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© Copyright 2001, Jim Loy
When you thin cut a ball on the head spot
into one of the closest corner pockets, where does the cue ball go? It goes
right into the other corner pocket, right? Very close, actually the center of
the cue ball ends up a ball's width from the center of the pocket (depending on
how accurate your cut shot was). Depending on the tightness of the pocket, you
may have scratched or not. The cut in the diagram on the left should scratch.
The cue ball rolls right over the spot.
What other thin cuts also scratch in the same way? The
answer to that is "Any time the object ball is on the gold colored circle in
the right diagram." These thin cuts all scratch. The principle comes straight
from Euclid's Elements (written about 300 BC).
If the object ball is inside the circle, then the cue ball hits the end rail. And if the object ball is outside the circle, then the cue ball hts the side rail. And if your cuts aren't so thin, then follow or draw curve the cue ball into or out of the pocket. On a normal spot shot, the cue ball has natural follow, and it curves after hitting the object ball, and ends up hitting the end rail. The circle in the diagram can be your guide to your choice of whether to use follow or draw.
There are identical circles for scratching in the side pockets. And there are larger circles (twice as big) for scratching in the far corner pockets. There are similar circles for banks. But you seldom thin cut a bank, and so you normally have plenty of opportunity to avoid scratches on these shots, with follow or draw. Of course you may misjudge your cue ball path after a bank (or any shot, for that matter) and still scratch.
Euclid's Theorem III-31 (Book III, proposition 31) states that an angle inscribed in a semicircle is a right angle. The principle that we are using here is the converse of that, that the right angle is on the semicircle. This follows from the proof of proposition 31. The path of a cue ball after contacting an object ball (and before it curves) is at right angles to the path of the object ball. If we can ignore the curve (either because of a thin hit or because of high speed), cut shots obey this theorem of Euclid's. This theorem is also called Thales' Theorem. Thales lived about 300 years before Euclid.