Finding roots and measuring areas by geometry
A square root drawn with a semicircle, and the areas of triangles and polygons
Practical geometry with figures in the margins. A semicircle over a diameter divided into ten spaces (labeled r, a, c, f, b, d) gives the square root of nine as the perpendicular a b. Further sketches show that a right triangle of base four and side eight has area sixteen, that a rectangle can equal the triangle, and that any many-sided figure can be measured by dividing it into triangles. A trapezoid with parallel sides 4 and 2 and slant sides each 10 closes the page.
On this page
The square root of nine drawn with a semicircle
To find a root geometrically, lay out nine equal spaces from c to b, add one more to make line c d of ten spaces, and draw a semicircle c r d on its midpoint f. The perpendicular raised between the ninth and tenth spaces meets the arc at a; line a b is the root, entering three times into c b.
Area of a right triangle (base 4, side 8)
To measure a right triangle of base four and other side eight, halve the base to two and multiply: two times eight is sixteen. That product is the area.
Any faceted figure measured through its triangles
A regular pentagon is joined from its center into five equal triangles; measure one triangle and the whole area follows. An irregular polygon is likewise cut into nine unequal triangles for measuring any body.
A trapezoid with its side measurements
The page ends with a trapezoid whose two parallel sides measure 4 and 2 and whose slant sides are each 10, drawn as a figure with its side lengths marked. It extends the area-measuring method to a four-sided figure.