HOW TO ATTACK THE EXERCISES
The old geometry, say of a century ago, usually consisted, as has been
stated, of a series of theorems fully proved and of problems fully
solved. During the nineteenth century exercises were gradually
introduced, thus developing geometry from a science in which one learned
by seeing things done, into one in which he gained power by actually
doing things. Of the nature of these exercises ("originals," "riders"),
and of their gradual change in the past few years, mention has been made
in Chapter VII. It now remains to consider the methods of attacking
these exercises.
It is evident that there is no single method, and this is a fortunate
fact, since if it were not so, the attack would be too mechanical to be
interesting. There is no one rule for solving every problem nor even for
seeing how to begin. On the other hand, a pupil is saved some time by
having his attention called to a few rather definite lines of attack,
and he will undoubtedly fare the better by not wasting his energies over
attempts that are in advance doomed to failure.
There are two general questions to be considered: first, as to the
discovery of new truths, and second, as to the proof. With the first the
pupil will have little to do, not having as yet arrived at this stage in
his progress. A bright student may take a little interest in seeing
what he can find out that is new (at least to him), and if so, he may
be told that many new propositions have been discovered by the accurate
drawing of figures; that some have been found by actually weighing
pieces of sheet metal of certain sizes; and that still others have made
themselves known through paper folding. In all of these cases, however,
the supposed proposition must be proved before it can be accepted.
As to the proof, the pupil usually wanders about more or less until he
strikes the right line, and then he follows this to the conclusion. He
should not be blamed for doing this, for he is pursuing the method that
the world followed in the earliest times, and one that has always been
common and always will be. This is the synthetic method, the building up
of the proof from propositions previously proved. If the proposition is
a theorem, it is usually not difficult to recall propositions that may
lead to the demonstration, and to select the ones that are really
needed. If it is a problem, it is usually easy to look ahead and see
what is necessary for the solution and to select the preceding
propositions accordingly.
But pupils should be told that if they do not rather easily find the
necessary propositions for the construction or the proof, they should
not delay in resorting to another and more systematic method. This is
known as the method of analysis, and it is applicable both to theorems
and to problems. It has several forms, but it is of little service to a
pupil to have these differentiated, and it suffices that he be given the
essential feature of all these forms, a feature that goes back to Plato
and his school in the fifth century B.C.
For a theorem, the method of analysis consists in reasoning as follows:
"I can prove this proposition if I can prove this thing; I can prove
this thing if I can prove that; I can prove that if I can prove a third
thing," and so the reasoning runs until the pupil comes to the point
where he is able to add, "but I _can_ prove that." This does not prove
the proposition, but it enables him to reverse the process, beginning
with the thing he can prove and going back, step by step, to the thing
that he is to prove. Analysis is, therefore, his method of discovery of
the way in which he may arrange his synthetic proof. Pupils often wonder
how any one ever came to know how to arrange the proofs of geometry, and
this answers the question. Some one guessed that a statement was true;
he applied analysis and found that he _could_ prove it; he then applied
synthesis and _did_ prove it.
For a problem, the method of analysis is much the same as in the case of
a theorem. Two things are involved, however, instead of one, for here we
must make the construction and then prove that this construction is
correct. The pupil, therefore, first supposes the problem solved, and
sees what results follow. He then reverses the process and sees if he
can attain these results and thus effect the required construction. If
so, he states the process and gives the resulting proof. For example:
In a triangle _ABC_, to draw _PQ_ parallel to the base _AB_,
cutting the sides in _P_ and _Q_, so that _PQ_ shall equal
_AP_ + _BQ_.
[Illustration]
=Analysis.= Assume the problem solved.
Then _AP_ must equal some part of _PQ_ as _PX_, and _BQ_ must
equal _QX_.
But if _AP_ = _PX_, what must [L]_PXA_ equal?
[because] _PQ_ is || _AB_, what does [L]_PXA_ equal?
Then why must [L]_BAX_ = [L]_XAP_?
Similarly, what about [L]_QBX_ and [L]_XBA_?
=Construction.= Now reverse the process. What may we do to [Ls]
_A_ and _B_ in order to fix _X_? Then how shall _PQ_ be drawn?
Now give the proof.
[Illustration]
[Illustration]
The third general method of attack applies chiefly to problems where
some point is to be determined. This is the method of the intersection
of loci. Thus, to locate an electric light at a point eighteen feet from
the point of intersection of two streets and equidistant from them,
evidently one locus is a circle with a radius eighteen feet and the
center at the vertex of the angle made by the streets, and the other
locus is the bisector of the angle. The method is also occasionally
applicable to theorems. For example, to prove that the perpendicular
bisectors of the sides of a triangle are concurrent. Here the locus of
points equidistant from _A_ and _B_ is _PP'_, and the locus of points
equidistant from _B_ and _C_ is _QQ'_. These can easily be shown to
intersect, as at _O_. Then _O_, being equidistant from _A_, _B_, and
_C_, is also on the perpendicular bisector of _AC_. Therefore these
bisectors are concurrent in _O_.
These are the chief methods of attack, and are all that should be given
to an average class for practical use.
Besides the methods of attack, there are a few general directions that
should be given to pupils.
1. In attacking either a theorem or a problem, take the most general
figure possible. Thus, if a proposition relates to a quadrilateral, take
one with unequal sides and unequal angles rather than a square or even a
rectangle. The simpler figures often deceive a pupil into feeling that
he has a proof, when in reality he has one only for a special case.
2. Set forth very exactly the thing that is given, using letters
relating to the figure that has been drawn. Then set forth with the same
exactness the thing that is to be proved. The neglect to do this is the
cause of a large per cent of the failures. The knowing of exactly what
we have to do and exactly what we have with which to do it is half the
battle.
3. If the proposition seems hazy, the difficulty is probably with the
wording. In this case try substituting the definition for the name of
the thing defined. Thus instead of thinking too long about proving that
the median to the base of an isosceles triangle is perpendicular to the
base, draw the figure and think that there is given
_AC_ = _BC_,
_AD_ = _BD_,
and that there is to be proved that
[L]_CDA_ = [L]_BDC_.
[Illustration]
Here we have replaced "median," "isosceles," and "perpendicular" by
statements that express the same idea in simpler language.
=Bibliography.= Petersen, Methods and Theories for the Solution
of Geometric Problems of Construction, Copenhagen, 1879, a
curious piece of English and an extreme view of the subject,
but well worth consulting; Alexandroff, Problemes de geometrie
elementaire, Paris, 1899, with a German translation in 1903;
Loomis, Original Investigation; or, How to attack an Exercise
in Geometry, Boston, 1901; Sauvage, Les Lieux geometriques en
geometrie elementaire, Paris, 1893; Hadamard, Lecons de
geometrie, p. 261, Paris, 1898; Duhamel, Des Methodes dans les
sciences de raisonnement, 3^e ed., Paris, 1885; Henrici and
Treutlein, Lehrbuch der Elementar-Geometrie, Leipzig, 3. Aufl.,
1897; Henrici, Congruent Figures, London, 1879.