More about proportion tables

Look again at the standard multiplication table :

${\displaystyle \times }$	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

By simply changing the '`xx`' to a `1` we have a proportion table.  The stubs no longer deserve special coloring because they have no special role.  This is generally true.  Any multiplication table can be converted to a proportion table in this way.

Conversely, any '`1`' in a proportion table can be treated as if it is the '`xx`' symbol in a multiplication table.  So, for instance, Hutton gives his multiplication table, which he also says is called Pythagoras's table, in the following form, although his is `12 xx 12`.

1	2	3	4	5	6	7	8	9	10
2	4	6	8	10	12	14	16	18	20
3	6	9	12	15	18	21	24	27	30
4	8	12	16	20	24	28	32	36	40
5	10	15	20	25	30	35	40	45	50
6	12	18	24	30	36	42	48	54	60
7	14	21	28	35	42	49	56	63	70
8	16	24	32	40	48	56	64	72	80
9	18	27	36	45	54	63	72	81	90
10	20	30	40	50	60	70	80	90	100

The '`1`' at the top left can serve as a multiplication sign.  This remains true for any '`1`' entry found in an arbitrary proportion table.

`5`	`8`	`2`	`1200`	`14" hours"`
`10" mph"`	`16" mph"`	`4" mph"`	`2400" mph"`	`28" miles"`
`2 1/2`	`4`	`1`	`600`	`7" hours"`
`22 1/2`	`36`	`9`	`5400`	`63" hours"`
`250`	`400`	`100`	`60000`	`700" hours"`
`20`	`32`	`8`	`4800`	`56" hours"`
`5/6`	`1 1/3`	`1/3`	`200`	`2 1/3" hours"`

It is readily apparent that the quantities directly below and directly to the right of the '`1`' serve as as stubs of a multiplication table, with the body of the table between these two arms, both below and to the right of the '`1`'. But the numbers directly above, and also directly to the left of the '`1`' can also serve as stubs. So, for instance, `8 xx 2 1/2 = 20`. On this view, here one has table bodies in all four quadrants.

It is better, however, to think of entire row containing the '`1`' as a row stub, and the entire column containing the '`1`' as a column stub. Any value in the table has a marker value on the same row in the column stub, and a marker value in the same column in the row stub. The '`1`', the arbitrary value in the table, and the two marker values lie at the corners of a rectangular box.

The four values

`\qquad\qquad\qquad{:(2 1/2\qquad, 1),(20\qquad, 8):}`

can be thought of as a `2 xx 2` subtable, with any value in this subtable equivalent to the product of its adjacent values divided by its diagonally opposite value :

`\qquad\qquad\qquad 20 = \frac{8 xx 2 1/2}{1}`.

In this form, we are using the proportion table version of the Rule of Three. Any `2 xx 2` subtable can serve in this way, and no '`1`' is required in the subtable, or indeed, anywhere in the proportion table :

`5`	`8`	`2`	`1200`	`14" hours"`
`10" mph"`	`16" mph"`	`4" mph"`	`2400" mph"`	`28" miles"`
`2 1/2`	`4`	`1`	`600`	`7" hours"`
`22 1/2`	`36`	`9`	`5400`	`63" hours"`
`250`	`400`	`100`	`60000`	`700" hours"`
`20`	`32`	`8`	`4800`	`56" hours"`
`5/6`	`1 1/3`	`1/3`	`200`	`2 1/3" hours"`

so that, e.g.

`\qquad\qquad\qquad 4" mph" = \frac{28" miles " xx  ""8}{56" hours"}.`

Here, we have used the top left entry of the subtable of the subtable as the target element, and we have continued to follow the convention of writing the same-row neighbor before the same column neighbor.

But we could use of these three same elements to find the fourth.  e.g. if the top right element is our target element, we can write :

`\qquad\qquad\qquad 28" miles" = \frac{4" mph  " xx" "56" hours"}{8}.`