Wherefore the sense of the Lemma is, that if the moments of any quantities A, B, C, &c., increasing or decreasing by a perpetual flux, or the velocities of the mutations which are proportional to them, be called a, b, c, &c., the moment or mutation of the generated rectangle AB will be aB + bA; the moment of the generated content ABC will be aBC + bAC + cAB; and the moments of the generated powers , , , , , , , , , will be 2aA, , , , , , , , , respectively; and, in general, that the moment of any power , will be . Also, that the moment of the generated quantity will be ; the moment of the generated quantity will be ; and the moment of the generated quantity or will be ; and so on. The Lemma is thus demonstrated. CASE 1. Any rectangle, as AB, augmented by a perpetual flux, when, as yet, there wanted of the sides A and B half their moments and , was into , or ; but as soon as the sides A and B are augmented by the other half moments, the rectangle becomes into , or . From this rectangle subduct the former rectangle, and there will remain the excess aB + bA. Therefore with the whole increments a and b of the sides, the increment aB + bA of the rectangle is generated. Q.E.D. CASE 2. Suppose AB always equal to G, and then the moment of the content ABC or GC (by Case 1) will be gC + cG, that is (putting AB and aB + bA for G and g), aBC + bAC + cAB. And the reasoning is the same for contents under ever so many sides. Q.E.D. CASE 3. Suppose the sides A, B, and C, to be always equal among themselves; and the moment aB + bA, of , that is, of the rectangle AB, will be 2aA; and the moment aBC + bAC + cAB of , that is, of the content ABC, will be . And by the same reasoning the moment of any power is . Q.E.D. CASE 4. Therefore since into A is 1, the moment of drawn into[Pg 263] A, together with drawn into a, will be the moment of 1, that is, nothing. Therefore the moment of , or of , is . And generally since into is 1, the moment of drawn into together with into will be nothing. And, therefore, the moment of or will be . Q.E.D. CASE 5. And since into is , the moment of drawn into will be a (by Case 3); and, therefore, the moment of will be or . And, generally, putting equal to , then will be equal to , and therefore equal to , and equal to , or ; and therefore is equal to b, that is, equal to the moment of . Q.E.D. CASE 6. Therefore the moment of any generated quantity is the moment of drawn into , together with the moment of drawn into , that is, ; and that whether the indices m and n of the powers be whole numbers or fractions, affirmative or negative. And the reasoning is the same for contents under more powers. Q.E.D.