Screw threads, a weighted bellows, and compass constructions
Friction of square vs. triangular threads; force in a bellows; bisecting lines and arcs
This mixed page pairs mechanical observations with geometric constructions. Cross-sections of square and triangular screw threads let Leonardo argue that more contact gives more resistance, so a triangular thread runs easier on its driver, while a bellows loaded with a 1000-pound weight illustrates how force is shared among its parts in proportion to the mouth's share of the volume. Alongside are compass constructions for finding the midpoint of line m n, the centre of an equilateral triangle, and for dividing an arc (a b c d) into an odd number of equal parts. Labels m, n, a, b, c, d and the weight 1000 mark the figures.
On this page
Square vs. triangular screw threads
Cross-sections of three thread profiles head the page: square protrusions, symmetrical triangles, and flat-topped right-triangle teeth. Where there is greater contact there is greater resistance, so the square thread traces a longer path, while the triangular thread, with less contact, is easier on its driver.
Finding the midpoint of line m n
A compass construction bisects the arc whose ends are m and n, locating the vertices for an equilateral triangle. The heading states simply the aim: to find the middle of line m n.
Finding the centre of an equilateral triangle
The previous figure is amplified into a completed equilateral triangle, its three construction arcs extended to semicircles that give the perpendicular bisectors of the sides. The bisectors meet at the triangle's centre.
Force in a bellows under a 1000-pound weight
A bellows is pressed by a weight of 1000 pounds, its nozzle mouth labelled a. As many times as the mouth fits into the whole volume of the bellows, into so many parts the weight is distributed: if the mouth fits 1000 times, only one pound acts at the mouth while the other 999 press on the remaining parts.
Dividing an arc into an odd number of equal parts
A construction divides a curved line into an odd number of equal segments, here three, delimited by points a, b, c, d with further points n2, o above and m below. The whole is done with only one opening of the compass.