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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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Thole are called Commensurable Surds, urtiiet
are 1o one anocher as Number eo Namber, as onc
Racional Ouanriey ro anorher; Or whiel are, orhen
reduced co cheir lea Verms, nuc Figurace Juan
riries of cheir onn kind.
Todiscorertheretore, Whecher cher are such ot
nOc: IFche Surdsare ot differenekinds, (Or FTetero-
geneat Surds a5 fome calL chem) cher must sirst be re.
duced ro one kind, and then dirided severally by
their grearest common Mcasure, for is then there
vill. Come our Rarional Duorients, che cirst Surds
are Commensurable: buc if che Ouorients are Irra-
nioval, or Surd Numbers or Guanricies, chen che
propoled Surds are Inncommenlurable.
V. er. T0 Examine. Whecher V: 12. andt V 3.
are Commensurable Surds; thes beins Llonose.
neal, L diride chem severally by cheir grearest Com
mon Divitor, Which is V: 3; and the Ouorients
ase v: 4. 2Bd V: c. Thac is 2 and r. Wheretore,
since 2 and r. are Rarional Numbers, L Pay rhaE
V: 12 and v: 3 are Commenlurable Surds; Or
are roone another, 25 2 20 r. Which is very plam:
for no douDr 12. 3:: 25 4. F. and Tis plain chat
as Squares are ro one another, (o are their ROOES)
Wheretore 12. 3. 25 V : 12. V 3. chac 15, 25.
1: 4. V 3 F. OF 25 2. E0 I.
Wthenerer imo Surds are divided by one com
mon Diritor, (cho noc the sreatest) it their Omori-
rienes come out Rational, or are to one another
as Number eo Number, thole Surds are cettain
ly Commenturable.
TE Fractional Surds ivere giren, noc harine a
common Denominaror, ibler must hirst be reduced
to their sinallest common Denominacor, and then
il cheir Numerators are Commenturable, vou may
conclude the Ars Surd Fractions evere (0.
Burit cicher the NVUICTAEOES OF DenominatOre
et rvo Surds, proper Fractions, or mixe Numbers
in the form oF Fractions (neolectine the Radica
Sien) be Povers oF rhar kind which che Radical
Sien expresses, then chiey will need no Reduction:
Fox ilrheir NIIMCTALOES Or Denominatorsare C’om
menturable, che Whole Surd Fractions propoled
are certainy (0. Thus il it vere enquired ivhe
ihe Radical sien expresses or Denores, omitrins che
HEche Surds proxoled be oF elic same Kinc, Mtul-
ciply chem one by anocher, and presix che common
Because r6 anc 25 are SGnaTCS, OF Su CH PORVerS 25
Sien V: vou necd ouls compareche Numerators
20 one anortier, 25  10 and consequenels, b7
Mulcipication of Simple Surd Roots.
Kadical sion ro che Produet; bur it che Surds are
Fererogeneat, ox oF ditterent kinds; cher must be
reduced Arst, (accordins ro Rule 2.) 10 Surds har-
es V:  and s:  are Commenturabie Sucds:
V: 50. and v: 72; Wrhich beins divided by cheir
ercarest common Dirilor, V: 2, che Ouorienrs
Will De 5 and 6 (i. c. V: 25 and 1 : 36) WVhere
tore che siren Surds are Commenlurable, and axe
he precedent Rule may be expressed chus, . S::
ins che lame Radieat Sion.
FhuS eo Mulrixis. V: 7bY Y: 8. che Produst
will bc V: 56.
Fon since in all Mulciplicarion, 25 r. is E0 opie
Factor, (0 is ehie ortier eo the Product; cherefore
here S: P. V: 7. :: V: 8. V: 56. Buc it 4 PCrHS
are Proporcionable, cheir Squares will be 10 100;
WhCretOre T. 7. 3: 8, 56. charis, 56 is rHC FIC
Square ot Y: 56. and V: 56. rhe crye R0Or ot
7 X 8 = 56.
Octrex Framples.
P. EEV: 8 were r0 BC Mulciplied into V: 4. DC-
cause cher are nor Floinogeneat Surds, etiey must
bereduced io such bY Rule 2. and cen chey vil
stand thus, V: 312 V: 16, wetich beine mudripli
ed inco each ortier, and the common Radical Sien
presix d, Will makie V: 8193; and chusche V: 27
mulcipied by V. 9. uoten reducec, and rixtuts
mulciplica, produces v: 531441.
AT. WVhen 2 Surd is ro be mulriplred by 2 Raric-
nal Ouanriey, chaE Rarional Guantiey OuDHE REs
tobe reduced ro a Surd ot like Nature wich chc
Erue Surd. Bue ris otcentimes conveniene on y ro
connect chem tosether, by prehxine che RationaL
Ouantier co the Letr Fland oF che Surd. AS sup
Pose V 27 vere ro bemulciplied by 6, cheProduct
mlay commodiouss be expressed chus, 6 v: 27
and Coit i: 9 vere co be mulriplied by r0, ic will
Aand cus, 10V: 9.
III. And when rro Bational Quanrities are
thus pretixd eo rro Surds ot the lame kind, rou
nar sind che Produet of chem, by mulciplvins ihe
Racional Parr, by the Rarional, and che Surd Pare
by che Surc chen chose joinedcosether, will bethe
Produet required. Thus, 6 Y: 7 mulciplied b7 5
V: 3 Produces 30 V: 21.
IV. IF anY SUIE LLOBL DE LO DC SAVIIPHCC MOA
Cel, or Involved accordins worhe Index OF iESPrO-
per PoweE, You necd only cast avay che Radica
Sign, and eheri the Ouancier, or Number remain-
ins, is alvars che Square, Cube, or Orhet POIeK
required; and will alivars be Rarional.
Thusche Square ot V: 1I, 15 F1.
The CobC OF V: 30, 15 50.
AIIO2 V: 3 mulciplica by 8 s: 3 = 48. and
3V: 5. mulriplied bY 2 Y: 5 = 30.
IV. F0 find Whetser tr3. Surd ROOtS siven, at
osa or notia
For an Instance in Species; suppose rhat it wet
enquired whecher v: 27 a a and V : 12 a a Wvet
Commensurable Surds; Diride caca Y che
compound Number srearer chan rwo, and tis re
quiredro square sueh a Surd: Phere necd omly a
Radical Sien, Whose Index is hait che Former, be
pretixd 10 rhe Cuatius, inAtead ot che formet
Componne one, and iis done.
greatest conmtent o V: 3 a a. And cheduo
tienEs V: 9 atid V : 4. Thal 15, 3 al Z. die Raris
ral Numbers; and consequeneiy, che propole
Surds are Commenturablc.
V. And if che Index of che Porver be any even