2. Let a be a number to be divided into three parts, having among themselves the same ratios as the given numbers m, n, and p. It is evident that the verification of the question would be as follows;

denoting the 1st part by x, we have

the three parts added together must make the number to be divided. We have then the equation

By reducing all the terms to the denominator m, it becomes

mx +nx +px = am,

and we deduce from this

am

x=

m+n+p

This result is nothing more nor less than an algebraic expression of the rule of Fellowship, (Arith. 124); for by regarding the numbers m, n, p, as denoting the stocks of several persons trading in company, m+n+p is the whole stock, a the gain to be divided, and the equation

shows that a share is obtained by multiplying the corresponding stock into the whole gain, and dividing the product by the sum of the stocks; which reduced to a proportion, becomes

the whole stock: a particular stock

:: the whole gain: to the particular gain.

15. To form an equation from the following question, requires an attention to some things, which have not yet been considered.

A fisherman, to encourage his son, promises him 5 cents for each throw of the net in which he shall take any fish, but the son, on the other hand, is to remit to the father 3 cents for each unsuccessful throw. After 12 throws the father and the son settle their account, and the former is found to owe the latter 28 cents. What was the number of successful throws of the net?

If we represent this number by x, the number of unsuccessful ones will be 12-x; and if these numbers were given, we should verify them by multiplying 5 cents by the first, to obtain what the father was bound to pay the son, and 3 cents by the second, to find what the son engaged to return to the father. The first number ought to exceed the second by 28 cents, which the father owed the son.

We have for the first number x times 5 cents, or 5x. With respect to the second, there is some difficulty. How are we to obtain the product of 3 by 12-x? If instead of a we had a given number, we should first perform the subtraction indicated, and then multiply 3 by the remainder; but this cannot be done at present, and we must endeavour to perform the multiplication before the subtraction, or at least, to give the expression an entire algebraic form, similar to that of equations that are readily solved.

With a little attention we shall see, that by taking 12 times the number three, we repeat the number 3 so many times too much, as there are units in the number x, by which we ought first to have diminished the multiplier 12, so that the true product will be 56 diminished by 3 taken x times or Sæ,

This conclusion may be verified by giving to x a numerical value. If for example x were equal to 8, we should have 3 to be taken 12 times 8 times, and if we neglect 8 times, we should make the result 8 times the number 3 too much; the true product then will be

--

This result agrees with that which would arise from first subtracting 8 from 12; for then

12 8=4, and 3 X 4 = 12.

This being admitted, since the money due from the father to the son is expressed by 5x, and that which the son owes the father by 36-3x, the second number must be subtracted from the first in order to obtain the remainder 28; but here is another difficulty; how shall we subtract 56-3x from 5x, without having first subtracted 3x from 36?

We shall avoid this difficulty by observing, that if we neglect the term 3x, and subtract from 5x the entire number 36, we shall have taken necessarily 3x too much, since it is only what

remains after having diminished 36 by 3x that is to be subtracted from 5; so that the difference 5x- 36 ought to be augmented by 3x in order to form the quantity that should remain after having taken from 5x the number denoted by $6- Sx. This quantity will then be

There have been then 8 successful throws of the net and 4

unsuccessful ones.

Indeed 8 throws at 5 cents a throw give 40 cents,

4 throws at 3 cents a throw give 12

difference

as required by the conditions of the question.

28

To render the solution general, let a represent the sum givea by the father to the son for each successful throw of the net, and b the sum returned by the son for each unsuccessful one, and the total number of throws, and d the sum received on the whole by the son. If x be put equal to the number of successful throws, CI x will express the number of unsuccessful ones; each throw of the former kind being worth to the son a sum a, x throws would be worth a × x or ax, and the unsuccessful throws would be worth to the father the sum b multiplied by the number c-z. The reasoning by which we have found the parts of the product of 3 by 12-x, applies equally to the general case. If we neglect in the first place in forming the product be of b by the whole of c, the sum b will be repeated a times too much, and consequently the true product will be bc-bx.

In order to subtract this product from the sum ax, it is necessary to observe, as in the numerical example, that if we subtract the whole of the quantity be we take the quantity be too much, by which the former ought to have been first diminished, and that consequently the true remainder is not merely ax-bc, but ax-bcbx.

As this sum is equal to d, we have the equation

As this general formula indicates what operations are to be performed upon the numbers a, b, c, d, in order to obtain the unknown quantity x, we may reduce it to a rule or carefully write instead of the letters a, b, c, d, the numbers given. This last process is called substituting the values of the given quantities, or putting the formula into numbers. Applying here those of the foregoing example, we have

Methods for performing, as far as is possible, the operations indicated upon quantities that are represented by letters.

16. FROM the preceding question it is evident, that in certain cases a multiplication indicated upon the sum or difference of several quantities cannot be separated into parts; and in art, 11, we have exactly the reverse, by resolving the quantity axbx+cx, which represents the result of several multiplications, followed by additions and subtractions, into the two factors a b+c and x, which indicate only a single multiplication preceded by addition and subtraction. The reasoning pursued in these two circumstances, will suggest rules for performing, upon quantities represented by letters, operations which are called algebraic multiplication and division, from the analogy which they have with the corresponding operations of arithmetic.

We have also by the same analogy two algebraic operations, which bear the names of addition and subtraction, in which the object is to unite several algebraic expressions in one, or to take one expression from another. But these operations, like the preceding, differ from those of arithmetic in this, that their results are, for the most part, only indications of the operations to be performed; they present only a transformation of the

operations originally indicated into others, which produce the same effect. All that is done, is either to simplify the expressions, or to give them a proper form for exhibiting the conditions that are to be fulfilled.

In order to explain these operations, we give the name of simple quantities to those which consist only of one term, as + 2a, 3ab, &c. binomials to those which consist of two, as a+b, a ·b, 5a · 2x, &c. trinomials to those which consist of three terms, quadrinomials to those which consist of four terms, and polynomials to those which consist of more than four terms. It may be observed also, that we call polynomials compound quantities.

Of the addition of algebraic quantities.

17. THE addition of simple quantities is performed by writing them one after the other, with the sign+ between them; thus, a added to b is expressed by a + b. But when it is proposed to add together several algebraic expressions, we aim at the same time to simplify the result by reducing it to as small a number of terms as possible by uniting several of the terms in one. This is done in articles 2 and 5, by reducing the quantity x+x to 2x, and the quantity x+x+x to 3x. It can take place only with respect to quantities expressed by the same letters, and which are for this reason called similar quantities. A literal quantity that is repeated any number of times is regarded as a unit, it is thus, that the quantities 2a and 3a considered as two and three units of a particular kind, form when added 5a, or 5 units of the same kind. Also 4ab and 5ab make 9ab.

In this case, the addition is performed with respect to the figures which precede the literal quantity, and which show how many times it is repeated. These figures are called coefficients. The coefficient then is the multiplier of the quantity before which it is placed, and it must be recollected, that when there is none expressed, unity is understood; for 1a is the same as a.

18. When it is proposed to unite any quantities whatever, as 4a+5b and 2c+3d,

the sum total ought evidently to be composed of all the parts joined together; we must write then

4a+5b+2c+ 3d.