The origins of mathematical thought represent a fundamental milestone in the cognitive evolution of Homo sapiens. Rather than emerging as a static, fully formed system of abstract logic, mathematics developed as an adaptive, socio-cultural mechanism designed to solve concrete environmental, economic, and administrative problems.
Table of Contents
The Cognitive Leap to Number
In early human history, man interacted with the physical world without a formal language for quantity. They possessed what modern cognitive scientists refer to as a “number sense”—an innate, biologically determined ability to approximate quantities, distinguish between large and small groups of objects, and perceive changes in a visual field. This Approximate Number System (ANS) is non-symbolic and relies on immediate perception. It operates on Weber’s Law, meaning that the ability to discriminate between two quantities depends on their ratio rather than their absolute difference.
The transition from this fuzzy number sense to symbolic exactness represents one of the most profound cognitive leaps in human history. It required an entirely new framework of thought:
- Reification: The mental transformation of an abstract property (quantity) into a concrete, manageable concept.
- One-to-One Correspondence (Bijective Matching): Pairing each item in a collection with a distinct token, mark, or physical counter.
- Sequential Ordinality: Realizing that numbers follow a rigid, invariable sequence where each step signifies a uniform increase in quantity.
This cognitive revolution did not occur overnight. It was a slow, cumulative process heavily dependent on material culture. Humans needed physical objects to think with—what material engagement theory describes as “cognitive extensions.” By externalizing memory through physical markers, early humans bypassed the natural limitations of working memory, paving the way for systemic calculation.
Ancient Mathematics: Tallying, Tokens, and External Memory Systems
The earliest archaeological evidence of proto-mathematical behavior dates back to the Upper Paleolithic period. These artifacts take the form of incised bones, stones, and antlers, bearing regular, deliberately carved notches. For decades, anthropologists debated whether these markings were merely decorative, accidental results of butchery, or intentional records. Today, the scientific consensus recognizes them as tallies—the earliest surviving external memory devices used to record quantitative data.
The Lebombo Bone and the Ishango Bone
The oldest known mathematical artifact is the Lebombo Bone, discovered in the Lebombo Mountains between South Africa and Eswatini. This object is a small section of a baboon’s fibula containing $29$ distinct, deeply carved notches. The number is highly suggestive; it corresponds precisely to the number of days in a lunar cycle ($29.53$ days) or a human menstrual cycle. This indicates that the bone likely served as a calendar or tracker, marking the first time a human externalized the passage of time into a physical, countable medium.
Even more complex is the famous Ishango Bone, unearthed in 1950 near the headwaters of the Nile River in the Democratic Republic of the Congo. The Ishango bone features a dark brown length of bone with a sharp piece of quartz fixed to one end, suggesting it was used for engraving. What elevates the Ishango bone above a simple tally stick is the deliberate, rhythmic grouping of its notches into three distinct columns:
- Column 1 contains groupings of $11, 13, 17$ and $19$ notches. Remarkably, these are all the prime numbers between $10$ and $20$.
- Column 2 displays groupings of $(3, 6), (4, 8), (5, 10)$ and $(5, 7)$. The first three pairs demonstrate an explicit understanding of doubling or multiplication by two.
- Column 3 shows groupings of $11, 21, 19$ and $9$. The sums of these columns are equally intriguing: Column 1 and Column 3 both sum to $60$, while Column 2 sums to $48$.
While some researchers argue that these groupings are coincidental or represent a lunar calendar based on a base-$10$ or base-$12$ system, the structural organization implies something deeper than random notation. It suggests an early human mind actively exploring the relationships between numbers, recognizing patterns, and categorizing numbers based on structural properties.
The First Revolution and the Clay Token System
In time, human societies shifted from nomadic hunting and gathering to sedentary agriculture and animal domestication. This monumental shift occurred independently across various cradles of civilization, most notably in the Fertile Crescent of the Near East.
Agriculture introduced an entirely new set of cognitive demands. Nomadic groups only needed to count what they could carry; agricultural communities needed to manage stockpiles of grain, track animal herds, calculate land areas, and plan for future seasons. Property, ownership, and trade required absolute precision. If a farmer deposited fifty bushels of barley into a communal granary, they required a guarantee that they could retrieve precisely fifty bushels later.
To solve this problem, communities in the ancient Near East developed a physical accounting system using clay tokens. Pioneer archaeologist Denise Schmandt-Besserat discovered that small, geometric clay shapes began appearing in thousands of archaeological sites across Syria, Iraq, and Iran. These tokens were not toys or ornaments; they were a sophisticated, standardized recording system.
CONE = 1 Small Measure of Grain
SPHERE = 1 Large Measure of Grain
DISK = 1 Flock of Sheep / Animals
CYLINDER = 1 Jar of Oil / Liquid Commodity
Each shape stood for a specific quantity of a specific commodity. This is an essential phase in mathematical history: the numbers were still tied directly to the objects themselves. There was no concept of an abstract number “five” that could apply universally to anything. Five sheep were represented by five clay disks; five jars of oil were represented by five clay cylinders. The arithmetic was purely additive, conducted by physically sorting, grouping, and matching these physical objects.
Mesopotamian Mathematics: The Birth of the Sexagesimal System and Algorithmic Logic
Over several millennia, the clay token system evolved directly into the world’s first written language and formal mathematical system. In the major Sumerian city-states like Uruk, the sheer volume of trade made managing thousands of individual clay tokens highly inefficient.
Accountants began enclosing these tokens inside hollow, sealed clay spheres called bullae. To ensure the contents of the bulla could be verified without breaking it, scribes used the tokens to press impressions into the wet exterior clay before sealing them inside.
Eventually, an intuitive realization occurred: if the impressions on the outside of the clay ball represented the tokens inside, the tokens themselves were redundant. The tokens were abandoned, and scribes began using a pointed stylus to draw the shapes directly onto flat, two-dimensional clay tablets. This marked the birth of Cuneiform writing and the complete separation of numerical symbols from physical counters.
Phase 1: Physical Tokens (Cones, Spheres)
Phase 2: Tokens sealed in Clay Bullae (Impressions on outside)
Phase 3: Bullae flattened into Tablets (Tokens abandoned)
Phase 4: Cuneiform Script (Abstract Base-60 Numerals)
The Base-60 (Sexagesimal) System
Unlike modern mathematics, which relies entirely on a base-10 (decimal) system, the Sumerians and their successors, the Babylonians, developed a sexagesimal (base-60) system. This choice was mathematically brilliant. While $10$ is only divisible by $1, 2, 5,$ and $10$, the number $60$ has twelve distinct divisors:
$$\text{Divisors of 60} = \{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}$$
This immense divisibility made everyday commercial arithmetic incredibly clean. It eliminated the messy, recurring fractions that plague base-10 systems when dealing with thirds, quarters, or fifths. For example, dividing a unit into three equal parts in decimal results in an infinite string ($0.333…$), whereas in sexagesimal, it yields an exact integer value of $20$ parts out of $60$.
The Babylonians constructed their base-60 numbers using a sub-base of 10. They utilized only two primary cuneiform marks: a vertical wedge ($\curlyvee$) representing the value $1$, and a corner-wedge ($\prec$) representing the value $10$. By grouping these marks, they could write any number from $1$ to $59$.
1 = 1 vertical wedge
10 = 1 corner wedge
43 = 4 corner wedges + 3 vertical wedges
The Place-Value Revolution
The Old Babylonian scribes achieved an extraordinary milestone: they developed a positional notation system (place-value system). In a place-value system, the absolute value of a symbol depends entirely on its position within the number sequence.
In our modern decimal system, the digit ‘5’ in $532$ represents $5 \times 10^2$ ($500$), because it sits in the hundreds place. In the Babylonian sexagesimal place-value system, positions increased by powers of $60$ moving from right to left:
$$\text{Positions} = \dots \mid 60^3 \, (216,000) \mid 60^2 \, (3,600) \mid 60^1 \, (60) \mid 60^0 \, (1) \dots$$
Therefore, a scribe writing the symbols for $1$ and $5$ side-by-side ($1, 5$) did not mean $1 + 5 = 6$. They meant:
$$(1 \times 60^1) + (5 \times 60^0) = 60 + 5 = 65$$
This structural breakthrough allowed the Babylonians to write exceptionally large numbers and perform highly complex calculations using the exact same set of basic symbols.
However, this early system faced a significant hurdle: the lack of a true zero symbol. If a position was empty—for instance, if a scribe wanted to write the number $3605$, which is $(1 \times 60^2) + (0 \times 60^1) + (5 \times 60^0)$—they simply left an empty space between the $1$ and the $5$. This caused massive ambiguity. Context alone told the reader whether a number was $1,5$ ($65$), $1,0,5$ ($3605$), or even a fraction like $\frac{65}{60}$. It wasn’t until the Hellenistic period, under Seleucid rule, that Babylonian scribes began using a double slanted wedge ($//$) as a placeholder to indicate an empty column, creating a proto-zero.
Babylonian Tables and Algorithmic Arithmetic
Babylonian mathematics was intensely computational and algorithmic. Scribes did not solve problems using abstract algebraic formulas; instead, they relied on highly structured, step-by-step numerical recipes found on instructional tablets. To facilitate these rapid calculations, scribes memorized and copied vast suites of pre-computed mathematical reference tables:
- Multiplication Tables: Giving products up to $60$.
- Reciprocal Tables: Essential because the Babylonians did not perform direct division. To divide a number by $x$, they multiplied that number by its reciprocal ($\frac{1}{x}$). The tables provided conversions for standard sexagesimal fractions: the reciprocal of $2$ was $30$ ($\frac{30}{60}$), the reciprocal of $3$ was $20$ ($\frac{20}{60}$), and so forth.
- Squares, Cubes, and Root Tables: Providing exact squares and square roots for hundreds of values.
Using these tables, Babylonian mathematicians developed a brilliant method for approximating square roots, commonly known today as the Babylonian Method or Heron’s Method. To find the square root of a non-square number $S$, they would start with a close approximation $x_0$ and iteratively refine it using the arithmetic mean of $x_0$ and $\frac{S}{x_0}$:
$$x_{n+1} = \frac{1}{2} \left( x_n + \frac{S}{x_n} \right)$$
The breathtaking accuracy of this algorithm is preserved on the famous clay tablet YBC 7289 (Yale Babylonian Collection). This tablet features a drawing of a square with its diagonals traced across it. Along the diagonal, a scribe inscribed cuneiform numbers that translate in base-60 to:
$$1; 24, 51, 10$$
To see how incredibly accurate this calculation is, we can convert this value into modern decimal notation:
$$1 + \frac{24}{60} + \frac{51}{3600} + \frac{10}{216000} = 1.41421296$$
When compared to the true value of $\sqrt{2} \approx 1.41421356$, we find that the Babylonian scribe calculated the square root of two to six decimal places of accuracy over $3,800$ years ago.
Plimpton 322 and the Foundations of Trigonometry
Perhaps the most intensely studied mathematical artifact in the world is Plimpton 322, a damaged clay tablet dating to roughly 1800 BCE. The tablet contains a table of fifteen rows and four columns of numbers written in a highly regular, neat script.
For decades, scholars believed Plimpton 322 was merely a commercial ledger tracking financial transactions or tax records. However, in 1945, historian Otto Neugebauer revealed that the numbers form columns of Pythagorean Triples—sets of integers $a, b,$ and $c$ that perfectly satisfy the fundamental geometric relationship:
$$a^2 + b^2 = c^2$$
For example, the very first row lists a short side of $119$ and a diagonal (hypotenuse) of $169$. If we apply the theorem to find the missing long side ($b$):
$$169^2 – 119^2 = 28561 – 14161 = 14400 = 120^2$$
The missing side is a clean integer ($120$). What makes Plimpton 322 truly profound is how these triples are arranged. As you move down the rows, the ratio of the sides changes systematically, causing the angle of the right triangle to decrease from roughly $45^\circ$ to $31^\circ$ in almost exact $1^\circ$ steps.
This has led modern researchers, such as mathematicians Daniel Mansfield and Norman Wildberger, to argue that Plimpton 322 is not just a list of number exercises, but the world’s oldest trigonometric table. Long before the Greek astronomer Hipparchus utilized circles and chords to measure angles, Babylonian scribes were using flat ratios of right-angle triangle sides to define angles, creating a unique form of rational trigonometry.
Egyptian Mathematics: Practical Geometry and Unit Fractions
While Mesopotamian mathematics developed along a highly computational, algebraic path, civilization along the Nile River approached mathematics through a practical, visual, and deeply geometric lens. The survival of Egyptian mathematics is tied directly to the unique geography of their home. Every summer, the Nile River burst its banks, flooding the valley. While this flooding deposited rich, fertile silt crucial for farming, it completely wiped out all physical property lines, landmarks, and boundary markers between fields.
Once the floodwaters receded, the Pharaoh sent out elite administrative specialists known as harpedonaptai (literally translated as “rope-stretchers”). These early surveyors used long, knotted ropes to measure out plots of land, re-establish agricultural boundaries, and recalculate property taxes based on precise area. Through this continuous, annual cycle of measuring and mapping, the Egyptians developed a robust, highly effective system of practical geometry.
Primary Sources: The Rhind and Moscow Papyri
Unlike the durable, baked clay tablets of Mesopotamia that survived buried in dry soil for millennia, the Egyptians wrote using organic ink on fragile papyrus scrolls. Because papyrus rots quickly when exposed to moisture, only a tiny handful of mathematical texts have survived to the modern day, preserved in the deepest, driest desert tombs.
The most extensive source is the Rhind Mathematical Papyrus, dating to approximately 1650 BCE. It was copied by a scribe named Ahmes from an older, lost Middle Kingdom document. The scroll measures over $5$ meters in length and contains $84$ distinct, practical mathematical problems covering linear equations, area calculations, volume measurements, and the distribution of food rations.
The second major document is the Moscow Mathematical Papyrus, dating to around 1850 BCE, which contains $25$ problems focusing even more intensely on advanced geometry.
Egyptian Numerals: Hieroglyphic and Hieratic Systems
For monumental stone carvings, the Egyptians used a base-10 hieroglyphic numeral system. This system was purely additive, lacking any place-value mechanics. They utilized distinct, symbolic icons for each power of ten from $1$ to $1,000,000$:
1 = A single vertical stroke
10 = A heel bone / arch symbol
100 = A coiled coil of rope
1000 = A sacred lotus flower
10000 = A pointing, single finger
To write the number $234$, a scribe would simply draw two coils of rope, three heel bones, and four single strokes. While visually beautiful, this repetitive system required drawing dozens of individual icons just to write a single complex number, making rapid calculation highly cumbersome.
To solve this inefficiency for everyday administrative work, scribes developed Hieratic script—a cursive, highly abstracted shorthand version of hieroglyphs. Hieratic script introduced distinct, single symbols for every number from $1$ to $9$, and every multiple of ten ($10, 20, 30 \dots$), hundred, and thousand. This process of “cipherization” drastically reduced the number of symbols required to write a value, allowing scribes to perform rapid scratch-pad arithmetic.
The Tyranny of Unit Fractions
The most distinctive and mathematically fascinating aspect of Egyptian arithmetic was their approach to fractions. Except for the isolated exception of $\frac{2}{3}$ (and occasionally $\frac{3}{4}$), the Egyptians could only write fractions as unit fractions—fractions where the numerator is strictly equal to $1$ (e.g., $\frac{1}{2}, \frac{1}{3}, \frac{1}{7}, \frac{1}{34}$).
To represent a unit fraction in hieroglyphs, a scribe placed the mouth symbol er ($\diamond$), which signified “part,” directly above a standard integer. In Hieratic shorthand, this was simplified to a dot placed over the number.
1/5 = A mouth symbol over the number five
1/10 = A mouth symbol over a heel bone
If an administrative calculation resulted in a complex fractional value like $\frac{5}{8}$, an Egyptian scribe was forbidden from writing “$\frac{5}{8}$”. Instead, they were required to decompose that fraction into a clean, additive series of distinct unit fractions:
$$\frac{5}{8} = \frac{1}{2} + \frac{1}{8}$$
Furthermore, they held a strict rule: you could never repeat the same unit fraction in a sequence. For example, writing $\frac{2}{7}$ as $\frac{1}{7} + \frac{1}{7}$ was completely illegal. It had to be broken down into entirely unique denominators:
$$\frac{2}{7} = \frac{1}{4} + \frac{1}{28}$$
To make this constant decomposition manageable, the beginning of the Rhind Papyrus contains a massive reference grid known as the $2/n$ Table. This table provides the exact unit fraction breakdowns for all fractions with a numerator of $2$ and odd denominators ranging from $n = 3$ to $n = 101$.
While this system appears incredibly tedious to modern eyes, it was highly effective for their specific economic reality: distributing concrete goods. If a scribe needed to divide $2$ sacks of grain equally among $7$ workers, giving each worker “$\frac{2}{7}$” of a sack provides no actionable guidance on how to physically cut the sacks. But if the scribe decomposes the fraction into $\frac{1}{4} + \frac{1}{28}$, the physical instructions become immediately obvious: open the sacks, give each worker exactly one-quarter ($\frac{1}{4}$) of a sack, and then divide the remaining grain into twenty-eight equal portions ($\frac{1}{28}$) for distribution. Everyone receives a visually identical, demonstrably equal share.
Advanced Geometric Achievements
Egyptian geometry was remarkably sophisticated, achieving approximations that rivaled those of much later eras. Problem 50 of the Rhind Papyrus outlines their method for calculating the area of a circle: the scribe asserts that a circular field with a diameter of $9$ khet has the exact same area as a square field with a side length of $8$ khet.
Let us analyze the mathematical implications of this rule. The true area of a circle is given by $A = \pi r^2 = \pi \left(\frac{d}{2}\right)^2$. According to the Egyptian scribe, this area is equal to the square of $\left(\frac{8}{9}\right)d$:
$$\pi \left( \frac{d}{2} \right)^2 \approx \left( \frac{8}{9} d \right)^2$$
$$\pi \left( \frac{d^2}{4} \right) \approx \frac{64}{81} d^2$$
$$\pi \approx \frac{256}{81} \approx 3.16049$$
This empirical approximation for $\pi$ ($3.1605$) is off by less than $0.6\%$ from its true value ($\approx 3.14159$). This represents an extraordinary level of accuracy for an era devoid of formal algebraic notation or limit theory.
Even more impressive is Problem 14 of the Moscow Mathematical Papyrus. This problem outlines a precise step-by-step algorithm to calculate the volume of a frustum—a truncated square pyramid whose top has been sliced off parallel to its base.
The text instructs the reader: square the base side length ($a$); square the top side length ($b$); multiply the base side length by the top side length ($a \times b$); add these three results together; then multiply this total sum by exactly one-third of the vertical height ($h$). Expressed as a modern formula, this is:
$$V = \frac{1}{3} h \left( a^2 + ab + b^2 \right)$$
This formula is completely accurate. It is not an approximation; it is the exact mathematical truth derived from calculus and solid geometry. How the Egyptians deduced this deep relationship without formal algebra remains one of the greatest and most fascinating mysteries in the history of archaeology.
Comparative Epistemology: East vs. West Cradles of Thought
To contextualize the achievements of Mesopotamia and Egypt, we must expand our view to the other two primary cradles of ancient civilization: the Indus Valley and Early China. Both regions independently arrived at strikingly similar mathematical breakthroughs to solve the universal challenges of the ancient state.
The Indus Valley Civilization
The Harappan culture of the Indus Valley (modern Pakistan/Northwest India) was a marvel of advanced, highly systematic urban engineering. While they did not leave behind long, readable mathematical scrolls, their mathematical genius is immortalized in the physical infrastructure of their cities, such as Mohenjo-daro and Harappa.
The Indus Valley civilization was the first in the ancient world to achieve complete standardization of weights and measures. Archaeologists have recovered thousands of small, highly polished chert stone cubes used as counter-weights for scales.
These weights follow a perfectly regular, uniform progression: first binary (doubling), then decimal. The smaller weights double in precise ratios: $1, 2, 4, 8, 16, 32, 64,$ while the higher weights jump to base-10 values: $160, 200, 480, 640, 1600, 3200, 6400, 12800$.
Furthermore, the bricks used to construct their massive public granaries, defensive walls, and advanced sewage systems were fired in an invariant, exact geometric ratio of dimensions:
$$\text{Bricks Ratio} = \text{Thickness} : \text{Width} : \text{Length} = 1 : 2 : 4$$
This exact ratio maximized structural interlocking, ensuring walls could withstand massive physical soil pressure. This demonstrates that mathematical precision was deeply integrated into every level of their daily production and craftsmanship.
Early Chinese Mathematics (c. 1600 – 200 BCE)
In the fertile valleys of the Yellow and Yangtze rivers, early Chinese mathematics developed around administrative efficiency, astronomical mapmaking, and divination practices. The earliest physical records of numbers survive as inscriptions carved into animal shoulder blades and turtle shells, known as Oracle Bones (dating to the Shang Dynasty, c. 1600–1046 BCE).
The Oracle Bone script reveals that from the very beginning, China utilized a true multiplicative decimal system. They maintained separate, unique symbols for the digits $1$ through $9$, and distinct symbols for the powers of ten ($10, 100, 1000, 10000$). To write a number like $500$, they did not write the symbol for one hundred five times in a row; instead, they wrote the symbol for $5$ directly combined with the symbol for $100$. This was a highly streamlined, compact system that laid the structural foundations for the invention of the decimal counting board and the abacus.
These early traditions culminated at the end of the ancient era in the legendary compilation of The Nine Chapters on the Mathematical Art (Jiuzhang Suanshu). This text codified advanced algorithms for solving simultaneous linear equations using matrices, centuries before Western mathematicians formulated similar systems.
Synthesis and Comparative Analysis
When we contrast the mathematical systems of the ancient world, we observe a striking phenomenon: diverse civilizations arrived at identical geometric truths using radically different cognitive frameworks and numerical scripts.
The table below summarizes the core epistemological and structural differences between the two most dominant ancient mathematical traditions:
| Feature | Mesopotamian Mathematics | Egyptian Mathematics |
| Primary Medium | Incised clay tablets (Durable, baked) | Ink on papyrus scrolls (Fragile, organic) |
| Number Base | Sexagesimal (Base-60) with Base-10 sub-cycles | Pure Decimal (Base-10) |
| Notation Style | Positional Place-Value System | Additive System (Hieroglyphic/Ciphered Hieratic) |
| Core Strength | Advanced numeric computations, roots, algebra | Practical geometry, volumes, surveying |
| Fractional Mechanics | Sexagesimal positional fractions ($60^{-n}$) | Strictly Unit Fractions ($\frac{1}{n}$) in additive rows |
| Key Achievement | Pythagorean Triples (Plimpton 322), $\sqrt{2}$ calculation | Exact volume of a frustum, $0.6\%$ accurate circle area |
The Absence of Proof
Despite their brilliance, it is essential to note a defining characteristic shared by all pre-classical mathematical systems: the absolute absence of deductive proof.
Nowhere in a cuneiform tablet or an Egyptian papyrus scroll will you find a generalized theorem statement, a formal definitions section, or a logical proof demonstrating why a particular formula works universally. Ancient mathematics was entirely inductive, empirical, and exemplary. Scribes taught by providing specific examples: “If a pyramid has a base of ten and a height of five, do this, multiply that, and behold, you have the answer.”
The concept of proving a mathematical truth purely through step-by-step deductive logic, independent of any physical application, would have to wait for the arrival of the Greek philosophical schools of Pythagoras, Thales, and Euclid in the Mediterranean world during the 1st millennium BCE.
Conclusion: The Enduring Legacy of Ancient Systems
Prehistoric and ancient mathematics was never a primitive, flawed precursor to modern mathematics. It was a highly sophisticated, fully realized response to the immediate operational needs of the world’s first large-scale human societies. The scribes of Babylon, the rope-stretchers of Egypt, the brick-makers of Harappa, and the administrators of Shang China successfully transformed numbers from an elusive, abstract aspect of nature into a powerful, highly structured tool of human empire.
Their deep breakthroughs remain woven into the fabric of our modern daily lives:
- Every time we look at a clock and divide an hour into 60 minutes, or a minute into 60 seconds, we are utilizing the sexagesimal place-value system developed by the astronomers of Mesopotamia.
- Every time we divide a compass or a circle into 360 degrees ($6 \times 60$), we trace our steps directly back to the clay tablets of the Fertile Crescent.
- The structural foundations of architectural engineering, land measurement, and fractional partitioning still rely heavily on the visual, geometric insights pioneered along the banks of the Nile River.
By studying this rich dawn of numbers, we gain a profound appreciation for mathematics not as an isolated, sterile textbook topic, but as a living, evolving, and deeply human language—a language forged in the dirt, written on clay, and stretched across ropes to bring order, clarity, and structure to the chaos of the ancient physical world.
Reference List
- Ahmes (c. 1650 BCE). The Rhind Mathematical Papyrus. British Museum, London. (Translation and commentary by Chace, A. B., 1927).
- Friberg, J. (2007). A Remarkable Collection of Babylonian Mathematical Texts. Springer.
- Mansfield, D. F., & Wildberger, N. J. (2017). Plimpton 322 is Babylonian exact rational trigonometry. Historia Mathematica, 44(4), 395-419.
- Neugebauer, O. (1957). The Exact Sciences in Antiquity. Brown University Press.
- Robson, E. (2008). Mathematics in Ancient Iraq: A Social History. Princeton University Press.
- Schmandt-Besserat, D. (1992). Before Writing: From Counting to Cuneiform. University of Texas Press.
- Struik, D. J. (1987). A Concise History of Mathematics. Dover Publications.