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PRO P. VI. THEO R.
mag. nitudes, and some parts taken away from them be equimultiples of she same magnitudes ; then the remainders will either be equal to the fame magnitudes, or equimultiples of ibem w.
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 say, the remainders G B, H D are either equal to E, F, or equimultiples of them.
For first let G B be eAT K к
qual to E: I say, HD is
equal to F; for put ck C
equal to F. C
Then because AG is
the same multiple of E,
H Н as ch is of F, and GB
is equal to E, and C.K to
i the same multiple fe, as B D E F B D E F KH is of F. But A B is put the same multiple of E, as CD is of F; therefore Kh is the same multiple of F, as cd is of F: Wherefore because K H, CD are each an equimultiple of F; KH will be equal to CD. Take away the common part c H, and then the remainder K c is equal to the remainder HD. But kc is equal to F: Therefore h D is equal to F. If therefore G B be equal to E, HD will be equal to F.
In like manner we demonstrate, whatever multiple GB is of E (as in the second figure) the same will HD be of F. ... Therefore if two magni udes be equimultiples of two other magnitudes, and some parts taken away from them be equimultiples of the same magnitudes ; then the remainders will be either equal to the fame magnitudes or equimultiples of them. Which was to be demonstrated.
This proposition being only in multiple proportion, is universally demonstrated at prop. 24. of all kinds of proportion.
PROP. VII, THEOR. Equal magnitudes bave the same ratio to the same magnitude, and the same magnitude has the same ratio to equal magnitudes *.
Let A, B be equal magnitudes, and c any magnitude whatsoever : I say, each of thele magnitudes A, B have the same ratio to c: and also c has the fame ratio to A;
For take D, E equimultiples of A and B, and take any other multiple F of c.
Then becaule D is the same 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, less. D
F 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, c, as B is to c.
· I say moreover that c has the same ratio to A or B.
For the construction remaining the same, we demonstrate in like manner that d is equal to E. other magnitude. Therefore if f exceeds d, it will also exceed E; if F be equal to D, it will be equal to E; if less, less. 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, so will c be to B.
Therefore equal magnitudes have the same ratio to the same magnitude, and the same magnitude has the same ratio to equal magnitudes. Which was to be demonstrated.
* This seventh, with the eighth, ninth, tenth, eleventh, and twelfth propofitions following, are taken by some to be mere àxioms requiring no demonstration at all. They are indeed very evident in numbers. But since they are applicable to all magnitudes in general, lines, planes, solids, commensurable, and incommensurable, Euclid could not but demonitrate them.
But F is any
greater than c.
PROP. VIII. THEOR. The greater of (two] unequal magnitudes has a greater
ratio to the same magnitude than the lefser ; and the same magnitude to the lesser of (two] unequal magnitudes bas 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
Also let D be any other magnitude whatJoever : I say, A B has a greater ratio to D, than c has to
D: and D has a greater ratio to FT
c, than it has to A B. TE
For because A B is greater than
c, make [by 3. 1.] BE equal to C; GH
then the lifler of thesetwomagni
tudes A E, E B being inultiplied, B
will at length exceed D [by 4. def. 5.] First let A E be less than EB, and multiply A E so often till it exceeds D : Ler FG be this multiple of Ae, which is greater than
D: Also make GH the same mulKHC D L MN is of A E. And take I double to
tiple of E B, and K of c, as F Gm D, M triple to it, and so 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 less than N, R: will not be esser than m. And since f g is the same multiple of A E as H G is of E B, [by 1. 5.) F G will be the same multiple of A E as FH is of A B. But F G is the same multiple of A E as k is of c: Therefore FH is the faine 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 same 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 G H is not less tha:2 m. But [by conftr.] F G is greater than D: Therefore the whole FH will be greater than
D, M together. But D, M to
FITA gether are equal to n; wherefore FH exceeds N; but K does not exceed n; and FH, K
E se equimultiples of A B, C; and n is any other multiple of D: Therefore [by 7. def. 5.]
B A B has a greater ratio to D,
+G than c has to D.
I say moreover that the ratio of D to c is greater than the ratio of D to A B.
For the same construction re- KHC D L MN maining, we demonstrate in the like manner, that n exceeds K, 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 oc multiplied, and let G H the multiple of E B be greater than b: And make fame multiple of A E, and K of c, as g h is of E B. Then by the like reason as before, we demonstrate that FH, K are equimultiples of AB, c. And likewise take n a multiple of D, [in the first place] greater than FG: Therefore again F G is not less than M.
But G H is greater than D. Therefore the whole FH exceeds d and m 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 demonstration.
Therefore the greater of (two] unequil magnitudes has a greater ratio to the fame magnitude than the leter i and the fame magnitude to the lesser of (two] unequal magnitudes has a greater ratio than it has to the greater of those (two] magnitudes. Which was to be demonstrated.
PROP. IX. THEOR. Magnitudes that have the same ratio to the sanre mag
nitude, are equal to one another ; and those mag, nitudes to which the same magnitude bas the same ratio, are also equal to one another.
For let each of the magnitudes A and B have the same satio to the fame magnitude c: I say, A is equal to B.
For if it were not equal, A and B would not [by 8. 5.] have the same ratio to c.
But it has. Therefore A is equal to B.
Again, let c have the same ratio both to A, B : I say, a is equal to B.
For if it were not equal, c would not have the same 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 those magnitudes to which the same magnitude has the same ratio, are also equal to one ano. ther. Which was to be demonstrated
PROP. X. THEOR. That magnitude of those magnitudes which have a
ratio to the same magnitude, is the greater which bas the greater ratio ; and that magnitude to which the same magnitude has the greater ratio, is the leffer.
For let a have a greater ratio to C, than B has to c: I say, 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
B, for then [by 7.5.] A and B would each have the same ratio to c. But they bave not the same ratio. Therefore A is not equal to B. Nor is a less than B, for then
[by 8. 5.] A would have a less ratio to c A
than B has to c. But it has not a less :
Wherefore a is not less than B : it is therefore greater.