BOOK V.
DEFINITIONS.
1
A magnitude is a
part of a magnitude, the less of the greater, when it measures the greater.
2
The greater is a
multiple of the less when it is measured by the less.
3
A
ratio is a sort of relation in respect of size between two magnitudes of the same kind.
4
Magnitudes are said to
have a ratio to one another which are capable, when multiplied, of exceeding one another.
5
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.
6
Let magnitudes which have the same ratio be called
proportional.
7
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.
8
A proportion in three terms is the least possible.
9
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.
10
When four magnitudes are <continuously> 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.
11
The term
corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.
12
Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.
13
Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.
14
Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.
15
Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.
16
Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.
17
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.
18
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.
BOOK V. 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.
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; 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; 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; 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.
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.
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.
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.
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.
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.
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]
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.
Let
CF be subtracted from each; therefore the remainder
GC is equal to the remainder
FD.
And, since
AE is the same multiple of
CF that
EB is of
GC, and
GC is equal to
DF,
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
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; 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.
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.
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.
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.
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.
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.
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
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.
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.
For let each of the magnitudes
A,
B have the same ratio to
C; I say that
A is equal to
B.
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.
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.
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.
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.
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.
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.
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.
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, 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.
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.
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.
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.
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.
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.
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.
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,
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; 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.
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.
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.
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.
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.
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.
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,
<that is, as A is to C, so is D to F>.
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.
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.
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]
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 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.
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.
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.
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.