It

of proportion is to be considered useful or essential, this arrangement is preferable on some accounts. does not lead the pupil to suppose the truth or answer is elicited, somehow, slyly, by virtue of those little dots, he puts among his numbers.

2. The method of solution upon general principles, is more philosophical; because in the operation, the mind is intent only on discovering the relation of the numbers; whereas, in the formality of a proportion at length, the attention is divided between circumstances and forms, which are of no importance to the solution, and those principles which are essential. That method cannot be called philosophical, which fixes the attention on a form, and induces neglect of the only part of the process, which is important. Besides, if the form must be presented, it is made more artificial and unphilosophical, in all the popular books, than is necessary. A common method is ; is; "State the question by making that number, which asks the question, the third term, or putting it in the third place; that which is of the same name or quality as the demand, the first term; and that which is of the same name or quality with the answer required, the second This rule gives explicit directions for a mechanical operation; for all the knowledge of the prin

term."

ciples of the rule, the pupil gets well have learned hocus pocus. and state it by the rule.

by it, he might as

Take an example,

"If 9lbs. of tobacco cost 6s. what will 25lbs.

Now "multiply the second and third terms together and divide by the first." Why? My readers can probably tell; but it is very certain, that the youth, who is just entering upon the subject, can assign no better reason for it, than because the rule says so. He has no more conception of what this step, in particular, has to do with obtaining the answer, than the natives had of Columbus' means of predicting an eclipse. And he ought to be as much astonished if he gets the true answer, as they were, when the event hap

I

pened according to his prediction. If this is philosophy, I do not understand what that term means. should call it catching the truth by legerdemain.

To assign as a reason for such statement, that the "first term has the same ratio to the second, as the third has to the fourth," is, if possible, more unphilosophical. It is not only ridiculous, but absurd.. A ratio, that is, any ratio, which relates to the rule of three, is the number of times one quantity is contained in another of the same kind. It is just as absurd to talk of the ratio of pounds weight, and shillings, as 'it would be to talk of buying a week of salt, instead of a bushel; or a yard of wine instead of a gallon. A ratio subsists between the figures, which express the number of units in one quantity, whatever be the unit of measure, and the figures, which express the number of units in another quantity, however different the unit of measure. That is to say, 5 is equal to 5, and is half 10. No one doubts this; but when the numbers are made concrete, by attaching to them particular denominations, it becomes absurd to say, 5 pecks are equal to 5 days; or that 5 pounds are half of 10 yards. This absurdity, which disgusts the learner, if he is sufficiently inquisitive to ask for reasons for what he is doing, is avoided, by a solution upon general principles.

The same question solved by analysis, would be reasoned upon thus. If 9lbs. cost 6s., 1lb. must cost

as much, or of a shilling; and if 1lb. cost of a shilling, 25lbs. will cost 25 times as much, or 25×6

9

equal to ' of a shilling. In this method, although essentially the same operations are performed upon the numbers, the pupil understands the reason of every step, and can tell, precisely, what approach he makes by it, to the true answer. Whereas, by the formality of a proportion, he does not know the object of any particular step. He only knows that by performing certain mechanical operations, he obtains an answer like the book. The proportion is a sort of crucible, into which he throws his numbers, and by a process altogether as unintelligible to him as shaking the crucible, he gets the desired result. He has no means of knowing whether the result is correct, but by comparing it with the book. But by analysis, he has intuitive knowledge at each step, and is as certain of his conclusion, as he is that two and two are four.

3. But one objection more can be anticipated to the system of arithmetick, which discards the formal rule of three. The doctrine of proportions has been considered very important, if not essential to the higher branches of mathematics. And all the books upon Geometry and Algebra, and all which treat of their application to the physical sciences, are filled with them. They are not only made the great instrument of reasoning, but they constitute of themselves, in all their modes and forms, a great part of all systems of arithmetick, geometry, and algebra. It is certain, a scholar would not be able to read the books on the higher branches of the pure and mixed

mathematics, without a knowledge of proportions, at least, sufficient to translate them into more intelligible language. But the French mathematicians, who have pursued the science more successfully than any others, for the last century, have long since pronounced the formal proportion unnecessary. Lacroix, who understands the subject of mathematicks, if he does not the best method of teaching it, after stating the doctrine of proportions in all their modes and forms; says, “This theory was invented for the purpose of discovering certain quantities by comparing them with others. Latin names were for a long time used to express the different changes or transformations, which a proportion admits of. We are beginning to relieve the memory of the mathematical student from so unnecessary a burden; and this parade of proportions might be entirely superseded by substituting the corresponding equations, which would give greater uniformity to our methods, and more precision to our ideas."*

Clearness and precision in our ideas are important on all subjects; on the subject of mathematics, they are essential. On moral questions, we balance probabilities, and found our belief on a preponderance of evidence; but in mathematics, we have demonstration or nothing. If one step in the process of demonstration comes short of intuitive knowledge, the demonstration is destroyed. Here, then, a want of clearness and precision, is a want of knowledge. And * Lacroix's Alg. Camb. Edit. p. 234.