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

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BOOK V.
DEFINITIONS.

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.

AXIOMS.

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.
PROPOSITION 1. 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.

PROPOSITION 2. 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.

PROPOSITION 3. 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.

PROPOSITION 5. 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 AEis 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.

PROPOSITION 6. 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.

PROPOSITION A. 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.

PROPOSITION B. 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.

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

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

PROPOSITION 7. 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.

PROPOSITION 8. 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.

PROPOSITION 9. 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.

Wherefore, magnitudes which &c. q.e.d.
PROPOSITION 10. THEOREM.

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.

Wherefore, that magnitude &c. q.e.d.
PROPOSITION 11. THEOREM.

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.

Wherefore, ratios that are the same &c. q.e.d.
PROPOSITION 12. 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.

PROPOSITION 13. 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.

PROPOSITION 14. 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.

PROPOSITION 15. 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.

Wherefore, magnitudes &c. q.e.d.
PROPOSITION 16. 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.

Wherefore if four magnitudes &c. q.e.d.
PROPOSITION 17. 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 AB.

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.

PROPOSITION 18. 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.

therefore AB is to BE as CD is to DF. [V. Definition 5. Wherefore, if magnitudes &c. q.e.d.
PROPOSITION 19. 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.

PROPOSITION 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.

PROPOSITION 20. 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.


PROPOSITION 21. 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.

Wherefore, if there be three &c. q.e.d.
PROPOSITION 22. 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.
PROPOSITION 24. 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.
PROPOSITION 25. 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.