The Greeks

History of Mathematics

• In 800-650 BC ,Greece consisted of Italy all the way to the edge of the Black sea. This colonization affected Greek mathematics in that all early Greek mathematics came from Asia minor, southern Italy, & Africa.

Greece

• Coinage affected the Greeks - stimulated trade, gave rise to a money economy based not only on agriculture but on movable goods, created a leisure class.

• Geography of Greece affected politics - no single ruler because each state separated by mountains or sea. Alexander eventually united Greece. (He died in 323 BC ruling over 2 million square miles – gave rise to the Hellenistic age

• Greek mathematics differ from the Babylonians and the Egyptians - more profound, more abstract, and more rational. Notion of seeking knowledge for its own sake.

• Greek civilization begins in Iron Age, flourishing in the 4th and 5th centuries BC. Most of the productive mathematics done in 350-200 BC.

Greece

• Thales is credited with inventing the modern proof.• Thales is known as the first of the Seven Sages of Greece; coined the

maxim “Know thyself” and when asked what was the strangest thing he had ever seen: “An aged tyrant.”

• Thales presumably got interested in geometry when he visited Egypt and wanted to measure the pyramids by means of shadows. Thales also determined how far a ship at sea is from the shore.

• It is said that Thales predicted a solar eclipse but this would not have been possible because astronomical records were not precise enough at that time.

Thales (625-547 BC)

Pythagoras• Pythagoras (580-500 BC) followed Thales. We know that he lived on the

Aegan island of Samos.• The distinctive feature of the school founded by Pythagoras was its aims

were political, philosophical, and religious.• The 300 or so students at the school studied the Quadrivium (number

theory, music, geometry, and astronomy). • What set the Pythagoreans apart from other sects was their belief that

knowledge is the greatest purification and to them knowledge meant mathematics.

• Mathematics played an essential part in life & religion. We saw last time that music provided the Pythagoreans with the best instance of their principle that “number was the cause of everything in nature.”

• Nichomachus’ Introductio Arithmeticae is the most complete exposition of Pythagoras and his successors work.

• The Pythagoreans did not have any symbols so it’s likely that they used pebbles to think of numbers geometrically – triangle numbers, square numbers, etc.

• The most important achievements of the Pythagorean school in its influence of the number concept was the discovery of the “irrational.”

PythagoreansPythagoreans Celebrate Sunrise by Fyodor Bronnikov

• This idea caused great consternation among the Pythagoreans for it challenged the adequacy of their philosophy that number was the essence of all things. They called these numbers “unutterable.

• Eventually Eudoxus (408-355BC) resolved this crisis. He revised the theory of proportion applicable to incommensurable as well commensurable quantities

• Hippias(b. c. 460BC) is credited with discovering that the side of a square and its diagonal are not commensurable.

• i.e., no matter how small one goes, there is no magnitude dividing evenly into both the square’s side and its diagonal. This discovery shattered all Pythagorean proofs using commensurability of all segment.

• Examples…

Commensurable

Eudoxus (408-355 BC)• Eudoxus eventually gave proofs about similar triangles that did not rely on this concept. Also it had an unsettling impact upon the supremacy of whole numbers, for if not all quantities were commensurable then, then whole numbers were somehow inadequate to represent geometric lengths. This firmly established the supremacy of geometry over arithmetic in all subsequent Greek mathematics. For example, in a square of side length 1, with diagonal root 2, then root 2 arithmetic character is mysterious so better to think of magnitudes as geometric entries!

• Eudoxus is best known for 3 things: his general theory of proportion, golden ratio, & method of exhaustion, which Archimedes refines.

Approximating root 2

• The Greeks were not the first to consider numerical values for root 2. • The Babylonians approximated the square root of 2.• Here the Babylonian clay tablet YBC 7289 (c. 1800–1600 BCE) gives an

approximation of in four sexagesimal figures, which is about six decimal figures:1 + 24/60 + 51/(60)^2 + 10/(60)^3 = 1.41421296

Hippocrates(460-380 BC)• The mathematician that dominated the 2nd half of the 5th C BC was

Hippocrates. • Hippocrates began his life as a merchant but ended his life as a teacher

(like Thales) . He was robbed of his money (he was robbed by either pirates or fraudulent customs officials) and went to Athens to prosecute the offenders. He stayed and attended the lecturers of many philosophers. He may have come in contact with some Pythagoreans.

• The major accomplishment of Hippocrates is that he was the first to write a systematically organized geometry textbook, called Stoicheia Elements, that is, basic theorems, or building blocks of mathematical theory.

• From then on, mathematicians from all over the ancient world could, at least in principle, build on a common framework of basic concepts, methods, and theorems, which stimulated the scientific progress of mathematics.

Special Problems from Antiquity• Three problems that Hippocrates faced when he arrived in Athens:

– Quadrature of the circle: Is it possible to construct a square whose area is equal to that of a given circle? Equivalent to constructing a line segment whose length is root pi times the radius of the circle.

– Duplication of the cube: Find the edge of a cube, having twice the volume of a given cube.

– Trisection of an angle.• Add a 4th problem: Construction of a regular heptagon.• Plato insisted 3 things with these special problems: the task be

performed with straightedge and compass only.– With the straight edge, a line can be drawn through 2 points and – With the compass, a circle with a given center and radius can be

drawn.

Squaring a Circle• Surprisingly, the impossibility proofs for the three problems: doubling the

cube, trisecting an angle, and constructing a regular heptagon, all fall into the same framework. Each depends on showing that roots of certain third degree polynomial equations are not constructible.

• The problem of squaring of a circle found itself in a completely different category. Carl Louis Ferdinand von Lindemann (1852 – 1939) was a German mathematician who was the first to prove that π is transcendental in 1882. Since π is NOT algebraic, it is automatically not constructible. The area of a circle with radius 1 is exactly π . If a square with such area were constructible, so would be the number itself. The areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with a compass and straightedge.

Doubling the Cube• Doubling the Cube, the most famous of the collection, is often referred to

as the Delian problem due to a legend that the Delians had consulted Plato on the subject.

• In another form, the story asserts that the Athenians in 430 B.C. consulted the oracle at Delos in the hope to stop the plague ravaging their country. They were advised by Apollo to double his altar that had the form of a cube. As a result of several failed attempts to satisfy the god, the pestilence only worsened and at the end they turned to Plato for advice.

• According to a letter from the mathematician Eratosthenes(276-195 BC) to King Ptolemy of Egypt, Euripides mentioned the Delian problem in one of his (now lost) tragedies.

• Euripides (480 BC – 406 BC) was the last of the three great tragedians of classical Athens (the other two were Aeschylus and Sophocles).

Statue of Euripides in Louvre, Paris

• To "double the cube" means to be given a cube of some side length s and volume V, and to construct a new cube, larger than the first, with volume 2V and therefore side length

• The problem is known to be impossible to solve with only compass and straightedge, because the cube root of 2≈ 1.25992105 is not a constructible number.

Eratosthenes

Trisecting an Angle• That there may be a difficulty trisecting

an angle is rather surprising. Bisecting is easy! The same easy approach applies to bisecting both straight line segments and circular arcs.

• The problem of trisecting a line segment is no more difficult than finding its n-th part for an arbitrary n. However, the general problem of trisecting an arbitrary angle is not solvable in a finite number of steps.

Two points must be made.– First, some angles are trisectable with a ruler and a compass. An obvious

example is supplied by the three quarters of the straight angle - 67.5o. (The angle itself is constructible as it is obtained by two consecutive angle bisections. Its third is obtained along the way.) Angles of 30o (draw a right triangle with a side 1 and hypotenuse 2) and 45o (bisect the right angle) are both constructible. Therefore, the latter also admits a classical trisection. However, and this is where the difficulty lies, an arbitrary angle can't be trisected with a ruler and a compass, as stated above.

– Secondly, consider the geometric series 1/4 + 1/16 + 1/64 + ... that adds up to 1/3. Once we know how to bisect an angle, we may also find its 2-n-th part, for any n. In particular, one may construct 1/4, 1/16, ... of any angle and, in principle, find its third after an infinite number of steps. This solution is universal but requires a forbidden (infinite) number of steps.

• The problem was settled in 1837 by Pierre Laurent Wantzel (1814-1848) who had proven that there was no way to trisect a 60o angle in the classical framework.

Constructing a Regular Heptagon• 3-, 4-, 5-, and 6-gons (i.e., regular triangles, squares, pentagons, and

hexagons) are all easily constructible. Octagons are constructible on the heels of squares with a single angle bisection. All polygons obtained from the above four by doubling the number of sides are also constructible. Not so for a heptagon, a 7-sided polygon.

• In 1796, at the age of 19, Gauss showed that a regular heptadecagon (a 17-sided polygon) is constructible. Gauss showed that the heptadecagon was only a particular case of a family of constructible regular polygons. Any N-gon, where N is in the form 2np1...pm, where pi's are distinct Fermat primes (i.e. primes in the form 22k + 1), is constructible. Wantzel later proved that no other regular polygon has this property.

The Lune of Hippocrates • Some apparent partial solutions gave false hope in the “Squaring the

Circle” Problem for a long time. In this figure, the shaded figure is the Lune of Hippocrates. Its area is equal to the area of the triangle ABC.

• This is not a complete solution of the task (obviously since the complete solution is proven to be impossible).

• Explanation….