Recto: a sheet of notes on the geometry of optics, illustrated with diagrams; a head, turned in profile to the right, squared for enlargement and at right angles to it a left profile, not by Leonardo. Verso: a sheet of notes on the geometry of optics, illustrated by diagrams; and a drawing of a vice; with a watermark of a lobed flower (Milanese).

The sheet is typical of Leonardo’s earliest scientific studies, an untidy combination of sketches, formal diagrams and irregular blocks of notes. At centre left he shows how the height of a mountain may be determined by measuring the angle of elevation of its summit from different points in the plain below. Other diagrams demonstrate the similar method of calculating the radius of the earth by measuring the elevation of the sun from different points on the earth’s curved surface. The diagrams show that Leonardo considered the sun to be relatively close to the earth, and the calculations would require one to know that distance; but the sun is in truth so distant that its rays may be considered parallel, simplifying the geometry – as surmised by the Greek mathematician Eratosthenes, who in the third century bc calculated the size of the earth using this method, by measuring midday shadows at different latitudes. The three diagrams at lower right study the shadows cast by spheres of different sizes, a subject of enduring interest to Leonardo. Two male heads at lower left with grids superimposed are a first glimpse of Leonardo’s proportional studies; the crude head alongside is not by Leonardo.

Text adapted from Leonardo da Vinci: A life in drawing, London, 2018