Euclid of Alexandria

Any two sides of a triangle are together greater than the remaining side.

Euclid's Elements, Book I, Proposition 20

In any right-angled triangle the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.

Euclid's Elements, Book I, Proposition 47 (Pythagorean Theorem)

If a number be measured by any prime number, and does not measure any number, it is prime.

Euclid's Elements, Book VII, Definition 11 (Implicit definition of prime)

A perfect number is that which is equal to its own parts.

Euclid's Elements, Book VII, Definition 22

If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and the sum multiplied into the last makes some number, the product will be perfect.

Euclid's Elements, Book IX, Proposition 36

To draw a straight line at right angles to a given straight line from a given point on it.

Euclid's Elements, Book I, Proposition 11

To draw a straight line parallel to a given straight line through a given point not on it.

Euclid's Elements, Book I, Proposition 31

Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

Euclid's Elements, Book V, Definition 4 (Archimedean Property)

The greatest common divisor of two numbers can be found by repeatedly subtracting the smaller from the larger.

Euclid's Elements, Book VII, Proposition 2 (Euclidean Algorithm)

If a straight line be cut in extreme and mean ratio, the whole is to the greater segment as the greater segment is to the lesser.

Euclid's Elements, Book VI, Definition 3 (Golden Ratio)

To construct a regular 15-gon in a given circle.

Euclid's Elements, Book IV, Proposition 16

The angles in the same segment are equal to one another.

Euclid's Elements, Book III, Proposition 21

In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle.

Euclid's Elements, Book III, Proposition 31

If two circles touch one another internally, and there be taken a point on the circumference of the inner circle, and from it there be drawn a straight line to the point of contact, and from the same point there be drawn another straight line to the circumference of the outer circle, the straight lines so drawn will be in the same ratio as the diameters of the circles.

Euclid's Elements, Book III, Proposition 11 (Implied, often paraphrased)

If a straight line touch a circle, and from the point of contact there be drawn across in the circle a straight line cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle.

Euclid's Elements, Book III, Proposition 32

To find the center of a given circle.

Euclid's Elements, Book III, Proposition 1

To cut a given straight line into two parts such that the rectangle contained by the whole and one of the parts is equal to the square on the remaining part.

Euclid's Elements, Book II, Proposition 11 (Golden Ratio construction)

Similar polygons are to one another in the duplicate ratio of their corresponding sides.

Euclid's Elements, Book VI, Proposition 20

The areas of circles are to one another as the squares on their diameters.

Euclid's Elements, Book XII, Proposition 2

The volumes of spheres are to one another as the cubes on their diameters.

Euclid's Elements, Book XII, Proposition 18

Quotes by Euclid of Alexandria