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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 numbers ; 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
upou 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 anotner purpose, which apparently has no immediate connexion with it, viz. to discover how many times one number is contained in another. In fractions the terins 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.
'í'he 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 prouciples, 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
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 dividend ; and also a proper idea of a fraction, that it is less than one, instead of saying, divide 40 by, 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 plied 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
mill by a mill, would produce a dollar, and though they are aware of the absurdity, cannot tell how to avoid the conclusion.
The above difficulties arise chiefly from not making a proper distinction between abstract and concrete numbers. Not one of these cases can ever occur in the manner here proposed. They are imperfect examples. When a perfect example is proposed, which involves one of the above cases, the difficulty is entirely removed.
It is not proper to speak of dollars being multiplied or divided by dollars or gallons.
At 5 dollars per barrel, what cost 3 barrels of flour ?
Instead of saying that 5 dollars is to be multiplied by 3 harrels, say 3 barrels will cost three times as much as 1 barrel, that is three times 5 dollars.
If 1 dollar will buy y lbs. of raisins, how many pounds may be bought for 4 dollars ?
Say 4 dollars will buy 4 times as many pounds as 1 dollar. In these two examples there is no doubt what the answer should be. In one it is dollars, and in the other it is pounds.
In a piece of cloth 5 feet long and 3 feet wide, how many
square feet ?
If it were 5 feet long and I foot wide, it would contain 5 square feet, but being 3 feet wide it will contain three times as many, or three times 5 feet.
In a certain town a tax was laid of 1 dollar upon every $150; how much did a man possess whose tax was 3 dollars ?
It is evident that he possessed three times $150.
At 1 cent each, how many apples may be bought for I cent?
Here the divisor is 1 cent and the dividend is 1 cent, and the result is an apple instead of a dollar.
How many gallons of wine at 2 dollars per gal., may be bought for 6 dollars ?
As many times as 2 dollars are contained in 6 dollars, so many gallons may be bought.
The truth is, the numbers are always used as abstract numbers, but a reference to particular objects is kept in view, and the nature of the question will always show to what the result must be applied.
It may however be established as a general principle, that
the multiplier and multiplicand are never applied to the same object, and in precisely the same way; and the product will be applied to the object which is mentioned in one denomination, as being the value of a unit in the other.
In division there are two numbers given to find a third, two of which will always be of the same denomination, and the other different, or differently applied.
If the divisor and dividend are of the same denomination and applied in the same way, the question is, to find how many times the one is contained in the other, and the quotient will be applied differently.
If the divisor and the dividend are of different denominations, or differently applied to the same denomination, the question is to divide the dividend into parts, and the quan tient will be applied in the same manner as the dividend.
When any difficulty occurs in solving a question, it is best to supply very small numbers, and solve it first with them, and then with the numbers given. If the question is in an abstract form, endeavour to form a practical one, which shall require the same operation, and the difficulty is generally very much diminished.
In all cases reason from many to one, or from a part to one; and then from one to many or to a part. If several parts be given, always reason from them to one part, and then to many parts, or to the whole.