It hence appears, that he must take an equal quantity of each.

By the same rule, we may find the quantity of alloy in any compound metal, for example bronze, which consists of copper and tin mixed together in a certain proportion. For this purpose, we must take three ingots of the same weight, one of bronze, another of copper, and a third of pure tin. These three bodies, when weighed in water, will each lose a different part of their weight the ingot of tin will lose more, and that of copper less, than the ingot of bronze. Let us suppose, that the loss of the bronze is 3 ounces, that of copper 21, and that of the tin 34. If these three numbers be arranged, according to the above formula, we shall have

21

3

31

2

The sum of the two differences, 2, shews that, in of bronze, there are one of copper and two of tin. This proportion being found, we may thence conclude that a mass of bronze similar to the ingot, weighing 150 pounds, would contain 100 pounds of tin, and 50 of

copper.

A CHRONOLOGICAL PROBLEM.

How many years, months, and days, elapsed between the Battle of Marignan, fought on the 3rd of September, 1515, and that of Fontenoi, fought on the 11th of May,

1745?

The period from the Christian era to the 3rd of September, 1515, comprehends 1514 years, 8 months and

3 days; and that from the same epoch to the 11th of May, 1745, comprehends 1744 years, 4 months, and 11 days.

Consequently, if we subtract the former from the latter, the difference, 229 years, 8 months, and 8 days, will express the interval of time between the battle of Marignan and that of Fontenoi.

This method may be employed for every other problem of the like kind, and especially when it is necessary, in calculating interest, to know how many years, months, and days, have elapsed between certain dates.

THE RULE OF TARE.

By tare is commonly meant the weight of the cask, box, or bag, in which goods are contained, and which being subtracted, when known, from the gross weight, leaves the real weight of the goods, called the net weight. In general, an allowance is made for it, at the rate of so much per hundred weight; and the quantity to be deducted is found by the Rule of Three, as in the following example:

A merchant purchases a bale of cotton weighing 7 cwt. including the package, and is allowed at the rate of 16 per cut. of tare: how much ought to be deducted on that account, from the gross weight of the bale of

cotton?

As the merchant purchases the goods by the net weight, the seller must give him 16 pounds over and above each cwt.; that is to say, for each 112 pounds he must give him 128. We must, therefore, make the following proportion:

128: 112 :: 700 : x. (23.)

The fourth term will express the number of pounds for which the merchant ought to pay.

DISCOUNT.

A merchant purchases goods to the amount of £1000. to be paid at the end of a year; but the vender offers to abate 10 per cent. for ready money; how much must down?

the buyer pay

It might here be supposed, that the abatement ought to be as many times £10. as 100 is contained in 1000; that is to say, that £100. ought to be deducted, so that the merchant would have to pay only £900.

But it is to be observed, that the vender ought to allow the purchaser only 10 per cent. on what he will really receive; that is to say, that every 110 pounds which the merchant has to pay, ought to be reduced to 100. We have, therefore, the following proportion:

110: 100 :: 1000 : x.

This is the only true method of estimating discount; for if the vender received only £900. ready money, this sum at 10 per cent. would produce, at the end of a year, no more than £990. consequently it would be much better for him to give a year's credit and receive £1000.

OF COMBINATIONS AND PERMUTATIONS.

Before we enter on this subject, it will be necessary to explain the method of constructing a kind of table, treated of by Pascal and others, called the arithmetical triangle; which is of great use to shorten calculations of this kind.

First form a band A B of ten equal squares, and below it another CD of the like kind, but shorter by one square on the left, so that it shall contain only nine squares; and continue in this manner, always making each successive band a square shorter. We shall thus have a series of squares, disposed in vertical

and horizontal bands, and terminating at each extremity in a single square, so as to form a triangle, on which account it has been called the arithmetical triangle.

The numbers with which it is to be filled up, must be disposed in the following manner:

In each of the squares of the first band inscribe unity, as well as in each of those in the diagonal A E.

Then add the number in the first square of the band CD, which is unity, to that in the square immediately above it, and write down the' sum 2 in the following square. Add this number, in like manner, to that in the square above it, which will give 3, and write it down in the next square. By these means we shall have the series of the natural numbers 1, 2, 3, 4, 5, &c. The same method must be followed to fill up the other horizontal bands; that is to say, each square ought al

ways to contain the sum of the number in the preceding square of the same row, and that which is immediately above it in the preceding. Thus, the number 15, which occupies the fifth square of the third band, is equal to the sum of 10, which stands in the preceding square, and of 5, which is in the square above it. The case is the same with 21, which is the sum of 15 and 6; with 35, in the fourth band, which is the sum of 15 and 20, &c.

The different series of numbers, contained in this triangle, have different properties; but we shall here speak only of those which relate to combinations and permutations, as the rest are of too abstract a nature to be employed in arithmetical recreations, the principal object of which is to afford easy and agreeable

amusement.

There are two principal kinds of combination. The first is that where the different arrangements of several things are sought, without any regard to their change of place.

The second is that where regard is paid to the different changes of place. For example, the three quantities A, B, C, taken two and two, without regard to the different changes of place, are susceptible of only three combinations A B, A C, B C; but if we pay regard to the changes of place, they are susceptible of six combinations; for, besides the three former, we shall have B A, C A, C B.

In combinations, properly so called, no attention is paid to the order of the things. If four tickets, for example, marked A, B, C, D, were put into a hat, and any one should bet to draw out A and D, either by taking two at once, or one after the other, it would be of no importance whether A should be drawn first or last; the combinations A D and DA ought, therefore, to be here considered only as one.

But, if any one should bet to draw out A the first time, and D the second, the case would be very different, and it would then be necessary to attend to the order in which these four letters may be taken, and arranged together, two and two: it may be readily

D