Balances and the excess of weights along the beam
A sequence of balance schemes reasoned through excess, doubling and proportional ratios
The verso continues the study of balances through a series of schemes that reason from the 'excess' of one weight over another to fix the spaces of the beam. Successive cases double and treble that excess, and one worked example applies sub-triple and sesquialterate proportions to the arms a, b, c and n. A red-outlined trapezoidal figure and several small beam diagrams accompany the fractional tables.
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Balance: the excess of one weight over another
The first scheme reasons that because 2 has an excess of 1 over 1, adding 1 to 3 makes 4; and since 4 has an excess of 3 over 1, then 3 against 1 will be the spaces of the balance. The argument converts a difference of weights into the lengths of the beam's arms.
Excess trebled along the beam
In a later case, because 4 over 1 has an excess of 3, the excess of 1 is trebled with respect to the first term above. The scheme (1 10 / 1 9 / 4 / 4) shows how enlarging the excess rescales the balancing spaces. It extends the doubling rule of the preceding figure.
Sub-triple and sesquialterate proportions (a, b, c, n)
Because n b is one third of a n, the note takes 4/6 of a and sets it against 12/3 of b; and because n c is in sesquialterate ratio with a n, it takes 2/6 of a against 3/3 of c. The reason given is that the excess of 4 over 1 is 3, so n b goes into a b three times. The worked ratios convert lengths on the beam into fractional weights.
Triangular balance scheme
A further small diagram, queried in the transcription as a triangular balance, is set out with the numbers 14 2 / 1 4 2 / 2 / 5. It tests the same excess reasoning on a three-armed arrangement. The figure is one of several compact schemes crowding the sheet.