Aristarchus of Samos Quotes

Proposition 16. The sun has to the earth a [volume] ratio greater than that which 6859 has to 27, but less than that which 79507 has to 216.

Proposition 10. The sun has to the moon a [volume] ratio greater than that which 5832 has to 1, but less than that which 8000 has to 1.

Proposition 13. The straight line subtending the portion intercepted within the earth's shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the moon move is less than double of the diameter of the moon, but has to it a ratio greater than [2.0] that which 88 has to 45; and it is less than 1/9th part of the diameter of the sun, but has to it a ratio greater than [1/10th] that which 22 has to 225. But it has to the straight line drawn from the centre of the sun at right angles to the axis and meeting the sides of the cone a ratio greater than [0.097] that which 979 has to 10125.

Proposition 5. When the moon appears to us halved, the great circle parallel to the circle which divides the dark and the bright portions in the moon is then in the direction of our eye; that is to say, the great circle parallel to the dividing circle and our eye are in one plane.

Proposition 9. The diameter of the sun is greater than 18 times, but less than 20 times, the diameter of the moon.

Proposition 8. When the sun is totally eclipsed, the sun and the moon are then comprehended by one and the same cone which has its vertex at our eye.

Proposition 17. The diameter of the earth is to the diameter of the moon in a ratio greater than [2.51] that which 108 has to 43, but less than [3.16] that which 60 has to 19.

Proposition 6. The moon moves (in an orbit) lower than (that of) the sun, and, when it is halved, is distant less than a quadrant from the sun.

Proposition 12. The diameter of the circle which divides the dark and the bright portions in the moon is less than the diameter of the moon, but has to it a ratio greater than [0.99] that which 89 has to 90.

Proposition 1. Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres.

Proposition 15. The diameter of the sun has, to the diameter of the earth a ratio greater than [6.3] that which 19 has to 3, but less than [7.2] that which 43 has to 6.

Proposition 18. The earth is to the moon in a [volume] ratio greater than [15.8] that which 1259712 has to 79507, but less than [31.5] that which 216000 has to 6859.

Proposition 14. The straight line joined from the centre of the earth to the centre of the moon has to the straight line cut off from the axis towards the centre of the moon by the straight line subtending the (circumference) within the earth's shadow a ratio greater than that which 675 has to 1.

[Hypotheses]
1. That the Moon receives its light from the sun.
2. That the earth is in the relation of a point and centre to the sphere in which the moon moves.
3. That, when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye.
4. That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant.
5. That the breadth of the (earth's) shadow is (that) of two moons.
6. That the moon subtends one fifteenth part of a sign of the zodiac.

Proposition 3. The circle in the moon which divides the dark and the bright portions is least when the cone comprehending both the sun and the moon has its vertex at our eye.