Notation. If you wish to realize the
advantages of a well chosen and well known notation to
try add a few not too small numbers with the condition that
you are not allowed to use the familiar Arabic numerals, 
although you may use, if you wish to write, roman numerals.
Take, for instance, the numbers MMMXC, MDXCVI, MDCXLVI,
MDCCLXXXI, MDCCCLXXXVII.
     We can scarcely overestimate the importance of mathematical
notation. Modern computers, using the decimal
notation, have great advantage over the ancient computers
who did not have such a continent manner of writing the
numbers. An average modern student who is familiar with 
the usual notation of algebra, analytical geometry and 
the differential and integral calculus has an immense 
advantage over a Greek mathematician in
solving the problems about areas and volumes which 
exercised the genius of Archimedes.
     1. Speaking and thinking are closely connected, the
use of words assists the mind. Certain philosophers and 
philologists went a little further and asserted that the use 
of words is indispensable to the use of reason.
     Yet this last assertion appears somewhat exaggerated. 
If we have a little experience of serious mathematical
work we know that we can do a piece of pretty hard
thinking without using any words, just looking at geometric
figures or manipulating algebraic symbols. Figures
and symbols are closely connected with mathematical
thinking, their use assists the mind. We could improve
that somewhat narrow assertion of philosophers and
philologist by bringing the words into line with other
sorts of signs and saying that the use of signs appears to 
be indispensable to the use of reason,
A     At any rate, the use of mathematical symbols is similar 
to the use of words. Mathematical notation appears as
a sort of language, une langue bien faite,  a language well
adapted to its purpose, concise and precise, with rules
which, unlike the rules of ordinary grammar, suffer no
exception.
     If we accept this viewpoint, SETTING UP EQUATIONS 
appears as a sort of translation, translation from ordinary 
language into the language of mathematical symbols.
2.    Some mathematical symbols, as +,-,=, and several
others, have a fixed traditional meaning, but other symbols,
as the small and capital letters of the Roman and Greek
alphabets are used in different meanings in different
problems. When we face a new problem, we must chose certain
symbols. we have to introduce suitable notation.  
There is something analogous in the use of ordinary language. 
Many words are used in different meanings indifferent contexts;
when precision is important, we have to chose our words carefully.
     An important step in solving a problem is to chose 
the notation. It should be done carefully. The time we 
spend now on choosing the notation may well be repaid
by the time we save later by avoiding hesitation and confusion.
Moreover, choosing the notation carefully, we 
have to think sharply of the elements of the problem
which must be denoted. Thus choosing a suitable notation
may contribute essentially to understanding the 
problem.
3.   A good notation should be unambiguous, pregnant,
easy to remember; it should avoid harmful second
meanings and take advantage of useful second meanings; the 
order and connection of signs should suggest the order
and connection of things.
4.    Signs must be, first of all, unambiguous it is inadmissible
that the same symbol denote two different objects
in the same inquiry. If solving a problem, you call
a certain magnitude a you should avoid calling 
anything else a which is connected with the same
problem.  Of course, you may set the letter a in
different meaning in a different problem.
     Although it is forbidden to use the same symbol for different objects
it is not forbidden to use different symbols for the same object.
Thus the product of a and b may be written as

          a X b         a * b     ab

In some cases, it is advantageous to use two or more different
signs for the same object, but such cases require
particular care. Usually it is better to use just one sign
for one object, and in no case should several signs be used
wantonly.
5.    A good sign should be easy to remember and easy to 
recognise; the sign should immediately remind us of the 
object and the object of the sign.
     A simple device to make signs easily recognisable is to use
initialsas symbols. For example, in section20 we used
r for rate t for time V for volume. We cannot use, however, initials
in all  cases. Thus in section 12  we had to
consider a radius but we could not call it r but cause this
letter was already taken to denote a rate. There are still
other motives restricting the choice of the symbols, and other 
means to make them easily recognisable which we are going 
to discuss.
6.   Notation is not only easily recognisable but
particularly helpful in shaping our conception when the
order and connection of the signs suggest the order and
connection of the objects  we need several examples
to illustrate this point.
(I)  In order to denote objects which are near to each
other in conception of the problem we use letters
which are near to each other in the alphabet.
     Thus, we generally use letters at the beginning of 
the alphabet as a,b,c, for given quantities or constants, and
letters at the end of the alphabet as x,y,z, for unknown
quantities or variables.
     In section8 we used a,b,c for the given length, width
and height of a parallelepiped. On this occasion, the
notation a,b,c, was preferable  to the notation by initials l,w,h.
The three lengths played the same role in the problem
which is emphasised by the use of successive letters
Moreover, being at the beginning of the alphabet, a,b,c
are, as we just said, the most usual letters to denote given
quantities. On some other occasion, if the three lengths.
play different roles and is important to know which 
lengths are horizontal and which one is vertical the 
notation, l,w,h, might be preferable.
(II)  In order to denote objects belonging to the same
category, we frequently chose letters belonging to the
same alphabet for one category, using different alphabets
for different categories.  Thus, in plane geometry we often
use:  
  

        Roman Capitals as A,B,C,........for points
        small Roman letters as a,b,c....for lines
        small Greek letters as a,b,y.....for angles

If there are two objects belonging to different categories
but having some particular relation to each other 
which is important for our problem, we may choose, to 
denote these two objects, corresponding letters of the
respective alphabets as A and a  B and b and so on. A 
familiar example is the usual notation for a triangle:

              A,B,C  stand for the vertices
              a,b,c  for the sides,
              a,b,c for the angles.

It is understood that a is the side opposite to the vertex A 
and the angle at A is called a.
(III)  Is section 20 the letters a,b,x,y, are particularly
well chosen to indicate the nature and connection
of the elements denoted. The letters a,b hint that the
magnitudes denoted are constants; x,y indicate variables;
a precedes b as x precedes y and this suggests that a is in 
the same relation to b as x is to y.  In fact, a and x are 
horizontal, b and y vertical, and a:b = x:y.
7.  The notation

         /\ABC ~ /\EFG

indication that the two triangles in question are similar. 
In modern book, the formula is meant to indicate that 
the two triangles are similar, the vertices corresponding
to each other in the order as they are written, A to E, B 
to F, C to G.  In older books, this provison about the order
was not yet introduced;  the reader had to look at the
figure or remember the derivation in order to ascertain
which vertex correspond to which.
     The modern notation is much preferable to the older
one.  Using the modern notation, we may draw consequences
from formula without looking at the figure
Thus, we may derive that 
 
                  /_A = /_E
                  AB:BC = EF:FG

and other relations of the same kind. The older notation 
expresses less and does not allow such definite consequences.
     A notation expressing more than another may be
termed more pregnant.  The modern notation for similitude
of the triangles is more pregnant than the older one,
reflects the order and connection of things more fully 
than the older one, and therefor, it may serve as basis
for more consequences than the older one.
8.   Words have second meanings Certain contexts in
which a word is often used influence it and add something
to its primary meaning, some shade, or second
meaning, or "connotation." If we write carefully, we try
to chose among the words having almost the same meaning
the one whose second meaning is best adapted.
    There is something similar in mathematical notation.
Even mathematical symbols may acquire a sort of second
meaning from contexts in which they are often used. If
we choose our notation carefully, we have to take this
circumstance into account. Let us illustrate the point.
     There are certain letters which have acquired a firmly
rooted, traditional meaning. Thus, e stands usually for
the basis of natural logarithms, i for sqrt(-1), the imaginary 
unit and pi for the ratio of the circumference of the 
circle to the diameter. It is on the whole better to use
such symbols only in their traditional meaning. If we use
such a symbol in some other meaning its traditional
meaning could occasionally interfere and be
embarrassing, even misleading. It is true that harmful second
meanings of this sort give less trouble to the beginner
who has not yet studied many subjects than  to the mathematician
who should have sufficient experience to deal
with such nuisances.
     Second meaning of the symbols can also be helpful,
even very helpful, if they are useful with tact.  A notation
used on former occasions may assist us in recalling some 
useful procedure;  of course, we should be sufficiently
careful to separate clearly the present (primary) meaning
of the symbol from its former(secondary) mean
meaning. A standing notation[as the traditional notation for
the parts of the triangle which we mentioned before, 6, (II)] has
great advantages; used on several former
occasions it may assist us in recalling various formerly
used procedures. We remember our formulas in some
standing notation. Of course, we should be sufficiently
careful when, owing to particular circumstances, we are
obliged to use a standing notation in a meaning some
what different from the usual one.
9.     When we have to chose between two notations,
one reason may speak for one, and some other reason for
the other. We need experience and taste to chose the 
more suitable notation as we need experience and taste
to chose more suitable words. Yet it is good to know the
various advantages and disadvantages discussed in the 
foregoing. At any rate, we should choose our notation
carefully, and have some good reason for our choice.
10.   Not only the most hopeless boys in the class but
also quite intelligent students may have an aversion for
algebra. There is always something arbitrary and artificial
about notation; to learn a new notation is a burden
for the memory. The intelligent student refuses to as
assume the burden if he does not see any compensation for
it. The intelligent student is justified in his aversion for
algebra if he is not given ample opportunity to convince
himself by his own experience that the language of 
mathematical symbols assists the mind To help him 
to such experience is an important task of the teacher,
one of his most important tasks.
     I say that it is an important task but I do not say that 
it is an easy one. The foregoing remarks may be of some
help. See alsoSETTING UP EQUATIONS.
Checking a formula by extensive discussion of its properties may be 
recommended as a particularity instructive exercise; see section
14 and CAN YOU CHECK THE RESULT