let A be taken away from CD as often as B is contained in EF, and let mD be the remainder; because Cm and EF are equi-multiples of A and B (1), and A is (1) Cnstr. greater than B (2), Cm is greater than EF (3), but EF (2) Hypoth. is equal to CD (2), therefore Cm is greater than CD, B. 5.

which is absurd.

(5) Ax. 3.

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 (5), and A is greater than B, therefore Cm is (5) Constr. greater than En (6), and also mD is greater than nF, (6) Ax. 3. therefore, the whole CD is greater than EF, but it is B. 5. also equal to it (7), which is absurd. In no case, (7) Hypoth. 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.

PROP. IV. THEOR.

If there be given two magnitudes (AB and CD), of Fig. 4. 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, è 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- B. 5. tained in AB oftener than in Am, and therefore oftener (2) Const. than in CD.

Fig. 4. & 5.

Fig. 4.

PROP. V. THEOR.

If there be given two magnitudes (AB and CD), and 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 as often as it is contained in CD, and let mB be the re(1) Hypoth. mainder (1); Am is equal to CD because they are equimultiples of e, therefore AB is greater than CD.

Fig. 5.

But if e be not a submultiple of CD, take it away from CD as often as possible, and let the remainder be nD less than e; take it away as often from AB, and since (2) Hypoth. it is contained in AB oftener than in CD (2), the remainder mB is greater than e, or equal to it, and therefore (3) Constr. greater than nD; but Am and Cn are equi-multiples (4) Ar. 1. of e (3), and therefore equal (4), and mB is greater than nD, therefore the whole AB is greater than CD.

B. 5.

Fig. 6.

PROP. VI. THEOR.

If there be given two magnitudes (a and AB,) of which 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)

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

P. 1. B. 5.

Fig. 7.

PROP. VII. THEOR.

If there be two magnitudes (a and b) equi-submultiples of two other magnitudes (AC and BD), and there be taken two others (x and z) 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

equal (1), and x is a submultiple of a (2), it must be an (1) Constr. equi-submultiple of Ao (3); in the same manner it is (2) Hypoth. proved that z is an equi-submultiple of b and Bn; but (3) Cor. 1. r and z are equi-submultiples of a and b (4), and there-Prop.1.B.5. (4) Hypoth. fore 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.

Cor. 1. If z be a submultiple of b, and be contained Fig. 8. in b as often as x is contained in an, and that there be see N. taken equi-multiples BD and AC of b and an, z is not contained in BD oftener than a in AC.

For if x be a submultiple of an, z and x are equi-sub- (1) Prop. 7. multiples of BD and AC (1), and therefore x is con- B. 5. tained in AC as often as z 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; since x and z are equi-submultiples of ao and b (2), as of- (2) Hypoth. ten as z 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 ao (3); therefore x is not contained (5) Ax. 3. in a multiple of ao oftener than in the equi-multiple of an B. 5. (4), and therefore is not contained in a multiple of (4) Cor. boftener than x in the equi-multiple of an.

Prop.3.B.5.

Cor. 2. Let b be a submultiple of BD and be con- Fig. 7. & 9. tained in BD as often as a is contained in AC, and let

z and x be equi-submultiples of b and a, z is not contained in BD oftener than ≈ in AC.

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

B. 5

But if not, take it away from AC as often as possible, Fig. 9. and let the remainder be oC; a and b are equi-submultiples of Ao and BD, and therefore since x and z are

equi-submultiples of a and b (2), as often as ≈ is con- (2) Hypoth. tained in BD, so often is a contained in Ao (3), and (3) Prop. 7. therefore z is not contained in BD oftener than x in B.3. AC.

Cor. 3. Let z be a submultiple of BD and be con- Fig 10. tained in it, as often as a is contained in the same BD, x is either equal to z or less than it.

(1) Ar. 1.

B. 5.

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

But if not, take away x from BD as often as z is contained in it, and let the remainder be mD; x and x are (2) Ax. 3. equi-submultiples of Bm and BD, and Bm is less than BD, therefore ≈ is less than ≈ (2).

B. 5.

Fig. 11. &

12.

Fig. 11.

PROP. VIII. THEOR.

If there be two unequal magnitudes (A and B) there is 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.

First let a be a submultiple of CD, take away b from CD, as often as a is contained in CD, and because bis (1) Hypoth. less than a (1), there must be a remainder mD; ifm D is greater than b, b is contained in CD oftener than in Cm, and therefore oftener than a is contained in CD.

& Ax. 3. B. 5.

But if mD be less than b, take ≈ a submultiple of b less than mD, and a an equi-submultiple of a; since r and z (2) Constr. are equi-submultiples of a and b (2), and a and b equisubmultiples of CD and Cm, x and x are equi-submulti(3) Prop. 7. ples of CD and Cm (3), but z is less than mD (4), thereforez is contained in CD oftener than in Cm, and therefore oftener than x is contained in the same CD.

B. 5.

Fig. 12.

& Ax. 3. B. 5.

2. Let a not be a submultiple of CD; take it away as often as possible from CD, and let the remainder be nD; take away b as often from CD, and let the re(4) Hypoth, mainder be mĎ; since Cm and Cn are equi-multiples of b and a, but b is less than a (4), Cm is less than Cn (5): take a submultiple of b less than mn, and ☛ an equi-submultiple of a, x and ≈ are equi-submultiples (6) Prop. 7. of Cm and Cn (6), but z is less than mn (7), therefore z is contained in Cn oftener than in Cm, and therefore (7) Constr. oftener than x is contained in Cn; but x is greater than 4.3.B.5. (8), therefore a is not contained in nD oftener than z

(5) Ax. 3. B 5.

B. 5.

(8) Constr.

in the same nD (9), but z is oftener contained in Cn (9) Prop. 3. than a in Cn, and therefore z is contained in CD of- B. 5. tener than a in CD.

Cor. In like manner, if there be four magnitudes, of which the first A is greater than the second C, but the third B equal to the fourth D; there can be taken a submultiple of C which is contained in D oftener than an equi-submultiple of A is contained in B.

PROP. IX. THEOR.

If there be given two magnitudes (B and A), and that Fig. 13. a submultiple (b) of one of them is contained in any third magnitude (CD) oftener than an equi-submultiple (a) of the other is contained in the same third, the latter (A) is greater than the former (B.)

First let b be a submultiple of CD, add a to itself as often as b is contained in CD, and let the sum be EF; a is contained in EF as often as b in CD, but b is contained in CD oftener than a in CD (1), therefore a is (1) Hypoth. contained in EF oftener than in CD, and therefore EF (2) Prop.5. is greater than CD (2); but a and b are equi-submul- B. 5. tiples of EF and CD (3), therefore a is greater than b (3) Const. & (4), and therefore, since A and B are equi-multiples of hypoth. a and b, A is greater than B (4).

2. Let b not be a submultiple of CD, take it away as often as possible from CD, and let the remainder be mD, make EF as before, and it shall be greater than CD, and therefore than Cm, but a and b are equi-submultiples of EF and Cm, therefore a is greater than b, and therefore, since A and B are equi-multiples of a and b, A is greater than B.

(4) Ax. 3.

B. 5.