Carrying out To conceive and plan and to carry it though are two 
different things.  This is true also of mathematical
problems in a certain sense; between carrying
out the plan of the solution, and conceiving it, there
are certain differences in the character of the work.
1.   We may use provisional and merely plausible arguments
when nerving the final and rigorous argument as
we use scaffolding to support a bridge during construction.
When however, the work is sufficiently advanced
we take of the scaffolding, and the bridge should be able
to stand by itself. In the same way, when the solution is
sufficiently advance, we brush aside all kinds of provisional
and merely plausible arguments, and the result
should be supported by rigorous argument alone.
    Devising the plan of the solution, we should not be too
afraid of merely plausible, heuristic reasoning. Anything
is right that leads us to there right idea. But we have no
change this standpoint when we start carrying out the 
plan and then we should accept only conclusive, strict 
arguments. Carrying out your plan to the solution, check
each step. Can you see clearly that the step is correct?
     The more painstakingly we check our steps when 
carrying out the plan, the more freely we may use heuristic
reasoning when devising it.
2.   We should give some consideration to the order in
which we work out the details of our plan, especially
if our problem is complex, we should not omit and detail, 
we should understand the relation of the detail before
us to the whole problem, we should not lose sight of
the connection of the major steps. Therefor, we should
proceed in proper order.
     In particular, it is not reasonable, to check minor details
before we have good reasons to believe that the
major steps of the argument are sound.  If there is a break
in the main line of the argument, checking this or that
secondary detail would be useless anyhow.
The order in which we work out the details of the
 argument may be very different form the order in which 
we invented them,;  and the order in which we write down 
the details in a definitive exposition may be sill different.
Euclid's ' elements present the details of the argument in
a rigid systematic order which was often imitated and
often criticised. 
3.   In Euclid's exposition all argument proceed in the 
same direction; from the data toward th unknown in
"problems to find," and from the hypothesis toward the
conclusion in "problems to prove" any new element, point, 
line, etc, has to be correctly derived form data
or from elements correctly derived in foregoing steps.
Any new assertion has to be correctly proved from the
hypothesis or from assertions correctly proved in 
foregoing steps. Each new element, each new assertion is
examined when it is encountered first, and so it has to be,
examined just once; we may concentrate all our attention
upon the present step, we need not look behind us, or 
look ahead. The very last new element whose derivation
we have to check is the unknown. The very last assertion
whose proof we have to examine, is the conclusion. If
each step is correct, also the last one, the whole argument
is correct.
      The Euclidean way of exposition can be highly recommended,
without reservation, if the purpose is to examine
the argument in detail. Especially, if it is our own argument
and it is long and complicated, and we have not
only found it but have also surveyed it on large lines so
that nothing is left but to examine each particular point
in itself, then nothing is better than to write out the 
whole argument in the Euclidean way.
      The Euclidean way of exposition, however, cannot be
recommenced, without reservation if the propose is to 
convey an argument to a reader or to a listener who 
never heard of it before. The Euclidean exposition is
excellent to show each particular point but no so good to 
show the main line of the argument. 
THE INTELLIGENT READER
can easily see that each step is correct but has 
great difficulty in perceiving the source, the purpose,
the connection of the whole argument.  The reason for this
difficulty is that the Euclidean exposition fairly often
proceeds in an order exactly opposite to the natural order
of invention. (Euclid's exposition follows rigidly the 
order of synthesis.; see PAPPUS
especially comments 3,4,5).
4.   Let us sum up. Euclid's manner of exposition, progressing 
relentlessly from the data to the unknown and
from the hypothesis to the conclusion is perfect for 
checking the argument in detail but far from being perfect
for making understandable the main line of the argument.
It is highly desirable that the students should examine
their own arguments in the Euclidean manner, proceeding
from the data to the unknown, and checking each
step although nothing of this kind should be too rigidly
enforced. It is not so desirable that the teacher 
should present  many proofs in the pure Euclidean manner, although
though the Euclidean presentation may be very useful
after a discussion in which, as is recommended by the
present book, the students guided by the teacher discover
the main idea of the solution as independently as possibly. 
Also durable seems to be the manner adopted by some
textbooks in which an intuitive sketch of the main
idea is presented first and the details in the
Euclidean order of exposition afterwards.
5. Wishing to satisfy himself that his proposition is 
true, th conscientious mathematician tries to see it
intuitively and to give a formal proof.  Can you see
clearly that it is correct? Can you prove that it is correct?  The
conscientious mathematician acts in this respect like the
lady who is a conscientious shopper.  Wishing to satisfy
herself of the quality of a fabric, she wants to see it and
to touch it. Intuitive insight and formal proof are two
different ways of perceiving the truth, comparable to the
perception of a material object through two different
senses, slight and touch.
    Intuitive insight may rush far ahead of formal proof. 
Any intelligent student, without any systematic knowledge
of solid geometry, can see as soon as he has clearly
understood the terms that two starlight lines parallel to
the same straight line are parallel to each other (the
three lines may or may not be in the he same plane). Yet
the proof of this statement, as given in proposition 9 of 
the 11th book of Euclid's Elements, needs a long, careful
and ingenious preparation.
      Formal manipulation of logical rules and algebraic
formula may get far ahead of intuition. Almost everybody
can see at once that 3 straight lines, taken at random
divide the plane into 7 parts(look at the only 
finite part, the triangle inclined by the 3 line). Scarcely
anybody is able to see, even straining his attention to the 
utmost, that 5 planes, taken at random, divide space into 26
parts. Yet it can be rigidly proved that the right number 
is actually 26, and the proof is not even long or difficult.
    Carrying out our plan, we check each step. Checking
our step, we may rely on intuitive insight or on formal
rules. Sometimes the intuition is ahead, sometimes
the formal reasoning. It is an interesting and useful exercise
to do it both ways. Can you see clearly that the step is 
correct? Yes, I can see it clearly and distinctly. Intuition
is ahead; but could formal reasoning overtake right?
 Can you also PROVE that it is correct? 
     Trying to prove formally what is seen intuitively and 
to see intuitively what is proved formally is an invigorating
mental exercise. Unfortunately, in the classroom
there is not always enough time for it.  The example
discussed in sections 12 and 14, is typical in this respect.