New-York daily tribune. [volume] (New-York [N.Y.]) 1842-1866, 1850-12-27

been called solids. | have a'so discovered that
the laws and operations Of geometry recognize
but one kiad Of uan, and that tue unit in every
possible application has the to7lq OF the perfect
cube | have therefore adopted definitions TO
agree with those discovered laws ofgeonetry.
NV wise antagonist Professor SAYS, w nd.
wise al reformers and progress men TO let the
principles Of geometry alone: aud he even ven
lures the bold assertion, that they cannot " oe
overturned by a modern poet" Whatever effect
that advice may have upon others, for one must
respectfully beg leave to decneit. have found
so much delight and satisfaction in digging among
chose principles during two la5orious wears, that
shall very likely be tempted to continue digging
among them as FOR " hid treasures " and when.
ever may chance to find uew one, for be-
]ieye there are many yet to be discovered will
endeavor to give 1t such definition the perfect
laws Of geometry require
Ny antagonist Professor again says, This we
know, tbathe the author Of the New Elements
mas not pursued geometry with any kind Of
cess" Ia answer to this pnsaiEg by the impor
tant principles and changes Of definitions already
mentioned. may be permitted TO ask him, if it
viii not tax too much his contempt to look
along in the volume of New Elements to the
sgd proposition, where this remarkable law of
fnangles is discovered and demonstrated, viz:
8 In all triangles whatever the whole crcumfer-
ence bears the same proportion to the base as the
perpendicular of the triangle bears TO the radius
of the inscribed circle When il is known that
triangles may assume an infinite variety OF forms,
and that this law applies to them all, shall not
hesitate to ask my antagonist Professor, what
geometer has ever discovered Il more beautiful or
more important principle in Geometry than this
Again, il my antagonist Professor can sufficient
jy overcome his contempt, ! would beg him to
do me the favor to look at the demonstration of
the 68sh proposition, which reads az follows:
" In every right angled triangle, the square OF the
hypothenuge equals four times tie Of the tri.
nil he u n q n
two sides" The truth of this proposition has
been known to mathernaticiens in the forms Of
arithmetic and algebra, but has never before, to
my knowledge, been presented in geometrcal
demonstration, though often and earnestly sought.
The diagram for the demonstratonis exceedingly
simple and clear Gentlemen Of high alathemati-
cal attainments pronounce the demonstration
Of the most beautiful to be found in the whole field
of geometry. / therefore, contrary to my antag
onist Professor, would "advise all reormers
and progress men not ito let the principles of
geometry alone but to examine carefully such
as have been discovered, and to search diligently
for more.
With REGARD to the main feature of my work,
shat the only amt in geometryis perfect cube, and
shat the laws of geometry make ali lines and sur
faces to consist of cubic quantity, requiring every
surface to have the thickness OF the cubic unit,
and every line to have the breadth and thickness
oi the cubic unit, have Just he'd an apparatus
prepared, and have written out lecture to accom-
It,
lally, aud in way to be readily understood even
by persons not acquainted with geometry
These demonstrations shown by measuring
era{e1 in various geometrical forms. By pouring
solids into surfaces and into lines, and pouring
surfaces into lines and into solids, can show III
crery instance that can be presented that where.
ever there is agreement IN Numbers between
solids, surfaces and lines there is always the
same agreement in positive cu6tc quantity. /
have in my apparatus spheres, cubes; tetrahe-
drons, octahedrons cylinders and cones, all made
on the scale Of an inch for the unit, that all the
surf ces in the demonstrations are one inch III
thickness, and the lines one inch in breadth
rnd one inch in thickness. The natare OF the de-
zonstrations will be better understood by stating
q particular case An octahcdron the solid with
eight equal triangular faces whose diameter 10
one inch does not contain quite one cuolc lnch
Calculating in Numbers the contents or solldlty IS
the decimal expression 866- +, or half the square
root offs. The diagonal of the same octahedron
calculated in numbers is l.7324, or the square
root of 8 Just double the soidty. BY the old de-
lintions OF geometry this diagonal is simply an
ideal icngA without magnitude, and cannot be
compared in quantity with the solid. Nmmbefs
say there is an agreement between them and that
the solid equals balJ the diagonal. The New
Elements say this diagonal is an inch In breadth
and an inch in thickness. Take such diagonal,
and the oCtahedron twice filed with water Will
lust fill the diagonal showing that the diagonal is
just double the sohdity in cubic quantity as is
IN numbers Now which is true to the law of
geometry, the cid definition or the new Nah-
iers which have never been convicted Of false,
hood since the world began, testify clearly and
unequiyocally that the definitions OF the New
Elements are true.
To vary the demonstration, take an octohedron
whose diameter is . The diagonal IN numbers is
IDOL. or tie square root OF IL. And the sofdity
in numbers is 6928 40; ,.s 9oH6le the diagonal!
And in this case the Diagonal twice flied WIZ
water lust fills the octahedron, agreeing perfectly
in cubic quantitv, as they are in numbers De
nonstrations of this kind, both Of lines and sur
faces, may be given without end, applying 70
hines in all situations and TO surfaces Oh soiids III
all forms.
the early foundations of geometry bad been
constructed upon the simple c@6e, instead OF tnree
Heterogeneous imaginary quantities, believe the
world would huve been saved the inilctlon Oh
thousands of tedious and formidable volumes OF
fnsthcmatical controversy, and the loss OF thou
sands of wears of valuable time, which have been
wasted in tTutless speculatlons ana VAN eI S
yors to make something agree with @9"'fHS. II Vnc
old definitions of geometry are t'Ne, am so huen
of n conservative that would be the last man III
+he world to lay Nandal hands upon the ancient
land. marks OF the science. Bat il they are 4ol
frue, | am Just enough of modern free IN,
ouirer' to be willing to see them swePG tW
like cobwebs | am ready tor trlai, MY WIZ
nesses (putpdcrSl are unimpeachable' Gentlemen
please let the case so to the jury
Respectfully yours, sEbA SMIIh'