Definition of a term is a statement of
its meaning in other terms which are supposed to be well
known.
1.      Technical terms in mathematics are of two kinds.
Some are accepted as primitive terms and are not defined.
Others are considered as derived centres and are defined 
in due form; that is their meaning is stated in primitive
terms and in formerly defined derived terms. Thus, we 
do not give a formal definition, or such primitive notions
as point, straight line, and plane(footnote: In this 
respect, ideas have changed since the time of Euclid and
his Greek followers who defined the point, and the straight
line, and the plane. Their "definitions" however are 
scarcely formal definitions, rather intuitive illustrations 
of a sort, illustrations, of course, are allowed, and even
very desirable in teaching.)  Yet we give formal definitions
of such notions as "bisector of an angle" or "circle" or 
"parabola."
     The definition of the last quoted term may be stated as 
follows. We call Parabola the locus of points which are 
at equal distance from a fixed point and a fixed straight
line.  The fixed point is called the focus of the parabola, the fixed 
line its directrix. It is understood that all elements
considered are in a fixed plane, and that the fixed
point(the focus) is not on the fixed line(the directrix).
    The reader is not supposed to know the meaning of
the terms defined: parabola, focus of the parabola,
directrix of the parabola. But he is supposed to know the
meaning of all the other terms as point, straight line,
plane, distance of a point from another point, fixed,
locus, etc.
2.  definitions and dictionaries are not very much different from 
mathematical definitions in the outward form
but they are written in a different spirit.
    The writer of a dictionary is concerned with the current
meaning of words. He accepts of course the 
current meaning and states it as neatly as he can in form
of a definition.
    The mathematician is not concerned with the current
meaning of his technical terms, at least not primarily
concerned with that. What "circle" or "parabola" or 
other technical terms of this kind may or may not denote
in ordinary speech matters little to him. The mathematical
definition creates the mathematical meaning.  
3.   Example Construct the point of intersection of a 
given straight line and a parabola of which the focus and 
the directrix are given.
     Our approach to any problem must depend on the 
state of our knowledge. Our approach to the present
problem depends mainly on the extent of our
acquaintance with the  of the parabola.  If we know
much about the parabola we try to make use of our
knowledge and to extract something helpful from it: Do
you know a theorem that could be useful? Do you know 
a related problem?  If we know little about parabola, 
focus and directrix, the terms are rather embarrassing 
and we naturally wish to get rid of them.  How can we 
get rid of them?  Let us listen to the dialogue of the 
teacher and the student discussing the proposed problem.
They have chosen already a suitable notation: P for any
of the unknown points of intersection, F for the focus, d 
for the directrix, c for the straight line intersecting the
parabola?
"And what is the unknown?"
"The point P"
"What are the data?"
"The straight lines c and d, and the point F."
"What is the condition?"
"P is a point of intersection of the straight line c and 
of the parabola whose directrix is d and focus F."
"Correct, You had little opportunity, I know, to study
the parabola but you can say, I think, what a parabola 
is."
"The parabola is the locus of points equidistant from
the focus and the directrix.":
"Correct. You remember the definition correctly. That
is right,but we must also use it; go back to definitions.
By virtue of the definition of the parabola, what can you 
say about your point P?
"P is on the parabola. Therefor, P is equidistant from
d and F."
"Good! Draw a figure."

   \      ~~-_                    /
    ~.  c     *-----____     P  .~
      ~                 *-__   ~
       ~           F    ....*-~-
        *~~        &....  ~~* . *---___
           *~~~~      ~~~~*   .
                *~~~~*        .
                              .
---------------------------------------
d                              Q

                   Fig.17

    The student introduces into Fig. 17 the lines PF and 
PQ, this later being the perpendicular to d from P
"Now, could you restate the problem?"
.....
"Could you restate the problem, using
the lines you have just introduced?"
"P is the point on the line c such that PF=FQ."
"Good. But please, say it in words: What is PQ"
"The perpendicular distance of P from d"
"Good. Could you restate the problem now? But please
state it neatly, in a round sentence."
"Construct a point P on the given straight line c at
equal distances from the given point F and the given
straight line d."
"Observe the progress from the original statement to
your restatement.  The original statement of the problem
was full of unfamiliar technical terms. parabola, focus
directrix; it sounded just a little pompous and inflated.
And now, nothing remains of those unfamiliar technical 
terms; you have deflated the problem. Well done~!"
4.  Elimination of technical terms is the result of the
work in the foregoing example. We started from a
statement of the problem containing certain technical terms
(parabola, focus, directrix)  and we arrived finally at a 
restatement free of those terms.
    In order to eliminate a technical term we must know
its definition; but is not enough to know the
definition, we must use it.  In the foregoing example, it was not
enough to remember the definition of the parabola. The
decisive step was to introduce into the figure the lines
PF and PQ whose equality was granted by the definition
of the parabola.  This is the typical procedure.  We
introduced suitable elements into the conception of the problem.
On he basis of the definition, we established relations
between the elements we introduced. If these relations
express completely the meaning, we have used the definition.
having used its definition, we have eliminated the
technical term.
    The procedure just described may be called Going back
to definitions. 
    By going back to the definition of a technical term, we 
get rid of the term but introduce new elements and new
relations instead. The resulting changing in our conception
of the problem may be important. At any rate, some
restatement, some VARIATION 
OF THE PROBLEM is bound to result.
5.   Definition and known theorems.  If we know the
name "parabola" and have some vague idea of the shape
of the curve but do not know anything else about it, our
knowledge is obviously insufficient to solve the problem
proposed as example, or any other serious geometric
problem about the parabola.  What kind of knowledge is
needed for such a purpose?
    The science of geometry may be considered as 
consisting of axioms, definitions, and theorems.  The parabola
is not mentioned in the axioms which deal only
with such primitive terms as point, straight line, and so
on. Any geometric argumentation concerned with the 
parabola, the solution of any problem involving it. must
use either its definition or theorems about it. To solve 
such a problem, we must know, at least, the definition 
but it is better to know some theorems too.
    What we said about the parabola is true, f course, of 
any derived notion.. As we start solving a problem that
involves a notion, we cannot know yet what we be
preferable to use, the definition of the notion, or some
theorem about it; but it is certain that we have to use
one or the other.
    There are cases, however, in which we have no choice
If we know just definition of the notation, and nothing
else, then we are obliged to use the definition. If we do
not know much more than the definition, or best chance
may be to go back to the definition. But if we know many
theorems about the notions, and have much experience
in its use, there is one chance that we may get a hold of 
a suitable theorem involving it.
6.    Several definitions. The sphere is usually defined as
the locus of points at a given distance from given point
(the points are not in space , not restricted to a plane)
Yet the sphere could also be defined as the surface
described by a circle revolving about a diameter. Still other
definitions of the sphere are known, and many others
possible.
    When we have to solve a proposed problem involving
some derived notion, as "sphere" or "parabola," and we
wish to go back to its definition, we may have a choice
among various definitions.  Much may depend in such a 
case on choosing the definition that fits the case.
    To find the area of the surface of the sphere was, at the
time Archimedes solved it, a great and difficult problem.
Archimedes had the choice between the definitions of the 
sphere we just quoted. He preferred to conceive the 
sphere as the surface generated by a circle revolving
about out a fixed diameter. He inscribes in the circle
a regular polygon, with an even number of sides, of 
which the fixed diameter joins opposite vertices.  
The regular polygon approximates the circle and, 
revolving with the circle, generates a convex surface 
composed of two cones with vertices at the extremities
of the fixed diameter and of several frustums of cones
in between. This composite surface approximates the 
sphere and is used by Archimedes, in computing the 
area of the surface of the sphere. If we conceive the
sphere as the locus of points equally distant from the
centre, no such simple approximation to its surface is
suggested.
7.   Going back to definition is important in inventing
an argument but it is also important in checking it.
    Somebody presents an alleged new solution of 
Archimedes problem of finding the area of the surface of the
sphere. If he has only a vague idea of the sphere, his
solution will not be any good. He may have a clear idea 
of the sphere but if he fails to use this idea in his argument
I cannot know that he had any idea at all, and his
argument is no good. Therefor, listening to the 
argument, I am waiting for the moment when he is going to
say something substantial about the sphere, to use its
definition or some theorem about it. If such a moment 
never comes, the solution is no good.
    We should check not only the arguments of others but,
of course, also our own arguments, in the same away
Have you taken into account all essential notions in
evolved in the problem How did you use this notion?
Did you use its meaning, its definition? Did you use
essential facts, known theorems about it?
8.     going back to definitions is important operation
of the mind. If we wish to understand why the definitions
 of words are so important, we should realize first that
words are important. We can hardly use our mind without
using words, or signs, or symbols of some sort. Thus, words
and signs have power. Primitive peoples believe that
words and symbols have magic power. We may understand
such belief but we should not share it. We should
know that the power of a word does not reside in its
sound, in the "vocis flatus," in the "hot air" produced
by the speaker, but in the ideas of which the word reminds
us and, ultimately, in the facts on which the ideas
are based.
    Therefor, it is a sound tendency to seek meaning and 
facts behind the words. Going back to definitions, the 
mathematician seeks to get hold of the actual relations
of mathematical objects behind the technical terms, as
the physicist seeks definite experiments behind his
technical terms, and the common man with some common
sense wants to get down to hard facts and not to be 
fooled by mere words.