Parallel Curved Lines and Concentric Circles
Circular parallels must be equidistant from one centre, with intersecting and concentric circle figures
Continuing the geometry of curves, the folio argues that lines forming circular parallels cannot share the same curvature, for on completing their circles they would touch, indeed intersect, in two places. True curved parallels instead require that each curve, part and whole, be everywhere equidistant from one single centre. Three figures accompany the notes: two parallel arcs at the top, two equal circles that are not concentric meeting at two points, and two concentric circles whose parallel circumferences are spanned by radial segments across the intervening corona.
On this page
Why circular parallels cannot have the same curvature
Lines that compose circular parallels cannot be of the same curvature, because in completing their whole circles they would make contact, or rather intersection, in two places. Two equal circles that are not concentric are drawn meeting at two points to show this.
Curved parallels must be equidistant from a single centre
For curves to compose true parallels, the part and the whole of each must, each for itself, be equidistant from one single centre. Only then do the two circumferences remain everywhere parallel.
Two concentric circles and the corona between them
At the foot of the page two concentric circles share one centre, their circumferences parallel. The distance between them, the corona, is marked by short segments drawn at two positions across the ring.