12..

10

may be expressed by 12, the title of the silver, and that of the other by 0, the title of the brass; after this observation, there is no longer any difficulty; we are to employ the same rules we have laid down above, and write the numbers thus ;

10

༡ །

0.. 2

According to the solution for 10 marks of silver, we are to put in two of brass; and consequently, with the 25 marks of silver, there must be mixed 5 of brass, which will make in all 30 marks, which, being melted and amalgamated, will produce silver of 10 penny-weights fine.

Fourth Question. Of which I shall only expose the 'solution.

A merchant purchases a quantity of strong brandy at the rate of 25d. the bottle: the market price is only 22d. a bottle; what proportion of water must he put with it, to gain two pence per bottle at the market price. For 20 bottles of brandy, he must put in 5 of

water.

25.. 20

20

5

NOTE. The question supposes, that, after the mixture, the brandy will still retain the qualities requisite to render it saleable.

RATIOS AND PROPORTIONS.

There are but two ways of comparing one number with another; the one, in considering how much the first surpasses, or is surpassed, by the second; and the second, in considering, how often the first contains or is contained, in the second. The first manner

of comparing two numbers, is called Arithmetical Ratio, and the second Geometrical Ratio, or simply Ratio.

Thus; suppose I have two numbers, such as 12 and 4, and that, in considering how much 12 surpasses 4, I take the difference between these two numbers; the difference is called the Arithmetical Ratio between 12 and 4.

But, according to the other manner, if I examine how many times 12 contains 4, or if I take the quotient of 12 divided by 4, three, which is the quotient, is called the Ratio of 12 to 4.

Thus each ratio is composed of two numbers, which we call terms; the first term of a ratio is called the antecedent, and the second the consequent.

A proportion is the equality of two Ratios; whence it follows there are two kinds of proportions, the one Arithmetical and the other Geometrical.

Suppose we have these four numbers, 12, 4, 14, and 6 the Arithmetical ratio, or the difference between 12 and 4, being equal to that between 14 and 6, these four numbers form an Arithmetical Proportion, which we write thus; 12..4::14..6; and which is pronounced in this manner, 12 is to 4, as 14 is to 6. This kind of proportion being of no use in commerce, we shall content ourselves with having indicated it, and we shall pass immediately to Geometrical Proportions, which we simply call propor

tions.

Let there be the four numbers, 12, 4, 21, and 7; the ratio of 12 to 4 being equal to that of 21 to 7, the four numbers form a proportion, which is thus writ ten; 12: 4::21 : 7, and which is pronounced, 12 is to 4 as 21 is to 7.

A proportion, therefore, is composed of four terms, the first and third are called the Antecedents; the second and fourth, the consequents; the first and fourth are denominated the extremes of the proportion; the second and third, are called the means.

The essential property of every Geometrical proportion, and the only one of which we shall speak here, is, that the product of the extremes, is equal to the product of the means. Suppose we have the proportion 12: 4::21 : 7; I say, that 12x7=4x 21; for the ratio of 12 to 4, which is the same thing as the quotient of 12 divided by 4, is equal, (since there is a proportion) to the ratio of 21 to 7, or to the quotient of 21 divided by 7; thus, from the equality of the ratios, we have equal to ; let us reduce these fractions to the same denominator, whereby we shall not change their value, and we shall still have 12×7 21 X 4 --equal to

;

but since these two fractions,

always equal, have the same denominator, it necessarily follows that their numerators are equal; there fore 12×7, the product of the extremes, is equal to 21 ×4, the product of the means. As in this reasoning the numbers of the proportion, are not particularly considered, we perceive very well, that this reasoning is general, and that demonstration, which we have just given, is applicable to every Geometrical proportion.

From this property results a very important consequence, it is, that, knowing three terms of a proportion, we can always find the fourth. The unknown term may be an extreme or a mean.

If it be an extreme, it is equal to the product of the two means, divided by the other extreme; for, if I have the proportion, 3; 9;;5; x (I call x the un

known term ;) since there is proportion, I shall have the product of the extremes equal to that of the means, which gives 3 xx=9×5=45, but since x taken three times, or three times x, is equal to 45, it follows,

that x alone is equal to the third of 45, or x=therefore, &c.

9×5

3

If it be a mean, it is equal to the product of the two extremes, divided by the other mean; for, suppose the proportion, 9:3::x:5; making the product of the means equal to that of the extremes, we shall

9x5 have 3 xx9 × 5, and consequently x,

3

It is necessary, in this place, to make an observation, from which we may often derive the greatest advantages to simplify the calculation of proportions; it is, that we can always multiply or divide two terms of a proportion by a same number, without destroying the proportion, provided we operate at the same time upon one extreme and one mean; it is easily conceived, that the product of the extremes, always remaining equal to that of the means, there is still proportion; but to demonstrate this truth in an evident manner, let us descend to some detail.

I say, in the first place, that we can divide the two first terms of a proportion, or the two second ones, by a same number, without destroying the proportion; suppose we have the proportion, 150: 50::24: 8; dividing the two first terms by 10, and the two last by 8, we have to prove that the quotients will also be in proportion, that is, that we shall have 15: 5:: 3: 1; since the four given terms are in proportion, we shall have the ratio of 150 to 50, equal that of 24 to 8, that is 18, but as we can always divide

the two terms of a fraction by a same number, without changing their value, if we divide the two terms of the first by 10, and the two terms of the second by 8, we shall then have, and these two ratios being equal, will then give the proportion 15: 5::

3:1.

I say secondly, that we can divide the two antecedents of a proportion, or the two consequents by a same number, without destroying the proportion. Suppose we have the proportion 24: 96::9: 36, dividing the two antecedents by 3, and the two consequents by 4, we have to prove that the quotients will also be in proportion, that is to say, that we shall have 8 24;:3:9. The above proportion gives

, but as it is plain, that two equal quantities may be divided by a same quantity, without changing the equality; therefore, if we divide these two equal fractions by 3, we shall have =; but as it is not less true that we can multiply each of these last fractions by 4 without destroying the equality, and that to multiply a fraction by 4, or to take the fourth of its denominator, is the same thing, we shall then have =; and these two equal ratios will give the proportion demanded 8 : 24::3 :9

THE RULE OF THREE.

The Rule of Three has for its object, to find out the 4th term of a proportion, three of which are given by the state of the question.

It may be either Simple or Compound; it is simple when the question contains but three known quanti ties; and it is compound when it contains more.

Arithmeticians divide it also into Direct and Inverse;