The Euclid's Elements of Geometry

Geometry is the study of figures. Euclid's Elements provides themost fundamental way of learning geometry geometrically.

Book IX: Number theory

Propositions

1.  If two similar plane numbers multiplied by one another make some number, then the product is square.
2. If two numbers multiplied by one another make a square number, then they are similar plane numbers.
3. If a cubic number multiplied by itself makes some number, then the product is a cube.
4. If a cubic number multiplied by a cubic number makes some number, then the product is a cube.
5. If a cubic number multiplied by any number makes a cubic number, then the multiplied number is also cubic.
6. If a number multiplied by itself makes a cubic number, then it itself is also cubic.
7. If a composite number multiplied by any number makes some number, then the product is solid.
8. If as many numbers as we please beginning from a unit are in continued proportion, then the third from the unit is square as are also those which successively leave out one, the fourth is cubic as are also all those which leave out two, and the seventh is at once cubic and square are also those which leave out five.
9. If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is square, then all the rest are also square; and if the number after the unit is cubic, then all the rest are also cubic.
10. If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is not square, then neither is any other square except the third from the unit and all those which leave out one; and, if the number after the unit is not cubic, then neither is any other cubic except the fourth from the unit and all those which leave out two.
11. If as many numbers as we please beginning from a unit are in continued proportion, then the less measures the greater according to some one of the numbers which appear among the proportional numbers. Corollary: Whatever place the measuring number has, reckoned from the unit, thesame place also has the number according to which it measures, reckoned from thenumber measured, in the direction of the number before it.
12. If as many numbers as we please beginning from a unit are in continued proportion, then by whatever prime numbers the last is measured, the next to the unit is also measured by the same.
13. If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is prime, then the greatest is not measured by any except those which have a place among the proportional numbers.
14. If a number is the least that is measured by prime numbers, then it is not measured by any other prime number except those originally measuring it.
15. If three numbers in continued proportion are the least of those which have the same ratio with them, then the sum of any two is relatively prime to the remaining number.
16. If two numbers are relatively prime, then the second is not to any other number as the first is to the second.
17. If there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then the last is not to any other number as the first is to the second.
18. Given two numbers, to investigate whether it is possible to find a third proportional to them.
19. Given three numbers, to investigate when it is possible to find a fourth proportional to them.
20. Prime numbers are more than any assigned multitude of prime numbers.
21. If as many even numbers as we please are added together, then the sum is even.
22. If as many odd numbers as we please are added together, and their multitude is even, then the sum is even.
23. If as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd.
24. If an even number is subtracted from an even number, then the remainder is even.
25. If an odd number is subtracted from an even number, then the remainder is odd.
26. If an odd number is subtracted from an odd number, then the remainder is even.
27. If an even number is subtracted from an odd number, then the remainder is odd.
28. If an odd number is multiplied by an even number, then the product is even.
29. If an odd number is multiplied by an odd number, then the product is odd.
30. If an odd number measures an even number, then it also measures half of it.
31. If an odd number is relatively prime to any number, then it is also relatively prime to double it.
32. Each of the numbers which are continually doubled beginning from a dyad is even-times even only.
33. If a number has its half odd, then it is even-times odd only.
34. If an [even] number neither is one of those which is continually doubled from a dyad, nor has its half odd, then it is both even-times even and even-times odd.
35. If as many numbers as we please are in continued proportion, and there is subtracted from the second and the last numbers equal to the first, then the excess of the second is to the first as the excess of the last is to the sum of all those before it.
36. If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.