CHAP. XVII.
OF FIGURES DEFICIENT.
[1.] Definitions of a deficient figure; of a complete figure; of the complement of a deficient figure; and of proportions which are proportional and commensurable to one another.—[2.] The proportion of a deficient figure to its complement.—[3.] The proportions of deficient figures to the parallelograms in which they are described, set forth in a table.—[4.] The description and production of the same figures.—[5.] The drawing of tangents to them.—[6.] In what proportion the same figures exceed a strait-lined triangle of the same altitude and base.—[7.] A table of solid deficient figures described in a cylinder.—[8.] In what proportion the same figures exceed a cone of the same altitude and base.—[9.] How a plain deficient figure may be described in a parallelogram, so that it be to a triangle of the same base and altitude, as another deficient figure, plain or solid, twice taken, is to the same deficient figure, together with the complete figure in which it is described.—[10.] The transferring of certain properties of deficient figures described in a parallelogram to the proportions of the spaces transmitted with several degrees of velocity.—[11.] Of deficient figures described in a circle.—[12.] The proposition demonstrated in art. 2 confirmed from the elements of philosophy.—[13.] An unusual way of reasoning concerning the equality between the superficies of a portion of a sphere and a circle.—[14]. How from the description of deficient figures in a parallelogram, any number of mean proportionals may be found out between two given strait lines.
Definition of a deficient figure.
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.
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.