PART III.

OF RATIOS AND PROPORTIONS.

CPAPTER I.

Elementary Considerations of Ratio.

$77. In the very outset we have shown, that quantity was all that is capable of increase or decrease, without regard to the nature or kind of things the number of which was increased or decreased. From the simple step of considering two things together, or adding them, and then successively more, or diminishing a certain number of things, both by the use of a determined system of numeration; we have arrived step by step at the principles of the combination of quantity, and conversely again, to the decomposition of a combination into its parts.

This process has led us to the four rules of arithmetic, that have been explained successively, two of which have been shown to be the opposite of the two others; each leading alternately to the decomposition of the composition of the other, as addition and subtraction, multiplication and division; the latter two of which have been shown to be the result of the continued repetition of the two first. By these means all the operations upon quantity in usual life, which depend merely on combinations, have become calculable, as shown by the application made of this theory in the second part.

This retrospective view of the part of arithmetic

hitherto treated of, appears proper to be taken here, in order to awaken appropriate reflections in reference to the whole of what has been done, and the means it has furnished for further progress. The scholar, atten- tive to what he has done hitherto, cannot but have acquired the faculty of reasoning upon quantity. The reflections which we shall have to make in future will be as simple as before, but the application of them will require that he have made himself acquainted with the tools, or means, which he has to use in the following parts of arithmetic, andacquired some dexterity in their use; he will do well therefore to cast back upon the whole a cursory view, in order the better to comprehend the general ideas that have directed it.

$78. The consideration which will be the foundation of the part of arithmetic to be now treated of, is the relation which the quantities may have to each other, whether they be combined in any way, or

not.

The relation of quantities to each other, in whatever way it may be, is called their ratio. As we have seen that the increase or decrease of quantities depends on their combination, so their relation to each other, that is, their ratio, must also depend on their possible combination, as it is determined by it. The ratio is, therefore, also considered in relation to these combinations; and, as we have had the two principal combinations, of addition or subtraction, and of multiplication or division, so we have also two kinds of ratio, corresponding to them; namely, by addition or subtraction, and this is called arithmetical ratio; and by multiplication or division, which is called the geometrical ratio. We evidently here again find the second a repetition of the first, as multiplication and division are the repetition of addition and subtraction; but we may omit going so far back into elementary considerations, and proceed forward

with the general idea, to render it fruitful for prac tical use.*

These two kinds of ratio take their mark of notation from the marks applied to the combinations or rules of arithmetic, on which they depend; thus: The arithmetical ratio of 7 to 3, is expressed by

The geometrical ratio of 7 to 3, is expressed by

7:3; or

7

3

They might be equally well expressed by the signs of addition and multiplication, if we were in the habit of generalizing the considerations on quantity to that extent; and we shall see hereafter, that their theory leads to it; that is to say, that when the ratio of two quantities by subtraction, or division, to which the above signs are appropriated, are given, their ratio by addition, or multiplication, is also given; or the one is a consequence of the other. In the habitual mode of writing, therefore, an arithmetical ratio expresses a difference between two quantities, and a geometrical ratio expresses the quotient arising from the division of the two quantities; this latter is called the index, when referred to the geometrical ratio.

$79. The simplest reflection leads to the idea: that two or more such ratios may be exactly equal to each other, as well as two quantities in general; such an equality of ratio is called a proportion.

This principle between two ratios is expressed very naturally by the sign of equality between them, as for example:

*The propriety of these denominations is not worth discussing; they are mere names, to which the idea above explained is to be attached, which forms what is called their definition.

An arithmetical proportion will be expressed thus:

7-312-8

This says the difference between 7 and 3 is equal to the difference between 12 and 8.

A geometrical proportion will be expressed thus:

And this says the quotient of 12 divided by 3 is equal to the quotient of 16 by 4, as it is evidently in both ratios 4; and this is therefore also the index of the two equal ratios.

The first term of a ratio is called the antecedent, the second the consequent; the first and last terms of a proportion are called the extreme terms, the second and third the mean terms.

A nearer investigation of the properties of these ratios will justify the assertion made above, for we shall find; that the arithmetical proportion, expressed as a difference, gives also an equality of sums ; and the equality of the quotients an equality of products; and that in this property lies their extensive utility in all calculations.

80. It may be easily seen that, while in the preceding part of arithmetic, grounded upon combination only, we were limited to things of the same kind. We obtain by this extension, or the consideration of the relation of two things to each other in respect to quantity, the means of forming conclusions by calculation from things of different nature mutually acting upon each other; by the condition, or simple consideration, of the equality of the ratio of two things of one kind, to two things of another kind, which we observe in nature in all things; for we may see a herd of cattle, as much, or as many times, larger than another herd of cattle, as the money owned by one man is as much, or as many times, larger than the money owned by another man; a mountain as much or as many times higher than a house, as the

amount of one bill of exchange is of as much, or as many times, a greater amount than another.

These considerations are daily made in common fife, by every one, and they need only be transferred into the language of arithmetic, to direct us in the principles of calculation derived from them.

The first of these ratios and proportions, namely the arithmetical, are naturally more limited in their application to practical purposes, as they are the result of a more limited scale of combination. The second, namely, the geometrical, are much more extensive, depending on a higher scale of combination; the geometrical proportion is the principle of what is called in arithmetic the rule of three.

81. I have thought proper to enter into these elementary deductions, though their aim is thereby kept back for a short time, because it is all-important in any study to conceive the fundamental ideas in their generalization, by which the explanation is so much facilitated, as ultimately to lead to a shortening of the task, both of teaching and of studying. To render these fundamental ideas useful, we shall in the first place show the consequences which lie in them, from the principles of combination upon which they are grounded, and the condition of equality, which forms the particular nature of a proportion. We may already, from the simple enunciation in signs, as it appears above, conclude: that their application to practice consists in the evident property, that any three of the quantities so conditioned being given, the fourth is necessarily determined; the manner in which this is done, thus rendered of practical use, will appear from the investigation of the properties resulting from the principles of proportion.