proportionably greater, as he had a greater distance to travel; but as he walks during a greater number of hours per day, in the second case, he will, for this reason, require less time; therefore, the operation relates in part to the rule of of Three direct, and to the rule of Three inverse.

We shall reduce it to a rule of Three simple, in considering that to walk during 30 days, employing 7 hours each day, is to walk during 30 times 7 hours, or 210 hours; thus, we can change the question to this other-it has required 210 hours to travel 230 leagues; how many will it require to travel 600 leagues? When we shall have found the number of hours which answers to this question, in dividing it by 10, we shall have the number of days demanded, since the man here spoken of employs 10 hours per day.

Therefore, we must seek the fourth term of a proportion, of which the three first are

We shall find that this fourth term is 547 hours and 1, which, divided by 10, the number of the hours that this man employs each day, gives 54 days and, or 54 days and 1, for the answer.

EXAMPLE III.

18

The foot in London being to the standard foot in France ::15:16, how many feet of France will equal 720 feet of London ?

It is clear, that to measure a determined length, it will require a less number of French than of English feet, in the same ratio that the first measure is, on the contrary, greater than the second; so that the question is reduced to this, viz. to calculate the fourth term of a proportion, which should commence with these three

16: 15: 720 :

multiplying 720 by 15, and dividing by 16, we shall have 675 for the number of French feet, equivalent to 720 feet in London.

EXAMPLE IV.

A convoy travelling 5 hours per day, can perform a certain journey in 18 days, but we would have it to arrive in 12 days, how many hours must it travel per day?

It is evident that this convoy should, each day, march during a number of hours as much more considerable than 5 hours, as the number 12 of the days which it must employ is, on the contrary, less than the number 18 of the days which it would have employed, if we had not forced the march. Thus the state of the question shows that we should calculate the fourth term of a proportion which should commence with these three:

12: 18:5: or these three, 2:3:: 5:

Multiplying 5 by 3, and dividing by 2, we have 7 for the number of hours during which the convoy should march each day.

OF THE RULE OF FELLOWSHIP.

197. The rule of Fellowship, or Society, is thus called, because it serves to divide among several associates the benefit or the loss resulting from their society.

Its end is to divide a proposed number into parts which have to each other given ratios.

The rule which we give to effect this, is founded upon what we have established (186 :) we shall deduce it from this principle in the following example :

EXAMPLE I.

Suppose, that it be required to divide 120, into three parts, which shall have to each other the same ratios as the numbers 4, 3, 2; the declaration of the question furnishes these two propositions :

4:3 : the first part is to the second.
4:2: the first part is to the third.

Or (182) these two others:

4 is to the first part :: 3 is to the second.
4 is to the first part: 2 is to the third.

So that we have these three equal ratios: 4 is to the first - part 3 is to the second: -2 is to the third.

Now, we have seen (186) that the sum of the antecedents is to the sum of the consequents, as any one antecedent is to its consequent: we can then say here, that the sum 9 of the three parts proportional to the parts that we seek, is to the sum 120 of these parts, as any one of the proportional parts is to the part of 120 which answers to it.

The rule then is, 1. to make a totality of the proportional parts given; 2. to make as many rules of Three as there are parts to find, and of which each one shall have for the first term the sum of the proportional parts given; for the second term the number proposed to divide ; and for the third term one of the proportional parts given: thus, in the question that we have taken for example, we should have these three rules of Three to perform:

9: 120: 4:

9: 120: 3:
9:1202:

of which we shall find (179) that the fourth terms are 551, 40, and 26, which have to each other the required ratios, and which compose in effect the number 120.

But it is easy to remark that it is not absolutely necessary to perform as many rules of Three as there are parts to find; we could dispense with the last, in subtracting from the proposed number the sum of the other parts, when we have found

them.

EXAMPLE II.

Three persons join in trade, the first puts in stock to the amount of 20000l.; the second, to the amount of 60000!.; the third, of 120000l. we demand what part each should have of the gain, which is 800000/., all expenses paid?

We see that it is required to divide 800000/. into parts which shall have to each other the same ratios as 20000, 60000, 120000; or (170) as 2, 6, 12, or 1, 3, 6, since each person should have a part in proportion to his fund; we must then add the three proportional parts, 1, 3, 6, and make the three following proportions, or only two :

10: 800000 :: 31.: the second part.

10: 800000: 67.: the third part.

These three parts will be 30000%., 240000l. and 480000l. The question might be more complicated, and yet be resolved by the same principles, as in the example which follows:

EXAMPLE III.

Three persons put into fellowship, the first 3000l., which has been during six months in the society; the second 40007. which has been in 5 months; and the third 8000l., which has remained during 9 months; how much should each have of the gain, which amounts to 12050l. ?

We shall reduce all these funds to the same time, in this

manner :

The principal of 3000l. ought to produce during 6 months as much as 6 times 3000l., or 18000l. during one month.

The principal of 4000/. should produce during 5 months ás much as 5 times 4000., or 20000l. during one month. Lastly, the principal of 8000l. should produce in 9 months as much as 9 times 8000l., or 720001. during one month.

Thus the question is reduced to this other: the funds of the three associates are 18000l., 20000l., 720007.: how much should each have of the gain 12050l. ?

In proceeding as in the above example, we shall find 1971. 16s. 4 d., 2190l. 18s 2 d., and 78871. 5s. 5d., for the answer.

mers.

EXAMPLE IV.

We would distribute to Albany, to Utica, and to Roches ter, a supply of utensils, namely, 4500 shovels, 4550 picks, 820 crow-bars, 800 axes, 2200 long drills, 2500 tamping irons, 2500 amalgamated priming wires, and 2500 stone hamThis distribution should be made for each kind of utensil, proportionably and conformably to a model of a supply, by which we see that of 85000 utensils of the same kind, Albany has had 6000, Utica 1400, and Rochester 1100. We demand how many there must be of each kind for each of these places?

Since each kind of utensil should be distributed proportionably to the numbers 6000, 1400, 1100, we shall find how many each place must have of one of the kinds; for example, of the shovels, in calculating the fourth term of each of these proportions:

8500: 4500 or 83: 45 or 17 : 9 :: 6000 :

17:9: 1400:

17:9: 1100 :

We shall take the same method for calculating the number of picks, of crow-bars, etc., which ought to be distributed to each place; and we shall find that the distribution should be made as follows:

Three wagoners have 1500l. to divide among them. The first was charged with 2 tons weight, which he conducted to 50 leagues distance; the second conducted 11⁄2 tons 75 leagues, and the third 3 tons 60 leagues. What is the share of each?

To solve this question by the preceding rule, we must re