What if I were to attach the chain from the saw to the sprocket in the centre (by the pedals) and then chain that to the rear sprockets?
And that, my friend, is the basis of the venerable shift kit.0
HOWEVER, one must be very careful of the sizes of
the drive sprockets and the driven sprockets.
One wrong calculation and the clutch would disintegrate
without nary any movement of the bicycle's rear wheel.
Basically driven sprockets (usually large) are divided by the
drive sprockets (usually much smaller).
The rare exception is where the drive sprocket is in the centre (by the pedals).
Normally on pedal bicycles, the drive sprocket in the centre is larger than the driven sprocket on the rear wheel.
This setup decreases the overall gear ratio.
If the pedal sprocket has 36 teeth and the rear wheel sprocket has 18 teeth, then 18/36 = .5:1.
If the gear ratio is 6:1 when reaching the pedal sprocket,
it'll total 6 x .5 = 3:1 at the rear sprocket.
In this case, your clutch would disintegrate and the bike would never move under its engine power.
However, if you had 18 teeth at the pedal sprocket and 36 teeth,
then you'd have 6 x (36/18) = 12:1.
Your bike would run sluggishly under engine power
and still burn out the clutch.
Now if you had 12:1 upon reaching the pedal sprocket,
then you'd have 24:1 at the rear sprocket.
Your engine would love these gears and scream at top end.
The trick is to attain 12:1 gearing, before you chain it to
the sprocket in the center.
Sooo, a 12-tooth engine sprocket linked to a 24-tooth jackshaft would yield 2:1.
Then a 9-tooth jackshaft chained to a 54-tooth sprocket in the centre gets 6:1.
2 x 6 = 12:1.
36/18 = 2:1.
12 x 2 = 24:1.
FYI, you want as small a jackshaft sprockets as you possibly can.
Better a tiny buzz saw at your ankle than a full-size buzz saw blade.