- Proposition 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.
- Proposition 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.
- Proposition 3.
- To find the greatest common measure of two given commensurable magnitudes.
Corollary. If a magnitude measures two magnitudes, then it also measures their greatest common measure.
- Proposition 4.
- To find the greatest common measure of three given commensurable magnitudes.
Corollary. If a magnitude measures three magnitudes, then it also measures their greatest common measure. The greatest common measure can be found similarly for more magnitudes, and the corollary extended.
- Proposition 5.
- Commensurable magnitudes have to one another the ratio which a number has to a number.
- Proposition 6.
- If two magnitudes have to one another the ratio which a number has to a number, then the magnitudes are commensurable.
Corollary.
- Proposition 7.
- Incommensurable magnitudes do not have to one another the ratio which a number has to a number.
- Proposition 8.
- If two magnitudes do not have to one another the ratio which a number has to a number, then the magnitudes are incommensurable.
- Proposition 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 in square also, but those commensurable in square are not always also commensurable in length.
Lemma. Similar plane numbers have to one another the ratio which a square number has to a square number, and if two numbers have to one another the ratio which a square number has to a square number, then they are similar plane numbers.
Corollary 2. Numbers which are not similar plane numbers, that is, those which do not have their sides proportional, do not have to one another the ratio which a square number has to a square number
- Proposition 10.
- To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line.
- Proposition 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.
- Proposition 12.
- Magnitudes commensurable with the same magnitude are also commensurable with one another.
- Proposition 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.
- Proposition 14.
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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.
Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 17.
- Lemma. If to any straight line there is applied a parallelogram but falling short by a square, then the applied parallelogram equals the rectangle contained by the segments of the straight line resulting from the application.
Proposition 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.
- Proposition 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.
- Proposition 19.
- Lemma.
Proposition 19. The rectangle contained by rational straight lines commensurable in length is rational.
- Proposition 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.
- Proposition 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.
- Proposition 22.
- Lemma. If there are two straight lines, then the first is to the second as the square on the first is to the rectangle contained by the two straight lines.
Proposition 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.
- Proposition 23.
- A straight line commensurable with a medial straight line is medial.
Corollary. An area commensurable with a medial area is medial.
- Proposition 24.
- The rectangle contained by medial straight lines commensurable in length is medial.
- Proposition 25.
- The rectangle contained by medial straight lines commensurable in square only is either rational or medial.
- Proposition 26.
- A medial area does not exceed a medial area by a rational area.
- Proposition 27.
- To find medial straight lines commensurable in square only which contain a rational rectangle.
- Proposition 28.
- To find medial straight lines commensurable in square only which contain a medial rectangle.
- Proposition 29.
- Lemma 1. To find two square numbers such that their sum is also square.
Lemma 2. To find two square numbers such that their sum is not square.
Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 33.
- Lemma.
Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 36.
- If two rational straight lines commensurable in square only are added together, then the whole is irrational; let it be called binomial.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.
- Proposition 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.
Lemma.
- Proposition 42.
- A binomial straight line is divided into its terms at one point only.
- Proposition 43.
- A first bimedial straight line is divided at one and the same point only.
- Proposition 44.
- A second bimedial straight line is divided at one point only.
- Proposition 45.
- A major straight line is divided at one point only.
- Proposition 46.
- The side of a rational plus a medial area is divided at one point only.
- Proposition 47.
- The side of the sum of two medial areas is divided at one point only.