Elements -7 - by Euclid

ELEMENTS

by Euclid

Translated by Thomas L. Heath
Book Seven

DEFINITIONS
An unit is that by virtue of which each of the things that exist is called one.
A number is a multitude composed of units.
A number is a part of a number, the less of the greater, when it measures the greater;
but parts when it does not measure it.
The greater number is a multiple of the less when it is measured by the less.
An even number is that which is divisible into two equal parts.
An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.
An even-times even number is that which is measured by an even number according to an even number.
An even-times odd number is that which is measured by an even number according to an odd number.
An odd-times odd number is that which is measured by an odd number according to an odd number.
A prime number is that which is measured by an unit alone.
Numbers prime to one another are those which are measured by an unit alone as a common measure.
A composite number is that which is measured by some number.
Numbers composite to one another are those which are measured by some number as a common measure.
A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.
And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.
And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.
A square number is equal multiplied by equal, or a number which is contained by two equal numbers.
And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.
Similar plane and solid numbers are those which have their sides proportional.
A perfect number is that which is equal to its own parts.

PROPOSITIONS

PROPOSITION I.
Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers will be prime to one another.
For, the less of two unequal numbers AB, CD being continually subtracted from the greater, let the number which is left never measure the one before it until an unit is left; I say that AB, CD are prime to one another, that is, that an unit alone measures AB, CD.
[Figure]
For, if AB, CD are not prime to one another, some number will measure them.
Let a number measure them, and let it be E; let CD, measuring BF, leave FA less than itself, let AF, measuring DG, leave GC less than itself, and let GC, measuring FH, leave an unit HA.
Since, then, E measures CD, and CD measures BF, therefore E also measures BF.
But it also measures the whole BA; therefore it will also measure the remainder AF.
But AF measures DG; therefore E also measures DG. [p. 297]
But it also measures the whole DC therefore it will also measure the remainder CG.
But CG measures FH; therefore E also measures FH.
But it also measures the whole FA; therefore it will also measure the remainder, the unit AH, though it is a number: which is impossible.
Therefore no number will measure the numbers AB, CD; therefore AB, CD are prime to one another. [VII. Def. 12] Q. E. D.
[p. 298]
PROPOSITION 2.
Given two numbers not prime to one another, to find their greatest common measure.
Let AB, CD be the two given numbers not prime to one another.
Thus it is required to find the greatest common measure of AB, CD.
If now CD measures AB--and it also measures itself--CD is a common measure of CD, AB.
[Figure]
And it is manifest that it is also the greatest; for no greater number than CD will measure CD.
But, if CD does not measure AB, then, the less of the numbers AB, CD being continually subtracted from the greater, some number will be left which will measure the one before it.
For an unit will not be left; otherwise AB, CD will be prime to one another [VII. 1], which is contrary to the hypothesis.
Therefore some number will be left which will measure the one before it.
Now let CD, measuring BE, leave EA less than itself, let EA, measuring DF, leave FC less than itself, and let CF measure AE.
Since then, CF measures AE, and AE measures DF, therefore CF will also measure DF.
But it also measures itself; therefore it will also measure the whole CD.
But CD measures BE; therefore CF also measures BE.
But it also measures EA; therefore it will also measure the whole BA.
But it also measures CD; therefore CF measures AB, CD.
Therefore CF is a common measure of AB, CD. [p. 299]
I say next that it is also the greatest.
For, if CF is not the greatest common measure of AB, CD, some number which is greater than CF will measure the numbers AB, CD.
Let such a number measure them, and let it be G.
Now, since G measures CD, while CD measures BE, G also measures BE.
But it also measures the whole BA; therefore it will also measure the remainder AE.
But AE measures DF; therefore G will also measure DF.
But it also measures the whole DC; therefore it will also measure the remainder CF, that is, the greater will measure the less: which is impossible.
Therefore no number which is greater than CF will measure the numbers AB, CD;
therefore CF is the greatest common measure of AB, CD.
PORISM.
From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure.
Q. E. D.
PROPOSITION 3.
Given three numbers not prime to one another, to find their greatest common measure.
Let A, B, C be the three given numbers not prime to one another; thus it is required to find the greatest common measure of A, B, C.
[Figure]
For let the greatest common measure, D, of the two numbers A, B be taken; [VII. 2] then D either measures, or does not measure, C.
First, let it measure it.
But it measures A, B also; therefore D measures A, B, C; therefore D is a common measure of A, B, C.
I say that it is also the greatest. [p. 301]
For, if D is not the greatest common measure of A, B, C, some number which is greater than D will measure the numbers A, B, C.
Let such a number measure them, and let it be E.
Since then E measures A, B, C, it will also measure A, B; therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.]
But the greatest common measure of A, B is D; therefore E measures D, the greater the less: which is impossible.
Therefore no number which is greater than D will measure the numbers A, B, C;
therefore D is the greatest common measure of A, B, C.
Next, let D not measure C; I say first that C, D are not prime to one another.
For, since A, B, C are not prime to one another, some number will measure them.
Now that which measures A, B, C will also measure A, B, and will measure D, the greatest common measure of A, B. [VII. 2, Por.]
But it measures C also; therefore some number will measure the numbers D, C; therefore D, C are not prime to one another.
Let then their greatest common measure E be taken. [VII. 2]
Then, since E measures D, and D measures A, B, therefore E also measures 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 E is not the greatest common measure of A, B, C, some number which is greater than E will measure the numbers A, B, C.
Let such a number measure them, and let it be F. [p. 302]
Now, since F measures A, B, C, it also measures A, B; therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.]
But the greatest common measure of A, B is D; therefore F measures D.
And it measures C also; therefore F measures D, C; therefore it will also measure the greatest common measure of D, C. [VII. 2, Por.]
But the greatest common measure of D, C is E; therefore F measures E, the greater the less: which is impossible.
Therefore no number which is greater than E will measure the numbers A, B, C; therefore E is the greatest common measure of A, B, C. Q. E. D.
PROPOSITION 4.
Any number is either a part or parts of any number, the less of the greater.
Let A, BC be two numbers, and let BC be the less; I say that BC is either a part, or parts, of A.
For A, BC are either prime to one another or not.
First, let A, BC be prime to one another.
Then, if BC be divided into the units in it, each unit of those in BC will be some part of A; so that BC is parts of A.
[Figure]
Next let A, BC not be prime to one another; then BC either measures, or does not measure, A.
If now BC measures A, BC is a part of A.
But, if not, let the greatest common measure D of A, BC be taken; [VII. 2] and let BC be divided into the numbers equal to D, namely BE, EF, FC.
Now, since D measures A, D is a part of A.
But D is equal to each of the numbers BE, EF, FC; therefore each of the numbers BE, EF, FC is also a part of A; so that BC is parts of A.
Therefore etc. Q. E. D.
[p. 304]
PROPOSITION 5.
If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.
For let the number A be a part of BC, and another, D, the same part of another EF that A is of BC; I say that the sum of A, D is also the same part of the sum of BC, EF that A is of BC.
For since, whatever part A is of BC, D is also the same part of EF, therefore, as many numbers as there are in BC equal to A, so many numbers are there also in EF equal to D.
[Figure]
Let BC be divided into the numbers equal to A, namely BG, GC, and EF into the numbers equal to D, namely EH, HF; then the multitude of BG, GC will be equal to the multitude of EH, HF.
And, since BG is equal to A, and EH to D, therefore BG, EH are also equal to A, D.
For the same reason GC, HF are also equal to A, D.
Therefore, as many numbers as there are in BC equal to A, so many are there also in BC, EF equal to A, D.
Therefore, whatever multiple BC is of A, the same multiple also is the sum of BC, EF of the sum of A, D.
Therefore, whatever part A is of BC, the same part also is the sum of A, D of the sum of BC, EF. Q. E. D.
[p. 305]
PROPOSITION 6.
If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.
For let the number AB be parts of the number C, and another, DE, the same parts of another, F, that AB is of C; I say that the sum of AB, DE is also the same parts of the sum of C, F that AB is of C.
[Figure]
For since, whatever parts AB is of C, DE is also the same parts of F, therefore, as many parts of C as there are in AB, so many parts of F are there also in DE.
Let AB be divided into the parts of C, namely AG, GB, and DE into the parts of F, namely DH, HE; thus the multitude of AG, GB will be equal to the multitude of DH, HE.
And since, whatever part AG is of C, the same part is DH of F also, therefore, whatever part AG is of C, the same part also is the sum of AG, DH of the sum of C, F. [VII. 5]
For the same reason, whatever part GB is of C, the same part also is the sum of GB, HE of the sum of C, F.
Therefore, whatever parts AB is of C, the same parts also is the sum of AB, DE of the sum of C, F. Q. E. D.
PROPOSITION 7.
If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder will also be the same part of the remainder that the whole is of the whole.
For let the number AB be that part of the number CD which AE subtracted is of CF subtracted; I say that the remainder EB is also the same part of the remainder FD that the whole AB is of the whole CD.
[Figure]
For, whatever part AE is of CF, the same part also let EB be of CG.
Now since, whatever part AE is of CF, the same part also is EB of CG, therefore, whatever part AE is of CF, the same part also is AB of GF. [VII. 5]
But, whatever part AE is of CF, the same part also, by hypothesis, is AB of CD; therefore, whatever part AB is of GF, the same part is it of CD also; therefore GF is equal to CD.
Let CF be subtracted from each; therefore the remainder GC is equal to the remainder FD.
Now since, whatever part AE is of CF, the same part also is EB of GC, while GC is equal to FD, therefore, whatever part AE is of CF, the same part also is EB of FD.
But, whatever part AE is of CF, the same part also is AB of CD; [p. 307] therefore also the remainder EB is the same part of the remainder FD that the whole AB is of the whole CD. Q. E. D.
PROPOSITION 8.
If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole.
For let the number AB be the same parts of the number CD that AE subtracted is of CF subtracted; I say that the remainder EB is also the same parts of the remainder FD that the whole AB is of the whole CD.
[Figure]
For let GH be made equal to AB.
Therefore, whatever parts GH is of CD, the same parts also is AE of CF.
Let GH be divided into the parts of CD, namely GK, KH, and AE into the parts of CF, namely AL, LE; thus the multitude of GK, KH will be equal to the multitude of AL, LE. [p. 308]
Now since, whatever part GK is of CD, the same part also is AL of CF, while. CD is greater than CF, therefore GK is also greater than AL.
Let GM be made equal to AL.
Therefore, whatever part GK is of CD, the same part also is GM of CF; therefore also the remainder MK is the same part of the remainder FD that the whole GK is of the whole CD. [VII. 7]
Again, since, whatever part KH is of CD, the same part also is EL of CF, while CD is greater than CF, therefore HK is also greater than EL.
Let KN be made equal to EL.
Therefore, whatever part KH is of CD, the same part also is KN of CF; therefore also the remainder NH is the same part of the remainder FD that the whole KH is of the whole CD. [VII. 7]
But the remainder MK was also proved to be the same part of the remainder FD that the whole GK is of the whole CD; therefore also the sum of MK, NH is the same parts of DF that the whole HG is of the whole CD.
But the sum of MK, NH is equal to EB, and HG is equal to BA; therefore the remainder EB is the same parts of the remainder FD that the whole AB is of the whole CD. Q. E. D.
PROPOSITION 9.
If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.
For let the number A be a part of the number BC, and another, D, the same part of another, EF, that A is of BC; I say that, alternately also, whatever part or parts A is of D, the same part or parts is BC of EF also.
[Figure]
For since, whatever part A is of BC, the same part also is D of EF, therefore, as many numbers as there are in BC equal to A, so many also are there in EF equal to D.
Let BC be divided into the numbers equal to A, namely BG, GC, and EF into those equal to D, namely EH, HF; thus the multitude of BG, GC will be equal to the multitude of EH, HF.
Now, since the numbers BG, GC are equal to one another, and the numbers EH, HF are also equal to one another, while the multitude of BG, GC is equal to the multitude of EH, HF, therefore, whatever part or parts BG is of EH, the same part or the same parts is GC of HF also; so that, in addition, whatever part or parts BG is of EH, the same part also, or the same parts, is the sum BC of the sum EF. [VII. 5, 6] [p. 310]
But BG is equal to A, and EH to D; therefore, whatever part or parts A is of D, the same part or the same parts is BC of EF also. Q. E. D.
PROPOSITION 10.
If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part will the second also be of the fourth.
For let the number AB be parts of the number C, and another, DE, the same parts of another, F; I say that, alternately also, whatever parts or part AB is of DE, the same parts or the same part is C of F also.
[Figure]
For since, whatever parts AB is of C, the same parts also is DE of F, therefore, as many parts of C as there are in AB, so many parts also of F are there in DE.
Let AB be divided into the parts of C, namely AG, GB, and DE into the parts of F, namely DH, HE; thus the multitude of AG, GB will be equal to the multitude of DH, HE.
Now since, whatever part AG is of C, the same part also is DH of F, alternately also, whatever part or parts AG is of DH, the same part or the same parts is C of F also. [VII. 9]
For the same reason also, whatever part or parts GB is of HE, the same part or the same parts is C of F also; [p. 311] so that, in addition, whatever parts or part AB is of DE, the same parts also, or the same part, is C of F. [VII. 5, 6] Q. E. D.
PROPOSITION 11.
If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder as whole to whole.
As the whole AB is to the whole CD, so let AE subtracted be to CF subtracted; I say that the remainder EB is also to the remainder FD as the whole AB to the whole CD.
Since, as AB is to CD, so is AE to CF, whatever part or parts AB is of CD, the same part or the same parts is AE of CF also; [VII. Def. 20]
[Figure]
Therefore also the remainder EB is the same part or parts of FD that AB is of CD. [VII. 7, 8]
Therefore, as EB is to FD, so is AB to CD. [VII. Def. 20] Q. E. D.
PROPOSITION 12.
If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.
Let A, B, C, D be as many numbers as we please in proportion, so that,
as A is to B, so is C to D;
I say that, as A is to B, so are A, C to B, D.
[Figure]
For since, as A is to B, so is C to D, whatever part or parts A is of B, the same part or parts is C of D also. [VII. Def. 20]
Therefore also the sum of A, C is the same part or the same parts of the sum of B, D that A is of B. [VII. 5, 6]
Therefore, as A is to B, so are A, C to B, D. [VII. Def. 20]
[p. 313]
PROPOSITION 13.
If four numbers be proportional, they will also be proportional alternately.
Let the four numbers A, B, C, D be proportional, so that,
as A is to B, so is C to D;
I say that they will also be proportional alternately, so that,
as A is to C, so will B be to D.
For since, as A is to B, so is C to D, therefore, whatever part or parts A is of B, the same part or the same parts is C of D also. [VII. Def. 20]
[Figure]
Therefore, alternately, whatever part or parts A is of C, the same part or the same parts is B of D also. [VII. 10]
Therefore, as A is to C, so is B to D. [VII. Def. 20] Q. E. D.
PROPOSITION 14.
If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.
Let there be as many numbers as we please A, B, C, and others equal to them in multitude D, E, F, which taken two and two are in the same ratio, so that,
as A is to B, so is D to E,
and, as B is to C, so is E to F; I say that, ex aequali,
as A is to C, so also is D to F.
[Figure]
For, since, as A is to B, so is D to E, therefore, alternately,
as A is to D, so is B to E. [VII. 13]
[p. 314]
Again, since, as B is to C, so is E to F, therefore, alternately,
as B is to E, so is C to F. [VII. 13]
But, as B is to E, so is A to D; therefore also, as A is to D, so is C to F.
Therefore, alternately,
as A is to C, so is D to F. [id.]
PROPOSITION 15.
If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth. [p. 315]
For let the unit A measure any number BC, and let another number D measure any other number EF the same number of times; I say that, alternately also, the unit A measures the number D the same number of times that BC measures EF.
[Figure]
For, since the unit A measures the number BC the same number of times that D measures EF, therefore, as many units as there are in BC, so many numbers equal to D are there in EF also.
Let BC be divided into the units in it, BG, GH, HC, and EF into the numbers EK, KL, LF equal to D.
Thus the multitude of BG, GH, HC will be equal to the multitude of EK, KL, LF.
And, since the units BG, GH, HC are equal to one another, and the numbers EK, KL, LF are also equal to one another, while the multitude of the units BG, GH, HC is equal to the multitude of the numbers EK, KL, LF, therefore, as the unit BG is to the number EK, so will the unit GH be to the number KL, and the unit HC to the number LF.
Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [VII. 12] therefore, as the unit BG is to the number EK, so is BC to EF.
But the unit BG is equal to the unit A, and the number EK to the number D.
Therefore, as the unit A is to the number D, so is BC to EF.
Therefore the unit A measures the number D the same number of times that BC measures EF. Q. E. D.
[p. 316]
PROPOSITION 16.
If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.
Let A, B be two numbers, and let A by multiplying B make C, and B by multiplying A make D; I say that C is equal to D.
[Figure]
For, since A by multiplying B has made C, therefore B measures C according to the units in A.
But the unit E also measures the number A according to the units in it;
therefore the unit E measures A the same number of times that B measures C.
Therefore, alternately, the unit E measures the number B the same number of times that A measures C. [VII. 15]
Again, since B by multiplying A has made D, therefore A measures D according to the units in B.
But the unit E also measures B according to the units in it;
therefore the unit E measures the number B the same number of times that A measures D.
But the unit E measured the number B the same number of times that A measures C;
therefore A measures each of the numbers C, D the same number of times.
Therefore C is equal to D. Q. E. D. 2. The numbers so produced.
PROPOSITION 17.
If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.
For let the number A by multiplying the two numbers B, C make D, E; I say that, as B is to C, so is D to E.
For, since A by multiplying B has made D, therefore B measures D according to the units in A.
[Figure]
But the unit F also measures the number A according to the units in it; therefore the unit F measures the number A the same number of times that B measures D.
Therefore, as the unit F is to the number A, so is B to D. [VII. Def. 20]
For the same reason, as the unit F is to the number A, so also is C to E; therefore also, as B is to D, so is C to E.
Therefore, alternately, as B is to C, so is D to E. [VII. 13] Q. E. D.
[p. 318]
PROPOSITION 18.
If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.
For let two numbers A, B by multiplying any number C make D, E; I say that, as A is to B, so is D to E.
[Figure]
For, since A by multiplying C has made D, therefore also C by multiplying A has made D. [VII. 16] For the same reason also C by multiplying B has made E.
Therefore the number C by multiplying the two numbers A, B has made D, E.
Therefore, as A is to B, so is D to E. [VII. 17]
PROPOSITION 19.
If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.
Let A, B, C, D be four numbers in proportion, so that,
as A is to B, so is C to D;
and let A by multiplying D make E, and let B by multiplying C make F; I say that E is equal to F.
For let A by multiplying C make G. [p. 319]
Since, then, A by multiplying C has made G, and by multiplying D has made E, the number A by multiplying the two numbers C, D has made G, E.
Therefore, as C is to D, so is G to E. [VII. 17]
But, as C is to D, so is A to B; therefore also, as A is to B, so is G to E.
[Figure]
Again, since A by multiplying C has made G, but, further, B has also by multiplying C made F, the two numbers A, B by multiplying a certain number C have made G, F.
Therefore, as A is to B, so is G to F. [VII. 18]
But further, as A is to B, so is G to E also; therefore also, as G is to E, so is G to F.
Therefore G has to each of the numbers E, F the same ratio;
therefore E is equal to F. [cf. V. 9]
Again, let E be equal to F; I say that, as A is to B, so is C to D.
For, with the same construction, since E is equal to F, therefore, as G is to E, so is G to F. [cf. V. 7]
But, as G is to E, so is C to D, [VII. 17]
and, as G is to F, so is A to B. [VII. 18]
Therefore also, as A is to B, so is C to D. Q. E. D.
PROPOSITION 20.
The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.
For let CD, EF be the least numbers of those which have the same ratio with A, B; I say that CD measures A the same number of times that EF measures B.
Now CD is not parts of A.
For, if possible, let it be so; therefore EF is also the same parts of B that CD is of A. [VII. 13 and Def. 20]
[Figure]
Therefore, as many parts of A as there are in CD, so many parts of B are there also in EF.
Let CD be divided into the parts of A, namely CG, GD, and EF into the parts of B, namely EH, HF; thus the multitude of CG, GD will be equal to the multitude of EH, HF. [p. 321]
Now, since the numbers CG, GD are equal to one another, and the numbers EH, HF are also equal to one another, while the multitude of CG, GD is equal to the multitude of EH, HF, therefore, as CG is to EH, so is GD to HF.
Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. [VII. 12]
Therefore, as CG is to EH, so is CD to EF.
Therefore CG, EH are in the same ratio with CD, EF, being less than they: which is impossible, for by hypothesis CD, EF are the least numbers of those which have the same ratio with them.
Therefore CD is not parts of A; therefore it is a part of it. [VII. 4]
And EF is the same part of B that CD is of A; [VII. 13 and Def. 20] therefore CD measures A the same number of times that EF measures B. Q. E. D.
PROPOSITION 21.
Numbers prime to one another are the least of those which have the same ratio with them.
Let A, B be numbers prime to one another; I say that A, B are the least of those which have the same ratio with them.
For, if not, there will be some numbers less than A, B which are in the same ratio with A, B.
[Figure]
Let them be C, D.
Since, then, the least numbers of those which have the same ratio measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, [VII. 20] therefore C measures A the same number of times that D measures B.
Now, as many times as C measures A, so many units let there be in E.
Therefore D also measures B according to the units in E. [p. 323]
And, since C measures A according to the units in E, therefore E also measures A according to the units in C. [VII. 16]
For the same reason E also measures B according to the units in D. [VII. 16]
Therefore E measures A, B which are prime to one another: which is impossible. [VII. Def. 12]
Therefore there will be no numbers less than A, B which are in the same ratio with A, B.
Therefore A, B are the least of those which have the same ratio with them. Q. E. D.
PROPOSITION 22.
The least numbers of those which have the same ratio with them are prime to one another.
Let A, B be the least numbers of those which have the same ratio with them; I say that A, B are prime to one another.
[Figure]
For, if they are not prime to one another, some number will measure them.
Let some number measure them, and let it be C. [p. 324]
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 C measures A according to the units in D, therefore C by multiplying D has made A. [VII. Def. 15]
For the same reason also C by multiplying E has made B.
Thus the number C by multiplying the two numbers D, E has made A, B; therefore, as D is to E, so is A to B; [VII. 17] therefore D, E are in the same ratio with A, B, being less than they: which is impossible.
Therefore no number will measure the numbers A, B.
Therefore A, B are prime to one another. Q. E. D.
PROPOSITION 23.
If two number be prime to one another, the number which measures the one of them will be prime to the remaining number.
Let A, B be two numbers prime to one another, and let any number C measure A; I say that C, B are also prime to one another.
For, if C, B are not prime to one another, some number will measure C, B.
Let a number measure them, and let it be D.
Since D measures C, and C measures A, therefore D also measures A.
[Figure]
But it also measures B; [p. 325] therefore D measures A, B which are prime to one another: which is impossible. [VII. Def. 12]
Therefore no number will measure the numbers C, B.
Therefore C, B are prime to one another. Q. E. D.
PROPOSITION 24.
If two numbers be prime to any number, their product also will be prime to the same.
For let the two numbers A, B be prime to any number C, and let A by multiplying B make D; I say that C, D are prime to one another.
For, if C, D are not prime to one another, some number will measure C, D.
Let a number measure them, and let it be E.
[Figure]
Now, since C, A are prime to one another, and a certain number E measures C, therefore A, E are prime to one another. [VII. 23]
As many times, then, as E measures D, so many units let there be in F; therefore F also measures D according to the units in E. [VII. 16]
Therefore E by multiplying F has made D. [VII. Def. 15]
But, further, A by multiplying B has also made D; therefore the product of E, F is equal to the product of A, B.
But, if the product of the extremes be equal to that of the means, the four numbers are proportional; [VII. 19] therefore, as E is to A, so is B to F.
But A, E are prime to one another, numbers which are prime to one another are also the least of those which have the same ratio, [VII. 21] and the least numbers of those which have the same ratio with them measure those which have the same ratio the same [p. 326] number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures B.
But it also measures C; therefore E measures B, C which are prime to one another: which is impossible. [VII. Def. 12]
Therefore no number will measure the numbers C, D.
Therefore C, D are prime to one another. Q. E. D. 1. their product.
PROPOSITION 25.
If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.
Let A, B be two numbers prime to one another, and let A by multiplying itself make C: I say that B, C are prime to one another.
For let D be made equal to A.
Since A, B are prime to one another, and A is equal to D, therefore D, B are also prime to one another.
[Figure]
Therefore each of the two numbers D, A is prime to B; therefore the product of D, A will also be prime to B. [VII. 24] [p. 327]
But the number which is the product of D, A is C.
Therefore C, B are prime to one another. Q. E. D. 1. the product of one of them into itself.
PROPOSITION 26.
If two numbers be prime to two numbers, both to each, their products also will be prime to one another.
For let the two numbers A, B be prime to the two numbers C, D; both to each, and let A by multiplying B make E, and let C by multiplying D make F; I say that E, F are prime to one another.
[Figure]
For, since each of the numbers A, B is prime to C, therefore the product of A, B will also be prime to C. [VII. 24]
But the product of A, B is E; therefore E, C are prime to one another.
For the same reason E, D are also prime to one another.
Therefore each of the numbers C, D is prime to E.
Therefore the product of C, D will also be prime to E. [VII. 24]
But the product of C, D is F.
Therefore E, F are prime to one another. Q. E. D.
[p. 328]
PROPOSITION 27.
If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another [and this is always the case with the extremes].
Let A, B be two numbers prime to one another, let A by multiplying itself make C, and by multiplying C make D, and let B by multiplying itself make E, and by multiplying E make F; I say that both C, E and D, F are prime to one another.
[Figure]
For, since A, B are prime to one another, and A by multiplying itself has made C, therefore C, B are prime to one another. [VII. 25]
Since then C, B are prime to one another, and B by multiplying itself has made E, therefore C, E are prime to one another. [id.]
Again, since A, B are prime to one another, and B by multiplying itself has made E, therefore A, E are prime to one another. [id.]
Since then the two numbers A, C are prime to the two numbers B, E, both to each, therefore also the product of A, C is prime to the product of B, E. [VII. 26]
And the product of A, C is D, and the product of B, E is F.
Therefore D, F are prime to one another. Q. E. D.
PROPOSITION 28.
If two numbers be prime to one another, the sum will also be prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers will also be prime to one another.
For let two numbers AB, BC prime to one another be added; I say that the sum AC is also prime to each of the numbers AB, BC.
[Figure]
For, if CA, AB are not prime to one another, some number will measure CA, AB.
Let a number measure them, and let it be D.
Since then D measures CA, AB, therefore it will also measure the remainder BC.
But it also measures BA; therefore D measures AB, BC which are prime to one another: which is impossible. [VII. Def. 12]
Therefore no number will measure the numbers CA, AB; therefore CA, AB are prime to one another.
For the same reason AC, CB are also prime to one another.
Therefore CA is prime to each of the numbers AB, BC.
Again, let CA, AB be prime to one another; I say that AB, BC are also prime to one another.
For, if AB, BC are not prime to one another, some number will measure AB, BC. [p. 330]
Let a number measure them, and let it be D.
Now, since D measures each of the numbers AB, BC, it will also measure the whole CA.
But it also measures AB; therefore D measures CA, AB which are prime to one another: which is impossible. [VII. Def. 12]
Therefore no number will measure the numbers AB, BC.
Therefore AB, BC are prime to one another. Q. E. D.
PROPOSITION 29.
Any prime number is prime to any number which it does not measure.
Let A be a prime number, and let it not measure B; I say that B, A are prime to one another.
For, if B, A are not prime to one another, some number will measure them.
[Figure]
Let C measure them.
Since C measures B, and A does not measure B, therefore C is not the same with A.
Now, since C measures B, A, therefore it also measures A which is prime, though it is not the same with it: which is impossible. [p. 331]
Therefore no number will measure B, A.
Therefore A, B are prime to one another. Q. E. D.
PROPOSITION 30.
If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.
For let the two numbers A, B by multiplying one another make C, and let any prime number D measure C; I say that D measures one of the numbers A, B.
[Figure]
For let it not measure A.
Now D is prime; therefore A, D are prime to one another. [VII. 29]
And, as many times as D measures C, so many units let there be in E.
Since then D measures C according to the units in E, therefore D by multiplying E has made C. [VII. Def. 15]
Further, A by multiplying B has also made C; therefore the product of D, E is equal to the product of A, B.
Therefore, as D is to A, so is B to E. [VII. 19]
But D, A are prime to one another, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore D measures B.
Similarly we can also show that, if D do not measure B, it will measure A.
Therefore D measures one of the numbers A, B. Q. E. D. [p. 332]
PROPOSITION 31.
Any composite number is measured by some prime number.
Let A be a composite number; I say that A is measured by some prime number.
For, since A is composite,
(5)
some number will measure it.
Let a number measure it, and let it be B.
[Figure]
Now, if B is prime, what was enjoined will have been done.
(10)
But if it is composite, some number will measure it.
Let a number measure it, and let it be C.
Then, since C measures B, and B measures A, therefore C also measures A.
(15)
And, if C is prime, what was enjoined will have been done.
But if it is composite, some number will measure it.
Thus, if the investigation be continued in this way, some prime number will be found which will measure the number
(20)
before it, which will also measure A.
For, if it is not found, an infinite series of numbers will measure the number A, each of which is less than the other: which is impossible in numbers.
Therefore some prime number will be found which will
(25)
measure the one before it, which will also measure A.
Therefore any composite number is measured by some prime number. [p. 333] 8. if B is prime, what was enjoined will have been done, 18. some prime number will be found which will measure.
PROPOSITION 32.
Any number either is prime or is measured by some prime number.
Let A be a number; I say that A either is prime or is measured by some prime number.
[Figure]
If now A is prime, that which was enjoined will have been done.
But if it is composite, some prime number will measure it. [VII. 31]
Therefore any number either is prime or is measured by some prime number. Q. E. D.
PROPOSITION 33.
Given as many numbers as we please, to find the least of those which have the same ratio with them.
Let A, B, C be the given numbers, as many as we please; thus it is required to find the least of
(5)
those which have the same ratio with A, B, C.
A, B, C are either prime to one another or not.
[Figure]
Now, if A, B, C are prime to one
(10)
another, they are the least of those which have the same ratio with them. [VII. 21]
But, if not, let D the greatest common measure of A, B, C be taken, [VII. 3] [p. 334] and, as many times as D measures the numbers A, B, C
(15)
respectively, so many units let there be in the numbers E, F, G respectively.
Therefore the numbers E, F, G measure the numbers A, B, C respectively according to the units in D. [VII. 16]
Therefore E, F, G measure A, B, C the same number of
(20)
times; therefore E, F, G are in the same ratio with A, B, C. [VII. Def. 20]
I say next that they are the least that are in that ratio.
For, if E, F, G are not the least of those which have the same ratio with A, B, C,
(25)
there will be numbers less than E, F, G which are in the same ratio with A, B, C.
Let them be H, K, L; therefore H measures A the same number of times that the numbers K, L measure the numbers B, C respectively.
(30)
Now, as many times as H measures A, so many units let there be in M; therefore the numbers K, L also measure the numbers B, C respectively according to the units in M.
And, since H measures A according to the units in M,
(35)
therefore M also measures A according to the units in H. [VII. 16]
For the same reason M also measures the numbers B, C according to the units in the numbers K, L respectively;
Therefore M measures A, B, C.
(40)
Now, since H measures A according to the units in M, therefore H by multiplying M has made A. [VII. Def. 15]
For the same reason also E by multiplying D has made A.
Therefore the product of E, D is equal to the product of
(45)
H, M.
Therefore, as E is to H, so is M to D. [VII. 19]
But E is greater than H; therefore M is also greater than D.
And it measures A, B, C: [p. 335]
(50)
which is impossible, for by hypothesis D is the greatest common measure of A, B, C.
Therefore there cannot be any numbers less than E, F, G which are in the same ratio with A, B, C.
Therefore E, F, G are the least of those which have the
(55)
same ratio with A, B, C. Q. E. D. 17. the numbers E, F, G measure the numbers A, B, C respectively,
[p. 336]
PROPOSITION 34.
Given two numbers, to find the least number which they measure.
Let A, B be the two given numbers; thus it is required to find the least number which they measure.
Now A, B are either prime to one another or not.
[Figure]
First, let A, B be prime to one another, and let A by multiplying B make C; therefore also B by multiplying A has made C. [VII. 16]
Therefore A, B measure C
I say next that it is also the least number they measure.
For, if not, A, B will measure some number which is less than C.
Let them measure D.
Then, as many times as A measures D, so many units let there be in E, and, as many times as B measures D, so many units let there be in F; therefore A by multiplying E has made D, and B by multiplying F has made D; [VII. Def. 15] therefore the product of A, E is equal to the product of B, F.
Therefore, as A is to B, so is F E. [VII. 19]
But A, B are prime, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore B measures E, as consequent consequent.
And, since A by multiplying B, E has made C, D, therefore, as B is to E, so is C to D. [VII. 17]
But B measures E; therefore C also measures D, the greater the less: which is impossible. [p. 337]
Therefore A, B do not measure any number less than C; therefore C is the least that is measured by A, B.
Next, let A, B not be prime to one another, and let F, E, the least numbers of those which have the same ratio with A, B, be taken; [VII. 33] therefore the product of A, E is equal to the product of B, F. [VII. 19]
And let A by multiplying E make C; therefore also B by multiplying F has made C; therefore A, B measure C.
[Figure]
I say next that it is also the least number that they measure.
For, if not, A, B will measure some number which is less than C.
Let them measure D.
And, as many times as A measures D, so many units let there be in G, and, as many times as B measures D, so many units let there be in H.
Therefore A by multiplying G has made D, and B by multiplying H has made D.
Therefore the product of A, G is equal to the product of B, H; therefore, as A is to B, so is H to G. [VII. 19]
But, as A is to B, so is F to E.
Therefore also, as F is to E, so is H to G.
But F, E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore E measures G.
And, since A by multiplying E, G has made C, D, therefore, as E is to G, so is C to D. [VII. 17]
But E measures G; therefore C also measures D, the greater the less: which is impossible. [p. 338]
Therefore A, B will not measure any number which is less than C.
Therefore C is the least that is measured by A, B. Q. E. D.
[p. 339]
PROPOSITION 35.
If two numbers measure any number, the least number measured by them will also measure the same.
For let the two numbers A, B measure any number CD, and let E be the least that they measure; I say that E also measures CD.
[Figure]
For, if E does not measure CD, let E, measuring DF, leave CF less than itself.
Now, since A, B measure E, and E measures DF, therefore A, B will also measure DF.
But they also measure the whole CD; therefore they will also measure the remainder CF which is less than E: which is impossible.
Therefore E cannot fail to measure CD; therefore it measures it. Q. E. D.
PROPOSITION 36.
Given three numbers, to find the least number which they measure.
Let A, B, C be the three given numbers; thus it is required to find the least number which they measure.
[Figure]
Let D, the least number measured by the two numbers A, B, be taken. [VII. 34]
Then C either measures, or does not measure, D.
First, let it measure it. [p. 340]
But A, B also measure D; therefore A, B, C measure D.
I say next that it is also the least that they measure.
For, if not, A, B, C will measure some number which is less than D.
Let them measure E.
Since A, B, C measure E, therefore also A, B measure E.
Therefore the least number measured by A, B will also measure E. [VII. 35]
But D is the least number measured by A, B; therefore D will measure E, the greater the less: which is impossible.
Therefore A, B, C will not measure any number which is less than D;
therefore D is the least that A, B, C measure.
Again, let C not measure D, and let E, the least number measured by C, D, be taken. [VII. 34]
[Figure]
Since A, B measure D, and D measures E, therefore also A, B measure E.
But C also measures E; therefore also A, B, C measure E.
I say next that it is also the least that they measure.
For, if not, A, B, C will measure some number which is less than E.
Let them measure F.
Since A, B, C measure F, therefore also A, B measure F; therefore the least number measured by A, B will also measure F. [VII. 35]
But D is the least number measured by A, B; therefore D measures F.
But C also measures F; therefore D, C measure F, so that the least number measured by D, C will also measure F. [p. 341]
But E is the least number measured by C, D; therefore E measures F, the greater the less: which is impossible.
Therefore A, B, C will not measure any number which is less than E.
Therefore E is the least that is measured by A, B, C. Q. E. D.
PROPOSITION 37.
If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.
For let the number A be measured by any number B; I say that A has a part called by the same name as B.
[Figure]
For, as many times as B measures A, so many units let there be in C.
Since B measures A according to the units in C, and the unit D also measures the number C according to the units in it, [p. 342] therefore the unit D measures the number C the same number of times as B measures A.
Therefore, alternately, the unit D measures the number B the same number of times as C measures A; [VII. 15] therefore, whatever part the unit D is of the number B, the same part is C of A also.
But the unit D is a part of the number B called by the same name as it; therefore C is also a part of A called by the same name as B, so that A has a part C which is called by the same name as B. Q. E. D.
PROPOSITION 38.
If a number have any part whatever, it will be measured by a number called by the same name as the part.
For let the number A have any part whatever, B, and let C be a number called by the same name as the part B; I say that C measures A.
[Figure]
For, since B is a part of A called by the same name as C, and the unit D is also a part of C called by the same name as it, therefore, whatever part the unit D is of the number C, the same part is B of A also; therefore the unit D measures the number C the same number of times that B measures A. [p. 343]
Therefore, alternately, the unit D measures the number B the same number of times that C measures A. [VII. 15]
Therefore C measures A. Q. E. D.
PROPOSITION 39.
To find the number which is the least that will have given parts.
Let A, B, C be the given parts; thus it is required to find the number which is the least that will have the parts A, B, C.
[Figure]
Let D, E, F be numbers called by the same name as the parts A, B, C, and let G, the least number measured by D, E, F, be taken. [VII. 36]
Therefore G has parts called by the same name as D, E, F. [VII. 37]
But A, B, C are parts called by the same name as D, E, F; therefore G has the parts A, B, C.
I say next that it is also the least number that has. [p. 344]
For, if not, there will be some number less than G which will have the parts A, B, C.
Let it be H.
Since H has the parts A, B, C, therefore H will be measured by numbers called by the same name as the parts A, B, C. [VII. 38]
But D, E, F are numbers called by the same name as the parts A, B, C; therefore H is measured by D, E, F.
And it is less than G : which is impossible.
Therefore there will be no number less than G that will have the parts A, B, C. Q. E. D.

NOTES

2. The numbers so produced. The Greek has oi genomenoi ex autôn, “the (numbers) produced from them.” By “from them” Euclid means “from the original numbers,” though this is not very clear even in the Greek. I think ambiguity is best avoided by leaving out the words.
1. their product. ho ex autôn genomenos, literally “the (number) produced from them,” will henceforth be translated as “their product.”
1. the product of one of them into itself. The Greek, ho ek tou henos autôn genomenos, literally “the number produced from the one of them,” leaves “multiplied into itself” to be understood.
8. if B is prime, what was enjoined will have been done, i.e. the implied problem of finding a prime number which measures A.
18. some prime number will be found which will measure. In the Greek the sentence stops here, but it is necessary to add the words “the number before it, which will also measure A,” which are found a few lines further down. It is possible that the words may have fallen out of P here by a simple mistake due to homoioteleuton (Heiberg).
17. the numbers E, F, G measure the numbers A, B, C respectively, literally (as usual) “each of the numbers E, F, G measures each of the numbers A, B, C.”

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