Figures are not only the object of geometric problems
but also an important help for all sorts of problems in
which there is nothing geometric at the outset. Thus, we 
have two good reasons to consider the role of figures
in solving problems.
1.     If our problem is a problem of geometry, we have 
to consider a figure. This figure may be in our imagination,
or it may be traced on paper. On certain occasions,
it might be desirable to imagine the figure without drawing
it. but if we have to examine various details, one 
detail after the other, it is desirable to draw a figure. If 
there are many details, we cannot imaine all of them
simultaneously, but they are all together on the paper.
A detail pictured in our imagination may be forgotten;
but the detail traced on paper remains, and, when we
come back to it, it reminds us of our previous remarks, 
it saves us some of the trouble we have in recollecting our
previous consideration.
2.     We now consider more specially the use of figures in 
problems of geometric construction.
    We start the detailed consideration of such a problem
by drawing a figure containing the unknown and the 
data, all these elements being assembled ads it is prescribed
by the condition of the problem. In order to 
understand the problem distinctly, we have to consider
each datum and each part of the condition separately; 
then we reunite all parts and consider the condition as a 
whole, trying to see simultaneously the various connections
required by the problem. We would scarcely be able
to handle and separate and recombine all those details
without a figure on paper.
    On the other hand, before we have solved the problem
definitively, it remains doubtful whether such a figure
can be drawn at all. It is possible to satisfy the whole
condition imposed by the problem? We are not entitled to
say Yes before we have obtained the definitive solution;
nevertheless we begin with assuming a figure in which 
the unknown is connected with the data as prescribed by
the condition.  It seems that, drawing the figure, we have
made an unwarranted assumption.
    No, we have not. Not necessarily. We do not act incorrectly
when, examining our problem, we consider the 
possibility that there is an object that satisfies the
condition imposed upon the unknown and has, with all the 
data, the required relations, provided we do not confuse
more possibility with certainty. A judge does not act in
correctly when questioning the defendant, he considers the 
hypothesis that the defendant perpetrated the crime
in question, provided that he does not commit himself to this
hypothesis. Both the mathematician and the judge may
examine a possibility without prejudice, postponing their
judgement till the examination yields some definite result
     The method of starting the examination of a problem 
of construction by drawing a sketch on which, supposedly,
the condition is satisfied, goes back to the Greek
geometers.  It is hinted by the short, some what enigmatic
phrase of PAPPUS:  Assume what is required to be done as
already done. The following recommendation is somewhat
less terse but clearer: Draw a hypothetical figure
which supposes the condition of the problem satisfied in all 
its parts.
     This is a recommendation for problems of geometric
construction but in fact there is no need to restrict us to
any such particular kind of problem. We may extend the 
recommendation to all "problems to find" stating it in
the following general form:  Examine the hypothetical 
situation in which the condition of the problem is 
supposes to be fully satisfied. compare 
PAPPUS.
3. Let us discuss a few points about the actual drawing
of figures.
(I)  Shall we draw the figures exactly or approximately,
with instruments or free-hand?
     Both kinds of figures have their advantages. Exact
figures have, in principle, the same role in geometry as
exact measurements in physics; but in practice, exact
figures are less important then exact measurements 
because the theorems of geometry are much more 
extensively verified than the laws of physics. The beginner,
however, should construct many figures as exactly as he
can in order to acquire a good experimental basis
and exact figures may suggest geometric theorems also to the 
more advanced. Yet, for the purpose of reasoning, carefully 
drawn free-hand figures are usually good enough.
And they are much more quickly done. Of course, the 
figure should not look absurd; lines supposed to be
straight should not look wavy, and so-called circles should
not look like potatoes.
    An inaccurate figure can occasionally suggest a false
conclusion, but the danger is not great and we can protect 
ourselves from it by various means, especially by
varying the figure. There is no danger if we concentrate
upon the logical connections and realize that the figure
is a help, but by no means the basis of our conclusion;
the logical connections constitute the real basis [This
point is instructively illustrated by certain well known
paradoxes which exploit cleverly the intentional inaccuracy
of the figure.]
     (II) It is important that the elements are assembled
in the required relations, it is unimportant in which
order they are constructed. Therefor, choose the most
convenient order. For example, to illustrate the idea of 
trisection, you wish to draw two angles, a and b, so that 
a=3b. Starting from an arbitrary a, you cannot
construct b with a ruler and compasses. Therefor, you chose a fairly
small but otherwise arbitrary b and, starting from
b, you construct a which is easy.
   (III) Your figure should not suggest any undue
specialisation. The different parts of the figure should not
exhibit apparent relations not required by the problem.
Lines should not seem to be equal or to be perpendicular
when they are not necessarily so. Triangles should
not seem to be isosceles, or right-angled, when no such 
property is required of the problem. The triangle having
the angles 45,60,75, is the one which, in a  precise 
sense of the word, is the most "remote" from both the 
isosceles, and from the right-angled shape(footnote: If the angles of a triangle are a,b,c and 90~>a>b>c then at least one of the differences 90-a, a-b, b-c is <15 degrees.  unless a=75, b=60, y=45. In fact
   (3(90-a) + 2(a-b) + (b-c) / 6) = 15
this, or a not very different triangle, if you wish to consider
a "general" trinagle.
(IV)   In order to emphasize the different roles of different
lines, you may use heavy and light lines, continuous
and dotted lines, or lines in different colours.  You
draw a line very lightly if you are not yet quite decided 
to use it as an auxiliary lines. You may draw the given
elements with red pencil, and use other colors to emphasize
important parts, as a pair of similar triangles, etc.
(V) In order to illustrate solid geometry, shall we use
three-dimensional models, or drawings on paper and 
blackboard?
     Three-dimensional models are desirable, but troublesome
to make an expensive to buy. Thus, usually, we must be satisfied
with drawings although it is not easy
to make them impressive. Some experimentation with
self-made cardboard models is very desirable for beginners.
It is helpful to take objects of our everyday surroundings
as representations of geometric notions. Thus, a box, a tile,
or the classroom may represent a rectangular parallelepiped,
a pencil, a circular cylindar, a lampshade, the frutsum of a
right circular cone, etc.
4.    Figures traced on paper are easy to produce, easy to 
recognize, easy to remember. Plane figures are especially
familiar to us, problems about palne figures especially
accessible. We may take advantage of this circumstance,
we may use our aptitude for handling figures in handling
non-geometrical objects if we contrive to find a suitable
geometrical representation for htose nongeometrical
objects.
     In fact, goemetrical representations, graphs and diagrams
of all sorts, are used in all sciences, not only in 
physics, chemistry and the natural sciences, but also in
ecconomics, and even in psychology. Using some suitable
gometrical represenation, we try to express everything
in the language of figures, to reduce all sorts of problems
to problems of geometry.
     Thus, even if your problem is not a problem of geometry,
you may try to draw a figure. To find Euclid geometric
representation for your non-geometrical problem
could be an important step toward the solution.