Reflections on Mathematical Reasoning. Ir the learner has studied he preceding pages attentively, he has had some practice in mathematical reasoning. It may now be pleasant, as well as useful, to give some attention to the principles of it. By attending to the objects around us, we observe two properties by which they are capable of being increased or diminished, viz. in number and extent. Whatever is susceptible of increase and diminution is the object of mathematics. Arithmetic is the science of numbers. All individual or single things are naturally subjects of number. Extent of all kinds is also made a subject of number, though at first view it would seem to have no connexion with it. But to apply number to extent, it is necessary to have recourse to artificial units. If we wish to compare two distances, we cannot form any correct idea of their relative extent, until we fix upon some length with which we are familiar as a measure. This measure we call one or a unit. We then compare the lengths, by finding how many times this measure is contained in them. By this means length becomes an object of number. We use different units for different purposes. For some we use the inch, for others the foot, the yard, the rod, the mile, &c. In the same manner we have artificial units for surfaces, for solids, for liquids, for weights, for time, &c. And in all there are different units for different purposes. When a measure is assumed as a unit, all smaller measures are fractions of it. If the foot is taken for the unit, inches are fractions. If the rod is the unit, yards, feet, and inches are fractions, and the smaller, being fractions of the larger, are fractions of fractions. It may be remarked, that all parts are properly units of a lower order. As we say sin gle things are units, so when they are cut into parts, these parts are single things, and consequently units, and they are numbered as such. When a thing is divided into eight equal parts, for example, the parts are numbered, one, two, three, &c. As we put together several units and make a collec tion which is called a unit of a higher order, so any single thing may be considered as a collection of parts, and these parts will be units of a lower order. The unit may be considered as a collection of tenths, the tenths as a collection of hundredths, &c. The first knowledge we have of numbers and their uses is derived from external objects; and in all their practical uses they are applied to external objects. In this form they are called concrete numbers. Three horses, five feet, seven dollars, &c. are concrete numbers. When we become familiar with numbers, we are able to think of them and reason upon them without reference to any particular object, as three, five, seven, four times three are twelve, &c. These are called abstract numbers. Though all arithmetic operations are actually performed on abstract numbers, yet it is generally much easier to reason upon concrete numbers, because a reference to sensible objects shows at once the purpose to be obtained, and at the same time, suggests the means to arrive at it, and shows also how the result is to be interpreted. Success in reasoning depends very much upon the perfections of the language which is applied to the subject, and also upon the choice of the words which are to be used. The choice of words again depends chiefly on the knowledge of their true import. There is no subject on which the language is so perfect as that of mathematics. Yet even in this there is great danger of being led into errors and difficulties, for want of a perfect knowledge of the import of its terms. There is not much danger in reasoning on concrete num bers; but in abstract numbers persons pretty well skilled in mathematics, are sometimes led into a perfect paradox, and cannot discover the cause of it, when perhaps a single word would remove the whole difficulty. This usually happens in reasoning from general principles, or in deriving particular consequences from them. The reason is, the general principles are but partially understood. This is to be attributed chiefly to the manner in which mathematics are treated in most elementary books, where one general principle is built upon another, without bringing into view the particulars on which they are actually founded. There are several different forms in which subtraction may appear, as may be seen by referring to Art. VIII. In order to employ the word subtraction in general reasoning, either of the operations ought readily to bring this word to mind, and the word ought to suggest either of the operations. The word division would naturally suggest but one purpose, that is, to divide a number into parts; but it is applied to another purpose, which apparently has no immediate connexion with it, viz. to discover how many times one number is contained in another. In fractions the terms multiplication and division are applied to operations, which neither of the terms would naturally suggest. The process of multiplying a whole number by a fraction (Art. XVI.) is so different from what is called multiplication of whole numbers, that it requires a course of reasoning to show the connexion, and much practice, to render the term familiar to this operation. These remarks apply to many other instances, but they apply with much greater force to the division of whole numbers by fractions. Arts. XXIII. and XXIV. are in stances of this. It is difficult to conceive that either of these, and more especially the latter, is any thing like division; and it is still more difficult to conceive that the operations in these two articles come under the same name. When a person learns division of whole numbers by fractions from general principles, where neither of these operations is brought into view, it is easy to conceive how very imperfect his idea of it will be. The truth is, (and I have seen numerous instances of it,) that if he happens to meet with a practical case like those in the articles mentioned above, any other term in the world would be as likely to occur to him as division. In an abstract example the difficulty would be very much increased. The above observations suggest one practical result, which will apply to mathematics generally, and it will be found to apply with equal force to every other subject. In adopting any general term or expression, we should be careful to examine it in as many ways as possible. Secondly, we should be careful not to use it in any sense in which we have not examined it. Thirdly, if we find any difficulty in using it in a case where we are sure it ought to apply, it is an indication that we do not fully understand it in that sense, and that it requires further examination. I shall give a few instances of errors and difficulties into which persons, not sufficiently acquainted with the principles, sometimes fall. Suppose a person has obtained a knowledge of the rule of division by a course of abstract reasoning, and that the only definite idea that he attaches to it is, that it is the opposite of multiplication, or that it is used to divide a number into parts. Let him pursue his arithmetic in this way, and learn to divide a whole number by a fraction. He will be astonished to find a quotient larger than the dividend; and if the divisor be a decimal, his astonishment will be still greater, because the reason is not so obvious. Let him divide 40 by according to the rule, and he will find a quotient 90. Or let him divide 45 by .03 and he will find a quotient 1500. This seems a perfect paradox, and he will be quite unable to account for it. Now if he had the idea intimately joined with the term division, that the quotient shows how many times the divisor is contained in the divi dend; and also a proper idea of a fraction, that it is less than one, instead of saying, divide 40 by 4, or 45 by .03, he would say, how many times is contained in 40, or .03 in 45; and all the difficulty would vanish. Innumerable instances occur, which show the importance of a single idea attached to a general term, which the term itself would not readily bring to mind, but which a single word is often sufficient to recal. The most important accessory ideas to be attached to the term division are. that the quotient shows how many times the divisor is contained in the dividend; and that it is the reverse of multiplication. Those for subtraction are that it shows the difference of the two numbers; and that it is the reverse of addition. Sometimes, it is asked if dollars and pounds, or gallons be multiplied together, what will they produce? If dollars be divided by dollars, what will they produce? If dollars be divided by bushels, what will they produce? &c. It is observed, in square measure, that the length multi pued by the breadth gives the number of square feet in any rectangular surface. It is sometimes asked, if dollars be multiplied by dollars, what will be produced? If 5s. 3d. be multiplied by 3s. 8d., what will be the result? It is observed in fractions, that tenths divided by tenths, hundredths by hundredths, &c. produce units; from this some have concluded, that a cent divided by a cent, or a |