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We have thus found a meaning for a" when n is a fraction from the fundamental theorem of powers.

We can with equal ease obtain from the same theorem an intelligible meaning for a" when ʼn is a negative quantity.

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a -".

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We have an × a" = an + Now let us assume We find a xa pn in order to interpret = an − n = ao = 1 (by p. 31).

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Or dividing by a”,

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that is to say, a" is the quantity which, multiplied by a", gives a product equal to unity. The former quantity is termed the inverse of the latter, or we may say that a " is the inverse of a". For example, what is the inverse of 4? Obviously 4 must be multiplied by in order that the product may be unity. Hence 4 is equal to 4. Or, again, since 4 = 22, we may say that 2 2 is the inverse of 4, or 22.

The whole subject of powers-integer, fractional, and negative-is termed the Theory of Indices, and is of no small importance in the mathematical investigation of symbolic quantity. Its discussion would, however, lead us too far beyond our present limits. It has been slightly considered here in order that the reader may grasp that portion of the following chapter in which fractional powers are made use of.

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CHAPTER IV.

POSITION.

§1. All Position is Relative.

THE reader can hardly fail to remember instances when he has been accosted by a stranger with some such question as: Can you tell me where the 'George' Inn lies?'' How shall I get to the cathedral ? '-'Where is the London Road?' The answer to the question, however it may be expressed, can be summed up in the one word-There. The answer points out the position. of the building or street which is sought. Practically the there is conveyed in some such phrase as the following: You must keep straight on and take the first turning to the right, then the second to the left, and you will find the George' two hundred yards down the street.'

Let us examine somewhat closely such a question and answer. 'Where is the 'George'?' We may expand this into: 'How shall I get from here' (the point at which the question is asked) to the George'?' This is obviously the real meaning of the query. If the stranger were told that the 'George' lies three hundred paces from the Town Hall down the High Street, the information would be valueless to the questioner unless he were acquainted with the position of the Town Hall or at least of the High Street. Equally idle

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would be the reply: The George' lies just past the forty-second milestone on the London Road,' supposing him ignorant of the whereabouts of the London Road.

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Yet both these statements are in a certain sense answers to the question: Where is the 'George'?' They would be the true method of pointing out the there, if the question had been asked in sight of the Town Hall or upon the London Road. We see, then, that the query, Where ? admits of an infinite number of answers according to the infinite number of positions-or possible heres—of the questioner. The where always supposes a definite here, from which the desired position is to be determined. The reader will at once recognise that to ask, "Where is the 'George'?' without meaning, Where is it with regard to some other place?' is a question which no more admits of an answer than this one: 'How shall I get from the 'George' to anywhere?' meaning to nowhere in particular.

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This leads us to our first general statement with regard to position. We can only describe the where of a place or object by describing how we can get at it from some other known place or object. We determine its where relative to a here. This is shortly expressed by saying that: All position is relative.

Just as the George' has only position relative to the other buildings in the town, or the town itself relative to other towns, so a body in space has only position relative to other bodies in space. To speak of the position of the earth in space is meaningless unless we are thinking at the same time of the Sun or of Jupiter, or of a star-that is, of some one or other of the celestial bodies. This result is sometimes

described as the 'sameness of space.' By this we only mean that in space itself there is nothing perceptible to the senses which can determine position. Space is, as it were, a blank map into which we put our objects; it is the coexistence of objects in this map which enables us at any instant to distinguish one object from another. This process of distinguishing, which supposes at least two objects to be distinguished, is really determining a this and a that, a here and a there; it involves the conception of relativity of position.

§ 2. Position may be Determined by Directed Steps.

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Where is the You must keep straight on

Let us turn from the question: 'George'?' to the answer: and take the first turning to the right, then the second to the left, and you will find the 'George' 200 yards down the street.'

The instruction to keep straight on' means to keep in the street wherein the question has been asked, and in a direction (straight on ') suggested by the previous motion of the questioner, or by a wave of the hand from the questioned. Assuming for our present purpose that the streets are not curved, this amounts to: Keep a certain direction. How far? This is answered by the second instruction: Take the first turning on the right. More accurately we might say, if the first turning to the right were 150 yards distant: Keep this direction for 150 yards. Let this be represented in our figure by the step A B, where A is the position at which the question is asked. At в the questioner is to turn to the right and, according to the third instruction, he is to pass the first turning to the left at c and take the second at D.

'We shall return to this point later.

More accurately we might state the distance B D to be, say, 180 yards. Then we could combine our second and third instructions by saying: From в go 180 yards in a certain direction, namely, B D. To determine exactly what this direction B D is with regard to the first direction A B, we might use the following method. If the strangedid not change his direction at B, but went straight on for 180 yards, he would come to a point D'. Hence if we measured the angle D'B D between the street in which the question was asked and the first turning to the right,

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we should know the direction of B D and the position of D exactly. It would be determined by rotating B D' about B through the measured angle D'B D. If we adopt the same convention for the measurement of positive angles as we adopted for positive areas on p. 133, the angle D'B D is the angle greater than two right angles through which в D' must be rotated counter-clockwise in order to take it to the position B D. Let us term this angle D'BD for shortness ẞ, then we may invent a new symbol {B} to denote the operation: Turn the direction you are going in through an angle ẞ counter-clockwise.

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