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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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U
Form of Fractions; cher must (it Etiey have HOE ( à = L e. We may conclude; That co Square any
one) be reducedro a common Denominator in ihe
least Perms; and then eo find our the Rational
Ouorients, rou necd oniy divide che rwo Nem
NumeraTOES, by cheir srearest common Dirilor
and chen sou must e0 on as abore, in Intecrat
Surds.
Thus, H ehe Sum and Diference oF V: 24 anc
y: = vere required: When reduced to 2 com
mon Denominaror, they will be V: ands:
and chese dirided by cheir greatest common Divi.
Sor V:  itie Ouores are V: and V: 95, Or
6Y:  and 5 V: =, Wotose Sum isY : 1L
and heis Disterence L V: 
NV. IE Etic Simple Surds xiven ro be added, or
Tubrracted, are Incommensurable, then ther can on
ly (generalls speakins). De added or subtracted b3
the Sions J and -: For neictier Sum nor Ditte.
renice can be exprels d by any, single ROOT. AnC
tromthis Addicion and Subrraction oE simple Surds
onty by che Siens, ariles wtrat ther call a Surd Bi
nomial, Ox Residual R0ot.
Thus, V: 6- V: 7, is a Dinomiat Surd, anC
1: 7 =-  : 615 a Residuat Surd.
BuE LrOm Prop. 4. and 7 OF Fuclid S second B0ek
there arises a Rule erhich helps us 1o sind che Sur
or Ditierence ot Incommensurable Square ROorS
Wrhich Rule is chis.
20 d siom the Sum of the sgantes s ie 
Surd Ro0tS, add, or subtract, their double Ret
angle. and the Square Root of the Sum, or Re
mainder, is else sum, or Disserence souxtra
. 81.
Fosind the Sum ana iHterence OL V. 14. ai
N: 12, cheir Squares beins r4 and 12, their Sum
will BC26, and etie Double Rectansle oF Y: 14
into V : 12, 152V : 168. WVherctore
SSum
V: 26 1 = 2 Y : 168. is che 3 DiRerence 5 te
quired.
es Compouna Surds.
The Aricturierick oE Compound surds, depende
on che Rules abore siren about simple Surds, aud
on therrue knomledse oF che Sisns :) and in
Algebraick Addirion, Suberaction, Mulciplicarion,
and Dirision; only fome parricular Directions maz
be oiven asro Binomials and Residuals: AS,
Y. Is any Binomiat be 20 be moleiplied by ile
correspondins Residual, the Difference ot cheir
Squares is the erue Product; and rheretore will
Come OuE ~ R arional Oiantity, 25 il ~: a - - 0
bemulriplied by V: a = e, cheProduet will bc:
Rarional Ouanirs, viZ. 4 a = e e.
and mont casily pertormed by a Table OE POrvers:
Binomial Whacloever, vou need on y ad d the dou
ble Rectangle oF che ParIS; t0 ihic Sum oF he
Squares ot those Parrs; or take thie double Rcct
aetote dedita lltal vlt i i DE a dichcllata
III. L’ox Dirition in Compound surds, ris conve-
nient, if nor necellary, to redisce them Arstto lOmE
berer, and when ir can be done, 10 2 RatiOnat
FORL. ANd,
(L.) IF a Binomial, consistins os rul M
Sguare ROOES, OF OF OnC Square ROOF and a LALI
onal Ouancier. be mulciplied by iis correspondine
Residuat, chie Produet ill alvays be. 2 Ratioral
Duancier.
ttnta tetaft maus ust ut eltt sebisus ett a
silt de eant idde destate lolo, e et e 
beins mulciphied, as is betore lata, b) us Binomial
ic produces à Lattoas uanrier.
(3.) IF a Prinomial, having chree Simple Squares
Roors, be multiplied by ic self, with one ot chic
Siens chansed; ihe Product will be cither a Bino-
mial, oe Rehidual, which beins mulciplied by is
correspondenr Relidual, or Binomial, Wvill sive im
che Product, a Ouanciry encixely Rarional.
Siduat; BUF COnJOYHCA IN CHC BLHOMITAL. Bγ =, and
tet ettu eetctta 1). ale altllabisut
al ox Kehdual, Will be a Omancies entirels Ra
TenCaS, ThC uLL bC FeduCCaIO a COMIMIOL
Radical Sign.
conneet chole parcicular Lroduets rogether by cheis
Siens; buc it che Birilorbe a Binomial, Lrinomi
al, or Ouacrinomial, 59c. ot such kind as betore
is specihed, reduce FHaE Biven DIrIIOE EO 2 new
Diritor thar may be a Simple Racional Ouantier
Reduce allo, the siren cene 1o a he Diri
dend, by multiplrins che tormer by ic Clantirie
chac vere Mulriplicarors, in reducins che siver
Dirilor ro a Racional Ouantier; chen diride che
new Diridend by che new Dirilor: Buc wtienctic
Diritor cannor be reduced ro a Simple Rationas
Ouancier, sec rhe Diridend 25 2 NuMETaTOI, OVet
che Dirilor as a Denominator.
Ouorient i5 4
V. : 33 V : 22 = V: 15 diri
Ae becaule we sec chac a a = 2 a c  c e = D ded by V: 3, chc Ouoriene is Y: 7  y:
(2.) IF 2 BIrOMIIAt COMMInS eF Euvo Biquadra
Ouanciey; if chis be mulriplied by ies correspond.
. clat, the Produet Wvis be a Residuat con
Sgnare RoOr a Rarional Ouanrier; Which Residuat
IV. IF a Binomial or Rehidual, consisting oF rvo
Simple Cubick or Biquadrarick ROOrS, (9C. Or
ot onc Cubick or. Biquadrarick ROOc, 69c. and a
Rational Onanus l proposed for a Divisor; sind
(0 many continual Proportionals in che Proporrion
OF the Patts O ie Binomial Or Relidual proposed
s chere be Vnices inthe Index of che Radical
Sten, ale luch whose Radical Sion may be che
Pame wich char ot eC Lales et  omial Or RC-
concraril), in che Proporriotials by -1-, and in che
Lelidual by -) and--; che Produet of thelalut
Proportionals (0 connexed, multiplied into che Bino.
rional. Attex che Pame manner mas a Binomin
Ox Rchdual, arB 566, Sc. o ene Mdex Ot 
Coon Radical Sign oF che Loots, De secuceu
a Duantiry enrirely Rarionalo
And Nore, thar when he RooS are oF Ditte
V. TCthe Divisor be a Simple Onanrieyr, divide
cach parr ot che Diridend by che Dirilor, ane
II. Involurion in Dinomials and Residuals, is BCA
ThuS, 12 -F= V : 63 divided by 3, ihe Ouoriere
15 4 - I V: 73 and 8 = V: 12 dirided by 2, che