Burlington free press. [volume] (Burlington, Vt.) 1827-1865, 1847-04-30

wi,osc diameter is L. and whose centre is G, AM
whose radi CA, CD, CD, &c. are consequently = ::
each. Now since the radius of a circle equals IN,
chord of l6 of its circumference, if the chord BID be
drawn equal to CB BID will be one side OF a rernla
Inscribed hexaig5on or poygon Of six sides and b d be
ing equal to CB 3, the whole perimeter of the in
scribed hexagon will be (5x6) g. Let the arc anc
chord BD be bsected in a and E. and through A
draw FAC parallel to BED; FG will be one side o
regular circumscribed hexagon OF which we wish tc
BE IS and BE a5 being hall of BD and be- BE- '
BEim or 3 - 25 IL 1873, and CT = .433Ol27
c q! and b EcR are s ss r'9se: and ssns,
IA but FIG =FAx2 = .2ss675lx2 = 57735O2 :
one side OF the circumscribed hexagon aud 57735Ol
X6 = 3i6flOl2 = its perimeter The periphery oil
the circle, therefore, exceeds 3, and is less than
Again; the BA being one half of BID, the chord
"a Is one side OF regular inscribed poiygon OF II
SIDES and IS also the hypo,enuse OF the right angled
triangle ABE, the two legs of which, AE and BE, are
hence AEaxBE5=ABa, or, IS o669s73s =AB@
23'sl9O. The perimeter Of the inscribed poygon o
ing the chord and are BA in and by the radius Co
and through drawing ba parallel to the chord BA
ana proceeding precisely as in the case OF the poygon
OF SIX sides, BIO is found TO be .l3391i51, and
qnenty ba=twice bo is 2679i9li =one side of a cir
xl2=3al5ss96s= the perimeter OF such poygon
Thus we find the periphery of the circle, lying be.
and less than 32l53s96s, but differing from them much
less than in the case of the hexagons.
Again the are BA being l-l2th OF the circunfer-
ence, the chord OF its half is the side OF regular
inscribed polygon Of "a Sides the length of which cal
culaed as in the case of the poygon of IL Sides i.
l3J526l, and then, bisectng Bo and proceeding as in
the case alluded to, the corresponding side OF the
uar circumscrbed polygon of 81 sides is found be
.l3l653!. These being each multiplied by - give
l3!626i tor the perimeter Of the inscribed aud
5616S Tor that Of the circumscrbed polygon of y
sides. BY repeating this process we should next OBJ
IS 4 -
polygons OF Is sides, and by repeating it again. thos,
OF 20 sides and so on; but, have carried the
process sufficiently for our purpose, We will here droI
ii and collect the results thus far
Inscribed, 100000 5oooooos6 : oSooooo
Z iC"cumscribcd, 57535o2 S,,a,o2s6 = 3i6,lola
9 Cmumscrbed, 2679i9li e6,9i9li.la=3 2la,s96
81 Inscribed, .i3o55l i3os26iaai=3 l:ta6e6i
Thus have shown by a process. which chile
can understand, and which our author acknowledge,
To be legitimate, that the perlmetcr OF n regular poly
gon OF - sides, circumscrbng circle whose dianete
is !, is 5J676S But author asserts that the cir
cumierence Of a circle, whose damater is !, il
3l7is02o, greater than the perimeer OF the poly
gon, which includes it by Ol5l25. reductio al ab
surdum which is sufficiently palpable
The process which we have described may be car
ried ON by increasing the number OF the sides of tn,
poygons, ill the two perneters and the periphery o
the circle shall coincide to any assgnabe extent but
on account of the great labor required in the opera
OF abrdgiug it, They all, however, TO the extent I,
which they carry the OPERATION agree in the result
with the exception OF our author and his result hay
ing been proved To be erroneous it does not
b. necessary 10 say any thing more respecting it
Indeed, an exposition OF the course he has pursuet
would require far room than it is worth. liav
ing sought out new track, invented, in part, new
notation and then, after boldly procaiming that her,
is legitimate geometry, excepting upon lines, h,
goes on through 30 or more pages OF rignarole re-
sprctng the approXimatng series OF polygonal perrin
eiers ill he reaches point where he inagines tina
the ptrmeers become merged in the periphery OF tn,
circle establishing his ratio, which we have alread,
reiuted and in support OF which we confess that W,
have not the acueress discover TINT he has advan
vanced even one plausible argument Not content
however, to coniine himself To the subject he is dis
cussing but true his unity purpose OF abusing 81
Isaac Newton he anticipates largely, in this chapter
the subjects at the succeeding ones, thus forming
tumble OF heterogeneous materials, and bringing al
OF things into unity, but such unity as exists onl,
in chaos.
/ The second chapter, occupying 100 pages OF MIR
Ioungs work, is devoted 10 n consideration vi tn,
/ lar Of grdpify.
The iap of grdrify,' says he, in this division o
the subject" referring to his division into theory an,
lar will simply consist in the rate intensity ELI
cacy which the force OF gravity exets, between tw'
given quantities OF matter, in proportion to the dis
lance between them at which il shall operate an,
hence to determine the numerical law OF gravity,
simply to determine the rate Oh intensity in proportion
to distance,
Having thus defined what he means by the law