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(1)

Euclid

(2)

• Proclus gave information about Euclid:

• “Not much younger then these (Hermotimus of Colophon and Philippus of Mende or Medma) is Euclid, who put together the Elements,

collecting many of Eudoxus’s theorems, perfecting many of Theaetetus’s and also bringing irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid; and further they say that

Ptolemy once asked him if there was in geometry any shorter way than that of the Elements, and he replied that there was no royal road to geometry. He is then younger than the pupils of Plato, but older than Eratosthenes and Archimedes, the letter having been contemporaries, as Eratosthenes somewhere says.” Proclus on Euclid.

(3)

• Proclus says that Euclid is a Platonist.

• It may have been but it is not certain.

• In any case, it is possible that he received his mathematical training in Athens from the

pupils of Plato.

(4)

• Euclid found his school at Alexandria.

• Euclid has always been known almost

exclusively as the author of the Elements.

From Archimedes onwards the Greeks

commonly spokes of him as the writer of the

Elements, instead of using his name.

(5)

• Euclid was the first philosopher to apply the bold remedy of laying down the indispensable principle of the theory in the form of an

indemonstrable postulate.

(6)

Translations of Elements

• Caliph al-Mamun obtained many books from Byzantine Emperor. Euclid’s book of Elements was among them. After that many Arabian

scholars translated it from Greek to Arabic.

• The known Latin translation begin with that Adelard of Bath; the date of it is about 1120.

• Gerard of Cramona (1114-1187) translated

book of Euclid from Arabic.

(7)

Elements

• Consisting of 13 books.

• The first book of the Elements necessarily begin with headings Definitions, Postulates and Common Notions. In calling the axioms Common Notions Euclid followed the lead of Aristotle, who uses the alternatives for

“axioms” the terms ‘common (things),

‘common opinions’.

(8)

• The propositions of Book I fall into three distinct groups. The first group consists of Propositions 1-26 dealing with triangles without the uses of parallels.

• The second group (27-32) includes the theory of Parallels.

• The third group of prepositions (22-48) deals

with parallelograms triangles and squares with

reference to their areas.

(9)

Axiomatic Method

• Starting definitions and propositions in a way

such that new term can be formally eliminated by the priorly introduced requires primitive notions to avoid infinitive regress. This way of doing

mathematics is called the axiomatic methods.

• Although many of Euclid’s results had been stated by earlier mathematicians, Euclid was the first to show these propositions could fit into a

comprehensive deductive and logical system.

(10)

• In the books, there are 131 definitions, five

postulates five common notions and 465

propositions.

(11)

Definitions

• 1. A point is that which has no part.

• 2. A line is breadthless length.

• 3. The ends of a line are points.

• 4. A straight line is a line which lies evenly with the point itself.

• 5. A surface is that which has length and

breadth only.

(12)

Postulates

• 1. To draw a straight line from any point to any point.

• 2. To produce a finite straight line continuously is a straight line.

• 3. To describe a circle with any center and radius.

• 4. That all right angles equal one another.

• 5. Any given point not on a line there passes

exactly one line parallel to that line in the same

plane.

(13)

Common Notions

• 1. Things which equal the same thing also equal one another.

• 2. If equals are added to equals, then the wholes are equal.

• 3. If equals subtract from equals, then the remainders are equal.

• 4. Things which coincide with one another equal one another.

• 5. The whole is greater than the part.

(14)

• Euclidean geometry was the dominant paradigm until the early 19th century.

However, later on, alternative geometries emerged with the work of Carl Friedrich

Gauss, Ferdinand Karl Schweikart, Bolyai, and

Lobachevski.

(15)

• Resource book:

• Thomas Heath, A History of Greek

Mathematics. Vol I.

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