# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/806933011:0790
# Permalink to this reading: https://leibnizlegible.com/page/806933011:0790?run=74
# Title: Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences (shelfmarks Leibn. Marg. 71; ZEN Leibn. Marg. 71)
# Folio - , page id 806933011:0790
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=806933011&page=790
# Source image: https://digitale-sammlungen.gwlb.de/content/806933011/jpgs/default/00000790.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 115 lines, status machine
# Lines: 115 recognised, mean confidence 0.88
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# After the first blank line: the recognised lines in reading order, one per line.

ies Cnbe Roorbeins requird, ic selF DC NOt L true
Cube, _c. Then ris impostible eo attign, cicher
in nhole Numbers or Fractions, any exact ROOr
Ot luch Numiber propoied. And wten ever His
trappens; tis usual in MachematickS ro mark HC
required Roors oF Tuch Numbersor Guantities, by
pretirins betore il cheproper Mark oF Radicalliry,
erhichis V: Fhus V: 2 sipnities the Square ROOr
METACOE atrel DenOminAtOT. Thus,is redu
ced to V
and
ihe Radieat Sien aftects Boch Numerator and DS.
nominaror alike.
OF2 and V: 16. Or V: (3) 26. Sienises che Cu TI. F0 reduce Simple Surds, havins different Radical
Bick ROOEOF 16: TVhich ROOES, because ther are
imporsible eo be expresed in Numbers exactlr (For
no ettable Nuniber, cirlier Inceger or Fraction MuI
ciplied into it felt can ever produce 2; Or beins
usciplied Cubically can ever produce 16) arc
verr propertr calld Surd RROOts.
Tlere is atso anorher Way OE Noration non muc
in ule, Wotierebr Roorsare expressed, Wirhour che
Radical Sigu, Dy cheir indexes: ThuS, 28 x. ~
x5. 3c. Cienitie the Square, Cube, and sch Porver
OF X; (0 x. v X5. C9c. signitic the Square ROOT
Cube RCOF, C9c. OF x. The Rcalon Ot wrtrich is
plain enoueh, For since V: x is a Creomerrical mear
Proporrional between r. and x. So = is an Aich.
metical. mean Proporrional betveen o and L. and
cheretore 25 2 is che Index ot chie Square ot X,
Wrill be cheproper Index of iis Square ROOr, C9c
Oblerre allo, char for Conrenience or Brevits
Pake, Juantiries or Numbers erhich are nor surds
are otien expressed in che Form ot Surd ROots.
Thus V: 4. Y: = V: 27, Oc. tieritie, 2,
C8c
Bur alcho chese Surd ROOrS (Wtien cruly such,
are inexpresible in Numbers, eliey are ver capable
et drichmerical Operations (Tuch as Addirion, Sub.
eraction, Musciplicarion, Dirision, esc. Which hon
readily eo pertorm che Alhebrist oughe nor to be
IBHOTAHE.
Surds are eictrex Simple, wtich arC expresed B
orie sinele Perm; or else compound, utrich are
Formed by the Addirion or Subrraction ot simple
Sueds: ASv: 5 =] V : 2: V : 5  V 5. 2. O1
: 7 =)- V: 2: Wrhich las is calledus an Uni-
versal R00t: And Signities, che Cubick ROOE oF thar
Number wbich is che result of addins 7. 10 the Squari
Root ef 2.
The Arahmerick. ef Surds consist sof ihese princt
cixal Parts.
I. Zo reduce Rational Ouantities 1o the Form of an
Surd Roots assiined.
Cuanrier accordins ro the Index of the PouerO
the Surd, andchen pretixins betore ir the Radica
Sien oFthe Surd provoled.
Thus reduce a = τ0to ite Form ot V: 15.
:b. Voumust Squarea == n0; and prefixing rh
Sign, it wvill Aand ctuS, V: 4 a = ~: 100
ertich is in che Form oE ihe Surd desired. Se
also if 3. verero be bronghero the FOnm OE : 12,
ron must raise 3 up 10 ies tourth Pover, and chen
prelixins che Nore oF Radicalies t0 ic, ic Wwill be
if: 81. Or 81 1, which is in che Came Form vict
: T2
Siens (wtrich are called Hlereroseneat Surds) t0
otbers that may have one common Radical Sign, or
mhich are Fomozeneal.
Diride the Index ot che P’overs ot che Surds by
etcir sreacest common Dirisor, aud ser che Ouct
enis under che Biridends; chen lalciply chole In-
dexes crols mays by cach biers MoCMIo, aHd
betore che Produets, Per hecommen Radical Sisn
V: Wichis proper Index: LhetrInrolrechePo
ets oFrhe siven Roors Alternatelr, accordins corhe
Index ot cach others OuoreM, and betore chose
Ptoducts, pretix che common Ladical sien be
lore houne.
And ehis ay may a Simple Surd Fraction, wwhose mas pt 2 N: 72. OF3 V: 32. Or 4 V: 18. Or
Radical Sion reters onty ro ore of ies Terms, be
_.. B. Dt tedtl l). L V. L ardet lnde telleae aetas
chansed inso anocher, wwhich Mall saleet boch Nu. done in Species.
10 V:: IVhere
Wrhich is pertormd By Involvins ehe Rationa
ToReduce V: a a and Y: bb
2d:
Lo Reduce V: 5 amt Y: 2.
25 V: 2407
III. Toreduce Surds 2o rheir lomest Terms possible.
Divide che Surd by che greacest Square, Cube
Biguadrate, C9c. Or any ocher higher Pover, wtict
rou can discorer iscontained in ic, and will Mea.
lure it wichour any Remainder, aud rhen prefix
The ROOE OF char Porver betore the OOTIeHE, OF
Surd s0 dirided, and ehis will produce a new
Surd ot che lame value with theFormer, but in more
Simple Terms. Thus, N: 162 ab, bY dividine
by 26 a a and pretixins che RoOr 4 a, vill BCrC.
duced tochis 4 a V: b and v: 12, willbe deprel
Acao2 V: 3. AIO V: cbor, vill be brouet
donnto bu: cr. And chis Reduction is OLGrCAE
Se Whenerer ir can be pertormed: Dut iF no sucH
Square, Cube, Bignadrare, (8c. can be Found for
a Dirilor; chenrou must findour all che Divilors
Of the Porer ot che Sure propold; and etien sec
Whiecher anr of them be a Sguare, Cube, C9c. O1
such a Porer as the Radical Sien denores; and il
ariy fuch can be found, lex thar be uled in the lame
macster as is abore Caid, ro tree the Surd Ouancits
in parr trom the Radical Sign. Thus, iF V: 288
beproposd; amons is Dirilors, will be found che
SGuares 4. 9. 16, 36. and 244. by which iF 288
be dirided, chere vill arise che Ouorients 72. 32
18. 8. and 2. Wheretore instend oF V: 288, 701