705 / 1042
G
F.
o
M
r:OROLLAl/r.
Heoee ?ny l:ig(lnometric.1 problem io ril!h, .lioed lI i–
~r.gles,
lI'ay
b~
rdo!l'eu
by
dclir,eatiog the triangle f, olll
whal is giveo cuoccroiog it, as in this propofilion.
The
uoknol'lIl fidcs
Uf
examined by a line of equal parrs. and
lhe angles by
~
protratlor or line of
chord~.
PR O P O
SIl'
ION :XXI!.
'!'he dinTllt'l" r
.¡
a cirele being givw. lo jlJd ill cír–
e/lO/ji:rma
neo/'(y.-
The ptriphe, y of any polygoo in–
fcribed in lhe cirele is lefs than lhe circumfe..ence• •nd
the periphery of any polygoo.defcribed about a cirele is
greater thao lhe eireumfaeoce. Wheoce Archimedes
firl! difcovered lhat the diameter was io proportioo lO lhe
,ircumfereoce, as
7
tO
22
nearly; which ferves for com–
n:oo ufe. But the moderns have eompuled the proponi–
on of the diamet« tO the cireumferenee to greater exaét·
oer,. Surpofing the diameta
100,
the periphery will be
moré tllln
314.
hut lefs thao
315 .
The di.meter is
more oearly tO tbe círeumference. as
In
to
355 . Bm
Ludolphus van Cudeo exceeded the labours of all; for
by immenfe nudy he fouod. that. fuppofing the diameter
J
00
I
000 ,000,000.000
,oeo
,000,000,000,000,000,
the p«iphery will be lefs Ihao
314. 159,265.358.979,323.846.264.338.327.951.
bu! greater than
314.159.265.3 58.979.323.846.264.;3 8•327.95°;
whence it will he
e.fy.any pan of the eircumfereoee he–
iog giveo io degrees and minutes. tO affigo. il in pans of
the diameter.
Of Sur'Vcyir.g n/Id Meafllring af
L AND.
HIT
HER
TOl'Ie have treated of Ihe meafuriog of ao–
gl~;
aod fides. wheoee il is abundandy eafy lO lay down
a field. a plane. or ao eotire eououy: for to this nOlhiDg
is requifi te bUI the protratlion of tliangles. and of olber
plain ligures. after having meafured their fides and an–
gles. But as this is elleemed ao importaol par! of prae–
tieal geometry. we (hall fu hjoin here ao aceouol of it
with all polliblehrevil)' ; fugge!liog witbal. lhat
á
furvey–
Or will improve himfc!f more by ooe day's praétiee. tban
by a greal deal of readiog.
PRO
P
O
S 1 T
ION
XXIII.
r oexplain 'What
jur~t)'i1/g
iI,
and 'Whot injlrunl<ntl
Sur.eJor¡
IIft.-FirU. ie is neee!!'ary tbal the furveyor
view Ihe {¡eld that is
10
be meafured. aod
inv~
nig.leies
fides ar.d angles. hy means of aniron ehaio (having a par–
lieular
m.rkal eaeh foot of !eogth. or al aoy number
of feet. as m.y be 01011 convenieot for redueiog lioes or
furfaees to the reeeived meafures). aod the gr:t?hometer
defcribed abol'e. Secondly.
It
is necerrary
10
d, hneate
¡he field
in plano,
or to form a map of it ; that is.
10
Jay dowo on p'per a fieuro firnilar
10
Ihe fidd ;. which is
dooe by the protraétor (or lioe of chord,) and of ri,e lioe
of equal pms. Thiroly. [t is omlr:"y 10 fiod OUI ,he
arca of Ihe fidd fo f""eyed 3nd reprefented by ama?
Of thi, lan we are 10 treal htlolV.
The fides aod
an~les
of fOl.1I fidds are rumyed hy the
help of a plaio table; which is genmlly of an ohlong
retlaogular figure. aod fupponcd by a
["ler,,"'.
fo as tO
turo every way by mc.o; of a ball and fo(kce.
lt
has a
E
T
y
1Il0vfal>le frame. which furrounds lhe board. and fervcs
~o
;;eep a
~Ic,¡n
paper pUl on th: board clofe and tight 10
11.
1
he
(Jd~s
of the (rame f"",g the p"per are divided
into equal pans el'ery v.oay. The board hath bcfides a
box with a magnelie oeedle. and moreover a large index
with 1\\'0 fights. On the eclge of the frame of the board
are marked degrees aou mir.ules. fo as 10 fupply the room
of a g"l'hometer.
PRO
P O S 1 T
r
O N XXIV.
Fr G.8.
r o
de~i1/eale
n
jeld
b~
!he {¡elp
of
a pfnÍ//–
loMe.fr.om~ne
flallon 'Whence
011
111
anglu IIJaJ he /ten,
and Ihelr dijlancu ",eajured hJ a chain.-Let
the field
Ihal is
ro
be b id dowo he ABCDE. At any eonvenieot
place F. let Ihe plain-Iable be ereéted; eover il with cleao
papero !n which let fome poiot oear lhe middle reprefeDI
Ibe Ilauon. T hen appl)'lOg at this place Ihe index with
lhe fighls. direét it fo as thal through the fights fome
malk may
h~
feeo at ooe of Ibe aogles. fUPFore
A;
aDd
f~om
the: p0101
F.
reprefenllOg lhe nalion, draw
a
faint
nghl Jioe a.long Ihe fide of .Ihe index: lheo. hy lhe help
of the chalO. let FA the ddfaoee of the nation from tbe
fo~efaid
angl
7
be meafured. Then lakiog what pan you
t1l1ok convement for a
~oot
.or pace from the lioe of equal
parls. fet off 00 Ihe f..ot IlOe Ihe pam cDrrefpoDdiog tO
the line F.A thal was Oleafured ; aod let there be a mark
made.reprefenling Ihe aogle?f tbe rield A. Keepiog the
lable rmmoveable. Ibe fame IS lObe dooe "'ilh the rel!
of Ihe aogles ; Ibeo ri&ht lines joioiog thofe marks (hall
inelude a figure Jike tO Ihe field. as is el'ideol from
S. 6.
EueJ.
COROLL A RT';
Tbe fame thing is done io like maDoer hy the grapbo–
meler : (or haviog obfmed io eaeh of the
tri~ogles
AFB
BFC. CFD.
ec.
Ihe aogle at Ihe natioo
F.
ando
havin~
me.fured
Ih~
lines.from Ihe {tatíoo tO the angles of the
field. lel Gm"ar
lrIangl~s
be
pr~traéted
on paper (by Ihe
21.
prop. of thls) havtng thm eommon vertex in lbe
poiot of {talion. AII the lines. exeeptiog thor, whieh
reprerent the lides 01 the field. are 10
be
drawn faiot or
obfeure.
Note
l.
Wheoa furvc)'or want, tO lay down a field
Jet him place' di!\inélly io a regiller 311 rhe
obf~rvatioos
of
the angles. and lhe meafures of Ihe fides. uotil. al time
aod place eoover,icot. he draw oue lhe figllre on papero
NOle
2
The obfervations made by Ihe help of the
graphOOleter are to he eX30lined : fvr aJl the aogles about
the poiol
F
onght 10 be equal 10 (our rigot ooes. (by coro
2.
arto
30.
of par!
1.)
PR O P O
S 1
T IO N
XXV.
F'G ' 9'
Tol.,)'d01vllafeld
hy
11I1an¡ 0(
1'W.jI,tior.I,
f~,m
(ach
.¡
'W~ich
nlllh. nl/glll (."" beIcen. hJ
mM[u–
r/llg 0,,1y the dijlallCl
.¡
Ihe /lal"'lI.
- Lel the in(hu_
ment be pl._eed at Ihe {latioo F: ,nd h.vil1g eholeo a
poinl
repl efentio~
it upoo che
p"p~r
wllld. is laid upon
Ihe plain táble. lel Ihe
i~Jex
b: aprlie? '1 Ihis poinI. fo
as to be Olov..ble abollt
rt
1
hen let '1 be dire8ed fuc–
C<'ffi vrly 10 lhe fever. langlos of :h:
r.dd:
and IIhen aoy
angl. is fcen,hrollgh Ihe
fi~hlS.
dr,¡w ao obfeure line a–
I""g the fide of che index o Lel the indexo lIilh the
liehts. be direéted afler the fame maoner
10
th: lI_tioo
(j :
on lb, obfeure
lin~
dw tn .:long ils fideo roin¡ing tO
,'l,