are known in circular measure, the other can be found at once by substracting the known one from the exterior angle.

8. All the exterior angles of any straight-lined figure, having all its angles salient or pointing outwards, are together equal to four right angles, and all the interior angles are equal to twice as many right angles wanting four, as the figure has sides. Hence the four angles of every four-sided figure, whatever are the lengths of its sides, will always exactly coincide with or cover the space round a point, if all their vertices are brought into contact at that point.

All the principles stated in this section are complete and simple; that is, they depend upon only one condition; therefore the truth of the converse or opposite of each of them follows as a matter of course. This is a very important general maxim, though it is one which, probably on account of its great simplicity, is rarely stated; but when it is admitted as general, it saves a great deal of unnecessary labour, and gets rid of a good deal of that perplexity which beginners feel in studying the elements of geometry. Before we apply it, however, we must be sure that we are in possession of all the conditions, each as single and simple; because if any one of them is compound, and we do not know its composition, the general maxim will not apply. A very simple case will show this:-that the sum of 2 and 1 is 3, is an absolute truth; but the converse-namely, 3 is the sum of 2 and 1, is not generally or absolutely true; for 3 is the sum of 1, 1, and 1, and also of an indefinite variety of other numbers.

That lines which are parallel, or have no inclination to or from each other, never can meet or cross one another in any point, is also a simple and general truth; and though the converse is not so palpable to our common understanding, yet it is equally true that every two lines which are not parallel must

meet and cross each other if produced far enough,-it being understood that the lines which are not parallel are in the same plane, or that neither of them is affected by a line crossing it at right angles, and thus as it were tying it down to a different plane from that of the other line. It is this possibility of the two lines which are not parallel to each other being situated in parallel planes, and thus in this particular case not meeting, which renders the converse "lines which are not parallel must meet each other" not absolutely true. When, however, the single condition of being in the same plane is added, the truth becomes as absolute as in the other case; and this is the only consideration of the position of lines which occurs in plane geometry. Parallel lines may always be considered as in the same plane, because it is easy to imagine a plane to be made to pass through any two parallels whatever.

SECTION XIII.

DOCTRINE OF PROPORTION.

THE doctrine of proportion, taken in its most extensive sense, is one of the most important, not only in the mathematical sciences, but in every science and every department of human knowledge. It is the general doctrine of the relations or ratios of all quantities and all subjects, whether existences, qualities, actions, or thoughts, whereof the comparison can be a source of knowledge. In the general sense of the term, it is closely allied to, and indeed nearly identical with, that principle which decides for us in all our judgments, and regulates

GENERAL IDEA OF PROPORTION.

265

us in all our actions,-namely, that "in circumstances which are exactly alike, like results may be predicted" with all the certainty which we can possess in any prediction.

When, however, we take only the mathematical view of this doctrine of proportion, it of course extends no further than mathematical subjects extend; but still when we thoroughly understand it in this simple and elementary point of view, we are masters of it as a general instrument of knowledge, inasmuch as the method of its application is the same in the most complicated cases as in those which are of the most simple and elementary nature.

The doctrine of proportion may be considered as in principle wholly a matter of arithmetic, though there are many cases in which even a relation that we understand perfectly in its nature, cannot be at all expressed in numbers; and there are others which cannot be expressed accurately in numbers, though we may approximate their truth to any degree that may be required in any one practical case. Thus, for instance, there is a relation between the state of the season -the Spring for instance, and the degree of development in the vegetable tribes; but there are no means by which we can state in two numbers the cause as operating in the season, and the effect as produced in the growing world. In like manner, the number 10, and the number 3, are both expressible, and in fact expressed in terms of the number 1; but if we were to attempt to get rid of this medium of expression, and try to express 3 in terms 10, there are no means by which this simple expression can be accomplished with perfect accuracy. The decimal 3 is the first step of the expression, but the relation cannot be accurately expressed unless this 3 is repeated without end.

These are simple instances, but they show us upon what

principle we must deal with more complicated ones, so as to get a relation between each and some third subject, when there is no relation expressible between the two subjects which we wish more immediately to compare. Thus, for instance, Farmer Gubbins has a load of excellent meadow hay over and above what is required for the supply of the horses employed on his farm; and Dame Gubbins and her daughters are in want of a certain web of printed cotton which lies in the shop of Mr. Tape the draper, in order that they may appear at church in seemly guise. Mr. Tape does not want the hay, and therefore there can be no possibility of instituting a ratio, so as to obtain an exchange, until recourse is had to a third element; and Gubbins first turns his hay into cash at the market-price, and then measures that cash against the web of printed cotton, according to the price set upon the latter, In this manner, though the commodity which we call cash, directly serves the necessities of nobody, yet it brings all the people into an exchanging condition with each other, so that each sells what he has to dispose of, and procures what he wants, and the whole are far better served than if time were wasted in vain attempts at finding relations among quantities which have no quality in common. This is the grand practical illustration; and it is this which renders the mathematical or general doctrines of proportion so very valuable.

The word proportion literally means "for or as a part,” that is, that inquiry shall be made as to what part, expressible by a number or by a fraction, the one of two quantities is of the other; and ratio means "the reason," that is, the inference or conclusion drawn from two quantities, which we can make available in acquiring a knowledge of other quantities.

There are two distinctions of relation, according to the object of the inquiry which we make respecting the two quanti

ties compared. The one alludes only to the difference, without any regard to the total magnitude or value of the quantities; and this is called arithmetical ratio or relation: the other, to the whole value of the one as compared with the whole value of the other; and this is called geometrical ratio or relation. These names are not correct, because both the one and the other may apply to quantities which are strictly arithmetical, strictly geometrical, or neither the one nor the other; and therefore the less that such names are used the better.

In those cases we do not, generally speaking, include quantities which are equal to each other, because from the comparison of such quantities no useful inferences can be drawn: consequently, the useful cases are those in which there is a less quantity and a greater one, and the difference between the arithmetical proportion and the geometrical one may be said to consist chiefly in this, that, in the arithmetical proportion, the ratio is a difference which, added to the less quantity, makes the greater; whereas the geometrical proportion has the ratio a multiplier, which, applied to the smaller quantity, in the common way of multiplication, produces the greater. Thus, if we take any quantity as a beginning, and add to it any equal quantity of the same kind by successive additions, the results which we obtain will be a series of quantities, of which any two following each other in order, how far soever the series may be extended, will have the same arithmetical proportion to each other. So also, if we take any quantity and multiply any number of times by the same multiplier, the series of products will be quantities of which any two immediately following each other will have the same geometrical proportion. The common difference which is added in the case of the arithmetical proportion is called the arithmetical ratio; and the common multiplier which is used in the case of the geometrical

it