The regularity of the increases to knit a flat circle intrigued me
to investigate a bit further, and I realized that the principle
could be stated more succinctly, and perhaps less explosively.
The number of stitches to increase each round (or decrease if
you are knitting bottom up) can be represented by the formula:
----------------- x 2 PI
or stitch gauge divided by row gauge times 2 times PI
In the example of 5 stitches per inch and 7 rows per inch,
the formula yields
5 / 7 X 2 X 3.14159 = 4.487 which we round up to 4.5.
If we decide to increase (or decrease) every other row, then
we would double that and get 9 stitches every other round.
If we wanted a flat circle with a diameter of 24 inches at that
gauge, we can start our calculation with that diameter and work
backwards to get all the figures:
24 inches x 5 stitches per inch = 120 stitches
Repeatedly subtracting 9 stitches every other row yields the sequence:
120, 111, 111, 102, 102, 93, 93, 84, 84, 75, 75, 66, 66,
57, 57, 48, 48, 39, 39, 30, 30, 21, 21, 12, 12
So, you could either start by casting on 120 stitches and decreasing
9 stitches every other row, or start by casting on 12 stitches and
increasing 9 stitches every other row.
I think that's a simpler way to look at it, and less arithmetic to perform.
If you knit a 4 inch (10 cm) swatch in whatever pattern stitch you
like, just plug the number of stitches and rows into the first formula
(doubled) to get the increases (or decreases) for your pattern stitch
in every other round:
Stitches divided by rows times 4 times 3.14159
(or stitches divided by rows times 12.57 )
Next topic: knitting a sphere :)