Here the perpendicular now found, and the sides cut, are the sides of the rectangle, whose halfe shall be the content of the Triangle: As here the Rectangle of 21 and 12 is 252; whose halfe 126, is the content of the triangle.
The second section followeth from whence ariseth the fourth rate or comparison.
7. If a right line be cut into two equall parts, and otherwise; the oblong of the unequall segments, with the quadrate of the segment betweene them, is equall to the quadrate of the bisegment. 5 p ij.
As for example, Let the right line ae 8, be cut into two equall portions, ai 4, and ie 4. And otherwise that is into two unequall portions, ao 7, and oe 1: The oblong of 7 and 1, with 9, the quadrate of 3, the intersegment, (or portion cut betweene them) that is 16; shall bee equall to the quadrate of ie 4, which is also 16. Which is also manifest by making up the diagramme as here thou seest. For as the parallelogramme as is by the [26 e x], equall to the
parallelogramme iu; And therefore by the [19 e x], it is equall to oy. For ou, is common to both the equall complements, Therefore if so be added in common to both; the ar, shall be equall to the gnomon mni: Now the quadrate of the segment betweene them is sl. Wherefore ar, the oblong of the unequall segments, with s the quadrate of the intersegment, is equall to iy the quadrate of the said bisegment.
The third section doth follow, from whence the fifth reason ariseth.
8. If a right line be cut into equall parts; and continued; the oblong made of the continued and the continuation, with the quadrate of the bisegment or halfe, is equall to the quadrate of the line compounded of the bisegment and continuation. 6 p ij.
As for example, let the line ae 6, be cut into two equall portions, ai 3, and ie 3: And let it be continued unto eo 2: The oblong 16, made of 8 the continued line, and of 2, the continuation; with 9 the quadrate of 3, the halfe, (that is 25.) shall be equall to 25, the quadrate of 3, the halfe and 2, the continuation, that is 5. This as the former, may geometrically, with the helpe of numbers be expressed. For by the [26 e x], as is equall to iy: And by the [19 e x], it is equall to yr, the complement. To these equalls adde so. Now the oblong au, shall be equall to the gnomon nju. Lastly, to the equalls adde the quadrate of the bisegment or halfe. The Oblong of the continued line and of the