In the first place, some curve joining the two points is involved in the above notion of a combined motion of the two points, or of two other points forming a figure congruent with the first two. For without some such curve, the two point-pairs cannot be known as congruent, nor can we have any test by which to discover when a point-pair is moving as a single figure[173]. Distance must be measured, therefore, by some line which joins the two points. But need this be a line which the two points completely determine? =167.= We are accustomed to the definition of the straight line as the _shortest_ distance between two points, which implies that distance might equally well be measured by curved lines. This implication I believe to be false, for the following reasons. When we speak of the length of a curve, we can give a meaning to our words only by supposing the curve divided into infinitesimal rectilinear arcs, whose sum gives the length of an equivalent straight line; thus unless we presuppose the straight line, we have no means of comparing the lengths of different curves, and can therefore never discover the applicability of our definition. It might be thought, perhaps, that some other line, say a circle, might be used as the basis of measurement. But in order to estimate in this way the length of any curve other than a circle, we should have to divide the curve into infinitesimal circular arcs. Now two successive points do not determine a circle, so that an arc of two points would have an indeterminate length. It is true that, if we exclude infinitesimal radii for the measuring circles, the lengths of the infinitesimal arcs would be determinate, even if the circles varied, but that is only because all the small circular arcs through two consecutive points coincide with the straight line through those two points. Thus, even with the help of the arbitrary restriction to a finite radius, all that happens is that we are brought back to the straight line. If, to mend matters, we take three consecutive points of our curve, and reckon distance by the arc of the circle of curvature, the notion of distance loses its fundamental property of being a relation between _two_ points. For two consecutive points of the arc could not then be said to have any corresponding distance apart--three points would be necessary before the notion of distance became applicable. Thus the circle is not a possible basis for measurement, and similar objections apply, of course, with increased force, to any other curve. All this argument is designed to show, in detail, the logical impossibility of measuring distance by any curve not completely defined by the two points whose distance apart is required. If in the above we had taken distance as measured by circles of _given radius_, we should have introduced into its definition a relation to other points besides the two whose distance was to be measured, which we saw to be a logical fallacy. Moreover,