figures would be drawn on a convex surface. Our three-dimensional geometry depends, therefore, on our modes of activity and the concepts with which it operates; points, straight lines, etc. are conceptual limits to those modes of activity. We can imagine a straight line only as a direction along which we can move without deviating to the right or the left, or up or down. But even if we draw such a line on paper with a fine pencil the trace would still have some width, and we can imagine ourselves small enough to be able to deviate to the right or the left within the width of the line drawn on the paper. We might make a very small mark on the paper, but no matter how small this mark is it would still have some magnitude; otherwise we should be unable to see it. If the straight line had no width and the point no magnitude they would have no perceptual existence. Our perceptual triangles are not figures, the angles of which are necessarily equal to two right angles. If we drive three walking sticks into a field and then measure the angles between them by means of a sextant we shall find that the sum is _nearly_ 180°, but in general not that amount. If we stick a darning needle into the heads of each of the walking sticks and then remeasure the angles by means of a theodolite we shall obtain values which are nearer to that of two right angles, but we should not, except by “accident,” obtain exactly this value. We do not, therefore, get the “theoretical” result, and we say this is because of the errors of our methods of observation; but why do we suppose that there is such a theoretical result from which our observations deviate, if our observations themselves do not in general give this ideal result? We might accumulate a great series of measurements of the angles of our triangle, and we should then find that these results would tend to group themselves symmetrically round a certain value which would be 180°. Some of the results would be considerably less than the ideal, and some of them would be considerably more; but these relatively great deviations would be small in number and most of the results would be a very little less than 180° or a very little more, and there would be as many which would be a little less as those that were a little more. We should have formed a “frequency distribution”[2] with its “mode” at 180°. [2] See appendix, p. 350.