The Euclid's Elements of Geometry

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

Book X: Classification of incommensurables, I

Definitions

1. Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.
2. Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.
3. With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square, or in square only, rational, but those that are incommensurable with it irrational.
4. And the let the square on the assigned straight line be called rational, and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.

Propositions

1.  Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out. And the theorem can similarly be proven even if the parts subtracted are halves.
2. If, when the less of two unequal magnitudes is continually subtracted in turn from the greater that which is left never measures the one before it, then the two magnitudes are incommensurable.
3. To find the greatest common measure of two given commensurable magnitudes.
   Corollary: If a magnitude measures two magnitudes, then it also measures theirgreatest common measure.
4. To find the greatest common measure of three given commensurable magnitudes.
   Corollary: If a magnitude measures three magnitudes, then it also measures theirgreatest common measure. The greatest common measure can be found similarly formore magnitudes, and the corollary extended.
5. Commensurable magnitudes have to one another the ratio which a number has to a number.
6. If two magnitudes have to one another the ratio which a number has to a number, then the magnitudes are commensurable.
7. Incommensurable magnitudes do not have to one another the ratio which a number has to a number.
8. If two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable.
9. The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number also have their sides commensurable in length. But the squares on straight lines incommensurable in length do not have to one another the ratio which a square number has to a square number; and squares which do not have to one another the ratio which a square number has to a square number also do not have their sides commensurable in length either.
   Corollary: Straight lines commensurable in length are always commensurable insquare also, but those commensurable in square are not always also commensurablein length. Lemma: Similar plane numbers have to one another the ratio which a square numberhas to a square number, and if two numbers have to one another the ratio which asquare number has to a square number, then they are similar plane numbers. Corollary: Numbers which are not similar plane numbers, that is, those which donot have their sides proportional, do not have to one another the ratio which asquare number has to a square number
10. To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line.
11. If four magnitudes are proportional, and the first is commensurable with the second, then the third also is commensurable with the fourth; but, if the first is incommensurable with the second, then the third also is incommensurable with the fourth.
12. Magnitudes commensurable with the same magnitude are also commensurable with one another.
13. If two magnitudes are commensurable, and one of them is incommensurable with any magnitude, then the remaining one is also incommensurable with the same. Lemma: Given two unequal straight lines, to find by what square the square on the greater is greater than the square on the less. And, given two straight lines, to find the straight line the square on which equals the sum of the squares on them.
14. If four straight lines are proportional, and the square on the first is greater than the square on the second by the square on a straight line commensurable with the first, then the square on the third is also greater than the square on the fourth by the square on a third line commensurable with the third. And, if the square on the first is greater than the square on the second by the square on a straight line incommensurable with the first, then the square on the third is also greater than the square on the fourth by the square on a third line incommensurable with the third.
15. If two commensurable magnitudes are added together, then the whole is also commensurable with each of them; and, if the whole is commensurable with one of them, then the original magnitudes are also commensurable.
16. If two incommensurable magnitudes are added together, the sum is also incommensurable with each of them; but, if the sum is incommensurable with one of them, then the original magnitudes are also incommensurable. Lemma: If to any straight line there is applied a parallelogram but fallingshort by a square, then the applied parallelogram equals the rectangle containedby the segments of the straight line resulting from the application.
17. If there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less but falling short by a square, and if it divides it into parts commensurable in length, then the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater. And if the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square on the less falling short by a square, then it divides it into parts commensurable in length.
18. If there are two unequal straight lines, and to the greater there is applied a parallelogram equal to the fourth part of the square on the less but falling short by a square, and if it divides it into incommensurable parts, then the square on the greater is greater than the square on the less by the square on a straight line incommensurable with the greater. And if the square on the greater is greater than the square on the less by the square on a straight line incommensurable with the greater, and if there is applied to the greater a parallelogram equal to the fourth part of the square on the less but falling short by a square, then it divides it into incommensurable parts.
19. The rectangle contained by rational straight lines commensurable in length is rational.
20. If a rational area is applied to a rational straight line, then it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied.
21. The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called medial. Lemma: If there are two straight lines, then the first is to the second as thesquare on the first is to the rectangle contained by the two straight lines.
22. The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied.
23. A straight line commensurable with a medial straight line is medial.
    Corollary: An area commensurable with a medial area is medial.
24. The rectangle contained by medial straight lines commensurable in length is medial.
25. The rectangle contained by medial straight lines commensurable in square only is either rational or medial.
26. A medial area does not exceed a medial area by a rational area.
27. To find medial straight lines commensurable in square only which contain a rational rectangle.
28. To find medial straight lines commensurable in square only which contain a medial rectangle. Lemma: To find two square numbers such that their sum is also square. Lemma: To find two square numbers such that their sum is not square.
29. To find two rational straight lines commensurable in square only such that the square on the greater is greater than the square on the less by the square on a straight line commensurable in length with the greater.
30. To find two rational straight lines commensurable in square only such that the square on the greater is greater than the square on the less by the square on a straight line incommensurable in length with the greater.
31. To find two medial straight lines commensurable in square only, containing a rational rectangle, such that the square on the greater is greater than the square on the less by the square on a straight line commensurable in length with the greater.
32. To find two medial straight lines commensurable in square only, containing a medial rectangle, such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater.
33. To find two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.
34. To find two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.
35. To find two straight lines incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them.
36. If two rational straight lines commensurable in square only are added together, then the whole is irrational; let it be called binomial.
37. If two medial straight lines commensurable in square only and containing a rational rectangle are added together, the whole is irrational; let it be called the first bimedial straight line.
38. If two medial straight lines commensurable in square only and containing a medial rectangle are added together, then the whole is irrational; let it be called the second bimedial straight line.
39. If two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial are added together, then the whole straight line is irrational; let it be called major.
40. If two straight lines incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational are added together, then the whole straight line is irrational; let it be called the side of a rational plus a medial area.
41. If two straight lines incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and also incommensurable with the sum of the squares on them are added together, then the whole straight line is irrational; let it be called the side of the sum of two medial areas.
42. A binomial straight line is divided into its terms at one point only.
43. A first bimedial straight line is divided at one and the same point only.
44. A second bimedial straight line is divided at one point only.
45. A major straight line is divided at one point only.
46. The side of a rational plus a medial area is divided at one point only.
47. The side of the sum of two medial areas is divided at one point only.