plane A B, along which the wheel has rolled, must be exactly equal to the circumference of the wheel. The arcs A C and C B, are each equal to two diameters of the wheel, and the cycloidal arc A C B is, of course, equal to four diameters. The superficial area of the rectangle A M N B, is exactly equal to 4 times the superficial area of the wheel or circle; and may be demonstrated to be so, either on a true or false value of P (P being taken to represent the number of times the diameter of a circle is contained in the circumference). Then, if the diameter of the wheel be 1, the length of the cycloidal arc will be 4, and there is an exact ratio between the two, of 1 to 4, both finite terms. These facts will be admitted by every mathematician who has examined the properties of this geometrical figure. The diameter and circumference of the wheel may be called the producing lines, and the cycloidal arc A CB and the plane A B, may be called the produced lines, and it appears to me to involve an absurdity, to suppose there can be an exact ratio between the diameter and one of the produced lines, which can be expressed in finite terms, and no definable relation between the diameter and the other produced line, the two lines being produced, pari passu, by the same revolution of the wheel. I venture to put the following question: If there be no definable relation between the diameter and circumference of the wheel at starting, is it conceivable that such a line as the arc A C D, could be described and have a relation to the diameter of the wheel, expressible in finite terms? To me, I confess the thing appears impossible. Let me now direct your attention to diagram No. 2, (see Fig. VI.) which may be new to you, as I have never met with any mathematician who has thought of con sidering this geometrical figure, with reference to the quadrature and rectification of the circle. You will perceive that the diameter of the Circle A B CD, is equal to twice the diameter of the inner circle, and consequently, the circumference of the Circle ABCD, must be equal to twice the circumference of the inner circle. Let the inner circle be supposed to revolve round the inner rim of the Circle A B C D, in the direction from A to B CD and on to A again, (it may be supposed to move in either direction). When the inner circle shall have made one revolution, the point O which rested on the point A at starting, will have passed from A to C, describing in its course the straight line A C. The point P of the inner circle which stood, at starting, in the centre of the Circle A B CD, will have passed from the centre to the point B, and back again to the centre. In the next revolution the point O will have passed from C to A, returning on the straight line C A, and the point P will have passed from the centre to the point D, and back again to the centre, and the inner circle will again stand at its original starting point, and in ten thousand revolutions the two points O and P, would go on passing backwards and forwards continually, along the straight lines A C and BD as just described, and could never, by any possibility, get off them. Then, the described lines A C and B D cut the Circle A B C D into 4 equal parts, each of which is exactly equal in superficial area to the superficial area of the inner circle, and the points O and P, have travelled a distance exactly equal to twice the diameter of the Circle ABCD, and 4 times the diameter of the inner circle, and the ratio of diameter of the inner circle, to the distance travelled by the points O and P, is exactly as I to 4. |