" look at the unknown" This is old advice; the corresponding
Latin saying is "respice finem."  That is, look a
the end. Remember your aim. Do not forget your goal.
Think of what you are desiring to obtain. Do not lose
sight of what is required. Keep in mind what you are working for. 
Look at the unknown.  Look at the conclusion. 
the last two versions of "respice finem" are specifically
adapted to mathematical problems, to "problems
to find" and to "problems to prove" respectively.
     Focusing our attention on our aim, and concentrating
our will on our purpose, we think of ways and means to
attain it.  What are the means to this end?  How can you 
attain your aim?  How can you obtain a result of this
kind? What causes could produce such a result? Where
have you seen such a result produced? What do people 
usually do to obtain such a result? And try to think of a 
familiar problem having the same or a similar unknown
and try to think of a familiar theorem having the same
or a similar conclusion. Again the last two versions are 
specifically adapted to "problems to find" and to "problems
to prove" respectively.
1.  We are going to consider mathematical problems,
"problems to find," and the suggestion:  try to think of a 
familiar problem having the same unknown. Let us compare
this suggestion with that involved in the question
Do you know a related problem?
     The latter suggestion is more general than the former
one.  If a problem is related to another problem, the two
have something in common; they may involve a few common
objects or notions, or have some data in common, or 
some parts of the condition, and so on. Our first suggestion
insists on a particular common point: The two
problems should have the same unknown. That is, the 
unknown should be both cases an object of the same
category, for instance, in both cases the length of a 
straight line.
     In comparison with the general suggestion, there is a 
certain economy  in the special suggestion.
     First we may save some effort in representing the problem;
we must not look at once at the whole problem but just at the 
unknown. The problem appears to us schematically, as

"Given............find the length of the line."

     Second, there is a certain economy of choice. Many,
many problems may be related to the proposed problem, 
Infinity have some point or other in common with it, But,
looking at the unknown, we restrict our choice; we take
into consideration only such problems as have the same
unknown. And of course, among the problems having 
the same unknown, we consider first those which are the
most elementary and the most familiar to us.
     2. The problem before us has the form:
 
     "Given...............find the length of the line."

     Now the simplest and most familiar problems of this
kind are concerned with triangles: Given three constituent
parts of a triangle find the length of a side. 
Remembering this, we have found something that may 
be relevant:  Here is a problem related to yours and solved
before. Could you use it? Could you use its result? In 
order to use the familiar results about triangles we must
have a triangle in our figure. Is there a triangle? Or
should we introduce one in order to profit from those
familiar results?  Should you introduce some auxiliary
element in order to make their use possible?
     There are several simple problems whose unknown is
the side of a triangle.(they differ from each other in the
data; two angles may be given and one side, or two sides
and one angle, and the position of the angle with respect
to the given sides may be different. Then, all these problems
are particularly simple for right triangles.)  With
our attention riveted upon the problem before us, we try
to find out which kind of triangle we should introduce,
which formerly solved problem (with the same unknown
as that before us)  we could most conveniently adapt to 
our present purpose.
     Having introduced a suitable auxiliary triangle it may
happen that we do not know yet three constituent parts
of it.  This, however, is not absolutely necessary; if we
foresee that the missing parts can be obtained somehow
we have made essential progress, we have plan of the 
solution.
3.     The procedure sketched in the foregoing (under 1 and 2)
is illustrated, essentially, by section 10. the illustration 
is somewhat obscured by the slowness of the students. It is 
not difficult at all to add many similar examples. In fact, 
the solution of almost all "problems to find" usually 
proposed in less advanced classes can be started by proper 
use of the suggestion:and try to think of a familiar 
problem having the same or similar unknown.
     We must take such problems schematically,
and look at the unknown first:
(1) Given.......find the length of the line.
(2) Given.......find the angle
(3) Given.......find the volume of the tetrahedron.
(4) Given.......construcst the point.

     If we have some experience in dealing with elementary
mathematical problems, we will readily recall some simple
and familiar problem or problems having the same
unknown. If the problem proposed is not one of those
simple familiar problems we naturally try to make use of 
what is familiar to us and profit from the result of those
simple problems. We try to introduce some useful well-known 
thing into the problem, and doing so we may get a
good  start.
     On each of the four cases mentioned there is an obvious
plan, a plausible guess about the future course of the
solution.
(1)   The unknown should be obtained as a side of 
some triangle. It remains to introduce a suitable triangle
with three known, or easily obtainable, constituents
(2)   The unknown should be obtained as an angle in
some triangle. It remains to introduce a suitable triangle
(3)   The unknown can be obtained if the area of the 
base and the length of the altitude are known. It remains
to find an area of a face and the corresponding
altitude.
(4)   The unknown should be obtained as the
intersection of two loci each of which is either a circle or a 
straight line. It remains to disentangle such loci from the 
proposed condition.
     In all these cases the plan is suggested by a simple
problem with the same unknown and by the desire to
use its result or its method. Pursuing such a plan, we may
run into difficulties, of course, but we have some idea to 
start with which is a great advantage.
    4. There is no such advantage if there is no formerly 
solved problem having the same unknown as the 
proposed problem. In such cases, it is much more difficult to
tackle the proposed problem.
     "Find the area of the surface of a sphere with given
radius."  This problem was solved by Archimedes. There
is scarcely a simpler, problem with the same unknown
and there was certainly no such simpler problem of which
Archimedes could have made use. In fact, Archimedes
solution may be regarded as one of the most notable 
mathematical achievements.
    "find the area of the surface of the sphere inscribed
in a tetrahedron whose six edges are given."  If we know
Archimedes' result, we need not have Archimedes' genius
to solve the problem; it remains to express the radius
of the inscribed sphere in terms of the six edges of the 
tetrahedron. This is not exactly easy but the difficulty
cannot be compared with that of Archimedes' problem.
     To know or not to know a formerly solved problem
with the same unknown may make all the difference between
 an easy and a difficult problem.
5. When Archimedes found the area of the surface of
the sphere he did not know, as we just mentioned, and
formerly solved problem having the same unknown. But
we knew various formerly solved problems having a similar
unknown. There are curved surfaces whose area is
easier to obtain than that of the sphere and which were
well known in Archimedes' time, as the lateral surfaces
of right circular cylinders, of right circular cones, and of
the frustums of such cones.  We may be certain that
Archimedes considered carefully these simpler similar
cases. In fact, in his solution, he uses an approximation
to the sphere a composite solid consisting of two cones
and several frustums of cones (see 
DEFINITION, 6).
     If we are unable to find a formerly solved problem having
the same unknown as the problem before us, we try to
find one having a similar unknown. Problems of the latter
kind are there less closely related to the problem before us 
than problems of the former kind and therefore less
easy to use for our purpose in general but they may
be valuable guidelines nevertheless.
6.   We add a few remarks concerning "problems to 
prove";  they are analogous to the foregoing more extensive
comments on "problems to find."
     We have to prove (or disprove) a clearly stated theorem.
Any theorem proved in the past which is in some 
way related to the theorem before us as a chance to be 
of some service. Yet we may expect the most immediate
service of theorems which have the same conclusion as
the one before us. Knowing this, we look at the 
conclusion  that is, we consider our theorem emphasizing
the condlusion. our way of looking at teh theorem can be 
expressed in writing by a scheme as:
 
      "If..................then the angles are equal."

     We focus our attention upon the conclusion before us
and try to think of a familiar theorem having the same 
or a similar conclusion. Especially, we try to think of very
simple familiar theorems of this sort.
     In our case, there are various theorems of this kind and
we may recollect the following: "If two triangles are congruent 
the corresponding angles are equal." Here is a 
theorem related to yours and proved before.  Could you
use it? Should you introduce some auxiliary element in 
order to make its use possible?
     Following these suggestions, and trying to judge the
help afforded by the theorem we recollected, we may
conceive a plan: Let us try to prove the equality of the
angles in question from congruent triangles. We see that
we must introduce a pair of triangles containing those
angles and prove that they are congruent.  Such a pan is
certainly good to start the work and it may lead eventually
ally to the desired end as in section 19.
7.     Let us sum up. Recollecting formerly solved problems
with the same or similar unknown (formerly
proved theorems with the same or similar condition)
we have a good chance to start if the right direction 
and we may conceive a plan of the solution. In simple cases,
which are the most frequent in less advanced classes, the 
most elementary problems with the same unknown (theorems
with the same conclusions) are usually sufficient.
Trying to recollect problems with the same unknown is
an obvious and common-sense device (compare what was 
said in this respect in section 4). It is rather surprising
that such a simple and useful device is not more widely
known; the author is inclined to think that it was not
even stated before in full generality. In any case, neither 
students nor teachers of mathematics can afford to ignore
the proper use of the suggestion Look at the unknown! 
And try to think of a familiar problem having the same 
or a similar unknown.