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Euclid's Elements

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Euclid's Elements

The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid. The Elements is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus, and Theaetetus, the Elements is a collection in 13 books of definitions, postulates, geometric constructions, and theorems with their proofs that covers plane and solid Euclidean geometry, elementary number theory, and incommensurability. These include the Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the construction of regular polygons and polyhedra. Often referred to as the most successful textbook ever written, the Elements has continued to be used for introductory geometry. It was translated into Arabic and Latin in the medieval period, where it exerted a great deal of influence on mathematics in the medieval Islamic world and in Western Europe, and has proven instrumental in the development of logic and modern science, where its logical rigor was not surpassed until the 19th century.

Infobox

Author
Euclid
Language
Ancient Greek
Subject
Euclidean geometry, number theory, incommensurability
Genre
Mathematics
Publication date
c. 300 BC
Pages
13 books

Tables

Summary Contents of Euclid's Elements (Heath edition) · Contents
Definitions
Definitions
Book
Definitions
I
23
II
2
III
11
IV
7
V
18
VI
4
VII
22
VIII
–
IX
–
X
16
XI
28
XII
–
XIII
–
Totals
131
Postulates
Postulates
Book
Postulates
I
5
II
–
III
–
IV
–
V
–
VI
–
VII
–
VIII
–
IX
–
X
–
XI
–
XII
–
XIII
–
Totals
5
Common Notions
Common Notions
Book
Common Notions
I
5
II
–
III
–
IV
–
V
–
VI
–
VII
–
VIII
–
IX
–
X
–
XI
–
XII
–
XIII
–
Totals
5
Propositions
Propositions
Book
Propositions
I
48
II
14
III
37
IV
16
V
25
VI
33
VII
39
VIII
27
IX
36
X
115
XI
39
XII
18
XIII
18
Totals
465
Book
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
Totals
Definitions
23
2
11
7
18
4
22
–
–
16
28
–
–
131
Postulates
5
–
–
–
–
–
–
–
–
–
–
–
–
5
Common Notions
5
–
–
–
–
–
–
–
–
–
–
–
–
5
Propositions
48
14
37
16
25
33
39
27
36
115
39
18
18
465
Euclid's } }common notions[19] · Contents
Let the following be postulated:
Let the following be postulated:
No.
Let the following be postulated:
1
1
No.
1
Postulates
To draw a straight line from any point to any point.
2
2
No.
2
Postulates
To produce a finite straight line continuously in a straight line
3
3
No.
3
Postulates
To describe a circle with any centre and distance
4
4
No.
4
Postulates
That all right angles are equal to one another
5
5
No.
5
Postulates
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles
No.
No.
No.
No.
Postulates
Common notions
1
1
No.
1
Postulates
Things which are equal to the same thing are also equal to one another
2
2
No.
2
Postulates
If equals be added to equals, the wholes are equal
3
3
No.
3
Postulates
If equals be subtracted from equals, the remainders are equal
4
4
No.
4
Postulates
Things which coincide with one another are equal to one another
5
5
No.
5
Postulates
The whole is greater than the part
No.
Postulates
Let the following be postulated:
1
To draw a straight line from any point to any point.
2
To produce a finite straight line continuously in a straight line
3
To describe a circle with any centre and distance
4
That all right angles are equal to one another
5
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles
No.
Common notions
1
Things which are equal to the same thing are also equal to one another
2
If equals be added to equals, the wholes are equal
3
If equals be subtracted from equals, the remainders are equal
4
Things which coincide with one another are equal to one another
5
The whole is greater than the part

References

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