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last. But the G.C.M. of the last but one and the last remainders is the last remainder, which is also the last divisor. Therefore the process being as above, the last divisor will be the G.C.M.

The same Prop. exhibited algebraically:-Let a, b, be the two numbers, of which suppose b the less; divide a by b, let the quotient be p; if there be no remainder, evidently b is the G.C.M.; but if there be a remainder, let it be c, so that a = pbtc;c=a-pl

Now every common measure of a and b measures a and pb, and ..a-pb, or c. Again, every common measure of b and c measures pb and and .. pb + c, or a. Hence every common measure of a and b is a common measure of b and c; and vice versa. Therefore the G.C.M. of a and b is a common measure of b and c, and cannot be greater than the G C.M. of b and c; nor can it be less than this, for if it were, then the G.C. M. of b and c being a common measure of a and b, a and b would have a common measure greater than the G.C.M., which is absurd. Hence the G.C.M. of a and b is equal to that of b and c. Let it then be tried to find the G.C.M. of b and c in the same manner as that of a and b. Divide b by c, let the quotient be q, the remainder d; then it may be shewn as before. that the G.C.M. of b and c is equal to that of c and d. Let then the division be continued till there is no remainder, by dividing each divisor by the remainder from each division, and it will appear by reasoning from step to step that the G.C.M. of a and 6 is equal to that of the last and last but one remainders, or to the last remainder or divisor.

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Prop. 23.— To shew that the G. C. M. of several numbers

may be obtained by finding first the G. C. M. of two, then of this and a third, next of this last and a fourth, and so on; the last so obtained being the G. C. M. of the whole.

Every common measure of two numbers is a measure of their G.C.M. therefore every common measure of three is a common measure of the G.C.M. of the first two, and of the third. Again, every measure of the G.C.M. of two numbers is a commou measure of the two, therefore every common measure of the G.C.M. of two, and of a third, is a common measure of the three. Hence it appears that the G.C.M. of three numbers is the G.C.M. of that of the first two, and of the third. Similarly it may be shewn that the G.C.M. of several numbers is the G.C.M. of that of all but one, and of that one. Whence the truth of the Prop.

The same Prop. otherwise :—The G.C.M. of two numbers contains all the prime factors common to the two, and no other. Hence the G.C.M. of this and a third number contains all the prime factors common to the three, and no other; and therefore is the G.C.M. of the three. In the same way it appears that the G.C.M. of this, and a fourth, is the G.C.M. of the four; and so on, how many soever numbers there may be.

Prop. 24.- To prove the Rule for finding the L.C.M. of

several numbers.

Every number is a measure of its multiple, therefore all its prime factors are factors of the multiple. Hence all the prime factors of several numbers are factors of their common multiple ; and conversely a common multiple of several numbers contains all their prime factors, as its factors. This therefore is the only condition of a number being a common multiple of several others, viz. that it contain all their prime factors, as its factors. Now evidently, of all the common multiples which may be formed subject to this condition, that will be the least, in which the condition is only just satisfied, in which therefore no prime factor is introduced any oftener than is absolutely necessary to satisfy the condition, and in which there is no other factor than the factors of the numbers. Therefore if all the prime factors were different, the L.C.M. would be formed by multiplying all together; but if any of the same factors appear once in two or more of them, it will be sufficient to introduce it once only as a factor, into the L. C. M.; if it appear twice, three times, &c. in one, and a less number of times in others, it must be introduced, as a factor, into the L.C.M. the number of times it appears in the first, aud no oftener; for, if it were introduced less often, all the factors of the first would not appear in the L.C.M., and if it were introduced oftener, there would be more factors than are necessary. All the different factors must be introduced as before. Hence the Rule for finding the L.C.M. of several numbers evidently is :-“Resolve them into their prime factors; write down all the different factors which appear, repeating each the greatest number of times, which it appears in the same number, and multiply all together."

Cor. 1. Hence the L.C.M. of two numbers may be found by dividing their product by their G.C.M., or by multiplying one of them by the quotient of the other divided by their G.C.M. For, the G.C.M. containing all the common prime factors, these are repeated twice in the product of the numbers, whereas in the L.C.M. they are only required to appear once. Therefore the product of the numbers is equal to the L.C.M. multiplied by the G.C.M.; or the L.C.M. is equal to the product divided by the G.C.M. And this quotient may be obtained by dividing one number and multiplying by the other.

Cor. 2. Every common multiple of several numbers is a multiple of the L.C.M. and conversely. For every common multiple must contain all the prime factors of the numbers, and the L.C. M. contains no other factors but these; therefore any other common inultiple can only be formed by introducing other factors, by which the multiple becomes a multiple of the L.C.M. The converse is evident.

Cor. 3. In forming the L.C.M. any numbers which measure any others may be omitted. For evidently if we introduce the factors of the multiples, we introduce also the factors of their measures, so that we shall not find any more factors to introduce, if we consider the measures, than if we omit them.

Prop. 25.-To shew that the L.C.M. of three or more

numbers may be obtained, by finding first the L.C.M. of two, then of this and a third ; next of this last and a fourth, and so on, the L.C.M, last found being that re

quired. The L.C.M. of the first two contains all the prime factors of the first two, and no others; and the L.C.M. of three must contain all the prime factors of the three, and no others, and may therefore be found by multiplying the L.C.M. previously found, by the prime factors of the third, which are not already factors of it, and which may be obtained by dividing the third by the G.C.M. of it and the former L.C.M. That is the L.C.M. of three numbers is the L.C.M. of that of the first two and of the third. Similar reasoning will apply to four or more numbers.

Prop. 26.— To explain the necessity, method, and meaning

of the system of Fractional Numeration and Notation. In order to render any magnitude a subject for Arithmetical calculation, it is necessary to denote it by a number or numbers. The general method of doing this is, by comparing it with some fixed magnitude of the same kind, called an unit, so as to discover how many of these units are contained in it: the magnitude is then properly denoted by the number of units, because, knowing the exact magnitude of the unit, we are able to form an idea of another magnitude of the same kind, if we know the number of units, which it contains,

If every magnitude contained some one of the units in ordinary use an exact number of times, all might be denoted by ordinary numbers. But it is evident that such is not the case; for there may evidently be magnitudes less than the least of the units employed in estimating magnitudes of the same kind. And such magnitudes cannot be avoided in Arithmetical questions; therefore it is necessary to denote them in some manner by numbers.

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 vnits, 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 11 in. denotes a magnitude, of which 13 compose an inch; 3 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

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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. expresses 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 mul. tiplying 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 off 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 ca not 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

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