You are correct that the difference between a 3/8 overlap and a 1/3 overlap can cause a miss on long shots. A fullness of hit of 3/8 results in a 38.7° cut, whereas a 1/3 hit is 41.8° (both ignoring throw). So the difference is about 3°, which is enough to cause a miss on long shots.
Your post (and others) prompts me to say more about Poolology.
Firstly, I had never before seen the methodology Brian Crisp presents in Poolology for determining the overlaps needed to pocket balls. It is quite clever, and I consider it a significant contribution to the literature. And the text is well written! Kudos to Mr. Crisp.
Poolology's arithmetic will yield overlaps that are "right on" in some cases and "close enough" in many others. But they are largely approximations to the geometrically correct overlaps or cut angles needed for center pocketing, not precise determinations. Because of pocket slop and cut-induced throw, those approximations seem to work well for a large percentage of shots.
That they are approximations is apparent from the fact that the overhang fraction (and its complement, the overlap fraction) is unchanged in Poolology when the cue ball is moved closer or farther from the OB on the line of centers between the two balls. Yet, with the OB fixed in place, the actual cut angle changes (straight-ins excepted) with each movement of the CB on that line of centers -- moving it closer to the OB increases the cut angle and moving it farther away reduces the cut angle.
Mr. Crisp is well aware of this. He has posted advice on AzB and in a YouTube video (but not in his book) to use a thinner aim point than the arithmetic would indicate, possibly along with some outside spin, when the two balls are quite close together. And, fortunately, the true cut angle reduces slowly with increasing separation when the balls are a decent distance apart, and that makes the method useful ("close enough") over a wide range of shots.
Here's an EXAMPLE to illustrate this change in cut angle. sixpack mentioned a shot for which Poolology arithmetic indicates a 1/3 hit, so I'll use one of those shots.
- • OB -- on the long string (the line that would run the length of the table through the head spot and foot spot) and 3 diamonds off the foot rail.
• CB -- at various positions on the long string
• Target -- one of the corner pockets at the foot of the table.
• 9-foot table.
Poolology's arithmetic says this shot is a 1/3-ball aim regardless of the distance between the balls. One-third is certainly between the quarters aim points, so Mr. Crisp might recommend a 3/8 aim. I don't think his book recommended going to anything finer than eighths. But let's consider both 3/8 and 1/3. As mentioned above, the geometric cut angles with those two hits are 38.7° and 41.8°, respectively. Cut-induced throw will reduce the actual cut angle a bit, and the amount depends upon a number of factors. But let's assume it takes 2° off the cut and that the resulting cut angles we'll end up with are about 37° with the 3/8 hit and 40° with the 1/3 hit.
With a 4½" pocket, and the OB located as specified, we have a margin of error (pocket slop) of about 1½° to each side of center pocket. With the CB on the long string at various distances from the OB, here is the cut angle we actually need to pocket each shot.
• CB ½ diamond away from the OB -- the cut angle needed is about 50° (±1½° with slop). Neither 3/8 nor 1/3 will work; a 1/4 hit (48.6° less an allowance for throw) might even be a bit too thick.
• CB 1 diamond away -- somewhat over 40° ± 1½°. The 3/8 hit (37°) will not work, but the 1/3 (40°) will.
• CB 2 diamonds away -- about 37° ± 1½°. The 3/8 hit works, the 1/3 does not.
• CB 3 diamonds away -- about 36° ± 1½°. The 3/8 hit works, the 1/3 does not.
• CB 4 diamonds away -- a bit over 35° ± 1½°. Neither 3/8 nor 1/3 works.
• CB against the head rail -- a bit under 35° ± 1½°. Neither 3/8 nor 1/3 works.
Now, these numbers just illustrate how this might work; they would change a bit with different amounts of throw. And that could affect whether the 3/8- or 1/3-hit would work, as would a different pocket size. But I think the example shows that "feel" is still important regardless of all the numbers. The cut angle needed for these CB locations changes by about 5½° as the CB moves from 1 diamond away from the OB to almost 5 diamonds away. That 5½° is too much for slop to accommodate using either a 3/8 overlap or a 1/3 overlap. The player's experience is what will allow him to make all those shots consistently rather than just following the numbers perfectly.
A few of my other thoughts on Poolology:
• Mr. Crisp calls the aim points on the quarters the "basic aim points" and those half way between the quarters the "in-between aim points." But 7 cut angles in each direction (plus straight-ins) are not enough for all shots when the OB is far from the pocket. So some tweaking (a little thicker, a little thinner, or maybe some spin) is needed on some shots.
• Even if Poolology's arithmetic produced an absolutely accurate overlap or fractional hit needed to pocket every shot, that still leaves the age-old problem with fractional aiming of applying that fraction accurately, i.e., accurately pointing the stick (finding the shot line) for fractions other than the easily seen 1/2. Mr. Crisp does provide some guidance as to where the center or edge of the tip should be pointed for each of the "eighths" fractions. I believe that the recommendations for 5/8 and 3/8 hits, using the edge of the tip or shaft, are fine for most cue sticks when the OB and CB are fairly close to each other (and appear approximately the same size) but not when they are far apart and the OB appears much smaller in relation to the tip. At long distances between the balls, the 5/8 recommendation will result in an undercut and the 3/8 recommendation will result in an overcut.
• On a YouTube video, for 1/8- and 1/4-ball aims, Mr. Crisp recommends aiming at the edge or center, respectively, of an imaginary piece of chalk butted up to the OB. Several brands of chalk are about 7/8" wide, which is quite close to the actual distance off the edge of the ball for a 1/8 hit (27/32"), so I think that is a good recommendation. But the 1/4-ball aim should be 9/16" off the edge of the OB -- a bit more than the 7/16" to the middle of the chalk.
• I feel the book should have included some of the material Mr. Crisp has presented in videos, particularly the adjustments needed to the arithmetic: (1) when the OB is on or close to the foot or head spot, (2) when the OB is on or close to a rail within about 1½ diamonds of a corner pocket, and (3) when the CB is close to the OB. Perhaps those will be added to a subsequent edition.
• Why does the cover graphic on the book show a half-ball aim resulting in a cut angle of over 40°?
Given those few caveats, I again congratulate Mr. Crisp on a terrific contribution to the material on aiming.