demonftrated by Propofitions 16, B, 18, 17, E, of the fame book.

Otherwise algebraically,

Let abcd, and therefore ad=bc,

Altern. a: cb: d
Invert. b: a::d : c

Divid. a-b: bc-d: d

Comp.

a+bbc+d: d

Convert. a: a-b:: c-d

For, fince ad=bc, it is obvious that in each of these cafes the product of the extremes is equal to the product of the means; the quantities are therefore proportionals, (prop. 1.).

Prop. IV. If four numbers be proportionals, according to Def. 5. B. V. of Euclid, they will be geometrically proportional, according to the preceding definition. ift, Let the four numbers be integers, and let them be a, b, c, d. Then, if b times a and b times c be taken, and also a times b and a times d, fince ba, the multiple of the firft, is equal to ab, the multiple of the fe

cond,

any

cond, bc, the multiple of the third, must be equal to ad, the multiple of the fourth. And, fince bead, by prop. 1. a, b, c, and d, must be geometrical proportionals. 2dly, If of the numbers be fractional, all the four being multiplied by the denominators of the fractions, they continue proportionals, according to Def. 5. B. V. Euclid, (by Prop. IV. of that book), and the four integer quantities produced being fuch proportionals, they will be geometrical proportionals, by the first part of this Prop.; and, therefore, being reduced by divifion to their original form, they manifeftly will remain proportionals, according to the algebraical definition.

СНАР.

CHAP. III.

Of Equations in general, and of the Solution of fimple Equations.

1.

A

DEFINITIONS.

N Equation may in general be defined to be a propofition afferting the equality of two quantities; and is expreffed by placing the fign =be

tween them.

II. When a quantity stands alone upon one fide of an equation, the quantities on the other fide are faid to be a value of it. Thus, in the equation xb+y-d, xftands alone on one fide, and b+j—d is a value of it.

III. When an unknown quantity is made to ftand alone on each fide of an equation, and there are only known quan

!

tities on the other; that equation is faid to be refolved; and the value of the unknown quantity is called a root of the equation.

IV. Equations containing only one un-
known quantity and its powers, are
divided into orders, according to the
highest power of the unknown quan-
tity to be found in any of its terms.

If the highest power of 1ft,
the unknown quanti-> 2d,
ty in any term be the 3d, &c.

The Equa- Simple, tion is cal-Quadrat.

led

Cubic,&c.

But the exponents of the unknown quan-
tities are fuppofed to be integers, and the
equation is fuppofed to be cleared of frac-
tions, in which the unknown quantity, or
any of its powers, enter the denominators.
3x-b
is a fimple equation;

Thus, xa

3x

5=

2x

12, when cleared of the fraction

3

by multiplying both fides by 2x, becomes 6x2-5=24x, a quadratic. x3-2xa—x* -20 is an equation of the 6th order, &c.

As