749. We have before observed that the geometrical ratio of two numbers is found by answering the question how many times is one of those numbers greater than the other, the quotient being the ratio required. Three things here present themselves for consideration. Firstly, the first of the two given numbers, which is called the antecedent; secondly, the other number, which is called the consequent; thirdly, the ratio of the two numbers, or quotient arising from the division of the antecedent by the consequent. Thus, if the relation of the numbers 18 and 12 be sought, 18 being the antecedent, and 12 the consequent, the ratio will be 81 whence we see the antecedent contains the consequent once and a half. Geometrical relations are generally represented by a point placed above another between the antecedent and the consequent. Thus ab signifies the geometrical relation of these two numbers, or the ratio of b to a. The sign just mentioned has, in a previous page, been given as indicating division, and it is on that account here used, because, in order to know the ratio, we divide a by b. The relation is merely read a is to b. On this account relation is expressed by a fraction whose numerator is the antecedent and denominator the consequent. It is hardly necessary to say that this fraction should, for perspicuity sake, appear in its lowest terms. Thus, if both terms be divided by 6, becomes .

750. Hence relations differ as their ratios are different; and there are, of course, as many kinds of geometrical relations as we can imagine different ratios. The first kind is that wherein the ratio becomes unity, which, of course, happens when the two numbers are equal, as in 3 3, 4: 4; a: a, and because the ratio here is 1, it is called the relation of equality. The relation then follows in which the ratio is another whole number; in 4 : 2 the ratio is 2, and is called a double ratio; in 12: 4, the ratio being 3, it is called a triple ratio; in 246 the ratio is 4, and is called a quadruple ratio. It is necessary, also, to notice those relations whose ratios are expressed by fractions, as 12: 9, where the ratio is , or 1; 18: 27, where the ratio is 3, &c. Those relations, as 6: 12, 5: 15, &c., wherein the consequent contains exactly twice, thrice, &c. the antecedent, are sometimes called subduple, subtriple, &c. ratios. The term rational is applied to ratios that are expressible numbers, the antecedent and consequent being integers, as 117, 8: 15, &c.; and that of irrational, or surd, is applied to ratios neither expressible by integers nor by fractions, 58, 4: √3.

as

n

751. If a be the antecedent, b the consequent, and d the ratio, d=%. Were the consequent b given with the ratio, we should find a=bd, for bd divided by b gives d. Finally, when the antecedent a is given, and the ratio d the consequent, b=2; for, dividing the antecedent a by the consequent (or its equivalent), we obtain the quotient d, that is, the ratio. In whatever way we multiply or divide the consequent and antecedent by the same number, every relation a; b remains the same, because the ratio is the same. Let d be the ratio of ab, we have then d=%: now the relation na; nb is still =d, and likewise a b is still =d. So, also, when a ratio has been reduced to its lowest terms, the relation is easily perceived and enunciated; for, let the ratio be reduced to its lowest terms 2, we say, a b=p: q, a: b::p : q, which is read a is to b as p is to q. Thus the ratio of the relation 6:3 being, or 2, we say, 6:3=2; 1. So, 18; 123: 2, and 2418: =43, &c. But if the ratio be not susceptible of abridgment, the relation does not become more evident, and we do not simplify a relation by saying 9:7=9:7. We may, however, change the relation of two large numbers into one more simple and evident by reducing both to their lowest terms, for we may say, 14484: 7242=2:1, or 15819 10546 3 2, or 57600 25200=167. All relations, therefore, should be reduced to the lowest possible numbers, which is readily done by dividing the two terms of the relation by their greatest common divisor. Thus, to reduce the relation 57600: 25200 to that of 16 7, we have only to perform the single operation of dividing the numbers 576 and 252 by their greatest common divisor, 36. The method of finding a common divisor of two given numbers will be given in the following subsection.

GREATEST COMMON DIVISOR.

752. There are many numbers whose only common divisor is unity, and where the numerator and denominator belong to this class, the fraction cannot be reduced to a more convenient form. Such is the case with the numbers 48 and 35; hence, as the division of 48 35 can only be divided by 1, their relation cannot be more simply expressed. But if two numbers have a common divisor, the greatest they have is found by the following rule. Divide the greater by the lesser number, and divide the preceding divisor by the remainder; the remainder resulting from the last division again becomes the divisor for a third division wherein the preceding divisor is to be the dividend. This operation being repeated till we arrive at a division to which no remainder is left, the last divisor will be found to be the greatest common measure or divisor of the two given numbers.

Now, let

us apply this to the two numbers 504 and 312, whereof we require the greatest common divisor.

312)504(1
312

192)312(1

192

120)192(1
120

72)120(1

72

48)72(1

48

24)48(2

48

Here we perceive that the last divisor is 24, and dividing 504 and 312 by it, we find that

the relation 504 312 is reduced to the form 21: 13.

Let the relation 456: 721 be given to find the greatest common divisor.

456)721(1
456

265)456(1
265

191)265(1

191

74)191(2

148

43)74(1

43

31)43(1

31

12)31(2

24

7)12(1

7

5)7(1

5

2)5(2

4

1)2(2

In this case 1 is the greatest common divisor, and we cannot express the relation 721 : 456 by less numbers, nor reduce it to less terms, than those in which it appears.

753. To demonstrate this rule, let a be the greater and b the less of the given numbers, and let d be one of their common divisors; it is evident that a and b being divisible by d, we may also divide the quantities a − b, a — 2b, a—3b, and, in general, a-nb, by d. Equally true must be the converse, that is to say, if the numbers b and a-nb are divisible by d, the number a will be also divisible by d. Farther, if d be the greatest common divisor of two numbers b and a-nb, it will also be the greatest common divisor of the two numbers a and b for if a greater common divisor than d could be found for the numbers a and b, it would also be a common divisor of b and a-nb, and consequently d would not be the greatest common divisor of these two numbers. But we have supposed d the greatest divisor common to b and a – nb, wherefore it must also be the greatest common divisor of a and b. With these considerations before us, let us, according to the rule, divide the greater number a by the lesser b, and let us suppose the quotient =n; the remainder will be a-nb, which must be less than b. This remainder a-nb having the same greatest common divisor with b as the numbers a and b, it is only necessary to repeat the division, dividing the preceding divisor b by the remainder a-nb; and the new remainder which is obtained will still have with the preceding divisor the same greatest common divisor, and so on. Proceeding in this way till we arrive at a division without a remainder, that is, in which the remainder is nothing, let p be the last divisor contained exactly a certain number of times in its dividend, which will therefore be divisible by p, and will have the form mp; so that the numbers p and mp are both divisible by p, and as no number can be divided by a number greater than itself, it is clear that they have no greater common divisor. Therefore the last divisor is the greatest common divisor of the given numbers a and b, and the rule laid down is thus demonstrated.

GEOMETRICAL PROPORTION

a C

754. When their ratios are equal, geometrical relations are equal, such equality of relations being called a geometrical proportion: thus we write a b = c; d, or ab::c: d, thereby indicating that the relation ab is equal to the relation e; d, which is expressed in language a is to b as c to d. Such a proportion is 4; 1= 12; 3, for the relation of 4; 1 is f, and this also is the relation of 12: 3. Thus, a b c d being a geometrical proportion, the ratio is the same on both sides, and ; and, reciprocally, if the fractions and are equal, we have a b c d. Hence, a geometrical proportion consists of four terms, such that the first divided by the second gives the same quotient as the third divided by the fourth; and hence, also, is deduced an important property common to all geometrical proportion, namely, that the product of the first and last term is always equal to the product of the second and third, or, in more simple language, the product of the extremes is equal to the product of the means.

c

755. To demonstrate this last named property, let us take the geometrical proportion abcd, so that = Multiplying both these fractions by b, we obtain a= a; and again multiplying both sides by d, we have ad=be. Now, ad is the product of the extremes, be that of the means, and these two products are found to be equal. Reciprocally, when a, b, c, d are such numbers that the product of the extremes a and d are equal to the product of the means b and c, we may be certain that they form a geometrical proportion. For, since ad=bc, we have only to divide both sides by bd, which gives us or, and therefore a b=c; d.

ad bc

=

bd bd

a C

756. The transposition of the four terms of a geometrical proportion, as a ; c=b; d, does not destroy the proportion, for the rule being that the product of the extremes is equal to the product of the means, or ad be, we may also say, 1st, b; a=d: c; 2d, a; c=b:d; 3d, d; b=ca; 4th, d: c=b; a. Besides these, some others may be deduced from the same proportions abcd; thus we may say a+b;a=c+d; e; that is, the first term added to the second is to the first as the third added to the fourth is to the third. So, also, a-ba-c-d: c. For, taking the product of the extremes, we have ac- -be-ac-ad, which leads to the equality ad=bc.

757. All the proportions deduced from a b c d may be generally represented as follows:

ma + nb: pa + qb = mc + nd: pc + qd;

For the product of the extremes is mpac + npbc + mqad + nqbd, which, because adbc, becomes mpac+npbc + mqbc + nqbd. Farther, the product of the means is mpac + mqbc + npad + nqbd; or, as ad=bc, it is mpac+mqbc + npbc + nqbd; so that the two products are equal. It is therefore evident that from any geometrical proportion an infinite number of others may be deduced: take, for example, 9: 318; 6, and we may have

besides many others.

3:9 6:18;

=

9:18 36; 12: 9=24: 18; 3:3 6:6; 12 24 3:6; 12: 3 = 24: 6;

758. Since in every geometrical proportion the products of the extremes and of the means are equal, we may, when the three first terms are known, find the fourth from them. Thus, suppose the three first terms to be 9: 318: the quantity sought. Now the product of the means is 3 x 18, or 54; the fourth term must therefore be 1, which multiplied into the first will produce that number; if, then, the product 54 of the means be divided by the first term 9, we shall have 6 for the fourth term, and the whole proportion will stand 9 318 6. In general, therefore, if the three first terms are a: b = c : . . . . . we put d for the unknown fourth letter; and since ad bc, we divide both sides by a, and have d; so that the fourth term, or is found by multiplying the second and third terms and dividing the product by the first term. This is the foundation of the celebrated RULE OF THREE in arithmetic, wherein three numbers are given to find a fourth in geometrical proportion, so that the first may be to the second as the third is to the fourth. And here we must note some peculiar circumstances which follow.

bc

bc

=

:

759. If in two proportions the first and third terms are the same, as in a b=c; d, and a: fc: g, then the two second and the two fourth terms will also be in geometrical proportion, and b: d=f: g. For the first proportion being transformed into a : c=b; d, and the second into a c=f: g, the relations b d and ƒ: g must be equal, since each of them is equal to the relation a: c. In numbers, if 5; 253: 15, and 5: 40-3: 24, we must have 25 40=15; 24. But if the two proportions be such that the means of both are the same, then the first terms will be in an inverse proportion to the fourth terms. Thus, if a; b=c; d, and ƒ: b=c : g, then a:f=g: d. In numbers, for example, 24: 8=9: 3, and 68 912, we have 24; 6=12; 3. And the reason is evident, for the first proportion gives ad bc; the second fg=bc; therefore ad=fg, and a : f=g: d, or a g::f: d.

760. If two proportions are given, a new one may always be produced by separately multiplying the first term of the one by the first term of the other, the second by the second, and so on with respect to the other terms. Thus, a b c d and e f g h will furnish ae bf=cg: dh. For the first gives ad=bc, and the second eh=fg, we have also adeh befg. But adeh is the product of the extremes, and bcfg is the product of the means, in the new proportion. So that the two products are equal, and the proportion is true. Let them, for example, be 8 2 20 5 and 6: 9=14: 21; the combination will be 6 x 82 x9= 20 × 14; 5 x 21, or 48: 18=280: 105.

761. Lastly, if two products are equal, ad=bc, the equality may be converted into geometrical proportion, for we shall always have one of the factors of the first product in the same proportion to one of the factors of the second product, as the other factor of the second product is to the other factor of the first product; that is, in the present case, a: cbd, or a b c d. In numbers, 3 x 8 = 4 × 6; and this proportion may be formed 84 63, or 3: 46: 8.

762. We do not think it necessary to pursue the subject here by examples of the use of proportion, without which the occurrences of common life could scarcely be carried on. Its basis is here explained, and the application must be obvious to the readers of this work.

COMPOUND RELATIONS.

763. If we multiply the terms of two or more relations, antecedents by antecedents, and consequents by consequents, compound relations are obtained; that is, the relation between the two products is compounded of the relations given. Thus the relations ab, ed, ef, give the compound relation ace: bdf. Each of these three ratios is said to be one of the roots of the compound ratio.

764. As a relation continues the same if both its terms are divided by the same number, in order to abridge it, we may greatly facilitate the above composition by observing whether among the first terms some are not found having common divisors with some of the second terms: for if so, those terms are destroyed, and the quotient arising from the division by that common divisor substituted, of which the following is an example. Let the relations given be 12: 25, 28: 33, and 55 : 56.

Whence we see that 2: 5 is the compound relation required.

765. The same operation is performed if we are calculating by letters; and a remarkable case occurs, when each antecedent is equal to the consequent in the preceding relation : thus, if the given relations are

a: b,

b:c,

c: d,

d: e,

e: a,

The compound relation is 1:1.

766. We may perceive the utility of these principles by applying them, for instance, to the relation between two rectangular fields, which is compounded of the relations of the lengths and breadths. Let one of them, A, be 500 ft. long and 60 ft. wide, and the other, B, be 360 ft. long and 60 ft. broad; then the relation of the lengths is 500: 360; that of the breadths 60: 100. Thus we have

Whence the field A is to the field B as 5 to 6.

767. So, again, if we wish to compare two rooms with respect to their space or contents, we are to recollect that here the relation between them is compounded of three relations, namely, that of the lengths, that of the breadths, and that of the heights. Let the room A be 36 ft. long, 16 ft. broad, and 14 ft. high; and the room B be 42 ft. long, 24 ft. broad, and 10 ft. high; we have the relations as follow:

For the length

For the breadth

For the height

So that the capacity of the room A is to that of the room B as 4 to 5.
768. When the relations thus compounded are equal, multiplicate relations result;
namely, two equal relations give a duplicate ratio or ratio of the squares. Three equal rela-
tions produce the triplicate ratio, or ratio of the cubes, and so on. Thus the relations ab
and ab give the compound relation aa: bb; whence we say that the squares are in the
duplicate ratio of the roots; and the ratio a: b multiplied thrice, giving the ratio a3: b3,
shows that the cubes are in the triplicate ratio of the roots.

769. From a knowledge of Geometry, we learn that two circular spaces are in the duplicate relation of their diameters; which means, that they are to each other as the squares of their diameters. Suppose A to be such a space, having a diameter = 45 ft. ; B another circular space, whose diameter = 30; then the first space will be to the second as 45 x 45 to 30 x 30, or, compounding the two equal relations,

770. Again, it is known that the solid contents of spheres are in the ratio of the cubes of the diameters. Thus, the diameter of a globe, A, being 1 ft., and the diameter of another globe, B, being 2 ft.; the solid contents of A will be to those of B as 13: 23, or as 1 to 8. If, therefore, the spheres are composed of similar substances, the sphere B will weigh 8 times as much as the sphere A.

a C

771. The ratio of two fractions may always be expressed in integer numbers, since we have only to multiply the two fractions by bd to obtain the ratio ad be, which is equal to the other; and if ad and be have common divisors, the ratio may be reduced to less For instance, 15 × 36: 24 × 25=9: 10.

terms.

772. Suppose we sought the ratio of the fractions and, it is evident we should have ba, which is expressed by saying that two fractions which have nominator are in the reciprocal or inverse ratio of their denominators.

=

a b C

unity for their de-
So when any two

fractions have a common enumerator, forb: a. When, however, two fractions have their denominators equal, as , they are in the direct ratio of their numerators, that is, as a b. Thus, } }=} } = 6:3=21 and 19:10: 15 or 2: 3.

773. It is upon the principles here laid down that we are enabled to resolve questions in what is called in books of arithmetic, THE RULE OF FIVE, as, for example, in the following question: If 25 pence per day be given to a labourer, and it is required to know what must be given to 24 labourers who have worked 50 days, we state it thus: