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space. A point is a thing which we can see and know, not an abstraction which we build up in our thoughts.

When we talk of drawing lines or points on a sheet of paper, we use the language of the draughtsman and not of the geometer. Here is a picture of a cube represented by lines, in the draughtsman's sense. Each of these so-called 'lines' is a black streak of printer's ink, of varying breadth, taking up a certain

FIG. 3.

amount of room on the paper. By drawing such ‘lines' sufficiently close together, we might entirely cover up as large a patch of paper as we liked. Each of these streaks has a line on each side of it, separating the black surface from the white surface; these are true geometrical lines, taking up no surface-room whatever. Millions of millions of them might be marked out between the two boundaries of one of our streaks, and between every two of these there would be room for millions more.

Still, it is very convenient, in drawing geometrical figures, to represent lines by black streaks. To avoid all possible misunderstanding in this matter, we shall make a convention once for all about the sense in which a black streak is to represent a line. When the streak is vertical, or comes straight down the page, like this, the line represented by it is its right-hand boundary. In all other cases the line shall be the upper boundary of the streak.

So also in the case of a point. When we try to represent a point by a dot on a sheet of paper, we

make a black patch of irregular shape. The boundary of this black patch is a line. When one point of this boundary is higher than all the other points, that highest point shall be the one represented by the dot. When however several points of the boundary are at the same height, but none higher than these, so that the boundary has a flat piece at the top of it, then the right-hand extremity of this flat piece shall be the point represented by the dot.

This determination of the meaning of our figures is of no practical use. We lay it down only that the reader may not fall into the error of taking patches and streaks for geometrical points and lines.

§ 2. Lengths can be Moved without Change.

Let us now consider what is meant by the first of our observations about space, viz., that a thing can be moved about from one place to another without altering its size or shape.

First as to the matter of size. We measure the size of a thing by measuring the distances of various points on it. For example, we should measure the size of a table by measuring the distance from end to end, or the distance across it, or the distance from the top to the bottom. The measurement of distance is only possible when we have something, say a yard measure or a piece of tape, which we can carry about and which does not alter its length while it is carried about. The measurement is then effected by holding this thing in the place of the distance to be measured, and observing what part of it coincides with this distance.

Two lengths or distances are said to be equal when the same part of the measure will fit both of them.

Thus we should say that two tables are equally broad, if we marked the breadth of one of them on a piece of tape, and then carried the tape over to the other table and found that its breadth came up to just the same mark. Now the piece of tape, although convenient, is not absolutely necessary to the finding out of this fact. We might have turned one table up and put it on top of the other, and so found out that the two breadths were equal. Or we may say generally that two lengths or distances of any kind are equal, when, one of them being brought up close to the other, they can be made to fit without alteration. But the tape is a thing far more easily carried about than the table, and so in practice we should test the equality of the two breadths by measuring both against the same piece of tape. We find that each of them is equal to the same length of tape; and we assume that two lengths which are equal to the same length are equal to each other. This is equivalent to saying that if our piece of tape be carried round any closed curve and brought back to its original position, it will not have altered in length.

How so? Let us assume that, when not used, our piece of tape is kept stretched out on a board, with one end against a fixed mark on the board. Then we know what is meant by two lengths being equal which are both measured along the tape from that end. Now take three tables, A, B, C, and suppose we have measured and found that the breadth of A is equal to that of B, and the breadth of B is equal to that of C, then we say that the breadth of A is equal to that of C. This means that we have marked off the breadth of A on the tape, and then carried this length of tape to B, and found it fit. Then we have carried the same length from B to C, and found it fit. In saying that the

breadth of C is equal to that of A, we assert that on taking the tape from C to A, whether we go near B or not, it will be found to fit the breadth of A. That is, if we take our tape from A to B, then from B to C, and then back to A, it will still fit A if it did so at first.

These considerations lead us to a very singular conclusion. The reader will probably have observed that we have defined length or distance by means of a measure which can be carried about without changing its length. But how then is this property of the measure to be tested? We may carry about a yard measure in the form of a stick, to test our tape with; but all we can prove in that way is that the two things. are always of the same length when they are in the same place; not that this length is unaltered.

The fact is that everything would go on quite as well if we supposed that things did change in length by mere travelling from place to place, provided that (1) different things changed equally, and (2) anything which was carried about and brought back to its original position filled the same space. All that is wanted is that two things which fit in one place should also fit in another place, although brought there by different paths; unless, of course, there are other reasons to the contrary. A piece of tape and a stick which fit one another in London will also fit one another in New York, although the stick may go there across the Atlantic, and the tape via India and the Pacific. Of course the stick may expand from damp and the tape may shrink from dryness; such non-geometrical circumstances would have to be allowed for. But so far as the geometrical conditions alone are concerned-the

These remarks refer to the geometrical, and not necessarily to all the physical properties of bodies.-K. P.

mere carrying about and change of place-two things which fit in one place will fit in another.

Upon this fact are founded, as we have seen, the notion of length as measured, and the axiom that lengths which are equal to the same length are equal to one another.

Is it possible, however, that lengths do really change by mere moving about, without our knowing it?

Whoever likes to meditate seriously upon this question will find that it is wholly devoid of meaning. But the time employed in arriving at that conclusion will not have been altogether thrown away.

§ 3. The Characteristics of Shape.

We have now seen what is meant by saying that a thing can be moved about without altering its size; namely, that any length which fits a certain measure in one position will also fit that measure when both have been moved by any paths to some other position. Let us now inquire what we mean by saying that a thing can be moved about without altering its shape.

First let us observe that the shape of a thing depends only on its bounding surface, and not at all upon the inside of it. So that we may always speak of the shape of the surface, and we shall mean the same thing as if we spoke of the shape of the thing.

FIG. 4.

Let us observe then some characteristics of the surface of things. Here are a cube, a cylinder, and a sphere.

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