Fowler's Binary and Ternary Tables
One of
Thomas Fowler’s
many professions was as Treasurer of the Poor Law Union. This was an
organization that collected fees (establishment
charges) from each parish in the Union to aid the poor in the community.
The amount of the fee was proportional to the value (assessment) of the
parish. Thus, Fowler had to calculate the value of each parish relative
to the Union as a whole, and then knowing the overall fees to be collected
by the entire Union, calculate the fraction that each parish owes.
This task was made much more complicated by the pre-decimal English currency (abbreviated as £sd after the symbols for pounds, shillings and pence). There were twenty shillings to a pound, twelve pence to a shilling and four farthings to a penny. When calculating proportions as Fowler needed to do, each monetary value would be converted into farthings before doing any calculations. With 960 farthings to the pound, these numbers became very large. The process was tedious, but decimal British currency was still over 130 years away.
When faced with this huge task, Fowler developed a system which used lower bases to ease his calculating burden. In 1838, he published his Tables for Facilitating Arithmetical Calculations with the subtitle Intended for Calculating the Proportionate Charges on the Parishes in Poor Law Unions. This book contains a table of binary numbers for values from decimal 1 to 130048, and a table of balanced ternary numbers from decimal 1 to 3985807. Fowler also describes in detail how he uses these tables to simplify the particular type of calculations required for Poor Law Unions. Details of Fowler’s method are given here.
A
sample page of the ternary tables is shown at the left. For each decimal
number in the leftmost column, the next two columns identify the powers
of three, both positive and negative, to express that number in balanced
ternary. The column at the far right is used to expand the range of
the table by a factor of 3^6. Each number in the center of the
block of three can be expressed in balanced ternary by adding six to
the value of the indicies shown to the left. The number directly above
and below indicate the range over which this value can be used. For
example, numbers between 1094 and 3280 can be determined by converting
the middle number, 2187, into balanced ternary,
and then looking up the remainder in the leftmost columns of the table,
keeping in mind that the remainder may actually be a negative value.
Fowler’s
publication of his book preceded his calculating machine by two years.
The machine is clearly designed as a mechanical aid to
this particular technique of calculation. Fowler’s technique has
definite advantages for the repetitive calculations required at the Poor
Law Union, but for less repetitive problems, the conversion to and from
balanced ternary might be more difficult than calculating the problem
directly in decimal.