Mathematics in Ancient Greece: Geometry, Measurements, Theorems | Early European History And Religion

MATH IN ANCIENT GREECE

Euclid's proof of the Pythagorean theorem

The Greeks had great success in the areas of mathematics, particularly geometry, borrowing heavily from the Egyptians (who were concerned primarily with practical applications) while raising the theoretical and intellectual bar to new heights. The Greek philosophers often equated beauty and mathematics. "Measure and commensurability," wrote Plato in “ Philebus”, "are everywhere identifiable with beauty and excellence." Aristotle wrote "the qualities of numbers exist in a musical scale, the heavens, and many other things. [Source: "The Creators" by Daniel Boorstin]

The Greeks invented or at least were pioneers in geometry and trigonometry. Diophantus's “Arithemtica” was an important work. Euclid is regarded as the father of geometry. Euclidian geometry is still widely studied. Many of his principals still form the basis of high school geometry textbooks. The Greeks and Romans were unable to make the breakthrough of the concept of zero. The first Old World Culture believed to have figured it out was the ancient Hindus.

A surprising number of ancient mathematics texts have survived to this day, including Euclid’s “Geometry”. The math and the sciences thrived during the Hellenistic period, especially in Alexandria where the Ptolemies financed a great library, quasi-university and museum. Fields of study included mathematics (Euclid's “Geometry”, 300 B.C.), astronomy (heliocentric theory of Arisrtarchus, 310 B.C., Julian calendar 45 B.C., Ptolemy's “Almagest” 150 A.D.), geography (Ptolemy's “ Geography” , world map of Eratosthenes 276-194 B.C.).

The ancient Greeks didn't make a distinction between philosophy and science, nor did they recognize the range of disciplines such as physics, chemistry, mathematics, astronomy, etc. that we do today. There simply wasn't the depth of knowledge and range of information that later made separate disciplines practical. In the Greek era, one individual could be an expert in several fields. Nowadays, with the tendency of specialists to know more and more about less and less (i.e. intensive knowledge about a rather limited field) the ability to keep abreast of detailed research in more than one area becomes almost impossible. But in the days of Thales, Pythagoras and Aristotle that was the norm. People expected an individual knowledgeable in one area to also be proficient in others. And many were. [Source: Canadian Museum of History]

Websites on Ancient Greece and Rome: Internet Ancient History Sourcebook: Greece sourcebooks.fordham.edu ; Internet Ancient History Sourcebook: Hellenistic World sourcebooks.fordham.edu ; BBC Ancient Greeks bbc.co.uk/history/ British Museum ancientgreece.co.uk; Illustrated Greek History, Dr. Janice Siegel, Department of Classics, Hampden–Sydney College, Virginia hsc.edu/drjclassics ; The Greeks: Crucible of Civilization pbs.org/empires/thegreeks ;Ancient-Greek.org ancientgreece.com; Metropolitan Museum of Art metmuseum.org/about-the-met/curatorial-departments/greek-and-roman-art; Cambridge Classics External Gateway to Humanities Resources web.archive.org/web

Influence of Egyptian Math on the Ancient Greeks

John Burnet wrote in “Early Greek Philosophy”: “That the Greeks learnt as much from them is highly probable, though we shall see also that, from. the very first, they generalized it so as to make it of use in measuring the distances of inaccessible objects, such as ships at sea. It was probably this generalization that suggested the idea of a science of geometry, which was really the creation of the Pythagoreans, and we can see how far the Greeks soon surpassed their teachers from a remark attributed to Democritus. It runs (fr. 299) : "I have listened to many learned men, but no one has yet surpassed me in the construction of figures out of lines accompanied by demonstration, not even the Egyptian arpedonapts, as they call them." [Source: John Burnet (1863-1928), “Early Greek Philosophy” London and Edinburgh: A. and C. Black, 1892, 3rd edition, 1920, Evansville University]

A fragment from Euclid's Elements (300 BC), regarded as the most influential mathematics textbook of all time

“Now the word arpedovaptês is not Egyptian but Greek. It means "cord-fastener," and it is a striking coincidence that the oldest Indian geometrical treatise is called the Sulvasutras or "rules of the cord." These things point to the use of the triangle of which the sides are as 3, 4, 5, and which has always a right angle. We know that this was used from an early date among the Chinese and the Hindus, who doubtless got it from Babylon, and we shall see that Thales probably learnt the use of it in Egypt. There is no reason for supposing that any of these peoples had troubled themselves to give a theoretical demonstration of its properties, though Democritus would certainly have been able to do so. As we shall see, however, there is no real evidence that Thales had any mathematical knowledge which went beyond the Rhind papyrus, and we must conclude that mathematics in the strict sense arose in Greece after his time. It is significant in this connection that all mathematical terms are purely Greek in their origin.”

Ancient Greek Measurements

A cubit, based on the length of a man's forearm, was the unit of measure throughout much of the ancient world. The measurement varied a great deal however. In ancient Egypt, for example, a cubit for a man was 17.72 inches while the cubit for a king was 20.62 inches.

In Greece, a cubit was 18.24 inches. One cubit equaled two spans. One span equaled three palms. Four cubits equaled one fathom and 400 cubits equaled one stadia, which was about 607 feet. An amphora held 44 “kotyles”. Some have said the Greeks introduced the foot based on the length of Hercules foot.

One talent was equal to 93.65 pounds. For measuring smaller weights, the ancients used grains of wheat or barley corns; the grain to this day is one of the smallest units of weight, 1/7000 of a pound. The carat, used in weighing gems, was derived from the tiny carob seed, prized during antiquity for its uniform weight from seed to seed.

Many of the ancient units of measurement were for practical reasons based parts of the body. The digit (width of a finger), the palm (the width of four fingers), the foot and the cubit (the distance from the tip of the middle finger to the elbow) were all measurements of length. The "pace" (the precursor of the yard and meter) was equal to one large step and "fathom" (roughly six feet) was the distance between two outstretched hands. ["The Creators" by Daniel Boorstin]

In ancient times a lot of the standard formulas that modern engineers and architects use to measure stress and balance had not yet been invented. Most building were but together with knowledge learned through trial and error. One of the first scientific principals of architecture — that a the base of column should be equal to one sixth of its height — was based on the observation that the size of a person's foot is one sixth of his height. [Ibid]

Euclid and Euclidian Geometry

Euclid(around 300 B.C.) was an ancient Greek mathematician and logician. Regarded as the "father of geometry", he is credited with producing for the Elements treatise, on which the foundations of geometry were established that dominated the field until the early 19th century. This system, now referred to as Euclidean geometry, combined and built on earlier theories by Greek mathematicians such as Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. Euclid, Archimedes and Apollonius of Perga are generally considered the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. [Source Wikipedia]

Euclid teaching students in Raphael's "The School of Athens"

"Euclid's genius," wrote classicist Lionel Casson, “lay in designing a superbly logical arrangement and in a presentation that was clarity itself...During the first two decades or so of third century B.C. [this] mathematician at Alexandria set himself the task of drawing up a beginner’s manual for learning geometry. He did the job so successfully that his textbook became the longest-lived in history: in the early years of this century English schoolboys were still taught geometry from what was more or less a translation of Euclid's “Elements”.

Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philosopher Euclid of Megara. It is now generally accepted that he spent his career in Alexandria and lived around 300 BC, after Plato's students and before Archimedes. There is some speculation that Euclid studied at the Platonic Academy and later taught at the Musaeum; he is regarded as bridging the earlier Platonic tradition in Athens with the later tradition of Alexandria.

“Element” is thirteen-book treatise. Much of its originates from earlier mathematicians and it is difficult to differentiate the work of Euclid from that of his predecessors. In the work, Euclid deduced the theorems from a small set of axioms. The classicist Markus Asper said “Euclid's achievement consists of assembling accepted mathematical knowledge into a cogent order and adding new proofs to fill in the gaps".

The Elements does not exclusively discuss geometry as is sometimes believed. It is traditionally divided into three topics: plane geometry (books 1–6), basic number theory (books 7–10) and solid geometry (books 11–13). The heart of the text is the theorems scattered throughout. Using Aristotle's terminology, these may be generally separated into two categories: "first principles" and "second principles". The first group includes statements labeled as a "definition" "postulate" or a "common notion".The second group consists of propositions, presented alongside mathematical proofs and diagrams. It is unknown if Euclid intended the Elements as a textbook, but its method of presentation makes it seem so.

Pythagorean Theorem

Pythagoras with a tablet of ratios in Raphael's "The School of Athens"

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. [Source Wikipedia]

The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: a² + b² = c²

The theorem is named for the Greek philosopher Pythagoras and has been proved numerous times by many different methods — possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

Even though the formalized Pythagorean rule is what students are taught in school experts have long known that the Greeks inherited mathematical teachings from Egyptians, and the Egyptians in turn from the Babylonians. See Below

Babylonians Used Pythagorean Theorem 1,000 Years Before it Was 'Invented' in Ancient Greece

A 3,700-year-old clay tablet has revealed that the ancient Babylonians understood the Pythagorean theorem more than 1,000 years before the birth of the Greek philosopher Pythagoras, who is widely associated with the idea. Ben Turner wrote in Live Science: The tablet, known as Si.427, was used by ancient land surveyors to draw accurate boundaries and is engraved with cuneiform markings which form a mathematical table instructing the reader on how to make accurate right triangles. The tablet is the earliest known example of applied geometry. [Source Ben Turner, Live Science, September 22, 2022]

A French archeological expedition first excavated the tablet, which dates to between 1900 and 1600 B.C in what is now Iraq in 1894, and it is currently housed in the Istanbul Archeological Museum. But it is only just now that researchers have discovered the significance of its ancient markings. "It is generally accepted that trigonometry — the branch of maths that is concerned with the study of triangles — was developed by the ancient Greeks studying the night sky," in the second century B.C., Daniel Mansfield, a mathematician at the University of New South Wales in Australia and the discoverer of the tablet's meaning, said in a statement. "But the Babylonians developed their own alternative 'proto-trigonometry' to solve problems related to measuring the ground, not the sky."

According to Mansfield, Si.427 is the Old Babylonian period's only known example of a cadastral document, or a plan surveyors used to define land boundaries. "In this case, it tells us legal and geometric details about a field that's split after some of it was sold off," Mansfield said. The tablet details a marshy field with various structures, including a tower, built upon it. The tablet is engraved with three sets of Pythagorean triples: three whole numbers for which the sum of the squares of the first two equals the square of the third. The triples engraved on Si.427 are 3, 4, 5; 8, 15, 17; and 5, 12, 13. These were likely used to help determine the land's boundaries.

Greeks and the Concept of Zero

J. J, O'Connor and E F Robertson wrote: The ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed. ”[Source: J. J. O'Connor and E F Robertson, St. Andrews University, December 2000]

Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favor the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden". Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number — it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.

Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.

Image Sources: Wikimedia Commons, The Louvre, The British Museum

Text Sources: Internet Ancient History Sourcebook: Greece sourcebooks.fordham.edu ; Internet Ancient History Sourcebook: Hellenistic World sourcebooks.fordham.edu ; BBC Ancient Greeks bbc.co.uk/history/; Canadian Museum of History, Perseus Project - Tufts University; perseus.tufts.edu ; MIT Classics Online classics.mit.edu ; Gutenberg.org, Metropolitan Museum of Art, National Geographic, Smithsonian magazine, New York Times, Washington Post, Live Science, Discover magazine, Natural History magazine, Archaeology magazine, The New Yorker, Encyclopædia Britannica, "The Discoverers" and "The Creators" by Daniel Boorstin. "Greek and Roman Life" by Ian Jenkins from the British Museum, Wikipedia, Reuters, Associated Press, The Guardian, AFP and various books and other publications.

Last updated September 2024