complement B H H D A, and consequently 2⁄3 of the whole parallelogram. In the same manner, if through I be drawn O P I Q parallel and equal to C D, it may be demonstrated that the proportion of O Q to O P, that is, of B C to B O, is triplicate that of O Q to O I, and therefore that the figure B I I D C is triple to its complement B I I D A, and consequently ¾ of the whole parallelogram, &c. Secondly, such three-sided figures as are in any of the transverse rows, may be thus described. Let A B C D (in fig. 3) be a parallelogram, whose diagonal is B D. I would describe in it such figures, as in the preceding table I call three-sided figures of three means. Parallel to D C, I draw E F as often as is necessary, cutting B D in G; and between E F and E G, I take three proportionals E H, E I and E K. If now there be drawn lines through all the points H, I and K, that through all the points H will make the figure B H D C, which is the first of those three-sided figures; and that through all the points I, will make the figure B I D C, which is the second; and that which is drawn through all the points K, will make the figure B K D C the third of those three-sided figures. The first of these, seeing the proportion of E F to E G is quadruplicate of that E F to E H, will be to its complement as 4 to 1, and to the parallelogram as 4 to 5. The second, seeing the proportion of E F to E G is to that of E F to E I as 4 to 2, will be double to its complement, and 4⁄6 or 2⁄3 of the parallelogram. The third, seeing the proportion of E F to E G is that of E F to E K as 4 to 3, will be to its complement as 4 to 3, and to the parallelogram as 4 to 7. Any of these figures being described may be produced at pleasure, thus; let A B C D (in fig. 4) be a parallelogram, and in it let the figure B K D C be described, namely, the third three-sided figure of three means. Let B D be produced indefinitely to E, and let E F be made parallel to the base D C, cutting A D produced in G, and B C produced in F; and in G E let the point H be so taken, that the proportion of F E to F G may be quadruplicate to that of F E to F H, which may be done by making F H the greatest of three proportionals between F E and F G; the crooked line B K D produced, will pass through the point H. For if the strait line B H be drawn, cutting C D in I, and H L be drawn parallel to G D, and meeting C D produced in L; it will be as F E to F G, so C L to C I, that is, in quadruplicate proportion to that of F E to F H, or of C D to C I. Wherefore if the line B K D be produced according to its generation, it will fall upon the point H. [Sidenote: The drawing of tangents to them.]