Balancing unequal arms by inverse proportion
The greater arm lightened by the lesser, with a worked subsesquialtera calculation
Leonardo opens with a methodological aphorism: a rule confirmed by two different reasonings and experiences is called general. He then treats the balance whose half-arms stand in inverse proportion to their powers, showing on a graduated beam that n S is double a n so the weight 8 is double 4. A detailed calculation follows for a lesser arm of 2 libbre opposing 6 libbre: reduced through subtriple nearness to the fulcrum, the greater arm is left at 5 and 1/3 libbre, converted to sixteen thirds and, by subsesquialtera proportion, matched by 8 whole placed at f. A closing rule explains how to load the end of the smaller arm to resist the descent of the larger.
On this page
When a rule becomes general
A brief statement of method opens the page: when a rule is confirmed by two different reasonings and experiences, that rule is then called general. It frames the balance demonstrations that follow.
Intersection of inverse proportions on the beam
So much of the greater arm is taken as equals the distance and the weight of the lesser arm, and the result is found by the intersection of the inverse proportions, as shown for the simple balance. Since n S is double a n, conversely the weight 8 will be double 4.
Reducing 6 libbre to 5 1/3 and matching it with 8
With the lesser arm at 2 libbre and its opposite at 6, the middle of the lesser lies at subtriple nearness to the fulcrum, so it is balanced by a subtriple weight and takes 2/3 of a libbra from the greater, leaving 5 and 1/3. Making thirds gives 16 thirds; because f m to m n stands in subsesquialtera proportion, the number in that proportion to 16 thirds is 24, that is 8 whole, which placed at f stands against 5 and 1/3.
Loading the smaller arm to resist the greater
To give the end of the lesser arm enough weight to resist the descent of its greater opposite, first subtract the weight of the lesser arm from the greater by the rule given. Then place at the end of the lesser arm whatever weight resists the load remaining to the greater arm.