ELEMENTS
by Euclid
Translated by Thomas L. Heath
Book Five
DEFINITIONS A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.
The greater is a multiple of the less when it is measured by the less.
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Let magnitudes which have the same ratio be called proportional.
When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.
A proportion in three terms is the least possible.
When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.
When four magnitudes are proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.
The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.
Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.
Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.
[p. 115]Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.
Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.
Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.
A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes;
Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.
A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.
PROPOSITIONS
PROPOSITION I.
If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.
Let any number of magnitudes whatever AB, CD be respectively equimultiples of any magnitudes E, F equal in multitude; I say that, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.
[Figure]For, since AB is the same multiple of E that CD is of F, as many magnitudes as there are in AB equal to E, so many also are there in CD equal to F.
Let AB be divided into the magnitudes AG, GB equal to E, and CD into CH, HD equal to F; then the multitude of the magnitudes AG, GB will be equal to the multitude of the magnitudes CH, HD.
Now, since AG is equal to E, and CH to F, therefore AG is equal to E, and AG, CH to E, F.
For the same reason
GB is equal to E, and GB, HD to E, F; therefore, as many magnitudes as there are in AB equal to E, so many also are there in AB, CD equal to E, F; [p. 139] therefore, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.
Therefore etc. Q. E. D.
PROPOSITION 2.
If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth.
Let a first magnitude, AB, be the same multiple of a second, C, that a third, DE, is of a fourth, F, and let a fifth, BG, also be the same multiple of the second, C, that a sixth, EH, is of the fourth F;
[Figure]
I say that the sum of the first and fifth, AG, will be the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F.For, since AB is the same multiple of C that DE is of F, therefore, as many magnitudes as there are in AB equal to C, so many also are there in DE equal to F.
For the same reason also, as many as there are in BG equal to C, so many are there also in EH equal to F; [p. 140] therefore, as many as there are in the whole AG equal to C, so many also are there in the whole DH equal to F.
Therefore, whatever multiple AG is of C, that multiple also is DH of F.
Therefore the sum of the first and fifth, AG, is the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F.
Therefore etc. Q.E.D.
PROPOSITION 3.
If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth.
Let a first magnitude A be the same multiple of a second B that a third C is of a fourth D, and let equimultiples EF, GH be taken of A, C; I say that EF is the same multiple of B that GH is of D.
For, since EF is the same multiple of A that GH is of C, therefore, as many magnitudes as there are in EF equal to A, so many also are there in GH equal to C. [p. 141]
Let EF be divided into the magnitudes EK, KF equal to A, and GH into the magnitudes GL, LH equal to C; then the multitude of the magnitudes EK, KF will be equal to the multitude of the magnitudes GL, LH.
[Figure]And, since A is the same multiple of B that C is of D, while EK is equal to A, and GL to C, therefore EK is the same multiple of B that GL is of D.
For the same reason
KF is the same multiple of B that LH is of D.
Since, then, a first magnitude EK is the same multiple of a second B that a third GL is of a fourth D, and a fifth KF is also the same multiple of the second B that a sixth LH is of the fourth D, therefore the sum of the first and fifth, EF, is also the same multiple of the second B that the sum of the third and sixth, GH, is of the fourth D. [V. 2]
Therefore etc. Q. E. D.
PROPOSITION 4.
If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.
For let a first magnitude A have to a second B the same ratio as a third C to a fourth D; and let equimultiples E, F be taken of A, C, and G, H other, chance, equimultiples of B, D; I say that, as E is to G, so is F to H.
[Figure]For let equimultiples K, L be taken of E, F, and other, chance, equimultiples M, N of G, H.
Since E is the same multiple of A that F is of C, and equimultiples K, L of E, F have been taken, therefore K is the same multiple of A that L is of C. [V. 3]
For the same reason
M is the same multiple of B that N is of D.
[p. 143]And, since, as A is to B, so is C to D, and of A, C equimultiples K, L have been taken, and of B, D other, chance, equimultiples M, N, therefore, if K is in excess of M, L also is in excess of N, if it is equal, equal, and if less, less. [V. Def. 5]
And K, L are equimultiples of E, F, and M, N other, chance, equimultiples of G, H; therefore, as E is to G, so is F to H. [V. Def. 5]
Therefore etc. Q. E. D.
PROPOSITION 5.
If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.
(5)For let the magnitude AB be the same multiple of the magnitude CD that the part AE subtracted is of the part CF subtracted; I say that the remainder EB is also the same multiple of the remainder FD that the whole AB is of the whole CD.
[Figure]
(10)For, whatever multiple AE is of CF, let EB be made that multiple of CG.
Then, since AE is the same multiple of CF that EB is of GC, therefore AE is the same multiple of CF that AB is of GF. [V. 1]
(15)But, by the assumption, AE is the same multiple of CF that AB is of CD.
Therefore AB is the same multiple of each of the magnitudes GF, CD;
therefore GF is equal to CD.
(20)Let CF be subtracted from each; therefore the remainder GC is equal to the remainder FD. [p. 146]
And, since AE is the same multiple of CF that EB is of GC, and GC is equal to DF,
(25)therefore AE is the same multiple of CF that EB is of FD.
But, by hypothesis,
AE is the same multiple of CF that AB is of CD; therefore EB is the same multiple of FD that AB is of CD.
That is, the remainder EB will be the same multiple of
(30)the remainder FD that the whole AB is of the whole CD.
Therefore etc. Q. E. D. 1
PROPOSITION 6.
If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them.
For let two magnitudes AB, CD be equimultiples of two magnitudes E, F, and let AG, CH subtracted from them be equimultiples of the same two E, F;
[Figure]
I say that the remainders also, GB, HD, are either equal to E, F or equimultiples of them.For, first, let GB be equal to E; I say that HD is also equal to F.
For let CK be made equal to F.
Since AG is the same multiple of E that CH is of F, while GB is equal to E and KC to F, therefore AB is the same multiple of E that KH is of F. [V. 2]
But, by hypothesis, AB is the same multiple of E that CD is of F; therefore KH is the same multiple of F that CD is of F.
Since then each of the magnitudes KH, CD is the same multiple of F,
therefore KH is equal to CD.
[p. 148]Let CH be subtracted from each; therefore the remainder KC is equal to the remainder HD.
But F is equal to KC; therefore HD is also equal to F.
Hence, if GB is equal to E, HD is also equal to F.
Similarly we can prove that, even if GB be a multiple of E, HD is also the same multiple of F.
Therefore etc. Q. E. D.
PROPOSITION 7.
Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.
Let A, B be equal magnitudes and C any other, chance, magnitude; I say that each of the magnitudes A, B has the same ratio to C, and C has the same ratio to each of the magnitudes A, B.
[Figure]For let equimultiples D, E of A, B be taken, and of C another, chance, multiple F.
Then, since D is the same multiple of A that E is of B, while A is equal to B,
therefore D is equal to E.But F is another, chance, magnitude. [p. 149]
If therefore D is in excess of F, E is also in excess of F, if equal to it, equal; and, if less, less.
And D, E are equimultiples of A, B, while F is another, chance, multiple of C;
therefore, as A is to C, so is B to C. [V. Def. 5]I say next that C also has the same ratio to each of the magnitudes A, B.
For, with the same construction, we can prove similarly that D is equal to E; and F is some other magnitude.
If therefore F is in excess of D, it is also in excess of E, if equal, equal; and, if less, less.
And F is a multiple of C, while D, E are other, chance, equimultiples of A, B;
therefore, as C is to A, so is C to B. [V. Def. 5]Therefore etc.
PORISM.From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely. Q. E. D.
PROPOSITION 8.
Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater. [p. 150]
Let AB, C be unequal magnitudes, and let AB be greater; let D be another, chance, magnitude; I say that AB has to D a greater ratio than C has to D, and D has to C a greater ratio than it has to AB.
[Figure]For, since AB is greater than C, let BE be made equal to C; then the less of the magnitudes AE, EB, if multiplied, will sometime be greater than D. [V. Def. 4]
[Case I.]
First, let AE be less than EB; let AE be multiplied, and let FG be a multiple of it which is greater than D; then, whatever multiple FG is of AE, let GH be made the same multiple of EB and K of C; and let L be taken double of D, M triple of it, and successive multiples increasing by one, until what is taken is a multiple of D and the first that is greater than K. Let it be taken, and let it be N which is quadruple of D and the first multiple of it that is greather than K.
Then, since K is less than N first, therefore K is not less than M.
And, since FG is the same multiple of AE that GH is of EB, therefore FG is the same multiple of AE that FH is of AB. [V. 1]
But FG is the same multiple of AE that K is of C;
therefore FH is the same multiple of AB that K is of C; therefore FH, K are equimultiples of AB, C.Again, since GH is the same multiple of EB that K is of C, and EB is equal to C,
therefore GH is equal to K.
[p. 151]But K is not less than M;
therefore neither is GH less than M.And FG is greater than D; therefore the whole FH is greater than D, M together.
But D, M together are equal to N, inasmuch as M is triple of D, and M, D together are quadruple of D, while N is also quadruple of D; whence M, D together are equal to N.
But FH is greater than M, D;
therefore FH is in excess of N,
while K is not in excess of N.And FH, K are equimultiples of AB, C, while N is another, chance, multiple of D;
therefore AB has to D a greater ratio than C has to D. [V. Def. 7]I say next, that D also has to C a greater ratio than D has to AB.
For, with the same construction, we can prove similarly that N is in excess of K, while N is not in excess of FH.
And N is a multiple of D, while FH, K are other, chance, equimultiples of AB, C;
therefore D has to C a greater ratio than D has to AB. [V. Def. 7][Case 2.]
Again, let AE be greater than EB.
Then the less, EB, if multiplied, will sometime be greater than D. [V. Def. 4]
Let it be multiplied, and let GH be a multiple of EB and greater than D; and, whatever multiple GH is of EB, let FG be made the same multiple of AE, and K
[Figure]
of C.Then we can prove similarly that FH, K are equimultiples of AB, C; and, similarly, let N be taken a multiple of D but the first that is greater than FG, so that FG is again not less than M. [p. 152]
But GH is greater than D; therefore the whole FH is in excess of D, M, that is, of N.
Now K is not in excess of N, inasmuch as FG also, which is greater than GH, that is, than K, is not in excess of N.
And in the same manner, by following the above argument, we complete the demonstration.
Therefore etc. Q. E. D.
PROPOSITION 9.
Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the same ratio are equal. [p. 154]
For let each of the magnitudes A, B have the same ratio to C; I say that A is equal to B.
[Figure]For, otherwise, each of the magnitudes A, B would not have had the same ratio to C; [V. 8] but it has;
therefore A is equal to B.Again, let C have the same ratio to each of the magnitudes A, B; I say that A is equal to B.
For, otherwise, C would not have had the same ratio to each of the magnitudes A, B; [V. 8] but it has;
therefore A is equal to B.Therefore etc. Q. E. D.
PROPOSITION 10.
Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.
For let A have to C a greater ratio than B has to C; I say that A is greater than B.
[Figure]For, if not, A is either equal to B or less.
Now A is not equal to B; for in that case each of the magnitudes A, B would have had the same ratio to C; [V. 7] but they have not;
therefore A is not equal to B.Nor again is A less than B; for in that case A would have had to C a less ratio than B has to C; [V. 8] but it has not;
therefore A is not less than B.But it was proved not to be equal either;
therefore A is greater than B.Again, let C have to B a greater ratio than C has to A; I say that B is less than A.
For, if not, it is either equal or greater.
Now B is not equal to A; for in that case C would have had the same ratio to each of the magnitudes A, B; [V. 7] but it has not;
therefore A is not equal to B.
[p. 156]Nor again is B greater than A; for in that case C would have had to B a less ratio than it has to A; [V. 8] but it has not;
therefore B is not greater than A.But it was proved that it is not equal either;
therefore B is less than A.Therefore etc. Q. E. D.
[p. 158]PROPOSITION 11.
Ratios which are the same with the same ratio are also the same with one another.
For, as A is to B, so let C be to D, and, as C is to D, so let E be to F; I say that, as A is to B, so is E to F.
[Figure]For of A, C, E let equimultiples G, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.
Then since, as A is to B, so is C to D, and of A, C equimultiples G, H have been taken, and of B, D other, chance, equimultiples L, M, therefore, if G is in excess of L, H is also in excess of M, if equal, equal, and if less, less.
Again, since, as C is to D, so is E to F, and of C, E equimultiples H, K have been taken, and of D, F other, chance, equimultiples M, N, therefore, if H is in excess of M, K is also in excess of N, if equal, equal, and if less, less.
But we saw that, if H was in excess of M, G was also in excess of L; if equal, equal; and if less, less; so that, in addition, if G is in excess of L, K is also in excess of N, if equal, equal, and if less, less.
And G, K are equimultiples of A, E, while L, N are other, chance, equimultiples of B, F;
therefore, as A is to B, so is E to F.Therefore etc. Q. E. D.
PROPOSITION 12.
If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
Let any number of magnitudes A, B, C, D, E, F be proportional, so that, as A is to B, so is C to D and E to F; I say that, as A is to B, so are A, C, E to B, D, F.
[Figure]For of A, C, E let equimultiples G, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.
Then since, as A is to B, so is C to D, and E to F, and of A, C, E equimultiples G, H, K have been taken, and of B, D, F other, chance, equimultiples L, M, N, therefore, if G is in excess of L, H is also in excess of M, and K of N, if equal, equal, and if less, less; so that, in addition, if G is in excess of L, then G, H, K are in excess of L, M, N, if equal, equal, and if less, less. [p. 160]
Now G and G, H, K are equimultiples of A and A, C, E, since, if any number of magnitudes whatever are respectively equimultiples of any magnitudes equal in multitude, whatever multiple one of the magnitudes is of one, that multiple also will all be of all. [V. 1]
For the same reason L and L, M, N are also equimultiples of B and B, D, F;
therefore, as A is to B, so are A, C, E to B, D, F. [V. Def. 5]Therefore etc. Q. E. D.
PROPOSITION 13.
If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth.
For let a first magnitude A have to a second B the same ratio as a third C has to a fourth D, and let the third C have to the fourth D a greater ratio than a fifth E has to a sixth F; I say that the first A will also have to the second B a greater ratio than the fifth E to the sixth F.
[Figure]For, since there are some equimultiples of C, E, and of D, F other, chance, equimultiples, such that the multiple of C is in excess of the multiple of D, [p. 161] while the multiple of E is not in excess of the multiple of F, [V. Def. 7] let them be taken, and let G, H be equimultiples of C, E, and K, L other, chance, equimultiples of D, F, so that G is in excess of K, but H is not in excess of L; and, whatever multiple G is of C, let M be also that multiple of A, and, whatever multiple K is of D, let N be also that multiple of B.
Now, since, as A is to B, so is C to D, and of A, C equimultiples M, G have been taken, and of B, D other, chance, equimultiples N, K, therefore, if M is in excess of N, G is also in excess of K, if equal, equal, and if less, less. [V. Def. 5]
But G is in excess of K; therefore M is also in excess of N.
But H is not in excess of L; and M, H are equimultiples of A, E, and N, L other, chance, equimultiples of B, F;
therefore A has to B a greater ratio than E has to F. [V. Def. 7]Therefore etc. Q. E. D.
PROPOSITION 14.
If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.
For let a first magnitude A have the same ratio to a second B as a third C has to a fourth D; and let A be greater than C; I say that B is also greater than D.
[Figure]For, since A is greater than C, and B is another, chance, magnitude, therefore A has to B a greater ratio than C has to B. [V. 8]
But, as A is to B, so is C to D;
therefore C has also to D a greater ratio than C has to B. [V. 13]But that to which the same has a greater ratio is less; [V. 10]
therefore D is less than B; so that B is greater than D.Similarly we can prove that, if A be equal to C, B will also be equal to D; and, if A be less than C, B will also be less than D.
Therefore etc. Q. E. D.
PROPOSITION 15.
Parts have the same ratio as the same multiples of them taken in corresponding order.
For let AB be the same multiple of C that DE is of F; I say that, as C is to F, so is AB to DE.
[Figure]For, since AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, so many are there also in DE equal to F.
Let AB be divided into the magnitudes AG, GH, HB equal to C, and DE into the magnitudes DK, KL, LE equal to F; then the multitude of the magnitudes AG, GH, HB will be equal to the multitude of the magnitudes DK, KL, LE.
And, since AG. GH, HB are equal to one another, and DK, KL, LE are also equal to one another, therefore, as AG is to DK, so is GH to KL, and HB to LE. [V. 7]
Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [V. 12]
therefore, as AG is to DK, so is AB to DE.
[p. 164]But AG is equal to C and DK to F;
therefore, as C is to F, so is AB to DE.Therefore etc. Q. E. D.
PROPOSITION 16.
If four magnitudes be proportional, they will also be proportional alternately.
Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D; I say that they will also be so alternately, that is, as A is to C, so is B to D.
[Figure]For of A, B let equimultiples E, F be taken, and of C, D other, chance, equimultiples G, H.
Then, since E is the same multiple of A that F is of B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is E to F.
But as A is to B, so is C to D; therefore also, as C is to D, so is E to F. [V. 11]
Again, since G, H are equimultiples of C, D, therefore, as C is to D, so is G to H. [V. 15]
But, as C is to D, so is E to F; therefore also, as E is to F, so is G to H. [V. 11]
But, if four magnitudes be proportional, and the first be greater than the third,
the second will also be greater than the fourth;
if equal, equal; and if less, less. [V. 14]Therefore, if E is in excess of G, F is also in excess of H, if equal, equal, and if less, less. [p. 165]
Now E, F are equimultiples of A, B, and G, H other, chance, equimultiples of C, D;
therefore, as A is to C, so is B to D. [V. Def. 5]Therefore etc. Q. E. D. 2
PROPOSITION 17.
If magnitudes be proportional componendo, they will also be proportional separando.
Let AB, BE, CD, DF be magnitudes proportional componendo, so that, as AB is to BE, so is CD to DF; I say that they will also be proportional separando, that is, as AE is to EB, so is CF to DF.
For of AE, EB, CF, FD let equimultiples GH, HK, LM, MN be taken, and of EB, FD other, chance, equimultiples, KO, NP. [p. 167]
Then, since GH is the same multiple of AE that HK is of EB, therefore GH is the same multiple of AE that GK is of AB. [V. 1]
But GH is the same multiple of AE that LM is of CF; therefore GK is the same multiple of AB that LM is of CF.
[Figure]Again, since LM is the same multiple of CF that MN is of FD, therefore LM is the same multiple of CF that LN is of CD. [V. 1]
But LM was the same multiple of CF that GK is of AB; therefore GK is the same multiple of AB that LN is of CD.
Therefore GK, LN are equimultiples of AB, CD.
Again, since HK is the same multiple of EB that MN is of FD,
and KO is also the same multiple of EB that NP is of FD, therefore the sum HO is also the same multiple of EB that MP is of FD. [V. 2]And, since, as AB is to BE, so is CD to DF, and of AB, CD equimultiples GK, LN have been taken, and of EB, FD equimultiples HO, MP, therefore, if GK is in excess of HO, LN is also in excess of MP, if equal, equal, and if less, less.
Let GK be in excess of HO; then, if HK be subtracted from each,
GH is also in excess of KO.But we saw that, if GK was in excess of HO, LN was also in excess of MP;
therefore LN is also in excess of MP,
[p. 168] and, if MN be subtracted from each,
LM is also in excess of NP;
so that, if GH is in excess of KO, LM is also in excess of NP.Similarly we can prove that, if GH be equal to KO, LM will also be equal to NP, and if less, less.
And GH, LM are equimultiples of AE, CF, while KO, NP are other, chance, equimultiples of EB, FD;
therefore, as AE is to EB, so is CF to FD.Therefore etc. Q. E. D.
PROPOSITION 18.
If magnitudes be proportional separando, they will also be proportional componendo.
Let AE, EB, CF, FD be magnitudes proportional separando, so that, as AE is to EB, so is CF to FD;
[Figure]
I say that they will also be proportional componendo, that is, as AB is to BE, so is CD to FD.For, if CD be not to DF as AB to BE, then, as AB is to BE, so will CD be either to some magnitude less than DF or to a greater.
First, let it be in that ratio to a less magnitude DG.
Then, since, as AB is to BE, so is CD to DG, they are magnitudes proportional componendo;
so that they will also be proportional separando. [V. 17]Therefore, as AE is to EB, so is CG to GD.
But also, by hypothesis,
as AE is to EB, so is CF to FD.Therefore also, as CG is to GD, so is CF to FD. [V. 11]
But the first CG is greater than the third CF;
therefore the second GD is also greater than the fourth FD. [V. 14]But it is also less: which is impossible.
Therefore, as AB is to BE, so is not CD to a less magnitude than FD. [p. 170]
Similarly we can prove that neither is it in that ratio to a greater;
it is therefore in that ratio to FD itself.Therefore etc. Q. E. D.
PROPOSITION 19
If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.
For, as the whole AB is to the whole CD, so let the part AE subtracted be to the part CF subtracted; I say that the remainder EB will also be to the remainder FD as the whole AB to the whole CD.
[Figure]For since, as AB is to CD, so is AE to CF, alternately also, as BA is to AE, so is DC to CF. [V. 16]
And, since the magnitudes are proportional componendo, they will also be proportional separando, [V. 17] that is, as BE is to EA, so is DF to CF, and, alternately,
as BE is to DF, so is EA to FC. [V. 16]But, as AE is to CF, so by hypothesis is the whole AB to the whole CD.
Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD. [V. 11]
Therefore etc. [
PORISM.From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.
] Q. E. D.PROPOSITION 20.
If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; and, if less, less. [p. 176]
Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, 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; and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and, if less, less.
[Figure]For, since A is greater than C, and B is some other magnitude, and the greater has to the same a greater ratio than the less has, [V. 8] therefore A has to B a greater ratio than C has to B.
But, as A is to B, so is D to E, and, as C is to B, inversely, so is F to E; therefore D has also to E a greater ratio than F has to E. [V. 13]
But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater; [V. 10]
therefore D is greater than F.Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.
Therefore etc. Q. E. D.
PROPOSITION 21.
If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less.
Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that,
as A is to B, so is E to F,
and, as B is to C, so is D to E, and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and if less, less.
[Figure]For, since A is greater than C, and B is some other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8]
But, as A is to B, so is E to F, and, as C is to B, inversely, so is E to D. Therefore also E has to F a greater ratio than E has to D. [V. 13]
But that to which the same has a greater ratio is less; [V. 10]
therefore F is less than D; therefore D is greater than F.
[p. 179]Similarly we can prove that,
if A be equal to C, D will also be equal to F;
and if less, less.Therefore etc. Q. E. D.
PROPOSITION 22.
If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali.
Let there be any number of magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two together 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 they will also be in the same ratio ex aequali,
A is to C, so is D to F>.
[p. 180]For of A, D let equimultiples G, H be taken, and of B, E other, chance, equimultiples K, L; and, further, of C, F other, chance, equimultiples M, N.
[Figure]Then, since, as A is to B, so is D to E, and of A, D equimultiples G, H have been taken, and of B, E other, chance, equimultiples K, L,
therefore, as G is to K, so is H to L. [V. 4]For the same reason also,
as K is to M, so is L to N.Since, then, there are three magnitudes G, K, M, and others H, L, N equal to them in multitude, which taken two and two together are in the same ratio, therefore, ex aequali, if G is in excess of M, H is also in excess of N; if equal, equal; and if less, less. [V. 20]
And G, H are equimultiples of A, D,
and M, N other, chance, equimultiples of C, F.Therefore, as A is to C, so is D to F. [V. Def. 5]
Therefore etc. Q. E. D.
PROPOSITION 23.
If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, they will also be in the same ratio ex aequali.
Let there be three magnitudes A, B, C, and others equal to them in multitude, which, taken two and two together, are in the same proportion, namely D, E, F; and let the proportion of them be perturbed, so that,
as A is to B, so is E to F,
and, as B is to C, so is D to E; I say that, as A is to C, so is D to F.
[Figure]Of A, B, D let equimultiples G, H, K be taken, and of C, E, F other, chance, equimultiples L, M, N.
Then, since G, H are equimultiples of A, B, and parts have the same ratio as the same multiples of them, [V. 15]
therefore, as A is to B, so is G to H.For the same reason also,
as E is to F, so is M to N.
And, as A is to B, so is E to F;
therefore also, as G is to H, so is M to N. [V. 11]Next, since, as B is to C, so is D to E, alternately, also, as B is to D, so is C to E. [V. 16]
And, since H, K are equimultiples of B, D, and parts have the same ratio as their equimultiples,
therefore, as B is to D, so is H to K. [V. 15]
[p. 182]But, as B is to D, so is C to E;
therefore also, as H is to K, so is C to E. [V. 11]Again, since L, M are equimultiples of C, E,
therefore, as C is to E, so is L to M. [V. 15]But, as C is to E, so is H to K;
therefore also, as H is to K, so is L to M, [V. 11]
and, alternately, as H is to L, so is K to M. [V. 16]But it was also proved that,
as G is to H, so is M to N.Since, then, there are three magnitudes G, H, L, and others equal to them in multitude K, M, N, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less. [V. 21]
And G, K are equimultiples of A, D, and L, N of C, F.
Therefore, as A is to C, so is D to F.
Therefore etc. Q. E. D.
PROPOSITION 24.
If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.
Let a first magnitude AB have to a second C the same ratio as a third DE has to a fourth F; and let also a fifth BG have to
[Figure]
the second C the same ratio as a sixth EH has to the fourth F; I say that the first and fifth added together, AG, will have to the second C the same ratio as the third and sixth, DH, has to the fourth F. [p. 184]For since, as BG is to C, so is EH to F, inversely, as C is to BG, so is F to EH.
Since, then, as AB is to C, so is DE to F,
and, as C is to BG, so is F to EH,
therefore, ex aequali, as AB is to BG, so is DE to EH. [V. 22]And, since the magnitudes are proportional separando, they will also be proportional componendo; [V. 18]
therefore, as AG is to GB, so is DH to HE.But also, as BG is to C, so is EH to F; therefore, ex aequali, as AG is to C, so is DH to F. [V. 22]
Therefore etc. Q. E. D.
[p. 185]PROPOSITION 25.
If four magnitudes be proportional, the greatest and the least are greater than the remaining two.
Let the four magnitudes AB, CD, E, F be proportional so that, as AB is to CD, so is E to F, and let AB be the greatest of them and F the least; I say that AB, F are greater than CD, E.
[Figure]For let AG be made equal to E, and CH equal to F.
Since, as AB is to CD, so is E to F, and E is equal to AG, and F to CH,
therefore, as AB is to CD, so is AG to CH.And since, as the whole AB is to the whole CD, so is the part AG subtracted to the part CH subtracted,
the remainder GB will also be to the remainder HD as the whole AB is to the whole CD. [V. 19]But AB is greater than CD;
therefore GB is also greater than HD.And, since AG is equal to E, and CH to F, therefore AG, F are equal to CH, E.
And if, GB, HD being unequal, and GB greater, AG, F be added to GB and CH, E be added to HD,
it follows that AB, F are greater than CD, E.Therefore etc. Q. E. D.
NOTES
1 let EB be made that multiple of CG, tosautaplasion gegonetô kai to EB tou GÊ. From this way of stating the construction one might suppose that CG was given and EB had to be found equal to a certain multiple of it. But in fact EB is what is given and CG has to be found, i.e. CG has to be constructed as a certain submultiple of EB.
2 “Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D.” In a number of expressions like this it is absolutely necessary, when translating into English, to interpolate words which are not in the Greek. Thus the Greek here is: Hestô tessara megethê analogon ta A, B, G, D, hôs to A pros to B, houtôs to G pros to D, literally “Let A, B, C, D be four proportional magnitudes, as A to B, so C to D.” The same remark applies to the corresponding expressions in the next
PROPOSITIONs, V. 17, 18, and to other forms of expression in V. 20-23 and later
PROPOSITIONs: e.g. in V. 20 we have a phrase meaning literally “Let there be magnitudes...which taken two and two are in the same ratio, as A to B, so D to E,” etc.: in V. 21 “(magnitudes)...which taken two and two are in the same ratio, and let the proportion of them be perturbed, as A to B, so E to F,” etc. In all such cases (where the Greek is so terse as to be almost ungrammatical) I shall insert the words necessary in English, without further remark.
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