Cor. 1. Hence it is evident, that if either of the given magnitudes be a multiple of A, the other is also an equi-multiple of A.

Cor. 2. If two magnitudes be equal, as often as one of them is contained in any third, so often is the other contained in the same third.

Cor. 3. If there be four magnitudes, the first equal to the second, the third equal to the fourth, as often as the first is contained in the third, so often the second is contained in the fourth.

PROP, II. THEOR.

If there be given two magnitudes (BC and DE), and Fig. 2. any submultiple whatsoever of a third magnitude (4) be always contained as often in the one as in the other, those given magnitudes (BC and DE) are equal.

For, if it be possible, let one of them BC be greater than the other, and let its excess be nC; take a submultiple a of A less than nC, a is oftener contained in BC than in Bn; but Bn is equal to DE, therefore a is contained in Bn as often as in DE (1), and therefore is con- (1) Prop. 1. tained in BC oftener than in DE; but it is contained as often (2), which is absurd; therefore neither of the given (2) Hypoth. quantities is greater than the other, and therefore they are equal.

Cor. If there be four magnitudes, of which the first is equal to the second, and any submultiple of the first be always contained in the third as often as an equisubmultiple of the second is contained in the fourth, the third is equal to the fourth.

PROP. III. THEOR.

If there be four magnitudes (A, B, CD and EF), of Fig. 3. which the first is greater than the second, and the third is equal to the fourth, the first is not contained in the third oftener than the second in the fourth.

1. Let B be a submultiple of EF; and, if it be possible, let A be contained in CD oftener than B in EF;

let A be taken away from CD as often as B is contained in EF, and let mD be the remainder; because Cm (1) Constr. and EF are equi-multiples of A and B (1), and A is (2) Hypoth. greater than B (2), Cm is greater than EF (3); but EF (3) Ax. 3. is equal to CD (2), therefore Cm is greater than CD, which is absurd.

2. Let B be not a submultiple of EF; and, if it be possible, let A be contained in CD oftener than B is contained in EF; take away B as often as possible from EF, and there shall remain nF less than B; take away A as often from CD, and since A is contained in CD oftener than B is contained in EF, the remainder mD is either greater than A or equal to it, and therefore greater than nF; but Cm and En are equi-multiples of A and B (1), and A is greater than B, therefore Cm is greater than En (3); and also mD is greater than nF, therefore the whole CD is greater than EF, but it is also equal to it (2), which is absurd. In no case therefore is A contained in CD oftener than B is contained in EF.

Cor. In like manner, if there be four magnitudes, of which the first is equal to the second, and the third less than the fourth, the first is not contained in the third oftener than the second in the fourth.

Fig. 4.

PROP. IV. THEOR.

If there be given two magnitudes (AB and CD), of which one (AB) is greater than the other (CD), there is a submultiple of any third magnitude (E), which is contained in the greater oftener than in the less.

Let mB be the excess of AB above CD; take e a submultiple of E less than mB; and since Am is equal to CD, e is contained in Am as often as it is contained (1) Prop. 1. in CD (1); but e is less than mB (2), therefore e is con(2) Constr. tained in AB oftener than in Am, and therefore oftener than in CD.

PROP. V. THEOR.

If there be given two magnitudes (AB and CD), and Fig. 4, 5. any third can be found (e) which is contained in one (AB) oftener than in the other (CD), the latter (CD) is less than the former.

If e be a submultiple of CD, take it away from AB Fig. 4. as often as it is contained in CD, and let mB be the remainder (1); Am is equal to CD because they are (1) Hypoth, equimultiples of e, therefore AB is greater than CD.

But if e be not a submultiple of CD, take it away Fig. 5. from CD as often as possible, and let the remainder be nD less than e; take it away as often from AB, and since it is contained in AB oftener than in CD (1), the remainder mB is greater than e, or equal to it, and therefore greater than nD; but Am and Cn are equimultiples of e(2), and therefore equal (3): and mB is (2) Constr. greater than nD, therefore the whole AB is greater than CD.

PROP. VI. THEOR.

If there be given two magnitudes (a and AB), of which Fig. 6. the former is a submultiple of the latter, and there be taken any third magnitude (x), which is a submultiple of the former (a), it is also a submultiple of the latter (AB).

(2) Cor.

Divide AB into parts Ao and oB equal to a; and since x is a submultiple of a (1), it shall be a submulti-Hypoth. ple of Ao (2) and also of oB; divide Ao and oB into Prop. 1. parts Am, mo, on, nB, equal to x; and the whole AB is divided into parts equal to x, and therefore a is a submultiple of AB.

PROP. VII. THEOR.

If there be two magnitudes (a and b) equi-submultiples Fig. 7. of two other magnitudes (AC and BD), and there be taken two others (x and 2) which are equi-submultiples of the former, they are also equi-submultiples of the latter.

Divide AC into parts Ao, oC, equal to a; and BD into parts Bn, nD, equal to b; because a and Ao are

(1) Constr. (2) Hypoth. (3) Cor. 1. Prop. 1.

Fig. 8,
See N.

equal (1), and x is a submultiple of a (2), it must be an equi-submultiple of Ao (3); in the same manner it is proved that is an equi-submultiple of b and Bn; but x and are equi-submultiples of a and b (2), and therefore of Ao and Bn, and therefore also equi-submultiples of oC and nD; whatever submultiple therefore x is of AC, the same is ≈ of BD.

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Cor. 1. If x be a submultiple of b, and be contained in b as often as x is contained in an, and there be taken equi-multiples BD and AC of b and an, ≈ is not contained in BD oftener than x in AC.

For if x be a submultiple of an, ≈ and x are equi-sub(1) Prop. 7. multiples of BD and AC(1), and therefore x is contained in AC as often as ≈ is contained in BD.

But if x be not a submultiple of an, take it away from an as often as possible, and let the remainder be on; (2) Hypoth. since x and x are equi-submultiples of ao and b (2), as often as is contained in any multiple of b, so often is x contained in an equi-multiple of ao; but an is greater than ao, and therefore any multiple of an is greater than the equi multiple of an (3); therefore x is not contained in a multiple of ao oftener than in the equi-multiple of an (4); and therefore & is not contained in a multiple of b oftener than x in the equi-multiple of an.

(3) Ax. 3.

(4) Cor. Prop. 3.

Fig. 7, 9.

Cor. 2. Let b be a submultiple of BD, and be contained in BD as often as a is contained in AC, and let ≈ and x be equi-submultiples of b and a; ≈ is not contained in BD oftener than x in AC.

If a be a submultiple of AC, it is evident that a and (1) Prop. 7. are equi-submultiples of AC and BD (1). Fig. 9.

But if not, take it away from AC as often as possible, and let the remainder be oC; a and b are equi-submultiples of Ao and BD, and therefore, since x and sare (2) Hypoth. equi-submultiples of a and b (2), as often as is contained in BD, so often is a contained in Ao (1), and therefore is not contained in BD oftener than x in AC.

Fig. 10.

Cor. 3. Let & be a submultiple of BD and be contained in it, as often as x is contained in the same BD; x is either equal to ≈ or less than it.

Let x be a submultiple of BD; and it is evident that x and ≈ are equal, because they are equi-submultiples of the same BD (1).

But if not, take away x from BD as often as x is contained in it, and let the remainder be mD; x and x are equi-submultiples of Bm and BD, and Bm is less than BD, therefore x is less than z (2).

(1) Ax. 1.

(2) Ax. 3.

PROP. VIII. THEOR.

If there be two unequal magnitudes (A and B), there is Fig. 11, 12, a submultiple of the less (B), which is contained in a third magnitude (CD) oftener than an equi-submultiple of the greater (A).

Take any submultiple a of A, and b an equi-submultiple of B.

1. Let a be a submultiple of CD; take away b from Fig. 11. CD, as often as a is contained in CD; and because b is

less than a (1), there must be a remainder mD: if mD (1) Hypoth. is greater than b, b is contained in CD oftener than in & Ax. 3. Cm, and therefore oftener than a is contained in CD. But if mD be less than b, take ≈ a submultiple of b less than mD, and x an equi-submultiple of a; since x and

are equi-submultiples of a and b (2), and a and (2) Constr. b equi-submultiples of CD and Cm, and are equisubmultiples of CD and Cm (3), but ≈ is less than mD (3) Prop.7. (2), therefore ≈ is contained in CD oftener than in Cm,

and therefore oftener than a is contained in the same CD.

2. Let a not be a submultiple of CD; take it away as Fig. 12. often as possible from CD, and let the remainder be nD; take away b as often from CD, and let the remainder be mD; since Cm and Cn are equi-multiples of b and a, but b is less than a (1), Cm is less than Cn (4): take a submultiple of b less than mn, and a an (4) Ax. 3. equi-submultiple of a; ≈ and x are equi-submultiples of Cm and Cn (3): but is less than mn (2), therefore ≈ is contained in Cn oftener than in Cm, and therefore oftener than x is contained in Cn; but x is greater than (5) Constr. ≈ (5), therefore x is not contained in nD oftener than ≈ & Ax. 3.

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