1. I call those _deficient figures_ which may be understood to be generated by the uniform motion of some quantity, which decreases continually, till at last it have no magnitude at all. [Sidenote: Definitions of a complete figure; of the complement of a deficient figure; and of proportions which are proportional & commensurable to one another.] And I call that a _complete figure_, answering to a deficient figure, which is generated with the same motion and in the same time, by a quantity which retains always its whole magnitude. The _complement_ of a deficient figure is that which being added to the deficient figure makes it complete. Four proportions are said to be _proportional_, when the first of them is to the second as the third is to the fourth. For example, if the first proportion be duplicate to the second, and again, the third be duplicate to the fourth, those proportions are said to be _proportional_. And _commensurable_ proportions are those, which are to one another as number to number. As when to a proportion given, one proportion is duplicate, another triplicate, the duplicate proportion will be to the triplicate proportion as 2 to 3; but to the given proportion it will be as 2 to 1; and therefore I call those three proportions _commensurable_. [Sidenote: The proportion of a deficient figure to its complement.] 2. A deficient figure, which is made by a quantity continually decreasing to nothing by proportions everywhere proportional and commensurable, is to its complement, as the proportion of the whole altitude to an altitude diminished in any time is to the proportion of the whole quantity, which describes the figure, to the same quantity diminished in the same time. Let the quantity A B (in fig. 1), by its motion through the altitude A C, describe the complete figure A D; and again, let the same quantity, by decreasing continually to nothing in C, describe the deficient figure A B E F C, whose complement will be the figure B D C F E. Now let A B be supposed to be moved till it lie in G K, so that the altitude diminished be G C, and A B diminished be G E; and let the proportion of the whole altitude A C to the diminished altitude G C, be, for example, triplicate to the proportion of the whole quantity A B or G K to the diminished quantity G E. And in like manner, let H I be taken equal to G E, and let it be diminished to H F; and let the proportion of G C to H C be triplicate to that of H I to H F; and let the same be done in as many parts of the strait line A C as is possible; and a line be drawn through the points B, E, F and C. I say the deficient figure A B E F C is to its complement B D C F E as 3 to 1, or as the proportion of A C to G C is to the proportion of A B, that is, of G K to G E.