1831.] 
An Essay on the Game of Billiards. 
117 
ties of their surfaces 4 ; while the centre of the ball not being in the line of direc- 
tion, the equilibrium of its parts is destroyed by the stroke, in a ratio expressed 
by the radius of a concentric circle, in the last Fig. to which it is a tangent ; and 
the ball, obedient to its vis inertia, undergoing less change by avoiding than sub- 
mitting to the violence that would necessarily ensue in a contention between the 
impetus on one side, and its own weight and resistance from the air, on the other ; 
turns backward round its centre, presenting the parts immediately behind, in 
quick succession to the cue’s surface. For all round bodies can with less force 
be turned about their centre, than driven in the progressive only ; because, in the 
first case, there is no resistance from the air, except what arises from attrition in 
its rotation — the ball still occupying the same place ; and the opposition from the 
vis inertia also, is very much diminished; but locomotion being an inseparable 
character of the second, the impediments are more considerable. 
This will appear in a stronger light, by extending the analysis a little further. 
Thus (see the line a, b, and its parallels, Fig. 5.) that part of the ball at a, which 
receives the impression, cannot move in the progressive only, otherwise than by 
forcing all the rest of which the body consists, to pass in conjunction with it, as 
it moves to b, through equal spaces, in equal times ; whereas, by turning round 
its centre it describes in the periphery of the ball, an arc of equal extent, (see a, 
b, Fig. 6.) while every part within the circle, or without its plane, whether the 
motion be concentrical or parallel, is carried with an inferior degree of velocity, 
in proportion to its distance from the axis on which it turns ; and may be ex- 
pressed by the sector, or mixed triangle a, b, c, constructed by lines radiating from 
the centre of the ball, to the extremities of the arc it describes ; each division 
of the intercepted portion, representing, in arithmetical progression, a consti- 
tuent particle. It also appears, by drawing the line of bisection c, d, and invert- 
ing the position of one half, viz. a, d, c, into c, e, b , a quadrilateral figure c, e, 
b, cl, is formed, the breadth of which, d, b, being equal to half e, b, is a mean 
proportional to the sum of its motion ; but the vis inertia excited, is as its 
motion, therefore the resistance is but half also. Thus, by obeying the impelling 
power, in moving through a space corresponding with the force impressed, and 
at the same time avoiding the difficulties opposed, by withdrawing from them ; a 
conflict between Nature and Art is most effectually prevented, while rotary motion 
seems to be employed as the means. 
Or, should a ball be struck above the middle, in a direction below the point 
resting on the table, as is represented (Fig. 7.) a similar effect may be produced. 
Here the progressive motion is impeded ; and a line drawn between both parts of 
contact, a lever of the 2d kind, in the order of mechanics, viz. the point touching 
the table, the fulcrum, that portion of the ball perpendicularly beyond it, the 
weight to be overcome, and at the part struck, the power, which, being in direct 
proportion with the distance between the line of direction and the fulcrum, con- 
sequently (if similarly applied) increases, as it is advanced towards that point 
of the semicircle to which it is perpendicular, as a ; where, after traversing an 
arc of ninety degrees, it loses its name — in other respects being still the same ; 
and as it passes beyond this boundary, (supposing the stroke made from the op- 
posite side,) acquires that of walking. But, when a ball is struck in this manner, 
to make it twist, as it cannot obey the impulse directly, it is apt to elude the effect 
hy reflection from the table ; and besides the inconvenience from attitude, the por- 
tion of the cue before the sight, is not only frequently very small, but to the view, 
4 It is best formed for this purpose, by sloping it, and usually prepared by chalk , * 
as a substance that does least injury to the cloth. 
