cp, or in the continuation of that line, how distant soever they might be from the point a. Hence it is evident that, supposing the line CD fixed to the plane of the paper on which it is drawn, we could not imagine the circle round a, to preserve that line, and be at the same time turned either to the right hand or to the left. But it is equally evident that we could turn it upon the line CD, either upwards in the direction of the top of the page, or downwards in the direction of the bottom; and that we could turn it round again and again in either of those directions as often as we chose, without in the least disturbing the position of the line CD.

It is further evident that the line CD bisects or divides into two parts exactly equal, the whole of that indefinite space through which it is supposed to be indefinitely drawn ; and that if we take any circle, as, for instance, the circle BEGF in the above figure, of which the centre a is any point in the line CD, and consider this circle as the representative of all space round that point, it follows that the line CD, that is, the part of it BG which passes through the centre and meets the circumference both ways, in в toward the one hand, and in G toward the other, divides the circle into two parts which are exactly equal, so that whatever can be proved as being true of the one of them must be equally true of the other, considered as a magnitude. They are, no doubt, of that description which we have named symmetrical magnitudes, that is, magnitudes of the same measure, but differing from each other in position; but then, from the very facts of the centre of the circle being in the straight line separating them and the circumference being everywhere at the same distance from this centre, we have every reason to conclude that they are perfectly equal in every respect, and no reason whatever to entertain even a suspicion that they are not equal.

Simple as it seems when analysed, this is a most important relation between the division of a circle by a straight line passing through its centre, and the division of all space in the ́plane of that circle; for, it follows almost immediately from this, that in what proportion or ratio soever two lines which meet at any point divide the circumference of a circle, they divide all space round that point, and in the plane of that circle, in exactly the same proportion or ratio. If they contain half the circle, in which case they are in the same straight line passing through the centre, they divide space into two equal parts. If they contain a fourth of the circle between their extremities which meet the circumference, they will also contain a fourth part of space round the point which is the centre of this circle; and as there is no reason from which we can even suspect that the ratio or proportion can be different in any other case, we may receive it as a general, and as nearly as possible a self-evident truth, that, as the entire circumference of any circle described round a point, is the measure or representative of all space round that point in the plane of the circle, so any portion whatever of the circumference of a circle, and any portion whatever of the space round the centre of a circle considered as a point, and contained between the two lines which divide off, or mark off that portion of the circumference, are mutually the measures of each other, and that, therefore, either of them may be used as the representative or expression for the other.

From this it follows, that we have only to apply our arithmetic to the circumference of a circle, in order to be in possession of a standard, or scale, for the measurement of any portion of the space round a point; and as any circle may be taken for this purpose, the scale may be expressed in terms of a circumference, and not in terms of any straight line as a standard,

as we require to do in the measuring of lines. There is much advantage even in this, for, as no part of a circle is a straight line, it is easy to see (though, in the mean time, we are not called upon to prove it) that no definite portion whatever of the circumference of any circle can be exactly equal in arithmetical proportion—that is, in a proportion fully and perfectly expressible by any numbers however large, to any straight line whatever.

When, therefore, we speak of circular measures, we speak of them generally, not in terms of a foot, a yard, a mile, or any other measure, but in terms of the whole circumference of which they are parts, and without any regard whatever as to whether this circumference, if we were to try to express it as nearly as possible in terms of straight lines, were equal to the thousandth part of an inch, to ten thousand millions of miles, or to the longest line which imagination could fancy to exist in space.

There is nothing either puzzling or new in this expression of portions of the space round a point by numbers which do not represent the lengths of any straight lines, for we meet with the very same thing in the most common use of arithmetic. If we use the name of any one thing which has existence or meaning, along with the name of a number, we tie the number down, as it were, to that particular kind of thing, so that we cannot, without contradiction and absurdity, regard it as being, in that case, the representative of anything else. Thus, when we say or write 5 MEN, 5 HOURS, or 5 anything else that we can name, we fix the number 5 to the men, the hours, or whatever else is named; and the number, in no one of these cases, expresses any part of the quantity which it expresses in the other. We number men in terms of a man; we number hours in terms of an hour; we number every thing in terms of one of that thing;

and therefore we are faithful to our arithmetic when we number, or count, or express (for these words have nearly the same meaning in arithmetic) circular measures in terms of ONE

CIRCUMFERENCE.

For this purpose, it is of little consequence what scale we take for the division of the circumference; but it is convenient to have larger and smaller denominations, just as we have in the case of money, of weight, of measure, and of all other things which admit of indefinite division into parts. The entire circumference is considered as divided into 360 equal parts, which are called degrees, and the short mark which is used in writing degrees is a small on the right hand of the figures, which may of course be any number not greater than 360°; but more than 360° would have no meaning in a single expression, because it would apply to more than one circumference, which could not, of course, be described round one centre.

Each degree is supposed to be divided into 60 equal parts, which are called minutes, and are marked in writing by a small dash over the right; thus, 24′ is read "twenty-four minutes."

For more minute division, the minute is again subdivided into 60 equal parts, which are called seconds, and are marked in writing by two dashes over the right; thus, 25′′ is read 66 twenty-five seconds."

From these definitions it follows that, as the whole circumference of a circle, or the whole measure of space round a point, is 360°, so half the circumference of a circle, or half the space round a point,—which last means the space on one side of a straight line passing through that point,-is equal to 180°; that one-fourth, or quadrant (which just means a fourth), of the circumference of a circle, or one-half of the space round a point, upon either side of any straight line passing through that point,

îs equal to 90°; and that, generally, any fraction whatever of the circumference of a circle, or of a space round a point, may be arithmetically expressed by the same fraction of 360°: a sixth will be 60°, a twelfth 30°, and so on for every other fraction; for if the fraction or part which the portion of the circumference, or of the space round the point, is of that circumference, or of that space, and the last is a constant quantity, not in any way affected by the lengths of lines, and the first is equally constant as to the number of degrees, is expressible arithmetically in terms of that circumference, or that space, considered as 1 whole, it is only the multiplying the value of this fraction by 360, to express it in degrees; by 60 again to express it in minutes; and a second time by 60, to express it in seconds.

It may not be amiss to bear in mind the extent of the circumference, or of the space round a point, in each of the three measures, the minutes being found by multiplying 360 by 60, and the seconds by multiplying this product by 60; and from these simple multiplications we have the following results :

One circumference, or all the space round a point is = 360°, or 21600', or 1296000".

In some cases it is convenient to use a greater degree of accuracy than seconds, and the modern way of doing this is by expressing whatever is less than seconds in decimals of a second.

The minutes and seconds which are here alluded to as expressing portions of the circumference of any circle, or of the measure or space round any point, must not be confounded with the minutes and seconds of time which we use as subdivisions of the hours of a day. There is the same number of minutes in the degree as in the hour, and the same number of seconds in the minute in both cases; but the meaning is not exactly