PROP. VI. THEOR.

If two magnitudes be equimultiples of two other magnitudes, and fome parts taken away from them be equimultiples of the fame magnitudes; then the remainders will either be equal to the same magnitudes, or equimultiples of them".

For let two magnitudes A B, C D be equimultiples of two other magnitudes E, F, and the parts AG, CH taken away from them be fome equimultiples of these other magnitudes E, F I fay, the remainders G B, H D are either equal to E, F, or equimultiples of them.

For first let G B be equal to E: I fay, H D is equal to F; for put c K equal to F.

LE

G

BDEF

Then because AG is the fame multiple of E, as CH is of F, and G B is equal to E, and C.K to F; A B [by 2. 5.] will be the fame multiple fe, as

BDEF KH is of F. But A B is put the fame multiple of E, as CD is of F; therefore KH is the fame multiple of F, as CD is of F: Wherefore because K H, CD are each an equimultiple of F; KH will be equal to C D. Take away the common part C H, and then the remainder K C is equal to the remainder H D. But KC is equal to F: Therefore H D is equal to F. If therefore G B be equal to E, H D will be equal to F.

In like manner we demonftrate, whatever multiple G B is of E [as in the fecond figure] the fame will HD be of F. .. Therefore if two magni udes be equimultiples of two other magnitudes, and fome parts taken away from them be equimultiples of the fame magnitudes; then the remainders will be either equal to the fame magnitudes or equimultiples of them. Which was to be demonstrated.

This propofition being only in multiple proportion, is univerfally demonftrated at prop. 24. of all kinds of proportion.

PROP.

PROP. VII, THEOR.

Equal magnitudes have the fame ratio to the fame magnitude, and the fame magnitude has the fame ratio to equal magnitudes *.

Let A, B be equal magnitudes, and c any magnitude whatsoever I fay, each of thefe magnitudes A, B have the fame ratio to c: and alfo c has the fame ratio to A;

or B

For take D, E equimultiples of A and B, and take any other multiple F of c.

Then becaule D is the fame multiple of A, as E is of B, and A is equal to B; D will be equal to E. But F is any other multiple of c: Therefore if D exceeds F, E will exceed F too; and if it be equal to F, E will be equal to F; if lefs, lefs. D But D, E are equimultiples of A, B, E and F is any multiple whatsoever of c: Therefore [by 5. def. 5.] a will be to £, as B is to c.

I fay moreover that c has the fame ratio to A or B.

B

For the conftruction remaining the fame, we demonftrate in like manner that D is equal to E. But F is any other magnitude. Therefore if F exceeds D, it will alfo exceed E; if F be equal to D, it will be equal to E; if lefs, lefs. But F is a multiple of c, and D, E are any equimultiples of A, B; Therefore [by 5. def. 5.] as c is to A, fo will c be to B.

Therefore equal magnitudes have the fame ratio to the fame magnitude, and the fame magnitude has the fame ratio to equal magnitudes. Which was to be demonftrated.

* This feventh, with the eighth, ninth, tenth, eleventh, and twelfth propofitions following, are taken by fome to be mere axioms requiring no demonftration at all. They are indeed very evident in numbers. But fince they are applicable to all magnitudes in general, lines, planes, folids, commenfurable, and incommenfurable, Euclid could not but demonftrate them.

PROP.

PROP. VIII, THEOR.

The greater of [two] unequal magnitudes has a greater ratio to the fame magnitude than the leffer; and the fame magnitude to the leffer of [two] unequal magnitudes has a greater ratio than it has to the greater [of these two magnitudes.

Let A B, C be two unequal magnitudes, and let A B be greater than c. Alfo let D be any other magnitude whatfoever I fay, AB has a greater ratio to D, than c has to D and D has a greater ratio to C, than it has to A B.

FLI

G+

A

B

For because A B is greater than c, make [by 3. 1.] BE equal to C; then the leffer of these two magnitudes A E, E B being multiplied, will at length exceed D [by 4.def. 5.] Firft let AE be less than EB, and multiply A E fo often till it exceeds D: Let FG be this multiple of AE, which is greater than D: Alfo make GH the fame multiple of E B, and K of C, as F Gˆ KHC DL MN is of AE. And take L double to D, M triple to it, and fo forwards greater by one, until the magnitude taken be a multiple of D, and [in the first place] greater that K. Let N be this magnitude, being four times the magnitude D, and [in the first place] greater

than K.

Then because K [in the first place] is lefs than N, K will not be leffer than M. And fince F G is the fame multiple of A E as H G is of E B, [by 1. 5.] F G will be the fame multiple of A E as F H is of AB. But F G is the fame multiple of AE as K is of c: Therefore F H is the fame multiple of AB as K is of C. And fo FH, K are equimultiples of A B and c. Again, because G H is the fame multiple of E B as K is of c, and EB is equal to C; GH will be equal to K. But K is not less than M. Therefore GH is not less than M. But [by conftr.] F G is greater than D: Therefore the whole FH will be greater than

D, M together. But D, M together are equal to N; wherefore FH exceeds N; but K does not exceed N; and FH, K re equimultiples of A B, C; and N is any other multiple of D: Therefore [by 7. def. 5.] AB has a greater ratio to D, than c has to D.

I fay moreover that the ratio of D to c is greater than the ratio of D to A B.

For the fame conftruction re- KHCDL MN maining, we demonftrate in the like manner, that N exceeds x, but does not exceed F H. But N is a multiple of D, and FH, K any other equimultiples of A B, C. Therefore [by 7. def. 5.] the ratio of D to c is greater than the ratio of D to A B.

Now let A E be greater than E B: Then [by 4. def. 5.] the leffer magnitude E B being multiplied, will at length become greater than D. Let it be multiplied, and let G H the multiple of E B be greater than D: And make F G the fame multiple of A E, and K of c, as G H is of E B. Then by the like reason as before, we demonftrate that FH, K are equimultiples of A B, C. And likewife take N a multiple of D, [in the first place] greater than FG: Therefore again F G is not lefs than M. But G H is greater than D. Therefore the whole F H exceeds D and м together, that is N. But K does not exceed N, because F G, which is greater than HG, that is, than K, does not exceed N. And after the like manner as before we finish the demonftration.

Therefore the greater of [two] unequal magnitudes has a greater ratio to the fame magnitude than the letter; and the fame magnitude to the leffer of [two] unequal magnitudes has a greater ratio than it has to the greater of thofe [two] magnitudes. Which was to be demonftrated,

PROP.

PROP. IX. THEOR.

Magnitudes that have the fame ratio to the fame magnitude, are equal to one another; and thofe magnitudes to which the fame magnitude bas the fame ratio, are alfo equal to one another.

For let each of the magnitudes A and B have the fame ratio to the fame magnitude c: I fay, A is equal to B.

B

A

C

For if it were not equal, A and B would
not [by 8. 5.] have the fame ratio to c.
But it has. Therefore A is equal to B.
Again, let c have the fame ratio both to
A, B: I fay, A is equal to B.

For if it were not equal, c would not have the fame ratio both to A and B. But it has : Therefore A is equal to B.

Wherefore magnitudes that have the same ratio to the fame magnitude, are equal to one another and thofe magnitudes to which the fame magnitude has the fame ratio, are alfo equal to one ano ther. Which was to be demonstrated

PROP. X. THEOR.

That magnitude of thofe magnitudes which have a ratio to the fame magnitude, is the greater which has the greater ratio; and that magnitude to which the fame magnitude has the greater ratio, is the leffer.

For let A have a greater ratio to C, than B has to C: I fay, A is greater than B.

For if it be not greater, it is either equal to it, or less than it. But A is not equal to B, for then [by 7. 5.] A and B would each have the fame ratio to c. But they have not the fame ratio. Therefore A is not equal to B. Nor is A less than B, for then [by 8. 5.] A would have a lefs ratio to c than B has to c. But it has not a lefs: Wherefore A is not less than B: it is therefore greater.