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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.

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Multiplying the first column together, and also the second,

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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.

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

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a: b::c:d, then also, a": b" :: c" : d"

= raising each of these equals to the power of n,

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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, surl: that

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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.

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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,

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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

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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.

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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

137. Equations are divided into degrees according to the highest power of the unknown quantity which they contain. Those which involve the simple power only of the unknown quantity, are called simple equations, or equations of the first degree; those into which the square of the unknown quantity enters, are called quadratic equations, or, equations of the second degree; so we have cubic equations, or, equations of the third degree; biquadratic equations, or, equations of the fourth degree; equations of the fifth, sixth, . nth degree. Thus,

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138. Numerical equations are those which contain particular numbers only, in addition to the unknown quantity. Thus, x3 + 5 x2 = 3 x + 17, is a numerical equation.

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139. Literal equations are those in which the known quantities are represented by letters only, or by both letters and numbers. Thus, x3 + p x2+ q x = r, x 1 − 3 p x 3 + 5 q x2 + 1 x = 5 are literal equations.

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140. Let us now pass on to consider the solution of equations, it being understood, that, to solve an equation, is to find the value of the unknown quantity, or to find a number which, when substituted for the unknown quantity in the equation, renders the first member identical with the second.

The difficulty of solving equations depends upon the degree of the equations, and the number of unknown quantities. We first consider the most simple

case.

ON THE SOLUTION OF SIMPLE EQUATIONS CONTAINING ONE UNKNOWN QUANTITY.

141. The various operations which we perform upon equations, in order to arrive at the value of the unknown quantities, are founded upon the following principles :—

If to two equal quantities, the same quantity be added, the sums will be equal. If from two equal quantities, the same quantity be subtracted, the remainders will be equal.

If two equal quantities be multiplied by the same quantity, the products will be equal.

If two equal quantities be divided by the same quantity, the quotients will be equal.

These principles, when applied to the two equal quantities which constitute the two members of every equation, will enable us to deduce from them new equations, which are all satisfied by the same value of the unknown quantity, and which will lead us to discover the value of that unknown quantity.

142. The unknown quantity may be combined with the known quantities in the given equation, by the operations of addition, subtraction, multiplication, and division. We shall consider these different cases in succession.

L. Let it be required to solve the equation,

x + a = b

If, from the two equal quantities + a and b, we subtract the same quantity e, the remainders will be equal, and we shall have,

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