Extending a proportion table

So far, we have rearranged proportion tables, we have deleted rows and/or columns, and we have cut proportion tables into smaller pieces.

Now we want to make larger tables.

One way to make a larger table is simply to duplicate a row,

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(4, 8, 10, 18),(6, 12, 15, 27)] `.

or a column,

`\qquad\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] \qquad->\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] `.

or both :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9, 5),(4, 8, 10, 18, 10),(4, 8, 10, 18, 10),(6, 12, 15, 27, 15)] `

Of course we can duplicate any row or column as many times as we like.  And we can do that to many distinct rows and many distinct columns.

Another way to augment a proportion table is by appending a row that is a multiple of another row.   A multiple of one row is necessarily a multiple (usually by different multipliers) of every row :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27),(60, 120, 150, 270)] `.

It should be clear that the result of appending a multiple row will always still be a proportion table, and mutatis mutandis for columns.

One can always append a column that is a sum of two or more other columns (and mutatis mutandis for rows) :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9, 5+9),(4, 8, 10, 18,10+18),(6, 12, 15, 27,15+27)] `,

`\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad =  \qquad [(2, 4, 5, 9, 14),(4, 8, 10, 18, 28),(6, 12, 15, 27, 42)]`.

More generally, the result of appending a sum or difference of existing  rows or of columns is another proportion table.

In the last posting, we dealt with subtables and amalgamation.  Amalgamating two rows is equivalent to appending their sum, and deleting the original two rows, and similarly for amalgamating two columns. Any amalgamation whatever of a table (i.e. over a tartanlike decomposition) can be constructed by a suitably chosen succession of amalgamations of pairs of rows and pairs of columns.