Auxiliary problem is a problem which we consider, 
not for its own sake, but because we hope that its
consideration may help us to solve another problem, our original problem.
The original problem is the end we wish to attain, the auxiliary problem 
a means by which we try to attain our end.
An insect tries to escape through the windowpane,
tries the same again and again, and does not try the next
window which is open and through which it came into
the room. A man is able, or at least should be able, to act
more intelligently. Human superiority consists in going 
around an obstacle that cannot be overcome directly, in
devising a suitable auxiliary problem when the original 
problem appears insoluble. To devise an auxiliary problem
is an important operation of the mind. To raise a 
clear-cut new problem subservient to another problem, 
to conceive distinctly as an end what is means to another
end, is a refined achievement of the intelligence. It is an
important task to learn (or to teach) how to handle 
auxiliary problems intelligently.
1.Example. Find x, satisfying the equation
  x^4-13x^2+36=0
If we observe that x^4=x^2^2 we may see the advantage of introducing
         
         y=x^2
We obtain now a new problem: Find y, satisfying the 
equation
          y^2-13y+36=0
The new problem is an auxiliary problem; we intend to
use it as a means of solving our original problem. The 
unknown of our auxiliary problem, y, is appropriately
called auxiliary unknown  .
2.  Example Find the diagonal of a rectangular parallelepiped
being given the lengths of the three edges drawn from the same corner
Trying to solve this problem(section 8) we may be 
led, by analogy (section 15), to another problem:
Find the diagonal of a rectangular parallelogram being
given the lengths of two sides drawn from the same
vertex. 
     The new problem is an auxiliary problem; we consider it 
because we hope to derive some profit for the original 
problem from its consideration.
3.   profit.  The profit that we derive from the
consideration of an auxiliary problem may be of various kinds
We may use to the results of the auxiliary problem. 
Thus, in example 1, having found by solving the quadratic equation
for y that y is equal to 4 or 9 we infer that x^2=4 or x^2=99 and 
derive hence all possible values of x.
In other cases, we may use the method of auxiliary
problem. Thus in example 2, the auxiliary problem is a 
problem  of plane geometry; it is analogous but simpler than,
the original problem which is a problem for it. In solid geometry it
is reasonable to introduce an auxiliary
problem of this kind in hope that it will be instructive
that it will give us an opportunity to familiarise our-
selves with certain methods, operations, or tools, which
we may use afterwards for our original problem.  In ex-
ample 2, the choice of the auxiliary problem is rather
lucky; examining it closely we find that we can use both
its method and its result. )(See section 15, and 
DID YOU USE ALL THE DATA?.)

4.    risk. We take away form the original problem the 
time and the effort that we devote to the auxiliary problem. 
If our investigation of the auxiliary problem fails
the time and effort we devoted to it may be lost. There-
for, we should exercise our judgement in choosing an
auxiliary problem. We may have various good reasons
for our choice. The auxiliary problem may paper 
more accessible than the original problem; or it may appear
instructive; or it may have some sort of aesthetic appeal
Some times the only advantage of the auxiliary y problem
is that it is new and offers unexplored possibilities; we 
chose it because we are tired of the original problem
all approaches to which seem to be exhausted.
5.   How to find one. The discovery of the solution of 
the proposed problem often depends on the discovery of
a suitable auxiliary problem. Unhappily there is no in
fallible method of discovering suitable auxiliary problems
as there is no infallible method of discovering the 
solution. There are, however, questions, and suggestions 
which are frequently helpful, as 
LOOK AT THE UNKNOWN. We are often led to useful auxiliary problems 
VARIATION OF THE PROBLEM.
6.      Equivalent problems. Two problems are  equivalent 
if the solution of each involves the solution to the other.
Thus in our example 1., the original problem and the
auxiliary problem are equivalent.
Consider the following theorems.
A. In any equilateral triangle, each angle is equal to 60 degrees.
B. In an equilateral triangle, each angle is equal to 60 degree.
These two theorems are not identical. They contain different notions; one is concerned with equality of the
sides, the other with equality of the angles of a triangle.
But each theorem follows from the other. Therefor, the 
problem to prove A is equivalent to the problem to 
prove B.   if we are required to prove A, there is a certain advantage
 in introducing, as an auxiliary problem, the problem 
to prove B. The theorem B is a little easier to prove
than A and, what is more important, we may foresee that B is easier than A, we may judge so, we may find
plausible from the outset that B is easier than A. In fact
the theorem B, concerned only with angles, is more
"homogeneous"than the theorem A which is concerned
with both angles and sides.
     The passage from the original problem to the auxiliary problem
is called convertible  reduction, or bilateral reduction, 
or equivalent , reduction if these two problems, the original and
the auxiliary and equivalent thus, reduction of A to B(see above)
is convertible and so is the reduction in example 1. Convertible
reductions are, in a certain respect, more important and more
desirable than other ways to introduce auxiliary problems,
but auxiliary problems which are not equivalent to the original problem may
also be very useful; take example 2.
7 .    Chains of equivalent auxiliary problems. are frequent
in mathematical reasoning. We are prepared  to solve a problem A;
we cannot see the solution, but we may find that A is equivalent
to another problem B. Considering B we may run into a third
problem C equivalent to B. Proceeding in the same way, we 
reduce C to D and so on until we come upon a last problem L
whose solution is known or immediate. Each problem  being 
equivalent to the preceding , the last problem l must be 
equivalent to our original problem A, thus we are able
to infer the solution of the original problem A from the 
problem L which we attained as the last link fin a chain
of auxiliary problems.
     Chains of problems of this kind were noticed by the 
Greek mathematicians as we may see from an important
passage of PAPPUS. For  an illustration, let us reconsider
our example 1. Let us call (A) the condition imposed upon the unknown x
A    x^4 - 13x^2 + 36 = 0
One way of solving the problem is to transform the pro-
posed condition into another condition which we shall
call (B);
(B)       (2x^2)^2-2(2x^2)13 +144=0
Observe that the conditions (A) and (B) are different
They are only slightly different if you wish  to say so,
they are certainly equivalent as you may easily convince 
yourself, but they are definitely not identical. 
The passage from A to B is not only correct but has a clear-
cut purpose, obvious to anybody who is familiar with
the solution of quadratic equations. Working further in the
same direction we transform the condition B into still
another condition C
C)        (2x^2)^2-2(2x^2)13+169=25
Proceeding in the same way, we obtain
D)        (2x^2-13)^2=25
E)        2x^2-13=+/-5
F)        x^2=(13+/-5)/2
G)        x=+/-sqrt((13+/-)5/2)
Each reduction that we made was convertible. Thus, the
last condition H is the equivalent to the first condition
A so that 3, -3, 2, -2 are all possible solutions of our
original equation.
     In the foregoings we derived from an original condition 
a sequence of conditions B,C,D,...
each of which was equivalent to the foregoing. This
point deserves the greatest care. Equivalent conditions
are satisfied by the same objects. Therefore if we pass
from a proposed condition to a new condition equivalent
to it, we have the same solutions. But if we pass from a 
proposed condition to a narrower one, we lose solutions,
and if we pass to a wider one we admit improper,
adventitious solutions which have nothing to do with  the pro-
posed problem If, in a series of successive reductions, we 
pass to a narrower and then again to a wider condition
we may lose track of the original problem completely. In
order to avoid this danger, we must check carefully the
nature of each newly introduced condition : Is it equivalent
to the original condition? This question is still
more important when we do not deal with a single equation
as there but with a system of equations, or when the 
condition is not expressed by equations, as for instance
in problems of geometric construction.
(Compare PAPPUS,especially comments 2,3,4,8. The 
description on p143, lines 4-21 is unnecessarily restricted;
it describes a chain of "problems to find each
of which has a different unknown. The example considered
here has just the opposite speciality: all problems
of the chain have the same unknown and different only in
the form of the condition. Of course no such restriction 
is necessary.)
8.    Unilateral reduction we halve two problems. 
A and B, both unsolved. If we could solve A we could hence
derive the full solution of B. But n not conversely; if we 
could solve B , would obtain, possibly, some information
about A,m but we would not know how to derived the
full solution of A from that of B, in such a case, more is
achieved by the solution of A than by the solution of B
let us call A the more ambitious, and B the less am ambitious 
to he two problems.
     If from a proposed problem, we either pass to a more 
ambitious or to a less ambitious auxiliary problem we
call the step a unilateral reduction there are two kinds
of unilateral reduction, and both are, in some way, or
other more risky than a bilateral or convertible reduction.
      Our example 2 shows a unilateral reduction to a less
ambitious problem. In fact if we could solve the original 
problem, concerned with a parallelepiped, whose length, 
width, and height are a,b,c, respectively, and we could move
on to the auxiliary problem putting c=0 and obtaining
a parallelogram with length a and width b. For another 
example of unilateral reduction to a less ambitions
problem see SPECIALISATION3,4,5.
These examples show that with some luck, we may be able to use a
less ambitious auxiliary problem as a stepping stone combining
the solution of the auxiliary problem with some appropriate
supplementary remark to obtain the solution of 
the original problem
      Unilateral reduction to a more ambitious problem may 
also be successful. (See GENERALISATION,
2 and the deduction of the first to the second problem considered in
INDUCTION AND MATHEMATICAL INDUCTION, 1,2) In fact, 
the more ambitious problem may be more accessible; this 
is the INVENTOR'S PARADOX.