New-York daily tribune. [volume] (New-York [N.Y.]) 1842-1866, 1851-06-14

cei,sd NTH three dlHeaS0Rs And again ne says IA
* evidently impossible 10 conceive any ertsRSpn a.
,o,,J se,-,,sd Of ant one Of The three ftlfcmetg!
diruensOnS." / might quote page amer p"5e HC (
Cornte directly in support of tins strange phosopAy
/ Of the 90 New Elements" when FfoI Looms sa's
"ls calculated to spread vel OF darkness over
/ subject which before was clear as MONDAY
Comte asserts that surfaces have thickness and the
New Elements" demonstrate rfdt taat tt-N'ieSs -
Comte asserts that lines have magnitude IN Oreadth
! and thickness as well as IN length- ana tne -Ne"
| Elements demonstrate the quantity OF thai naghl-
tude Comics CLASSIFICATION Of the different ordll Hes
If mathematical science Is emnently clear anu pm
losophica And according to that classlhcauon NE
! lines and surfaces which are demonstrated N Ihe
New Elements and which are there sOHetIli re-
| called mathematical lines and surfaces >houiu "
called geOrpefricaj lines and surfaces lor tiey do ~
| fact purely geometncal. They are the lines anu
surfaces of Measurement recognized by the universal
laws of geometrica operation. THINK therefore, j.
In he nroner 10 restrict the definitions 10 geometr
cal lines and surfaces and more especially so since
the poverty Of language seems TO leave US no hloie
convenient expression for the abstract idea OF lehs
or distance between two points, than to cal II an ay-
,tract ltne. Custom and CONVENIENCE VIII pIoDaoy
require the continuance Of THIS improper use OF tle
word line Therefore let the true line OF geometry
be known As A gcOmetrlcai line. Ahu lhe sdlle io-
mark applies to surface.
But must notice some particular points IN Profess
sor LOomiss lecture. The first half OF the lecture
is devoted mainly 10 prollX statement OF the com
monIdeas Of Geometry and having no bearing upon
the principles OF the New Elements, does not re-
Qure a single remark. In the second nan me
learned Professor attempts 10 show up some Things
III the " New Elements which ne presents as pro-
found errors and perhaps thinks they are 50 and ne
seems 10 hug himself with great comp.acency al ne
idea of having given them such keen and satlrlcal
| handling Let slft some OF these polnts ne
| Professor quotes the seventh proposition OF the
New Elements" which Is TINS. The diameter OF
a circle inscribed in an eqmateral triangle equals
two-thirds of the perpendicular Of The triangle Inc
Professor says, THIS proposition IS true as the
terms arc defined by EuChdbut il Is not true as the
terms are defined by Mir Smith For he has dlscoy-
\ 4ffgff IN a a
circle according TO the New Elements IS Shorter
than the diameter according to Euclid. The dame
| ter according 10 the " New Elements has breadth
OF one 1t is reCtangle therefore quOth the frO-
fessOr the ends OF the diameter must form chords OF
the circle and such diameter cannot be so long aS
Euchds Profound critic! The nineteenth dehnl-
ton Of the New Elements" reads thus " The d.
ameier OF circle IS a straight line passing through
It's center, and extending length 20 The erTe1le H
us If the cucie." Does Euchds diameter extend
something beyond the extreme limits OF the circle, IN
! order thai 1t may beat the diameter OF the New
| EomenisW
The Professor has found a flaw a note appended
70 the New Elements" where 1t Is stated that IN
any given quantity of matter or space the relation OF
.Ohdliy diameter, ana surface Is an INFINITELY vary
| iIg relation. changing with every change OF form IN
the given quantity" The circle and the eQu lateral
triangle are mentioned as the two opposite extremes
of this varying relation IN plane hgureS ana the sphere
and ctrshcdrOn the two opposite extremes in solid
! figures And II is added ID this connection that
/ sphere whose surface, for instance, equals SIX
square inches, contains the greatest amount OF bulk
sohdlty and the tetrahedron the least amount OF
bulk sOlldlly, that is possible TO be contained un-
| der the given quantity of surface. The Professor
doaches the substance Of this last remark from the
context and from the condition iFtch IL IS entirely
! based, and prOnOunceS 1t altogether falSei' In
! proof of his assertion, he SAYS A square prism
whose surface Is square inches, and altitude one
tenth of an inch, has capacity less than half that
of the above mentioned tetrahedrOn.' Profound
critic Tie Professor Is emInently right IN this
sertion And If he had reduced the altitude from
One-t Cnth Of inch down TO nothing, he would not
only have obtained capacity less than half the
tetrahedron, but he would have excluded every YES
uge Of capacity altogether. But he seems to have
forgotten, or not to have COmprehended, what Is
clearly stated, that the whole matter in question
the relation of sohdlty, diameter and surface"
Everywhere In the New Elements A" the diameter
OF solid IS the diameter OF an inscribed Sphere-a
sphere touching every face OF the solid. The acute
Professor presents SOLID which has such diameter,
and cannot have, and tTlumphanty eXclalmS, your
law Of relation is altogether false" Does Euclid
anywhere teach how the relation between three
varying quantities can be continued and shown
uhen enc u tho quantities Is entirely eTct1ied Jror
The CGSei
particular proposition in the New Elements
the SlXiyeghth, reads as follows: In every right
anglCd triangle, the square of the hypOthenuse equals
four times the Of the triangle, plus the square OF
the difference Of the other two sides" The demon
slrailon OF tins proposition with the Diagram Is gwen
as III original discovery, which with me II truly was
did not suppose IT had ever been demonstrated by
geometucal diagram before But Professor Looms
says the proposition was demonstrated by Hlndoo
be found m Huton's Mathematical Tracts. 1t may
be SO, for ! never saw d" Hutton's Mathematical
Tracts II tls So II only proves, for the thousandth
minds have originally and independent of each other,
made the same discovery
Another important and original proposition in the
New Elements the fitv-second Is as follows;
" In al tnangleS whatever, The whole circumference
bears The proportion TO the base the perpen-
dlcuar OF the triangle bears to the radius of the
scrbed circle" The learned Professor charges thus
proposition with want of originality but he fails
entirely To show II. lie cites TWO propositions OF
Geometry which he says are equivalent to IT, The
frst is the very simple and common one" The area
OF triangle is equal TO half the product of It's base
by It's altitude" The second IS this; THE area Of
triangle * equal 10 hall The product OF ITS permetcr
by the radius of the inscribed circle" This last
not quite simple and common the other PROPOSE
uon b II Is denucaly the same principle, presented
IN another form For We take any triangle and
draw lines from the center of the inscribed circle to
the three angles We divide the triangle into three
new tuangleS, each of which has side of the Orgl-
nal triangle for lis base, and the radius of the
scrlocd circle for Its altitude orperpendcul&r Then
each OF the three tuangles comes under the simple
proposition OF nall the base 10y The altitude or radius
Ana the three bases make the permcter of the
original triangle, OF course halt the peumeer by the
radius viii give the area of The original triangle
Now 10 vii readily be seen that the hftysecOnd
proposition ot the New Elements has no reference
whatever 10 Gred but expresses universal law OF
lauon between the lines of all trangleS And vet
Professor lOOmS discovers an equivalent to IN he
simple proposition that the area OF triangle EQUALS
hall the base by The altitude. l think nobody else
III the world but this learned Professor would have
been capable OF making the discovery
cannot spend the tune, or take up the printers
time. 10 notice few other points brought up by the
Professor, net Of which has the slightest sub
stance as an argument against the principles and
theory OF the New Elements" | therefore beg to
take leave Of Professor LoOmS with all due respect
and IN perfect good nature for after ALL hope he
didn't mean any HARM and am he has done
none.
do rot feel. however. that should discharge nv
fan duty To the public IN tins matter without adding
he following remark by the distinguished Gcome,ox
and Mathematician, Professor PlayaIr: Even m
matters purely Intellectual and m Inch the abstract
truths of Arithmetic and GeOmcty seem alone con-
eelned the preludlces, the SelhShneSS or the vanity
OF those tho pursue them not unfreQuenty combine
10 resist Improvements, and often engage no neon