Modern heuristic endeavours to understand the 
process of solving problems, especially the mental operations
typically useful in this process. It has various sources of
information none of which should be neglected. A serious
study of heuristic should take into account both the 
logical and psychological background, it should not
neglect what should older writers Pappus, Descartes,
Leibnitz, and Bolzano have to say about the subject, but
it should least neglect unbiased experience. Experience in
solving problems and experience in watching other people
solving problems must be the basis on which heuristic
is built. In this study, we should not neglect any sort of
problem, and should find out common features in the 
way of handling all sorts of problems; we should aim at
general features, independent of the subject matter of
the problem. The study of heuristic has "practical" aims; 
a better understanding of the mental operations typically
useful in solving problems could exert some good influence
of teaching, especially on the teaching of mathematics.
     The present book is a first attempt toward the realisation
of this program. We are going to discuss how the
various articles of this dictionary fit into the program.
1.  Our list is, in fact, a list of mental operations typically
useful in solving problems; the questions and suggestions
listed hint at such operations. Some of these
operations are described again in the Second Part, and
some of them are more thoroughly discussed and illustrate
in the he First Part.
     For additional information about particular questions
and suggestions of the list, the reader should refer to 
those fif***** articles of the Dictionary whose titles are 
the first sentences of the fif***** paragraphs of the list:
WHAT IS THE UNKNOWN? IS IT POSSIBLE TO SATISFY THE CONDITION?
IT IS POSSIBLE TO SATISFY THE CONDITION?  DRAW A FIGURE
...CAN YOU USE THE RESULT?
      The reader, wishing information about a particular item
of the list, should look at the first word of the paragraph
in which the item is contained and then look up
the article in the Dictionary that has those first words as 
title. For instance, the suggestion "Go back to definitions"
is contained in the paragraph of the list whose first
sentence is : COULD YOU RESTATE THE PROBLEM? Under this
title, the reader finds a cross-reference to DEFINITION  in
which article the suggestion in question is explained and
illustrated.
2.   The process of solving problems is a complex process
that has several different aspects. The twelve principal
articles of this Dictionary study certain of these aspects
at some length; we are going to mention their titles in
what follows.
     When we are working intensively, we feel keenly the 
progress of work; we are elated when our progress
is rapid, we are depressed when it is slow.  What is essential
to PROGRESS AND ACHIEVEMENT in solving problems?
The article discussing this question is often quoted in
other parts of the Dictionary and should be read fairly
early.
     Trying to solve a problem, we consider different
aspects of it in turn, we roll it over and over incessantly in 
our mind; VARIATION OF THE PROBLEM is essential to our 
work. We may vary the problem by DECOMPOSING AND RECOMBINING
 its elements, or by going back to the DEFINITION of
certain of its terms, or we may use the great resources of
GENERALISATION, SPECIALISATION and ANALOGY. 
Variation of the problem may lead us to AUXILIARY ELEMENTS 
or to the discovery of a more accessible AUXILIARY PROBLEM.
     We have to distinguish carefully between two kinds of 
problems, PROBLEMS TO FIND, PROBLEMS TO PROVE.
Our list is specially adapted to "problems to find."  We have 
to revise it and change some of its question and suggestions
in order to apply it also to "problems to prove."
     In all sorts of problems, but especially in mathematical 
problems which are not too simple, suitable NOTATION
and geometrical FIGURES are a great and often indispensable help.
     3.  The process of solving problems has many aspects
but some of them are not considered at all in this book
and others only very briefly. It is justified, I think to 
exclude from a first short exposition points which could 
appear too subtle, or too technical, or too controversial.
     Provisional, merely plausible HEURISTIC REASONING 
is important in discovering the solution, but you should not 
teach it for a proof; you must guess, but also EXAMINE YOUR GUESS.
The nature of heuristic arguments is discussed in SIGNS OF PROGRESS
but the discussion could go further.
     The consideration of certain logical patterns is important
in our subject but it appeared advisable not to
introduce any technical article. There are only two articles
predominantly devoted to psychological aspect, on 
DETERMINATION,HOPE,SUCCESS, and on SUBCONSCIOUS WORK.
There is incidental remark on animal psychology; see WORKING BACKWARDS
     It is emphasised that all sorts of problems, especially
PRACTICAL PROBLEMS, and even PUZZLES, are within the 
scope of heuristic.  It is also emphasised that infallible 
RULES OF DISCOVERY are beyond the scope of serious re-search.
Heuristic discusses human behaviour in the face 
of problems; this has been fashion, presumably, since
the beginning of human society, and the quintessence of
such ancient discussions seems to be preserved in the
WISDOM OF PROVERBS.
4.    A few articles on particular questions are included
and some articles on more general aspects an expanded
because they could be, or parts of them could be, of
special interest to students or teachers.
     There are articles discussing methodical questions
often important in elementary mathematics, as PAPPUS, 
WORKING BACKWARDS(already quoted under 3),
 REDUCTIO AD ABSURDUM AND INDIRECT PROOF
INDUCTION AND MATHEMATICAL INDUCTION,
SETTING UP EQUATIONS,TEST BY DIMENSION,
and WHY PROOFS A few articles address themselves more particularly 
to teachers as ROUTINE PROBLEMS and 
DIAGNOSIS, and others to students somewhat more 
ambitious than the average,
as THE INTELLIGENT PROBLEM SOLVER,THE INTELLIGENT READER,
and THE FUTURE MATHEMATICIAN.
     It may be mentioned here that the dialogues between 
the teacher and his students, given in sections 8,10,18,19,20 
and in various articles of the Dictionary ay serve
as models not only to the teacher who tries to guide his
class but also to the problem-solver who works by himself.
To describe thinking as "mental discourse," as a sort
of conversation of the thinker with himself, is not in appropriate.
The dialogues in question show the progress
of the solution; the problem-solver, talking with himself
may progress along a similar line.
5.   We are not going to exhaust the remaining titles;
just a few groups will be mentioned.
     Some articles contain remarks on the history of our
subject, on DESCARTES,LEIBNITZ,BOLZANO, on HEURISTIC,
 on TERMS,OLD AND NEW,
and on PAPPUS(this last one has 
been quoted already under 4).
     A few articles explain technical terms: CONDITION, COROLLARY,
LEMMA
     Some articles contain only cross-references (they are 
marked with daggers [+] in the Table of Contents).
     6. Heuristic aims at generality, at the study of
procedures which are independent of the subject-matter and
apply to all sorts of problems. The present exposition,
however quotes almost exclusively elementary mathematical
problems as examples. It should not be overlooked
that this is a restriction but it is hoped that this
restriction does not impair seriously the trend of our
study. In fact, elementary mathematical problems present
all the desirable variety, and the study of their solution
is particularly accessible and interesting. Moreover,
non-mathematical problems although seldom quoted as
examples are never completely forgotten. More advanced
mathematical problems are never directly quoted but
constitute the real background of the present exposition.
The expert mathematician who has some interest for this
sort of study can easily add examples from his own
experience to elucidate the point illustrated by elementary 
examples here.
7.   The write of this book wishes to acknowledge 
indebtedness and express his gratitude to a few modern
authors, not quoted in the article on HEURISTIC
they are the physicist and the philosopher Ernst mach,
the mathematician Jacques Hadamard, the psychologists
William James and Wolfgang Kohler''. He wishes also to
quote the psychologist K. Duncker and the mathematician
F. Krauss whose work (published after his own research 
was fairly advanced, and partly published) shows certain
parallel remarks.