Let a b::c: d, then also, a + b : b : : c + d : d ̧

X. If four quantities be proportionals, they will be proportionals also divi dendo, that is, the difference of the first and second will have to the second the same ratio that the difference of the third and fourth has to the fourth.

Let a b c d, then also, a bb::c d: d

subtract unity from each of these equals, then

C

1

XI. If four quantities be proportionals, they will be proportionals also convertendo, that is, the first will have to the difference of the first and second the same ratio that the third has to the difference of the third and fourth.

Let abc: d, then also, a: a b::c:c- d

с

b

с

= then by prop. VIII. = and hence subtracting these

XIL If four quantities be proportionals, the sum of the first and secon: will have to their difference the same ratio that the sum of the third and forrth has to their difference.

Let a b c d, then also, a + ba b::cd: c — d

a

Since =
b

we have

XIII. If there be any number of quantities more than two, and as many others, which, taken two and two in order, are proportionals, (ex æquali,) the firs! will have to the last of the first rank the same ratio that the first of the second rank has to the last.

Multiplying the first column together, and also the second,

XIV. If there be any number of quantities more than two, and as manu others. which, taken two and two in a cross order, are proportionals, (ex æquali pertarbatâ,) the first will have to the last of the first rank the same ratio that the Arst of the second rank has to the last.

XV. If four pantities he proportionals, any powers or roots of these quantities will also be proportionals.

XVI. If there be any number of proportional quantities, the first will have to the second the same ratio that the sum of all the antecedents has to the sum of au the consequents.

Let

a, b, c, d, e, f, g, h, be any number of proportional quantities, such that

and. a (b+ d + f + h) = b (a+c+e+g)

XVII. If three quantities be in continued proportion, the first will have to the hird the duplicate ratio of that which it has to the second.

XVIII. If four quantities be in continued proportion, the first will have to the fourth the triplicate ratio of that which it has to the second. Let a, b, c, d, be four quantities in continued proportion, so that, a:b::b:c::c: a, then also, &:d: ¡¤1‚3

Since,

ON EQUATIONS.

PRELIMINARY REMARKS.

134. AN equation, in the most general acceptation of the term, signifies two alge braic expressions which are equal to each other, and are connected by the sign =. Thus, a x = b, c x2 + d x = e, c x3 + g x2= h x + k, m x 1 + n x3 + p x2 +qx+r=o, are equations.

The two quantities separated by the sign equation, the quantity to the left of the sign

are called the members of the is called the first member, the quantity to the right the second member. The quantities separated by the signs + and are called the terms of the equation.

135. Equations are usually composed of certain quantities which are known and given, and others which are unknown. The known quantities are in general represented either by numbers, or by the first letters in the alphabet, a, b, c, &c.; the unknown quantities by the last letters, s, t, x, y, z, &c.

136. Equations are of different kinds.

1o. An equation may be such, that one of the members is a repetition of the other, as, 2 x 5=2x

5.

2o. One member may be merely the result of certain operations indicated in the other member, as, 5x + 16 = 10 x − 5 — (5 x — 21), (x+y) (x−y) = x2+ xy + y2.

Ꮖ y 3

y

3o. All the quantities in each member may be known and given, as, 25 = 10 + 15, a+b=c—d, in which, if we substitute for a, b, c, d, the known quantities which they represent, the equality subsisting between the two members will be self-evident.

In each of the above cases the equation is called an identical equation.

4. Finally, the equation may contain both known and unknown quantities, and be such, that the equality subsisting between the two members cannot be made manifest, until we substitute for the unknown quantity or quantities certain other numbers, the value of which depends upon the known numbers which enter into the equation. The discovery of these unknown numbers constitutes what is called the solution of the equation.

The word equation, when used without any qualification, is always understood to signify an equation of this last species; and these alone are the objects of our present investigations.

x + 4 = 7 is an equation properly so called, for it contains an unknown quantity x, combined with other quantities which are known and given, and the equality subsisting between the two members of the equation cannot be made manifest, until we find a value for x, such, that when added to 4, the result will be equal to 7. This condition will be satisfied, if we make x = 3, and this value of x being determined, the equation is solved.

The value of the unknown quantity thus discovered is called the root of the equation