5. A strait line may be drawn so as to touch the crooked line of the said figure in any point, in this manner. Let it be required to draw a tangent to the line B K D H (in fig. 4) in the point D. Let the points B and D be connected, and drawing D A equal and parallel to B C, let B and A be connected; and because this figure is by construction the third of three means, let there be taken in A B three points, so, that by them the same A B be divided into four equal parts; of which take three, namely, A M, so that A B may be to A M, as the figure B K D C is to its complement. I say, the strait line M D will touch the figure in the point given D. For let there be drawn anywhere between A B and D C a parallel, as R Q, cutting the strait line B D, the crooked line B K D, the strait line M D, and the strait line A D, in the points P, K, O and Q. R K will therefore, by construction, be the least of three means in geometrical proportion between R Q and R P. Wherefore (by coroll. of art. 28, chapter XIII.) R K will be less than R O; and therefore M D will fall without the figure. Now if M D be produced to N, F N will be the greatest of three means in arithmetical proportion between F E and F G; and F H will be the greatest of three means in geometrical proportion between the same F E and F G. Wherefore (by the same coroll. of art. 28, chapter XIII.) F H will be less than F N; and therefore D N will fall without the figure, and the strait line M N will touch the same figure only in the point D. [Sidenote: In what proportion the same figures exceed a strait-lined triangle of the same altitude and base.] 6. The proportion of a deficient figure to its complement being known, it may also be known what proportion a strait-lined triangle has to the excess of the deficient figure above the same triangle; and these proportions I have set down in the following table; where if you seek, for example, how much the fourth three-sided figure of five means exceeds a triangle of the same altitude and base, you will find in the concourse of the fourth column with the three-sided figures of five means 2⁄10; by which is signified, that that three-sided figure exceeds the triangle by two-tenths or by one-fifth part of the same triangle.