1000000030

69 8

In \\'hat follows, a fi gurc is

o~ten

to be j¡id clown on

p3p~r,

li!:e

another

figu,r~

given;

~nJ b~eauf~ t111~

lik~nefs

eonr.Hs

th~

equahty of ,hm ,ngb,aotl,n the

fides having lhe fame proportion

eaeh ot;,er (hy lhe

d~·

finilions of lhe 6th of Ellel ) we

~re

now

Ihc'M what

melhods praétical geomelricians ufe for making on p,'per

an angle equal to

given

angl~,

and how they eonlbtule

thefides in the lame proportion, For this purpofe Ihey

make ufe of a-prométor, (or, when it is wanting, a lioe

of chords), and of a line of eqllal part!,

PRO P O

S 1

T ION

XX,

FIG,2,

3. 4'

í,

.nd

d'fcribtlhe<OI/Jlrutli,~

and

11ft

~/lh,

prolraflor, ollhe

line~(<hordJ.

and o( Ih,

01 c·

qualparlr,·.The

protraét~r,is

afmall fem,icir,c1e ofbrafs, or

fuchfolid maner. The femlclrcumferencelS dlYldcd Into 180

degrees. The ufe nf it is, to draw angles

0.0

any plane,

0.0,

paper, or to examine the extent of angles

al~eady

laid down. For this lan purpofe, let the [,uall pOlOt

Ihe ceolre of Ihe prolTaélor be placed aboye the angular

point, and lel the fide AB coincide with one of Ihefides

that contain the angle propofed; the nllmber of degrees

cut off by the other fide, compuling

0.0

the protraétor

from B, will fhow the quantity of the angle that is to be

meafured,

But if ao angle is to be made of a given quantity on a

giveo line, aod at a given point of Ihat line, let AS eo·

incide wilh the given line, and let tbe centre A of the

ionrumeot be applied

that poiot. Then let there be a

rnark made at the given number of degrees ¡ and a righ!

lioe drawn from that mark tO the given point, will eoo(li·

tute ,an angle with the giveo right lioe of the quantity

required; as is manifell.

This is the moll natural and eafy method, either for

the ment of an angle

paper, or for deferibing bo pa·

per ao angle of a given quantilY,

But when there i, feareity of inllruments, or beeaufe a

line of ehords is more eaGly carried about, (being de·

fcribed on aruler on whieh there are many other lines be–

fides), praétical geometrieians frequently make ufe of it,

is madethus: let the quadrant of

eirele be divided

into

degrees; (as in fig. 4') The line AS is the

ehord of 90 degrm¡ the ehord of every are of the qua–

draot is transferred tO tbis lioe AB, whieh is always

rnarked with the number of degrees in {he eorrcfponding

are.

Note, that the ehord of 60 degrees is equal to the ra–

dius, by eorol.

4th Euel.

now agiv<n angle EDF

is to be meafured by the line of ehords from the centre

with the dinanee DG, (the chord of 60 degrees,)

deferibe the areh

¡ and let the points G and F be

marked where lhis areh interfeéts the fides of the angle,

Then if the dinanee

GF,

ap~lied

on the line of ehords

from A to

gives (for ex, mple) 2í degrees, this ¡hall

be the meafure of the angle propofed.

When an obtufe angle is to be meafured with this line,

let its eomplement to afemieirele be meafured, and Ihenee

it will be known, lt were eafy

transfer to the diameter

eirele Ihe ehords of all arches tO the extent of a fe.

rnieirele¡ but fueh are rarely found matked upon rules,

Hut now, ,if an angle of a given quantily, fuppofe of

í,O

degrces, IS to be made at a giveo point

of the right

hoe

(r.g,

6.)

From the centre

ano the di-

E T R Y.

llanee

MN,

equal to the-chord of 60 degrees, d&riLe

the are

Qll,

Take off an are

NR.

whof.. ehord is e.

qu,t!

tlm of

íO

degrm on the line nf ehorus : join

the po:nts

aod

and it is pl. in that

Mil

Iball

contaio an angle

01'

íO

degrm wirh the line

propo.

Ced ,

fometimes we eannot produce the fides, til! Ihey

be of tbe length of

ehord of 60 uegrees

0.0

our feale;

in whieh eafe it is fit tO work by a tirele of proponians

(that is a feétor), by whieh an are may be made of a

given number of degrees,to any radius,

The quantities of angles ,are likewife determined

other lines ufually markedupon rules. as Ihe linesof fines,

tangents, and fecants; but, as thefe methoos are not fo

eafy or fo proper in this place, we omit them,

To ddineate Ggures fimilar or like to others gim,

befides the equality of the angles, the [ame proportirn

is tO be preferved among the fides of the figure that is

to be delinmed, as is among the fides of the figures gi–

ven, For whieh purpofe, on the rules ufed by anills,

there is a line divided into equal parts, more'or lefs in

number, and greater or lelfer in quaotity, aecording to

Ihe pleafure of Ihe maker,

foot is divided inlO inches ¡ and an ineh, by meaDs

of tranfverfe lines, into 100 equal parts: fo that with

this feale, aoy number of inehes, helow twelve, with

any par! of ao ioeh, cao be laken by the eompafl'es, pro·

viding lueh pan be greater thao the one hundredth par!

of ao ineh. And this exaétnefs is very neeelfary iD de–

lineating the plans of houfes, and in other eafes,

PRO P O

S 1

T ION

XX!.

FIG,

r olay down on paper, hy Ihe pro/raa"

lin, o[ chordJ, QlId /ine

'qual par/J, a righl.limd

figure like lo onegiven, providing Ihe angla andjda

.¡

/h, figure given b, knlYWn hy Oé/trvalion or mm/uralion.

-For example, fuppofe tbat it is knowo that in a qua–

drangular figure, one fide is of

23í

feet, that the angle

eontained by it and the [eeood fide is of

84°,

the feeond

fideof 288 feet, the angle eontained by it and the third

fide of

72°,

and that the third fide is

294

feet, There

things being given, a figure is

be drawn on paper like

to tltis quadrangular figure. On your paper" at a proper

point A, let a right line be drawn, upon whieh take 235

equal pam, as AS. The par! reprefentiog a foot is

tao

ken greater or ltlfer, aecording as you would have yoor

figure greater or lefs. In the adjoining figure, the tooth

part of ao ¡neh is taken for a foot. And accordiogly ao

ineh divided into 100 pam, and aonexed to Ihe figure,

is ealled a feale of 100 feet. Let there be made al the

poiot B (by the preeeding prop.) an angle ABC of

8S',

and let BC be taken of 288 pam like to the former,

Then let the angle BCD be made of 72°, and the fide

CD of

294

equal pam. Then let the Gde ADbedrawn¡

and it will eompleat the figure like to the figure given,

The meafures of the anole

and D eao be known by the

protraétoror lineof cltords, and the fide AD by the line of

~qllal

parts ¡ whieh will exaétly anfll'er to the eorrefpond–

Ing angles and to the fide of the primary fignre.

After the very fame manner, from the fides and aD–

gles given, which bound any right,linrd ligUr<, a figure

like to it may be drawn, "nd the rcfl of its fides and

ao–

gles

~e ~lIown.

ROL·