# Read PDF The Beginnings of Greek Mathematics

One of the oldest surviving mathematical works is the Yi Jing, which greatly influenced written literature during the Zhou Dynasty — BC. For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching contained elements of binary numbers. Mantras from the early Vedic period before BCE invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots.

Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus ca. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of BC, which probably occurred while he was in his prime. Despite this, it is generally agreed that Thales is the first of the seven wise men of Greece.

The two earliest mathematical theorems, Thales' theorem and Intercept theorem are attributed to Thales. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed. Although it is not known whether or not Thales was the one who introduced into mathematics the logical structure that is so ubiquitous today, it is known that within two hundred years of Thales the Greeks had introduced logical structure and the idea of proof into mathematics.

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Babylonian mathematics also known as Assyro-Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in BC. Babylonian mathematical texts are plentiful and well edited. When Chinese, Islamic, and Indian mathematicians had been in ascendancy, and Europe fell in Dark Ages, almost all mathematics and intellectual endeavor stagnated.

With the possible exception of this theorem, all of Aristotle's original mathematics may be found in his arguments against infinity and on motion in the Physics iii-vii and De caelo i, many of which use proportion theory. In his attempt to work out theorems about ratios and infinite magnitudes, Aristotle makes important mathematical observations about infinite magnitudes and may have been the first to attempt them.

Supplement to Aristotle and Mathematics Aristotle and Greek Mathematics This supplement provides some general indications of Aristotle's awareness and participation in mathematical activities of his time. Greek mathematics in Aristotle's Works Here are twenty-five of his favorite propositions the list is not exhaustive. In a given circle equal chords form equal angles with the circumference of the circle Prior Analytics i. The angles about a point are two right angles Metaphysics ix 9; Eucl. If two straight-lines are parallel and a straight-line intersects them, the interior angle is equal to the exterior angle Prior Analytics ii.

If two straight-lines are parallel and a straight-line intersects them, the alternate angles are equal possibly, but not likely Prior Analytics ii.

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If a straight-line intersects two straight-lines and makes interior or exterior angles equal to two right angles on the same side with each, then the lines are parallel possibly Posterior Analytics i. Prior Analytics i. To find the mean proportion of two lines De anima ii.

## History Of Mathematics

If from two points equal lines are drawn to meet and form angles, the locus of points at the angles forms a circle Meteorologica iii. The circle encompasses the greatest area for a given circumference, possibly Posterior Analytics i. To square a lunule, a figure shaped like a crescent formed by the intersection of two circular arcs Prior Analytics ii 25, Sophistici Elenchi 11, Physics i. A spiral which sort? The shape of a square is unaltered when a gnomon is added, but a rectangle's shape is altered, where a gnomon has the shape of a carpenter's square; about a unit you add three units to get a 2 by 2 square, and about two units you add four units to get a 3 by 2 square Categories 14, Physics iii.

Two spheres rotating in different directions presumably about axes which do not coincide , with one carrying the other, produce a non-uniform motion De gen. Metaphysics iii. In a parallelogram, a line drawn parallel to a side through two sides cuts the area and the side in the same ratio Topics viii. Posterior Analytics i. Olympus, and for reasons evident in the proof, which is otherwise based on a false theory of reflection Meteorologica iii.

De anima iii. The side and diagonal of a square are incommensurable from showing that odd numbers would otherwise be equal to even numbers ver frequent, but cf. Peculiar Claims Not every claim in Aristotle's corpus would we regard as felicitous. Original Mathematics Few today would credit Plato with original mathematics. Open access to the SEP is made possible by a world-wide funding initiative.

Mirror Sites View this site from another server:. The traitor was thrown into deep waters and drowned. This episode is sometimes referred to as the first martyr of science. However, we could also think about this person as one of the many martyrs of superstition, since it was not the scientific aspect of irrational numbers that was the root cause of this homicide, but rather its religious extrapolations that were seen as a threat to the foundation of Pythagorean mysticism. The crisis of irrational numbers encouraged the creation of clever methods of approximation of the value of the square root of 2.

One of the best examples of these is the method described in the following chart:.

### Aristotle and Greek Mathematics

After many unsuccessful attempts in finding the value of the square root of 2, the Greeks had no choice but to accept that arithmetic could not be the basis of mathematics. They had to look somewhere else, so they looked into geometry. Euclid c. He was familiar with all of the Greek mathematical work that had preceded him, so he decided to organize all this knowledge in a single coherent work. This work has come down to us known as The Elements , and is the second best selling book of all times, surpassed only by the Bible. The Elements is remembered mostly for its geometry.

The opening of Book I begins with different definitions on basic geometry:. A point is that which has no part. A line is breadthless length. The extremities of a line are points. A straight line is a line which lies evenly with the points on itself. A surface is that which has length and breadth only.

The extremities of a surface are lines. There is nothing original to Euclid in the contents of The Elements he was just a compiler.

It is without a doubt one of the most important and influential books ever written and a masterpiece of the Greek intellectual tradition. From the standpoint of modern scientific knowledge, The Elements has some flaws. First, it relies solely on deduction building conclusions on an assumed set of self-evident generalizations , not a trace of induction starting with observations of particular facts and deriving generalizations from them is to be found in it. Second, it follows a logical sequence by which all theorems in it could be proved through the use of theorems previously proven.

This logical sequence leads us to a set of initial assumptions that cannot be proved. These assumptions are presented by Euclid as unquestionable, which means that they are so obvious that no proof is needed. An analogy of this structure would be a chain where each link is required to be connected to another link, but the initial links are simply hanging, connected nowhere. In addition to the value of the square root of 2, there was also another famous problem that occupied the Greeks: the duplication of the cube. The legend says that:. The oracle of Apollo told the people of Delfos that, to be freed from a plague, they should build him an altar twice the size of the existing one.

Architects had no idea on how to solve this. The right way to approach this problem is to ask: What length should each of the sides of the new altar be if we want to make the volume twice as large as the volume of the original altar?

This is about determining the value of the cube root of 2, which is also an irrational number. This issue caused in geometry the same perplexity that the square root of 2 caused in arithmetic. Greek mathematicians, including Plato, took the issue up and worked on it for centuries producing a large amount of admirable work. The central issue here is being able to determine the cube root of 2.

The Greeks understood something that somehow eluded the Egyptians: the importance of mathematical rigour.

## Greek mathematics - Wikipedia

This is a very accurate calculation around half percent error , but mathematically incorrect. For the purposes of Egyptian engineering, however, this half percent error was not actually important, otherwise their impressive monuments would have collapsed long ago. However, ignoring this half percent error neglects a fundamental property of the true value of pi, which is that no fraction can express it.

It is also an irrational number. By using rounded up values, the irrational nature of these numbers was not noticed by the Egyptians. The Greeks were obsessed with mathematical rigour; for them rounding up was not good enough.