Geometry History File

Virginia Rekemeyer 1

MME 518

Geometry – History File

Thales of Miletus

Born: about 624 BC in Miletus, Asia Minor (now Turkey)Died: about 547 BC in Miletus, Asia Minor (now Turkey)

Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer. He is believed to have been the teacher of Anaximander (611 BC - 545 BC) and he was the first natural philosopher in the Milesian School.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Thales.html

The Theorems Attributed to Thales

Five Euclidean theorems have been explicitly attributed to Thales, and the testimony is that Thales successfully applied two theorems to the solution of practical problems. Thales did not formulate proofs in the formal sense. What Thales did was to put forward certain propositions which, it seems, he could have 'proven' by induction: he observed the similar results of his calculations: he showed by repeated experiment that his propositions and theorems were correct, and if none of his calculations resulted in contrary outcomes, he probably felt justified in accepting his results as proof. Thalean 'proof' was often really inductive demonstration. The process Thales used was the method of exhaustion. This seems to be the evidence from Proclus who declared that Thales 'attacked some problems in a general way and others more empirically'.

DEFINITION I.17: A diameter of the circle is a straight line drawn through the centre and terminated in both directions by the circumference of the circle; and such a straight line also bisects the circle (Proclus, 124). >

PROPOSITION I.5: In isosceles triangles the angles at the base are equal; and if the equal straight lines are produced further, the angles under the base will be equal (Proclus, 244). It seems that Thales discovered only the first part of this theorem for Proclus reported: We are indebted to old Thales for the discovery of this and many other theorems. For he, it is said, was the first to notice and assert that in every isosceles the angles at the base are equal, though in somewhat archaic fashion he called the equal angles similar (Proclus, 250.18-251.2).

PROPOSITION I.15: 'If two straight lines cut one another, they make the vertical angles equal to one another' (Proclus, 298.12-13). This theorem is positively attributed to Thales. Proof of the theorem dates from the Elements of Euclid (Proclus, 299.2-5).

PROPOSITION I.26: 'If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle equal to the remaining angle' (Proclus, 347.13-16). 'Eudemus in his history of geometry attributes the theorem itself to Thales, saying that the method by which he is reported to have determined the distance of ships at sea shows that he must have used it' (Proclus, 352.12-15). Thales applied this theorem to determine the height of a pyramid. The great pyramid was already over two thousand years old when Thales visited Gizeh, but its height was not known. Diogenes recorded that 'Hieronymus informs us that [Thales] measured the height of the pyramids by the shadow they cast, taking the observation at the hour when our shadow is of the same length as ourselves' (D.L. I.27). Pliny (HN, XXXVI.XVII.82) and Plutarch (Conv. sept. sap. 147) also recorded versions of the event. Thales was alerted by the similarity of the two triangles, the 'quality of proportionality'. He introduced the concept of ratio, and recognized its application as a general principle. Thales's accomplishment of measuring the height of the pyramid is a beautiful piece of mathematics. It is considered that the general principle in Euclid I.26 was applied to the ship at sea problem, would have general application to other distant objects or land features which posed difficulties in the calculation of their distances.

PROPOSITION III.31: 'The angle in a semicircle is a right angle'. Diogenes Laertius (I.27) recorded: 'Pamphila states that, having learnt geometry from the Egyptians, [Thales] was the first to inscribe a right-angled triangle in a circle, whereupon he sacrificed an ox'. Aristotle was intrigued by the fact that the angle in a semi-circle is always right. In two works, he asked the question: 'Why is the angle in a semicircle always a right angle?' (An. Post. 94 a27-33; Metaph. 1051 a28). Aristotle described the conditions which are necessary if the conclusion is to hold, but did not add anything that assists with this problem.

It is testified that it was from Egypt that Thales acquired the rudiments of geometry. However, the evidence is that the Egyptian skills were in orientation, measurement, and calculation. Thales's unique ability was with the characteristics of lines, angles and circles. He recognized, noticed and apprehended certain principles which he probably 'proved' through repeated demonstration.

http://www.utm.edu/research/iep/t/thales.htm

Eratosthenes of Cyrene

A versatile scholar, Eratosthenes of Cyrene lived approximately 275-195 BC. He was the first to estimate accurately the diameter of the earth. For several decades, he served as the director of the famous library in Alexandria. He was highly regarded in the ancient world, but unfortunately only fragments of his writing have survived. Eratosthenes died at an advanced age from voluntary starvation, induced by despair at his blindness.

http://www.math.utah.edu/~alfeld/Eratosthenes.html

A consummate Greek scholar whose status as second best in each field earned him the nickname "Beta." He served as librarian at the great library in Alexandria, and wrote works of mathematics, geography, philosophy, and astronomy. He also wrote a poem called Hermes which described the fundamentals of astronomy in verse! Although most of Eratosthenes' writings are lost, many are preserved through the writings of commentators.

Among Eratosthenes' accomplishments was the accurate measurement the diameter of the Earth by observing that, on the day of the summer solstice, the Sun was directly overhead in Syene while it was 7° from the zenith in Alexandria, which he assumed was due north of Syene (Dunham 1990). Unfortunately, since the original work On the Measurement of the Earth was lost, the details of Eratosthenes' procedure are not known. Eratosthenes also determined the obliquity of the ecliptic, prepared a star map containing 675 stars, suggested that a leap day be added every fourth year, tried to construct an accurately-dated history, and developed the "sieve of Eratosthenes " method of finding prime numbers. At the age of 80, blind and weary, he died of voluntary starvation.

Euclid of Alexandria

Born: about 325 BCDied: about 265 BC in Alexandria, Egypt

Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements. The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. However little is known of Euclid's life except that he taught at Alexandria in Egypt.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html

Euclid and his Elements

One of the most influential mathematicians of ancient Greece, Euclid flourished around 300 B.C. Not much is known about the life of Euclid. One story which reveals something about Euclid's character concerns a pupil who had just completed his first lesson in geometry. The pupil asked what he would get from learning geometry. So Euclid told his slave to give the pupil a coin so he would be gaining something from his studies. Included in the many works of Euclid is Data, concerning the solution of problems through geometric analysis, On Divisions (of Figures), the Optics, the Phenomena, a treatise on spherical geometry for astronomers, several lost works on higher geometry, and the Elements, a thirteen volume textbook on geometry. [1]

The Elements, which surely became a classic soon after its publication, eventually became the most influential textbook in the history of civilization. In fact, it has been said that apart from the Bible, the Elements is the most widely read and studied book in the world. [2] It has also been said that the Greeks used to post over the doors of their schools the inscription: ``Let no one come to our school who has not learned the Elements of Euclid.'' [3] Probably every great Western mathematician to arise in the last two thousand years has studied Euclid's Elements.

In writing the Elements Euclid collected and extended many of the ideas of other Greek mathematicians before him. The Elements is basically a chain of 465 propositions encompassing most of the geometry, number theory, and geometric algebra of the Greeks up to that time. [4] Book I contains twenty-three definitions, five common notions (axioms), five postulates, and forty-eight propositions of plane geometry.

The definitions of Book I include those of points, lines, planes, angles, circles, triangles, quadrilaterals, and parallel lines.

The five postulates may be translated into the following:

1. Two points determine a straight line.

2. A line segment extended infinitely in both directions produces a straight line.

3. A circle is determined by a center and distance.

4. All right angles are equal to one another.

5. If a straight line falling an two straight lines forms interior angles on the same side less than 180 degrees, the two straight lines, if produced indefinitely, will meet on that side.

The last of these, commonly known as the ``parallel postulate,'' is by far the most important of the five. Through manipulation, the following statement may be derived: ``The sum of the angles in a triangle is equal to 180 degrees.'' Changing ``equal to'' to ``less than'' or ``greater than'' results in entirely different geometries -- non-Euclidean geometries. In spherical geometry, for example, this would read: ``The sum of the angles in a triangle is greater than 180 degrees.'' In hyperbolic geometry it would read: ``The sum of the angles in a triangle is less than 180 degrees.'' Hyperbolic geometry was invented by the Russian mathematician Nicolai Ivanovitch Lobachevsky. [5]

Postulates, by definition, are not and cannot be proven. However, some mathematicians have claimed that postulate four can be proven; [6] and many have believed that postulate five, partly because of its length and complexity, can be proven. [7] Lobachevsky's geometry grew out of his unsuccessful attempts to prove Euclid's parallel postulate. [8] Zeno of Sidon in the first century B.C. believed that Euclid's list of postulates was incomplete. He claimed that one must postulate that two distinct straight lines cannot have a segment in common. Without this, he claimed, some of the propositions in Book I are fallacious. [9]

Unlike the specialized nature of the postulates, the five common notions, or axioms, were essentially taken to be universal truths in all of mathematics and the sciences. The fifth axiom breaks down when exposed to the concept of infinite sets. For example, the set of all integers is not larger than the set of all even integers. [10]

The final section of Book I includes the forty-eight postulates. Included in these are the familiar results on triangles, such as proposition 5 [that the angles at the base of an isosceles triangle are equal], as well as the four congruence theorems for triangles: side-angle-side (prop. 4), side-side-side (prop. 8), angle-side-angle (prop. 26), and side-angle-angle (prop. 26, also). The last two propositions are the Pythagorean theorem and its converse.

http://www.obkb.com/dcljr/euclidhs.html

Pythagoras of Samos

Born: about 569 BC in Samos, Ionia
Died: about 475 BC

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

Pythagoras (fl. 530 BCE) must have been one of the world's greatest men, but he wrote nothing, and it is hard to say how much of the doctrine we know as Pythagorean is due to the founder of the society and how much is later development. It is also hard to say how much of what we are told about the life of Pythagoras is trustworthy; for a mass of legend gathered around his name at an early date. Sometimes he is represented as a man of science, and sometimes as a preacher of mystic doctrines, and we might be tempted to regard one or other of those characters as alone historical. The truth is that there is no need to reject either of the traditional views. The union of mathematical genius and mysticism is common enough. Originally from Samos, Pythagoras founded at Kroton (in southern Italy) a society which was at once a religious community and a scientific school. Such a body was bound to excite jealousy and mistrust, and we hear of many struggles. Pythagoras himself had to flee from Kroton to Metapontion, where he died.

It is stated that he was a disciple of Anaximander, his astronomy was the natural development of Anaximander's. Also, the way in which the Pythagorean geometry developed also bears witness to its descent from that of Miletos. The great problem at this date was the duplication of the square, a problem which gave rise to the theorem of the square on the hypotenuse, commonly known still as the Pythagorean proposition (Euclid, I. 47). If we were right in assuming that Thales worked with the old 3:4:5 triangle, the connection is obvious.