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# Title: Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences (shelfmarks Leibn. Marg. 71; ZEN Leibn. Marg. 71)
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Index oF 100 150; then may che ROOE be casily ) Hnovn Ouantiries, the Ouanciries chemsclres vill
Found by etie folloine Merhod.
AII Equations OF ehis Ranle wwill be in one o
chese chree FOrms.
an =)= a d = R. 2 + Some male four Forms.
a a - - a d = R.  but at lons run it comes
a5 = a a = R.tothesame rhins.
En all vohiict Forme, R, rhe abloture Number
giren, is a Rectanele or Produet made out oF rhC
vo Ouanciries or Roors Coushe, a Crearer and:
L eNCF.
OF wtich, inthe First Form, where all is AHir-
matire, the Coefficient d is the Disserence Between
(oon be bnown.
Thus in che Ars FOrM, let a a - d a = R.
Here is given R, the Rectansle OF the ROOES,
a cheir Ditlerence; and ris hnoun thac a repre
Cents the lesser oF them. Lec S Cand Forthe Summ
Eo bc Coughr.
Lec a V c = Sand a = c == d, Then a a -]
2 a c -) - e e == 55, and a a = 2 a c =) e0 = d
Substract the lacter trom the former, and there re-
mains only 4a c = 4R. Wheretore SS-d d = 4R
Tou may cheretore by Simple Alsebra And cha
4R == 55= d d, and consequeney chac 4 R -1-
a d == 5S, and cheretore S istnown; and then
those tuvo Ouanticies or Roors; and a is the Lesser havins S and d, a the lesser Roor will be Hnoon t00,
OF chem, a5 is plain is von suppose the EWO ROOES FOF =S = = d =
(25 Oushered doch) ro be a the greater, and c the
Lesser. Porthen lec d = X be che Ditterence Be-
tveen chem: Sochat e -  X = a. it then vou
Mulciply cach Parr by e, ic will bC e c -F e
ac; trom whence ic appears allo plainly, chac a c
is equal ro R, the absolure Number siren, oi c
a
Aeain, in che second FOrM. Let aa = a d = R.
Eere d aud R (as betore) che Ditterence and
Rectanse oF The LWO ROOTS are siren; arid a the
crearer ot chem; Wheretore ris easiero tind S che
qual co che Rectansle oF the rvo unknonn ROOrS Summ, and chen =5  Z d = a.
a and c, ot which in this Form, the Coefficiene a
Or dis equal coche Difference betveen them, and e
isthe Lesser os chem.
In rhe Second Form, The Coeficiene d is the Dit.
Ference ot the trvo Roors as betore, Dur a there re
In che chird Form, iotere 5 a = aa = R.
There is siven the Coefficien S = Summ ofche
untnoun Rooes, R che Rectansle berween them;
and a max be cichierrhe bisser or thelesser of them;
prelemts the Creater oFthem, a5 is plain by purtine Llere cheretore ro tind d che Ditterence
(because the Sion is Negatire) a -- x = e, and
Musciptrins each Parr bY a, ic produces a a = a X
a e, che second Form, Where x Or d che C0.
efficiene is the Difference oF the rvo untenovn
Roors; and a represenis the greater oFthem.
Inthe Thira Form, Where che highest Pouer is
Negarive, the Coesficientsis the Sum of the EVO,
Ouanticies ox RoorS Toughr; and a the AfFirmatire
Roor (oughr, may be cirher itre Bigger or theLes-
Because SS- d d = 4R, cheretore SS - ] 4 F
=a d, and consequener d is known; and then
25-  Zd = BrCacer, and ZS = = d = leNer.
II. The Solution os Adfected Dudraricte Equations,
by the Method of Compleatins the Squâre.
Wrhich is BY DIP. Parriot, ehus: Since in every
onc ot che chree Forms of Ouadraricks; one quar-
rer of the Square of the Coefficient will make the
sex oFrhem. For lec (Because chehighest Porver is untenoun Side ofthe Equarion a C ompleat Square,
TFegatire) Z - a = e, Then Mulriplyins Boch by a wtose erue Roor will be a -1 - = d (or Whaterer
it Wvill Z a = a a = a e== R. Or it Z ehad been po
equalro an then it ould hare been 2e = ec=
ac, by Mulciplrins all by c.
So That CIS Method MeS vou IHC OIISIIaL
Consticution ot chese Forms, and the Nature and
Ofice oFcach Member et them.
From all Which may be found chis general Canor
For che Solurion oF Ouadrarick Equarions, accord.
ins eo chis Method.
the Produet add. the Square ot the Coeticient
then Extract che Square Roo OEthat Sum: Whict
RoOF Mallbe the Sum oF the trvo Numbers Touxti
Then e0 ox From the halE OF TaE RROOF, Add anc
Substract hals the Coessicient, and the Sum ane
Remainder are the two Roors required.
For rhe parricular Solurion os Adfected Ouadr
cicks there are chrec WayS.
Lerrer esse be the Coefficient.) Tis plain by chis
means, an Adfected duadratick Equation, may bc
reducedro a simple one.
WVheretore,
In che Arst Form, Wrhere all the Species are AHir-
matire.
be a pertect Square a a - d a- = d d, Whosetrue
ROOE 1S 4 =
a perfect Square Number and Hnowmn: Whose
Square ROOF extracted in Numbers, will be equat
t0a-) = d; and consequener, a will be equal 10
chac Roor, When = d is caten trom ic, and s0 a will
Let aa t a a  L
IF = d d be added ro the unkenovn Side, ic will
Add Fhen, = d d EO R, and R - = d d will be
be known.
Musriply che absolure Number By 4, and 10
Lde Llalleat cille 2d latdus
I. That of Oushered, uho proceeds in its
tdet tobco
In all che chree Forms, there is giren cithet che
Foche Absoture Number, add : ot he square
Rectanete ad Summ, Or the Rectansa and Diffe- OE Coefficient, (or the Square of half the Coessici
rence ot the to unknovn Onantiries; Whence ris enF) and Fxrract che ROOE oF the Sum; then FrOM
very eahe eo End cither the Differance in che For- chac ROOr found in PTumbers, Subrract ; che CoeF-
mer, ox the summ in che larter Cafe: And then) ficient, and che Remainder is a, the letser ol che
harius ihte summ aud Dissererce ot auy tuo uEvO ROOO, Oi Values e a.
F Natiole.