A very interesting figure in Col. C. M. Western's book "The Practical Science of Billiards," is Fig. 18, page 57, which shows graphically how a small error in the direction of the cue ball, when aiming half-ball, makes but little difference in the direction of the cue ball after contact with the object ball. It also shows that when playing fine, or nearly full to follow on, a very small error in the aim, and consequent contact, has a very large and serious influence on the direction and divergence of the cue ball after contact with the object ball. (See also Fig. 15, page 49).
We all know, of course, that among the advantages of the half-ball are: 1, A definite something, i.e., the outer edge of the object ball, to aim at; 2, A definite angle to get into the eye and also into the head (you should be able to imagine and see the half-ball angle over all the billiard table, with your eyes shut), not only to play it, when you can, but also as a standard whereby to judge other angles.
These two advantages are obvious to anyone. Most of us have also been told that when the shot is a half-ball shot, error in aim, if only small, will not much affect the direction of the cue ball after contact. But no one tells us how much, or how little, it will affect it. Our own experience may convince us that a small error in a half-ball loser is not important, but though we get to feel this and to know it, we don't know why—which may not matter—and we don't know how much—which does matter.
These figures 15 and 18 of Col. Western's exactly show the "how much." Anyone can see them and understand them. Some elementary knowledge of trigonometry and geometry is required to understand how they are arrived at, but there are the results in Figs. 15 and 18 for anyone.
Perhaps a brief sketch of Col. Western's method may be of interest, and perhaps some idea of it may be conveyed without bringing in mathematics.
A cue ball, struck in the centre, with medium force, and aimed half-ball at the object ball strikes it and goes off at the half-ball angle—where? and why there? But first let us follow the object ball. Fortunately its direction is easy and universally admitted. A line drawn through the centres of the two balls, and through the point of contact at the instant of contact gives the direction. And this is always true for any contact—fine, full, or medium.
The direction which the object ball, when struck half-ball, makes with the original direction of the cue ball is 30 degrees. This is quite certain, but for those who don't like degrees, I hasten to add that if you imagine you play from six o'clock on a clock face half-ball on the right of an object ball at the centre of the clock, the direction the object ball will take will be 5 minutes to 12. This is so—absolutely, theoretically, and practically; and it is nice to find, sometimes, that theory and practice agree.
But where is the cue ball going? It is hard to say theoretically, and here, fortunately, the practical billiard players step in and help us.
It will be readily admitted that all players recognise certain positions on the table as the half-ball angle. See, for instance, Mr. Mannock's book "Billiards Expounded," or the Badminton billiards, or Charles Roberts's book, "The Complete Billiard Player," or John Roberts's book, "The Game of Billiards." Perhaps we shall all agree that John Roberts is a practical billiard player. We find certain closely approximate half-ball angles duly recognised such as red on spot and cue ball at a top or middle pocket. We know the distance of the spot from the top cushion, the length of cushions, etc (or we can measure these matters), and now practice must hearken to theory.
Theory does not dispute what practical men say are half-ball losers, but with facts and measurements theory can easily and indisputably arrive at the half-ball angle of departure of cue ball after contact. Theory would give it exactly if the practical players could state the circumstances exactly; but the angle varies with the substance of the balls and the strength of the stroke. We all know that the half-ball angle varies slightly with ivory, crystalate and bonzoline balls, but from the different recognised positions of the half-ball and the variations for the nature of the balls, Col.
Western deduces, and rightly, that 35 degrees is a fair average angle for the departure of the cue ball from its original direction, after a half-ball contact with the object ball.
Again, to avoid degrees, I hasten to add that playing halfball from 6 o'clock on a clock face, on the object ball at the centre of the clock, the cue ball would go to nearly about 6 minutes past 12 and the object ball to 5 minutes to 12.
You may say, to include the various classes of balls, front about 5 min. 35 sec. past 12 for ivory, to about 6 minutes past 12 for bonzoline. I grant the angle looks more like 7 or 7½ minutes past 12 to many people, but it is not so. However, as Col. Western shows, we have now the two directions for the two balls after a half-ball contact. As regards the relative distances the balls will travel, we all know that if you hit the object ball pretty full, it will go far and the cue ball will travel but little, and if you hit the object ball fine, it will not travel far, but the cue ball will. And, for most of us, this is sufficient, but Col. Western pursues the subject for those who may be interested to follow it.
Having got the half-ball directions and a given force.
Col. Western, with the assistance of the parallelogram of forces, is able to give two similar semi-circles, on opposite sides of the original line of direction of the cue ball, so that when we know the direction and distance of the travel of the object ball after an impact by the cue ball (and we can always find this direction by a straight line through the centres of the balls at the moment of impact) we can also find the direction and distance of travel of the cue ball on the circumference of the opposing semi-circle.
We see, then, from these diagrams, that, playing halfball, an error of one-eighth diameter of the ball in direction will make but little difference in the direction of cue ball after contact, but one-eighth will make a very great difference when playing fine or full, so the nearer you can keep to half-ball, the less do your errors matter for losers and cannons.
On the other hand, as regards the direction of the object ball, the diagrams, etc., in Col. Western's book show that your error matters least when playing nearly full, and goes on increasing in an increasing ratio, until a very small error in a fine cut has a very serious effect on the direction.
W.