Now if it appears that the magnitude in question contains an exact number of any equal parts of an unit, (i.e. of such parts as are contained an exact number of times in the unit,) it might be denoted by this number, the part of the unit being called by some name, as another unit. But this method would involve the necessity of having a different name for every different exact part of an unit, which would manifestly be very inconvenient, and ineffective, as it would be impossible to remember the relative value of an infinite number of units. Instead therefore of giving to them names, as to other units, let them be called by those which may be said to be their natural names, viz. the parts of the unit, which they may be. Thus instead of giving to the thirteenth part of an inch any arbitrary name, let it be called "the thirteenth of an inch; and so for all other similar quantities. There will thus be no difficulty in conceiving an idea of the magnitude, expressed as it will be, in terms of known units, and of parts of units.

Now, as in ordinary numbers, so here, it will manifestly conduce much to brevity, if we denote these parts of units (which may be called subordinate units) by some fixed method of notation. This of course is purely arbitrary, being determined only by convenience.

The method adopted is that of writing underneath a line the number, denoting the particular part of the unit, and over the line the number, which shews how many of these parts there are in the magnitude, the name of the unit being written after, before, or over, the whole. Thus in. denotes a magnitude, of which 13 compose an inch; in. denotes two such parts of an inch, and similarly other quantities are denoted.

Such magnitudes being formed by the repetition of equal parts of units, are called" Fractions."

From the above it appears that the meaning of such an expression as inch, is two fifths of one inch. It may also mean one fifth of two inches, for these may be shewn to be equivalent, as follows:-Take a line A B, equal in length to two inches; bisect it in C; and divide

A C, C B, each into five equal parts, then it is seen that in A B there are ten of the same parts, of which there are five in A C; and therefore that in the fifth part of A B there are two of the same parts:—that is, one fifth of A B, or of two inches, is equal to two fifths of A C, or of one inch. Similar reasoning may be used respecting other quantities. Hence it appears, that the fractional notation may be properly used to express the quotient of a concrete by an abstract number. For a part of any quantity is obtained by dividing by the number of parts, therefore inch, which means the fifth part of two inches, is obtained by dividing two inches by five; or in. ex

presses the result of the division of 2 inches by 5. The same may be shewn of any other quantities.

From what has preceded it is evident, that the numerical part of a fraction, as (read two-fifths, or 2 upon 5), must be considered as the exponent of two operations, viz.-multiplication and division, the upper number being the multiplier, the lower the divisor. For inch is obtained by multiplying 1 inch by 2, and dividing the product by 5, or vice versa. This is the only meaning which can be attached to the fractional notation, abstracted from any concrete unit. We cannot speak of as a number in the sense, in which we speak of 2 and 5 as numbers. For abstract numbers being words or signs, used to convey an idea of how many articles there are in a collection, without any reference to the nature of the articles, and the word one, or sign 1, being used to denote the number, when an article stands by itself, and two, three, four, &c. denoting successive degrees of number differing by one, we cannot conceive of any abstract numbers other than these.

For if there be a collection at all, we cannot conceive of their being fewer than one article in it; and if there be more than one, there must be either two, or three, or more, since the least difference that we can make in a number of articles by addition or subtraction is one. Therefore all abstract numbers being denoted by the figures 1, 2, 3, &c. and such a symbol as being unknown in this notation, and all similar expressions cannot be abstract numbers. Nor can they be concrete numbers, there being no concrete unit mentioned. But it has been shewn, that they may be considered as combinations of two abstract numbers, the upper being a multiplier, the under a divisor.

Hence also will denote the part or parts which 2 units are of 5 units; for in order to form the 2 units, we must evidently divide the 5 units into 5 equal parts, and repeat one of them twice; i.e. we have to divide by 5 and multiply by 2, which operations are represented by the symbol .

Again if we extend the meaning of the expression "a time" to signify not only such a repetition of the unit as takes place in Multiplication, but also such a repetition of the subdivision of the unit, as occurs in forming a part of the unit, then the symbol & may denote the "number of times" that 5 units are contained in 2 units.

Using the expression "a time" in the above sense, the fractional notation may be used to express the quotient of one abstract number by another. To recapitulate the results arrived at in this Prop. :

1. The fractional notation is necessary in order to represent numerically quantities, which are not expressible, as an exact number of any standard units, but which are equivalent to some number of equal parts of units. 2. The number under the line denotes the number of parts into which

the unit is divided; that above the line 'shews how many of these parts there are in the given quantity.

3. The expression & inch means 2-fifths of 1 inch, or 1-fifth of 2 inches. 4. It also denotes the quotient of 2 inches divided by 5.

5. The expression is the symbol of the multiplication by 2, and division by 5.

6. It also denotes the part (or fraction) which 2 is of 5.

7. Or the "number of times" which 5 units are contained in 2 units. 8. Or the quotient of 2 divided by 5.

Cor. Hence is apparent the method of converting an improper fraction into a mixed number. For since such an expression as 23 represents the quotient of 23 divided by 8, let this division be performed as far as possible, from which we obtain the integral quotient 2, with a remainder 7, which has still to be divided by 8; but this division not being capable of being actually performed, must be denoted by the fraction. Hence the whole quotient is 2+, written 23: or 23 = 27.

Prop. 27.-The numerator and denominator of a fraction may be both multiplied or divided by any the same number, without altering its value.

Take a line A B, divide it into 3 equal parts, and each third part into

4 equal parts; then the whole A B will be divided into 12 equal parts. And it appears that A C, which is equal to of A B, contains 8 of the twelfth parts of A B; or of A B = 12 of A B: i. e. the multiplication of numerator and denominator by 4 has not altered the value of the fraction. The same may be shewn of any fraction, and any multiplier. Also since 1 of A B of A B, the division of numerator and denominator by 4 has not altered the fraction. The same may be shewn in every case. The same Prop. otherwise:

8

12

- of an unit 8 units 12 (8 units 4) 3 (Prop. 15)

Cor. 1. Hence the correctness of the Rule for reducing fractions to their lowest terms is evident. For numerator and denominator may be divided by any the same number without altering the value of the fraction, and therefore by their G.C.M., or by their common factors in succession.

Cor. 2. Hence also the Rule for conversion of an integer into an improper fraction. For every integer may be expressed in the fractional notation with denominator 1; thus 3, because this expresses a quantity composed by 3 of such parts, of which 1 forms the unit, i. e. 3 units.

Prop. 28. To prove the Rule for reducing fractions to their L.C.D.

Since the terms of a fraction may be both multiplied by the same number without altering its value, therefore if multipliers can be found, which shall convert all the denominators of several fractions into the same number, and we multiply the terms of each fraction by the proper multiplier, we shall have converted all the fractions into others of equivalent value, but having the same denominator. Now it is evident that the multipliers in question cannot be found, unless the common denominator be a number capable of exact division by each of the denominators, i.e. unless the C.D. be a multiple of all the denominators. Hence it follows that the least possible common denominator to which several fractions can be reduced is the L.C.M. of all the denominators. And the multipliers, which shall convert each denominator into the L.C.D. are the quotients of the L.C.D. by each.

Prop. 29-To explain the Rule for Addition of frac

tions.

It is an axiom that it is impossible to add together concrete numbers of different denominations, i.e. to express their sum by a concrete number of either denomination equal to the sum of the abstract numbers. Thus it would be absurd to say that £2 and 2s. added together are equal to £4 or 4s. Now the denominator of a fraction denotes the kind of sub-unit, (or part of unit,) of which the numerator denotes the number: therefore fractions with different denominators are of different denominations; therefore it is impossible to add them, as they are. They must therefore be expressed as fractions with the same denominator, and then their sum may be obtained by adding the numerators, and retaining the same common denominator which only shews the denomination: for just as we say £4+ £2 = £6 ; soor 4-twelfths+2-twelfths 6-twelfths.

If any of the fractions be mixed numbers, we may separate them into their parts of integers and fractions, add each separately, and then add the

sums.

Cor. Hence the correctness of the Rule for conversion of a mixed number into an improper fraction is evident. For the mixed number is the sum of an integer, and a fraction, indicated but not performed; but if the integer be expressed in the same denomination as the fraction, the addition may be actually performed. Therefore the integer is expressed as a fraction with the denominator of the given fraction by multiplying it by the denominator, and the addition is performed in the ordinary way.

Prop. 30.-To explain the Rule for Subtraction of fractions. As it is impossible to add numbers of different denominations, so is it impossible to subtract them; therefore fractions must be expressed with the same denominator, before they can be subtracted. This being done, the difference may be obtained by subtracting the numerator of the subtrahend from that of diminuend. This is evident, if we bear in mind that the denominator of a fraction shews the denomination of the numerator. Thus -, or 5-sevenths - 3-sevenths 2-sevenths.

If the diminuend be a mixed number, or both diminuend and subtrahend, then we may subtract in any parts we please (Prop. 5). Let then the fractions be reduced to their L.C.D., and let the fraction of the subtrahend be subtracted first, which may be done by subtracting it from the fraction of the diminuend; but if the fractional subtrahend be less than the fractional diminuend, this is not possible; therefore increase the numerator of the diminuend by the C.D. and the integer of the subtrahend by 1, (thus increasing both numbers by 1, which does not alter the difference,) and subtract. Having subtracted the fraction, the integer may now be subtracted from that of the diminuend.

Prop. 31.-To prove the Rule for Multiplication of a fraction by an integer.

Multiplication being only a short method of addition, may be performed on the same principle. Therefore as in Addition of Fractions with a C.D. the numerators are added for the numerator of the sum, and the denominator left the same, so in forming the product of a fraction by an integer, or in forming the sum of a fraction repeated as often as there are units in an integer, we must find the sum of the numerator repeated the same number of times, i.e. find the product of the numerator and integer, for the numerator of the product, and leave the denominator untouched.