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V. The ninth Propofition of Euclid.

Quantities that have an equal Ratio to the same Quantity, or to equal Quantities, are equal.

If A and B have the same Ratio to C, or D; A and B are equal. Also, if A has the fame Ratio to C, as B to D; if C=D, A-B.

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The tenth Proposition of Euclid. That Quantity which has the greater Ratio to a third Quantity, or to equal. Quantities, is the greater Quantity. And, of iwo Quantities, that, to which a third Quantity has a greater Ratio, is the lesser Quantity....

If C has a greater Ratio to B, than A has to B, C is greater than A.

Also, if A has a greater Ratio to D than to C, D is less than C.

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VII. The fifteenth Proposition of Euclid.

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Quantities have an equal Ratio to their Equimultiples ; or to their equal Parts.

1 3 Let A and B be two Quantities, in any Ratio whatever; and let them he taken an equal number of times.' Or, let them be divided into the same

B. number of equal Parts; The wholes of A and B

I 2 3 are equimultiples of those Parts, A:2A or 3A, ::B:2B or 3B. i.e. 1:2 or 3:1:2 or 3. A 14? Or, as : A, or A, :A:::B, or B, :B.

B_ube Also, A:A a-({ A) or A 1,::B:B6 (:B) or Bi. VIII. Quantities are in the fame Ratio, to each other, as their Equimultiples; or, as their equal Parts,

Let

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Let A and B be any Quantities, divided into equal Parts, as before, at a and b, or 1, 2, 3; also, let C and D be any equimultiples of A and B.

Then, A:B::2A:2B, or as 3A:3B; i. e. as C:D.

For Cand D are any equimultiples of A and B; wherefore, as often as A contains B, or is contained in B; so often 2 A contains 2 B, or, is contained in 2B; i.e. as C contains D, or is contained in D; consequently, A:B::C:D.

Also, as A:B :: Aa (A): Bb (1 B) or, as AI to Bs. ( ; A to B.)

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D IX. If two Quantities be divided into Parts of equal а d

magnitude (consequently equal amongst themselves)

then, one Quantity is in the fame Ratio to the B

other, as the number of Parts, in one, to the number of Parts, in the other.

Let A and D be two Quantities, divided into equal Parts, a, b, and c; d, e, f, and g;

Then, A:D:: 3:4; or, as 6 to 8, &c.

i.e. as a +b+is to d tetf+g. A B C D

X. If Quantities are proportional; the first to the second,

as a third is to a fourth, &c; then, if the first be any multiple or equal part of the second, the third is an equimultiple or equal part of the fourth, &c.

For Quantities are to each other, as the number of Parts in one, to the number of Parts in the other.

by the gth. Therefore, if A:B::C:D; A contains, or is contained in B, as often as C contains, or is contained in D.

Def. 6. otherwise, the Ratios are not equal, or analogous.

Axiom XI. In four proportional Quantities, i. e. when any one is to another, as a third is Ato the fourth, (Def. 6.) whether the first be equal Bto, greater, or less than the second, the third is a also equal to, greater, or less than the fourth. D

If A:B::C:D; then, if A be equal to B, C=D;
and, if A be greater, or less than B; C is, also, A
greater, or less than D.
For, A has the same Ratio to B, as C to D;

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And, a greater Quantity cannot have the same
Ratio to a less, as a less to a greater.

Def. 6.

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XII. The fourteenth Proposition of Euclid.

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If four Quantities are proportional ; then, if the Antecedent of one Ratio be greater than the Antecedent of the other, the Consequent of the first is also greater than the Consequent of the other; if the first are equal, the other are also equal; and, if less, less.

For, A:B::C:D; and, if A be equal to, greater, or less, than B, C is also equal to, greater, or less than D.

Wherefore, if A=C, B=D (Ax. 5.) and consequently, if A be greater or less than B, C is, necessarily, greater or less than D.

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XIII. The eleventh Proposition, of Euclid.

Ratios that are equal to the same Ratio, or to equal Ratios, are equal between themselves.

This follows from the third Axiom of Book ist, by substituting Ratios for Quantities.

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XIV. The thirteenth Proposition of Euclid.

If four Quantities are proportional, the first to the second, as the third to the fourth; and if the Ratio of the third to the fourth, be greater or less than a fifth Quantity has to a fixth; the Ratio of the first to the second, is also greater or less than the fifth to the sixth.

If A:B::C:D; and, if the Ratio of C to D be either greater or less than E to F; the Ratio of A to B is also greater or less, than E to F.

For, the Ratio of A to B is equal to the Ratio of C to D; therefore, their Ratios are the same, in respect to any other.

Note. By mistake, the 7th and 8th Axioms are misplaced. But,

as either is the other, alternately, it is of no consequence ; fave only, that the first part of the 8th is, properly, Euclid's fifteenth Proposition.

POSTULAT E S.

1. Grant, that equal Ratios may be taken, one for the other. 2. Grant, that any Quantity may be divided into any

number of Parts, equal to one another; or as any other

Quantity is divided. 3. Grant, that a Quantity may be taken or assumed, in any

Ratio to any given Quantity.

The assumption of this last Axiom is not allowed by some Geometers; notwithstanding, to me, it appears full as possible, as that equimultiples, or equal parts of Quantities may be taken; by one or other of which Suppofitions, they demonstrate the whole fifth Book.

THE

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If Quantities, which are equimultiples or equal parts

of other Quantities, be added into one Sum, or Quantity; the same multiple or part, which one Quantity is of the other, respectively, the whole is of the whole.

A B C D E F Let A, B, and C be equimultiples, or equal parts, of D, E, and F, respectively; i. e. whatever multiple, or part, A is of D, let B be the same multiple, or part, of E, and C of F.

Then, A, B, and C, added together, into one Sum, is the same multiple or part of D, E, and F, added together, as A is of D, B of E, or C of F.

Dem.For, let A be equal to 2D, or 3D; then,B=2E,or3E,&c.

conf. A+B+C=D+E+F, taken twice, or thrice, &c. But, Quantities are to each other, as the number of Parts, in one, is to the number of equal Parts, in the other;-Ax.9. wh. as A:D, B:E, and C:F, ::A+B+C:D+E+F. Consequently, 'as often as A contains D, or is contained in D, &c. so often the whole of At-B+C contains, or is contained in, D+E+F. Therefore, &c. Q. E. D.

For, if either be multiples of the other; the latter is, consequently, equal parts of the former.

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