Burlington free press. [volume] (Burlington, Vt.) 1827-1865, 1847-12-10


between the area OF such polygon, the area OF like
polygon the diameter Of whose inscribed circle is L.
That because 1887 Tor cube root of 81 is the farther
Of two mean proportonals from the area Of poygoI
the diameter of whose inscribed circle is J. to the di
ameter of the inscribed circle of like polygon who's
is .5-and from the area of poygon the diane
ter OF whose circumscrbcd circle is to the diameter
Of the circunscrbed circle of like polygon who's
area is 5. Hence wherever the area OF the circle
whose diameter is may be situated 1887 is the far
thcr OF two mean proportiolals from such area to th'
diameter Of the circle who's, area 8 consequently
if the Of the circle whose diameter is /, is 1879
the diameter of the circle whose area is 3 is 1887
and for the that the popular area OF the circle
whose diameter is |, is assigned at 7s33's, &c henc'
the popular diameter of the circle whose area Is D Is
assigned at 1882, &c.
That when one fourth of the pcrmeter OF regular
poygon is the area is equal the diameter OF th'
inscribed and is the reciprocal OF the area OF
like polygon the diameter of whose inscribed circle is
land the diameter OF the circumscribcd circle Of
such polygon is the reciprocal of oncfouh of the pc
rmeter of like poygon the diameter of whose cir
cunlscrbed circle is l. And hence that necessary
conception Of the relative lengths Of the periphery Of
the circle and of the utmost final pcmetcr OF its in
StTlDeu anu w '''''
mence with trgon, square, or any other scres0 is
tina, the periphery of the circle is the farther Of three
mean proportonals from the final inscribed, TO the
final circumscribed perimetcr.
He might have stated the fact, and made his attempt
to account for the phenomenon otherwise than as
have accounted that when the area of a right angled
soscees triangle is l. onctourh fits circumference i:
equal to the area of the octagon whose crcumerenc,
4 A.. l.ll.. For
sidcration viz THAT although of two pcrmeters o
equal length but of different poygols that having tho
greatest number of sides will circumscribe the mos
area and consequently the periphery of the circle vii
circumscribe more area than the pcrimctcr OF AN!
polygon of equal length, (Which property is general.
called the permctrc capacity Tor clicuscrblng area,
and although there is actuai increase IN the peri
metric capacity of all the permetcrs situated between
the square and its inscribed circle 1T has nevertheless
been suspected by geometers and mahcma'clails
that such capacity is increased in greater ratio iron
the square its inscribed octagon, than from the oc
t1gon to it, inscribed circle in proportion to the prox
unity or length OF such perimcter to the pcrinctcr
it, crcumscribed square Whereas the popular mehoC
of obtaining the circle increases the permctrc capacit,
from the square to its inscribed circle 10y the rate
in proportion to its proximity to the square
IT, then, we would obtain the length of the perphe
ry of the circle whose diameter is |, by means OF
progression through an infinite series consisting OF it
crcumscribed square, octagon, and so on, by con
slant doubling of the number OF sides of the poygon
of the series, process must be so continued tina
from the octagon onward to the circle the permetr
capacity shall increase by diminished ratio in pro
portion to the proximity OF the permetcr to the perrin
etcr of its circumscribed square: which dminishin!
ratio, as well the law by which it dminishes
(Which ought to have been noticed long, long ago. il
expressly dictatcd or pointed ont in the simple fact
that mean proportional between the area Of the oc
tagon, the diameter OF whose inscribed circle is nn
the dinmctcr of the inscribed circle of the octa goi
whose is lis proportonaiyj Just half as far abov
the area of the polygon OF sixteen sides the diancic
OF whose inscribed circle is !, as proportiona
between the OF the square the diameter Of who's
inscribed circle is and the diameter OF the inscribed
circle OF the square whose area is >, is above the
of the octagon the diameter OF whose inscribed cir
Gle is l. And this law, maniicstly denoted die
taed, places the area OF the circle whose diameter
|, in 1881 also the diameter OF the circle who's
area is D. And without this law to govern and direc
us, polygonal economy and phenomena is wholly in
explicablc and the quGdraturc of the circle who'll
impracticable.
But this article is not designed to give exposition
of the quadlature but only to suggest what the re
viewer might, perhaps, have considered as pausibl
argument, had he not been welded or would rattle
say strangely attached, an object, the beauty an
projections OF which, however well endorsed by lon
train OF lovers, might, nevertheless, be in danger
reputation Were he permitted to expatiate, con
template other objects
But | will cnnclndp This branch of tbo srbect l.
suggesting that when the popular method and fins
result shall be compared with my own i, will be oun
to corroborate and establish my ratio: And whc
mine shall be accepted and acknowledged as the tru,
then the architect the machins, and mechanic, (wh
are always law-lovng people, and desire fulfill it
will cease TO wonder why they could never fulfill IN
law-why large or heavy cylinder (ns of iron f.
insanceO is more ponderous in proportion to its popu
lar measure OF bulk, than that OF square bar: wh
the architects geometric ratio, while it makes the be
rphery greater in proportion to the diameter than
given by the popular rule should still be found litt
too small Tor practical use: And then the world wi
wonder why is that the circle which is tn
most simple and beautiful OF all geometric figure
should, i'll all its numerical or mathematical appica
tons, be the most homely and complex
/ will hence pass a consideration of the review
ers comments upon the idp Of grarify and as h
review las mnttcr OF coursc) consists only OF III
Ncwtonian exposition of the law drawn from his CO.,
sdera,ion OF the law of falling bodies, shall in tn
article, far pursue the OF the reviewer as
draw my deductions, conclusions and differences in r,
; gard to the law of gravity from the law OF ian
bodies, Notwithstanding there are other more simp
and satisfactory methods, which have noticed an
} explained in my work-to which work | must RET,
for full and satisfactory exposition OF the la
of gravity than can be presented in this brei article
and although IN my work have altogether oyerbu
dcncd the subject with evidence, for which | hay
that whoever shall read the reviewers exposition
t the law Of gravity, will fully appreciate and acceI
! my apology.
} | will hence proceed to set forth the premises, wi
; Me lar Of falling bodies from which Sir Isaac New
ton inferred that the rate or intensity of the force
gravity varies invcrsy the square of the distance
and from which inter that it varies nycrsely as ii
distance from the centre OF gravity from which
will be that II is not the premises upon which
differ, but only upon the deductions, conclusions an
Segment of the law: And with view to such assigr
ment, | will proceed TO set forth the premises rol
which for the sake OF the argument at least we shn
all agree that our deductions and iniercnccs are to
drawn;
f FIRST The attraction of the earth retains the moo
in its orbit and heavy bodies to fall upon IL
surface OF the earth and the centre OF the earth in It