned out to be irrational. However, it was possible to find cases where all three terms were integers, such as the well-known Pythagorean triples like 3, 4, 5 and 5, 12, 13. (These solutions are often referred to as Pythagorean triples.) An ancient tablet held in the Columbia University Collection lists 15 such triples. The ordering of the lines in this list becomes evident from another column that presents the values of d^2/h^2. These values form a continuously decreasing sequence, indicating that the angle formed between the diagonal and the base increased from just over 45° to just under 60°. This sequence suggests that the scribe was acquainted with a general procedure for finding all such number triples. This procedure is based on the relationship 2d/h = p/q + q/p, where p and q are integers. The values of p and q implied by this table are regular numbers from the standard set of reciprocals, as mentioned earlier in the context of multiplication tables. The construction and intended use of this table are still subjects of scholarly debate, but it undeniably demonstrates a high level of mathematical expertise among Babylonian scholars.
Mathematical astronomy
The sexagesimal system developed by the Babylonians possessed a much greater computational capacity than what was initially required for their early mathematical problem texts. However, with the emergence of mathematical astronomy during the Seleucid period, this method became indispensable. Astronomers needed to predict future occurrences of significant celestial events, such as lunar eclipses and critical points in planetary cycles (such as conjunctions, oppositions, stationary points, and the first and last visibility). They devised a technique for calculating these celestial positions, expressed in terms of degrees of latitude and longitude, measured relative to the apparent annual path of the Sun. This involved successively adding appropriate terms in an arithmetic progression. The results were then organized into tables that could extend far into the future, depending on the preferences of the scribe. (While this method is fundamentally arithmetic, it can also be interpreted graphically, as the tabulated values create a linear “zigzag” approximation to what is essentially a sinusoidal variation.) The astronomers required observations spanning centuries to determine the necessary parameters, such as periods and the angular range between maximum and minimum values. However, it was the computational tools available to them that made it possible to engage in such forecasting efforts.
In a relatively short span, possibly within a century or less, elements of this mathematical system made their way into the hands of the Greeks. Although Hipparchus, a prominent Greek astronomer from the 2nd century BCE, favored the geometric methods of his Greek predecessors, he adopted the sexagesimal style of computation and incorporated parameters from the Mesopotamians into his work. Through the Greeks, this mathematical tradition passed to Arab scientists during the Middle Ages and eventually reached Europe. It remained influential in mathematical astronomy during the Renaissance and the early modern period. Even today, it endures in the use of minutes and seconds to measure time and angles.
Certain aspects of Old Babylonian mathematics may have reached the Greeks even earlier, possibly in the 5th century BCE during the formative period of Greek geometry. Scholars have identified several parallels between the two traditions. For instance, the Greek technique of the “application of area” corresponds to Babylonian quadratic methods, albeit in a geometric rather than arithmetic form. Additionally, the Babylonian rule for estimating square roots was widely applied in Greek geometric calculations, and there may have been shared technical terminology nuances. While the exact timing and manner of this transmission remain unclear due to the lack of explicit documentation, it is evident that Western mathematics, while primarily rooted in Greek contributions, owes a substantial debt to the older Mesopotamian mathematical heritage.
Mathematics in ancient Egypt
The introduction of writing in ancient Egypt during the predynastic period, around 3000 BCE, led to the emergence of a distinct class of educated professionals known as scribes. These scribes, skilled in the art of writing, assumed various responsibilities within a civil service framework. Their duties encompassed tasks such as record-keeping, tax accounting, overseeing public construction projects, and even managing the logistical aspects of warfare, including military supplies and payrolls. Young men enrolled in scribal schools to acquire the essential skills of the trade, which not only included literacy but also covered the fundamentals of mathematics.
During the New Kingdom period in the 13th century BCE, one of the texts commonly used as a practice exercise in scribal schools was a satirical letter where a scribe named Hori ridicules his rival, Amen-em-opet, for his perceived incompetence as an advisor and manager. At one point, Hori taunts, “You are the clever scribe at the head of the troops,” referring to his rival’s inability to solve challenging problems presented in the letter. These problems, including three others like them, required additional information for a solution. However, the humor lies in the fact that they were designed to be tough, yet typical, challenges.
Our knowledge of ancient Egyptian mathematics aligns well with the kind of tests presented by the scribe Hori. This knowledge primarily stems from two extensive papyrus documents that once served as textbooks within scribal schools. The Rhind papyrus, housed in the British Museum, is a copy created in the 17th century BCE, based on a text that dates back two centuries earlier. It contains a lengthy table of fractional parts designed to aid in division, followed by solutions to 84 specific problems in arithmetic and geometry. The Golenishchev papyrus, located in the Moscow Museum of Fine Arts and dating from the 19th century BCE, presents 25 problems of a similar nature. These problems provide valuable insights into the practical functions performed by scribes, as they address issues like the distribution of wages in the form of beer and bread, methods for measuring field areas, and techniques for calculating the volumes of pyramids and other geometric solids.
