Whence three times the least part, plus twice the excess of the middle part above the least, plus also the excess of the greatest above the middle one, will be equal to the number to be divided.

Whence three times the least part will be equal to the number to be divided, minus twice the excess of the middle part above the least, and minus also the excess of the greatest above the middle one.

Whence in fine, the least part will be equal to a third of what remains after deducting from the number to be divided twice the excess of the middle part above the least, and also the excess of the greatest above the middle one.

7. The signs mentioned in article 2 are not the only ones used in algebra. New considerations will give rise to others, as we proceed. It must have been observed in article 2, that the multiplication of x by 2, and in articles 5 and 6 that of a by s and that of b by 2, is denoted by merely writing the figures before the letters x and b without any sign between them, and I shall express it in this manner hereafter; so that a number placed before a letter is to be considered as multiplied by the number represented by that letter. 5x, 5a, &c. signify five times x, five times a, &c. 3x or &c. signifies of x or three times 4'

x divided by 4, &c.

3.x

In general, multiplication will be denoted by writing the factors in order one after the other without any sign between them, whenever it can be done without confusion.

Thus the expressions ax, bc, &c. are equivalent to a xx, bx c, &c. but we cannot omit the sign when numbers are concerned, for then 3 x 5, the value of which is 15, becomes 35. In this case we often substitute a point in the place of the usual sign, thus, 3.5.

Equations.

8. If the solution of the problems in articles 3 and 6 be examin ed with attention, it will be found to consist of two parts entirely distinct from each other. In the first place, we express by means of algebraic characters the relations established by the nature of the question between the known and unknown quantities, from which we infer the equality of two quantities among themselves; for instance, in article 3 the quantities 2x + b and a, and in article 6 the quantities 3x + 2b + c and a.

We afterwards deduce from this equality a series of consequences, which terminate in showing the unknown quantity x to be equal to a number of known quantities connected together by operations, that are familiar to us; this is the second part of the solution.

These two parts are found in almost every problem which belongs to algebra. It is not easy, however, at present to give a rule adapted to the first part, which has for its object to reduce the conditions of the question to algebraic expressions. To be able to do this well, it is necessary to become familiar with the characters used in algebra, and to acquire a habit of analyzing a problem in all its circumstances, whether expressed or implied. But when we have once formed the two numbers, which the question supposes equal, there are regular steps for deducing from this expression the value of the unknown quantity, which is the object of the second part of the solution. Before treating of these I shall explain the use of some terms which occur in algebra.

An equation is an expression of the equality of two quantities. The quantities which are on one side of the sign = taken together are called a member; an equation has two members. That which is on the left is called the first member, and the other the second.

In the equation 2x + b = a, 2x + b is the first member, and a is the second member.

The quantities, which compose a member, when they are separated by the sign+ or -, are called terms.

Thus, the first member of the equation 2x + b = a contains two terms, namely, 2x and + b.

The equation 3x+7= 8x 12 has two terms in each of its members, namely,

Although I have taken at random, and to serve for an example merely, the equation 3x+7=8x-12, it is to be consid ered, as also every other of which I shall speak hereafter, as derived from a problem, of which we can always find the enunciation by translating the proposed equation into common language. This under consideration becomes,

To find a number x such, that by adding 7 to 3x, the sum shall be equal to 8 times x minus 12.

Also the equation ax + bc cx ac bx, in which the letters a, b, c, are considered as representing known quantities, answers to the following question;

To find a number x such, that multiplying it by a given number a, and adding the product of two given numbers b and c, and subtracting from this sum the product of a given number c by the num ber x, we shall have a result equal to the product of the numbers a and c, diminished by that of the numbers b and x.

It is by exercising one's self frequently in translating questions from ordinary language into that of algebra, and from algebra into ordinary language, that one becomes acquainted with this science, the difficulty of which consists almost entirely in the perfect understanding of the signs and the manner of using them.

To deduce from an equation the value of the unknown quantity, or to obtain this unknown quantity by itself in one member and all the known quantities in the other, is called resolving the equation.

As the different questions, which are solved by algebra, lead to equations more or less compounded, it is usual to divide them into several kinds or degrees. I shall begin with equations of the first degree. Under this denomination are included those equations in which the unknown quantities are neither multiplied by themselves nor into each other.

of the resolution of equations of the first degree, having but one unknown quantity.

9. WE have already seen that to resolve an equation is to arrive at an expression, in which the unknown quantity alone in

one member is equal to known quantities combined together by operations which are easily performed. It follows then, that in order to bring an equation to this state, it is necessary to free the unknown quantity from known quantities with which it is connected. Now the unknown quantity may be united to known quantities in three ways;

1. By addition and subtraction, as in the equations,

2. By addition, subtraction, and multiplication, as in the equations,

3. Lastly, by addition, subtraction, multiplication, and division, as in the equations,

The unknown quantity is freed from additions and subtractions, where it is connected with known quantities, by collecting together into one member all the terms in which it is found; and for this purpose it is necessary to know how to transpose a term from one member to the other.

10. For example, in the equation

7x

it is necessary to transpose 4x from the second member to the first, and the term 5 from the first member to the second. In order to this, it is obvious, that by cancelling + 4x in the second member, we diminish it by the quantity 4x, and we must make the same subtraction from the first member, to preserve the equality of the two members; we write then 4x in the first member, which becomes 7x - 5 4x and we have

7x-5- - 4x 12.

To cancel 5 in the first member, is to omit the subtraction of 5 units, or in other words, to augment this member by 5 units ; to preserve the equality then we must increase the second member by 5 units, or write + 5 in this member, which will make it 12+5; we have then

By performing the operations indicated there will result the equation

Sx = 17.

From this mode of reasoning, which may be applied to any example whatever, it is evident, that to cancel in a member a term affected with the sign +, which of course augments this member, it is necessary to subtract the term from the other member, or to write it with the sign; that on the contrary when the term to be effaced has the sign minus, as it diminishes the member to which it belongs, it is necessary to augment the other member by the same term, or to write it with the sign+; whence we obtain this general rule;

To transpose any term whatever of an equation from one member to the other, it is necessary to efface it in the member where it is found, and to write it in the other with the contrary sign.

To put this rule in practice, we must bear in mind that the first term of each member, when it is preceded by no sign, is supposed to have the sign plus. Thus, in transposing the term cx of the literal equation ax-bcx + d from the second member to the first, we have

transposing also-b from the first member to the second, it be

11. By means of this rule, we can unite together in one of the members all the terms containing the unknown quantity, and in the other all the known quantities; and under this form the member, in which the unknown quantity is found, may always be decomposed into two factors, one of which shall contain only known quantities, and the other shall be the unknown quantity by itself.

This process suggests itself immediately, whenever the proposed equation is numerical and contains no fractions, because then all the terms involving the unknown quantity may be reduced to one. If we have, for example, 10x + 7x +7 by performing the operations indicated in each member, we shall have in succession

2x=25

and 15a is resolved into two factors 15 and a; we have then