# The Elements of Euclid for the Use of Schools and Colleges/Book V

*BOOK V*.

1. A less magnitude is said to be a part of a greater magnitude, when the less measures the greater ; that is, when the less is contained a certain number of times ex- actly in the greater.

2. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less; that is, when the greater contains the less a certain number of times exactly.

3. Ratio is a mutual relation of two magnitudes of the same kind to one another in respect of quantity.

4. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.

5. The first of four magnitudes is said to have the same ratio to the second, that the third has to the fourth, when any equimltiples whatever of the first and the third being taken, and any equimultiples whatever of the second and the fourth, if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth, and if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth, and if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.

6. Magnitudes which have the same ratio are called proportionals. When four magnitudes are proportionals it is usually expressed by saying, the first is to the second as the third is to the fourth.

7. When of the equimultiples of four magnitudes, taken as in the fifth definition the multiple of the first is greater than the multiple of the second, but the multiple of the third is not greater than the multiple of the fourth, then the first is said to have to the second a greater ratio than the third has to the fourth ; and the third is said to havo to the fourth a less ratio than the first has to the second.

8. Analogy, or proportion, is the similitude of ratios.

9. Proportion consists in three terms at least.

10. When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.

[The second magnitude is said to be a *mean propor- *
tional* between the first and the third.] *

11. When four magnitudes are continued proportionals, the first is said to have to the fourth, the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c. increasing the denomination still by unity, in any num- ber of proportionals.

*Definition of compound ratio*. When there are any
number of magnitudes of the same kind, the first is said to
have to the last of them, the ratio which is compounded of
the ratio which the first has to the second, and of the ratio
which the second has to the third, and of the ratio which
the third has to the fourth, and so on unto the last mag-
nitude.

For example, *A*, *B*, *C*, *D* be four magnitudes of the
same kind, the first *A* is said to have to the last *D*, the
ratio compounded of the ratio of *A* to *B*, and of the ratio
of *B* to *C*, and of the ratio of *C* to *D* ; or, the ratio of *A* to
*B* is said to be compounded of the ratios of *A* to *B*, *B* to
*C*, and *C* to *D*.

And if *A* has to *B* the same ratio that *E* has to *F*;
and *B* to *C* the same ratio that *G* has to *H* ; and *C* to *D*
the same ratio that *K* has to *L* ; then, by this definition,
*A* is said to have to *D* the ratio compounded of ratios which
are the same with the ratios of *E* to *F*, *G* to *H*, and *K* to *L*.

And the same thing is to be understood when it is more
briefly expressed by saying, *A* has to *D* the ratio com-
pounded of the ratios of *E* to *F*, *G* to *H*, and *K* to *L*.

In like manner, the same things being supposed, if *M*
has to *N* the same ratio that *A* has to *D* ; then, for the
sake of shortness, *M* is said to have to *N* the ratio com-
pounded of the ratios of *E* to *F*, *G* to *H*, and *K* to *L*.

12. In proportionals, the antecedent terms are said to be homologous to one another ; as also the consequents to one another.

Geometers make use of the following technical words, to signify certain ways of changing either the order or the magnitude of proportionals, so that they continue still to be proportionals.

13. *Permutando*, or *alternando*, by permutation or
alternately; when there are four proportionals, and it is
inferred that the first is to the third, as the second is to
the fourth. V. 16.

14. *Invertendo*, by inversion; when there are four
proportionals, and it is inferred, that the second is to the
first as the fourth is to the third. V. *B*.

15. *Componendo*, by composition ; when there are four
proportionals, and it is inferred, that the first together
with the second, is to the second, as the third together
with the fourth, is to the fourth. V. 18.

16. *Dividendo*, by division ; when there are four pro-
portionals, and it is inferred, that the excess of the first
above the second, is to the second, as the excess of the
third above the fourth, is to the fourth. V. 17.

17. *Convertendo*, by conversion; when there are four
proportionals, and it is inferred, that the first is to its
excess above the second, as the third is to its excess above
the fourth. V. *E*.

18. *Ex aequali distantia*, or *ex aequo*, from equality of
distance ; when there is any number of magnitudes more
than two, and as many others, such that they are propor-
tionals when taken two and two of each rank, and it is
inferred, that the first is to the last of the first rank of
magnitudes, as the first is to the last of the others. Of this there are the two following kinds, which arise
from the different order in which the magnitudes are taken,
two aniftwo.

19. *Ex aequali*. This term is used simply by itself,
when the first magnitude is to the second of the first rank,
as the first is to the second of the other rank; and the
second is to the third of the first rank, as the second is to
the third of the other ; and so on in order ; and the inference
is that mentioned in the preceding definition. V. 22.

20. *Ex aequali in proportione perturbata seu inordinata*,
from equality in perturbate or disorderly proportion. This
term is used when the first magnitude is to the second of
Ihe first rank, as the last but one is to the last of the second
rank ; and the second is to the third of the first rank, as the
last but two is to the last but one of the second rank ; and
the third is to the fourth of the first rank, as the last but
three is to the last but two of the second rank ; and so on
in a crogs order ; and the inference is that mentioned in the
eighteenth definition. V. 23.

1. Equimultiples of the same, or of equal magnitudes, are equal to one another.

2. Those magnitudes, of which the same or equal mag- nitudes are equimultiples, are equal to one another.

3. A multiple of a greater magnitude is greater than the same multiple of a less.

4. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other

magnitude. *THEOREM*.

*If any number of magnitudes be equimultiples of as many, each of each; whatever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other*.

Let any number of magnitudes *AB*, *CD* be equimultiples of as many others *E*, *F*, each of each: whatever multiple *AB* is of *E*, the same multiple shall *AB* and *CD* together, be of *E* and *F* together.

For, because *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 are there in *CD* equal to *F*.

Divide *AB* into the magnitudes *AG*, *GB* each equal to *E*; and *CD* into the magnitudes *CH*, *HD*, each equal to *F*.

Therefore the number of the magnitudes *CH*, *HD*, will be equal to the number of the magnitudes *AG*, *GB*.

And, because *AG* is equal to *E*, and *CH* equal to *F*, therefore *AG* and *CH* together are equal to *E* and *F* together;

and because *GB* is equal to *E*, and *HD* equal to *F*, therefore *GB* and *HD* together are equal to *E* and *F* together. [*Axiom* 2.

Therefore as many magnitudes as there are in *AB* equal to *E*, so many are there in *AB* and *CD* together equal to *E* and *F* together.

Therefore whatever multiple *AB* is of *E*, the same multiple is *AB* and *CD* together, of *E* and *F* together.

Wherefore, *if any number of magnitudes* &c. q.e.d.

*THEOREM*.

*If the first be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; the first together with the fifth shall be the same multiple of the second that the third together with the sixth is of the fourth*. Let *AB* the first be the same multiple of *C* the second, that *DE* the third is of *F* the fourth, and let *BG* the fifth be the same multiple of *C* the second, that *EH* the sixth is of *F* the fourth: *AG* the first together with the fifth, shall be the same multiple of *C* the second, that *DH*, the third together with the sixth, is of *F* the fourth.

For, because *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 in *BE* equal to *F*.

For the same reason, as many magnitudes as there are in *BG* equal to *C*, so many are there in *EH* equal to *F*.

Therefore as many magnitudes as there are in the whole *AG* equal to *C*, so many are there in the whole *DH* equal to *F*.

Therefore *AG* the same multiple of *C* that *DH* is of *F*.

Wherefore, *if the first de the same multiple* &c. q.e.d.

Corollary. From this it is plain, that if any number of magnitudes *AB*, *BG*, *GH* be multiples of another *C*; and as many *DE*, *EK*, *KL* be the same multiples of *F*, each of each; then the whole of the first, namely, *AH*, is the same multiple of *C*, that the whole of the last, namely, *DL*, is of *F*.

*THEOREM*.

*If the first he the same multiple of the second that the third is of the fourth, and if of the first and the third there he taken equimultiples, these shall he equimultiples, the one of the second, and the other of the fourth*. Let *A* the first be the same multiple of *B* the second, that *C* the third is of *D* the fourth; and of *A* and *C* let the equimultiples *EF* and *GH* be taken: *EF* shall be the same multiple of *B* that *GH* is of *D*.

For, because *EF* is the same multiple of *A* that *GH* is of *D*, [*Hypothesis*.

as many magnitudes as there are in *EF* equal to *A*, so many are there in *GH* equal to *C*.

Divide *EF* into the magnitudes *EK*, *KF*, each equal to *A*; and *GH* into the magnitudes *GL*, *LH*, each equal to *C*.

Therefore the number of the magnitudes *EK*,*KL*, will be equal to the number of the magnitudes *GL*, *LH*.

And because *A* is the same multiple of *B* that *C* of *D*, [*Hypothesis*

and that *EK* is equal to *A* and *GL* is equal to *C*; [*Constr*

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*.

Therefore because *EK* the first is the same multiple of *B* the second, that *GL* the third is of *D* the fourth,

and that *KF* the fifth is the same multiple of *B* the second that *LH* the sixth is of *D* the fourth;*EF* the first together with the fifth, is the same multiple of *B* the second, that *GH* the third together with the sixth, is of *D* the fourth. [V. 2.

In the same manner, if there be more parts in *EF* equal to *A* and in *GH* equal to *C*, it may be shewn that *EF* is the same multiple of *B* that *GH* is of *D*. [V. 2, *Cor*.

Wherefore, *if the first* &c. q.e.d.

PROPOSITION 4. *THEOREM*.

*If the first have the same ratio to the second that the third has to the fourth and if there be taken any equi-* *multiples whatever of the first and the third, and also any equimultiples whatever of the second and the fourth then the multiple of the first shall have the same ratio to the multiple of the second, that the multiple of the third has to the multiple of the fourth*.

Let *A* the first have to *B* the second, the same ratio that *C* the third has to *D* the fourth; and of *A* and *C* let there be taken any equimultiples whatever *E* and *F*, and *B* and *D* any equimultiples whatever *G* and *H*: *E* shall have the same ratio to *G* that *F* has to *H*.

Take of *E* and *F* any equimultiples whatever *K* and *L*, and of *G* and *H* any equimultiples whatever *M* and *N*.

Then, because *E* is the same multiple of *A* that *F* is of *C*, and of *E* and *F* have been taken equimultiples *K* and *L';*
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 because *A* is to *B* as *C* is to *D*, [*Hypothesis*.

and of *A* and *C* have been taken certain equimultiples *K* and *L*, and of *B* and *D* have been taken certain equimultiples *M* and *N*;

therefore if *K* be greater than *M*,*L* is greater than *N*-, and if equal, equal; and if less, less. [V. *Definition* 5.

But *K* and *L* are any equimultiples whatever of *E* and *F*, and *M* and *N* are any equimultiples whatever of *G* and *H*; therefore *E* is to *G'* as *F* is to *H*. [V. *Definition* 5.

Wherefore, *if the first* &c. q.e.d.

Corollary. Also if the first have the same ratio to the second that the third has to the fourth, then any equimultiples whatever of the first and third shall have the same ratio to the second and fourth: and the first and third shall have the same ratio to any equimultiples what- ever of the second and fourth.

Let *A* the first have the same ratio to *B* the second,
that *C* the third has to *D* the fourth ; and of *A* and *C* let
there be taken any equimultiples whatever *E* and *F*: *E*
shall be to *B* as *F* is to *D*.

Take of *E* and *F* any equimultiples whatever *K* and *L*,
and of *B* and *D* any equimultiples whatever *G* and *H*.

Then it may be shewn, as before, that *K* is the same
multiple of *A* that *L* is of *C*.

And because *A* is to *B* as *C* is to *D*, [*Hypothesis*.

and of *A* and *C* have been taken certain equimul
-tiples 'K* *
and *L*, and of *B* and *D* have been taken certain equimul-
tiples *G* and *H*;

therefore if *K* be greater than *G*, *L* is greater than *H*; and
if equal, equal ; and if less, less. [V. *Definition* 5.

But *K* and *L* are any equimultiples whatever of *E* and *F*.
and *G* and *H* are any equimultiples whatever of *B* and *D*,

therefore *E* is to *B* as *F* is to *D*. [V. *Definition* 5.

In the same way the other case may be demonstrated.

*THEOREM*.

*If one magnitude he the same multiple of another that a magnitude taken from the first is of a magnitude taken from the other, the remainder shall be the same multiple of the remainder that the whole is of the whole*.

Let *AB* be the same multiple of *CD*, that *AE* taken
from the first, is of *CF* taken from the other : the remain-
der *EB* shall be the same multiple of the remainder *FD*,
that the whole *AB* is of the whole *CD*.

Take *AG* the same multiple of *FD*, that *AE*is of *CF*;

therefore *AE* is the same multiple of *CF* that *EG* is
of *CD*. [V. 1.

But *AE* is the same multiple of *CF* that *AB* is of *CD*;

therefore *EG* is the same multiple of *CD* that *AB* is
of *CD*;

therefore *EG* is equal to *AB*. [V. *Axiom* 1.
From each of these take the common magnitude *AE*; then the remainder *AG* is equal to the remainder *EB*;

Then, because *AE* is the same multiple of *CF* that *AG* is of *ED*, [*Construction*. and that *AG* is equal to *EB*;

therefore *AE* is the same multiple of *CF* that *EB* is of *ED*.

But *AE* is the same multiple of *CF* that *AB* is of *CD*; ['*Hypothesis*.

therefore *EB* is the same multiple of *FD* that *AB* is of *CD*.

Wherefore, *if one magnitude' &c q.e.d.*

*THEOREM*.

*If two magnitudes he equimultiples of two others and if equimultiples of these he taken from, the first two, the remainders shall he either equal to these others, or equi-multiples of them*.

Let the two magnitudes *AB*, *CD* be equimultiples of the two *E*, *F*; and let *AG*, *CH*, taken from the first two, be equimultiples of the same *E*, *F*: the remainders *GB'*, *HD* shall be either equal to *E*, *F*, or equimultiples of them.

First, let *GB* be equal to *E*: *HD* shall be equal to *F*.

Make *CK* equal to *F*.

Then, because *AG* the same multiple of *E* that *CH* is of *F*, [*Hyp*.

and that *GB* is equal to *E*, and *CK* is equal to *F*;

therefore *AB* is the same multiple of *E* that is *CH* is of *F*.

But *AB* is the same multiple of *E* that *CD* is of *F*; [*Hypothesis*.

therefore *KH* is the same multiple of *E* that *CD* is of *F*;

therefore *KH* is equal to *CD*. [V. *Axiom* 1.

From each of these take the common magnitude *CH*; then the remainder *CK* is equal to the remainder *HD*. But *CK* is equal to *F*; [*Construction*.

therefore HD is equal to F.

Next let *GB* be a multiple of *E*: *HD* shall be the same multiple of *F*.

Make *CK* the same multiple of *F* that *GB* is of *E*. Then, because *AG* is the same multiple of *E* that *CH* is of *F*, [*Hypothesis*.

and *GB* is the same multiple of *E* that *CK* is of *F* [*Constr*.

therefore *AB* is the same multiple of *E* that *KM* is of *F*. [V. 2.

But *AB* is the same multiple of *E* that *CD* is of *F*; [*Hyp*.

therefore *KH* is the same multiple of *F* that *CD* is of *F*; [*Hyp*

therefore *KH* is equal to CD. [V. *Axiom* 1.

From each of these take the common magnitude *CH*; then the remainder *CK* is equal to the remainder *HD*.

And because *CK* is the same multiple of *F* that *GB* is of *E*, [*Construction*.

and that *CK* is equal to *HD*;

therefore *HD* is the same multiple of *F* that *GB* is of *E*.

Wherefore, *if two magnitudes* &c. q.e.d.

*THEOREM*.

*If the first of four magnitudes have the same ratio to the second that the third has to the fourth, then, if the first be greater than the second, the third shall also be greater than the fourth, and if equal equal, and if less less*.

Take any equimultiples of each of them, as the doubles of each.

Then if the double of the first be greater than the double of the second, the double of the third is greater than the double of the fourth. [V. Definition 5.

But if the first be greater than the second, the double of the first is greater than the double of the second therefore the double of the third is greater than the double of the fourth,

and therefore the third is greater than the fourth.

In the same manner, if the first be equal to the second, or less than it, the third may be shewn to be equal to the fourth, or less than it.

Wherefore, *if the first* &c. q.e.d.

*THEOREM*.

*If four magnitudes he proportionals, they shall also be proportionals when taken inversely*.

Let *A* be to *B* as *C* is to *D*: then also, inversely, *B* shall be to *A* as *D* is to *C*.

Take of *B* and *D* any equimultiples whatever *E* and *F*;

and of *A* and *G* any equimultiples whatever *G* and *H*.

First, let *E* be greater than *G*, then *G* is less than *E*.

Then, because *A* is to *B* as *C* is to *D*; [*Hypothesis*.

and of *A* and *C* the first and third, *G* and *H* are equimultiples; and of *B* and *D* the second and fourth, *E* and *F* are equimultiples;

and that *G* is less than *E*;

therefore *H* is less than *F*; [V. *Def*. 5.

that is, *F* is greater than *H*.

Therefore, if *E* be greater than *G*, *F* is greater than *H*.

In the same manner, if *E* be equal to *G*, *F* may be shewn to be equal to *H* and if less, less.

But *E* and *F* are any equimultiples whatever of *B and D, and G and H are any equimultiples wliatever of *A* and *C*; [*Construction
therefore

*B*is to

*A*as

*D*is to

*C*. [V.

*Definition*5.

*if four magnitudes*&c. q.e.d.

*C. THEOREM*.

*If the first he the same multiple of the second or the same part of it, that the third is of the fourth, the first shall he to the second as the third is to the fourth*.

First, let *A* be the same multiple of *B* that *C* is of *D*: *A* shall be to *B* as *C* is to *D*.

Take of *A* and *C* any equimultiples whatever *E* and *F*; and of *B* and *D* any equimultiples whatever *G* and *H*.

Then, because *A* is the same multiple of *B* that *C* is of *D*; [*Hypothesis*.

and that *E* is the same multiple of *A* that *F* is of *C*; [*Construction*

therefore *E* is the same multiple of *B* that *F* is of *D*; [V. 3.

that is, *E* and *F* are equimultiples of *B* and *D*.

But *G* and *H* are equimultiples of *B* and *D*; [*Construction*.

therefore if *E* be a greater multiple of *B* than *C* is of *B*, *F* is a greater multiple of *D* than *H* is of *D*;

that is, if *E* be greater than *G*, *F* is greater than *H*.

In the same manner, if *E* be equal to *G*, *F* may be shewn to be equal to *H*; and if less, less.

But *E* and *F* are any equimultiples whatever of *A* and *C*, and *G* and *H* are any equimultiples whatever of *B* and *D*; [*Construction*.

therefore *A* is to *B* as *C* is to *D*. [V. *Definition* 5.

Next, let *A* be the same part of *B* that *C* is of *D*: *A* shall be to *B* as *C* is to *D*.

For, since *A* is the same part of *B* that *C* is of *D*,

therefore *B* is the same multiple of *A* that *D* is of *C*;

therefore, by the preceding case, *B* is to *A* as *D* is to *C*;

therefore, inversely, *A* isto *B* as *C* is to *D*.

*if the first*&c. q,e.d.

*D. THEOREM*.

*If the first he to the second as the third is to the fourth, and if the first he a multijyle, or a part, of the second, the third shall he the same multiple, or the same part, of the fourth*.

Let *A* be to *B* as *C* is to *D*.

And first, let *A* be a multiple of *B*: *C* shall be the same multiple of *D*.

Take *E* equal to *A*; and whatever multiple *A* or *E* is of *B*, make *F* the same multiple of *D*.

Then, because *A* is to *B as *C* is to *D*, [*Hypothesis*.*
and of

*B*the second and

*D*the fourth have been taken equimultiples

*E*and

*F*; [

*Construction*. therefore

*A*is to

*E*as

*C*is to

*F*. [V. 4,

*Corollary*.

But *A* is equal to *E*; [*Construction*.

therefore *C* is equal to *F*. [V. *A*.

And *F* is the same multiple of *D* that *A* is of *B*; [*Construction*.

therefore *C* is the same multiple of *D* that *A* is of *B*.

Next, let *A* be a part of *B*: *C* shall be the same part of *D*.

For, because *A* is to *B* as *C* is to *D*; [*Hypothesis*.

therefore, inversely, *B* is to *A* as *D* is to *C*. [V. *B*.

But *A* is a part of *B*; [*Hypothesis*.

that is, *B* is a multiple of *A*;

therefore, by the preceding case, *D* is the same multiple of *C*;

that is, *C* is the same part of *D* that *A* is of *B*.

Wherefore, *if the first* &c. q.e.d.

*THEOREM*.

*Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes*. Let *A* and *B* be equal magnitudes, and *C* any other magnitude: each of the magnitudes *A* and *B* shall have the same ratio to *C*; and *C* shall have the same ratio to each of the magnitudes *A* and *B*.

Take of *A* and *B* any equimultiples whatever *D* and *E*; and of *C* any multiple whatever *F*.

Then, because *D* is the same multiple of *A* that *E* is of *B*, [*Construction*.

and that *A* is equal to *B*; [*Hypothesis*.

therefore *D* is equal to *E*. [V. *Axiom* 1. Therefore if *D* be greater than *F*, *E* is greater than *F*; and if equal, equal; and if less, less.

But *D* and *E* are any equimultiples whatever of *A* and *B*, and *F* is any multiple whatever of *C*; [*Construction*. therefore *A* is to *C* as *B* is to *C*. [V. *Def*. 5.

Also *C* shall have the same ratio to *A* that it has to *B*.

For the same construction being made, it may be shewn, as before, that *D* is equal to *E*.

Therefore if *F* be greater than *D*, *F* is greater than *E' '; and if equal, equal; and if less, less.*

But *F* is any multiple whatever of *C*, and *D* and *E* are any equimultiples whatever of *A* and *B*; [*Construction*.

therefore *C* is to *A* as *C* is to *B*. [V. *Definition* 5.

Wherefore, *equal magnitudes* &c. q.e.d.

*THEOREM*.

*Of unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less than it has to the greater.*

Let *AB* and *BC* be unequal magnitudes, of which *AB* is the greater; and let *D* be any other magnitude whatever: *AB* shall have a greater ratio to *D* than *BC* has to *D*; and *D* shall have a greater ratio to *BC* than it has to *AB*. If the magnitude which is not the greater of the two *AC*, *CB*, be not less than 'D*, take *EF*, *FG* the doubles of *AC*, *CB* (Figure 1).*

But if that which is not the greater of the two *AC*, *CB*, be less than *D* (Figures 2 and 3), this magnitude can be multiplied, so as to become greater than *D*, whether it be *AC* or *CB*.

Let it be multiplied until it becomes greater than *D*, and let the other be multiplied as often.

Let *EF* be the multiple thus taken of *AC*, and *FG* the same multiple of *CB*;

therefore *EF* and *FG* are each of them greater than *D*.

And in all the cases, take *H* the double of *D*, *K* its, triple, and so on, until the multiple of *D* taken is the first which is greater than *FG*. Let *L* be that multiple of *D*, namely, the first which is greater than *FG*; and let *K* be the multiple of *D* which is next less than *L*.

Then, because *L* is the first multiple of *D* which is greater than *FG*, [*Construction*.

the next preceding multiple *K* is not greater than *FG*;

that is, *FG* is not less than *K*.

And because *EF* is the same multiple of *AC* that *FG* is of *CB*, [*Construction*.

therefore *EG* is the same multiple of *AB* that *FG* is of *CB*; [V.l.

that is, *EG* and *FG* are equimultiples of *AB* and *CB*. And it was shewn that *FG* is not less than *K* and *EF* is greater than *D*; [*Construction*.

therefore the whole *EG* is greater than *K* and *D* together.

But *K* and *D* together are equal to *L*; [*Construction*

therefore *EG* is greater than *L*.

But *FG* is not greater than *L*.

And *EG* and *FG* were shewn to be equimultiples of *AB* and *BC*;

and L is a multiple of *D*. [*Construction*.

Therefore *AB* has to *D* a greater ratio than *BC* has to *D*. [V. *Definition* 7.

Also, *D* shall have to *BC* a greater ratio than it has to *AB*.

For, the same construction being made, it may be shewn, that *L* is greater than *FG* but not greater than *EG*.

And *L* is a multiple of *D*, [*Construction*.

and *EG* and *FG* were shewn to be equimultiples of *AB* and *CB*.

Therefore *D* has to *C* a greater ratio than it has to *AB*. [V. *Definition* 7.

Wherefore, *of unequal magnitudes* &c. q.e.d.

*THEOREM*.

*Magnitudes which have the same ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the same ratio, are equal to one another*.

First, let *A* and *B' have the same ratio to *C*: *A* shall be equal to *B*.*

For, if *A* is not equal to *B', one of them must be greater than the other; let *A* be the greater.*

Then, by what was shewn in Proposition 8, there are
some equimultiples of *A* and *B*, and some multiple of *C*, such that the multiple of *A* is greater than the multiple of *C*, but the multiple of *B* is not greater than the multiple of *C*.

Let such multiples be taken; and let *D* and *E* be the equimultiples of *A* and *B*, and *F* the multiple of *C*; so that *D* is greater than *F*, but *E* is not greater than *F*.

Then, because *A* is to *C* as *B* is to *C*; and of *A* and *B* are taken equimultiples *D* and *E*, and of *C* is taken a multiple *F*;

and that *D* is greater than *F*; [*Construction*

therefore *E* is also greater than *F*.[V. *Definition* 5.

But E is not greater than F; [*Construction]]*
which is impossible.

Therefore *A* and *B* are not unequal; that is, they are equal.

Next, let *C* have the same ratio to *A* and *B*: *A* shall be equal to *B*.

For, if *A* is not equal to *B*, one of them must be greater than the other; let *A* be the greater.

Then, by what was shewn in Proposition 8, there is some multiple *F* of 'C*, and some equimultiples *E* and *D* of *B* and *A*, such that *F* is greater than *E*, but not greater than *D*.*

And, because *C* is to *B* as *C* is to *A*, [*Hypothesis*.

and that *F* the multiple of the first is greater than *E* the multiple of the second, [*Construction*.

therefore *F* the multiple of the third is greater than *D* the multiple of the fourth. [V. *Definition* 5.

But *F* is not greater than *D*; [*Construction*.

which is impossible.

Therefore *A* and *B* are not unequal; that is, they are equal.

*magnitudes which*&c. q.e.d.

*That magnitude which has a greater ratio than another has to the same magnitude is the greater of the two; and that magnitude to which the same has a greater ratio than it has to another magnitude is the less of the two*.

First, let *A* have to *C* a greater ratio than *B* has to *C*: *A* shall be greater than *B*.

For, because *A* has a greater ratio *A* to *C* than *B* has to *C*, there are some equimultiples of *A* and 'B*, and some . multiple of *C*, such that the multiple *C* of *A* is greater than the multiple of *C*, but the multiple of *B* is not greater than the multiple of *C*. [V. *Def*. 7.*
Let such multiples be taken; and let

*D*and

*E*be the equimultiples of

*A*and 'B

*, and*F

*the multiple of*C

*; so that*D

*is greater than*F

*, but*E

*is not greater than*F

*;*

therefore

*D*is greater than

*E*.

And because *D* and *E* are equimultiples of *A* and *B*, and that *D* is greater than *E*,

therefore *A* is greater than *B*. [V. *Axiom* 4.

Next, let *C* have to *B* a greater ratio than it has to *A*: *B* shall be less than *A*.

For there is some multiple *F* of *C*, and some equimultiples *E* and *D* of *B* and *A*, such that *F* is greater than *E*, but not greater than *D*; [V. *Definition* 7.

therefore *E* is less than *D*.

And because *E* and *D* are equimultiples of *B* and *A*, and that *E* is less than *D*, therefore *B* is less than *A*. [V. *Axiom* 4.

*that magnitude*&c. q.e.d.

*Ratios that are the same to the same ratio, are the same to one another*.

Let *A* be to *B* as *C* is to *D*, and let *C* be to *D* as *E* is to *F*: *A* shall be to *B* as *E* is to *F*.

Take of *A*, *C*, *E* any equimultiples whatever *G*, *H*, *K*; and of *B*, *D*, *F* any equimultiples whatever *L*, *M*, *N*.

Then, because *A* is to *B* as *C* is to *D*, [*Hypothesis*.

and that *G* and *H* are equimultiples of *A* and *C*, and *L* and *M* are equimultiples of *B* and *D*; [*Construction*.

therefore if *G* be greater than *L*, *H* is greater than '*N*;

and if equal, equal; and if less, less. [V. *Definition* 5.

Again, because *C* is to *D* as *E* is to *F*, [*Hypothesis*.

and that *H* and *K* are equimultiples of *C* and *E*, and *M* and *N* are equimultiples of *D* and *F*; [Construction.

therefore if *H* be greater than *M*, *K* is greater than *N*; and if equal, equal; and if less, less. [V. *Definition* 5.

But it has been shewn that if *G* be greater than *L*, *H* is greater than *M*; and if equal, equal; and if less, less.

Therefore if *G* be greater than *L*, *K* is greater than *N*;

and if equal, equal; and if less, less.

And *G* and *K* are any equimultiples whatever of *A* and *E*, and *L* and *N* are any equimultiples whatever of *B* and *F*. Therefore *A* is to *B* as *E* is to *F*. [V. *Definition* 6.

*ratios that are the same*&c. q.e.d.

*THEOREM*.

*If any number of magnitudes he proportionals, as one of the antecedents is to its consequent, so shall all the ante-cedents he to all the consequents*.

Let any number of magnitudes *A*, *B*, *C*, *D*, *E*, *F* be
proportionals ; namely, as *A* is to *B*, so let *C* be to *D* , and
*E to *F*: as *A* is to *B*, so shall *A*, *C*, *E* together be to *
*B*, *D*, *F* together.

Take of *A*, *C*, *E* any equimultiples whatever *G*,*H*,*K*,
and of *B*, *D*, *F* any equimultiples whatever *L*, *M*, *N*.

Then, because *A* is to *B* as *C* is to *D* and as *E* is to *F*,
and that *G*, *H*, *K* are equimultiples of *A*, *C*, *E*, and *L*, *M*,*N*
equimultiples of *B*, *D*, *F*; [*Construction*.

therefore if *G* be greater than *L*, *H* is greater than *M*,
and *K* is greater than *N* and if equal, equal ; and if less,
less. [V. *Definition* 5.

Therefore, if *G* be greater than *L*, then *G*, *H*, *K* together
are greater than *L*, *M*, *N* together ; and if equal, equal ;
and if less, less.

But *G*, and *G*, *H*, *K* together, are any equimultiples
whatever of *A*, and *A*, *C*, *E* together ; [V. 1.

and *L*, and *L*, *M*, *N* together are any equimultiples what-
ever of *B*, and *B*, *D*, *F* together. [V. 1.

Therefore as *A* is to *B*, so are *A*, *C*, *E* together to
*B*, *D*, *F* together. [V. *Definition* 5.

Wherefore, '*if any number* &c. q.e.d.

*THEOREM*.

*If the first have the same ratio to the second which the third has to the fourth, but the third to the fourth a greater * *ratio than the fifth to the sixth, the first shall have to the second a greater ratio than the fifth has to the sixth*.

Let *A* the first have the same ratio to *B* the second
that *C* the third has to *D* the fourth, but *C* the third a
greater ratio to *D* the fourth than *E* the fifth to *F* the
sixth: *A* the first shall have to *B* the second a greater
ratio than *E* the fifth has to *F* the sixth.

For, because *C* has a greater ratio to *D* than *E* has to *F*,
there are some equimultiples of *C* and *E*, and some equi-
multiples of *D* and *F*, such that the multiple of *C* is greater
than the multiple of *D*, but the multiple of *E* is not greater
than the multiple of *F*. [V. *Definition* 7.

Let such multiples be taken, and let *G* and *H* be the equi-
multiples of *C* and *E*, and *K* and *L* the equimultiples of
*D* and *E*;
so that *G* is greater than *K*, but *H* is not greater than *L*.

And whatever multiple *G* is of *C*, take *M* the same mul-
tiple of *A* ; and whatever multiple *K* is of *D*, take *N* the
same multiple of *B*.

Then, because *A* is to *B* as *C* is to *D*, [*Hypothesis*.

and *M* and *G* are equimultiples of *A* and *C*, and *N* and
*K* are equimultiples of *B* and *D* ; ['*Construction*.

therefore if *M* be greater than '*N*, *G* is greater than *K*;

and if equal, equal ; and if less, less. [V. *Definition* 5.

But *G* is greater than *K* ; [*Construction*.

therefore *M* is greater than *N*.

But *H* is not greater than L ; [*Construction*.

and *M* and *H* are equimultiples of *A* and *E*, and *N* and *L*
are equimultiples of *B* and *F* ; [*Construction*.

therefore *A* has a greater ratio to *B* than *E* has to F,

Wherefore, *if the first* &c. q.e.d. Corollary. And if the first have a greater ratio to the second than the third has to the fourtli, but the third the same ratio to the fourth that the fifth has to the sixth, it may be shewn, in the same manner, that the first has a greater ratio to the second than the fifth has to the sixth.

*THEOREM*.

*If the first have the same ratio to the second that the third has to the fourth, then if the first he greater than the third the second shall he greater than the fourth; and if equal, equal; and if less, less*.

Let *A* the first have the same ratio to *B* the second that *C* the third has to *D* the fourth: if *A* be greater than *C*, *B* shall be greater than *D*; if equal, equal; and if less,

First, let *A* be greater than *C*: *B* shall be greater than *D*. For, because *A* is greater than *C*, [*Hypothesis*.

and *B* is any other magnitude;

therefore *A* has to *B* a greater ratio than *C* has to *B*. [V. 8.

But *A* is to *B* as *C* is to *D*. [*Hypothesis*.

Therefore *C* has to *D* a greater ratio than *C* has to *B*. [V. 13.

But of two magnitudes, that to which the same has the greater ratio is the less. [V. 10.

Therefore *D* is less than *B*, that is. *B* is greater than *D*.

Secondly, let *A* be equal to *C*: *B* shall be equal to *D*.

For, ^ is to ^ as *C*, that is *A*, is to *D*. [*Hypothesis*. Therefore B is equal to D. [V. 9.

Thirdly, let *A* be less than *C*: *B* shall be less than *D*.

For, *C* is greater than *A*.

And because *C* is to *D* as *A* is to *B*; [*Hypothesis*.

and *C* is greater than *A*;

therefore, by the first case, *D* is greater than *B*;

that is, *B* is less than *D*.

Wherefore, *if the first* &c. q.e.d.

*THEOREM*.

Magnitudes have the same ratio to one another that their equimultiples have.

Let *AB* be the same multiple of *C* that *DE* is of *F*: *C* shall be to *F* as *AB* is to *DE*.

For, because *AB* is the same multiple of *C* that *DE* is of *F*, [*Hypothesis*.

therefore as many magnitudes as there are in *AB* equal to *C*, so many are there in *DE* equal to *F*.

Divide *AB* into the magnitudes *AG*, *GH*, *HB*, each equal to *F*; and *DE* into the magnitudes *DK*, *KL*, *LE*, each equal to *F*. Therefore the number of the magnitudes *AG*, *GH*, *HB* will be equal to the number of the magnitudes *DK*, *KL*, *LE*.

And because *AG',' *GH*, *HB* are all equal; [*Construction*.*
and that

*DK*,

*KL*,

*LE*are also all equal;

therefore

*AG*is to

*DK*as

*GH*is to

*KL*, and as

*HB*is to

*LE*. [V. 7.

But as one of the antecedents is to its consequent, so are all the antecedents to all the consequents. [V. 12.

Therefore as

*AG*is to

*DK*so is

*AB*to

*DE*.

But

*AG*equal to

*C*, and

*DK*is equal to

*F*.

Therefore as

*C*is to

*F*so is

*AB*to

*DE*.

*magnitudes*&c. q.e.d.

*THEOREM*.

*If four magnitudes of the same kind be proportionals, they shall also he proportionals when taken alternately*.

Let *A*, *B*, *C*, *D* be four magnitudes of the same kind which are proportionals; namely, as *A* is to *B* so let *C* be to *D*: they shall also be proportionals when taken alternately, that is, *A* shall be to *C* as *B* is to *D*.

Take of *A* and *B* any equimultiples whatever *E* and *F*, and of *C* and *D* any equimultiples whatever *G* and *H*.

Then, because *E* is the same multiple of *A* that *F* is of *B*, and that magnitudes have the same ratio to one another that their equimultiples have; [V. 15.

therefore *A* is to *B* as *E* is to *F*.

But *A* is to *B* as *C* is to *D*. [Hypothesis.

Therefore *C* is to *D* as *E* is to *F*. [V. 11.

Again, because *G* and *H* are equimultiples of *C* and *D*, therefore *C* is to *D* as *G* is to *H*. [V. 15.

But it was shewn that *C* is to *D* as *E* is to *F*.

Therefore *E* is to *F* as *G* is to *H*. [V. 11.

But when four magnitudes are proportionals, if the first be greater than the third, the second is greater than the fourth; and if equal, equal; and if less, less. [V. 14.

Therefore if *E* be greater than *G*, *F* is greater than *H*; and if equal, equal; and if less, less.

But *E* and *F* are any equimultiples whatever of *A* and *B*, and *G* and *H * are any equimultiples whatever of *C* and *D*. [*Construction*.

Therefore *A* is to *C* as *B* is to *D*. [V. *Definition* 5.

*if four magnitudes*&c. q.e.d.

*THEOREM*.

*If magnitudes, taken jointly, he proportionals, they shall also he proportionals when taken separately; that is, if two magnitudes taken together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these*.

Let *AB*, *BE*, *CD*, *DF* be the magnitudes which, taken jointly, are proportionals; that is, let *AB* be to *BE* as *CD* is to *DF*: they shall also be proportionals when taken separately; that is, *AE* shall be to *EB* as *CF* is to *FD*.

Take of *AE*, *EB*, *CF*, *FD* any equimultiples whatever *GH*, *HK*, *LM*,*MN*;

and, again, of *EB*, *FD* take any equimultiples whatever *KX*, *NP*.

Then, because *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*, [*Constr*.

therefore *GK* is the same multiple of *AB* that *LM* is of *CF*.

Again, because *LM* is the same multiple of *CF* that *MN* is of *FD*, [*Construction*.

therefore *LM* is the same multiple of *CF* that *LN* is of *CD*. [V. 1.

But *LM* was shewn to be the same multiple of *CF* that *GK* of *A*B.

Therefore *GK* is the same multiple of *AB* that *LN* is of *CD*;

that is, *GK* and *LN* are equimultiples of *AB* and *CD*.
Again, because *HK* is the same multiple of *EB* that *MN* is of *FD*, and that *KX* is the same multiple of *EB* that *NP* is of *FD*, [*Construction*.

therefore *HX* is the same multiple of *EB* that *MP* is of *FD*; [V. 2.

that is, *HX* and *MP* are equimultiples of *EB* and *FD*.

And because *AB* is to *BE* as *CD* is to *DF*, [*Hypothesis*.

and that *GK* and *LN* are equimultiples of *AB* and *CD*, and *HX* and *MP* are equimultiples of *EB* and *FD*,

therefore if *GK* be greater than *HX*, *LN* is greater than *MP*; and if equal, equal; and if less, less. [V. *Def*. 5.

But if *GH* be greater than *HX*, then, by adding the common magnitude *HK* to both, *GK* is greater than *HX*;

therefore also *LN* is greater than *MP*; and, by taking away the common magnitude *MN* from both, *LM* is greater than *NP*. Thus if *GH* be greater than *KX*, *LM* is greater than *NP*.

In like manner it may be shewn that, if *GH* be equal to *KX*, *LM* is equal to *NP*; and if less, less.

But *GH* and *LM* are any equimultiples whatever of *AE* and *CF*, and *KX* and *NP* are any equimultiples whatever of *EB* and *FD*; [*Construction*.

therefore *AE* is to *EB* as *CF* is to *FD*. [V. *Definition* 5,

Wherefore, *if four magnitudes* &c. q.e.d.

*THEOREM*.

*If magnitudes, taken separately, he proportionals, they shall also he proportionals when taken jointly; that is, if the first he to the second as the third to the fourth, the first and second together shall be to the second as the third and fourth together to the fourth*. Let *AE*, *EB*, *CF*, *FD* be proportionals; that is, let *AE* be to *EB* as *CF* is to *FD*: they shall also be proportionals when taken jointly; that is, *AB* shall be to *BE* as *CD* is to *DF*.

Take of *AB*, *BE*, *CD*, *DF* any equimultiples whatever *GH*, *HK*, *LM*, *MN*;

and, again, of *BE*, *DF* take any equimultiples whatever *KO*, *NP*.

Then, because *KO* and *NP* are equimultiples of *BE* and *DF*, and that *KH* and *NM* are also equimultiples of *BE* and *DF*; [*Construction*.

therefore if *KO*, the multiple of *BE*, be greater than *KH*, which is a multiple of the same *BE*, then *NP* the multiple of *DF* is also greater than *NM* the multiple of the same *DF*; and if *KO* be equal to *KH*, *NP* is equal to *NM*; and if less, less.

First, let *KO* be not greater than *KH*;

therefore *NP* is not greater than *NM*.

And because *GH* and *HK* are equimultiples of *AB* and *BE*, [*Construction*.

and that *AB* is greater than *BE',*
therefore

*GH*is greater than

*HK*; [V.

*Axiom*3.

but

*KO*is not greater than

*KH*; [

*Hypothesis*.

therefore

*GH*is greater than

*KO*.

In like manner it may be shewn that *LM* is greater than *NP*.

Thus if *KO* be not greater than *KH*, then *GH*, the multiple of *AB*, is always greater than *KO*, the multiple of *BE*;

and likewise *LM*, the multiple of *CD*, is greater than *NP*, the multiple of *DF*. Next, let *KO* be greater than *KH*; therefore, as has been shewn, *NP* is greater than *NM*. And because the whole *GH* is the same multiple of the whole *AB* that *HK* is of *BE* [*Construction*.

therefore the remainder *GK* is the same multiple of the remainder *AE* that *GH* is of *AB*; [V. 5.

which is the same that *LM* is of *CD*. [*Construction*.

In like manner, because the whole *LM* is the same multiple of the whole *CD* that *MN* is of *DF*, [Construction. therefore the remainder *LN* is the same multiple of the remainder *CF* that *LM* is of *CD*. [V. 5.

But it was shewn that *LM* is the same multiple of *CD* that *G* is of *CD*.

Therefore *GK* is the same multiple of *AE* that *LN* is of *CF*;

that is, *G* and *LN* are equimultiples of *BE* and *CF*.

And because *KO* and *NP* are equimultiples of *BE* and *DF*; [*Construction*.

therefore, if from *KO* and *NP* there be taken *KH* and *NM*, which are also equimultiples of *BE* and *DF*, [*Constr*.

the remainders *HO* and *MP* are either equal to *BE* and *DF*, or are equimultiples of them.

Suppose that *HO* and *MP* are equal to *BE* and *DF*. Then, because *AE* is to *EB* as *CF* is to *FD*, [*Hypothesis*. and that *GK* and *LN* are equimultiples of *AE* and *CF*; therefore *GK* is to *EB* as *LN* is to *FD*. [V. 4, *Cor*.

But *HO* is equal to *BE*, and *MP* is equal to *DF*; [*Hyp*

therefore *GK* is to *HO* as *LN* is to *MP*. Therefore if *GK* be greater than *HO*, *LN* is greater than *MP*; and if equal, equal; and if less, less. [V. *A*.

Again, suppose that *HO* and *MP* are equimultiples of *EB* and *FD*.

Then, because *AE* is to *EB* as *CF* is to *FD*; [*Hypothesis*.

and that *GK* and *LN* are equimultiples of *AE* and *CF*, and *HO* and *MP* are equimultiples of *EB* and *FD*;

therefore if *GK* be greater than *HO*, *LN* is greater than *MP*; and if equal, equal; and if less, less; [V. Definition 5.

which was likewise shewn on the preceding supposition.

But if *GH* be greater than *KO*, then by taking the common magnitude *KH* from both, *GK* is greater than *HO*;

therefore also *LN* is greater than *MP*;

and, by adding the common magnitude *NM* to both, *LM* is greater than *NP*.

Thus if *GH* be greater than *KO*, *LM* is greater than *NP*.

In like manner it may be shewn, that if *GH* be equal to *KO*, *LM* is equal to *NP*; and if less, less.

And in the case in which *KO* is not greater than *KH*, it has been shewn that *GH' is always greater than *KO* and also *LM* greater than *NP*.*

But *GH* and *LM* are any equimultiples whatever of *AB* and *CD*, and *KO* and *NP* are any equimultiples whatever of *BE' 'and *DF*, [*Construction*.*

*AB*is to

*BE*as

*CD*is to

*DF*. [V.

*Definition*5. Wherefore,

*if magnitudes*&c. q.e.d.

*THEOREM*.

*If a whole magnitude he to a whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder shall he to the remainder as the whole is to the whole*.

Let the whole *AB* be to the whole *CD* as *AE*, a magnitude taken from *AB*, is to *CF*, a magnitude taken from *CD*: the remainder *EB* shall be to the remainder *FD* as the whole *AB* is to the whole *CD*.

For, because *AB* is to *CD* as *AE* is to *CF*, [*Hypothesis*.

therefore, alternately, *AB* is to *AE* as *CD* is to *CF*. [V. 16.

And if magnitudes taken jointly be proportionals, they are also proportionals when taken separately; [V. 17.

therefore *EB* is to *AE* as *FD* is to *CF*;

therefore, alternately, *EB' is to *FD* as *AE* is to *CF*. [V. 16.*
But

*AE*is to

*CF*as

*AB*is to

*CD*; [

*Hyp*.

therefore

*ED*isto

*FD*as

*AB*is to

*CD*. [V.ll.

Wherefore, *if a whole* &c. q.e.d.

Corollary. If the whole be to the whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder shall be to the remainder as the magnitude taken from the first is to the magnitude taken from the other. The demonstration is contained in the preceding.

*E*.

*THEOREM*.

*If four magnitudes he proportionals, they shall also be proportionals by conversion; that is, the first shall be to its excess above the second as the third is to its excess above the fourth*.

Let *AB* be to *BE* as *CD* is to *DF*: *AB* shall be to *AE* as *CD* is to *CF*.
For, because *AB* is to *BE* as *CD* is to *DF*; [*Hypothesis*.

therefore, by division, *AE* is to *EB* as *CF* is to *FD*; [V, 17.

and, by inversion, *EB* is to *AE* as *FD* is to *CF*. [V. *B*.

Therefore, by composition, *AB* is to *AE* as *CD* is to *CF*. [V. 18.

Wherefore, *if four magnitudes* &c. q.e.d.

*THEOREM*.

*If there me three magnitudes, and other three, which have the same ratio, taken two and two, then, if the first be greater than the third, the fourth shall he greater than the sixth; and if equal, equal; and if less, less*.

Let *A*, *B*, *C* be three magnitudes, and *D*, *E*, *F* other three, which have the same ratio taken two and two; that is, let *A* be to *B* as *D* is to *E*, and let *B* be to *C* as *E* is to *F*: if *A* be greater than *C*, *D* shall be greater than *F*; and if equal, equal; and if less, less.

First, let *A* be greater than *C*: *D* shall be greater than *F*.

For, because *A* is greater than *C*, and *B* is any other magnitude,

therefore *A* has to *B* a greater ratio than *C* has to *B*. [V. 8.

But *A* is to *B* as *D* is to *E*; [*Hypothesis*.

therefore *D* has to *E* a greater ratio than *C* has to *B*. [V. 13.

And because *B* is to *C* as *E* is to *F*, [*Hyp*.

therefore, by inversion, *C* is to *B* as *F* is to *E*. [V. *B*.

And it was shewn that *D* has to *E* a greater ratio than *C* has to *B*;

therefore *D' 'has to *E* a greater ratio than *F* has to *E*; [V. 13, *Cor*.*
therefore

*D*is greater than

*F*, [V. 10. Secondly, let

*A*be equal to

*C*:

*D*shall be equal to

*F*.

For, because *A* is equal to *C*, and *B* is any other magnitude,

therefore *A* is to *B* as *C* is to *B*. [V. 7.

But *A* is to *B* as *D* is to *E*, [*Hypothesis*.

and *C* is to *B* as *F* is to *E*, [*Hyp*. V. *B*.

therefore *D* is to *E* as *F* is to *E*; [V. 11.

and therefore *D* is equal to *F*. [V. 9.

Lastly, let *A* be less than *C*: *D* shall be less than *F*.

For *C* is greater than *A*;

and, as was she^vn in the first case, *C* is to *B* as *F* is to *E*;

and, in the same manner, *B* is to *A* as *E* is to *D*;

therefore, by the first case, *F* is greater than *D*;

that is, *D* is less than *F*.

Wherefore, *if there be three* &c. q.e.d.

*THEOREM*.

*If there be three magnitudes, and other three, which the same ratio, taken two and two, but in a cross order, then if the first he greater than the third, the fourth shall he greater than the sixth; and if equal, equal; and if less, less.*

Let *A*,*B*, *C* be three magnitudes, and *D*, *E*, *F* other three, which have the same ratio, taken two and two, but in a cross order; that is, let *A* be to *B* as *E* is to *F*, and let *B* to *C* as *D* is to *E*: if *A* be greater than *C*, *D* shall be greater than *F*; and if equal, equal; and if less, less.

First, let *A* be greater than *C*: *D* shall be greater than *F*.

For, because *A* is greater than *C*, and *B* is any other magnitude,

therefore *A* has to *B* a greater ratio than *C* has to *B*. [V. 8.

But *A* is to *B* as *E* is to *F*; [*Hypothesis*.

therefore *E* has to *F* a greater ratio than *C* has to *B*. [V. 13.

And because *B* is to *C* as *D* is to *E*, [*Hypothesis*.

therefore, by inversion, *C* is to *B* as *E* is to *D*. [V. *B*.

And it was shewn that *E* has to *F* a greater ratio than *C* has to *B*;

therefore *E* has to *F* a greater ratio than *E* has to *D*; [V. 13, *Cor*.

therefore *F* is less than *D*; [V. 10.

that is, *D* is greater than *F*.

Secondly, let *A* be equal to *C*: *D* shall be equal to *F*.

For, because *A* is equal to *C*, and *B* is any other magnitude,

therefore *A* is to *B* as *C* is to *B*. [V. 7.

But *A* is to *B* as *E* is to *F*; [*Hyp*.

and *C* is to *B* as *E* is to *D*; [*Hyp*. V. *B*.

therefore *E* is to *F* as *E* is to *D*; [V. 11.

and therefore *D* is equal to *F*. [V. 9.

Lastly, let *A* be less than *C*: *D* shall be less than *F*.

For *C* is greater than *A*;

and, as was shewn in the first case, *C* is to *B* as *E* is to *D*;

and in the same manner, *B* is to *A* as *F* is to *E*;

therefore, by the first case, *F* is greater than *D*;

that is, *D* is less than *F*.

*if there be three*&c. q.e.d.

*THEOREM*.

*If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall ham to the last of the first magnitudes the same ratio which the first of the others has to the last*.

[This proposition is usually cited by the words *ex aequali*.]

First, let there be three magnitudes *A*, *B*, *C*, and other three *D*, *E*, *F*, which have the same ratio, taken two and two in order; that is, let *A* be to *B* as *D* is to *E*, and let *B* be to *C* as *E* is to *F*: *A* shall be to *C* as *D* is to *F*.

Take of *A* and *D* any equimultiples whatever *G* and *H*; and of *B* and *E* any equimultiples whatever *K* and *L*; and of *C* and *F* any equimultiples whatever *M* and *N*.

Then, because *A* is to *B* as *D* is to *E*; [*Hypothesis*.

and that *G* and *H* are equimultiples of *A* and *D*,

and *K* and *L* equimultiples of *B* and *E*; [Construction.

therefore *G* is to *K* as *H* is to *L*. [V. 4.

For the same reason, *K* is to *M* as *L* is to *N*.

And because there are three magnitudes *G*, *K*, *M*, and other three *H*, *L*, *N*, which have the same ratio taken two and two,

therefore if *G* be greater than *M*, *H* is greater than *N*, and if equal, equal; and if less, less. [V. 20.

But *G* and *H* are any equimultiples whatever of *A* and *D*, and *M* and *N* are any equimultiples whatever of (7 and F.

Therefore *A* is to *C* as *D* is to *F*. [V. *Definition* 5.

Next, let there be four magnitudes, A, B, C, D, and
other four *E*, *F*, *G*, *H*, which have the same ratio taken two and two in order; namely, let *A* be to *B* as *E* is to *F*, and *B* to *C* as *F* is to *G*, and *C* to *D* as *G* is to *H*: *A* shall be to *D* as *E* is to *H*.

For, because *A*, *B*, *C* are three magnitudes, and *E*, *F*, *G* other three, which have the same ratio, taken two and two in order, [*Hypothesis*.

therefore, by the first case, *A* is to *C* as *E* is to *G*.

But *C* is to *D* as *G* is to *H*; [*Hypothesis*.

therefore also, by the first case, *A* is to *D* as *E* is to *H*.

And so on, whatever be the number of magnitudes.

Wherefore, *if there be any number* &c. q.e.d.

PROPOSITION 23. *THEOREM*.

*If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in a cross order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last*.

First, let there be three magnitudes. *A*, *B*, *C*, and other three *D*, *E*, *F*, which have the same ratio, taken two and two in a cross order; namely, let *A* be to *B* as *E* is to *F*, and *A* to *C* as *D* is to *E*: *A* shall be to *C* as *D* is to *F*.

Take of A, B, D any equimultiples whatever G, H, K; and of C, E, F any equimultiples whatever L, M,N.

Then because *G* and *H* are equimultiples of *A* and *B*, and that magnitudes have the same ratio which their equimultiples have; [V. 15.

therefore *A* is to *B* as *G* is to *H*.

And, for the same reason, *E* is to *F* as *M* is to *N*.

But *A* is to *B* as *E* is to *F*, [*Hypothesis*.

Therefore *G* is to *H* as *M* is to *N*. [V. 11.

And because *B* is to *C* as *D* is to *E*, [*Hypothesis*.

and that *H* and *K* are equimultiples of *B* and *D*,

and *L* and *M* are equimultiples of *C* and *E*; [*Constr*.

therefore *H* is to *L* as *K* is to *M*. [V. 4.

And it has been shewn that *G* is to *H* as *M* is to *N*.

Then since there are three magnitudes *G*, *H*, *L*, and other three *K*, *M*,*N*, which have the same ratio, taken two and two in a cross order;

therefore if *G* be greater than *L*, *K* is greater than* *N*; and if equal, equal; and if less, less. [V. 21.*
But

*G*and

*K*are any equimultiples whatever of

*A*and

*D*, and

*L*and '

*N*are any equimultiples whatever of

*C*and

*F*;

therefore

*A*is to

*C*as

*D*is to

*F*. [V.

*Definition*5.

Next, let there be four magnitudes *A*,*B*,*C*, *D*, and other four *E*, *F*, *G*, *H*, which have the same ratio, taken two and two in a cross order; namely, let *A* be to *B* as *G* is to *H*, and *B to *C* as *F* is to *G*, and *C* to *D* as *E* is to *F*:*

*A*shall be to

*D*as

*E*is to

*H*.

For, because *A*, *B*, *C* are three magnitudes, and *F*, *G*, *H* other three, which have the same ratio, taken two and two in a cross order; [*Hypothesis*.

therefore, by the first case, *A* is to *C* as *F* is to *H*.

But *C* is to *D* as *E* is to *F*; [*Hypothesis*.

therefore also, by the first case, *A* is to *D* as *E* is to *H*.

And so on, whatever be the number of magnitudes.

Wherefore,*if there he any number*&c. q.e.d.

*THEOREM*.

*If the first have to the second the same ratio which the third has to the fourth, and the fifth have to the second the same ratio which the sixth has to the fourth, then the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth*.

Let *AB* the first have to *C* the second the same ratio which *DE* the third has to *F the fourth; and let *BG *the fifth have to *C* the second the same ratio which *EH* the sixth has to *F* the fourth: *AG*, the first and fifth together, shall have to *C* the second the same ratio which *DH*, the third and sixth together, has to *F* the fourth.*

For, because *BG* is to *C* as *EH* is to *F*, [*Hypothesis*.

therefore, by inversion, *C* is to *BG* as F is to EH [V. *B*.

And because *AB* is to *C* as *DE* is to *F*, [*Hypothesis*.

and *C* is to *BG* as *F* is to *EH*; therefore, ex aequali, AB is to BG as *BE* is to *EH*. [V. 22.

And, because these magnitudes are proportionals, they are also proportionals when taken jointly; [V. 18.

therefore *AG* is to *BG* as *DH* is to *EH*. But *BG* is to *C* as *EH* is to *F*;[*Hypotheseis*.

therefore, ex aequali, *AG* is to *C* as *DH* is to *F*. [V.22

Wherefore, *if the first* &c. q.e.d.

Corollary 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second as the excess of the third and sixth is to the fourth. The demonstration of this is the same as that of the proposition, if division be used instead of composition.

Corollary 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to the fourth magnitude; as is manifest. *THEOREM*.

*If four magnitudes of the same kind be proportionals, the greatest and least of them together shall he greater than the other two together*.

Let the four magnitudes *AB*, *CD*, *E*, *F* be proportionals; namely, let *AB* be to *CD* as *E* is to *F*; and let *AB* be the greatest of them, and consequently *F* the least: [V.*A*, V.14.

*AB* and *F* together shall be greater than *CD* and *E* together.

Take *AG* equal to *E*, and *CH* equal to *F*.

Then, because *AB* is to *CD* as *E* is to *F*, [*Hypothesis*.

and that *AG* is equal to *E*, and *CH* equal to *F*; [*Construction*.

therefore *AB* is to *CD* be *AG* is to *CH*. [V. 7, V. 11.

And because the whole *AB* is to the whole *CD* as *AG* is to *CH*;

therefore the remainder *GB* is to the remainder *HD* as the whole *AB* is to the whole *CD*. [V. 19.

But *AB* is greater than *CD*; [Hypothesis.

therefore *BG* is greater than *DH*. [*V. A*.

And because *AG* is equal to *E* and *CH* equal to *F*, [*Constr*. therefore *AG* and *F* together are equal to *CH* and *E* together.

And if to the unequal magnitudes *BG*, *DH*, of which *BG* is the greater, there be added equal magnitudes, namely, *AG* and *F* to *BC*, and *CH* and *E* to *DH*, then *AB* and *F* together are greater than *CD* and *E* together.

Wherefore, *if four magnitudes* &c. q.e.d.