1000000030

10 havc 19, : I

he .

(celia a pole, and

1('0

(lIeh po!es

" 10 ao .el<; I'.hereds. by Ihe lIalulc, ll>-!-lcet

m.ke

"p~le.

In (lIch cafes the aero is gre,'l\!( io tho duph .

" eale mio 01' II,e number

01'

rert '",a pole.

.. They \Vho me.,!'ure

I.nd

io Seodand by ao

" Enghlh ioches,

IlIJke

the aCLe lefs thao the true Seots

.. ac,e by

59,;a

(qudff Eogl'llt f,el, or

y .bou:

~'r

(he

ael

.. Ao h9(bano.laod coolaios 6 acres of (oek aod fYlhe.

" laod, th at is, uf laod that may be till,d with a plou¡:h,

" aod molVu with a fythe;

acres of arable laod Rla!..e

.. ao oxgaog or oXéog'te ; (our oxengale make a pouoe!·

" laod 01' olJ exteot (by a deeree of Ihe Exehequer,

"Mareh

11 .

1585), and is ' ealled

lihratn lur".

" forty.O\illiog land of ole! éXlenl eontaios eight oxgaog,

.. or

acres.

" Thearpent, about PJfis, eonl.ios 32400 (quare P.·

" ris feet, and is equal 102

Seots roods, or

,.\~7<s

Eng.

" lifh roods

"The

a{/ul quadrnlul,

aeeordiog to Varro, Collu·

"mella,

ce.

lI'as a C,¡uare of 120 Romao fcet . T he

" jllgnum

was ¡he dcublc of this.

is to the Seols a·

"ere as 10 000 to 20,456, aod to tbe Eoglifh aCle as

" 10,000 10,6,097,

was diyiood (1 ke the

ai)

ioto

" la

uncid,

aod the

ullcia

inlo 24

[crIJ/,ula."-Tbis,

with Ihe' Ih,ce precediog paragraphs, are laken (rom ao

iogenious maou(eript, wrilteo by Sir Hoben Stowart

pro(e{for o( natur.1 philoCophy. l'he grmefl pan of

tl,~ ta~le

io p. 693 . was takeo from it J'kewiCe.

PRO P O S

T I ON XXVIII.

FIG.

12 .

7 .'pJ/ioUllh,a" • •[a

n{/anglll'lr taral.

I,hgram

ABCD --Let the (¡de AB, for example, be

5 1m long, 20d

(whieh eonl!:lutcs I, i,h

/lA

ri~h[

.ngle at

feet. Let

be mul,iplieJ by

20.!

Ihe pr,duél8 5 IVi!1 betbe number of C'Iuare

(ee!

in the arca

o( the figll re AI3CD. Rut if the parallclQgram propo¡, d

is not 1(I1.ngIl12r

BEFC, its bafe

Oluhiplted into

its perpeodieul. r hei),t

(not ioto its lide BE) wllI

give

lIS

area. Tftis is

~",d<nt

from art o68. of

p3rt

O P O S I T 1

XXIX.

F,r; .

13,

jiIllJ If.,

arta

,eiv/TI Ifl""gl,.-

·I.el

th, triangle B.".C bo given, \\'I,ofe bale /lC is [uppcfed

loo~ : I~ '

the pt rpendieuldf

Al)

bedralVn from the

aogle A oppofite

the b,re, ano Jet

fUl'pofé

(o~r

(,el.

Let ,he

h.lf

of the perpenJ" ular be mul·

tiplieo iota Ihe

h.Ce,

oi the haleof the bJ(e into the pero

f,odieular, er takc the half of the produtl vf the "hale

bale ioto the pe'pendieu lar,

t ~e

pr<l¿Lét ¡;il'ls 18 fquare

(<et (or lhe arra of the gi>co triaogl,.

Dut if

~r,ly

lhe liJ,:! are giY< n, the petpendirlllar is

found ti'hel by protraéling th¿

iao~I<,

or by IZtll aod

13th

Eucl or by triJooometry [J.u how the 3t<a uf

tri,ngle mal' may be found (rclII the gi"eo

fi jes

0nly,

Jilall

fhe"'o io Ihe

tll p"'p.

O p O S I T I ON ;;XX.

Flc. 14'

jinJ

11.,

afla.r .ny

"":I¡¡'" alf.~urt

¡Jte fig'lf< be irregu!.,r,

I~t

it hc

f"I,(t1 ioto t".,ogles;

aod t'raw,og

rrt

penJicul.rs

10 tite

~.f"

in c,I,;h

(,1', 1"

111 ,

I~t

tf,e al'( ,

e.,eh

tri"n~le

1011".1

by the

1'",.

·:hng

l'rop. aou

ti,"

(um

~(tldc

a«.s

w,1I

ve thc .. lo 01 t::o

gUIe.

lt .

SClIúL I V/II

l o

OIe.Ir

."ing b03lds, I'lallks, aotl glals,

tb.ir

rode! are

to be IIlcalu rrd by a (oot·rlllediviJed into 100

c~lIal

pam;

and arter mllltipl yiog the lides. the decimal (ral1 ions are

e.lily redueed tOlerrer deoo:n,oatioos. The OIcnCuration

of thde is eaCy, wheo they are rellangul... paralle/o.

grams.

Sell O L/ Vlli 3.

a field is to be mealured, let it (¡I'II be plolteo 00

p'per, by (oOle o( the method! above

de(eri~cu,

anolet

the figure (o

l.iJ

down be

divid~d

ioto triaogles, as was

010100 io the p.recedi'!e propo(¡tioo.

The ba(e 01 aoy tnaoglé, or the perpeodicular upon

the ba(e, or the olltaoce of aoy t\Vo poiolS o( the field, is

moafu red by applyiog it

the fcale aeeording to wbien.

the map is dra\Vll .

S C HOLfV/11 3.

Bnt if the field giveo be OOt io a horizootal plane, but

uneven aod mouotainous, the feale gives the horizootal

lioo betweeo any two poiots, btlt nut their dillanee mea–

(ured on the uoeveo (url aee o( the field. And iodeed it

",ould appear, that the horizootal plaoe is to be aeeouot·

ed thé area o( ao uoeyeo aod hilly eouotry. For if fueh

r.rouod is I,id out (or buildiog 00, or (or plaotiog with

trees, 01' bearing eorn, Goce thefe lIanu perpeodieular to

the horiwn, it is plain, that a OIountainous eountryean–

not be eoofidered as of greatu extent (or tlto(e ufes thao

the horizootal plane; nay, perhap' , for oouriOling of

plants, the horizontal plane mal' Ge pre(eraSle.

hOIVever, the area o(

(¡gure, as it lies regul. rly on

the (ur(oce o( the eanh, i! tO be OIed(ured, this may

ea(¡ly done bl' re(oh'iog it ioto

tri~ogles

as it lies. The

fum o( their areas will be the area (ought; Vlh ich ex–

eeeds the area

the horizootal figure OIore or leCs, ae–

cordiog as the

ficld

is OIOre or I, fs unel·cn.

P R

P O S [ T

I ON XXXI.

FIG.

13,

Th'jid" .jatrialI/il,bciJJggivr1l, t.¡.dth,a–

f(a, .vith

,/JI

jil ..

ding

1/'(

f upmdicular.

Let.1I the (¡des of

the .tr idn~le

be cnlleéled ¡titO ooe rum; from the hale of

wh ieh let the fides be (epar?rely (ubm8ed, that time

diff<fenecs may be (uunJ betlVlxt the forcf,id half fum

ano each (¡de; titen

I~[

the(e three difierenecs and the

t,alf CUOt be DI"lt ipllod into ooe another, aod ¡Jte fquare

100t o( the proJull will givc the are. of lhe tridngle. Fur

txal\ll'le, let the fides be \0, 17, 2\ : the half

their

lum is

24 ;

the tllfee drtrerenm he," ixt this hal( (um

and the tltlel fid,;s, are 14,

alld

T ite fir(J being

multiphetl hy tk feeond, and th..ir prOl!ltél by the thirJ,

\Ye h.,c

294

for the

I'roc<l~1 ~f

tbe di

R'tfl

,es , ","ieh

multiplicd by the (OlrlCod hal( C<ltlI

=4,

gilt s 7056 ; the

[qu:tfe

rout nf whieh 84 is the;(fe. nI' the

I, i~ogle

Tho

d,rn"nllt ' t,oo

th,s, lor Ihe

¡;,ke

r<lil y,

IYe

omito

It 's tO be fOlLnd io I"'eral t, eatires, p.trticulolfly in Cla–

"ius's P"u'I,d C;"onlctry.

P K O P

1 T

I ON XXXII.

F' G. ' 5,

rl" "rdll~f'lj("·di,,.I"kUI(A BE FGH

19"111

Ih,

pr.drdl

11"

l'oIlj ClI'(uJJi[,rrlJ((

Ih,

/,.I

,g.JJ

1iI',lti/l"d i/Jlo

1/"

f'a f'wdl:lllar

u'roOJJ1/

/'-1.111

/,"' Cdllr6

ol lh~' (i"/llIIfi·,il~(.J

"·,,,;Ir 1(;

til;

fd,

ti,,·

p" !I:~ ""-F,,r

,lte

(\rJina'c (¡I'"re fan be reCol·

, ~~ Jllt~

dJ 111.111)

l..'

(jI.JJ

;1l;lók~.

élS tllll l..

are {¡des uf the

{¡sule