New-York daily tribune. [volume] (New-York [N.Y.]) 1842-1866, 1850-12-02
NEW PUBLICATIONS.

8vo. PIP. 500. Neyr-York: George P Putnam
The attempt to present the elements OF Geon.
etry under new aspect-appying the principles
OF modern free inquiry to science whose claims
to rigorous exactness have stood the test of ages
calling in question, without the slightest defer
ence to authority the essential definitions on which
the imposing structure has been erected as on a
basis of primeyal granite-may be regarded as
signal proof Of Intellectual boldness or in the opin
ion of many, of scientific audacity. Indeed, go
firmly have the established doctrines OF Geometry
been fixed in the general mind, that no one could
presume even to doubt their correctness, without
mourning the suspicion of charlaanry superficial
Investigations The author of this volume, how.
ever, is not liable to either Of these charges
Judging from the intrinsic character OF the work,
he is not person OF crotchety mind but firm
and cautious thinker; with no love Of paradox, or
of novelty, for its own sake devoted to the pur-
suit of truth with an uncommon degree of mental
honesty and arriving at his convictions by the
sheer force of inquiry and evidence. Whatever
opinion may be formed to the business of his
concusions-and this is question which should
be submitted to q jury Of eXperts-no fair. minded
reader can fail to recognize in this treatise high
degree of ability, devoted study of the subject,
and facility in the expression OF mathematical
results, which should bespeak attention to bis
claims and insure them the chance of al unpreju-
diced examination.
Whether the views Of practical geometry which
he advances are founded in the nature Of things,
or are merely the fruits OF ingenious speculation
point which we acknowledge our incompeteacy
to decidc-he certainly has presented striking
collection of the harmones of quantity, of the pro-
portions and relations Of various geometrica fig
ures, which have escaped the notice OF previous
students, and which afford curious subject of
mathematical inquiry.
The main principle of the New Elements Of
Geometry" is that in the measurement Of extern
sion, the unit Of reference is invariably CUBE.
Rejecting the usual division OF quantity into lines,
surfaces and solids, incapable OF measuring each
other, the author maintains that there is but one
kind of geometrical quantity, and that lines, sur
faces and solids, being identical in their nature.
are always referred to the same unit, and are
hence perfect measures of each other a mathe-
NAUTICAL line accordingly is not filmY, airy thread,
or pure ideal conception, reduced, In the last
analysis, to zero, but real magnitude, positive
quantity, used to measure and compare positive
quantities. Hence the unit employed in geo.
metrical measurement not only represents mag
nitude, but magnitude of a definite FORM mag
nitude with an extension in every direction from
its center not only one in length, but also one in
breadth and one in thickness.
Having established the unit Of comparison, the
author pursues his expostions in regard to the
bptyy he Oily m diameter
and area of plane figures. The area OF plane
figure is determined by its diameter, line pass
ing through the center of the figure, and its cir
conference, line outside of the figure, and en-
closing it. The law which is set forth by the
author as universally controlling the relation in
question is that the diameter being one the area
equals one fourth Of the circumference, the diane
ter being two, the area equals twotourths OF the
circumference, and so on, until when the diameter
is four, the area and the circumference become
equal. This law applies to all regular plane fig
ures, the circle, equatera triangle, square, PEN
tagon, hexagon, and all regular polygons OF any
number of sides The diameter, in these cases,
is always understood to be the diameter Of the
inscribed circle.
similar law exists, it is maintained, in the
measurement OF ali bOlu AlSuACS "ltiA Pi'i1~ SNA
faces, the diameter in this case being that of an
inscribed sphere. The difference in the two kinds
of figures is that while in plane figures the point
Of equality is four, the corresponding point IN solid
figures is six
From these principles Mr.. Smith develops
system of geometry, which cohering in all its
parts presents the whole subject of magnitude in
new light evolving relations of beautiful and
unexpected harmony, and suggesting views of the
most curious interest to the lover of mathemati-
cal investigation. The originality of the work
will commend it to the attention of those who are
in quest of intellectual novelties while its bold
ness and freedom are adapted to pique the pro-
fessors of the established science to diisent en-
deavor for the detection of its fallaces.
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DDnNIyJ N~N ITEMS