| 1 | 2 3 4 5 6 7 Parallelogram | 1 | : : : : : : +----+ Strait-sided triangle | 1/2| : : : : : : |----+ Three-sided figure of 1 mean | 2/3| : : : : : : |----+----+ Three-sided figure of 2 means | 3/4| 3/5| : : : : : |----+----+----+ Three-sided figure of 3 means | 4/5| 4/6| 4/7| : : : : |----+----+----+----+ Three-sided figure of 4 means | 5/6| 5/7| 5/8|5/9 | : : : |----+----+----+----+----+ Three-sided figure of 5 means | 6/7| 6/8| 6/9|6/10|6/11| : : |----+----+----+----+----+----+ Three-sided figure of 6 means | 7/8| 7/9|7/11|7/11|7/12|7/13| `` : |----+----+----+----+----+----+----+ Three-sided figure of 7 means | 8/9|8/10|8/11|8/12|8/13|8/14|8/15| +----+----+----+----+----+----+----+ [Sidenote: Description & production of the same figures.] 4. Now for the better understanding of the nature of these three-sided figures, I will show how they may be described by points; and first, those which are in the first column of the table. Any parallelogram being described, as A B C D (in figure 2) let the diagonal B D be drawn; and the strait-lined triangle B C D will be half the parallelogram; then let any number of lines, as E F, be drawn parallel to the side B C, and cutting the diagonal B D in G; and let it be everywhere, as E F to E G, so E G to another, E H; and through all the points H let the line B H H D be drawn; and the figure B H H D C will be that which I call a three-sided figure of one mean, because in three proportionals, as E F, E G and E H, there is but one mean, namely, E G; and this three-sided figure will be 2⁄3 of the parallelogram, and is called a _parabola_. Again, let it be as E G to E H, so E H to another, E I, and let the line B I I D be drawn, making the three-sided figure B I I D C; and this will be 3⁄4 of the parallelogram, and is by many called a _cubic parabola_. In like manner, if the proportions be further continued in E F, there will be made the rest of the three-sided figures of the first column; which I thus demonstrate. Let there be drawn strait lines, as H K and G L, parallel to the base D C. Seeing therefore the proportion of E F to E H is duplicate to that of E F to E G, or of B C to B L, that is, of C D to L G, or of K M (producing K H to A D in M) to K H, the proportion of B C to B K will be duplicate to that of K M to K H; but as B C is to B K, so is D C or K M to K N, and therefore the proportion of K M to K N is duplicate to that of K M to K H; and so it will be wheresoever the parallel K M be placed. Wherefore the figure B H H D C is double to its