Three Balances of Equal Nature
Levers and counter-levers proved equivalent through the proportion 6, 18, 36
Across the top of the sheet Leonardo draws three balance-and-lever figures and declares that all three are "of equal nature." Each pairs a long lever arm with a shorter counter-lever carrying a hanging weight, and each is labelled with the same proportion — 6, 18, 36. In the prose below he argues that because the counter-lever a n fits three times into the arm n t, a load of 6 at c needs 18 at the far end a to stay level, and that the six weights of the beam multiplied by 6 — its root — yield 36 at the extreme counter-lever.
On this page
Three figures declared "of equal nature"
A single line heading the argument states that the three figures drawn above are all of equal nature — mechanically equivalent despite their differing arrangements. This is the claim the rest of the page sets out to demonstrate.
First balance a b: why 6 at c requires 18 at a
The balance a b has six spaces of lever against one of counter-lever. Because the weight of the whole lever n b acts as if hung at an imagined line at t, a load of 6 sits at c; and since the counter-lever a n fits three times into n t, three times the weight — 18 — is required at the end a to keep the balance level.
Second and third balances reach the same result
The balance m d is of the same substance as the first even though its counter-lever rises upward, since its end is pulled by the 18 pounds at a right angle. The last balance e f likewise gives the same result: its pole o K sustains 24 pounds between the force at e and the weight of the beam o f, matched by 24 on the opposite side of the round balance h K.
Multiplying the beam's weight by its root to reach 36
The demonstration closes with an arithmetical rule: the six weights of the beam are to be multiplied by 6 — called the root (radice) — which yields 36 at the extreme counter-lever. The same trio of numbers, 6, 18, 36, is written against every figure (b t n a on the first, m r u d c on the second).
The three lever-and-pulley drawings
Three horizontal lever beams are drawn one above another across the upper half of the sheet, each running to a pulley or wheel from which a labelled weight hangs. Faint concentric circles beside the first figures suggest the round balances the text names h K.