Feb. 8, 1913 
FOREST AND STREAM 
177 
The Toxopholist’s Paradox 
By E. J. RENDTORFF 
A PARADOX is a statement of fact appar¬ 
ently opposed to common sense. Why is 
it that an arrow will fly along its aimed 
course instead of several degrees toward the left, 
when it is held on the side of the bow with the 
string pushing it further to the left as the nock 
approaches the bow? The fact that it does fly 
true, instead of shifting radically off its course, 
as we would naturally expect, constitutes the 
toxopholist’s paradox. 
Ford, Butt and Walrond have explained’ 
this phenomenon to us, and for several decades 
their ideas have been accepted and passed un¬ 
challenged. It takes some little nerve for a 
novice to state publicly his unbelief in the 
classical explanation. In order to explain my 
heresy I shall attempt to discuss the various 
phenomena that enter, give my conclusions, 
and then let our archery friends act as the jury. 
gravity an accelerated motion toward the left. 
The inertia of the arrow thus produces the 
force C, which tends to prevent the angular 
displacement. 
This force C becomes constantly greater 
until the arrow leaves the string, because, first, 
the arrow accelerates in velocity; and second, 
the angle of displacement constantly increases. 
The force C produces the two balancing forces 
B against the bow, and A against the string. 
While the center of gravity lies within the 
bow, the force C will produce a slight bending 
of the arrow, with the concave side toward the 
left. It is the force B which prevents the arrow 
from falling away from the bow, and not a 
problematical twisting of the string, as some of 
our English authorities have stated. 
In Figure 2 are illustrated the conditions 
applying when the center of gravity of the 
string approaches the bow. When the nock 
leaves the string, with the customary clearance 
of about 7 inches, the force C would be about 
10 inches beyond the bow for my 29-inch arrows. 
The equation for equilibrium Would be 
GE 10 
AXNE=CXGE, or Force A =-C = — C 
NE 7 
At this moment the angle D N G is in¬ 
creasing at its most rapid rate, while the ve¬ 
locity of the arrow has reached its greatest 
value. In other words, the force C is now at 
its maximum value, while the force A is 10/7 
times larger than C. What happens? 
Effect 3. The force A moves the bow 
string toward the left, thus relieving the bend¬ 
ing of the arrow and decreasing the shifting of 
the arrow out of its aimed course. 
These three effects are operative, but are 
they sufficient to explain the paradox? If they 
were sufficient it would follow that a bow held 
rigidly in a frame, with the arrow released 
either by hand or by some trigger contrivance, 
would project the arrow at least as well as 
when shot by an archer. Let us see what Dr. 
Figure i represents an arrow full drawn 
and ready for release. The center of gravity of 
the arrow is at G. On release the force F ex¬ 
erted through the agency of the string moves 
the arrow forward. The arrow starts from rest 
and accelerates in velocity until it leaves the 
string, after which it travels with a uniform 
velocity, except as it is slowed up by the fric¬ 
tion of the air. 
Now, every body at rest has a tendency to 
remain at rest. This tendency is called inertia. 
The existence and action of inertia can be illus¬ 
trated as follows: 
Attach two strings with a breaking strength 
of about eleven pounds to the top and bottom 
of a ten-pound weight. Suspend the weight 
from one string, with the other one hanging 
loosely below. A slowly acting force of but 
one pound exerted on the lower string will 
suffice to break the upper one; but if the lower 
string be jerked rapidly the weight will remain 
hanging while the lower string breaks. 
Effect I. When the arrow is released and 
the bow string moves forward, the angle D N G 
constantly increases. This gives the center of 
arrow has passed the bow. The forces C and 
B remain in direction as before, but they have 
increased in magnitude. The force A is zero 
when G is opposite the bow, and then changes 
in direction and increases in magnitude until 
the nock leaves the string. The force C now 
bends the arrow toward its aimed course, viz., 
with the concave side of the flexure toward the 
right. This corresponds exactly with the state¬ 
ments of Walrond, Ford and Butt, with the 
following exceptions: They do not mention the 
forces A and C, but assume the existence of B 
only (which Ford calls a blow) and ascribe to 
it the various bendings of the arrow. Now, 
as the force B always pushes the arrow directly 
against the bow, no turning moment can exist 
and the force is therefore not operative in any 
of the motions of the arrow as described. 
Effect 2. They also mention the effect of 
the feathers throwing the nock to the left, as 
the feathers pass the bow, thus helping the 
alignment of the shaft. 
Giving our English friends full credit for 
the two reasons so far deduced, let us proceed 
with our inquiry. The force A increases as the 
Elmer has to say about the subject. I shall 
quote from a recent letter: “What do you 
think of a mechanical support and trigger for a 
bow as a method for testing arrows? I made 
one. and it gave results that were very surpris¬ 
ing. For instance, the arrows flew rather wild 
and showed a very marked tendency to buckle. 
What astonished me still more was that they 
veered off to the left. This made it seem likely 
that ordinarily the bow is slightly pushed to the 
right by the arrow.’’ 
Although we have already added one new 
reason to those formerly accepted as sufficient, 
we are apparently not yet entirely on the cor¬ 
rect track, so let us continue the study of our 
force diagram. 
Effect 4. From the instant the center of 
gravity of the arrow leaves the inside of the 
bow until the nock leaves the string, the force 
A offers a considerable turning moment, tend¬ 
ing to turn the plane of the string and bow to¬ 
ward the left. The magnitude of this turning 
moment is the product of the force A multi¬ 
plied by tbe distance N E. The distance N E 
varies from 17 to 7 inches, but as it decreases. 
