12. The truth of that proposition, which I demonstrated in art. 2, which is the foundation of all that has been said concerning deficient figures, may be derived from the elements of philosophy, as having its original in this; _that all equality and inequality between two effects, that is, all proportion, proceeds from, and is determined by, the equal and unequal causes of those effects, or from the proportion which the causes, concurring to one effect, have to the causes which concur to the producing of the other effect_; and that therefore the proportions of quantities are the same with the proportions of their causes. Seeing, therefore, two deficient figures, of which one is the complement of the other, are made, one by motion decreasing in a certain time and proportion, the other by the loss of motion in the same time; the causes, which make and determine the quantities of both the figures, so that they can be no other than they are, differ only in this, that the proportions by which the quantity which generates the figure proceeds in describing of the same, that is, the proportions of the remainders of all the times and altitudes, may be other proportions than those by which the same generating quantity decreases in making the complement of that figure, that is, the proportions of the quantity which generates the figure continually diminished. Wherefore, as the proportion of the times in which motion is lost, is to that of the decreasing quantities by which the deficient figure is generated, so will the defect or complement be to the figure itself which is generated. [Sidenote: An unusual way of reasoning concerning the equality between the superficies of a portion of a sphere and a circle.]