HISTORY NOTEBOOK
MME 518
Dannette B. Daniels
Thales of Miletus
Born: about 624 BC in Miletus, Asia Minor (now Turkey)Died: about 547 BC in Miletus, Asia Minor (now Turkey)
Little is known of Thales. He was born about 624 BC in Miletus, Asia Minor (now Turkey) and died about 546 BC in Miletos, Turkey
Some impression and highlights of his life and work follow:
· Thales of Miletus was the first known Greek philosopher, scientist and mathematician. Some consider him to be the teacher of of Pythagoras, though it may be only that he advised Pythagoras to travel to Egypt and Chaldea.
· From Eudemus of Rhodes (fl ca. 320 B.C) we know that he studied in Egypt and brought these teachings to Greece. He is unanimously ascribed the introduction of mathematical and astronomical sciences into Greece.
· He is unanimously regarded as having been unusally clever--by general agreement the first of the Seven Wise Men, a pupil of the Egyptians and the Chaldeans.
· None of his writing survives; this makes it is difficult to determine his philosophy and to be certain about his mathematical discoveries.
· There is, of course, the story of his successful speculation in oil presses -- as testament to his practical business acumen.
· It is reported that he predicted an eclipse of the Sun on May 28, 585 BC, startling all of Ionia.
· He is credited with five theorems of elementary geometry.
Eratosthenes of Cyrene
Born: 276 BC in Cyrene, North Africa (now Shahhat, Libya)Died: 194 BC in Alexandria, Egypt
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The man who first measured the world, the Greek astronomer Eratosthenes (c. 276-196 B.C.), lived in Alexandria during the 3rd century B.C. He noticed that on the first day of summer in Syene (now Aswan), Egypt, the Sun appeared directly overhead at noon. At the same time in Alexandria, however, the Sun appeared slightly south (about 7 degrees) of the zenith. Knowing the distance between Syene and Alexandria and assuming that the Sun’s rays were parallel when they struck the curved Earth, he calculated the size of our planet using simple geometry. His result, about 25,000 miles for the circumference, proved remarkably accurate.
Eratosthenes wasn’t the only Greek who tried to measure the Earth. About a century later, Posidonius copied this feat, using the star Canopus as his light source and the cities of Rhodes and Alexandria as his baseline. Although his technique was sound, he had the wrong value for the distance between Rhodes and Alexandria, so his circumference came out too small. Ptolemy recorded this smaller figure in his geography treatise, where it was seized upon by Renaissance explorers looking for a quicker way to the Indies. Had Ptolemy used Eratosthenes’ larger figure instead, Columbus might never have sailed west.
Euclid of Alexandria
Born: about 325 BCDied: about 265 BC in Alexandria, Egypt
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Greek geometer who wrote the Elements , the world's most definitive text on geometry. The book synthesized earlier knowledge about geometry, and was used for centuries in western Europe as a geometry textbook. The text began with definitions, postulates ("Euclid's postulates "), and common opinions, then proceeded to obtain results by rigorous geometric proof. Euclid also proved what is generally known as Euclid's second theorem: the number of primes is infinite. The beautiful proof Euclid gave of this theorem is still a gem and is generally acknowledged to be one of the "classic" proofs of all times in terms of its conciseness and clarity. In the Elements , Euclid used the method of exhaustion and reductio ad absurdum. He also discussed the so-called Euclidean algorithm for finding the greatest common divisor of two numbers, and is credited with the well-known proof of the Pythagorean theorem.
Neither the year nor place of his birth have been established, nor the circumstances of his death, although he is known to have lived and worked in Alexandria for much of his life. In addition, no bust which can be verified to be his likeness is known (Tietze 1965, p.8).
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. Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [1] or [9] or many other sources):-
Pythagoras of Samos
Born: about 569 BC in Samos, IoniaDied: about 475 BC
Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.
Of Pythagoras's actual work nothing is known. His school practised secrecy and communalism making it hard to distinguish between the work of Pythagoras and that of his followers. Certainly his school made outstanding contributions to mathematics, and it is possible to be fairly certain about some of Pythagoras's mathematical contributions. First we should be clear in what sense Pythagoras and the mathematikoi were studying mathematics. They were not acting as a mathematics research group does in a modern university or other institution. There were no 'open problems' for them to solve, and they were not in any sense interested in trying to formulate or solve mathematical problems.
Rather Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof.
Heath [7] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.
(i) The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans knew the generalisation which states that a polygon with n sides has sum of interior angles 2n - 4 right angles and sum of exterior angles equal to four right angles.
(ii) The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square.
(iii) Constructing figures of a given area and geometrical algebra. For example they solved equations such as a (a - x) = x2 by geometrical means.
(iv) The discovery of irrationals. This is certainly attributed to the Pythagoreans but it does seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number.
(v) The five regular solids. It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two.
(vi) In astronomy Pythagoras taught that the Earth was a sphere at the centre of the Universe. He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star
Apollonius of Perga
Born: about 262 BC in Perga, Pamphylia, Greek Ionia (now Murtina, Antalya, Turkey)Died: about 190 BC in Alexandria, Egypt
The mathematician Apollonius was born in Perga, Pamphylia which today is known as Murtina, or Murtana and is now in Antalya, Turkey. Perga was a centre of culture at this time and it was the place of worship of Queen Artemis, a nature goddess. When he was a young man Apollonius went to Alexandria where he studied under the followers of Euclid and later he taught there. Apollonius visited Pergamum where a university and library similar to Alexandria had been built. Pergamum, today the town of Bergama in the province of Izmir in Turkey, was an ancient Greek city in Mysia. It was situated 25 km from the Aegean Sea on a hill on the northern side of the wide valley of the Caicus River (called the Bakir river today).
While Apollonius was at Pergamum he met Eudemus of Pergamum (not to be confused with Eudemus of Rhodes who wrote the History of Geometry) and also Attalus, who many think must be King Attalus I of Pergamum. In the preface to the second edition of Conics Apollonius addressed Eudemus (see [4] or [5]):-
If you are in good health and things are in other respects as you wish, it is well; with me too things are moderately well. During the time I spent with you at Pergamum I observed your eagerness to become aquatinted with my work in conics.
The only other pieces of information about Apollonius's life is to be found in the prefaces of various books of Conics. We learn that he had a son, also called Apollonius, and in fact his son took the second edition of book two of Conics from Alexandria to Eudemus in Pergamum. We also learn from the preface to this book that Apollonius introduced the geometer Philonides to Eudemus while they were at Ephesus.
We are in a somewhat better state of knowledge concerning the books which Apollonius wrote. Conics was written in eight books but only the first four have survived in Greek. In Arabic, however, the first seven of the eight books of Conics survive.
Apollonius of Perga was known as 'The Great Geometer'. Little is known of his life but his works have had a very great influence on the development of mathematics, in particular his famous book Conics introduced terms which are familiar to us today such as parabola, ellipse and hyperbola.
René Descartes
Born: 31 March 1596 in La Haye (now Descartes),Touraine, FranceDied: 11 Feb 1650 in Stockholm, Sweden
René Descartes was a philosopher whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry.
We may consider Descartes as the first of the modern school of mathematics. René Descartes was born near Tours on March 31, 1596, and died at Stockholm on February 11, 1650; thus he was a contemporary of Galileo and Desargues. His father, who, as the name implies, was of good family, was accustomed to spend half the year at Rennes when the local parliament, in which he held a commission as councillor, was in session, and the rest of the time on his family estate of Les Cartes at La Haye. René, the second of a family of two sons and one daughter, was sent at the age of eight years to the Jesuit School at La Flêche, and of the admirable discipline and education there given he speaks most highly. On account of his delicate health he was permitted to lie in bed till late in the mornings; this was a custom which he always followed, and when he visited Pascal in 1647 he told him that the only way to do good work in mathematics and to preserve his health was never to allow anyone to make him get up in the morning before he felt inclined to do so.
Pierre de Fermat
Born: 17 Aug 1601 in Beaumont-de-Lomagne, FranceDied: 12 Jan 1665 in Castres, France
Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- de- Lomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. Although there is little evidence concerning his school education it must have been at the local Franciscan monastery.
While Descartes was laying the foundations of analytical geometry, the same subject was occupying the attention of another and not less distinguished Frenchman. This was Fermat. Pierre de Fermat, who was born near Montauban in 1601, and died at Castres on January 12, 1665, was the son of a leather-merchant; he was educated at home; in 1631 he obtained the post of councillor for the local parliament at Toulouse, and he discharged the duties of the office with scrupulous accuracy and fidelity. There, devoting most of his leisure to mathematics, he spent the remainder of his life - a life which, but for a somewhat acrimonious dispute with Descartes on the validity of certain analysis used by the latter, was unruffled by any event which calls for special notice. The dispute was chiefly due to the obscurity of Descartes, but the tact and courtesy of Fermat brought it to a friendly conclusion. Fermat was a good scholar, and amused himself by conjecturally restoring the work of Apollonius on plane loci.
Except a few isolated papers, Fermat published nothing in his lifetime, and gave no systematic exposition of his methods. Some of the most striking of his results were found after his death on loose sheets of paper or written in the margins of works which he had read and annotated, and are unaccompanied by any proof. It is thus somewhat difficult to estimate the dates and originality of his work. He was constitutionally modest and retiring, and does not seem to have intended his papers to be published. It is probable that he revised his notes as occasion required, and that his published works represent the final form of his researches, and therefore cannot be dated much earlier than 1660. I shall consider separately (i) his investigations in the theory of numbers; (ii) his use in geometry of analysis and of infinitesimals; and (iii) his method for treating questions of probability