The Historical and Mathematical Significance of Pythagorean Triples

Around 2000 BC, ancient Babylon and Egypt made significant contributions to mathematics with their recognition of Pythagorean triples. These sets of three positive integers $(a, b, c)$ satisfy the equation $a^2 + b^2 = c^2$, encapsulating the foundational concept of the Pythagorean theorem. This theorem, critical in geometry, states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

In Babylon, clay tablets such as Plimpton 322 provide evidence of advanced mathematical understanding. This artifact, dated to around 1800 BC, contains a list of integer pairs that form Pythagorean triples. Scholars believe these were likely used for solving geometric problems, constructing accurate architectural designs, or for educational purposes in scribal schools. The level of precision in their calculations underscores the sophistication of Babylonian mathematics.

Similarly, in ancient Egypt, the Berlin Papyrus 6619, dating to approximately 1800 BC, includes problems suggesting knowledge of Pythagorean triples. Egyptians applied these principles in practical contexts, such as surveying and pyramid construction. For example, the "rope-stretchers" (harpedonaptae) used knotted ropes to form precise right angles during land measurement, implicitly relying on Pythagorean principles.

The early awareness and application of Pythagorean triples in these civilizations illustrate their advanced intellectual achievements and problem-solving capabilities. These early explorations formed the foundation of later mathematical disciplines, including geometry and number theory. The work of ancient scholars like Thales and Pythagoras centuries later built upon this knowledge, further formalizing and disseminating these ideas.

Today, Pythagorean triples remain central to mathematics, extending to modern applications such as cryptography, computer science, and physics. The enduring interest in their properties highlights not only the timeless ingenuity of ancient mathematicians but also the interconnectedness of historical and contemporary mathematical thought.
The Historical and Mathematical Significance of Pythagorean Triples

Ancient Egyptian Math Skills

Around 6000 BC, the ancient Egyptians settled in the fertile Nile valley, where they began observing lunar phases and seasonal patterns for agricultural and religious reasons.

Within their society, mathematics played a pivotal role, being applied to tasks such as measuring time, establishing straight lines, evaluating Nile flood levels, computing taxes, surveying land areas, managing financial transactions, and even in culinary endeavors.

The numerical system employed by ancient Egyptians endured from about 3000 BC until the early first millennium AD. This system relied on multiples of ten, often rounded to higher powers, and was expressed through hieroglyphs. Distinct symbols represented one unit, ten, hundred, thousand, ten thousand, hundred thousand, and one million.

In contrast to the positional notation of the decimal system, the ancient Egyptians did not embrace such a concept. Instead, their proficiency lay in working with unit fractions, where the numerator was consistently one. These fractions were skillfully applied to express volumes of irregularly shaped objects and to address problems related to areas and volumes.
Ancient Egyptian Math Skills

Geometry in ancient history

Geometry, one of the most important branches of mathematics, has its roots in countries such as Egypt, Babylon, China, Greece, and Vedic India. Around 2100 BC, the concept of area is first recognized in Babylonian clay tablets, and 3-dimensional volume is discussed in an Egyptian papyrus. This begins the study of geometry.

In Egypt, ancient Egyptians developed geometry from the ‘Age of Pyramids’. The evidence of usage of geometry is seen on the walls of temples and written on papyrus. The Moscow Mathematical Papyrus is a well-known mathematical papyrus containing various problems in arithmetic, geometry, and algebra.

The Babylonians of 2,000 to 1,600 BC knew much about navigation and astronomy, which required knowledge of geometry. Ancient Babylonians used studies of triangles techniques 1500 years before Greeks. The ancient Babylonians were using geometrical calculations to track the biggest objects in space.

The Babylonians were also responsible for dividing the circumference of a circle into 360 equal parts. They also used the Pythagorean Theorem (long before Pythagoras), performed calculations involving ratio and proportion, and studied the relationships between the elements of various triangles.

Beginning about the 6th century BC, the Greeks gathered and extended this practical knowledge and from it generalized the abstract subject now known as geometry, which is derived from Ancient Greek words – ‘Geo’ means ‘Earth’ and ‘metron’ means ‘measurement’.

Herodotus in 5th BC credits the Egyptians with inventing surveying in order to reestablish property values after the annual flood of the Nile. Similarly, eagerness to know the volumes of solid figures derived from the need to evaluate tribute, store oil and grain, and build dams and pyramids.

Alexandria became one of the important centers of Greek learning and this is where Euclid who is often referred to as the “Father of Geometry”, wrote perhaps the most important and successful mathematical textbook of all time, the “Stoicheion” or “Elements”.
Geometry in ancient history

History of ancient fraction

The word fraction actually comes from the Latin "fractio" which means to break. Many cultures have developed fractions (and decimals) independently.

Around 1800 BC: Fractions were first studied by the Egyptians in their study of Egyptian fractions.

Very few scrolls remained which found their way to antique collectors. The Lahun papyrus forms part of the Kahun Papyri and dates back to c. 1825 B.C. It contains an incomplete table of Egyptian fractions 2/n (n=3,…,21) (a complete version is in the Rhind Papyrus).

The huge disadvantage of the Egyptian system for representing fractions is that it is very difficult to do any calculations.

Around 300 BC the Greeks were writing fractions using their alphabet to represent the numbers. The number 2 was written as β and the number 5 was written as ε .

In Ancient Rome, fractions were only written using words to describe part of the whole. They were based on the unit of weight which was called the as. One "as" was made up of 12 uncia so fractions were centered on twelfths.

The Babylonians had one of the oldest written records of fractions and decimals, dating from around 2000 BC. While the current number system uses base 10 (that is, there are 10 digits that make up all our numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), the Babylonian’s number system was organized around the number 60, so current system say it is base 60.

By about 500AD, the Indians had developed a system from a way of writing called brahmi, which had nine symbols and a zero. The knowledge of fractions in India can be traced to ancient times. The fractions one-half (ardha) and three-fourths (tri-pada) occurred already in one of the oldest vedic works the R. gveda (circa 1000 BC). In mathematical works Sulba-s´utras (circa 500 BC), fractions were not only mentioned, but were used in statements and solutions of problems.
History of ancient fraction

Discovery of Pythagorean Theorem

The Pythagorean theorem states that for a right triangle the square of the length of the hypotenuse equals the sum of the squares of the two remaining sides.

Among the many classical Greek schools of mathematics and philosophy the Pythagorean was the oldest and most venerable.

Pythagoras was born around 570 BC. Tradition has it that he came from the island of Samos and traveled widely before he established his school in Crotona in Southern Italy. He learned mathematics from Thales (624-547 BC), traditionally regarded as the founder of Greek mathematics.

In Egypt and Babylonia he absorbed the lore of mysticism and also learned the laws of numbers and geometry.

A legend says that Pythagoras discovered his theorem by observing a tiles floor in the place of Polycrates, the tyrant of Samos. However, many said that he did not discover the theorem.

The story of Pythagorean theorem begins long before that, at least as far as back 1800 BC in Babylonia. Pythagoras’s name became attached to the theorem because he was reputedly he first person to provide such a proof.
Discovery of Pythagorean Theorem

Origin of mathematics

The Cro-Magnon people arrived in Europe around 40,000 BC. By 30,000 BC, they had colonized the entire globe, a tremendous accomplishment requiring sailing technology and celestial navigation.

The first recognizable models were numbers; counting and ‘writing’ numbers is documented since 30,000 BC. These people developed a system of mathematics adequate to determine the cycle of phase of the moon, engraving a primitive graph of this cycle in a bone.

They devise a method of counting. Mathematics was born when ancient man learned how to count.

Archeologists have found a bone with tick marks carved into it. The ticks are arranged into 11 groups of five, suggesting that the bone is used as a tally. One of the artifacts is a wolf bone with many straight notches cut into it.

Between 10000 and 8000 BC in Mesopotamia, people used sets of pebbles as modeling sets.

It is well known that by 2000 BC at least three cultures (Babylon, Egypt, India) had a decent knowledge of mathematics and used mathematical model to improve their every-day life.
Origin of mathematics

Ancient history of calculus

Calculus comes from the Latin word for ‘pebble’ the primitive method of counting whose influence is also visible in the English word ‘calculation’.

The calculus had its origin in the logical difficulties encountered by the ancient Greek mathematicians in their attempt to express their intuitive ideas on the ratios or proportionalities of lines, which they vaguely recognized as continuous, in terms of numbers, which they regarded as discrete.

Archimedes (287 BC – 212 BC) developed a significant part of the calculus. The fundamental notion of the calculus - that of a limit - was well understood by Archimedes, although he did not call it by name.

The first known definition of continuity according to Aristotle (383-322 BC): A thing is continuous when of any two successive parts the limits at which they touch are one and the same and are, as the word implies, held together.

Indian mathematicians produced a number of works with some ideas of calculus. The formula for sum of the curve was first written by Aryabhata in 500 AD, order to find the volume of a cube, which was an important step in the development of integral calculus.

Indian astronomers also came very close to creating what now call calculus. They had advanced to the point where they could apply ideas from both integral and deferential calculus to derive the infinites series expansion of the sine, cosine and arctangent functions.

Early applications for calculus included the study of gravity and planetary motion, fluid flow and ship design and geometric curves and bridge engineering.
Ancient history of calculus

Snell’s law of refraction

Snell’s Law was discovered by various investigators over the centuries. In antiquity, optics began when Egyptians and Mesopotamians developed the first lenses with impressive mechanism based on reflection. Perhaps the first person to understand the basic relationship expressed by Snell’s Law was the Arabian mathematician Ibn Sahl in the year 984.

Abu Sa’d al-Ala Ibn Sahl 

Based on catoptrics and the study of burning mirrors and lenses, Abu Sa’d al-Ala Ibn Sahl (940-1000) devised a systemic elaboration of the fundamentals of dioptrics.

It is believed that Ibn Sahl established a principle akin to the so-called ‘Snell’s Law’ of refraction. Ibn Sahl shows that every transparent medium, including the ‘celestial sphere’ has a certain degree of opacity.

Ibn Sahl described a law of refraction in his treatise ‘On burning mirrors and lenses.’

In 1602, English astronomer and mathematician Thomas Harriot discovered the law but he did not publish his work.

Snellius Willebrord

In 1621, Snellius Willebrord discovered the law. It expresses the relationship between the path of a ray of light passing through the boundary of two adjacent substances and their respective refractive indices.  His unpublished notes on the subject were discovered by the Dutch scholar and manuscript collector Isaac Vossius around 1662.

Christiaan Huygens read unpublished equation by Snell and discussed the law in his Dioptrica, published in 1703.
Snell’s law of refraction

Ancient Ideas of Technology and Sciences

Ancient Idea of Technology and Sciences
There remain many unsolved mysteries about the technologists of the ancient world. There is evidence that in Mesopotamian, craftsmen knew how to use electrical currents for electroplating metals.

The ancient Polynesian, without the aid of compasses or charts, navigated the Pacific.

The Egyptians not only constructed the pyramids but also were able to lift massive stone obelisks onto their ends by some unknown method.

The ancient Egyptians built a canal to link the Red Sea with the Mediterranean and other technological and mathematical innovations took place in India, China and central Asia.

In pre-Bronze Age Britain and on the continent of Europe, builders somehow moved heavy stones to build monuments with apparent astronomical orientations such as at Stonehenge.

The ancient Greeks used a complicated navigational device that was a sort of early geared analog computer to locate the positions of the stars and planets know as Anikythera computer.

The workings of that strange machine, found by a sponge fisherman off the Greek island of Syme in 1900, were partially unraveled by 1974 by a historian of technology, Derek de Solla Price.

In Americas, the Mayans, Toltecs, and subjects of the Inca knew about wheeled pull toys, but they never used wheels for vehicles or even wheelbarrows.

Yet the Mayans used the concept of the mathematical zero several centuries before the Europeans.

The fact that wisely dispersed nations and races came upon the same idea, in cases of parallel invention, rather than diffused invention , leave another set of tantalizing.
Ancient Idea of Technology and Sciences

Mathematics of Three Civilizations

Mathematics of Three Civilizations
There is history to mention the mathematics of three ancient civilizations, interesting in themselves, but of little or no influence on the further course of mathematics: the Minoans-Mycenaeans, the Mayas and the Incas.

Their science is not that of the beginning, but belongs rather to the category of the Ancient Orient.

Mathematical symbols used in administration have been found on the ruins of the Minoan-Mycenaean civilization of Crete and the Greek mainland.

They belong to the scripts called Linear A and B and belong to the period of 1800-1200 BC.

Numbers are represented, as in Egypt (but with different symbols), by special symbols for 1, 10, 100, 1000 in an additive way.

There are also symbols for simple fractions not all unit fractions. Since the scribes did not bake the clay tablets on which they wrote, only those that were baked in the final conflagration of their cities have been preserved, so that we have adequate knowledge of the extend of the mathematical knowledge of this civilization; it may have been comparable to that of Egypt.

The Mayas of Central America, mainly in what is now Yucatan and Guatemala, established a civilization that lasted for a millennium and a half, but reached its height in the so-called classical period, about 200-900 of our era.

The arithmetic of Mayas, mainly deciphered from inscribed stone monuments, some codices, and Spanish chronicles, and closely related to their astronomy, notably their calendric system, was vigesimal (it still is), represented by dots for the units up to 4, and horizontal bars for the fives up to 15.

For larger numbers they used a position system with base 20, powers of 20 being represented by the same symbol as 20, the unit symbol.

There were some modifications for calendric purposes.

This position system required a symbol for zero, often a kind of shell or half open eye sign.

This system with its calendric connections, spread to other peoples of Central America.

The Incas built a large empire in and west of the Andes of South Amreica from the middle of the thirteenth century of our era on, their capital being Cuzco.

Its vast bureaucracy, strong in administration, crafts, and engineering used for communications and information, no writing but so-called quipos.

The simplest quipo has a main cord of colored cotton or sometimes wool, from which knotted cords are suspended with the knots formed into clusters at some distance from each other.

Each cluster has a number of knots from 1 to 9, and a cluster of say, 4 followed by one of 2 and one of 8 knots represents 428.

This is therefore a position system , in which our zero is indicated by a greater distance between the knots.

The colors of the cords represent things: sheep, soldiers, etc; and the position of the cords, as well as additional cords suspended from the cords, could tell a very complicated statistical story to the scribes who could “read” the quipos.
Mathematics of Three Civilizations