sible to get a graphic picture of this crowding, difficult to get a mental picture, and still more difficult to get a verbal description of it; due partly to mental parvitude, partly to verbal poverty. If the path of the hound and hare be circular, it will emphasize the unity of the two problems.

At the point there must occur apparently a limited point continuum. By the conditions of the problem, the point X cannot be reached so long as any interval is left between C and B. But there comes a time when there is no interval; C overtakes B and passes beyond. But no interval implies a

continuum.

Another illustration of the crowding down of a series of discrete points into a single point and past each other, would be given by the intersections A, B, C. . . as the line AE is moved to the right beyond the point X. At X they cease to be discrete, but how far in front of X this discreteness ceases I don't know. If we regulate the approach of B and C by the revolution of the line AE around the point A, governed by the movement of G along the parallel GH, the approach will have a "limit," zero; another illustration of limit or no limit according to the process adopted.

Perhaps a series of osculating curves of various degrees of osculation, all tangent at the same point, and cut by a secant moving across the point of tangency would give another illustration of a series of discrete points jamming down into a single point of no discreteness with a limited continuum before and after.

Another illustration of the dependence of "limit" upon the process adopted is the straightening of a circular arc. If straightened by the lengthening of the radius, the straightening arc has a limit-a straight line. If, however, we use a Peaucellier linkage, there is no limit to the straightening, and the arc straightens out into a straight line and over into a curve on the other side. In both cases it is a line of constant curvature, one of the set being the straight line, a line of zero

curvature.

Arranged at finite intervals this sum has the "limit" 2, which exceeds the real sum by ". But regulate the arrangement of the ordinates, not by the finite spacing, but by the constant slope of the line of ends, as shown in the diagram, and by reason of the similar tri- 2 angles the sum is easily seen to be exactly 2; there is no unattainable and evasive limit with its defending term. In each case we are dealing with exactly the same

ordinates; in one case arranged so as to have an inaccessible "limit" to the sum, in the other case not. In the one case the spacing is finite, but the "fall" of the ends decreases to infinite slowness; in the other case the "fall" is regular, but the spacing crowds to infinite closeness.

A forceful illustration of the avoidance of a "limit" is found in the historic problem of squaring the circle. In numbers this is impossible, because among other reasons is the limit of an infinite series of discrete terms. In geometry, with a rule and compass the length of the circumference is also the limit of an infinite series of operations, and is, therefore, unattainable. But change the process by using the integraph, and what was before a "limit," and just out of reach, becomes attainable, and we get a straight line equal in length to the circumference.

Let us define, for the time being, an "n-point" as the figure made by a stretched elastic around 2-points; a "regular npoint" being the case where the n-points are symmetrically arranged around a center; a "semi-regular n-point" being the case where the points are symmetrical around a center, but in two sets, an illustration of which would be the figure made by moving the mid points of the sides of a square outward equal amounts. The perpendicular from the center onto the sides of the semi-regular n-point we will call a "pseudapothem," a special case of which is the apothem. Evidently the semi-regular n-point is equilateral, and its area equals its perimeter multiplied by its pseud-apothem. I introduce this

idea of semi-regularity and pseud-apothem so as to preserve the continuity of surface change, as I introduced C in the hound and hare case to preserve the continuity and regularity of surface change.

Now starting with the inscribed square, 4-point, I am going to devise a verbal mechanism for changing the inscribed square into a circumscribed square, and vice versa. (a) Imagine the center of each side to be moved outward until it assumes a position symmetrical with the vertices, the undisturbed points, and then the process repeated, and so on. In analogy to the hound and hare we will call the points of time, or epochs of configuration, when the moving points become symmetrical with the others, "contact points."

If we regulate this process by a finitely timid succession of events, contact points, we get the ordinary mode of swelling an inscribed polygon toward the circumscribing circle by doubling the number of sides. Like the hound and hare pursuit, this process stretches out to infinity, and the circle is the "limit" of the inscribed n-point.

(6) Imagine the vertices of the circumscribed 4-point (square) moved inward toward the center until the vertices evanesce in straight lines, the new vertices being symmetrically arranged. The result is a circumscribed 8-gon. The finitely timed repetition of this process tends toward the circle as a "limit" of the successive n-points.

Suppose, however, we regulate the process not by the finitely timed succession of "contact points," but by the finite speed of surface change. Starting with the inscribed 4-point, and adopting any rate of surface change you please, regular or variable, we have a rather close analogy to the hound and hare problem. In each case a regulation by finite speed of surface change sweeps the pursuing figure (point or inscribed n-point) through the limit and beyond it. For exactness of mechanism we might say, use process (a) until the limit is reached and process (6) beyond, or vice versa; use pursuit to the overtaking point and flight beyond. The speed of surface change is finite, but the recurrence of events, contact points, is increasing to infinite rapidity, and the contact points themselves are

crowding to infinite closeness. At the limit-which has of course ceased to be a limit now-in analogy to the hound and hare, minute and hour-hand cases, the contact points form a continuum. A finite rate of surface change necessitates the sweeping of the inscribed polygons through the circle, just as the finite rate of surface change in the hour and minute hand compelled the minute hand to sweep past the hour hand. The area of any one of the n-points, whether interior, exterior or neither, is its perimeter by half its pseud-apothem. The n-points are semi-symmetrical and symmetrical; one of them is a symmetrical continuum. But a symmetrical continuum is a circle. That it is a continuum is evidenced by the fact that continuous finite increase of area necessitates sweeping over from the inner to the outer polygons, else the area could not increase indefinitely; and such sweeping over cannot occur so long as there is any discreteness of vertices; hence the transition from inner to outer must occur by reason of a continuum of points.

In other words, the broken line continuum which is symmetrical in sets of points is changed into a curved line continuum, symmetrical throughout. This curved line coutinuum is merely a phase of the broken line continuum, just as the straight line is a phase of the constant curvature lines of the Peaucellier linkage. Any proposition true of the general forms is true of the phase.

Now, if anyone desires to restrict the word polygon to a figure with finite sides and discrete vertices, there is no law against so doing except the law of generalization; that law which demands that if the investigator is to get a complete grasp of the subject he shall take the broadest view possible, and not restrict himself by unnecessary limitations. Under the restricted definition the circle is not a polygon. But under the broader definition of n-point, the circle is one of the set of regular n-points, and every general mensurational proposition which applies to n-points applies as well to the regular ∞-point, the circle. Some of the propositions may be unmanageable by reason of the introduction of infinite values; as for example, the sum of the angles of an n-point is (n-2) straight angles. But this unmanageableness does not in the slightest affect their

validity. And when we say the area of a semi-regular n-point is the perimeter by the pseud-apothem, we have a proposition which applies to all the n-points of the double set, including the transition one between the inner and outer sets, the regular ∞-point, the circle. As a proposition in continua, the area of the circle equals a rectangle whose sides are the perfectly determinable lines, perimeter and radius.

Under the current use of limits in the elementary text-books, the implication is that the circle cannot be rectified; a most vicious error, and one fraught with many evil consequences.

There is no fundamentally distinctive reason, when we have proved a proposition for regular n-points, for excluding the regular -point from its application. To exclude it because its points are ∞, and its sides in length is arbitrary and unreasonable, for there are countless millions of finite polygons which are absolutely indistinguishable in any way, shape or manner from the regular ∞-point by any verbal or mental process whatever, except the one denominative one of using the terms finite and infinite; a verbal distinction, which means, as far as our concepts are concerned, absolutely nothing, and in trying to make which distinction we are merely fooling ourselves with words. The finite polygons merge imperceptibly into the -point; there is no line of demarkation, no electric signal to mark the point where we leave the finite to enter the infinite, where the singly discrete contact points merge into the co-ple overtaking point X, where the broken line continuum merges into the curved line continuum. If so why should we halt the proposition and let the form go on?

That we have no material mechanism for producing these semi-regular n-points is of no consequence. Previous to 1864 we had no mechanism for drawing a line of zero curvature. But that does not affect the fact that a straight line is one of the series of lines of constant curvature, though it did affect our apperception of the fact previous to that time. In a similar way the absence of a material mechanism for drawing polygons tends to affect our apperception of the fact that a regular ∞-point is one of the set of regular n-points.

The use of limits in proving the equality of the ratios between