An extensive use of models is, however, unwise. The pupil must learn
very early how to visualize a solid from the flat outline picture, just
as a builder or a mechanic learns to read his working drawings. To have
a model for each proposition, or even to have a photograph or a
stereoscopic picture, is a very poor educational policy. A textbook may
properly illustrate a few propositions by photographic aids, but after
that the pupil should use the kind of figures that he must meet in his
mathematical work. A child should not be kept in a perambulator all his
life,--he must learn to walk if he is to be strong and grow to maturity;
and it is so with a pupil in the use of models in solid geometry.[88]
The case is somewhat similar with respect to colored crayons. They have
their value and their proper place, but they also have their strict
limitations. It is difficult to keep their use within bounds; pupils
come to use them to make pleasing pictures, and teachers unconsciously
fall into the same habit. The value of colored crayons is two-fold: (1)
they sometimes make two planes stand out more clearly, or they serve to
differentiate some line that is under consideration from others that are
not; (2) they enable a class to follow a demonstration more easily by
hearing of "the red plane perpendicular to the blue one," instead of
"the plane _MN_ perpendicular to the plane _PQ_." But it should always
be borne in mind that in practical work we do not have colored ink or
colored pencils commonly at hand, nor do we generally have colored
crayons. Pupils should therefore become accustomed to the pencil and the
white crayon as the regulation tools, and in general they should use
them. The figures may not be as striking, but they are more quickly made
and they are more practical.
The definition of "plane" has already been discussed in Chapter XII, and
the other definitions of Book VI are not of enough interest to call for
special remark. The axioms are the same as in plane geometry, but there
is at least one postulate that needs to be added, although it would be
possible to state various analogues of the postulates of plane geometry
if we cared unnecessarily to enlarge the number.
The most important postulate of solid geometry is as follows: _One
plane, and only one, can be passed through two intersecting straight
lines._ This is easily illustrated, as in most textbooks, as also are
three important corollaries derived from it:
1. _A straight line and a point not in the line determine a plane._ Of
course this may be made the postulate, as may also the next one, the
postulate being placed among the corollaries, but the arrangement here
adopted is probably the most satisfactory for educational purposes.
2. _Three points not in a straight line determine a plane._ The common
question as to why a three-legged stool stands firmly, while a
four-legged table often does not, will add some interest at this point.
Public-domain text, read in full here on John Shaqi.
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