SECTION II. CHAPTER X. THE STRAIGHT LINE. Now it If we now refer to the beginning of our studies we shall find that the inquiry before us is very similar to the first which we had to make. The question then was,- "What is a line?" is," What is a straight line?" And the best way to obtain an answer will evidently be to place ourselves in the position which we then occupied. This position was as follows:-We had our two objects A and B, and the line between them AB, which was the measure of the distance from one object to the other. Afterwards we had a third object C, and the line AC, and the two lines together forming the angle BAC (1.) A But from what has been said above, each of these two lines AB, AC, may be con 1. -B 2 sidered, and indeed are, made up of a series of points as close together as possible (2.) A B In what order, then, do these points follow each other? This is the problem we have to solve. Now, conceive that AB remains fixed in a horizontal position, while AC revolves without alter ing its form or length (3, 4.) c For example, conceive one leg of a pair of compasses to remain fixed in a horizontal position while the other revolves. 3 By 4 B B this means we make AC assume a number of different directions, and in so doing find that the order of each three consecutive points (whatever that order may be) remains unchanged in all cases except that of the point A, and the point on either side of it (4.) Let us call the point next to A in the line AB, P1, and the corresponding point in the line AC, P,; then the order of all the points remains unchanged, except that of P,, A, P ̧. Now we may remark that these points enclose the angle at A: consequently we see that an angle depends upon the order of three consecutive points. 2 To proceed. As AC revolved, the path of P, gradually increased, and the angle P, AP, increased along with it (5, 6, 7,) until suddenly in the end AC arrived at AB and the angle P1, AP, apparently disappeared altogether (8.) 2 But next let us imagine AC as it were, to retrace its steps: (i.e. to revolve in the opposite direction.) A similar result will take place to that in the previous case, except that as AC passes Now then, if we consider AC in any one fixed direction, there will be evidently two angles where there is apparently but one (12, 13.) For as AC might have arrived in its present position by revolving from AB either upwards at first or downwards, the angle may be conceived to be the result of either motion, and therefore to be measured in either direction. And this is true not merely in any one direction of AC¡ but of every direction in which AC can lie. In the position of AC then, there are two angles to be taken into account, both measured from A B, one upwards and the other downwards. Let us call the former the Superior or Positive, the latter the Inferior or Negative Angle. These two angles are in this relation to each other. Each starts from zero, and attains ultimately the same size, but the negative angle is zero when the positive is at its "maximum" or greatest size, and vice versa. Hence there must be some position in which the positive angle is equal to the negative (14.) And in this case A C is in a straight C 14. B line with A B. And this is the required position. We obtain therefore the following definition of a straight line. A Straight Line is the locus of a point which moves in such a manner that the positive angle between each three consecutive points is equal to the negative. And this is what is meant by the definition : "A straight line is that which lies evenly between its extreme points." СНАРТTER X I. THE STRAIGHT LINE CONTINUED. 、 SINCE the law of motion in a straight line fixes the third and all successive points, while it leaves the first two arbitrary, four results of the utmost importance immediately follow: : 1. Through any one point an infinite number of straight lines can be drawn, since the second point may be fixed anywhere. Thus if in shooting, the right elevation being obtained, one "sight" only be brought to bear upon the object, the barrel may yet point to the right or left, and thus the shot may miss. 2. Through any two points one straight line, and one straight line only, can be drawn, since all the points are determined in relation to the first two by the law of motion. Hence if both sights are brought to bear, the barrel is in a direct line, and down comes the quarry. 3. Through more than two points no straight line can be drawn, unless the points are in such a position as to fulfil the above condition. 4. No straight line can have more than one point in common with another straight line, for if it have two it will have all in The same example will serve here as before; common. the meaning being that if two points of the barrel are in a straight line with the object, every point of the barrel is so also. Each of these facts is so simple and common as to require no further illustration here. When then two points are given the path of a point moving in a straight line is determined in direction. But this of course does not by any means fix any particular position in which the moving point is to be arrested: to determine which a further condition will be required. How is it then that if a straight line be drawn from one given point to another it can only be of a certain length? It is because the second point being fixed not only in direction but also in position is equivalent to two fixed points; or, to put it more simply, in that case the point which fixes the direction and that which fixes the length coincide. *Hence a law of motion which depends upon the length of a given straight line, is equivalent to one depending upon three given points. And as above, a line, the law of which depended upon two points, could be made to pass through any two points, so a line the law of which depends upon three points, can be made to pass through any three points. That is to say, through any three points whatsoever a line can be made to pass, such that its law of motion shall depend upon a certain straight line of fixed length. But this will be clearer when we have considered the circle. Ir in a surface any two points whatsoever (A and B) be taken, and a straight line be drawn from A to B, then A B will be either completely or partially in the surface according as the latter is |