Lever and Counter-Lever: Weights Carried by an Angled Cord
Triangles and a semicircle of chords resolving how a hanging load loads the arms that hold it
The sheet works out how a weight hung from an angled cord divides its load between the two arms that hold it, treating each arm as a lever or counter-lever and each cord as the hypotenuse of a triangle. Numbered diagrams show that the axis of the triangle, compared with its hypotenuse, gives the proportion between the hanging weight and the pull felt at the ends, so that a load of 4 may register as 8, 12 or 16 as the lever shortens. A semicircle strung with chords collects several such cases, and a squared grid in the top corner sets up the geometry, all carried through in ratios and pounds.
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The axis and hypotenuse of a loaded triangle
When the axis m c is one third of its hypotenuse c f, a weight of 4 hung on the axis produces a third of what the two hypotenuses feel, that is 4 against 12. This agrees with the check on the counter-lever m n of 3 and its lever n o of 2, so that the doubled load resolves to 4 against 6 on the arm.
Lever and counter-lever bearing the middle weight
With a b as counter-lever and a c as lever standing three to four, a load of 3 in the lever resists 4 in the counter-lever, while n m and n o stand two to three. Because the counter-lever's 2 is doubled to 4, the lever's 3 must become 6, giving 12 in the hypotenuses against 4 in the axis: the shorter axis m c doubles the weight carried by its hypotenuses.
A cord angled by a hanging weight
If a cord in the position of equality is bent in the middle by a weight hung from it, draw the straight line between its ends and a perpendicular from that line's midpoint onto the angle, forming a triangle. The proportion of the hung weight to the weight it delivers to the cord's ends is then as the axis of that triangle to its hypotenuse.
Semicircle strung with chords of varying lever
A semicircle collects chords a b / b c, c d / d e, r p / p n and o h / h L as paired counter-levers and levers. Where the lever is half the counter-lever the weight 4 becomes 8, where it is a quarter it becomes 16, so the same central load of 4 registers ever larger as the lever shortens.
Reckoning the shares of the load
When b c axis is two thirds of its hypotenuse, the hypotenuses feel two thirds of 4 and 4, that is 8, of which 4 falls to each. The check sets 3 of counter-lever b a against 4 of lever a c, wanting 4 in counter-lever against 3 in lever.