By the same process, as in the former example, is obtained :

x=

1000 X 3 X 100 x 2,5 × 20

47,5 × 142

= 2223,87 rub.

§ 94. In the activity which nature presents to us, as well as in all our actions, we observe this principle: that the product of any cause into the time of its action is equal to the effect of it. Or, the product of any means whatsoever, into the time of their action, or the power which acts upon them, or the conventional law of their action, produces a determined effect; that is, it is equal to it. Thus we have seen, that a capital loaned on interest renders as the product of the rate of interest into the time; that a man's labour is the result of the product of his strength (or power) into the time he exercises this strength (or power.) In all this therefore, we see nothing but the simple multiplication of certain factors, and their product; as has been quoted in the remarks to 71 and 72. In the same manner as products in arithmetic may be the result of a continued multiplication, so may an effect in nature be the combined product of a number of causes, means, powers, or times; and the effect itself may be represented by a combined product; as occurs, for instance, in higher mechanics, where these quantities often appear as multiplied by themselves, or in the square, cube, &c.

§ 95. If we now consider the relation of two such effects; that is to say, their ratio to each other, we find, as we have done in simple numbers, that: the same ratio must take place between the products of cause into time (as it will be simplest to call that by a general name) as that existing between the effects. We have now for some time made use of letters to denote quantities, before we knew the numbers which would correspond to them; we shall here

extend the advantage derived from it, in order to present this idea at one glance in its full connexions, and with the arithmetical operations connected with it. For that purpose we shall designate the objects of calculation, or the quantities of them, by their initial letters, and call

the cause = C

the time

Tfor one of the objects;

the effect =E

and for the other, which is compared to it in the compound proportion, we shall call the same objects by the corresponding small letters, as:

the cause = с

the time = t

the effect = e

We then obtain, by the principles stated already in the remarks to § 71:

СХТ E; and cxt = e

and for the proportion arising from this, in a manner exactly similar to what had been done in common numbers, we obtain the statement:

which corresponds, as simple products expressed by their factors and their results, to a statement similar to

3 X 47 X 9. = 12: 63

It evidently follows from this, by the division of the corresponding terms of the proportion, that we have also:

96. As we have seen in the preceding application of geometric proportion to the rule of three, that whatever term of the proportion be unknown, if the three others are given, this fourth is determined by the principles of the proportion; so in the present case, whatever may be the quantity unknown in such a compound rule of three, whether a cause, a time, or an effect, or a part of the one or the other of them, this quantity will be determined by the others, and obtained by the appropriate mutations of the proportion, or the operations of arithmetic resulting from it.

By this consideration and process all the complication, often resulting from combinations of direct and inverse proportions, in a compound rule of three, which are apt to lead young calculators into mistakes, are avoided, because every quantity, in any way concerned, is by its nature placed as factor in its proper place, by the simple reflection of its acting as either cause, time, or effect.

It may be easily seen that it will solve with ease questions upon combined actions of capitals during different times, as well in interest, as in shares of profit or loss, that is, in partnership, in complicated

*The teacher who will take the trouble to speak with his scholar upon this principle, or the attentive reader, who will compare it with the circumstances that surround him, will have no diffculty in explaining this simple idea; its correctness and generality will prove a great facility to the intelligent arithmetician. My own experience has proved to me that it meets no difficulty with boys of about 12 or 14 years, as scholars usually are, when in common schools they are thus far advanced in arithmetic, and that they made the statements appropriated to it very readily, and with peculiar satisfaction. It furnishes the best exercise of the mind for the appropriate application of common arithmetic. The examples which follow are worked out, and will, I hope, lead the way to its proper and easy application.

questions upon combined works, and all similar cases, as the following examples will show.

Example 1. A capital of $6200 produces in 5 years, at 17 per cent. $2170, amount of interest; what will a capital of $9300, at 4 per cent. produce in 9 years? Here the statement is extremely simple, thus:

C

X T

с

x t = E

: e

6200 × 0,07 X 5: 9300 × 0,04 × 9 = 2170: x This proportion may evidently be much reduced. 1st by dividing by 100, it becomes,

62 × 0,07 X 5: 93 X 0,04 × 9 = 2170: x Dividing by 2,

31 X 0,07 X 5: 93 × 0,02 × 9 = 2170 : ≈ Dividing by 70,

31 × 0,001 × 5 : 93 × 0,02 × 9 =

Dividing by 31,

31: x

0,001 X 5: 93 X 0,02 × 9 =

1: x

That is, the capital of $9300, at 4 per cent. produces, in 9 years, $3348 interest.

Example 2. A capital of $9500, at 6 per cent. interest, annually produced $4560, in 8 years, at what rate of interest must a capital of $12000 be lent out, which shall render $4800 in 5 years?

Statement:

9500 X 0,06 X 8: 12000 × 5 × x = 4560 : 4800 Reducing as above, by dividing the first by 500, and the second by 40;

19 × 0,06 x 8: 24 x 5 x x = 114 : 120

Dividing the first and third term by 6:

19 × 0,01 X 8: 24 X 5 x x = 19: 120 Dividing the first and third by 19, and the second and fourth by 24.

Dividing the second and fourth term by 5, and executing the multiplication indicated in the first term, we obtain : 0,08 : x =1:1

Or, the rate per cent. = x = 0,08 or 8 per cent.

Thus the simple reductions of the proportion given, has furnished the result. It is evident, that if we had at the first outset of this and the preceding example, expressed the term in which x is, by the other three, we would have reached the same results by the compensations in the numerator and denominator, and the factors of x with the opposite numerator.

Example 3. Two men, in partnership, contribute as follows: A puts in 7521 dollars, which he withdraws after 5 years and a half. B puts in 9772 dollars, which act in the company during 6 years, before which time the accounts cannot be settled. It is required to determine the share of each in the general result of all the operations, (which are taken together,) amounting to a net profit of 15472 dollars?

The sum of the products of the stocks into the times of their acting, are here to be compared to each single product of stock into the time of its acting, as cause and time; the whole benefit evidently represents the effect, corresponding to the whole stock, and its time of action.

Thus we obtain the two following statemens :

7521X5,5+9772×6: 7521×5,5 = 15472: share of A 7521X5,5+9772 × 6: 9772×6 = 15472: share of B

Or 99997,5 41365,5 = 15472 : share of A And 99997,5: 58632,0 = 15472: share of B Here we evidently obtain, as in the case of a bankrupt, treated in a former example, a constant fraction from the third term divided by the first, with which the second, or the product of the stock into the time of each partner is to be multiplied, to obtain his share in the profit; or we have:

The share of A=

15472

99997,5

X 41365,5 =