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# Title: Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences (shelfmarks Leibn. Marg. 71; ZEN Leibn. Marg. 71)
# Folio - , page id 806933011:0776
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Then mulciply che Mra Rigure inco che le-
cond, wrtich takten cogerher make a; and write
S. Square the chird Fipure, and subleribe its
Square 25 Betore in (Step. 3.) chen ad d all inco one
in Produet dont rvice as betore (in SceP. 2.)
The Frample Collonvs.
133225 (365
133225 == Square OF 365.
From which Mechod oF compotine a Square
crill be easie eo understand the way t0 Fxtract
rhe Square RooroF ans Number siven,
AS SupPOlC 133225. FOF,
2. Beginnine ac Cheriphr Fand, make a Pointover
che Arst Figure (5); and aster chat over every ocher
place alternatety. Which Poins me rou che
Number oF Places in che ROOE.
2. Find che nearest. Square ROot to the srst parE
13, Wrhich is 3, urite ic in the Ouorient, and
then place irs Square (9) under, and subtract i
From 13. Ihis Roor is a, and che Square a a.
3. Toche Remainder (4) brine dovon che nexr
Square 32, wrhich will make 432, and will be c.
qual eo 2 a b -) = b b, in the last Scep oF all, els
alvars bisger, and is called che Resolvend. Them
double che Fisure (2) in che Ouorienc, and make
it a Dirisor 20 all ihe Resolvend, excepe the last
Figure rowardsche rishr Lland, enquirins hour of
ren you can hare 6 in, 42, and place che Antver 6
(FOr cho you can have ic 7 eimes, yer  must noc
bo caken, because ris 100 large) in che duorient,
andalso atrer 6 in che Divisor. Phen multiply che
thus augmented Dirisor by 6, and vrire the PrO-
duct under, and Subtract ic trom che Resolvend
Torhe Remainder 36, bring donn che next Square
25, and chen rou hare 3625 For a new Relolvend
Fhen, as betore, double rhe Ouorienc 35, wtich
akes 72 == 2 a, For a new Dirisor, and enquire
ho otren rou can have 72 in 362, vrrire che An
PIVer 5 = a in che Ouorient, and alo afrer the Di
visor; and mulciplyins che ausmented Dirisor by
ic, ron will hare a Product 36255 = 2 a b- 1 bb
wtrich suberacted From che Relolrend, leares no-
ching and (oche Work is over, and the true ROOt
tound robe 365. TE chere had been more Points.
che Work in chis chird Scep must hare been repea
red ril all ras done.
432 = Relolrend.
396 = 2 2b -- b
3625 = Besolvend.
3625 = 24b.- b
0000
Dut when a whole Number hach nor 2 Square
Roor exactly exprestible by any rational Number,
chan eo approach intinirels near che exaet ROOI
Proceed chus.
Place a5 many pairs oE Cyphers on che riehe
Fland of che Remainder, as rou vould have Deci
mal Places in che ROOr, and work as betore, di
Rinsuishrins ehem trom che Incesers by a Comma,
thus: IF FhC SguarE ROOEOF r2 be desir d tochree
places in Decimals; annex 6 Cy phers 10 12, and
cwvill stand etrus:
12, 000000
Whose Square Roor beine extracted, is sound ie
be3, 464, OF 3. ∞686 bur because of che Remain-
der, ic must be chac 3, 464 is sels chan rhe crue
TOOr, and 3, 465 greater than iE; Vet nOc wanuins
the 200S OF an Vnit OF rhe true ROOF.
The SquAsE ROBE OF a Vulgar Fraction, is
thus found.
Firs, HE che Praction be nor in ils least Terms;
ler ic be reduced ro che least Perms; chen exrract
tHC Square ROOF oF the DonOminaTOF FOF a neW
Denominator, and of che Numerator for a new
Numerator; ehis nem Fraction Mall be che Square.
Root ol the Fraction propoled. Thus the Square
ROOE OF 2 18 2, OF. = 15 3.
HFche Vulgar Fraction siven, be incommensura.
ble FO irS Square ROOL, boch in che siren PerMSs
and also in any other Terms chac it is reducibleto;
chen reduce the Paid Vulsar Fraction into Decis
mals, consictins ot an eren Number ot Places; and
chen extract che Square ROOr thereoF bY Approxi
mation, accordins ro the Preceprs already delir
rered.
Thus: IFthC Squaro ROOF oF = De required, irs
equiralent Decimal is 75y Wrhose Square ROOE,
86602, 690. Theretore che Square ROOT OF 3 15,
86602 ferc.
HEa Mix d Number wete given For Fvrraction,
Wohose Fractional Parr is exprest in Terms ot a
Vulgar Fraction; ReduCCiE IntO an ImPFOPeF Fra-
ction, and (it commensurable, extract the Square
RoOE oF the Numerator and Denominator, (25 bC
fore;) obserrins to reduce the Fractional parE OF
the Mix d Number, (Or che improper Fraction e
quivalene eo che Mixd Number) inco ies least
TerInS.
ThuS: THC Square ROOF OF 5 = I5 2 2, For etie
improxer Fraction equiralene 105 3, is 13, and the
Square ROOF OF 13, 183, Or21.
22.  b = 66,
2 a. F b = 125
Sum, and repeat ehis Procels as otten as there is
Equied.
occation; che last Sunt e au l M tue Square
Thus.
160.