It may from hence be collected, that when the velocity of a body is increased in the same proportion with that of the times, the degrees of velocity above one another proceed as numbers do in immediate succession from unity, namely, as 1, 2, 3, 4, &c. And when the velocity is increased in proportion duplicate to that of the times, the degrees proceed as numbers from unity, skipping one, as 1, 3, 5, 7, &c. Lastly, when the proportions of the velocities are triplicate to those of the times, the progression of the degrees is as that of numbers from unity, skipping two in every place, as 1, 4, 7, 10, &c., and so of other proportions. For geometrical proportionals, when they are taken in every point, are the same with arithmetical proportionals. [Sidenote: Of deficient figures described in a circle.] 11. Moreover, it is to be noted that as in quantities, which are made by any magnitudes decreasing, the proportions of the figures to one another are as the proportions of the altitudes to those of the bases; so also it is in those, which are made with motion decreasing, which motion is nothing else but that power by which the described figures are greater or less. And therefore in the description of _Archimedes' spiral_, which is done by the continual diminution of the semidiameter of a circle in the same proportion in which the circumference is diminished, the space, which is contained within the semidiameter and the spiral line, is a third part of the whole circle. For the semidiameters of circles, inasmuch as circles are understood to be made up of the aggregate of them, are so many sectors; and therefore in the description of a spiral, the sector which describes it is diminished in duplicate proportions to the diminutions of the circumference of the circle in which it is inscribed; so that the complement of the spiral, that is, that space in the circle which is without the spiral line, is double to the space within the spiral line. In the same manner, if there be taken a mean proportional everywhere between the semidiameter of the circle, which contains the spiral, and that part of the semidiameter which is within the same, there will be made another figure, which will be half the circle. And to conclude, this rule serves for all such spaces as may be described by a line or superficies decreasing either in magnitude of power; so that if the proportions, in which they decrease, be commensurable to the proportions of the times in which they decrease, the magnitudes of the figures they describe will be known. [Sidenote: The proposition demonstrated in art. 2 confirmed from the elements of philosophy.]