Line divisions and figures of the Pythagorean theorem
'Tenth of the second': segments a-c-b with squares raised on a right isosceles triangle
Headed 'Tenth of the second,' the page divides a straight line into equal parts (a, 6, c, 6, b) and extends it with a further part (b, 4, d), then raises a right isosceles triangle bisected on that base. The construction is linked to the figure of the theorem of Pythagoras, with a rectangle and a second right triangle built onto it and a concluding figure at the left. Columns of arithmetic (100 + 36 = 136; 256 + 16 = 272) run along the top, tallying the squares on the sides. A new figure of the Pythagorean theorem is drawn at the foot of the page.
On this page
A line divided and extended, points a c b d
A straight line is divided into equal parts marked a, 6, c, 6, b, then prolonged by a further part b, 4, d, with the whole labelled 16. This extended base carries the triangles and squares constructed above it. The lettering fixes the points used in the later proof.
Right isosceles triangle and the Pythagorean figure
A right isosceles triangle is divided in half and set on the preceding line, tied to the figure of the theorem of Pythagoras (a, 6, c, 6, b, d). A rectangle is then formed on the same figure and a further right triangle added (b, a - c). A concluding figure is placed at the left.
New figure of the theorem of Pythagoras
At the foot of the page a fresh drawing of the Pythagorean theorem is laid out, labelled b, a - c. It restates the relation among the squares on the sides of a right triangle. It closes the sequence begun at the top of the sheet.
Sums of the squares on the sides
Two columns of addition head the page: 100 plus 36 giving 136, and 256 plus 16 giving 272, with a further sum 256 + 16 = 272 at the upper left. The figures tally the areas of the squares raised on the triangle's sides. They give the numerical side of the geometric proof.