much of the other commodity, at the rate proposed, may be had for that sum.

"2. If the quantities of both commodities be given, and it should be required to find how much of some other commodity, or how much money should be given for the inequality of their values: -Find the separate values of the two given, commodities, subtract the less from the greater, and the remainder will be the balance, or value of the other commodity.

"3. If one commodity is rated above the ready money price, to find the bartering price of the other-Say, as the ready money price of the one, is to the bartering price, so is that of the other to its bartering price."

Now under which of the above cases does the question come? Here I leave my readers in the midst of the difficulty ;-and without wasting time in tracing analogies, solve the question without any reference to a rule. Twelve cords of wood, at three dollars per cord, will cost thirty-six dollars, and it will take as many barrels of flour, at six dollars per barrel, as there are sixes in thirty-six. There is no great trouble in arriving at an answer.

I shall not regret obliging my readers to learn so much mathematicks, at such an expense of patience, if I convince them by example of the trouble of it; and may assure them at the same time, this is precisely what thousands and thousands of learners in our schools, are doing every day. An example involv

ing small numbers, only, was selected to make the illustration more plain. The reasoning would be the same, however large the numbers. The same difficulties are experienced in numerous rules, but this single example will suffice to expose the difficulties and suggest the remedies.

Could I enter into a detailed examination of the execution of the inductive system of Mr. Colburn, much would be found, to show a profound knowledge of the subject, as well as of the powers and principles of mind, to which it is adapted. A few faults might be detected by a vigilant and scrutinizing eye. But as I am obliged, by the circumstances under which I write, to confine myself to general principles, and forbear to enlarge upon the excellencies in execution, justice requires me to abstain from the faults.

LETTER VIII.

THERE is one result from the arrangement of arithmetick by general principles, so important, that it demands particular consideration. The Rule of Three is entirely omitted. Those, who first learned arithmetick mechanically, and have never thought

of it except in connexion with its forms, will start at so bold an innovation; and think of course, that a rule, which has been dignified with the name of the Golden Rule, and which takes up with all its modes, no inconsiderable portion of their books, cannot be omitted, without omitting something essential to the subject. This is not the fact. The omission is an essential improvement. But this is being positive without proof. Objections will, no doubt, be started. So far as they can be anticipated, they shall be met under the two heads of the possibility, and the expediency of the omission.

First. It will be possible to dispense with the rule, if all questions which are now solved by it, can be solved by other rules, or by general principles. This is a position pretty easily sustained. I offer four examples, which present all the variety that can occur under the Golden Rule.

The first is an example of the "Rule of Three Direct" the second, of the "Rule of Three Inverse;" the third is an example of direct proportion in "Double Rule of Three ;" and the fourth, of "Inverse Proportion," of the same rule.

1. If a family consume of a barrel of flour in 3 weeks, how many barrels would they consume in 15 weeks?

Analysis. If they consume of a barrel in 3 weeks, they will consume one third as much, or a barrel, in one week; and if they consume

of

of a

barrel in one week, they will consume 15 times as much, or, equal to 4 barrels, in 15 weeks.

2. If 3 men do a piece of work in 7 days, how long will it take 5 men to do the same work?

Analysis. If 3 men do the work in 7 days, it will take one man three times as long, or 21 days; and if it take 1 man 21 days, 5 men will do the same work in

of the time, or of a day, equal to 4 days. 3. If the interest of $50 for 2 months is $3, what will be the interest of $30 for 5 months?

Analysis. If the interest of any sum of money for 2 months, is 3 dollars, the interest of the same sum for 1 month will be as much, or of a dollar; and if of a dollar is the interest of $50, the interest of 1 dollar will be as much, or of a

3

IOO

I

3

dollar; if of a dollar is the interest of 1 dollar for 1 month, the interest of $30 will be thirty times

90

as much, or of a dollar, and for 5 months it will

be 5 times as much, or 45%, equal to$4,50.

4. If 8 dollars' worth of provision serve 7 men 5 days; how many days will 16 dollars' worth of vision last 4 men?

pro

Analysis. If any quantity of provision will serve 7 men 5 days, it will serve one man 7 times as long, or 35 days; if 8 dollars' worth serve one man 35 days, one dollar's worth will serve him but as long, or 35 of a day, 16 dollars' worth will serve him 16 times as long, or 16x35, equal to 70 days; and the same provision can serve 4 men but as long or 2, equal to 17 days.

These examples of analysis, which are spread out to their full length, demonstrate the entire practicability of solving, upon general principles, every question, which can occur under the rule of single or compound proportion. Small numbers were selected, only, because the analyses would be better understood; the reasoning would be the same, hower large the numbers.

Secondly. It will not be expedient to omit the form of the rule of three, unless the substitute offered is more expeditious, more philosophical, and better adapted to the future progress of the learner, in the higher branches of mathematicks. 1. While the numbers involved in questions of the rule of three are small, the calculation will always be carried on in the mind, without any reference to the form the rule prescribes. If the numbers are large, the question must be examined in the same manner, and when it is sufficiently understood, to know what operations are necessary to discover the relation of the numbers, the learner may as well proceed, forthwith, to the solution, as to make a parade of proportion; for every step in the solution is as essential after the statement as before. Placing the numbers in a line with a certain number of points among them, is altogether arbitrary. It would be just as well to place the numbers in the corners of the slate or paper, and then multiply the numbers in diagonal corners, and divide by the odd number, and put the quotient, or answer, in the other corner. Indeed, if the form