New-York tribune. [volume] (New York [N.Y.]) 1866-1924, 1881-10-04
NEW PUBLICATIONS. |

Predecessor of Mathematics United States Navy Svo'
400 PIP. New-York Henry Holt d 20
Tho bodv of truth in regard to the relations OF
oagnitudes and forms In space which the school,
bows of to-day are required to learn as the fouhu
ton of mathematical science. was demonstrated
two thousand years ago with such accuracy anu
elegance and wrought into system OF su'h n''-
monious proportions as to render any attemp' W I'M
prove upon the old scheme of deveiopmeht q n"H
ardous undertaking. Indeed it would oe oinoui'
matter at this age of the world TO oter any
of method IN a treatise on elementary geometry.
The whole field has been so thoroughly pioughed
over. so many acute thinkers have devoted them
selves TO the investigation of the nature and source
of our fundamental conceptions, and have bent all
their powers of imagination ingenuity and logic 10
discover the most direct and rigorous lines OF reas-
oning in the premises, that one can hardly look LOT
any originality in the treatment of the SUBJECT
That the Elements" of Euclid still hold high, IL
not tho highest. rank a text book and authority
sturdily refusing to be supplanted by any modern
rival proves how thorough and complete was ms
mastery of the science and how skilful was his
ordering of its details. Eucld's work has Lem
shes as his most reverent followers admit. But
they claim that most of it's defects owing to the
nature of the subject rather than 10 faulty treat
ment and that no system which departs materially
from his method will begin to endure the searching
scrutiny which has been trained upon the " ble-
ments" for these hundreds of years The evident
advantage of having universally recognized
standard of reference is also urged IN favor OF
Euclid so that no scientific treatise is likely to re-
ceive a inhospitable welcome than Elementary
Geometry.
Professor Newcomb holds that the Euclidlan
system needs to be remodellcd in order to meet
modern requirements; but he claims TO adhere 10
the vital principle of the ancient model namely,
that of strictly following clearly enunciated dehn
tons and axioms, and stating the steps OF each
course of reasoning in their completeness ms
means that Professor NewcOmb realizes that the
function of pure geometry in scheme OF education
is not primarily to discover and apply mathematical
truth but to furnish a certain kind of mental
discipline. knowledge of bare geometrlcai facts
can be more readily attained by other processes
than the rigid logic of the Eucidan method but
in no other way can habits of close discrimination
be cultivated so effect vely, the power of logical
reasoning be developed certainly. Aud 1t Is
necessary that the basis of this discipline should DC
laid in the elementary study of pure geometry-for
in later stages of mathematical progress the analytic
methods of algebra and the calculus will natur-
ally be employed furnishing weapons of greater
force and compass. 1t is on this account that a
treatise on pure geometry prepared for beginners
should be developed with the utmost care. and 1t is
fortunate that the work has been undertaken by
mathematician of such sound judgment and wide
attachments Professor Newcomb.
Among his adopted innovati ons on the old method
may be mentioned the doctrine of symmetry, both
in regard to an axis and to point. These are
purely geometrical conceptions and at the outset
they make an important addition to the students
mental equipment. 1t is no objection to them that
they involve the idea of motion. for the same idea
is indispensable to the Euclidian method of super
position aud indeed without the idea of motion
there could be no comparison of magnitudes. The
generation of lines, surfaces and solids by the mo-
tion of points lines and surfaces also valuable
elementary conceptions, and they are properly intro
duced in the first chapter and might have been en-
ployel more frequently to advantage in subsequent
demons.-..-s'is. The measurement OF an angle by
the circular distance traversed by one of its sides
conceived to revolve about the angle as centre is
modern. The ad option of the term, straight
angle" the author eems to consider great advan-
tage but the gain is hit obvious. The treatment OF
parallel lines is quite do logical that of Euclid
which involves the questionable twelfth ION. The
new axiom emplo yed is open to less objection. he
subject of proportion has been treated III accordance
with the conceptions already formed by the stu
dent and therefore has advantages. Euclids famous
definition involves and unfamiliar ideas. But
it has the merit of being universal and the reason
ing based on it III the Fifth Book cannot be assaied
One other striking difference between the ancient
Euclid nothing is supposed to be done until the way
how TO do it has been set forth. Even in the con-
structon of figures every line drawn in accordance
with some postulate previously solved prob
lem In the present work theIe is a freer latitude
of assumption. The modern plan is clearly logical
for line which is assumed to be the bisectOr of
angle can be used for purposes of demonstration
with the same effect if the problem Of bsecting
the angle had been solved There is, however.
&C ofap in ha Md
way which is missed in the more flexible method
adopted by Professor NewcOmb.
Altogether there is much to commend in the book
even where it DIVORCES most widely from the beaten
track. The strong point in favor of these changes
is that they tend TO make the study of pure
geometry lead up more naturally and consist
emily to the processes of the higher mathemat-
les. The arrangement is excellent, and especially
to be commended is the practice of breaking up the
demonstrations with each step IN separate para
graph Numerous exercises to afford practice in
geometrical deduction and analysis are scattered
through the book aud will greatly increase its
usefulness. The free of algebraic language and
other symbols is no disadvantage, so long AS they
are employed solely tor purposes ot distnctness and
brevity. But sometimes and eve1 IN the earliest
theorems, the attention is directed rather TO the
symbols than to the object sgnifed, and results are
obtained by operating with equations while the
objects represented are dismissed from tne COIL
seriousness 1t Is the distinction of pure geometry
that it directly concerns things and not Symbols
and the peculiar discipline which it imparts is only
attained when reasoning is conducted with con-
slant reference to the figures themselves.
For those who are prepared TO abandon Euclid no
more satisfactory work covering this ground need
be looked for. The question remains whether an
edition of the Elements" of the Alexandrian
Geometer, with additions embodying some moro
recent ideas would not be preferable to any system
radically different from the ancient one.
TEARS FOR THE UNREQUITED DEAD.