breadth implies (that is necessitates) the existence of thickness also, but at the same time the thickness may be smaller than we can perceive or appreciate. 1. In this case, that is, when the dimension is too small to be perceptible, it is said to be in its "Limit." Hence our definition of a surface becomes, "That which has length and breadth appreciable and thickness in its limit," i.e. smaller than we can appreciate. Treating the surface in a similar manner, that is, considering its length to remain fixed while its breadth decreases, we find it approach more and more nearly to a line (2, 3,) until at last when the breadth becomes smaller A than we can perceive, the distance alone remains appreciable, and the surface in its limiting form becomes a line (4.) But just as the thickness would not entirely disappear unless both length and breadth were taken away also, so here both thickness and breadth exist, but both in their limit. As an example of this consider a single hair. What is there to prevent both its breadth and its thickness from being different in different men? Yet who can measure either? C Lastly, imagine that the same line, we will call it A B, has one end fixed towards which the other object (B) approaches (5, 6.) Then the distance itself diminishes also, and at last when B comes up to A, length itself is in its limit as well as breadth and thickness (7.) A 2. D B 3. D B 4. A B 5. Here, then, we have another idea,entirely new-a kind of magnitude namely, which has neither length, breadth nor thickness appreciable, yet every one of them existing in an inappreciable degree. Let us call it a "Point," then its definition is already given. A Point is that which has all three dimensions in an inappreciable degree. Or, A Point is the limiting form of geometical magnitude. Hence we have the following definitions of the four kinds of magnitude. All space and therefore all magnitude can vary in three directions only. A Solid has 3 dimensions appreciable, and 0 inappreciable. The solid, therefore, when shrunk up as much as possible in any one dimension is reduced to a surface. The surface in the same way becomes a line; the line dwindles into a point. Nor let it be said that these are opposed to Euclid's definitions, for they only differ from him in the particular in which the definitions of Euclid himself are somewhat hard to reconcile. "The extremities of a solid,” says he, "are surfaces." If so, by continual division the solid may be cut up into surfaces, and inversely the surfaces will make up the solid. If then not one of these surfaces have any thickness, how shall they altogether produce a solid? Let the thickness be inappreciable, and the difficulty is explained. To proceed. Since the line when reduced becomes a point, the point by its motion will produce or generate a line. Hence a line may be regarded as the path in which a point moves, or more strictly as the "Locus" of a point, locus meaning the path which is formed by the motion of anything. So again, a line by its motion will "generate" a surface-a moving surface generates a solid; and thus a surface is the locus of a line, and a solid the locus of a surface. Hence in addition to the above definitions, we have the following relations between the four varieties of magnitudes:— And under the latter form (especially the line as the locus of a point) it is often very convenient to consider them. There are some fountains at the Crystal Palace which afford a pretty illustration of the last observations. A single drop of water becomes a jet, the jet expands into a surface, the surface rises into a column, the column sinks back into the basin, and disappears by the little opening from which the first drop came forth. LET us now pause for a moment, and observe how closely the definitions we have obtained correspond with natural processes. The point of a pen or pencil tracing out a line as it moves over the paper, the fiery circle formed by rapidly whirling round a stick, one end of which is alight, the furrows of a plough, the orbits of the stars are all lines formed by moving points; the wake of a ship and the track of a roller are surfaces produced by the motion of lines; the mass of water in a basin is a solid generated by the rising surface of the water. So perfectly general indeed is this method of generation that it is as impossible to generate a line without moving a point, as to reach a place without moving our body. Having thus obtained our general ideas as to " Motion,” “ Loci," and "Limits," let us now confine ourselves to those particular cases of which Euclid treats. First of all then we may dismiss at once two out of the four great varieties of magnitude which we have before us,-viz. a solid and a surface; as of these Euclid says very little or nothing in the parts with which we are at present concerned. He does certainly mention a "plane surface" once or twice, but this merely so far as it regards a line, in the consideration of which therefore we may briefly notice it hereafter. With the two remaining magnitudes we have at present to deal, viz. a Line and a Point. And they are defined as follows: A Point has all three dimensions in their Limit. A Line has one dimension appreciable, and two in their Limit. And they stand in this relation to each other. A Point is the Generator of a Line-i.e. is that by the motion of which a line is formed. A Line is the Locus of a Point, i.e. is the path in which a point Hence a line may be considered as a series of consecutive moves. points. From this it is evident that before any geometrical magnitude can be produced, a point must be given. "The world stood upon the elephant, and the elephant stood upon the tortoise, but what did the tortoise stand upon?" says the old fable. "Give me a "Let A stand-point and I will move the world," said Archimedes. be a given point, and I will prove my proposition," says Euclid with equal necessity. Every line then that can be described is described by the motion of a point, and varies in its form according to the mode in which the point moves. Thus a billiard ball moves in a straight line, a stone in a sling in a circle before it is let go, in a parabola afterwards; the planets have elliptic orbits, the comets hyperbolic; the motion of the point of a corkscrew again is spiral; and each of these lines comes under our present definition. Of the greater number however even of the six we have mentioned, Euclid takes no account-indeed, his first six books are confined to two of them, the straight line and the circle, even these being by no means exhausted. But these two lines are, as may be conjectured, the foundation of all geometrical measurement; and the forty-seventh proposition of the first book of Euclid would almost seem to be that very stand-point of which we were just now speaking, the destruction of which would involve in its ruin every calculation hitherto made in geometry, trigonometry, and astronomy. Our first object now will be to discover the law of motion by which these two lines are formed, the names of which we have anticipated; the next will be to trace the various results which follow from their method of generation. |