Euclid – World History Encyclopedia
Euclid of Alexandria (lived around 300 BC) systematized ancient Greek and Near Eastern mathematics and geometry. he wrote the elements, the most widely used mathematics and geometry textbook in history. older books sometimes confuse him with euclid of megara. modern economics has been called “a series of footnotes to adam smith,” who was the author of the wealth of nations (ad 1776). Similarly, much of Western mathematics has been a series of footnotes to Euclid, either developing his ideas or questioning them.
the life of euclid
almost nothing is known about the life of euclid. around 300 B.C. c., he directed his own school in Alexandria, Egypt. we do not know the years or the places of his birth and death. he appears to have written a dozen books, most of which are now lost.
the philosopher Proclus of Athens (AD 412-485), who lived seven centuries later, said that Euclid “put the elements together, brought together many of Eudoxus’ theorems, perfected many of Theaetetus’s, and brought to irrefutable proofs of things his predecessors proved only somewhat vaguely he collected Greek manuscripts that were in danger of being lost he told a story about Euclid that rings true:
someone who had started to [study] geometry asked euclid: “what will I get by learning these things?” Euclid called his slave and told her: “Give him [some money], since he must make a profit.” what he learns’.
(heather, 1981, location 8625)
geometry before euclid
in the elements, euclid collected, organized, and tested geometric ideas already in use as applied techniques. With the exception of Euclid and some of his Greek predecessors, such as (624-548 B.C.), Hippocrates (470-410 B.C.), Theaetetus (417-369 B.C.), and Eudoxus (408-355 B.C.), hardly anyone had attempted to calculate. why the ideas were true or applied generally. Tales even became a celebrity in Egypt because he was able to see the mathematical principles behind the rules for specific problems and then apply the principles to other problems, such as determining the height of pyramids.
The ancient Egyptians knew a lot about geometry, but only as applied methods based on evidence and experience. For example, to find the area of a circle, they made a square whose sides were eight-ninths the length of the diameter of the circle. the area of the square was close enough to the area of the circle that they couldn’t detect any difference. his method implies that pi has a value of 3.16, slightly below its true value of 3.14… but close enough for simple engineering. Most of what we know about ancient Egyptian mathematics comes from the Rhind Papyrus, discovered in the mid-19th century AD. and which is now kept in the British Museum.
The ancient Babylonians also knew a lot of applied mathematics, including the Pythagorean theorem. Archaeological excavations at Nineveh uncovered clay tablets with number triplets that satisfied the Pythagorean Theorem, such as 3-4-5, 5-12-13, and with considerably larger numbers. as of 2006 CE, 960 of the tablets had been deciphered.
the elements
euclid did not originate most of the ideas in the elements. his contribution was fourfold:
- gathered important mathematical and geometric knowledge in one book. The Elements is a textbook rather than a reference book, so it doesn’t cover everything that was known.
- gave definitions, postulates and axioms. he called the axioms “common notions”.
- presented geometry as an axiomatic system: every statement was either an axiom, a postulate, or was proved by clear logical steps from axioms and postulates.
- gave some of his own original discoveries, such as the first known proof that there are an infinite number of prime numbers.
The Elements has 13 chapters (often called “books”), divided into three main sections:
chapters 1-6: plane geometry. chapters 7-10: arithmetic and number theory. chapters 11-13: solid geometry.
each chapter begins with definitions. Chapter 1 also includes postulates and “common notions” (axioms). examples are:
definition: “a point is that which has no part”. postulate: “draw a straight line from any point to any point”. (That’s Euclid’s way of saying that there are straight lines.) common notion: “things equal to the same thing are also equal to each other.”
if the ideas seem obvious, that’s the point. Euclid wanted to base his geometry on ideas so obvious that no one could reasonably doubt them. From his definitions, postulates, and common notions, Euclid deduces the rest of geometry. its geometry describes the normal space we see around us. modern ‘non-Euclidean’ geometries describe space at astronomical distances, at speeds close to light, or deformed by gravity.
other works by euclid
About half of Euclid’s works have been lost. we only know of them because other ancient writers refer to them. lost works include books on conic sections, logical fallacies, and “porisms”. we are not sure what the porismos were. Euclid’s works that still exist are the elements, data, division of figures, phenomena and optics. In his book on optics, Euclid defended the same theory of vision as the Christian philosopher St. Augustine.
the influence of euclid
From ancient times to the late 19th century, people regarded the elements as a perfect example of correct reasoning. Over a thousand editions have been published, making it one of the most popular books after the Bible. The 17th-century Dutch philosopher Baruch de Spinoza modeled his book on the ethics of the elements, using the same format of definitions, postulates, axioms, and proofs. In the 20th century, the Austrian economist Ludwig von Mises adopted Euclid’s axiomatic method to write about economics in his book Human Action.
