PAPPUS an important Greek mathematician, lived probably 
around AD 300. In the seventh book of his Collections 
PAPPUS reports about a branch of study which 
he calls anaalyomenos.  We render this name
in English as "Treasury of Analysis." or as "art of
solving problems," or even as "heuristic"; the last term
seems to be preferable here. A good English translation
of pappus's report is easily accessible 7(7:T.L. Heath, the thir***** books of Euclid's elements, Cambridge, 1908, vol 1, p138) what follows is a 
free rendering of the original text:
     "The so-called Heuristic is, to put it shortly, a special
body of doctrine for the use of those who, after having
studied the ordinary Elements, are desirous of acquiring
the ability to solve mathematical problems, and it is
useful for this alone. It is the work of three men, Euclid
the author of the elements, Apollonius of Perga, and
Aristaeus the elder. It teaches the procedures of analysis
and synthesis.
     "In analysis, we start from what is required, we take it
for granted, and we draw consequences from it, and 
consequences from the consequences, till we reach a point
that we can use as starting point in synthesis. For in
analysis we assume what is required to be done as all ready
done (what is sought as already found, what we have to 
prove as true). We inquire from what antecedent the 
desired result could be derived;  then we inquire again
what could be the antecedent of the antecedent, and so
on until passing from antecedent to antecedent, we come
eventually upon something all ready known or admittedly
true. This procedure we call analysis, or solution back-
wards, or regressive reasoning.
     "but in synthesis, reversing th the process, we start from
the point which we reached last of all in the analysis,
from the he thing already known or admittedly true. We
derive  from it what preceded it in the analysis and go
on making derivations until, retracing our steps, we
finally succeed in arriving at what is required. This
procedure we call synthesis, or constructive solution, or
progressive reasoning.
     "Now analysis is of two kinds; the one is the analysis
of the "problems to prove' and aims at establishing true
theorems; the other is the analysis of the 'problems to 
find' and aims at finding the unknown.
     "If we have a 'problem to prove' we are required to 
prove or disprove a clearly stated theorem A. We do not
know yet whether A is true or false; but we derive from
A another theorem B form B another C and so on, until
we come upon a  last theorem L about which we have
definite knowledge. If L is true, A will also be true,
provided that all our derivations are convertible. From L
we prove the theorem K which preceded L in the analysis
and, proceeding in the same way, we retrace our steps;
from C we prove B, from B we prove A, and so we attain
our aim. If, however, L is false, we have proved A false.
     "If we have a "problem to find" we are required to find
a certain unknown x satisfying a clearly stated condition.
We do not know yet whether a thing satisfying such a 
condition is possible or not; but assuming that there is
an x satisfying the condition imposed we derive from it
another another unknown y which has to satisfy a related
condition; the we link y to still another unknown, and so
on, until we come upon a last unknown z which we can
find by some known method. If there is actually a z
satisfying the condition imposed upon it, there will be 
also an x satisfying the original condition, provided that
all our derivations are convertible. We must first find z; then
knowing z,we find the unknown that preceded z in the
analysis; proceeding in the same away, we retrace our
steps, and finally, knowing y, we obtain x, and so we 
attain our aim. If however, there is nothing that would
satisfy the condition imposed upon z, the problem
concerning x has no solution."
     We should not forget that foregoing is not a literal
translation but a free rendering a paraphrase Various
difference between the original and the paraphrase deserve
serve comment, for PAPPUS' text is important in many
ways.
1.  Our paraphrase uses a more definite terminology
than the original and introduces the symbols A,B, ...,
L,x,y,...,z which the original has not.
2.   The paraphrase has (p141, line 330) "mathematical 
problems" where the original means "geometrical
problems?" This emphasises that the procedures described by
Pappus are by no means restricted to geometric problems;
they are, in fact, not even restricted to
mathematical problems. We have to illustrate this by examples.
since in these matters, generality and independence from
the nature of the subject are important( see setion 3)
3.  Algebraic illustration  Find x satisfying the
equation
 
             8(4^x+4^-x)-54(2^x+2^-x)+101=0

This is a "problem to find," not too easy for a beginner.
He has to be familiar with the idea of analysis; not with
the word "analysis" of course, but with the idea of at-
attaining the aim by repeated reduction.  Moreover, he has 
to be familiar with the simplest sorts of equations. Even
with some knowledge, it takes a good idea, a little luck, 
a little invention to observe that, since 4^x=2^x^2 and
4^-x = 2^x^-2 it may be advantageous to introduce

                        y = 2^x

Now, this substitution is really advantageous, the equation-
obtained for y

         8(y^2+1/y^2) - 54(y+1/y) +101 = 0

appears simpler than the original equation; but our task
is not yet finished. It needs another little invention, an-
other substitution
  
                 z = y + 1/y

which transforms the condition into

             8z^2 - 54z + 85 = 0

Here the analysis ends, provided that the problem-sovler
is acquainted with the solution quadratic equations
     What is the synthesis? Carrying through, step by step
the calculations whose possibility was foreseen by the
analysis. The problem-solver needs no new idea to finish
his problem, only some patience and attention in calculated
the various unknowns.  The order of calculations is
opposite to the order of invention; first z is found
(z=5/2, 17/4), then y (y=2, 1/2, 4, 1/4), and finally
the originally required x (x=1,-1,2,-2). The synthesis
retraces the steps of the analysis, and it is easy to
see in the present case why it does so.
4.  Non-mathematical illustration.  A primitive man
wishes to cross a creek; but he cannot do so in-the usual
way because the object of a problem; "crossing the
creek'; is the x of this primitive problem. The man may
recall that he has crossed some other creek by walking
along a fallen tree. He looks around for a suitable fallen
tree which becomes his new unknown, his y. He cannot
find any suitable tree but there are plenty of trees standing
along the creek; he wishes that one of them would
fall.  Could he make a tree fall across the creek? There is
a great idea and there is a new unknown; by what means
could he tilt the tree over the creek?
     This train of ideas ought to be called analysis if we 
accept the terminology of Pappus. If the primitive man
succeeds in finishing his analysis he may become the
inventor of the bridge and of the axe.  What will be the
synthesis? Translation of ideas into actions. The
finishing act of the synthesis is walking along a tree across the 
creek.
     The same objects fill the analysis and the synthesis;
they exersize the mind of the man in the analysis and his
muscles in the synthesis; the analysis consists in thoughts,
the synthesis in acts. There is another difference; the
order is reversed. Walking across the creek is the first
desire from which the analysis starts and it is the last
act with which the synthesis ends.
5.     The paraphrase hints a little more distinctly than
the original the natural connection between analysis and
synthesis. This connection is manifest after the foregoing
examples. Analysis comes naturally first, synthesis afterwards;
analysis is invention, synthesis, execution; analysis is
devising a plan, synthesis, carrying through the plan
6.     The paraphrase preserves and even emphasises certain-
curious phrases of the original: "assume what is 
required to be done as all-ready done, what is sought as
found, what you have to prove as true." This is paradoxical;
is it not mere self-deception to assume that the 
problem that we have to solve is solved? This is obscure;
what does it mean?  If we consider closely the context and
try honestly to understand our own experience in solving
problems, the meaning can scarcely be doubtful.
     Let us first consider a "problem to find."  Let us call the 
unknown x and the data a,b,c.  To "assume the problem 
as solved" means to assume that there exists an object x
satisfying the condition; that is, having those relations
to the data a,b, which the condition prescribes. This
assumption is made just in order to start the analysis it
is provisional, and it is harmless.  For, if there is no such
object and the analysis leads us anywhere, it is bound
to lead us to a final problem that has no solution and 
hence it will be manifest that our original problem has
no solution. Then, the assumption is useful. In order to
examine the condition, we have to conceive to represent
to ourselves, or to visualise geometrically the relations
which the condition prescribes between x and a,b,c;
how could we do so without conceiving, representing or
visualising x as existent?  Finally, the assumption is natural.
The primitive man whose thoughts and deeds we
discussed in comment 4 imagines himself walking on a 
fallen tree and crossing the creek long before he actually
can do so; he sees his problem "as solved"
     The object of a "problem to prove" is to prove a certain-
 theorem A. The advice to "assume A as true" is just
an invitation to draw consequences from the theorem A
although we have not yet proved it. People with a certain
mental character or certain philosophy may shrink
from drawing consequences from an unproved theorem;
but such people cannot start an analysis.
Compare FIGURES,2.
7.    The paraphrase uses twice the important phrase
"provided that all our derivations are convertible"; see
p142 line 33 and p143 lines 14-15. This is an
interpolation; the original contains nothing of the sort and
the lack of such a provision was observed and criticised in
modern times. See AUXILIARY PROBLEM,6
for the notion of convertible reduction.
8.    The "analysis of the problems to prove" is explained
in the paraphrase in words quite different from those
used by the original but there is no change in the sense;
at any rate, there is no intention to change the sense.
The analysis of the "problem to find," however, is
explained more concretely in the paraphrase than in the
original. The original seems to aim at the description of
a somewhat more general procedure, the construction of
a chain of equivalent auxiliary problemswhich is described
in AUXILIARY PROBLEM,7.
9.   Many elementary textbooks of geometry contain a 
few remarks about analysis,  synthesis, and "assuming the
problem as solved." There is a little doubt that this almost
ineradicable tradition goes back to Pappus, although
there is hardly a current textbook whose write would
show any direct acquaintance with Pappus. The subject
is important enough to be mentioned in elementary textbooks
but easily misunderstood. The circumstance alone that it is 
restricted to textbooks of geometry shows a current lack of
understanding; see comment 2 above. If the foregoing comments
would contribute to a better understanding of this matter
their length would be amply justified. For another example,
a different viewpoint, and further comments see 
WORKING BACKWARDS
     Compare also REDCUTIO AD ABSURDUM AND INDIRECT PROOF.