An Example of Calculating With Fowler's Ternary Tables
In Fowler’s book, Tables for Facilitating Arithmetical Calculations, the section "Application of the tables &c" gives an example of his use of the ternary tables to simplify the Poor Law Union calculations. Here Fowler states that the assessment of the whole Union is £7416, and the establishment charges for the whole are £177/2s/1.5d. He also states the individual assessments for several of the parishes. I will use Fowler’s example throughout this explanation.
The problem Fowler needs to solve is: if the individual parish assessments are known, and the assessment for the whole Union is known, given the total establishment charge for the entire Union, how much should each individual parish pay?
Let A = the individual parish assessment (value of the parish)
Let B = the assessment of the whole Poor Law Union (value of the entire
Union)
Let C = the establishment charge the individual parish must
pay (fee for the parish)
Let D = the total establishment charge for the whole Union (total fees
for the entire Union)
Each parish should pay proportionally, according to how large its assessment is. The quantity "A/B" is a measure of how much an individual parish is worth as compared to the whole. This fraction should equal the amount the individual parish pays as compared to the total charges, as measured by the quantity "C/D". In other words, if a parish is worth one-quarter of the value of the whole Union, then it should pay one-quarter of the total establishment charges for the Union.
In mathematical terms,
A/B = C/D
To simplify
the equation and solve for C, multiply both sides of the equation by "D"
C =
D x (A/B)
Due to the difficulties of calculating in the old British monetary system, all monetary values were converted to farthings before any other calculations were done. After the calculations were complete, the values were converted back to £sd.
Here are the steps to perform this calculation using the decimal system:
- convert
each value of "A" to farthings (calculate once for
each parish)
- convert "B" to farthings (single value – only performed
once)
- convert "D" to farthings (single value – only performed
once)
- calculate "A/B" (calculate once for each parish)
- multiply "A/B" times "D" to obtain the value for "C" (calculate
once for each parish)
- convert "C" from farthings to £sd (calculate once for
each parish)
There is a lot of multiplication and division involved, and since there are 960 farthings per pound, calculating in farthings means the numbers are usually very large.
Fowler probably had to perform these calculations repeatedly, so any reduction in the number of steps, or simplification of the individual steps, would be a great help. This is where his ternary tables come into play.
Fowler describes making a chart of the charges that would apply to parishes if they were assessed at exact powers of three (i.e. £1, £3, £9, £27, £81, £243, £729, £2187 etc). He calls this chart the "New Series of Corresponding Value With the Like Terms of the Ternary Scale". To build this chart, Fowler begins with a calculation for a hypothetical parish, which has been assessed at value equaling an exact power of three, in this case, the 6th power of three, or £729 (this should be some suitably large power of three – I believe the choice of which power to use is somewhat arbitrary, larger powers require more calculation to make the initial table, smaller powers may not have enough resolution at the lower end).
Let A =
the hypothetical parish assessment, £729
Let B = the assessment of the whole Union (£7416)
Let C = the amount of establishment charge for a single hypothetical
parish (unknown)
Let D = the total establishment charge for the whole Union (£177/2s/1.5d
or 170022 farthings)
We are
still calculating in decimal, so Fowler uses the equation described
above to calculate
the value of "C" for the hypothetical parish
C = D x (A/B)
The
resulting value of C for an assessment of £729 is 16713.33 farthings,
or £17/8s/2.25d with a remainder of 33/100ths of a farthing.
This number forms the entry in the table for the sixth power of three.
The other values in the chart are formed by successively multiplying
or dividing this value by three. The chart is shown on page xvi, and
you can confirm that £17/8s/2.25d is three times £5/16s/0.75d
which is three times £1/18s/8.25d, and so on.
Given this table, how does one calculate the establishment charges of a parish if its assessment is not an exact power of three?
First, one must convert the assessment value to signed ternary, a task made easier by the fact that the assessments are rounded to the nearest pond. For the Great Torrington parish, assessed at £1207, Fowler’s ternary table shows positive indicies for 7 and 0, and negative indicies for 6, 5 and 2. This is simply a different way of expressing the signed ternary number "+--00-0+". Another way of expressing the same thing:
1207 = 2187 – 729 – 243 – 9 + 1
It then follows that the establishment charges for a parish assessed at £1207 would be as follows:
PLUS the
charge for a parish assessed at £2187 (positive index 7)
MINUS the charge for a parish assessed at £729 (negative index 6)
MINUS the charge for a parish assessed at £243 (negative index 5)
MINUS the charge for a parish assessed at £9 (negative index 2)
PLUS the charge for a parish assessed at £1 (positive index 0)
------------------------------------------------------------
the total will equal the charge for a parish assessed at £1207
The values of the charges for each of the powers of three have already been calculated in the chart on page xvi, shown at right.
On
the top of page xviii, shown below right, the example for the Great
Torrington shows the calculation
of the establishment charges for that parish. Reading the values from
the chart on page xvi, one adds the values for indicies 7 and 0 and
then subtracts the values for indicies 6, 5 and 2. The result is £28/16s/6d
with a remainder of 13/100ths of a farthing. Note that no multiplication
or division is needed, and that the calculations are all performed
directly in £sd.
How much of a simplification is this new system? The advantages are greater as the number of parishes in the Poor Law Union increases. Fowler’s system adds the step of calculating the table shown on page xvi, but once the table is made, the individual establishment charges can be calculated by simple addition and subtraction. Also, the results are already in £sd, so no additional conversion from farthings is needed.
For Fowler’s example of a Poor Law Union with 23 parishes, his ternary system would require:
- 9 conversions
to/from £sd to/from farthings
- 2 divisions of large values
- 6 divisions by 3
- 2 multiplications by 3
- 23 sums (adding and subtracting up to seven terms, once per parish)
Compare to the same example in decimal:
- 48 conversions
to/from £sd to/from farthings
- 23 divisions of large values
- 23 multiplications of large values