Sword and shield. (Clinton, Miss.) 1885-1888, 1885-04-11

gulsh the several varieties, what
are these but series of observations?
Not only in the fundamental con-
ceptions of form, does observation
play an important part, but the very
genius of geometry, as a method of
reasoning, system of investigation,
is that it is but a series of obserya-
tions. The diagram before the eye
in actual representation, or before
the mind IN conception, is so closely
scrutinlzed, that all its distinctive
features are perceived; auxiliary
lines are drawn (the Imagination
leading in this), and a new series Of
inspections is made; and thus by
means Of direct, simple observations,
the investigation proceeds. This,
method Of investigation by means of
successive observations is so essen
tial characteristic Of geometry,
that it has by some eminent writers
upon the philosophy of mathematics
been classed among the natural sci
ences, as to its methods. Moreover,
when we consider applied mathe-
MATES, we need only to notice that.
the exercise of the faculty of the
faculty of observation is go essential,
that the basis of all our reasoning,
the very materials with which we
build our demonstrations or calcula-
tions, have received the name obser-
udti0ns.
Were we to consider separately
the whole range of the mental facul-
ties, for most of them we would find
abundant exercise in mathematical
studies. The memory surely is not
neglected. The very first steps in
number, such as counting, adding,
multiplying, etc, make large de
mands on the exercise of this facul-
ty. All the tables give scope to the
exercise of memory. The higher
branches, algebra, geometry, trigo-
nometry, analytical geometry and
calculus, all require the memoriz-
ing Of formulas which are simply APP
palling to the uninitiated. They
can only be retained by closely OBJ
serving analOgies and differences, by
giving to them that fixed attention,
which psychologists tell us is an in
dispensable prerequisite to memory.
Not even the intricacies of the
Greek verb afford better facilities
for the exercise and cultivation of
memory than do the formulas which
embrace the results of all the more
important theorems and demonstra-
tions in the higher mathematics,
both pure and mixed.
The Imagination, TOO, the creative
faculty of the mind, has constant ex-
ercise in all original mathematical
investigation, from the solution Of
the simplest problem to the discovery
Of the most recondite principle for
it is not, as many suppose, by sure,
consecutive steps, that we advance
from the known to the unknown.
The imagination, not the logical fac
ulty, leads the way. In fact, pract
cal observatlo:. is often in advance
Of logical exposition. Thus, in the
discovery of truth, the imagination
presents hypothesis, and observa-
ton supplies the fact from which it
may require the tardy reason to
construct IA bridge spanning the
chasm logically between the known
and unknown. OF this truth, math-
ematics, IN common with other sci
ences, affords abundant illustrations.
Whether tl.at sublimest branch of
mathematics, the Inhnitesimal Cal
culus, rests upon an empirical fOun-
dation or not, admits of serious oues-
ton, mathematicians not being
agreed as to its logical basis. But
that the imagination leads, and not
the logical faculty, in all original in.
vestigation, no one who has ever
succeeded in producing an original
demonstration of a simple PROPOSE
ton in geometry, can entertain any
doubt.
Induction, analogy, the search for
premises, and the balancing of prob
abilities are spheres Of mental ac-
ton not unknown to mathematics.
We would not claim a pre-eminence
for mathematical studies in all these
departments Of intellectual culture,
but scarcely any branch of science
affords discipline to so many of our
faculties and no other presents so
complete a gradatIon in its exercise
of these faculties, from the first
principles thereof to the farthest
verge of its application, as mathe-
matics. But special pre-eminence
may Justly be claimed for mathemat-
les, in training the mind to form
clear and definite conceptions and tC
clothe these conceptions in exact
and perspicuous language. This is
due doubtless to the fact, that in
mathematics the terms employed
convey exactly the same meaning tC
all minds. The conceptions of 5, 8
s, a straight line, a circle, ETC., art
precisely the same as entertained by
different persons. So, too, the
statements that 3 and 5 are s, ant
that the sum Of the angles ola plane
triangle is two right angles, arr
alike apprehended by all, This j
not so in other sciences. We cal
not be sure that all have the sam
conception of the metaphysical
terms idea, perception, reason nor Q
the distinctions in the meanings Q
dOapny, hirsirt, and pubescelat. Tn
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