Geometry of tangent points, with sketches of springs and devices
Lowering points proportionally to find the angle of incidence, amid columns of figures
A large inscribed triangle spans arcs at the centre of the sheet, part of a study of tangent lines and the angle of contingency on a circle. A note explains that when the highest point lies too high for the tangent construction to intersect, the points must be lowered proportionally to their first heights so the construction succeeds. Columns of figures and multiplications fill the lower right, and an inverted note restates a rule of proportion between wholes and halves. Small sketches of coiled springs and mechanical parts appear at the top and along the left margin; these are drawn but are not transcribed.
On this page
Lowering points to complete the tangent construction
When the highest point lies so high that the line from it to the point of contact of the circle leaves the other point beneath it, the intersection that yields the angle of incidence cannot be made. Leonardo instructs that the points be lowered along their central lines, in proportion to their first heights.
Proportion of wholes and halves
An inverted note states that the same proportion a whole has with a whole, a half has with a half, and so the halves of both wholes stand to the other half.
Sketches of coiled springs and mechanical parts
Along the top and the left margin are small studies of coiled springs and mechanical fittings, drawn faintly among the geometry. They are not included in the transcription and carry no accompanying text.