If we consider that the principal and interest added together must give the amount, and if we put A to represent the amount, then we shall have

Prt+P A (2)

Equations (1) and (2) embrace all the conditions in relation to interest, and furnish all the rules for computations.

For instance, equation (1) gives

That is, when any problem requires the finding of the principal, observe the following rules:

RULE 1.-Divide the interest by the product of the rate and time.

RULE 2.-Divide the amount by the product of the rate and time, increased by unity.

I

=1

Pr

That is, to find the time, we have the following rule: RULE.-Divide the whole interest by the interest for one

To find the rate per cent., take the following rule:

RULE. Divide the interest by the product of the principal and time.

FELLOWSHIP.

Two men united capital to engage in a certain enterprise, the first put in a dollars, the second 6 dollars, and they gained g dollars. Give a rule for the equitable division of this gain.

Let x represent the portion belonging to that one which paid in a dollars, and y the portion of the other.

But their portions of the gain should be in just the same proportion as their capital paid in,

That is,

x:y=a:b, or bx=ay

Multiply the first equation by a, then

ax+ay=ag, or ax+bx=ag

Hence, we have the following rule to find each man's share. RULE.-Multiply the gain by each man's stock, and divide the product by the whole capital invested.

Again, suppose three persons, A, B, and C, enter into partnership, and furnish capital in proportion to a, b, and c, and they gain a sum, g, what is each man's share of it? Let x= A's share, y= B's share, and z= C's share,

Then,

x+y+z=g (1)

x:y=a:b, also, y:z=a:c

And,.

These values of y and z, put in equation (1), give

This value of a put in (2) gives y, and in (3) gives z.

Here, again, we find that each man's share of the gain is equal to the whole gain multiplied by his particular portion of the stock, and that product divided by the whole stock.invested.

The same results would be obtained in relation to any number of partners. Observe, that g can be of any value, positive, negative, or zero. When it is zero, each numerator is zero; and, thus, x, y, and z become zero, as they ought in that case. When g is negative, it denotes loss, and losses. must be shared in the same proportion.

It is not necessary that a, b, and c should designate the actual stock of each partner if they represent their due proportional parts.

In taking up a book on common Arithmetic, we find the following rule for fellowship:

As the whole amount of stock or labor

Is to each man's portion,

So is the whole property, loss or gain,

To each man's share of it.

These four lines express either one of the equations, (4), (5), or (6); for, by resolving (4), for example, into a proportion, we have (a+b+c): a=g:x

Thus, we perceive that this, like most other arithmetical rules, is the result of algebraic investigation.

Let us now consider the case in which time is an element, and for the sake of clearness we will suppose an example.

Two men, A and B, hired a pasture, for which they agreed to pay g dollars. A put in a cows 3 weeks, B put in b cows for 5 weeks; what shall each pay?

Consider that, a cows for three weeks would consume as much as 3a cows for one week. Also, b cows for five weeks would consume as much as 56 cows for one week. Thus, we reduce all action to some unit of time. To be more general, we will consider 3 and 5 as t and t, any number of weeks or days whatever, then the action will be at and bť, and it is

evident that the partners must pay in proportion to this action, or in this case, to the consumption of the cows.

Now, let x what A must pay, and y= what B must pay,

This value of y put in equation (1), gives

Equation (4) will furnish the following proportion:

(at+bt'): at=g: x

Equation (5), at+bt': bt'=g: y

Taking up a work on Arithmetic, I found the following rule for computing results in compound proportion.

RULE.-Multiply the active agents by the time each was in action. Then by proportion.

As the sum of the products
Is to each particular product,
So is the whole gain or loss

To each man's share of it.

Now, it is evident that the words of this rule were dictated by the preceding proportions.

THE END.