Suspended rods: squaring the simple rod for its load
144 pounds from 12 x 12; upper supports and unequal angles bear more weight.
The leaf continues the science of weights with a rod e d held by a cord and a hatched support, followed by two smaller beams propped between supports at the foot of the page. Applying the rule of the sixth book on composite versus simple rods, Leonardo squares the simple rod (12 x 12 = 144), so a 12-pound rod suspended by cord a e demands 144 pounds at a. He warns against averaging: the middle line b d is not simply half-way between 6 and 144, because the lever b e must be worked against the rod's length. Two further notes show that a beam slung between an upper and lower support gives the upper support more than its natural weight as the support's angles grow more unequal and it sits nearer the lower one.
On this page
Square the simple rod to find the cord's load
By the rule of the sixth book on the composite rod against the simple, multiply the simple rod by itself. If o e goes 12 times into e d, and rod e d is 12 pounds and 12 braccia, then 12 x 12 = 144, so cord a e must carry 144 pounds to hold e d suspended.
Why the middle line is not the average load
One might think the middle line b d bears half-way between the 6 pounds of line c d and the 144 pounds of line a d, i.e. 69 pounds. Not so: this rule fails, because you must work the lever of line b e against the length of the simple rod e d.
More unequal angles load the support more
A beam whose ends hang between an upper and a lower support of equal height gives the upper support more than the beam's natural weight in proportion as the support's line departs from the beam at more unequal angles.
Upper support nearer the lower bears more
The upper support will bear the more weight the more unequal the angles it forms and the nearer it is placed to the lower support.