ample similarly stated in his book on Mercantile Calculation; but the form nearest to the present mode of stating may be found in a work of Achacium Dörinck, arithmetician at Hamburg, which was published at Magdeburg in 1608.

It was not until the middle of the last century that the Chain Rule became more known, and as De Rees of Holland, a different person from Adam Ries, wrote on the subject, the Chain Rule has sometimes been called the Reesian Rule.

That those who follow the Chain Rule may acquire a practical knowledge of it, a variety of examples has been given to accustom learners to the proper way of stating and working.

It is desirable to have a fixed and convenient mode of treating the usual combinations of numbers that arise in business calculations; and which should answer for many purposes, without being obliged to go over a long series of Rules, to find out the most advantageous method for obtaining the result. The Chain Rule is superior to the Rule of Proportion, as it fully embraces and expedites that operation and lends assistance to a variety of its branches, besides offering the best means of instructing the scholars in the readiest method of properly

stating arithmetical questions. Notwithstanding the facility of calculating by the Chain Rule, it must not be considered a mere mechanical medium for forming an arithmetical scholar. The teacher's intelligence will enable him to use it as an efficient auxiliary in the stating of exercises, continuing, along with it, to inculcate the theory of the science to his pupils.

It is hoped that this little contribution to the school-room and counting-house, will not be considered unnecessary, or uncalled for, and that it may be the means of leading to the production of some more elaborate and more valuable work on the subject, based on science and on extended commercial experience.

SECTION I.

STATING AND WORKING.

IT

Ir may be safely observed, at the outset, that scarcely any example in arithmetic can be calculated with so much ease and certainty, or by a shorter process, than by the Chain Rule, wherever it can be applied. The following points must, invariably, be attended to:

A. State the question properly, by placing the figures in two columns, thus: | | B. Change into fractions numbers of different kinds, such as £14, instead of £1 10s. C. Equalize and arrange those fractions when formed, thus, £1 are equal to 3 halfpounds sterling.

D. Cancel or contract the numbers of each column, thus,

E

Multiply the numbers of each column, and divide the right by the left, which gives the answer. thus,

The proper statement of the question is the principal point in the Chain Rule, and, though it is easy, it requires attention. Suppose the following example:

If 4 yards of velvet cost £3, what will be the cost of 12 yards?

x£
4

12 yards
3£

The number of the kind

known, is expressed by x. top, left column, with x,

required, being unCommence at the which, as in the

example, is coin, or the number of £s wanted.

Opposite to it, forming the commencement of the right column, place the corresponding number of the equivalent, being here 12 yards.

Recommence at the left column with the number of the same kind as stated at the right, 4 yards.

Place opposite to it the corresponding number of the equivalent, 3£. Finish at the right with the same kind as stated at the commencement, expressed by x, which is here £.

This, and every other example, may be written in various ways:

1.

x£ = yards 4 =

12 yards

3£

Using as the sign for the question, and connecting it with the opposite number by = (equal to), and placing a similar sign between the successive numbers.

Using x, as before, and connecting the columns by a dash, thus :