On the Nature of the Surface: Boundary of Body and Air
Twin definitions arguing that surface, line and point are all equal to nothing
Two facing columns of mirror-script develop the same argument: the surface is nothing but the boundary where a body meets the surrounding air, so it has 'being' but occupies no place and is therefore equal to nothing. From this Leonardo concludes that surface, line and point are all equal to one another, each being the dimensionless limit of the next higher magnitude. A short column of figures sits at the lower right, and faint geometric and framework sketches (including a triangle) appear at the lower left; further untranscribed matter is present on the sheet.
On this page
The surface as the boundary of bodies, having being but no place
The surface is defined as the boundary between a body and the air that encloses it; where air and body touch, no room remains for another body, so the surface has no body and needs no place. Since whatever has being must have place, and the surface has none, it is equal to nothing. Leonardo therefore concludes that surface, line and point are equal among themselves, and that each alone equals the other two joined together.
Left-column restatement: the surface has the name but not the being
The left column recasts the same idea: the surface is the very contact two bodies make, belonging to neither the surrounded nor the surrounding body. If they touch continuously, nothing lies between them, so the surface enclosed there is nothing. It has the name and not the being, and being like nothing, its part equals the whole, so point and line are equal to the surface.
Column of loose figures
A small vertical arrangement of numbers stands at the lower right of the sheet, apart from the two prose columns. It records isolated calculations rather than continuous text.
Geometric and framework sketches (untranscribed)
At the lower left of the leaf are faint pen sketches: a triangular figure with drawn lines, a rectangular frame, a small boxed letter, and a tiny pyramid-like solid. These diagrams carry no transcription in this edition and are described only from what is drawn.