History of mathematics in ancient times. When I elaborated the article entitled Background of the computer, history of the informatica ancient age and middle ages and comparing other posts that I had elaborated so far, I understood that I could perfectly reconstruct a history of the so-called queen of sciences.

## Older mathematical texts

### Plimpton 322

Plimpton 322 is a remarkable artifact of ancient Babylonian mathematics. It is a clay tablet containing a table of 15 rows and 4 columns of numbers, written in cuneiform. Numbers are sexagesimal fractions (base 60) that represent the sides and diagonals of right triangles. The table is ordered by the ratio of the short side to the long side of each triangle, which is equivalent to the secant of the angle opposite the short side.

The purpose and origin of Plimpton 322 have been the subject of debate for decades. Some have suggested that it was a school textbook, a writing exercise, or a collection of problems. Others have argued that it was a trigonometric table, a precursor to the chord table developed by Greek astronomers in the second century B.C. However, there is no evidence that the Babylonians had an angle or secant concept, and their mathematical culture was different from that of the Greeks.

A more recent interpretation is that Plimpton 322 was a tool for generating Pythagorean triples, which are integer solutions of the equation a²=b²+c². The Babylonians were interested in finding exact solutions to quadratic equations, and had methods for constructing right triangles with rational sides and diagonals. Plimpton 322 may have been part of a larger tablet that contained instructions on how to use the table to find such triangles. The table could also have practical applications in surveying, architecture or astronomy.

### Moscow Mathematical Papyrus

The Moscow Mathematical Papyrus is one of the oldest and most important sources of ancient Egyptian mathematics. It contains 25 problems and solutions covering topics such as arithmetic, geometry, and algebra. The papyrus was written in hieratic script around 1850 BC, during the Thirteenth Dynasty of Egypt. It was acquired by the Russian Egyptologist Vladimir Golenishchev in the late nineteenth century and is now preserved in the Pushkin Museum of Fine Arts in Moscow. The papyrus demonstrates the high level of knowledge and mathematical skills possessed by the ancient Egyptians, such as calculating the area of a circle, the volume of a truncated pyramid, and solving linear equations.

### Rhind mathematical papyrus

The Rhind mathematical papyrus is one of the most important sources of ancient Egyptian mathematics. It contains 84 problems covering various topics, such as arithmetic, fractions, geometry, and algebra. The papyrus was written by a scribe named Ahmes around 1550 BC, based on an older document from the Twelfth Dynasty. The papyrus was acquired by a Scottish scholar named Henry Rhind in 1858 and is now preserved in the British Museum. The papyrus demonstrates the high level of knowledge and mathematical skills possessed by the ancient Egyptians.

### Sutras Shulba

The Śulbasūtras are ancient Indian texts dealing with geometry applied to the construction of fire altars for Vedic rituals. These texts are the only testimonies of the mathematical knowledge of the Vedic era, and contain formulas for calculating areas, perimeters, diagonals, angles and square roots. The Śulbasūtras also show the use of the string as a measuring and drawing instrument, and the knowledge of some properties of geometric figures such as the right triangle, the circle and the square.

The four most important Śulbasūtras from the mathematical point of view are those attributed to Baudhāyana, Manava, Apastamba and Katyayana. These texts were written in late Vedic Sanskrit, between the eighth and second centuries B.C., and are considered appendices to the Vedas, the oldest sacred texts in India. It is unknown whether these texts predate or postdate the mathematical discoveries of Pythagoras in Greece, or whether there was any mutual influence between the two cultures.

The Śulbasūtras are an example of the practical application of mathematics to religion and architecture in ancient India. These texts demonstrate the interest and ability of ancient Indians to solve geometric problems with precision and rigor.

Ancient Egyptian and Mesopotamian/Babylonian mathematics was developed by Greek and Hellenistic mathematicians, with Egypt as the center of Hellenistic learning.

## Introduction

The history of mathematics in antiquity is a rich and diverse subject that spans diverse civilizations and cultures. Ancient civilizations such as Mesopotamia, Egypt, India, China, and Greece made important contributions to the development of mathematics, laying the foundation for modern mathematical concepts and techniques. Here is a brief summary of the history of mathematics in Antiquity:

## Mesopotamia (3000 BC-500 BC):

The Mesopotamians, who lived in the region between the Tigris and Euphrates rivers (present-day Iraq), were one of the first known civilizations to develop mathematical concepts. They developed a sexagesimal (base 60) numbering system, which is still used today to measure time and angles. They also made important advances in geometry, which they used for surveying and measuring land.

#### Numbering system

The Mesopotamians developed a sexagesimal (base 60) numbering system that is still used to measure time and angles. They used a combination of symbols to represent numbers, with a special symbol for zero.

This system allowed them to perform arithmetic operations such as addition, subtraction, multiplication and division. Mesopotamian mathematicians also developed tables for multiplication and division, known as multiplication tables, which were widely used for practical purposes such as trade.

#### Geometry and measurement

The Mesopotamians were experts in geometry and used it mainly for surveying and measuring land. They developed methods for calculating field areas, container volumes, and other geometric shapes.

They also used geometry to solve practical problems such as determining property boundaries, constructing canals and irrigation systems, and constructing buildings. Mesopotamian clay tablets containing geometric problems and their solutions have been discovered, allowing a better understanding of their mathematical knowledge and practices.

#### Mathematical texts

Several mathematical texts from Mesopotamia have been discovered that provide valuable data on their mathematical knowledge and practices. Among them are the mathematical tablets of the ancient cities of Sumer, such as Nippur, Ur and Larsa, dating from approximately 2000 BC. These tablets contain mathematical problems, calculations, and tables related to various aspects of everyday life, such as commerce, agriculture, and construction.

#### Notable mathematicians

Although the names of individual mathematicians from Mesopotamia are not well documented, some mathematical texts from the region mention notable scholars who made significant contributions to mathematics. For example, the tablet Plimpton 322, discovered in present-day Iraq and dated to around 1800 BC, contains a list of Pythagorean triples, indicating knowledge of the Pythagorean theorem. Another famous mathematical text from Mesopotamia is the Yale tablet, which contains a quadratic equation and its solution, and dates from around 2000 BC.

#### Influence and legacy

The mathematical knowledge developed in Mesopotamia had a lasting impact on the development of mathematics in other civilizations. For example, the sexagesimal numbering system developed by the Mesopotamians influenced other ancient cultures, such as the Greek and Babylonian. It also served as the basis for the modern system of measuring time and angles. In addition, Mesopotamian methods of geometry and measurement influenced later cultures, such as the ancient Egyptians and ancient Greeks.

In conclusion, Mesopotamia was a civilization that made important contributions to the field of mathematics during Antiquity. His development of a sexagesimal numbering system, his knowledge of geometry and measurement and the creation of mathematical texts and tables laid the foundations of mathematical knowledge that influenced later civilizations and that mathematicians continue to study and appreciate today.

## Egypt (3000 BC-300 BC)

The ancient Egyptians developed a mathematical system primarily for practical purposes, such as architecture, construction, and astronomy. They used a decimal numbering system and were experts in arithmetic and geometry. The construction of the pyramids is a testament to his advanced knowledge of geometry and measurement.

#### Hieroglyphic numbers

The ancient Egyptians used a decimal system of numbering and had a system of hieroglyphic numerals to represent numbers. Hieroglyphic numbers were symbols representing powers of 10, from 1 to 1,000,000. This system allowed the ancient Egyptians to perform basic numbering operations. This system allowed the ancient Egyptians to perform basic arithmetic operations such as addition, subtraction, multiplication, and division.

#### Practical geometry

The ancient Egyptians had a practical approach to geometry, using it for various applications such as land measurement, building construction, and surveying. They used knotted strings to measure lengths and angles and used simple geometric shapes, such as squares, rectangles, and triangles, in their calculations. They were especially skilled in calculating field areas and vessel volumes, essential for agriculture and construction.

#### Practical applications

The ancient Egyptians applied their mathematical knowledge in various practical ways. For example, they used geometry and measurement in the construction of pyramids, temples and other structures. They used sophisticated surveying techniques to determine land boundaries, calculate taxes, and plan irrigation systems. They also used their knowledge of astronomy to develop a calendar based on the cycles of the Nile River, essential for agricultural planning.

The mathematical knowledge developed in ancient Egypt greatly influenced other civilizations, especially in the fields of geometry and measurement. For example, Greek mathematicians such as Thales and Pythagoras were influenced by Egyptian geometry and incorporated it into their own mathematical works. The Rhind Mathematical Papyrus and other mathematical texts of ancient Egypt also had a lasting impact on the development of mathematics in other cultures.

## India (2000 B.C.-500 A.D.):

Ancient India made important contributions to the field of mathematics, especially in the development of the decimal number system. Indian mathematicians invented the concept of zero and developed the place value system, which laid the foundation for modern positional notation. They also made important advances in algebra, geometry, and trigonometry, as evidenced by texts such as the Vedas, the Sulba Sutras, and the famous mathematical treatise called Aryabhatiya.

The history of mathematics in ancient India dates back to the Indus Valley civilization (2500-1900 BC), where evidence of practical mathematics such as weights, measurements, and geometry have been found. However, the earliest surviving mathematical texts date from the Vedic period (1500-500 BC) and contain rules for the construction of altars and fire rituals using geometry and algebra. The Sulba Sutras, composed during this period, are among the oldest sources of geometric constructions in the world.

The classical period of Indian mathematics (400-1200 AD) witnessed the emergence of influential mathematicians such as Aryabhata, Brahmagupta, Bhaskara II and Varahamihira, who made important contributions to various fields of mathematics, such as arithmetic, algebra, trigonometry, astronomy and calculus.

They developed the decimal number system, the concept of zero and negative numbers, solutions of linear and quadratic equations, the approximation of irrational numbers, sine and cosine functions and their series expansions, and the calculation of pi and other constants. They also transmitted their mathematical knowledge to other civilizations, such as China, Arabia and Europe.

## China (1100 BC-200 AD):

Chinese mathematicians made important contributions in the field of algebra, especially in solving equations. They also developed methods for calculating areas and volumes, and were adept at using mathematical techniques for practical applications such as calendaring, currency exchange, and terrain measurement.

The history of mathematics in ancient China is a fascinating subject that spans thousands of years and reveals many aspects of the development of human civilization. Mathematics in China emerged independently in the eleventh century B.C., and the Chinese invented or discovered many important concepts and methods, such as the real number system, algebra, geometry, number theory, trigonometry, binary numbers, Pascal’s triangle, and the evaluation of polynomials.

The dating of the first Chinese mathematical text, known as The Nine Chapters on the Art of Mathematics, is uncertain, with some estimates placing it in 1200 B.C. and others suggesting it could be more than a thousand years later.

This text contains detailed procedures for solving various mathematical problems of everyday life, such as measuring land, constructing buildings, calculating taxes, and dividing goods. The text also shows the use of counter rods, a decimal value system, inverse elements, Euclidean divisions, Gauss elimination, and Horner’s method. Another influential text from antiquity is The Book of Numbers and Calculus, which deals with arithmetic operations, fractions, square and cubic roots, and linear equations.

Ancient Chinese mathematics reached its apogee during the Han Dynasty (202 BC-220 AD), when many scholars made important contributions to various fields of mathematics. For example, Zhang Heng (78 – 139 AD) was the first to use negative numbers and to approximate pi to 3,146; Liu Hui (220 – 280 AD) was the first to use Cavalieri’s principle to find the volume of a sphere; and Zu Chongzhi (429 – 500 AD) was the first to calculate pi to seven decimal places using a method similar to Archimedes.

Ancient Chinese mathematics also influenced other cultures, such as India and Japan, through trade and cultural exchanges. The history of mathematics in ancient China demonstrates that the Chinese were innovative and creative in their mathematical endeavors, and that they developed a rich and sophisticated mathematical tradition that deserves recognition and appreciation.

## Greece (600 BC-300 BC):

Ancient Greece is known for its contributions to the development of theoretical mathematics. Greek mathematicians, such as Pythagoras, Euclid, and Archimedes, made revolutionary advances in geometry, number theory, and measurement. Euclid’s works, particularly his “Elements”, laid the foundations of modern geometry and were widely used as a textbook for centuries.

Greek mathematics began with the works of Thales of Miletus and Pythagoras of Samos, who traveled to Egypt and Babylon and learned from their mathematical traditions. They brought new ideas and methods, such as geometry, proof, and number theory, and applied them to various fields of knowledge, such as astronomy, music, and philosophy.

Greek mathematics reached its apogee in classical times, with contributions from Plato, Euclid, Archimedes, Apollonius and many others. They laid the foundations of geometry, algebra, calculus, mechanics and optics, among others. They also developed sophisticated calculation and measurement tools, such as the abacus, sundial, and astrolabe. Greek mathematics was characterized by its emphasis on logic, deduction, and rigor, as well as the use of symbols and diagrams to represent abstract concepts.

Greek mathematics declined after the Hellenistic period, due to the political and cultural changes that affected the region. However, it was preserved and passed down by later civilizations, such as the Romans, Arabs and Byzantines. Greek mathematics had a lasting impact on the history of science and culture, and inspired many generations of mathematicians and philosophers.

## Ancient Rome

The ancient Romans, known for their architectural and engineering marvels, also made notable contributions to the field of mathematics. Although the Romans did not develop new mathematical concepts or theories, they applied and expanded on mathematical knowledge inherited from earlier civilizations, especially the Greeks and Egyptians. Here is a summary of the history of mathematics in ancient Rome:

#### Practical applications

The Romans were known for their practical approach to mathematics, using it primarily for engineering, construction, and commerce. They applied mathematical concepts in various fields, such as surveying, architecture, and accounting. For example, they used geometric principles in the construction of roads, aqueducts and bridges, and applied arithmetic in trade and financial transactions.

#### Roman numerals

The Romans are famous for their numeral system, known as Roman numerals. Roman numerals were used to count and record numbers in various contexts, such as on monuments, inscriptions, and official documents. The Roman numeral system used a combination of letters from the Latin alphabet, such as I, V, X, L, C, D, and M, to represent numbers.

#### Roman abacus

The Romans also used an abacus, a counting device used to perform arithmetic operations. The Roman abacus was a simple counting board with grooves or lines where chips or pebbles were placed to represent numbers. It was used for addition, subtraction, multiplication and division, and was widely used in trade and accounting.

#### Mathematical treatises: History of mathematics in ancient times

Although the Romans did not produce original mathematical treatises, some Roman scholars and writers, such as Boethius (A.D. 480-524), wrote about mathematics and translated Greek mathematical works into Latin. These translations preserved and transmitted important mathematical knowledge to later generations, especially during the High Middle Ages, when the Roman Empire went into decline and European mathematics went through a period of stagnation.

Other valuable resources on our site in this regard: History of robotics, timeline, timeline, AI; Relationship between logic and mathematics; What is the importance of logic and critical thinking

External resource: Britannica

Editions: 2018-20-23