How Did Early Mathematics Evolve?
Mathematics today profoundly affects our societies and has had a lasting impact on science and technology. However, before Greek mathematicians developed many wonderful achievements in this area, what are the origins of mathematics and how did this field evolve in its early history?
The Early Origins
While it is clear that early arithmetic developed in early prehistoric periods of human societies, more complex mathematics have developed via many different routes. Perhaps one of the most significant contributors to mathematics was the development of agriculture and cities. With the rise of these two factors, societies increasingly needed to calculate division of property, sharing of resources, accrual of wealth, and track the motion of the planets and stars, which all required more complex mathematical developments.
Most likely, the oldest documents that are mathematical in nature generally deal with accounting, which developed when writing first develops in the late 4th millennium BC in ancient southern Mesopotamia (now southern Iraq). By the 3rd millennium BC, calculations for interest, loans, and division of goods seem to be common among the Sumerians, who lived in southern Mesopotamia.  This included dividing field property and determining dimensions of land. In fact, this aspect of property division, related to agricultural land, created more complex problems such as determining the area of a triangle or calculating what the dimensions should be for given sides of a triangle. Because of this, we know by about 2000 BC, but likely earlier, the Pythagorean theorem had been invented, nearly 1500 years before Pythagoras, the man who has given his name to the theorem, was born. This innovation of the Pythagorean theorem seems to have happened both in Mesopotamia and Egypt (Figure 1).
The Egyptians had also developed by the 3rd millennium BC a base 10 numeration system. By around 1650 BC, Egyptian mathematicians had developed a system of binary numbers, while also being able to determine prime numbers and calculations for arithmetic, geometric, and harmonic means (Figure 2). Both Egyptian and Mesopotamian sources indicate a system for developing fractions and providing their notation. 
The Mesopotamian system of mathematics was also sexagesimal, or based on the division of 60. While this may seem an odd division-based system, the numeric system is very convenient for calculation of coordinates or rotation in a circle (e.g., 360 degrees is easily divided by 60). The divisions of time we use today (i.e., 60 seconds, minutes, etc.) derive from this Mesopotamian system. Additionally, we know that by the early 2nd millennium BC, estimates for π (3.125 estimated by the Babylonians) existed and the Babylonians, who lived in southern Mesopotamia and succeeded the Sumerians, understood how to measure volume and area in circular objects.
Other mathematical principles already understood by the early 2nd millennium BC were concepts of square roots/powers, inverse numbers, quadratic equations, exponential growth, a form of Fourier analysis, and other types of calculation used in geometry and algebra. Concepts of zero were also well understood by the early 2nd millennium and were represented as a space or symbol in Mesopotamia and Egypt respective. Very recently, a paper was published that showed the Babylonians had already developed an early forerunner of calculus, by around 2000 BC using abstract mathematics for estimating distances plants travel in space. Previously, some of the forms of Mesopotamian calculations used for complex shapes and distances were assumed not to have been invented until the 14th century AD.
The Role of Astronomy
What was key to the development of sophisticated calculations such as quadratic equations or forms of Fourier analyses was Astronomy. The Babylonians, in part out of religious devotion and need to observe the movement of the planets and stars, developed many sophisticated calculations to predict where and when plants and stars would be, including the timing of eclipses. The need to develop complex calculations to track orbits of celestial objects necessitated the need to develop far more sophisticated analyses that eventually became the foundation of modern day calculus. In fact, by approximately 300 BC, Babylonian mathematicians had developed sophisticated techniques in calculating distance and time together for estimating the movement of celestial objects.
Sometime by the 1st millennium BC, Chinese mathematicians began using negative numbers. This seems to have been used to calculate taxes or other monetary values. In fact, we know from Leibniz, who was one of the developers of calculus, that ancient Chinese mathematicians had developed a binary numeric system that influenced his own development of binary calculus. The system of the Chinese, in effect, influences our own use of binary calculus. Developments in India include a decimal systems that became the foundation of our own decimal system used today. Additionally, many other similar achievements noted for Egypt and Mesopotamia, such as in geometry, algebra, and trigonometry, have also been found in India and China.
What is clear is long before major contributions by Greek mathematicians, we see numerous contributions by Egyptian, Mesopotamian, Chinese, and Indians sources unfolding. These contributions were critical for later mathematics and sciences, enabling such fields as modern cartography, prediction of celestial movements, calculations of areas and volumes, and creation of accurate time measurements and calendars. In some cases, such as the early forerunners to calculus, have been entirely unexpected discoveries that have forced our re-evaluation of what people knew sometimes thousands of years before techniques were though to have developed. This shows that many mathematical discoveries were likely made and may have been forgotten before their rediscovery in Classical or Renaissance works. However, foundations of our own mathematics and sciences can be now certainly traced back to far older sources. Given the large volume of unpublished ancient texts, it is highly likely even more discoveries and surprises in mathematical sciences wait.
- For further information on factors leading to the rise of early mathematics, see: Bell, Eric Temple. 1992. The Development of Mathematics. New York: Dover Publications.
- For information on early written texts and their role in mathematics, see: Campbell-Kelly, Martin, ed. 2003. The History of Mathematical Tables: From Sumer to Spreadsheets. Oxford ; New York: Oxford University Press.
- For a history on the development of the Pythagorean theorem, see: Katz, Victor J., and Annette Imhausen, eds. 2007. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton: Princeton University Press.
- For information on early Egyptian mathematics and key developments in numerical systems, prime numbers, and means, see: Clagett, Marshall. 1989. Ancient Egyptian Science: A Source Book. Memoirs of the American Philosophical Society, v. 184, 214, 232. Philadelphia: American Philosophical Society.
- For information on early fractions, see: Katz, Victor J., and Annette Imhausen, eds. 2007. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton: Princeton University Press.
- For information on the Babylonian numeric system and calculations of volume and area, see: Friberg, Jöran. 2007. A Remarkable Collection of Babylonian Mathematical Texts. Sources and Studies in the History of Mathematics and Physical Sciences 1. New York: Springer.
- For information on Babylonian Fourier analysis, see: Prestini, Elena, and Elena Prestini. 2004. The Evolution of Applied Harmonic Analysis. Boston: Birkhauser. For information on quadratic equations and square roots, see: Kline, Morris. 1972. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, pg. 8.
- For a history of zero, see: Bunt, Lucas N. H., Phillip S. Jones, and Jack D. Bedient. 1988. The Historical Roots of Elementary Mathematics. New York: Dover Publications.
- For this paper, see: Ossendrijver, M. 2016. “Ancient Babylonian Astronomers Calculated Jupiters Position from the Area under a Time-Velocity Graph.” Science 351 (6272): 482–84. doi:10.1126/science.aad8085.
- For information on how the astronomy played a role with mathematical developments, see: Hunger, Hermann, and David Edwin Pingree. 1999. Astral Sciences in Mesopotamia. Handbuch Der Orientalistik. Erste Abteilung, Der Nahe Und Mittlere Osten, Handbook of Oriental Studies. The Near and Middle East, 44. Bd. =. Leiden ; Boston: Brill.
- For a history of ancient Chinese mathematics, see: Cullen, Christopher. 2007. Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing. Cambridge Univ Pr.
- For more information on the Indian decimal system, see: Bhanu Murthy, T. S. 2009. A Modern Introduction to Ancient Indian Mathematics. New Delhi: New Age International.