1 2 3 4 5 6 7
The triangle 1 : : : : : :
A three-sided fig. of 1 mean 1⁄3 : : : : : :
The ex- cess of A three-sided fig. of 2 means 2⁄4 1⁄5 : : : : :
A three-sided fig. of 3 means 3⁄5 2⁄6 1⁄7 : : : :
A three-sided fig. of 4 means 4⁄6 3⁄7 2⁄8 1⁄9 : : :
A three-sided fig. of 5 means 5⁄7 4⁄8 3⁄9 2⁄10 1⁄11 : :
A three-sided fig. of 6 means 6⁄8 5⁄9 4⁄10 3⁄11 2⁄12 1⁄13 :
A three-sided fig. of 7 means 7⁄9 6⁄10 5⁄11 4⁄12 3⁄13 2⁄14 1⁄15

A table of solid deficient figures described in a cylinder.

7. In the next table are set down the proportion of a cone and the solids of the said three-sided figures, namely, the proportions between them and a cylinder. As for example, in the concourse of the second column with the three-sided figures of four means, you have 5⁄9; which gives you to understand, that the solid of the second three-sided figure of four means is to the cylinder as 5⁄9 to 1, or as 5 to 9.

1 2 3 4 5 6 7
1 2 3 4 5 6 7
A cylinder 1 : : : : : :
A cone 1⁄3 : : : : : :
A three-sided fig. of 1 mean 2⁄4 : : : : : :
A three-sided fig. of 2 means 3⁄5 3⁄7 : : : : :
The sol- ids of A three-sided fig. of 3 means 4⁄6 4⁄8 4⁄10 : : : :
A three-sided fig. of 4 means 5⁄7 5⁄9 5⁄11 5⁄13 : : :
A three-sided fig. of 5 means 6⁄8 6⁄10 6⁄12 6⁄14 6⁄16 : :
A three-sided fig. of 6 means 7⁄9 7⁄11 7⁄13 7⁄15 7⁄17 7⁄19 :
A three-sided fig. of 7 means 8⁄10 8⁄12 8⁄14 8⁄16 8⁄18 8⁄20 8⁄22

In what proportion the same figures exceed a cone of the same altitude and base.

8. Lastly, the excess of the solids of the said three-sided figures above a cone of the same altitude and base, are set down in the table which follows:

1 2 3 4 5 6 7
The Cone 1 : : : : : :
The exces-ses of the sol-ids of these three- sided fig- ures above a cone. Of the solid of a three-sided figure of 1 mean 6⁄12 : : : : : :
Ditto ditto, 2 means 12⁄15 6⁄21 : : : : :
Ditto ditto, 3 means 18⁄18 12⁄24 6⁄30 : : : :
Ditto ditto, 4 means 24⁄21 18⁄27 12⁄33 6⁄39 : : :
Ditto ditto, 5 means 30⁄24 24⁄30 18⁄36 12⁄42 6⁄48 : :
Ditto ditto, 6 means 36⁄27 30⁄33 24⁄39 18⁄45 12⁄51 6⁄57 :
Ditto ditto, 7 means 42⁄30 36⁄36 30⁄42 24⁄48 18⁄54 12⁄60 6⁄66

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.

9. If any of these deficient figures, of which I have now spoken, as A B C D (in the [5th figure]) be inscribed within the complete figure B E, having A D C E for its complement; and there be made upon C B produced the triangle A B I; and the parallelogram A B I K be completed; and there be drawn parallel to the strait line C I, any number of lines, as M F, cutting every one of them the crooked line of the deficient figure in D, and the strait lines A C, A B and A I in H, G, and L; and as G F is to G D, so G L be made to another, G N; and through all the points N there be drawn the line A N I: there will be a deficient figure A N I B, whose complement will be A N I K. I say, the figure A N I B is to the triangle A B I, as the deficient figure A B C D twice taken is to the same deficient figure together with the complete figure B E.

For as the proportion of A B to A G, that is, of G M to G L, is to the proportion of G M to G N, so is the magnitude of the figure A N I B to that of its complement A N I K, by the [second article] of this chapter.