9. wolf rotivib sdɔ iw nab mb dode of Proportion.

BY Y the preceding operations, quantities By the preceding op may be compared

of the fame

together.

may

The relation arising from this comparifon is called Ratio or Proportion, and is of two kinds. If we confider the difference of the two quantities, it is called Arithmetical Proportion; and, if we confider their quotient, it is called Geometrical Proportion. This laft being moft generally ufeful, is commonly called fimply Proportion.

1. Of Arithmetical Proportion.

Definition. When, of four quantities, the difference of the first and second is equal to the difference of the third and fourth, the quantities are called arithmetical proportionals.

Cor. Three quantities may be arithmetically proportional, by fuppoiing the two middle terms of the four to be equal.

Prop. In four quantities arithmetically proportional, the fum of the extremes is equal to the fum of the means.

Let the four be a, b, c, d.

Therefore from Def. a-b-c-d; to thefe add b+d and a+d=b+c.

Cor. 1. Of four arithmetical proportionals, any three being given, the fourth may be found.

Thus, let a, b, c, be the ift, 2d, and 4th terms, and let x be the 3d which is fought. Then, by definition, a+c=b+x, and x=a+c-b.

Cor. 2. If three quantities be arithmetical proportionals, the fum of the extremes is double of the middle term; and hence, of three fuch proportionals, any two being given, the third may be found.

2. Of Geometrical Proportion.

Definition. If, of four quantities, the quotient of the first and second is equal to the quotient of the third and fourth, these quantities are faid to be in geometrical proportion. They are alfo called proportionals.

Thus,

Thus, if a, b, c, d, are the four quantities, then, and their ratio is thus de

noted, a b c : d.

Cor. Three quantities may be geometrical proportionals, viz. by supposing the two middle terms of the four to be equal. If

the proportion is expreffed thus, a : b : c.

Prop. I. The product of the extremes of four quantities, geometrically proportional, is equal to the product of the means: and conversely.

and multiplying both by bd, ad=bc.

If ad bc, then dividing by bd,

is, a: b::c: d.

'that

b

Cor. 1. The product of the extremes of three quantities, geometrically proportional, is equal to the fquare of the middle

term.

Cor.

Cor. 2. Of four quantities geometrically proportional, any three being given, the fourth may be found.

Ex. Let a, b, c, be the three firft; to find the 4th. Let it be x, then a : bcx, and by this proposition,

ax-bc
bc

and dividing both by a, x= a

This coincides with the Rule of Three in

arithmetic, and be confidered as a de

may

monftration of it. In applying the rule to

tą

any particular cafe, it is only to be obferved, that the quantities must be so connected, and so arranged, that they be proportional, according to the preceding definition.

Cor. 3. Of three geometrical proportionals, any two being given, the third may be found.

Prop. II. If four quantities be geometrically proportional, then if any equimultiples whatever be taken of the first and third, and alfo any equimultiples whatever of the fecond and fourth; if the multiple of the first be greater than that of the fecond, the

multiple

multiple of the third will be greater than that of the fourth; and if equal, equal; and if lefs, lefs.

For, let a, b, c, d, be the four proportionals. Of the firft and third, ma and mc may represent any equimultiples whatever, and alfo nb, nd, may reprefent any equimultiples of the fecond and fourth. Since a:b::c:d, ad bc; and hence multiplying by mn, mnad=mnbc, and therefore (Conv. Prop. 1.) ma: nb :: mc: nd; and from the definition of proportionals, it is plain, that if ma is greater than nb, mc must be greater than nd; and if equal, equal; and if lefs, lefs.

Prop. III. If four quantities are proportionals, they will also be proportionals when taken alternately, or inverfely, or by compo fition, or by divifion, or by converfion. See def. 13. 14. 15. 16. 17. of book V. of Euclid, Simfon's edition.

By Prop. II. they will also be propor tionals, according to Def. 5. book V. of Euclid; and therefore this proposition is

demonftrated