Time, Geometry, and Definitions of Line and Angle
Point, instant, surface and the horizontal lines of the world
Two facing pages continuing Leonardo's meditation on the common ground of geometry and time. He repeats that time, though counted among continuous quantities, is incorporeal and so touches geometry only through the point (the instant) and the line (a span of time). He then rehearses several formulations of the point, the line, the angle and the surface, defining the surface as the motion of a line and noting that surfaces are the boundaries of bodies. A closing list names the horizontal "lines of equality" whose ends stay equidistant from the centre of the world.
On this page
The point, line, angle and surface defined
The point has no part; the line is the transit of the point, terminated by two points; the angle is the concurrence, breaking or contact of two lines at a point. The surface is the motion of the line, and lines are the boundaries of the surface, just as surfaces are the boundaries of bodies.
Time as a continuous quantity, joined to geometry only in point and line
Though numbered among continuous quantities, time is invisible and bodiless and so does not fall wholly under geometry, which divides other quantities into figures. It agrees with geometry only in its first principles: the point in time is the instant, and the line answers to the length of a span of time terminated between two instants.
The horizontal lines equidistant from the centre of the world
A power moves a body so much space in so much time. Leonardo lists the line of equality, the line of the horizon, the lying line and the equi-lying line, defining them as those lines whose extremities stand equidistant from the centre of the world.