Finding circle centres, an octagon, and a divided triangle
Locating the centre of an arc, building an octagon on a line, and dividing a triangle proportionally
This geometry sheet sets three problems. At the upper left, arcs struck about an oblique line S f show how to recover the centre of a circle from a small piece of its arc. At the right a compass construction inscribes a regular octagon in a circle raised on a given line a m. Below, a tall isosceles triangle divided by horizontal lines demonstrates a proportional rule: as many times as a b enters d e, so many times b c enters c e, in equal number.
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Recovering the centre of a circle from an arc
Working from the oblique line S f, the construction finds the centre of a circle given only a small part of its arc. The lines arising between equal angles are made to converge, in their meeting, upon the place from which they set out — the cause of their origin.
Inscribing an octagon on a given line a m
On the given line a m, swinging the compass from a makes the arc S t and from m the arc v x; the line p r joins their intersections. Dividing the half-line a p into three parts and setting one part above the upper intersection gives the centre of a circle whose circumference will hold eight of the given lengths — a regular octagon.
Dividing a triangle into equal proportional parts
A tall isosceles triangle is cut by horizontal lines into equal parts. The rule that follows is proportional: as many times as a b enters into d e, so many times b c enters into c e, and a b enters b c the same number of times that d e enters e c.