BOOK X.
DEFINITIONS I.
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 which are incommensurable with it irrational.4
And 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 I—47.
PROPOSITION 1.
Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out. Let AB, C be two unequal magnitudes of which AB is the greater: I say that, if from AB there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the magnitude C. For C if multiplied will sometime be greater than AB. [cf. v. Def. 4] Let it be multiplied, and let DE be a multiple of C, and greater than. AB; let DE be divided into the parts DF, FG, GE equal to C, from AB let there be subtracted BH greater than its half, and, from AH, HK greater than its half, and let this process be repeated continually until the divisions in AB are equal in multitude with the divisions in DE. Let, then, AK, KH, HB be divisions which are equal in multitude with DF, FG, GE. Now, since DE is greater than AB, and from DE there has been subtracted EG less than its half, and, from AB, BH greater than its half, therefore the remainder GD is greater than the remainder HA. And, since GD is greater than HA, and there has been subtracted, from GD, the half GF, and, from HA, HK greater than its half, therefore the remainder DF is greater than the remainder AK. But DF is equal to C; therefore C is also greater than AK. Therefore AK is less than C. Therefore there is left of the magnitude AB the magnitude AK which is less than the lesser magnitude set out, namely C. Q. E. D. And the theorem can be similarly proved even if the parts subtracted be 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, the magnitudes will be incommensurable. For, there being two unequal magnitudes AB, CD, and AB being the less, when the less is continually subtracted in turn from the greater, let that which is left over never measure the one before it; I say that the magnitudes AB, CD are incommensurable. For, if they are commensurable, some magnitude will measure them. Let a magnitude measure them, if possible, and let it be E; let AB, measuring FD, leave CF less than itself, let CF measuring BG, leave AG less than itself, and let this process be repeated continually, until there is left some magnitude which is less than E. Suppose this done, and let there be left AG less than E. Then, since E measures AB, while AB measures DF, therefore E will also measure FD. But it measures the whole CD also; therefore it will also measure the remainder CF. But CF measures BG; therefore E also measures BG. But it measures the whole AB also; therefore it will also measure the remainder AG, the greater the less: which is impossible. Therefore no magnitude will measure the magnitudes AB, CD; therefore the magnitudes AB, CD are incommensurable. Therefore etc. [X. Def. 1]PROPOSITION 3.
Given two commensurable magnitudes, to find their greatest common measure. Let the two given commensurable magnitudes be AB, CD of which AB is the less; thus it is required to find the greatest common measure of AB, CD. Now the magnitude AB either measures CD or it does not. If then it measures it—and it measures itself also—AB is a common measure of AB, CD. And it is manifest that it is also the greatest; for a greater magnitude than the magnitude AB will not measure AB. Next, let AB not measure CD. Then, if the less be continually subtracted in turn from the greater, that which is left over will sometime measure the one before it, because AB, CD are not incommensurable; [cf. X. 2] let AB, measuring ED, leave EC less than itself, let EC, measuring FB, leave AF less than itself, and let AF measure CE. Since, then, AF measures CE, while CE measures FB, therefore AF will also measure FB. But it measures itself also; therefore AF will also measure the whole AB. But AB measures DE; therefore AF will also measure ED. But it measures CE also; therefore it also measures the whole CD. Therefore AF is a common measure of AB, CD. I say next that it is also the greatest. For, if not, there will be some magnitude greater than AF which will measure AB, CD. Let it be G. Since then G measures AB, while AB measures ED, therefore G will also measure ED. But it measures the whole CD also; therefore G will also measure the remainder CE. But CE measures FB; therefore G will also measure FB. But it measures the whole AB also, and it will therefore measure the remainder AF, the greater the less: which is impossible. Therefore no magnitude greater than AF will measure AB, CD; therefore AF is the greatest common measure of AB, CD. Therefore the greatest common measure of the two given commensurable magnitudes AB, CD has been found. Q. E. D.PORISM.
From this it is manifest that, if a magnitude measure two magnitudes, it will also measure their greatest common measure.PROPOSITION 4.
Given three commensurable magnitudes, to find their greatest common measure. Let A, B, C be the three given commensurable magnitudes; thus it is required to find the greatest common measure of A, B, C. Let the greatest common measure of the two magnitudes A, B be taken, and let it be D; [X. 3] then D either measures C, or does not measure it. First, let it measure it. Since then D measures C, while it also measures A, B, therefore D is a common measure of A, B, C. And it is manifest that it is also the greatest; for a greater magnitude than the magnitude D does not measure A, B. Next, let D not measure C. I say first that C, D are commensurable. For, since A, B, C are commensurable, some magnitude will measure them, and this will of course measure A, B also; so that it will also measure the greatest common measure of A, B, namely D. [X. 3, Por.] But it also measures C; so that the said magnitude will measure C, D; therefore C, D are commensurable. Now let their greatest common measure be taken, and let it be E. [X. 3] Since then E measures D, while D measures A, B, therefore E will also measure A, B. But it measures C also; therefore E measures A, B, C; therefore E is a common measure of A, B, C. I say next that it is also the greatest. For, if possible, let there be some magnitude F greater than E, and let it measure A, B, C. Now, since F measures A, B, C, it will also measure A, B, and will measure the greatest common measure of A, B. [X. 3, Por.] But the greatest common measure of A, B is D; therefore F measures D. But it measures C also; therefore F measures C, D; therefore F will also measure the greatest common measure of C, D. [X. 3, Por.] But that is E; therefore F will measure E, the greater the less: which is impossible. Therefore no magnitude greater than the magnitude E will measure A, B, C; therefore E is the greatest common measure of A, B, C if D do not measure C, and, if it measure it, D is itself the greatest common measure. Therefore the greatest common measure of the three given commensurable magnitudes has been found.PORISM.
From this it is manifest that, if a magnitude measure three magnitudes, it will also measure their greatest common measure. Similarly too, with more magnitudes, the greatest common measure can be found, and the porism can be extended. Q. E. D.PROPOSITION 5.
Commensurable magnitudes have to one another the ratio which a number has to a number. Let A, B be commensurable magnitudes; I say that A has to B the ratio which a number has to a number. For, since A, B are commensurable, some magnitude will measure them. Let it measure them, and let it be C. And, as many times as C measures A, so many units let there be in D; and, as many times as C measures B, so many units let there be in E. Since then C measures A according to the units in D, while the unit also measures D according to the units in it, therefore the unit measures the number D the same number of times as the magnitude C measures A; therefore, as C is to A, so is the unit to D; [VII. Def. 20] therefore, inversely, as A is to C, so is D to the unit. [cf. V. 7, Por.] Again, since C measures B according to the units in E, while the unit also measures E according to the units in it, therefore the unit measures E the same number of times as C measures B; therefore, as C is to B, so is the unit to E. But it was also proved that,PROPOSITION 6.
If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable. For let the two magnitudes A, B have to one another the ratio which the number D has to the number E;I say that the magnitudes A, B are commensurable. For let A be divided into as many equal parts as there are units in D, and let C be equal to one of them; and let F be made up of as many magnitudes equal to C as
there are units in E. Since then there are in A as many magnitudes equal to C as there are units in D, whatever part the unit is of D, the same part is C of A also; therefore, as C is to A, so is the unit to D. [VII. Def. 20]
But the unit measures the number D; therefore C also measures A. And since, as C is to A, so is the unit to D, therefore, inversely, as A is to C, so is the number D to the unit. [cf. V. 7, Por.]
Again, since there are in F as many magnitudes equal to C as there are units in E, therefore, as C is to F, so is the unit to E. [VII. Def. 20] But it was also proved that,
therefore, ex aequali, as A is to F, so is D to E. [v. 22] But, as D is to E, so is A to B; therefore also, as A is to B, so is it to F also. [V. 11] Therefore A has the same ratio to each of the magnitudes B, F;
therefore B is equal to F. [V. 9] But C measures F; therefore it measures B also. Further it measures A also; therefore C measures A, B.
Therefore A is commensurable with B. Therefore etc.
PORISM.
From this it is manifest that, if there be two numbers, as D, E, and a straight line, as A, it is possible to make a straight line [F] such that the given straight line is toit as the number D is to the number E. And, if a mean proportional be also taken between A, F, as B, as A is to F, so will the square on A be to the square on B, that is, as the first is to the third, so is the figure on the first
to that which is similar and similarly described on the second. [VI. 19, Por.] But, as A is to F, so is the number D to the number E; therefore it has been contrived that, as the number D is to the number E, so also is the figure on the straight line A to the figure on the straight line B. Q. E. D. 1
