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Full text of "The Whetstone of Witte"

oftoitte, 

to5icl)e is tDe rcronne patte of 

Arithm€ti1^e:fonta<nrng i\)vxtttit^ 

tion of Itootcs: SCbe o>fiiks p;tiai(C} 

iottb tbe ruU of Sjuatiomann 

t\)t ll)00;kCS( of Surde 

Komhers. 

Though MMyflenes d^e Itareireate^rke, 
Ihe "ftfhetltonc hfor ettrfice 
jfs neadifull,andin T^^prhj asftraunge: 
Dtillethingei dndharde it yulljo chaitn^Cj 
jfmdmdh^ thnnfharpeyti right good i' ft: 
Mlartefmen l^ot^ejheican mtchu/e, 
$ut v/f hu helpe.yet as menfee, 
K»eJhMrpeneJefemeth1n it to hee. 

Ttf groun:lc ofartes didhrede thUJlonc 
His \fc itgreate,andtnoar( then one. 
Her I if y ou lijiyour \¥ittes t§ t^hette, 
Mochejhir^enejfe therhyfhally otigette, 
1)ulle Hfittes hereby d(K gredtely mtnde^ 
Shdrpe tetttes arefintdto their fulle inde, 
Ko Uproue,tm4praife,it4you dotjinde, 
jindtoyourfeljbenot^nkjnde. 



^'EMctt IBookfs are te hit rolDc,at 
tlbe®ilrl!eD0o;cofpoulc0, 




ucrner^, CcnfuUcja;, anD t^e rede of tbe com* 

paitteofbenturerdintoMofcouia, Kobcrt i^e^ 

tQiht ^atitfittan , iDtfin^et^ I)ealtt)e hiitif 

(onttnualk increafc of (ommoDt^ 

ti( ^ bp tljetr tuo;it^(e ano 

famous traadU 

iDDilnot>not^tousl^t3l 
fbeutUptotuDseof mp 
countc(e,t!jatlmrnpng 
l^erecanf^auenoUbei:^ 
tte:butbpaiDeoffrcnDc* 
ft<ppe,o?ftccn8t^ofpo^ 
toer* jf 0? aie( Cnglanoe 
QiDneuerbDanteieameO 
tDittctf,fo at tWtvmtJi Doubt not,biu tl^ete 
be a great multituDe,tl)at Deftrouflp embjtace 
alBinDeiS of fenotoleoge, anD fcetiDelpare af* 
fecteD toiuatD tl)e furtherance of it*3lnD ttjec* 
fo^c 3 Dare fate,tt|e( can not malice me, \3)W 
tlft am fo toiUpng to l)elpe tl)e tgno^taunte^ac^ 
cozDrngto mp gtfte anD fimpletalete^'V^^er^ 
bp alfo tW mocl?epzntfe3 ma<e iuftlp craue, 
to \}nm tl)e commenDatton anD retoarDe of a 
foUiciter in tW caufe* fot tl)oug!) mp traucll 
can not moc^ep;tofite tl)em,tl)at be tDcUlear^ 
ncD, pet Doett) it eiccite t!ic befte learneD, to re^ 
member t^cir Duette to t^etr countrte : anD tt 
beaftameD,tl)ntt^eil)aupng fo greateftabfe 
litte,C^all be founDe moare flaci^e to aiDe tf^eit 
cootrie.ti^en ^e tl^at ^atl^ fmalier i^notvIeDge, 



The Epiftle 

anti IctTe occation otftertwaiciaf* atccojtD^ng a^at 
meti ^aue recemcD^fo nee t^i bounoeto pelD^ 
'Ctjcfc ejccellente qifm are not letite tnf o me, 
to be !)iDDen»2KnD ttjere are a great muUituDe 
tl)att^nrft,anD!ongmocl)efo;tfocl)eaiD^fo? 
bot^e i^efe tmicfsi 31 faic, tijat naturaUe boDe 
to our countne oeet^ cbalcnge tt:anD fo^i tl^af 
f ^e i^onefle oeftreiS of Co manp gooD natures 
fo motive requtretl) it, 3 e]Cl)o;ttetl)em tl^at be 
bedc^able , to taj^e from me t\)isi ci)argeable 
ittoozHe^anotofari^rttieircountriemen^aai 
cquittc timulO«3iuD in ti^e meaae ceafon tattle 
3 fee ti^em Co llacfee>let tl)em not bee offenDeD 
iBitft me,fo;tp^tttentpttg tijem* jf ojt better it isf 
tl)dt a Bmple Coiie Doe prepare t^p b^el^efaa» 
t^en tW tl^ou CboulDea goe a l^ungereD to 
t)eDDe« peabf ttet ix i^ to l^aue fome grotfe re^ 
pafle,tl)en to detue fojt^ongier.l^nO t^e com ^ 
mon Co^te lotll fmDe fmaltefaoiteof mante> 
as;iDnga:dti^ei(l^anp manferuetil^etreirpec^ 
tatton. feo ttjatfo? tftt$ caufe alfo , tl^at mp 
paine)if &ja ttme.ooet^ercufe ot^er finer b3(t' 
tc0 , ti^i ourtt to rent^ me fbme tljanfce^ a^ 
gain.But ift^ei ftate fo^ feate of taunteiBf,anD 
barferngof ttifre0,tli$tr co^ge ijBffmaUe* 3f 
tl)ei mif Doubte ti^ gcatetwil acceptation of 
tlietrihiote^^etooe tmiirieto t^eic countne- 
f o; i0D^oe C4 ooubt but Co ctutle a contrte^tjotll 
tJianJieMp receiue>anO mode ftenDlprecom^ 
pemfe t\^ trtttelle > of focl^e t0 ftiiDte fo^ tijcir 



J>eJtcatork 

bemfttcdnDreruet^tQcirneceOarte common 
tiiticiS.W^ pecftoaOott maM\) mz fo bolDc, 
t^ftt^l tm mt t\)mUit neadefuU, to feke anp 
p^oteiio^^fo^ tbi0ojanpliketewo;ifee» ^it^ 
eueiggooC) manijpiiloffier l)Fmfelf>to DefienD? 
tftat,tx):)€cb? l)i£(natiue countcie ij^ benifiUD. 
€mptt at fwm tpme, bp eiccitatioii of ti)c fu- 
rie0,foine naugbtie natures Doe p;iactice tb^ir 
fcauiK , to iHuitt tiK realine of fomc fingu^ 
Lire commDDitit ♦ l5ut a;J 3 feare no fort>e,ro 
at tl^i^i tpme 3i felic no iotht aiDe agamft tbe* 
get foi teftcfipng of frenocfljippe, auD gcate^ 
fulle tmtmbxmntt.Ji coulo Doe noe leOe^but 
fenDet^i.8^23oobeto rocbea03l tboui5bt,not 
on*lF to Defetue it , but aUo tooult) glaDipre- 
ceiuc jt*3lnD if 3 mate ptrceiue. ttjat ppu Do? 
accepte it ( a3( S Doubte not ) tx)itl) aaf gooD a 
b3ille,afl( 31 DooefcnDnt,31 t»illfo;i pour pica ^ 
rure>to pour coumfoitte^nD fojpoxir common 
Ditie,fiboitlp ff t fo^tftfe forije a bool^e of Saiu^ 
gation,ajJ3DareC*iC3Qjallpartl?fatifrieanD 
contente, not onelp pour crpectation, but alfo 
t^e Defire of a greate nomber btfiDe^ soobere* 
in 3 tJDillnot fo^tgcti CpeciaUp to toucl)e,bot^0 
t\fz oIDe attemptc fot t^e 0o^tblie 04utgati* 
on0,anD tbe later gooDaDuenture, txjitbtlje 
fortunate fucceiTe in Difcouerpng t^at tjoiage, 
to^icbenoemen bcfo^te pouDurftc attempte, 
fitbtbe tpme of fepng 3lureDei^i0 reigne* 31 
meane bp tlje fpace of,7 o o^pere* i^totbcr euei; 

a.(fi» anp 



TkEpiJlk 

«np before ti)^t tvtm , ^aD pafleD tW Mmt, 
occeptc ouelp Ohthcrc, ti^t Dtjaclte in Hal= 
golandettol^oc tepo^tcD tftat lo^nep to t\)t no* 
ftle fepng 3llateDe:a0 itDoetlj vtt ttmmt in 
aunctente reco^e of c^e o!De ^aicon tongue^ 
&o tljatif ?ou continue toit^ cozage , aj8f pou 
l)aue toeUbegon , pouC^all notonelp toinne 
greate ticlje^ to pout feluej8f,anD b;t j^ng tuon^ 
nerftill eommoDttie$ to pour coutricBut pou 
O^all pur cbafe tbcreluitft immojttall faine,ana 
be p^atfeO fo^ tmtM rcafon tooulo : fo;t opc^ 
npng tbat palTagc, (batC^all p^ofitc Co manp* 
3h tbat Bolie alfo 3 uoill (betoe certain mea^ 
ne 0,boto toitbout greate biffimltte,pou maie 
fatle to tbe lie^tbe dnttt JnDitsi, ^inDfo to 
Camin 5 C^incbital i atiD 23alo^ tobicbe bee 
coutrie$ of greate commoOitie^.a^ fo^ Cba<» 
tai lietb Co fatretoitbin tbe lanDe, tolxjarD tbe 
Swjutbe^nDian feaj8( , tbattbe io^neie i;0f not 
to be att^mptcb.bntiU pou be better acquaint 
eeDlu(tbtbefecountrie;8;,tbatpoumu(tfirftar^ 
tiueat . i5ut tbefe tbpngejSf come in tbi^ place 
bntimelp* 3 p^taie pou accepte ftenOelp in tbe 
meane ceafon tbt025oofee,tobicbe toil! bee a 
greate aioe to tbe ^ell bnoerOanopng of tbe 
telle tbati$ bebinoe* 3flnD ajgJftallbnOer^ 
aande pour Oeftre, fo toili 3 bade tbe otber* 
<15oD ppfpere toeUpour enDeuoure,anD fenbe 
pou focbe gooD fuceelTe , n$ fo trsoitbte aourn^ 
ture boetb beferuejaoc^iclje 3 boubte not twin 

ijifue. 



f)eJicatorte. 

mfue ,<f canfeercD maHce of fome (pittfuU fto^ 

macUe^ Doe not p;tcuaile,aj9; tl^ti can not c cafe 

to p^ctice,to^mDcrrout commoDiticanD 

Deface pour trauelBut a0 it i$ eucc feen,anD 

tf)ecfo;te commonip bno\»en,t^at cnuic Doetl^ 

aill repine at glo;itte, fo ousl)t all Ijonefte l^a r^ 

tej8^,to p^ofecute tf)cir gooD attempted, nnD 

contcmpne tl)c ballpngeof DosgeD 

turret* S)o fere pou toclL 3lno 

Ioiie^pmagame,tl)atDc^ 

ligl)tetl)anDlluDiett) 

tofartl)erpour 

tomoDitie, 

3it)LonDont!)c.)t:n.Daieof 
^ouembcr^iyyz* 



THE ^%EVJCE 
to thegmtle Reader. 




iLtl^ougl) nomber be m^ iifetxui 

finite in incrcafpng : fo tfeat Uncie^f 
t^cre iB not in all t^c tuo^loe, nmber. 
anp t^tng t^at can exctnc t^e 
quantity of it: ij^ottcc t^c 
graOTc on tbe grounti,notbcr 
tt)e D;oppc0 of tuatcc tn tbe 
tea, no not tl^e fmall graincs 
^^_^_ _ __^ of ^anDe tb^ugb tbe lubole 
malle ofYft reartlj: ?et mate it feme bv gooD rcafon, 
tbat noe man is fo erpecte in /rHhmeti{e,tlfnt can no^ 
bcctbecommooitiesof it. Ml^erefo^e 5 maietcuelp 
faie, tljat tf anp imperfection bee in nomber , it is hU 
caufe tbat nomber, can fcarfelp nomber, tbc common 
Cities of it fclf. fo; tbe moare tbat anp erperte man, 
Ijoetb tocigb in bis mpnoe tbe bcnifitcs of it,tbe mo;e 
of tbem flj8" be fee to remain bcbmoe-ano fo (ball be 
toell pecceuie , tbat as nomber ts infinite , fo are tbe 
commooitit'S of it as infinite, and if anp tbpng Doe oj 
mate crceaoe tbe twbolc too^e,it is nomber, Ujbicbc 
fo farre furmountetb tbe meafure of tbe lDo;lDe, tbat 
If tbere toere infinite iDOjloes, it InoulD at tbe full c6> 
p^cbenD tbem all. iCbis nomber alfo batb otber p^e^' 
rogatiues,abouc all naturalle tbpngcs/oj ncubcr is 
tberc ccrtatntfe in anp tbpng tuitboat it, notber gooo 
agrcmente Uiberc it loantctb* HSl Uercofno man can 
noubte , tbat batb been accuStomeo in Hit Bootes of 
Plato, Arhlotell , ano otber auncientepbilofopbers, 
iDbcrc be fljall fce,bolD tbci fearcbe ail fecrete hnoUi^ 
leogc ano bio mtftcries , bp tbe aioc of nomber. fo; 
not onelp tbe conHitution of tbe lubole Uio^lDe , Dooe 
tbcirefcrre to nomber, but alfo tbe compofittonof 

b.j. maH) 



totjicbc tl)f i p;ote0'c w knoiue m moarc, tljcn tbci ca 
fcp tlje bcn(6tc of nombcr attantc. i^urtbci'mo^c, foi 
fcnotolcDgcanDccctamticia aivv otijcr tO^'Ugc , tljat 
manitc3 tattte can cecJ)e l3ntt^,tl)?re 10 nor poUtbtUtic 
imtOouhtomber. Sjt^s eonfeafclltmoiigcile all meir, 
tftatSnouje Uji)aftte«!'>n)»rtg woawert) , ti)at bcfioe tf)c 
^©at^rattealfeae^c^itftc^eu iKwtjitfaUiblc feirolDca 
leDgC:^*ccpteif bi^c bo;jo®eo of tijcni. aiiD cmoiigcfte 
tbrm,ttis fu(ficicmlv^HOU>cn, aitouicltocclarcobp 

Kicsmachus , ailD DtttCrfe OtljCC \OlitCtS , tl)at Jrithmcf 

fikeis tbe founfatne ofall tije otber,anD tijcir grouna 
anD bcnDe,a0 b^calletb it» J^'anp man luill faie,tbat 
E)iatntttc,iUtwe,a!tD|3l)prtb0, inaie be ban Uiitbout: 
It: 0; tbat tbci take tt«c aisk nmh^* actbougb 3; bauc 
before tbt« tpiiw aurtfluercD tljemo , pet itoto jB fatt? 

f>iuinit;e, again : tbatirt Ditilnttictbeceare greate biODefccre^ 
tes In nomb0r0. ^0 tbat Wu^tfr evxcllcntc Dtuinc0, 
!>aue I»?itten tiH)ol0 15ooSe« of tbe miUmts of nom* 
berg. i^lnD fome of tbcic Booftes tntttuled : r-^f S>iaU 
nit'ie ofKomhers, liSut tubalt Cb^tttett tttaime 10 tgno^ 
rainite,tl)at b^\3)m^TnriUU ano >»i>/V,ooetb cotififte 
tbefiiUgrwinbe of al S:)tumi«e:'Mberefo;e 3? ncaoe 
not to allege tbe otb^r bollteaiio facreD Romberg, 
^aue tbat.7» toiU not pmnittc me to paflTe it luitb fi^ 
lence. gn U>bicbe <0 contatneD,not onelp tbe fecrcte0 
oftbefKatwnofalltbpnge0ianotb2fonfummation 
of tbe Uibole U)o;ilDe agamc , twtb tbe ttate of etcrnt- 
tie: y5ut alfo bp it w tbe ^bbctbe0 relic, anb tbtrbp 
ti)e fulUife anb conuerfationofgoolie perfones, re^ 

t/^t. pjefenteo ano mffnuate. sinJlatoe ttwoe Iti?n5e0of 
BmOice are tbe fomme of tbe ftubit :/«/2kf <Difirilnu 
ttucj nnnlHftitiCmmnfatiui , Xub^^be ternw0 3|t)fe, 
a0 befte fenoiuen m tbat arte: iBut Uibat <0 anp of tbr 
botbe tortbout jpiombe rf 3f ftane fa(D m an otber place 
{u 31 Uarneo of tbat noble labtlofopber Arijhtell) 



THE PR.EFBCE, 



anu M'tthmeticall p^opc;tton bceuotlrcU obfcrucD, 
mxt tan noe juftice lucll bee crccutco. ilnD tjclu or 
ten t!)e winttters of ttje Laluc Ufe aiDc of /;ior.;ljer . j: 
tieaoe not repcte,btcaufe none but maooc men Doubt 
of it. anDasfo^pbpfifee, tuitbout bnotulcoge ano Ttyjif;j. 
alDc of nomber is notbrng^ ♦ ^^^ ^^c ti)at nature m 
generation,b£t!)e of manne ano beatteSj^ea anD of al 
tbpnges el6 Doctb obfcrue nomber erartlp. as tyell m 
tbe tvme of formation, as in tbe inonctbcs of quicUc^ 
nFng,anD of bictbcCbe mttteries of tlje fcuentb anD 
iiinctb monctbcs are fqffutente tcftimonies tbf rem. 
iSefioe tbat from tbe fourtbe wonetbe tU tbe feuentb 
ttianp thv^its bee pcrmitteD, tbat els bee not ronuc^ 
ntcnte. #o; tbe tfc of tbe puire,anD fo; crttkalle Da^ 
tec , befibc tbe p;opo;rion m Degrees tn Dmple meDt^ 
tines , anD mirture of e ompounoe mebictnes , anD o^ 
tbec mflntte inaters,lubat nomber can Doe ano tobat 
aiDe It giuetb, onel^ tbe tgno;auntc Doe Doubte. 

^ut Ujbere can tberc bee anp better tetttmonlc fo; j^jtmimfu 
<iombcr , tbcn tbat tbe ccleftialle boDtea Doe Uepc an 
"bnfallible nomber.in all tbetr icoiiDerfulle motions.-' 
tspmranes U}bereof,mannesU)ttte is bableD to at^ 
tainetbeUnolulcDge of tbem.as bp.tbe Jrithmtticallt 
tables, of tbeir motions it is eafilp bnoluen. s:bcre:^ 
fo^e ano fo? tbat lue fee tbe pere , anD all tbe DiQindi^ 
on of times,bcQDe tbe common tjte of traebe bettuene 
menne,to Depcnoe of nomber, luee mufte neaDes not 
onelp confcCTc ii to bee , as it loere tbe onelp ttaie of 
all natures Ujoo;hes, anD of all ciuilitic: but luemud 
alfo bonoarc aitt> rcuercnce it, as often as tuee Duclp 
remember tbe errellencie anD bentfite of it. taas not 
j^omber, tbinbepou, toonDerfnllie bonourcD,tobe« 
ttoe name i»as tbougbt moare meter fo^ (25oD , tben 
tbenamcofi^ombcr;'3Bmeane. i. ano.^. tbe name of 
tbeffl^riniti^f iPuttocomc to moare familiarema^ 

b.ii. ters, 



mo bono»Ttf^f 4Tft>4/> Anthmtti^f^tififh mcafure nud^tteigh* 
tes,fr»m all other urtes, anitberejle thgtremainetb is buP 
hAfe.andofnoeefttmdtion, WS3i\iZXZ alttjougb flato O0OC 
name tl).:ce tbtnges m appearaunce.tbat 10 j^omber, 
fl^eafure , ano Meigtjte. tMbat ace S^eafurcana 
^I2te!g!)te, hwt nombcr applied to feucraUetjfes:' jpo;^ 
UedCure, Meafure is but tfte noinb;tyng of tbepartes of lengtbe, 
vrelgbte, b;eDtbe, 0? Deptbe. ano fo ytreigbteiaa ijere it 10 ta ton) 
t0 tbe nomber^ng of tbe bcuineflTe of anp tbpng. ^0 
tbat if nombec ioere iuitbo^aUjen , no manne coulD 
eitbec meafure^o; tueigb anp quantitie. ano ttjerfo^ 
it ntoft foUoUie : tbat nomber onel^ maUetb all artes 
perfette, ano too^tbte eftimation:fepng tbat toitbout 
if,all artes are but bafe,anD toitbout commcnDatton, 
2tbt« mate futfice fo; tbe tufte comenoation of jritbf 
mti(e, isut ?et one commoDitie moace , tobicbe all 
menne tbatffuoie tbat arte,boe fele,3l can notomitte. 
Cbat is; tbe filing , (l^arpen^ng , and quickening of 
l^t Uiitte,tbat bp practice of ^r/V^we/%Doetb infue. 
gtteacbetb menne anba(;cuaometbtbem,ro certain^ 
l^ to remember tbpnge0 paOe: ^0 circumfpettlvta 
conCDer tb^ngeg pjefente: Zno fo p;iouiDentlp to fo^^ 
fee tbpnges tbat follolDe: tbat it mate truclie bee cal^ 
leD tbe ViUofxtfittt, Idea itmaie aptlp bee named tbe 
ScbMmfc ofreafou, SDbe Itfee tudgcmcnte bad fUto of 
tt , as appearetb bp bw tuoo?de0 in tbe feuentb boofee 
Dcre pubhca.Mbere be faietb tlmsiTbeitbat beapte •/ 
nAtare t9 jfrithmetik^ ^ bee readieavd^iuicl^etodttaineaU 
f^indes oflearnyng.jfnd tbei tbat bee dnlie yfHted,dndyet bee. 
piftruitedand exerctftd in it.tbougb theigette nothyng els, yet 
tbts/halhhei all gbuin tbat theifball bee moarejharpe t^ttf. 
ted, then tbei t^ere before, Wiijnt a benifite tbat onel? 
tb?ngi«,to bane tbe ^ittt lubetted and fl^arpened,3i 
neafde not tcaaeU to declare, Ut\) all menne confeffe ft 
ta be ai» greate u$ male be* C-jccepte anj? toitlelTe pec^ 



THE pxefa;c2, 

rontt\)inlitl^tmait htttQ h)iCe. Wuthcth^tmottz 
fearetJj tljat, is kattc iii Daungcr of it* OJbcrefo^c to 
concluDc, 31 fee moare mcniie to acfenoiulcDgc ti)c bc> 
mate of nomber , tben iJ can efpie loiltpng to ttuote, 
to attamc ti^c bentfitcs of it. ^anp p^atfe it, but fcUic 
fiooc grcatelp p;jattifc it: onlctte it bee foa tlje bulgare 
pjactUe, concempitg ^crcljaunDes trace, ^bcrein 
t^eocfire anD Ijope of gam, mabctb inanp imllpng to 
fuftaine fome traucll. fo; atoe of luljom , 3J DiD fette 
fo;tl) tbe ficftc parte of jfrtthmeti^e,)i5ut if tbei knetoe 
IjotDfarretbts feconoe parte, Dooetberceil tbeftrSc 
parte, tbet UioulD not accoumpte anp t^mc iolle, tbat 
loere imploteo mtt. ^eatbetUioulo nottbtnUeanp 
t^me UjcU bcftotucD, till tbei bao gotten focbe babiUf 
tie bp tt,tbat it migbt be tbcir aioc m al otbcr ttuDiea. 
janD If !P/tf^o Doe require Mtbmetil(e^as a fpecialle auD 
a neceOTarie qualitic m bpin^Uibom b^ tuoulD aomittc 
as a cite^^ein m W politibetoune: ^oId maieUiee 
tbinke of our felues.tbat DeCre to gouerne otbcr,ano 
pet can fcante ffeiUe of commonnomber:'&o farrearc 
manp , pea moftc parte of t3s fromcunnpug m nom? 
btt.fUt* tbinketb noe manne bable to bee a gooo ca> 
pitaine,ercepte be bee f bilfuUc in tbis arte: anb toee 
accoumpte it noe parte of tbofe qualities, tbat bee rc^ 
quircD in anp focbe manne. ]$o\3)bat fo;j tbe better 
triallc tbcrcof , 3B baue m tbis BooUe framcD fome of 
tbe quellions in focbe fojte, as tb«i maie app;ioue t\:)c 
tfe of tbis arte,not onelp gooo fo; capitaincs, but ab 
fo mollc nccelTaric foztbeim.^o tbat tuitbout it,tbei 
can not 95ar(ball tbcir battaile , wotber Ijeloe tbcir e? 
nemies campe o^ fo2te. ^nD if 3 Iball faie as 3 tbinfee, 
Xuitbout itacapitaineis noecapitaine.gn tbis boohe 
lubat J baue luzittcn, fo<: tbe aioe of all mcnne , an> 
namelp focbe of mp countric mcnne, tbat bnoerttano 
notbpng but GDnglifije,^ ncaDc not to rcpetc pcrtuu-' 
larelp, but remtttc tbcm to tbe boohe it fclf.to fee it at 

b.iu. large 



THE PKEFAGE, 

largc« ^nti^ tW niaCe 3; fate: tW a^ 3; K^auc torn In 

fitl)er artf B,fo isttW 3 am tl3e firft tjcnturo:, <n tbrfe 

datkc mamB*mi)tttQ;t a trutt tbe< tUatbe Uarngo, 

autJ Ijappcn to featrc thin too^fee, tuU bcarc tte moarc 

%nith me, if tl)£i fittoe ant tbt»t0*tK?at tJjci Dee mifli&c: 

^Ijercm iftW toill tjfe tIjiB turtcfie , eitijec bp U);r^ 

t^nge to aDmonillje me ttiercof, citbcr tfteim fclfca to 

fcttc foitl)£ a moare pcrfcacc Ujoo;be , 3 luiU tb^nii; 

t^etii piaife lijoatljtf . mut if anp mannc iDill be fo ia* 

gie, otDcr to blame t^at , "^Uif 6e Ije w not iable to a^ 

iitinDe,0; to conDcmime tfiat,tubtrbf \}c did neucr t)n» 

scrttanoe : M fome ofte t^mcB Ooe of a fonoe curiofiv 

tie,3 ^iil toirt^e bpm a better luitte, ano moare mo^ 

fieaic. anoto paeucnteall foclje feuere guDges , 31 

tl30ugl)t it gooD to aDmontri^e pou bcfo;:e, tbat b? oe^ 

cafion of trouble tjpon trouble, 31 tuas bin^ereo from 

accompltfl^png tbis luo;fee, as 3 om (ntcnoe. But pet 

t0 bcre moare ,t!jcn anp mattne mlgbt tuell loobe fp; 

atm? banO£5, if tbct mo Knoiue ano ronfiurr m^ne 

cHate. 2nJ} tbf g motbe moare 3i fate : tbat tf 31 mate 

perceluctbat this lIBoofee bee as Uiell recci uco,aB tbe 

firlfe parte foaj8,3l toUl Crlue mocbe,to dele from mp 

troubles fo moe^,tpme,as to fet out tbe refte of 

tl)t0 arte , moare rompletelp m engffi^e, 

tben euer 31 fatue it tn anp toungue, 

fjetberto Ooen; truft tbereto ao^ 

fureDlp. anulDiflbeb^m 

gooD,tbattrauelctI) 

fo;tbBbetiifite« 



Of the rule ofCofe, 

One thyng is iidthyn^ythe prouerhe isj 
"^hiche injome cafes doeth not miffe, 
Tkt here by ^jorkyng -^ith one thyng, 
Soche knowledge doeth from one rootejpryn^, 
That one thyng mate l^ith right good fkille^ 
Com j^ are '^tth all thyng: And yon ^ill 
The praSlice learn ^jy on p?all fine fee, 
lohat thynges by one thyng Knolom mate bet. 



To thecurhufefcanntr. 

If you ought findeycts fome men makj 
That you can mmde^l [hall you praie, 
To take fome paine-fo grace ma'iefende. 
This Iporke togrol^etoperfeSie ende, 

(But tfyou mande not thatyou blame, 
lUfinnethe praife^ajidyouthe/hame, 
Therfore be "^ifcjandlearne before^ 
SithJJaunderhurtes itfelfmo/ie/ore. 




^ThefeconJe parte ofjrlthmet'tkej 

€0ntainyngtbe extra^ion of^rtd ->, ^^ 

Utrfc l{indts,'i)titb the Arte ofCojf^^e 

nombe rs, drill of Surdenomhci 

alfo , in fond rie fortes. 

fTbe mtcrlmitors, MaJier.Sdolar. 

^ccjJourDcfitc cannot 

bee fattfficD, ncitfjev vour re- 
qiicft CatctJ, tntill J nmc tu* 
fflp aunflurrc rou,tI)at 3; can 
tcacljc t?ou no mo;c : luljtcbe 
aunftucre male date rour re^ 
qucff,altl)ougl) it content not 
VourDcfirc. 
^f !)0lar. 3: hs:(cc\K CoD of 
his mercic, to luttbffanuc all fucl)c occafion:crcept it 
mate be mo;je to pour olunc contcntation nno p;jofitr, 
tljen It UjouId be plcafaunt to tt)c loners of learning, 
i^affcr* |0etaiuffe ercufcmatc 0anDc fo;rmp Dc^ 
claration : asif tgno^aunce Doc info;cc me to 0atc 
mv trauell. 

^c!)olar. |9our otunc fgno^auce,^? trult, rou UiUl 
not allegc:anD as fo? t\)t Ignorance of otljer,it ougtjt 
to bee no (fate : fitb tijc tgno?aunte multituoc ooetlj, 
but as It luaseucc loonte,cnu(e tftat fenotueleOgc-, 
tDljUlje tf)et can not attainc,anD ioinje all men igno - 
taunt,libet)nto t!)emfelf,but all gentle nature6,con 
tentnetlb fucbe malice : ano ocfpifctl) tfjeim as blmoc 
tDo;mes,tD!)om nature Doetlj plague,to ftap tl}t yoU 
fone of tftetr tenemous flange. 

fl^after* Wit l^all not ncDc to ffantie on t^is tal&r, 
but trauell Voith bnotuleogeto tanqutlljc igno;:auce: 
2inn beleuetl)attfte/>rHf of bnoUjelcoge, is mo^cof 
ft^ce t^en tfjcffpnge of tgno;auncc:pca,tiei'o/«^f m 

n«l« Ccometrtc^ 



The fecondc parte 

Gsmetrie.ati^ tltzMitic lit jfnthmetikeJhOH^h botlie 
b:t>iioiuifible,lioejitakc (treatei* U)oo?fees,t mcfcafc 
crrcaccc mtimtiiciejet* tUeu t&e totii^e banoe of ignoi^ 
caiutce 10 l)abie to iDtthftaitne. 

&ci)olar» ^ut tsXkt grotuetlj luell to our mater* 
fnttff. 3|bcfekcpoutl)crfo;e,luitl)tljatt)mtie faeginne,anD 
bmlDc on it pour tuo?ke,a3 a fo;te agauiU tguo^ame 
iDattcr* mnitit 10 of it fclf tjitoiuifible , ano pet 10 
it lit al partes of tbc luo;ilDr,ano in cucrp tt}mg»;pea> 
t^e txjo^loc tt felf conftUetl) of tjnttie,t0 nameo of tjni^ 
tie, lua0 maoe fap tjititie , attD is p.iefcrueD bp tinitie, 
aitD omlv igno.:aurtfc Ujitl) l)cr b;ooOe fccluoeo from 
tmiticfo tijat of it to repete tbe fulle fo:ce, tuoulD oc^ 
cupie muclje time, ano malte greate tjolumes. 

^i)olar. ^ittj tjitftie is fo migljtic , anu of fucfje 
fo.ice^as pou raie)U)l)at maie betljougftttljen of itom* 
Kmher. hct , twljiclje containetl) amultituDc of tiiities ^ ^m 
is notljpng els hut a collection of tjnities* 

£©atter, tanitieistlje fountaincanoo;iginalleof 
itomber,pea tjnttie bp aomtion onclp tl^allmaUca 
greater nombcr , tben anp numbers can ooc bp mul^ 
tiplicaciontliSut t^is is maruciloufejtbat no nombec 
rti^imt\i again!! otutfio, till it come to aittjnitie:ano 
t1:)m Mil It permit no fartber Jjiuifio.^no tberfo.ie it 
is raiD,tbat tmitie uoctlmcilber multiplie no; oiuioe. 

0nti as al nombers maie be mo;e 0; lelTejfo tbe lef- 

.yf^Arte. fer Is euer apar^ o^ partes of tbe greater* 

Tartes. ^s^Vjnto ioisaparte,nameoabalfe:butt)nto7.f. 

is not a parte, out partes,anD is called S.&o 8 to,24. 

is a parte tbat is4;but t nto, ]6* it is partes>tbat is ^* 

^cyoiar* 31 pcrceute,pou call it a parte > tuben tbe 
nunrerato,: in tbe fraction ( rebuceo f the fmallelfe) 
i0 an tjnitie. 0nti tuben tl>e numerato;5 is a nomber, 
tbcn tbat fractio betobenet!) partes of a nomber» 

1113 ut 31 p;taie ^ou , tubat tjarieties of nombers bee 
tbere p^incipallp to be confioeretJ in tW arte^ 

05aftcr* 



of Jrlthmt'ikt^ 
(Ba0cr. ^ombcr (stitiHDcD into tiiunTc Innfcrs, j/^r.^r ?( Y. 

fO.J fomc are y^hok nomhen , ailD ttjCt onclv of t:.cltdcy -u ihon 'oj vom 

^<jfrrw,antiotI)crgooDU?;itcrsnrf fallcDnoiiil'fi's. hns. 
£)tl)cr are brohjnmmbnsjnm arcfcirnicnlvfalicD 
frdBims. ^f tWc hotijt S hmt U);tttcn tn tljcftiftc 
anD fcronoe partes of J'rithmeH^e : ^o tbat j nngtjtc 
feme to curioure,to repctc mn^ parte of it agamr 

:ll5ut nolD tn erfte UinDe of tljcfr , there arc rci tame Tie Ccconit 
nombcrjs nameo AhfiraUe : aiiD otljcr callciD ncnih is dmifvu of 

ContrdHe, nomhers] 

AhJiraHe nombcrsaretljofe, iuibtcfie banc no Ueno' JbjlracU. 
mmation annereD tnto tliem.:^nti tliofc tljat tiaue a 
iir Denomination lo^neo to tijelm^are caitcc ContrdH^ Conhaae. 
ttombers. 

^c!)oIar. SCljisj^reetobcareafDiiablc tJimnftio, 
ano agrealilc to tbe fignifiration of tijc namf0. 

fo; as ttjat nomber is rotracte, from l)js gcncrall 
libertie of fignificaticn,ii)l)tcl)e is bouDc to one Oeno: 
jnination>asinfaivng. lo. grctes :lul}cre. lo. isrc- 
CraineDfro tbc libertie of fealobvng anp otijer t!)ing 
fcut grotes ) fo tf it l)aD no Oenomination aDiotneD, it 
i3i(g!jt tbai fignific tlje nomber of Dates, o;: of miles, 
02 anp Iihe tfj^ng^as Icell as of grotcs . f o: totien ^ 
fafe»io.atiD Doe not Iimitte anp Dcnominatio,tl)en ts 
t!)at«io, abCraeteanDfeuereDfroalirpectalties, ano 
ftanDctl) free to anp name of t!)(ng» 

5i5uttI)ts(metl)inl{etl)tnDervourro;reftiDn)fam;^rf/>^r3;c 
notertcnD to b;ofeen nombers:bI)ifbe cuermo;e rar^ Jf» no7nhns 
tV luttl) tl^em tljelr Denomtnanon:rcrnn: tljei ronfme be contrafir 
Bf a numerator anD a Denominator. ^^.^^^ ' 

£paften pou fentc to fate lueU.anD tl}c iil-.c (uDge^ 
tnrtc Doetl) appere to be m fome U);jiters of t!)is arte. 
llPnt ret fe^ng tfjat fractions mate Ijaiie all otber ar 
tiftdall Denominations, tl)at iuf^ole ncmbersmaie 
rceetue : anD maiealfo bee Uiitbout tbeim : tljerefe^e 
niiiSti\}tttit\)txmt^Ui[mo}t cutiouk Dmmctionof 

^Ai> til at 



The feconJe parte 

tW name orDenommatton : o; els toee mutt ferfuDe 
fractious, fco tl)t neceCfitte of tljat name:o; cla tl)\t^ 
lp,toat!OieDcontcntioiT,caltbemnomfacr0coiuract8 

^cbolan 31 alTentc thereto as reafon luoulo. 
y»hyfranioi ptt one tb^ng mo^c J mutt DemaunDe of pou,U)6p 
if not called iuclidc^mi} tl\c otl)cr learncD men,refufe to accompte 
nomberspm fractions emongeft nombers. 
pcrly, ^after.HStcaufe all nombers DOC conirae of a muU 

titu^t of tjnttics : anDcucrg proper fraction is leOfc 
tben an tsnitie^ano tbercfo^e can not fractions eract^ 
ip Ijc callcD nombcrs: but mate bee called rather frap 
tionsofnomucrs. 

^cl)olar. 3n Dceoe notu tbat J Ooc loaie t\jt matct 

mo;c cractlr J tt apperctl) tbat a fractio is not pjoper^ 

Ip a nombccbut a conncrion ano conference of nam^ 

bcrs. Declaring tbe partes of an tjnitie . fo; tbe nu* 

inerato? DoctI) fi^niUt one nobcr,anD tbe oenomina^ 

to; an otl)cr:2Ct)e Dcnominato;j declarpnge into l)ol» 

nranp partes tl)e tjnttie is uuiiDeD , ano tbe numeral 

to.: fignifii^ng tbat of tbofc partes, not all, but fo ma^ 

np onelp are to be taUr,as tbe nunierato;^ inipo;itetb^ 

T3e diutfion idaUet. C^cU,t!)cn to p;oceDe,nombcrs abliracte 

•fnmhers are conGDercD in. .\p;incipall ljarietics:n^bat is,firft 

AhflraSh, iuttbijut eomparifon to anp otber nomber o; figure. 

Nomhers ^"^ *^^^ nomber maiC tocll be callcD nomh^r ahfolutt. 

Abfelute ^econDartly , fome nombcrs bee ijfcD onelp in re* 
Kmher's ^^tton to otbcr, anotberftj^e otigbt to bee calico nom^ 

Helatme ^^^^ reljjiue. 

^ ' S^birDlp , mani' nombers are referred to fome fi^ 
gure,tbat mate rife bp multiplicacion of tbeir partes 
togetbcr,anu tbat Dtuerflr.^nD tbofe nombcrs tber? 
fbie mate bee calleii/^«r-i//r mmhen, 
H^mhers ^cbolar, 3!f 33 conceiue pour too^Dcs ngbtlp , tb(« 
fiiunlU. is rour mcanpnge : tbat tuben 3! fate, lo, i^, loo . o;» 
200, ic, tbefc nombci^s Cano abfolwte from all Oeno^ 

mtnaetoa 



of^rithmetike. 

nimacioit,anD cletc from all rclatio anD companToH. 
l!5ut lubcn 3; faic. 6.10 ijalfc cif*i2.o^i).is triple to 

5 . Ijerctbe numbers beepngcompareo together, arc 
aptlp rallcD nombtrs reUiiuci^Q if 3 faiejtljat . 16. 15 a 
/^w4>^e»ow6fr,bicaureittsmaDeof.4.niultiplicDbp.4 
tdcnis.i6»l)erctobecallcDa/^M>'4//c»o»i^fy. 

SBaOcr. iPoutaUeitUjcll.2:berfo;cUjill3 b^icflp 
toucl)c tbc mcmb;t& of euerp Umoe. 

^trU of abfolute nomb;e0 , fomc arc (uen nomhers, 
aiio fomc are odde, 

^cljolar. ail men bnoluc tljat ♦ ant> farther, tbat Kmhers, 
euen nomben arc tbofc, lubicbc ntaic be DiulDcD into C' r«f»,cr oddt 
qualle balfcs : ano fo can not odd: nomhen , luitljout a 
fraction. 

Matter. £Df tbls plaine eafie tbpng , marttc lubat 
feloU)etb:a greater DoubtcDlflTolucD. ^o^fan oDDc 
nombcr(as.7.fo;crample) can not bee partes lnto.2. 
rqualle nombcr^ , ccbc bcepng balfe of. y. tbcno.i. 
iubicbc is commonlp callcD tbc balfc of.y.ts no nobcc 

^cbolar. 3;t can not be oenico. 0no fo {^ fee nolu) 
no fraction can bee a nombcr. 2^bis greatc ooubte in 
platnlp DiflToliieD , bp a tjcrp'certainc ano mofte Uno- 
Ujcn principle. 

i^atter. 4^otu fartber. £)ffaotbc tbcfe I?inOcs at KomVers qq, 
nombersjfome bee comboundc,^m fomc bafim^le ann pouude,ind 
vncompouHde . Comfounde nombcrs are maOe bp miilti- y^w^j/r ' 
pUcacion of.2.nomb:es togetbcr,ano not bp aDOitio, 
tbougl) tbc name mfgbt feme to feruc to botbe. 

^cbolar. ^o3!percciuC:,tbat5'.isno«»f/'oj^n</fno.' 
ber,altbougb it becmaoe bp aDortton of.2. ano. 5. but 

6. tubtcbc is maoc bp multiplication of. 2. ano. 3. 
ILibetuatcg. 9. is compoundc, bicaufc tbat. 5. multi^ 

plieD bp.3«ooetb make. 9. 

ano. 15-. alfo ts compounde bp multiplipng. ^ ano. 5. 
togetber* One it h$ 

0nD bcrcbp 31 fc tbat i.is not to be calico a nombcr nember, 

a. ill. fo; 



The feconde parte 

fn tfeeit nil tiobers about rt, tnuH netJCB be impounde, 
incauft tijci confift all of Unities* 

fatter. X5utvctb?wultiplicationof»Kno otlbcr 
itonib<rt0"w/ow;</f» 

^cljolar. )bv tijofe Xoo^ijes 3| am taufibt to fenolue 
mo;ic,anD fpcaUe better » 

^BaUcr. Httfttnowl-crjaretet otuerflttobe coii0tie' 

ret) in tlietr Diutfions«i^o? altbougl) tlje greate multt- 

Tr^oo is v»-'ttioeofcuenno»iber£;beefo»»/'o«n</f,^ct.2»!jEiaceontp.' 

£onjt>outtdc. teD truelv an twtn nomber^ojiginall^ano yncom^ouh, 

^ortiBtitlimie mafee otJber iTomber0, t w maDe of no 

ncBerilbut of tjnitieg onel^jW aLoODe nonibcrs are. 

j^ll dtber euen nomlbertf are cmptunde , atib are nu 

nerCi^ DlUlDeO , fO J fomc are euen nomhers euevly , anD 

^uenmmi fonte aree««»«owi'mo</ii</>,anDfomearef//e»wow^m 

hrs,€ucnly Ixitljt eueiily and odddy.Euen nomhers ftten/j? , are fucl}e 

nombers as mate bee parteD fontmiiallp into turn 

!)alfe0,till ?ou tome to an tjnltte ^ jas fo? erample. 32. 

fird tjs Dtuttjeo <nto, 16. as l)t0 enen Ijalfe : ano again, 

i6anto«8« as]b<sl)alfe:0nD«8.agamebr»4* tsparteo 

mto.2«e«en partei5:2Cl)e4T«4«alIJ3 b?«2«anD tf^atiA^ 

Dtutoeti tnto.2.t3ntttes,a0 tts futte Balfes* 

Eue nomhers HBut euen nomhers "ifnevenly , are fucbe nottibers 30 

t>neuenly. male bee OliiiOeD into. 2. equalle partes : toljifije are 

xjODe nottiters.0s.i8.(s otufoeo (nto. 9. anU-p.ajj Ijfs 

lialfes, anotlbetareoooe. ^o«io,tsmutocubv.^-^nO 

3o.bv.i^tutt^a0reatenottTbernio?eoffucl}ero?te. 

tuen nomt N«»j Jers f»rtt tuenly andoddelyy bee conwnonl^ calleO 

lers euenh f«f^J^ nontbers 5 as mate bee Diutocu tnto* 2. equaUe 

andoMi ^"^ ^»^*^" ^^alnes : but befojetou r ome to an tjnttie, 

tbe Ijalfcs tutil be oDDe nombers. jas*6o,inate be firft 

partcti into. 5c, airo.^o. tubifljeare euen:BnD tbeta^ 

gatne istuttjeo b^.Ts-.tubiclje fs oDOe. 

3ltfeetuaies-24.ts DiuioeO firft b^ J2.0no tljat I2.bp 

6. $ laltlf* 6* IS otuiOeD bp, ^.Wiclje is an ouoe nober. 

^^3. 28. male bee tiuiUjeu Into. 2 . equalle ano enen 

Values, 



of jfrlth.mtlke* 

!)aluc5,tliat 10 into 14. artt) that 14.i1tto.7- tufticbc is 
tl)0 Ijalfc of. H.but 15 ait oDOc nombtr. 

^ct)olac» 2C;)i0 31pcrcciucUieU. Hno, asjlinisge, 
tl)c DiftiiTctloH irtto ttjofe. ]. UinDes, 10 not onel? ceap 
fonabIe,lJut alfo n:DcfuU. ano pet vou feme to fpcaUc 
DoubtruUp,of tl)i0 uae mcmb:c. I13icaufc31 rcmcm^ 
bcc not tbat pou tfc tbi0 toojoe commolp, but tobcfc 
pou glue place ratbcr to cudome, tbcn to reafon. 

^pafter. iD:cl3tocuftomcof tbe common fo^tc of 
tu;ltci:0 , catbcr tljcn to tbc luogemcntc of tijc moftc 
aun£tentetu:ttcr0. 

anD fo in tl)t0 cafe ?«c//</f ooctb not feme to mnittt 
tl)i3 t'oiroo member. :5l3ut accomptetb it tmder tbe fc^ 
conDc kinDe. as ma(c tuell appcre in !)i0»9-bokc,anD 
;4.p:opofition,ltJbci-c be caUetl) fucbe a no?nbcr,f«c» 
Ij eusnytndetienlyoddh aUoy tubichc p!ace cofcrreD U)Itl) 
tbeDcfinition0 intbefame booHc, Doctbapp:ouctn 
manv Ujifc menne0 opinion0,tbat ^ttclidc mmDco but 
2. oncli»kinoe0 of tbofcnobcrs.ano pet in tbistbinjf 
(I tljinUe ) be did ratbec app;joue. 5 ♦ Varieties bp bi0 
p,joporttto0,tbeneaabliajeonelH.2.ro;te0 bp bi0 firtt 

Deftnition0. 

"13iit beretrt 3| luill fpenoe no mo:c tpme . But faie 
b;icap tbat tbe Oiatnctio of. 5. Umoc0,reruetb to gooo 
tfcano eafe in teacbpng. 

auD nolu fo I fautber knolulcDge of nombers, fomc 
are calico mmhersperfethj% fome attnomhers im^erfdl, 

ferfelhyiomlen arcfucbe ones , luboreparte0 top^' Homhen 
itcD togctber,tuill maUe eractlp t't)t lubole nomber. pfrfeSfc, 

anJ) tberfozc are.6.anD.2S. accomptctJ perfecte no* 
bers : btcaafc tbe parte0 of ecbe of tbcim aDDeo toge* 
tber>Doe maUc tlje ful anD tnterc nomber, tobofe paro- 
les tbei bee. ^0 cf.6.tbe balfe 10. ?.tbe tbiroe parte 10 
2.tbe firte parte i0. i. ^0 fo.: a quarter, ano fifte parte 
it batb not in tubole nomber. Jiotu put togctber.i.2, 
nnD. V ano tbct make inft^. 6. tubcfc partc0 tW bee. 

a no 



Komiirs 



7f.9mhrs 
fu^erfmvfe 



The feconde parte 

0nD tl)etf0aef0*6.a perfecte nomber* 
28« Etketuates* 28vliatii fo; l^us f)alfe. 14. to^ W qnar^ 
ter*7*fo? f^iB (tutntt) pacte*4* ano fo^ W folucrtctit^ 
parte.2.anDfo;t 1)10*28* parte* i* allluijtcljeputtoge^ 
t|)er,t!)at U3I* i*2*4»7.anD.i4*0oe malje.28.of t^)W fo?t 
rtcre are terp fetoe mo;e tit copartfa. jano fo? an era^ 
pie, 31 fett ibere, a» manp as are tjnDer.6ooooooooo. 
niiD tUi are tljefe . 6. 28. 496. 8128. 150816. 2096128. 
35S'So356.n68H5'28. 

315ut nolo of tbe contrary feinO, imferfeHenomhers ht 
fuctie, tDl)brc partes aDDcD together, ooe make citljec 
mo^e oa lelTe , tl[)en tl^e U)lroIe nomber it felf : lut)ore 
partes tbet bee. 

0nD If tbe partes mafee nTo;re tljen tbe lol^ole nom* 

i£r,tl)Cn is tjiat nober tailtHfuferfluoufe^Qi dhtmdaunt, 

as 12. loftofe partes are. 1.2. ^4, 1.6. U)Wcl)enTabe i6« 
^o« 20 ♦ t)atl)fo;6t0 partes, i. 2. 4. y. 10. tu^icbe 
tna&e.22.ilt^lDates» i2o*l^atb tbefe partes* 

'• 60. 4^0. st A. ';. f^ \t.}^mtm,U24o. 

SIiiDtftbe partes mabeleflTc tbentbetDbole iiom^ 
ber,twt)ore partes thti be, tben ts tijat nomber called 
Kmhers S>mmute,o^S^efeBiue, jas.8.ibatl^tbefepartes.l.2.4» 
Sf minute. ^W^t make buty* 

^0. 16. ijatb tbefe partes. 8. 4. 2. i. am> tijei ma&t 
finely. ly. 

3Life£lua(cs.32. tubofe partes are. 1. 2. 4. 8. 16. anft 
make but. 31* 

^cbolar. ^n all tbefe itombers 3 note tbat pou re^ 
ken one, fo; a parte of ecbe one of tbetm j tu^iicbc be^ 
fo;!e 3 tl)ougbt pou l)aD oenieo* 

$@alter. i. canne neltl)er multfplte no;i DeutDe^ano 
tberfo^e compounoett) no nomber. 15 ut one male m^ 
ereafe aODttton, anotijerefo^c tubere partes be aODeD 
to g[etbcr,tbere.i.m ate tueille f alMa ^arte. 
'SnottjisQiallfumFe foa tbe ttmUon of eucn nom^ 

bew 



of Jrkhmetih* 

ji50to to fpcaUC Q^oUenomhers,^omt Of tl)f arc cm Oddt mhers 
foundc,% fotrc Vnconjpoude,'Sl'i)tl arC compoi:n(Ic,\jo\ni.\}i: Comj)OunHc. 

maic Ucc DiuitJcD into anp otftcr partes tljcii initics. 
::!s.9.1ul)td)cifii:cpounDeDf.v ilno. i).tljati;emaDc 
of.vanD.^ ^iro.2i.isfonipounDcof.7. aiiD.^. I'jiiD 
fofiutbc. 13nto. v7, lu i^\ 17. 19. 2> :9.aii0fu:l;c 

UllCbCC ODDC noilltcrs Vncompounde, fo; ti)tl atC nor Vncom^ouiic. 

maoc of anv otbcr tljcn of bniticsj. 

i')tTC muft V'OU IjllDcrUaitDC IV compojition, t\)t mill 

ttplifation of tbc partes of noiubcrs to0ctt)er,as you 
renTeml:e,bcfo;c tuas D:e(areD. 

S'cholar. 3: confiDer it fo. ano 5 remenib<jc all tfjat 
I'ou bauc taugbt me^fo^j t tjc Dtntfio of nobers ab/itdfh 
ano abfolutf . CGbat faic vou noUi of nobers reUttue: Xomlen 

{Batter. ^onietV'incs tl)cir reUthn j^atf) rrn.aroe ff/tf^/^^ 
to tljeir partes, nanuip, Uiljetljer tl)efc.:.thnt bee fo ^ 
compartDjljaneanvronimon parte, tbatUniloiniDc 
thcnn botbe. f-a; if tbei bane fo , tben are thei ealleD 

nomhcrscommenfurahle, •3S.12. anD.:i. bctnombcrscoin^ Commenfiis 

»»fw/ttr4^/f:fo>. 5. tuill DuuDc ecbe of tbeun. rabU. 

J^lUclualeS. 20. anDo6.br commenfurahle,k\miX. 4.ts 

n commo Diuifo;ifo; tbcun botbe. L^ut if tbei ba'ne no 
fucbc common Dunfo2,tbcn are tbet ralleD imommeufuc 
rahle^^s i8anD 2v i^o;2jean bee DiutoeD up no nom- hicommei^. 
htt mo^t tben bp.y. <anD. iS.can not be DttuDcD br it. furahk 

3n Iihcmaner.^6.anD.49.aremcomwfn/tt;4^/f: j^q? 
49.batb no Diuifo; bnt.7. ano 7. can not DiuiDe. ;6. 

S)cboIar. Doc vou mt^nt tbcn,tbat incommenfura-. 
llenomhen,])mc no COparifon n02 proportion toctetbct:' 

SBafter, ip^aicnotbpnglcflrc. j-o:anp.2.nombers 
mate bauccomparifcn t/ro/Jorf/Wtogctbe r,aItbon(rb 

tbet be mcommenfurahU, ^s. '^, ailD. 4. arc incommenfn ■- 

rdhleyunti pet arc tbci (n aproportien togctbcr: as fl;aU 
appeareanon. 
Mut firftjj Ml Declare tntopou,tbctanetie0of 

15.1. proporiitn 



The feconde parte 

fr^trt'm. priportm, tufjerefn tftcre maic be Double coitferece: 3| 
meant of tbe Icffer to tfte greater, o^ of t(je greater to 
tbc IcflTer ♦ 
tXJbe tf)c greater is copareo to tlje learer,tt iB calleO 

Of greater a froporthn cfthe greater inc^UdUtie. Za 6 to 2.0; ) tO ?♦ 
mejudlitie. 3»o luljcn t!jc leflTer is confcrreD to tbe greater , it 
OfUJferim i& talleD a proportion of the lejfer ine^tialitie . ^5. 3^ t0» 7. 
ewslitie, 0^2.to,6« 

^!)olar» anu tobat if 31 tuoulD copare ttuo equalle 
Mombers togetber;? 

Rafter. Cbat 15 accoumpteb alfo a p;iopo;jtioit of 
nraitv men:anD is caUeb tl^t proportion ofequAlltie, 0nD 
tben ougbt tbe firft DiuiCon of p?opo;cion to be,tbtt« 

CCqualitie* 

id.:opo:c(on of -s ^^t. 

^ } CSCbe greater. 

C3;iTeqnaIitie» ^ 

CS^bcIeiTer. 

^0 p?opo.2ci6 of tbe greater ittequalitie, is DiuiDeir 
mto,^reueraUkinDcs:U)bereof»5» btfmple, anD.2.0^ 
tt)tt compounder ^IjtfttQc iJiitDe ts , Uibcn a greater 
nomber contatnetb tl)c leifer Diucttc times:as tUjife, 
oHf)n(c, o^oftener. ^0. 6. containctb»^» ttutferano 
itcontaiitetb»2.tb?ire. SCbis p?opo?cion is callcD 
Multiplex, generally , multiplex, tljat is to faic, manp folue: hut 
fpeciallp it is nameO , accoiDvng to tbe tmcs tbat it 
fonteinetb tbe leflTer. ^0 tbat if it contein b?m ttuife, 
tben is it irameo dupla,o; oouble. 0s 2 to i ano 4 to.i. 

anDifitcontaineittb;ifetas.3»to.Lanb.6.to*2*it 
is talleD tripU, 0; triple* 

Jf it containe it ♦ 4 ♦ tmt^ ? tben is it itiAdrupU, q^ 
quadruple, 

i©f tbefe ano of Diuerfe otber ro;ites in tbis feinb al^ 
ro,bere are tbe names b;ieflH r^t ooune loitb eraples. 



of Jrhhmetlke. 

iiufU. 4.to-2t6.t0o:io. to. ^:i8.to.9J4 trouble. 

TripU, 6.to.2:9.to.:?n2.to.4: I8. to,6. i' ^x\^\t. 

QuadrupU. 4,to.i:S'.to-2:i2-to.;: 16. to.4. -) foU'crfoloe 

QumtupU, ^.to.i:io.to.2:if.to. 5:20.10.4. 7 j'uirfoltc. 

Scxtupls. 6.to.i:i2.to.5:i8.to..:24.to.4. K S'lrcfolDc. 

StptupU, 7.to.i:i4.to.2:2i.to.;:28.to.4. 7 ^cucnfolDc* 

OciupU. 8.to.i:i6.to.2:24.to.5:52.to.4. ' GiC(l)tfolDc. 

KoncupU, 9. to. 1:18.10.2:27.10.5:^6.10.4. j ;^ j^incfolcf. 

§)ecupta. 10101:20.102:50.105:40.104. 'f SI^inncfolDC. 

VndccupU. ii.lo.i:22.to.2:55.to.5, •-;- i^kucnfolDc 

f>uodecupld. 12.10, 1:24.10.2: j6.tO,5« ^ 2CUjflUCfolO. 

;anOfcinf.uitdp. 

^cfiDc tW tbcrc in an olljcc fettiDc of p;opo;ttorT, 
tubcii the greater c onlainctb tlje Icflfcr , mo;jc llicn 
oncs,anD not iU)irc:anD tl)al mait farm 2 fo;tC53. fo^ 
tflbc greater rontatne tl)c Icflfer^anoanp one parte 
of l;rni :, ttat p;cpo;cion ts ealleo SuperpArtUuUre. supertarti- 
i^o;ierample,iabc.y.lo.4. ^tl!).^t)oel6contatitc.4. ^J^^^ 
ano l)iB quarter. Itkekaies. 6 . to. f . 10 in l^efamc 
bmoe of p;opo;lton: alttjouglj , not of ibefainc fpeci^? 
all fojlci^ o<i 6.(cmp;jel)c noitfj.f.ano !)iu fifie parte. 

^c ll^ai fc? a n*o?c fpr eiall mflinetion,ef l)c of tbcfc 
cno ma nv Dil)er,l)anc tverr fait ral names-, arco:Ding 
to tl;at parte, luiiiele itei toe ccnratrc. ^s if it rcn^ 
tame ibe l-alfe nic?e,it is namcD Sef^uUhera.'sn \hhu sen^utakeYA 
cljcpjopojlionar: tl)efcnonib£ii5folbli[-r.g. ^^ 

5.10,2. 6.10.4: 9.1c. 6: 12.10.8: l)MG.lo, |i!. 
But If t^e greater (Dirp;e!)enDe the le(rer,ant)*l)ij5 sefmttrti^ 
tl)trDcparte,tl)ent0tl)atnameD5f/;tt/ffr//dp:opo;ti* 
on.i^slnt^cfc. 

4,10,5: 8.10.6: 12.10.9: i6.to.12: 20.to.15', ! i:, 
i^nD kljen tl)c fifte,firtc,feuentlj,02 eigfjt part tioetfj 
mafee l!]e paopo;icicn,D; anf otber part el£5,tl)e name 
U taljen of tfjal fame parte, ^s fo^j baefncOTc 3; luiU 
|ercfettee):dmple0» 



SefqiiiijHdna. 

Sefquiquinta. 

SefquifixU. 

Sefquiftptima, 

SefquioHauA. 

Sefqtiinona, 

Sefquidecima, 

Sefquiundecima, 

Sg/qutduodteima. 



The fecondt parte 



f, to»4:io,to»8:i)'.to.i2. 
6.to.)-:i2,to. lo: iS» to» i). 

7, to. 6:i4'to/i2»2i:to,i8. 

8. to.7:i6.to,i4:24.to.2i, 
9.to.S:iS. to.i6:27»to.24. 
i?.to, 9:2o.to.i8:yo.to.27. 
iLto.io:22.to.2o:5;. to^c. 
i2.to.ii:24.tc.22:;6.to. ^^\ 
i>to.i2:26»to.24:59,to»56. 



I t!0 quarter mo;ic» 

i:iafifrcmo;c» 

i:|afirrcmo.:c. 

ir,afcucntI)mo?c. 

ic'aiuigbtmo^c* 

I'lanmctljmo^c. 

i7'j'atrnt^mo;c, 

i,-rjaIcurntl)mo;c» 

I ,', a tiueliictl) nio;e 



antj fo as farrc as pou Iiffc to p?ofcDc in fucl)c p;o^ 
ponioiKiDbcrc one parte of tbe leflTer, is tl)e luftc Dif 
ference anD crccOrcbetUjcnc it ano tl)e greater* 

15ut If tbe Difference be.2.parte0.^.partcs,o^mo;jc 
Suptrpartirs partes: tbC p;iopo;tl0 is mmc^fu^erpartiente. ^s.y.to 

3.anD.ic.to.6. i^o2as.r.containettj»5.anD.4^,ofit;ro 
io,l)olDctb»6.anD.fofm 

^cljolar. .^olti3lperreiuefomet3feairo, oftfjcDt^ 
Otnction faetUiene a parte and partes in nomber : £Df 
tobicbeattbe beginning ^oti DiDfpeake. 15utl)olu 
manp Uindes are ttjerc of tbis fo:tef 

{palter. SDIjereare in8nitefemDes intbisfoztcof 
^j:opo.:cion , as toellasmtbeotbcr. i3utfo;jeram' 
pies fafee^i VoiU fct fnrtbe fome of tbe motte common 
nombers: tbattbcrbi'pou maicgatbertOefomiesof 
ti)e relic, j^ud tbcfc be tbct , tuitb tl)eir names. 



,TertiaS, k.tO.^: Io.t0.6:r5'.t0.9:2o.tO. 12. 

Quinias. 7t0f:i4.t0 Io:2I.tO I^:28.t0.2o 

Stiptrbipan'wts.'<^^eptima4, 9 to 7:iS to i4:27.to,2i:56»to 28. 

Kon^. hi to 9:22 to m]]Ao 27:44.to 36 

Vttdecim4S.\i],tQ W26 to 22.$9 tO 53:)2 to 44 



I 



of jirlthmetike, 

^Qu^rtas. 7.to.4:i4.to.S::i,to.f:i:S.t!:.r6. 
\Quintas. 8.to.^:i6.to.i .:24.toj);-2.to 2c 

JSefitimas. lotOyilCtO 14:y5t02i:4^. to ^S. 

/,t,r.f,ri4 °JlZ-, ;!-f.S:2:.w..6:;;.to.:4. 

J £ r I <jjecim4i, I). tO . I o : 26 . tO ♦ 2 ) : -9. tO. Jo. 

Vndecimai, I4 . tO ♦ 11 ; 2S . to . 22 : 42. tO. V^, 

'Decim.utertidj. 16 . tO ♦ H : ;2 . tO . 26 : 4X. tO. 39. 
'Decmoiquartas. 17 . tO ♦ I4 : ^4. to. 2S: ) I» tO . 42, 

(-Quint^i. 9.ro.):iS\to.r:;7.to.h-:;6.to.2o 

\5f;//m.ti. nto7:22tof4:;^.to.2i:44.ta2S» 

St^ferquAdru JKoaas. i;to 9:26. to. lS:;9.tO. ^-:r.tO ]6, 

prtiens. J Fndeamai. 1 f . to. I I .S o. tO. 2 2: 4 f. tO. 5 ^ 

/ Dectmastertias. 1 7. to. I ,: ^ 4. to. 2 6: )' I. to. 3 9. 

y^7)ecimas.;juint4S.i 9, tO. I )-: ; S. to. 5 o; > 7. tO . 4^ 

/*5fx/<^. iito6:22toi2:ato.iS:44.t024. 4- 

\ S^phmas. 1 2. to. 7: 2 4. tO. I4: ;6. tC. 21.^ 

JOUau^s. 13. to. 8: 2 6. to. 16: ;9.to.24 

5«/;.ri«/n^« 3 ??'*'• ^ 4* ^0* 9^ 2 8. to. I8: 42. tO. 27. 

plJn ^ ^^'^T''' ^ 6. to. 1 1 : 32. to. 22: 4v to. 5^ 
f4nims. Cq),,odecm4S. 17, to . 12: 34- to. 24: fl. to. 11 

lDeam44Urtia4. i^, to, U: 36. tO. 26: ^4. tO. 39. 
Decm,uquartas,l<^, to. 14: ^S. tO. 2S: p. tO. 4^ 

iDecima4fcxt^. 21, tO. 16: 42. tO. 32: 63^0.48. 

/-Sr^j/m^. 15^0.7: 26. to. 14: 39.tO.2L 

\Vndectm^. 17^0. I i: 34. tO.^-: n.tO XX 

5«/>fr/?>r/«»J2)fr//w4i/er//^. I9.to.n: r>. to. 26: 57. tO.Vo 
f aniens. 1 '^ecm^./epimaszx, to. 17: 46. to. U: 69. to. Vi; 
/ 'Decnn^noTm. 2), tO . 19: p. to. ^S: 7^ to. O, 
Vs.r/rf//»^fa//^.29.to.23: f8.to,46: 78. to. 69. 

$>coojar, 3 DiiDcraanDc bp tbrfc ctampks , fome^ 
Jujatof t()eicceafon5:but3I pcrcciuc.pou Doc notfo^ 
louje t|)eir naturaUc o;ocr, luitoout interruption,!!! 

WAi). t|)c& 






The feconde parte 



tticfeofti&elaffefeinje, 

Rafter. Ko tftmtc ntc tcu mate t%t httttt ttitim 
ffantie gooD grcuno m tl)atctmfii0ir,3! Ittlfttfuttlje 
tcrc tljofc omittco nontberistSEtiat tcu traie fee fjctu 
tl)ci tooulo erp?cire fome cti^er p?opo?tic Ijere itameO 
0itD tl)ccfD?e tl)ci tioe feme ratbtc to le oimtttO,t|jen 
m oeeDe fo to be. 

^arke ttctm tuelL 



^ SecundiU, 4, tO ♦ 2: 8 . to. 4,] 4 
S^4^^^. 6. to. 4: 12. 10. 8.IT 
5;//f^'^/J4m*f»^<'5rv/4f. 8. to. 6: 16. to. 12. i-f 

JOaauoi, Io.tO.8: 2O.t0.l6.|l:5- 

^Decimat. I2.t0.lo:24.t0.2o.ilf 



^c^olar. 3'n oecfie tiereffee^, tl^efirfiefigDcubte 

p;j0p0 Jtton.^fjC ktonJitfe/^uiaUera,t\}C tWtitftf^ui:} 
tenia jt\}t t0lji3Cttf}fefijuiquartaj<i tfec fktitftj^uipinta^ 

jailer* ^o maclie tijefe otfter. 



y- SftunddS, y, to. 2 : J o, tP . 4 . IZt 

SXrr/w. 6. t0. ?r 12 :t, 6. 1 r 

/upcriri^4Yiiens ^Sfxias. 9.tD* 6t I^.U. irJl^ 

^ (DkUecimM. iy.tO.I2oO.10.24. ;i:y 



^cfjolar. SCljefirfieoftljefelfeitctcctiof, tut all 
tlje otfter are mmtn before* 

Sailer. SHIbe ftrfle it a r cmfrunte p?cpo?tton (as 
anon 3 luiU DeciarejanD (0 name t? dupUjeJquialtera, 

3i3ut nolu iuHl I aite futt^e ail t|jc otbei cnritteo 
nante0. 






fttrtiens. 



f Secundii. 

\ Tertias, 
QaarUi, 
Sextas^ 
OHauas, 
^ecimas. 
(DHodecimds. 



of JrtthmHih. 



6. to.2:i2.to»4* 
7.to, i:i4.to.6. 
8.to,4:i6.to»S. 
io,to,6:2o,to.i2 
i2,to.8:24«to.i6 
i4,toio:2S,t0 2o 



;- IripU. 

2y ^uplafef^uitertia. 
Y ^upU. 

I] fu^biparties teriias 
I [fefquialtera. 
I ] fu^hipartih qulntas 
ly Sefquitertid. 



I 



^„, i6.to.i2:52»to.24 , .. 

IDecimas ^uartas.]^, tO, 1 4- 56- to, 28 j I ]- fu^htpartiesfeptmai 
^ecimasfextas. 2c,t0 16:40 tO $2. ! I 4 ScfjuirjUarta. 



fuptr^uin^ 
tiipartms. 



r Secmdau 7 to 2 : 1 4 to 4 ♦ \Vt Tripla fefquialtera . 

\ Tertias. 8 tO 5 : I 6 tO 6 . \ly 1>uplafuperhipartiens ttrtias, 

J QuartHS, 9 tO 4: I 8 tO 8 ♦ i 2 ;- (Duplafcfquiquarta. 

"^Quintas. Iot0p2ot0loj ^'DupU. 

/ ^nimai. I) tO lo: y^ tO 2oJ \\ SefquUltera. 

L^DecimasqumUs. 2otO I^HotO 50,, H SefquittrtU, 



'Secundas, 8t0 2: I6t0 4» 

Tertids, ptO?: I8t06. 

Qudrtis. Iot04t 2ot08, 

Qi^nUs. 1 1 toy: 22 to 10. 

^extas, 12 to 6: 24tOI2. 

\OSiaudS, 14 to 8: 28 to 16. 

rnpcrfcxiH. ^ 2),,/«,45. 16 to lo:32.tO 20. 

piniens. \pa9decmAS, 18 to 12: ?6 to 24. 
fiecimdsquartas. 20 to 14:40 to 28 
'Decmaiqu'mtns. 21 tO I^": 4^ tO ;o. 
Tieiimaifextas, 22 tO I6:44 to ^2- 
<Dec'masoHams. 24 tO 18:48 to 36- 
Vicefimau I6 tO 2o:)2 tO 40- 

yigefmasfecfid4S.z?> tO 22:^6 tO 44. 



4 QuddrupU, 

2t ^Duplafefjuidltera 
2y dupUfcfquiqu'mta. 

I ]- fupertriparticm ^uirtM. 

It fuperbipartieui tertU$> 

1 5' fupertripartiens^mitas. 

\\ fefquialtera. 

I--- fupertripartiemfeptimAS. 

I '- ftiperhipariiens qtiintas. 

I'^r fupertriparttens oitauas. 

If fefquitertia. 

It: fupertripartjens decJTtias. 

I IT fupertripdrticns Vndechnd}. 



^cljolac* 9feetDclltl)attI)Cfcp2opo.2tton$, bcea^- 
gccable tDitb fomc otljcr name : anD tbcrfo;ic migljt 
fv»mc fuperfluoufc in tl)(0 place, 



The feconde parte 

leafier, ijiot onclp fuperfiuoiiff Ip, fcut alfo falfcv 
Ijj (IjoulD tl)ci bee placet) licre: fernge tbei Doe belong 
to otljcr: places of rigljt 

&cl)olar. m\)v Doerou not name tljemt all bp 
(ii;ngltft)c names:' 

£l5aaer» Bicaufe t\)ctt are no focfje nameg in t\)t 
C-nglifije tongue, ano if 3 (IjoulD gimtJjam ncUic 
names , manj' iooulD jnnfee a quarrelle agamft me, 
fojobfcurrng ttje olDeartc tuitb neUic names: as 
fome m otfter cafes all rcop baue Docn. 

^cljolar, #et 3 p;aie ^ou Declare tljofe Doubtfull 
names of compounoc proportions. 

cpaffer. 0s tbcre ts one femoe of propo?tton,tI)at 
(s nameD multiplex^ o? mani?folDe, tuljicbe Doctt) con^ 
tame tbe lelTer Dtucrfe times eractlf ♦ 0nD tUio otljec 
tDl)tcl)e Doe contame tfte leCfer ones , anD fome parte 
or partes of tftefame : g>o tftofe feuiDes male be com«^ 
pounDeo togetber. jas tul^en tbe greater number con 
tametb tbe leirer,ttDiTe, 0? tbrife, oj oftenertanD ret 
more ouer fome parte 0; partes of tbefame . ^0 , 8 ♦ 
containetfi ? ttDife,anD bis Y»^nD lo comprcbenDetb 
3 . tbrife ano bts \. 

%\\t firfte erample (s generally calleD multiplex fw, 
perpaniensibUauCetl^e greater eontametb tbe leflTec 
tertame tpmes , anD fome partes of it befiDes* ^ut 

more fpeciallp it is called dupUfuperhipartkm tertUs, 

tbat is^Double iuitb t more* 

2Dbe feconDe erample is generally referreD to mult 
tipUx fuperparticuUris : btcaufe itt It tbe greater com^ 
prebcnDetb tbe lelTer oftentrmes, (as bere tbrife)anD 
bis -f more.anD tberfore fpeciallp it is calleD trlpUfejl 

^ujtfrtia. 

^ut as 31 Doe intenDe briefly to ouer runne tbis 
parte : fo iuill 3 bp tables fet fortbe tbe femoes of tbe 
iDttbtbeir examples* 

2Cbe 



of^rithmetike. 

The tabte of proportion of the 
greater Inequcditle, 

C^ohle. 
a5ani.'folDC« ^ Tnple. 

(^OuAdriple. 
O-c. 

SSeJ^uialter, 
,. , . „, ,_,, .„,, Sefjuiterce, 

, j ^ Sefquiquint, 

\ I Cirf. 

parti ',^^^P''^''P'*''^'^^^- 
I ^r«|f ^^uperquadrupdrtiete 

the greater ^ 

mqnalitfc. \ C^oUe. S^^M'^^^'^"* 



O'C. 



(opounde. 



^Mdnifolde. J^^'jpi^ ^Sefquitierce. 
ftt^partmlartf ' '^Sefquiquintf.O'e, 

' Sefquialtcr. 
Sefquifexte.&c, 



,QmdripU.S^'fP''l^''' 

2 



Coolie, S^^P^^^'ip'irtiente, 

\ (^Supripartietets'Q, 

Manifolds, J-.. . ^Superhipartiente. 

'Su£partient(. y^F^^- ISupripMrtiete.t^'c. 

Luadriplclpf'T^^^^^^ 
^■* ^ iSu^fextupartiente, 



Thefeconde parte 
Examples ofeche compounde kindly 

mentioned in the former tahlc. 

r c ^efl^t'juarte. 9 tO 4* 

Mmfolde SuperpartiiJ ^^. , ^ ^efquitierce. I o to ^ 

C J '^r {Sefquiaher, Qt0 2. 
r^^^n^^^ Ise/ciuifexte, 2yt0 6. 

^$uble. iSw;'fri/;4r//>«/f^/rrw. gtO^ 
C C Superipartiente quart as \ i to.4 

-., r^*. i.'-*. y T- 'i / SSuperhjpartiente tierces, u to U 
mmfoUefilt'rMe < Tr,flc. ^ s/pn/m,e„n partes. If to 4I 

"^ nnMple ^ l"mt''i"t?t"li"'Jt'" Z9 to 6 
cJi^^jextupartieteJeptnuas. l^tO"/ 

^cijolar. tClljat nTo;jc is tljcre tol'ee IcarncDof 
t^ffe p^opo^tioitg; j^o; b?' tftefefo^juics, 3: juaic cafe - 
Ip gather tl)c tjalurc oj rate of ainp p;jopo;tion. 

£l5aftcn Sl)ismatc0anocfonbctr numcratioiTt 
faue tijat mofte aptlv tljci ougbt to bcc fcttc as fraiti^ 
ons,m tljeir Icaft^ tcarmcs:as5 pou Ijaitc Ijcrc Dtucrfc 
cramplcg. 

^^!ar« |^o«mcaiTCtI)at tioubIcyf/^;//W^f^-mu!l 
bt tujttten tl)us i, ano fo of tljc rcfte* 

S9a&er* fD? els tljuien-^ ♦ ano fo triple fef^ui^imntc 
m tte fo;te:s ' :.d.: u)us faun ftj of all otljcr, 

HiiDfo; fartljec luoojUc , ^ow fl)all tnoctHanDe 
tljat.p^opo?ttGn«maicbtcaDDeO, fufatractcD, \xiw\tU 
pitco anD oiuiDeO; ano tjaic J^rawuge iuoplicfi tbcrbp 

occbiucD: 



ofyfrithmettke. 



aaljuicD. / oioftbc arte of p;op,:otiog,DcpcnDcntb 
all the rubfilt!::5,an:» fine U)o:Uc5,not oiiclv of Jrith^ 
weti/{e, but i\Uo otGeometre'bdiiJts farther matcr tuat 
ao iiouj jS luill net toucbe. 15ut as fo: tl)r Uic:i;rj3 of 
TropQnio)ts,^\xi\\[ oniittctl)em til an otiicr t\unc:cc:i 
llDt^r^'ngrnotcnch'tbctroublrronicfonDitton, qiiuv 
Unqmctcellatc: bntalfotl)cconiicnicnto:Dcr often 
i:l)Vi\^c, U)l)crebp u as rcqmrcD ttiat the ertraaion of 
rootcSi QionlD go o:Dcrlr bcfo:e tl)e arte of p^oponi 
oni3:lnl)!ci)c lurt'oout ttiofe othcr,:a not be lu;oiigbt. 
i::^iKn*fo:e luilljCnoluonelv Declare tbcfc kniDes 
orp:opo:tion,U)lncl)e pet are not fpoUen Df:to tbe n;: 
tente tlnu pon mate ijaue here , the generall Diulfion 
of ni:iubrr5,fomcliihat fufftcientlp lOucheD. 

L^3 vo:i fee that betluenc anp tluo numbers, there 
male be a conference of p,:opo:tion: roifanvonep:o^ 
pouianbccontutueDnimo^ethen.:, nombers, there 
male be then a conference alfoofthefe p^opoutons, 
\n their fcuerall tcrmee: anD that conference o: coin. 
parifon(£{ \\amct> Malogie : Uihiche fomcoclighteto y^ndogk. 
rail p,:opo^tionalitie: Cls \n this eraniple.Uihere ; no. 
bers bearclikcpjopouion m their p.:o(Treirton: 4. 6« 
9.l0oufcethat6.to4J£iinp2opo:tion7^'^/«^''i/^^'':nnD 
fo is 9.tD 6.anD thcrfo:c 13 there a liU: p;opo:tion bes 
tU)encthe.2Jaftc,asther0isbetluenethe,2.ftrae« 
^cholan %{)\& 3; confioer Uiell bp pzcgreflfion in 

jfnihmeti{f. 

oaatter. Uikctualcs lul)cre foluer termes! 0; mojc 
be fet m o:Der of p:opo;tton,a3 here 2.6. is^'a. 

Scholar* 31 pcrccuie thi3 lucl:fo2 here the p:opo:' 
tio t0 triple. But luhat faic pon to this foinic of cont^ 
parifon in p;ropo:t!onf £!5 6.13 to.2:^0o .13 to ,20. 
31S it not all one UtnDc t^tAnakgia 

a^ader. Jtts one lunDeof analogic gencralle, 

toljicbe maie ht rallcD dlreHe Analogic: bicaufe the firit 'TiireHe am- 
is wmpareo to Ijim that tjocth foiouie nerte; 1 fo cche logie. 

Cai. other 



The feconde parte 

fltlTcr \& m\ rcferrcti to tbat,tl)at foIoUitt!) ncrtciBut 

tljts w tljc Difference: tbat tn tt)c fiitte, tijcrc is a coit^ 

tmuauncc of collation : ano one tcnnc 13 compareo 

iuitl) tUjoo nomfaei*0 : But in tljat fo?mc of erauiple, 

lo!)tf be von put,tf)ere 10 no nomber r ompareD ttutfe: 

i*^o; tbe fira ts referred to tbe feconoe, ano tlje fee ono 

to tbe tbiroe, 0nD fo banc tbei feucraUe namenj to W 

Ginttctbetmafonoer, 

Cmtinuall tClberfo.:ie tube tbe firtf nombcr (s referred to tbc 

Tra^orhon. (econDe,anD tbat feconoe to tbe tbirde : tbe piopo^ti^ 

on is f alleD continiulU : ano it mate confifte betiucnc 

vtenneg. as 5:, 1 5'.4 j-.iioe p>ocetie in a continuall trb 

^.r . .Plep3opo?tion.j^o;a5 5^a0to b-:fo(si5^.to4vasvou 

mjcontinuai j,oc fee, "idwi tuben 3 faie ftimx^s j-as to i ^fo 6*10 to 

rro^ortm. iS»l^eret0atriplep;5opo?t<on, butnotcontmualle* 

i?02tbe feconDc temie be^ngei^, 10 not comparer 

Hjitb tbc tbiroe ternte,tbat 10 6. ;3nD tbcrfo^e 10 it cal 

leD a p.20po;Jtlon difcont'mualh, 

s^olan jfjioiu 3? pcrceiue certatnlp tbctr DfUmc^ 
tton:i^o; in tluoo poinctes tbefe ej:ample0 Due agree, 
anD niffcr m a tbiroe poinde . 

i'^irfte tbei agree m tbat(a0 pou faico)tbat tbe fo j^ 
moftcisreftrrcDtotbe otbertbat foloUietbit nerte: 
j9nD feconolp , tW agree in ms alfo , tbat botbe arc 
comparer in a triple p:opo:tion»l!3ut in tbi0 tbei oif^ 
fer , tbat tbe feconoe terme,&oetb not facare like pjo^ 
T^mxm to tbc tbirocas tbc tbiroe ooetb ta tbe fourth 
0; tbe firfte to tMt feconoe* 

falter, i^artber ino^ie tbere (0 to bee noteU , tbat 
in oifcontf nualle p;:opo2tion , tbere can htt no fctoer 
tben fotoer termes, 0? number0 : ano fo bv cum fo;:^* 
me0 fttU,a0,6.o^8.anofo fo^tbe»^bere as in conti^ 
nuall p2flpo?t(on, poor termes maie bee of anp nnm*^ 
ber,euen 0; o00e:aboue«2* 

J3110 altbongb 3 Jn<3l)t fafe mo^e of tbe tifuerffties 
of paopo;tion;a0 Qf^roportim (onutrfido^tudireHe^fm 

portion^ 



of ^rithmetike. 

perthn intei%hdungtd,comj^9undefyo^orti9nipdrttd Tropin 
tion, reuirfcd Troportton, atlD froportion by equalitie, \^ct 
3! ttiinUc better to p^oceDe fo: tins tmt y to tl)c otl>er 
iinocsofnombcr , anoto referuetbc explication of 
p;opo:tion3 to tijeir pcculiare place. 

S>cbolan 35 ^ou Unolue tbc beC o JDcr,ro It l^albc 
mete t)$at pou ooe life ^our olunc iuDgcmcnt tOcrciit. 

Offi^uraUe nombers. 
95atten 

l)Cne]Ctefetnticofnom^ 

bcrs are callCD figuralU mm^ 

hers\ bicauretl)eiDoe,o;maic 
rep;iefcntc fome figure : Z\Xt^ 
arccuerconfiDercDtn rclati^ 
ontotljofcfo?me0. 

$)ome of tbcm bauc a corns 
partfon ano lelatio to lengtb 

. onel^ , anD ti)erefo;c are na^ 

meo linedHenomhersi lutjtclje nanie> altljongb it maie Kmhmlh 
htc rcfcrrcD motte aptlp to fucbc numbers , as tuiU ncarU. 
mafee no otl)er fo:mc DuclM'ct mate it alfo be npplieD 
to an? number abaratt.^ttl) all focbe numbers mate 
be confiDcreD as tl)e noes of otbcr figurallc numbers* 
^conDl^,numfaers mate be con(iDereD,acco.:Dmgf 
to fodje fo:me0 as thci make otljer in p:ogreirion, o? 
in multipUcation: ^nD tljofc maic luell be nameo Su^ 

perficiallno7nberSyO; VUttenomhers, cli^Oereof tl>crc bce Superficiall 

as manp tariettes, av tberc bee Diner fittes of figures nombers, 
in Geometric, 0s mv.XXbctsTrhniuJarejQuadrateyCml^e. FUttenttm' 
sttgeledj Sifean^eUd anD fo furtljC.iltiro numbers circu. hers^ 

Ure,dimetrille,\ li\eflattesy all tul)t(be nombers bauc 
botfje lengtbe anD b^eaotbe ; anD tbcreof bee nameD 
fuptr/iciallnQmhers, 




Thefecondt parte 

liBefiDc ti)cfe tbccc ace otljer niimbcrg, tofticljc are 
matt cf ntan^ mtilttpltcatiQnB , atiD thti arc called 
s#tttt</e founde nomhers'Mcai\(^ tl)at a5 bp tl)c firftc multipUfa- 
numbers, noit,tl)ci talte Icngtljc auo fa;caDtl)c, liUc flatte num- 
bers, fo bj> tlje fee oiiD multiplicatioit,tbei tafec ocptbe 
alfor^no tljcieof be tljei nauieO hodtlj noml;m,o;found 

nomhers, 

2Dt)c leaffe of tljem all is commonip calleD a CM^r^o: 
a Cubil{€ nomhen^nn tije otl)cr in tl)cir Degrcend fcucral 
Ip namcD, as tbei bee maoe h\> fcuerallc multipUcati> 
ons.j^o; Of CO jDf nge to tl)c number of tljcir multiplt* 
eations,tt)et taUe tbeir nameSt 0nD all tbat baue libe 
number of mMltipltcations,arc »f one binDcand bere 
one nametas luell in flatte nurtiber0,as m fouuDc, 

i5ut ccnCoer^ng ttje infinite htultituoc of tbofe fi^ 
gurallc numbers , 3 t!)inbe belle to fpeafee oftbcim 
onelv m tljis place, Voljicbe ftaae tnucbe profitable tife 
in tl)is (LVtt $!nD,of tbofe^emong infinite flatte num-- 
bers,3l luiU taUe onelp foloer * Cfjat ts to fm.fjuare 

nomhers, longefquares , didmetralli nmUts^ and likjflattei^ 
Suture Square nomberstLtttWC) U)b<c!)etnaiebeOiuiDeO b? 

nlmifers fome one number , ano fjaue tbefame number fo;t tbe 
quotiente : tbat w to faie^tbat a fquare nober id maoe 
bp tbe muItipUcatton of an^ mxmbzt into it felf, as i o 
muItipMeo bp. lojiiafeetb* loo. Cbat loo, is a fquare 
riumber:tDl)icbc. ioo,if 3 boe oiuioe bp« io« tbe quoti^ 
enteU)tllbe.io,alfo. 

^cbolar» ^o,4Jnultiplieb b^4Jnafeetb.i6 : ano 
tbat mutt be a fquare number bp lifee reafon. 

spatter* ^oitts, 

^cbolar* <2lni)if3( multiplier 9. bp*4* isnotttjatg 
fquare number^^e^ng fotoer femetb to mafee all no^ 
bers fquare b^ multiplicatiom 

OJ^attcr. CoufiDer tbis tuell i tbat a fquare nutm 
berboetb make fucbe a fquare ui number , tidVLiuftt 
f^uaretiQah maUe in GvometYiix%\^^t\^ fucbeaone 

U)bofe 



of Jrithmetike, 
lubofc fi^cs are cqualle.i^o; ano if tljc our fioc be Icn* 
gcr then tljc otijcc , tljat figure \\\ Geometries calico a 

fmgf^Hare^mn fo It 10 namcD m number, a longfquare 

alfo, 

j'iotDif^jfcttctiounc tbcfigurc of rour number, 
a5routcrmeOit,anDrcttc.4. ^ ^ ,,»♦♦♦ ♦ 
fo:tl)c one ftDc, anD.9. fo: ♦♦♦♦♦♦♦♦♦ 

ti)coti)cr,tbisujmti)c figure ::::;:::: 

tCl bcre rou fc a plant longfquare: 
^et 15 tl)cU){)olc number tljatamoun^ : J : ♦ t ♦ 
tetl) of tl)is multiplication: truelrna- ♦♦♦♦♦♦ 
mcD a fquare number, as bcrc rou ♦♦♦♦♦♦ 
male fee. i6uttl)entstl)c(iDco;rootc :::::: 
efiv,neitlier. 4. no:. 9. but. 6 ♦ 

^rbolar. jj^otu 3 t)n9crftanDe it: anu tl)e better br 
ti)i5 figuralle evample. 2nD bcrc alfo j^ t)auc learnea 
tuljata'R?*^' tfi:fo; V'ouftuieto erpounoc it, to bectbc jfroon, 
fiDc of a figuralle number. 

a^aiter. Cuerp flatte n.omber , auD euer^ founoe 
number alfo t)aue tbeir fiDes: i5ut no fiattc number, 
faueonely fquare» baue a'u^jotte : bicaufea rooteut 
flatte numbers, 13 a number multiplier bp it felf. 

0rrtJ in founoe numbers , tbeionelr bauerootes* 
fcbiclje bee imaoe b^manp multiplications, of fomc 
one nuber bj? it felf: otber bg tbat,tel^Klje rifetb of it. 

ii^ tuben 3 faie,tUjoo t?me!3,ituoo rtuife, maketl) 
8 . that number is a fountje numle r : ano 10 nameo a 
Caie.!ani»fo.5.trmcs,2.tb?ife,DoeinmaUe.27,U}brfbc 
13 alfo a Cube, 

anO generally, anp number tbat is made bpfucbc 
2,inultiplication0, iscaUcDaCa^e, oKw^/^enuurber. ^^f'^^^ 
0notl)e number of tliat multiplication, lubiebecoim 
monlf IS naufeDtliemultiplier, is mtljispoincte cal' ^ftt^% 
IcB tbc CubiUc roote of tbat number. roote. 

vX3 berfo?e,tl}us alfamaie ?cu Define a CubiUe no* AcuhiHe 

beCj nomhtr* 



The feconde parte 

bcrt to bee fuclje a number , as becftrg Dfufoct bp b<0 
rootc, fljallljaiic fo^tl^e Quotients t^e fquarcoftbe^ 
famcrootc* 

^cljolan l^crebr 3 perccf ue, tijat one multipara^ 
tion, of an? number bi? it fclfe , ooetb make a fquarc 
number* 0nD tluoo mnltipliratlons in tbat fo?te,ooe 
maUe a Cubike number* 

(m bat If 3;ooe multiplte anp number t\^^i^t^ ojfo^ 
luer ttmeie; , o? oftcner m tbat fojte y are tbere p^^opec 
names fo; fucbenumbersf 
Rafter* ^es in Deedetas 3i tuill Declare anon. 
i&ixt firftc before iue attempte tlje otber founDc no^ 
bers, it fijall bee mete, tbat lue Doe Declare tbofe ttuoo 
fortes of flatte numbers, iublcbe 3? nameD befo;e:tijat 
ts Diametralle numbers,anD lifee flattes* 
jf diametral ^ diametralle mmher, is rucl)e a numbci aS fjatl) 
nomhr. ttDoo partes of ttjat naturejtljattf tbei beemuItipUco 
together, tbei tuiU make tbefaicD dUmetralU nomben 
^nn tbe fquares of tbofe tluoo partes , beepng aODeD 
togetber, tuill make a fquare nober alfo: iobofe rootc 
j£dlamtter* is tbe diameter to tbat diametrallenomier, 

BS 12 iiS nameD a diametralle nomber^fo^ tbat be batl^ 

ttuoo partes,tbat iB* ^ ano. 4,tuUtcbe beepng multi^ 
plieD togetber,Doe make 12* tbat is tbe firlle number* 
janD if tbeir fquares be aODeD togetber,tbei lotl make 
a tbirDe fquaretano tbe roote of tbat number iuill bee 
tbe diameter to tbat platte fojmc of i2.as in tbis eram^ 
plepoufee* 

2Dbeone(iDeis,4* 
anDtbeotberfioeis 
3lubicbefaotbemul^ 
tiplieD togetber,Doe 
make 12. a^ben take 
tbefquareof fotuer 
tubtcbe is 16 anD tbe 
fquare of* 3rtubicbe 
iB»g, anDputtbem 

togeti^ec 





of jfyithmetlke, 
togftOcr,anati)n jjillmafecij-. lul)ofcroote,bctng 

f.lB tl}C diameter ol tljat plattC fo^llir, 

^d)oIar» Cljat Doc J pcicciuc tocll , btraufe it is 
ronflrntcD hy> tt}c. ; > tl)co;cmc of tJje patbcluaic. 

cpallcr. i0cttai%cait 12. 

otlbereramplc. 3ntl)is 
platte fo;mc of.6o . f ou 
fee t!)c one fiDe to bee. 5^. ^* 
atiD tl)c otftcr fiDc to bcc 
12. j^oU) taUc tl)e fquarc 
nombcr of. 1 2. tul)tcljc tjs. 144. aiiD tl)c fqiiarc of,)-, 
lDl)icl)ci£!,25-.anUputtl)cmtogetl)cr:roU)iUitmakc 
1 6 9* U)l)icl)c 10 a fquarc iTombcr:aiiD l;atlj, i> fo^ bis5 
rootc. 

JLiUctuaies, 1 2 o . ijs to be accoiimptcD a Jiametialle 
nomber , fo} fo mucl)c as It \)cLtl) tUioo pactcs: tljat \& 
8 » anD , I y . tubicbc bccpitg inultiplicD togctbcr. Doc 
make tbe firOc nombcr. 1 2 o. 0nD tbc fquarc of tljofc 
tU)oopartC£;(tl)atis.64.fo;,8:anD.225-.foMf,)befng 
botljc aDDcD togctbcr , Doe maUc ,289. Mnthc is a 
fquare nobectaiiD batb fo.2 fjis rootc, 1 7. 0nD tberfo^c 

tbat, 1 7* is tbc diameter to t^^^tMametralle nomher, 120, 

iltUe eramplc0 infinite nngbt 3! ^iwt t^ou. I5ut 
tijefe fo;?crpIication of tbc namcnraic fuffife. 

$)CboIar, 3 Doe UjcU tinDcrUaitDc t\)t cramplcs: 
fauetbat3liinoU)e not boluto finDetbe rooteoftljc 
lafte fquare nomber, tubicljc amountctl) bj? tl)e aDDt^ 
tion of t^cfo;imer ttuoo fquarcs togctbcr, 

jailer. 2Cl)at arte luill 31 tcacljcfou anon, But 
tue maic not fo^gette ftrttc to cnDe all tl)c Diftnitions 
of focfte names, as 3 minDc to tu.:itc of, 

^Ijercof pet tijere react!) Uieflattes: tuljictje maic nig Mattes 
bee as iDcIl taken fo? trtangulcr figures , as fo;5 qua* ^ 
2i;{atc figures, 

^0 tt>at of anp of tljem, tuben tbc fiDcs of one plat 
topntM&tzth like p?opoition togetber , as tbe fiDcs 

D»;. of 



The feconde parte 




of aiv? otftec flatte fo^me of tOcfame felnoe Doetlj^ttjcit 

are tljofe fo;j^ ^ «9 

incfi callthlikf 

flatten , as tit 

tbefc*!* ionge 5. 

Squares: bt^ 

caufctbcfiDcs 

of tl)em faotl)e,arc in one 

pjopo^tion(fo?.6>i$to 

pleto*2:astDeUas,9a0 

triple to 3.)2Cl)erfo;jearc 2. 

tfteiubote figureacaUeo 

lihj flams ♦ 

and fo of Due conuenfendc , tl>efrnombers ( tbat 
erpaelTetljetr quantities, iul)icfte!)ereare*27»ani) 12) 
be calleo bv tl)e like names, /rit^)?4«M. 

#artljermo;e in triangles (as liere ^ou re)if tlw {v 






I* 6* 

ues of t6e one bearc lfeep;iopo?tion togetljer , as tht 
fioes of t1)eotl)cr ooe:tl)cn are tljei calleo Uieflattes aU 
to* 0nD tbeir nombers,tliat oeclare tijeir quantities, 
tn lilic fojte are nameo Ukeftaites, 

&cbolar» 31 pearcetue tjere: as 4 is to 2: fo,6»is to 
3,botbe bepng m a Double p;topo;tion. ano t})tttm 6 
anD*24»are to be calleD hieflattes, 

£©aller» ;^ ou tmoerftanDe it iuell. 

anD tl)us Ijaue tue b;iicflp ouer runne tbe Diuiftort 
of nomber,into Ijis pjincipalle kinoes : ano ^aue fct 
fo^tlje tire Definittosof w!ie of tliem, l»it^ ejrampl^s* 



cf Jrlthinetikc, 

l^^t tfe of tljcm ^ou ajall fc largely iir tljc pMrtifc of 
t^isartc. ^ , 

• )5ut to the intent vou mafc tijc better obfcriie ann 
rcgarDc ti^cfc tUico laftc femoes of nombers : lubictjc 
arcconimonip ncglcrteDofartesmen , 3 U)iU(^cluc 
{?ou fome ijfc of tljein^feitl) tljcir p:operties. 

i^irftc, aiUiametralltnomUts tJOefcttcfo^tljeatn' Ofdiamf; 

angle, Ijau^ng all tI);cefiDe6ljnotuen:U)l)tc!)etl)rng^w/,„,j;; 
a0itDoet!)fcructo manpanD tuonDcrfulIpurpofcB: i,ers. 
fo can It be founD m no ottjcr nombers, tljcn onclp nt 

d'tametrall nvnihers, 

fo; altbougl) in figures Geometrkalle, pou mafee^ 
«cr mo:e tjnfalliblp ftnoe one line , tbat luiU maUe a 
fquarcequctll to tbe tluoo fquarcs of anj' otber tujoo 
lincs(as in tbe pattl)e luaie pou Doe fee it taugbt) pcjt 
tbc meafure certame of tbofc Goes, are not bnokien. 

CiElberfojc in nomber tbatisnotpoflfiblcalluaies 
to be tjoen : ncitber can it be Doen tuitb anp otber no^ 
berg , tijen oncl? diametricallnombers ♦ lEJct maie otljec 
nombers go \jcrv nigb* ^jb namely? m tbefe eramplcs 




•f fqaace nmhtmm^U 5quble,3! tafec fo? tDe fqua. 

^♦^« rc0 



The feconde parte 




rcisoftljeliDcs, 
ijicaufetftciare 
equalltantitbet 

make* 8. $-0,288. 
1 6 8 2 . 98 o o. 

57I22.I. 552928. 

all iDfticbeOif^ 
feconelp bpan 
bnltie, from a 
fquarenombec. 
i^ojmneiisa 
fquare nombcr 
ano fo arc tljcfe 
otljcrfoloiDpng* 

U)l)oferootegbe. 7^ 17* 4^ 99* 239. 577. 

mimf)t eramplcis if von Doe coitfiDer luell Uteuf* 
ter, tljet toai bcipe ^ou to geOe at tl)e itiglieac rootcs 
of nomber0 t^at be not fquare ♦ anD alfo fo? Doblpitg 
of fquares , in a fquare fo?me : iDitljin air tnfpcaUe^ 
able itereneflTe. 

fo; as tit ooblpirg of tl)is greater fquare. 166464. 
tljere rifet^. :? 3 2 9 2 8 . iuijicbe tuairtetlj one of a tuftc 
fquare.iPou feeafelp^tftat as tljatorte i$ but a fmalle 
portion to tbe loljole fquare ; ^0 pet , tbat one iuan^ 
tm not in tbe roote, but at tlje toljole fquare: Uiljere 
bV ro« mate pert eiue,tl)at it 10 a tjerp fmalle ano tin.- 
fenfible parte of one, tbat luantetb in t\)t roote. 

^cbolar. atmuftfeme bp reafon of multiplicatt- 
omtm it i$ fcarfe tije. i o o o o.parte of one. 

jailer. s^oufaietrutJje. 

|>cbolar. But^oUi fijalli finoe tlje ufametcrof 
focbe nombersf 

Rafter. %Utis eamrDoen,tfpoufenotoefirlle 

certatnip tbat pour nomber is a oiametrall nomber. 

anofeconoarilp, tfpou^notwe tfi^ true partes of 

<tj 



of Arithmetike. 

<t:lul)icf)e pou (!)OulD bfe in tl)i3 cafe. 

s>c!)olar» c'a III not anp tluoo focbc partes ftruc, 
lul)icl)e bp multiplication tuill malte tljc Uiljolc nom- 
bet:' 

a^affcr. ,^011 mm by tljc fo;mcr cramplc5,eafilp 
fc ii^t contrary. J^o,2 12 is a diametrallmmhenanti Ijatf) 
tl)cfe partc3(a3 it is fonc percciucD). 2. ?» 4» 6, pet if 
poll taUc . 2 ♦ anD . 6 . fo: tlje fioes of it , tijci Uiill not 
inaltc a diameter in UnoUien noinbcr. 

^cbolar. S^tjat 31 t3nDcrllanDc:fo; tt)c fquarc of 2. 
bcvng,4.aDDcDtOo6.U)l)icbcistl)cfqnarcof6.Doetl) 
maUc. 40. luljofe rootc mull bcc greater ti)cn,6> aiiD 
leflfc tben. 7* ^no ti)crfo,:e« 4 o, can baue no rootc m 
iul)olenombrr. 

spatter. .<ieitlicr ret in broken nombcrs: fo^ tbat 
i$ a gcncralie rule: tbat if anp lubole nomber baue a 
roote, tbat rootc (tjall be a lubole nomber, ^0 tljat if 
tljc rootc can not bee founoc intubole nomber: j?ou 
fl;aU neuer finoe it in broken nombcrs* 

ianDfo;tmo;ccertainticof tbatSl faicD before, tbat 
all partes be not apte fo.j tbc fioes of a diametralU norm 
hereto finDc out tl)c diameter : marUc tucll tt^c fcconDc 
eFamplc,U)I)icbcis.6o.anD^atbtberc partes. 

2. 5. 4. 5'. 6. 10. 12. ly. 2o. p. 

^0 tbat faeginnpng tuitb the ttuo crtremettc, tbat 
ts.2.ano.5o.tbeiU)illbpmultlplicationmake.6o. 

0no likeluaies anp tU)o nombcrs, equally oillant 
frcmtl)orecrtremes:0s.3.anO2o.iliUeluaics.4.ano 
I s; t otber. 5'.anD. 12. 3ni3 in like maner.6.anD. 10, ^11 
tbofe couples bp multiplication doc make . 6 o ♦ pet 
none of tbcm are aptefiDcs tofinDc tbe ^/^meffr by, 
but onely s anD. u.j^o? of tbe otber fioes btvng mnb 
ttplicD fquarely (tbat is by tbe fclfcs) ano tbofe fqua- 
res bepng aODcu togctber, tbere tuil not rife a fquarc 
nomber* a?«poutl)aU better tjnoeraanoj, tobenpou 

^.ii^, Ijauc 



The feconde parte 

i^aaf learnen to bnolue Tquarc nombcrjf, bp cxttactia 
of^^ctr rootes* 

^et m tl)c mcane ccafon 31 toiU fct fo;tfte attaint 
notc0,to buotuc tl)c «;i4mfffr3anti tlje aptc fitics,tii all 

diametralk mmlters, 

u 0nD firae 3 fatettbat ais tftci are tl);tee nomfaerg in 
all ( 3 mcaitc tl)e ttooo fiotg , ano tlje <//4mf /er ) fo all 
luaie0iftl)efirfieo? leafiefttie bee oDOe , tfteitl^all 
tl)cix be ttuoo of tbem oDOe nombers ♦ 0nD tlje dfame^ 
ur ^all euec bee tbe otlicc of tl)e otue nomberg : t^at 
ts to fate^tljc greatclle of tljem* 

:♦ ^econOanl?. Jt t0 true tftat all diametrsll nmhers 

are euen noinberB* 0rto no oODe nomber can bee a </i^ 

ametralle nomher* 

3. 2ni)trDl^»3i faie,tl)at all oODe ttombers aboue one, 

ntaic be tlje lelTer ftOe tn fOClje diametralhnombers, 

TBixt euen nontbers Doe not ferue fo generally: foj 
tljet onelp mate llano tn focbe place, tJoWht be grea^^ 
tertl)cn»4:atf*6.8.io«i2«i4»i6.i8«2o. ic, janOnonc 
otl)er euen^iDmbers tben foclje as mate be otutoeo b^ 
4* mate be tlic greater fiDe tn an? Mmetrsllenomher, 
A, ^ourtlil?* 3|f tibe leflfer ftoe bee an ofiOe nomber, 
tben o;(Dtnartl? , tbe fquare of tt ts tulle equalle tutt^ 
tbattbatamountetbbp tbe aoottton of tbe diameter ^ 
to tlje greater nomber « 00 tn tbe firfte erample,^, td 
tbe lelTer nomber, ano. 4. isi tbe greater : tinto tbem 
botbc tbe diameter t$ *5: * /^oto * 3 • batb fo; bt0 fquare 
9,ano fo mocbe t« maoe b? tbe aotrttton of«4.ano.^ 

^gatn tn tbe feeontje erample,tl)e lelTer nomber t£( 
y^ano biss fquare tss 25'.2Cbe greater nomber ts i2,anD 
tbe diameter A ?♦ put 1 2* ano. I ?♦ togetber, ano tbci 
ma^e . 2 > « tubtcbe ts equalle luttb tbe fquare of tbe 
ielTer* 

iltfeetuate0«7* anD 24«multtplteti togetber mafeetl^ 
1 6 8* lubtcbe t0 a dimttralU nmUr, j^no btcaufe tbe 
fqstare of tbe lelTer ftoe(tDb^(!^^ t)^^^ ^$«4 9*)tnttft bee 

eqnalle 



ofXrlthmetike. 

cqualle to the greater ftoc , aiiD ttfc diameter aDDco to^ 
gcti)er:ti)crfo:c fcimg.z >♦ aoDcu to. 2 4» 11 wUctb. 419* 
tf)at. 2 ^ mutt neDc0 bee tbe diameter ttuijt fo^cfaieD 
nornber. 

13p tbefe rule0 ( tf pou Doe marlte tbcm luell ) pou 
mate foite pcrcemcbotu to make anp diametralUnom; 
ber : if tbe leflTer fioe bee giueii tjnto vou , anD bee an 
oDUe rtombcr,i0et fo: pour eafe, 3 tuiU gtue pou tbid 
plaiite rule. 

?!Ktlbenatt? oDDe nombcr (0 p;topounDed : astde 

lelTer flOe of a diametrdllenomber, ailD poil UjOUlD ftnOe 

tl)t otber fiDe , ano tbe diameter alfo : ozels tbe <//4»if ; 
trallenomher , tl)atmaie Ijaue focbe a fiDe: multipUc 
tbat pzoponcD nomber bp it felfe , aiiD it tuill mafee a 
fquarc nombcr, ano UiUl be an oDDe nomber : fo tbat 
of It poti (ball finoe no tulle balfe. 2Dberfo:e take pou 
tbofe tU)oo nombcr5, tbat are ncrte Unto tbe balfe of 
it:2^be leflTer fljall alloaies bee an etien nombcr, anD 
(ball be tbe fCCOnOC flDC of tbe diametrallenmben %\)t 

otber nomber lubtcbe 10 tbe greater^fi^all aliuate0 be 
an ODDe nomber: ano (ball bee tbe diameter of tbat nom- 
ber Uibtcbe pou Defire. #o? erarttple martie U^el tbefe 
fo;me0 tbat Doe folotue* 

^f tb;jce bee pjopounoeD a0 tht one fiDe of a diam; 
tralle nomber : ^nD pou luoulD knoUie, Uibat male bee 
tbe otber fiOetanD tubat (0 tbe dtametralle mmbert 0nD 
tbirDlp,Uibat is tbe diameter to tbat nomber : 2Doe,a0 
31 fateD befo;e:mu(ttp(p* ^* bp it felf,anD it luill mabe 
9 . U)btcbe (0 a fquare nomber, anD an oDDe nomber: 
anD tberfo:e batb no iutte balfe. But tbe nigbeft no^ 
mber0 to tbe balfe,are.4, anD» y. 

2Cberfo.2e 3 fate, tbat 4» lobicbe is tht leflTer of tb? 
(tDOO,t0 tbe feConDe flDe of tbe dtametralle nomhen ano 

j-.faepng tbe greater of tlrem, is tbe diameter it felf. 

^cbolar. jjiouitsitligbtmougbtoperceiuetbat 
1^ dimetralU nomber i0» j 2 J fecpng . 3* multiplieD bp 

fotoec 



The feconde parte 

0gatit,tf» ff be aflTignet) fo^ one fttjc of a diametralle 
nomher^ anDfouobfeructI}cfo;mcr luo^Uc tou mate 

tafllV finOe tftC Otljcr flDe,anD tljC diameter, 

iPtrat:Oufcc,tl)attl)cfquarcof^i6.25^,anD(t!)atl) 
nol)alfe»)15uti2.anD.i ^.aretftc.z, nontbcrc nigbcll 
l)is l)alfc:U)l)crfoae«i 2« fljall bcetljc feconDc fiDc:anD 
I^muft be tbe ^//tfwf^er, 0iiD t^e dimetralU nol>cris,6o» 

aiiheluategjif ♦ y* be fet fo? tlje Icffer fiOctbc grea- 
ter fioe (!)all be.2 4. anD tl)C diameter,! y, 

^clbolar* SToucftiitg tbis 31 ircDe no mo;e inttruc^ 
t(oit:tbe tbfirg 10 To manifelle* 

matt* Cben (lietoe rour fertotulege bp ait eram 
ple,o;tttuoo. 

•a no ficft 31 appoincte 9 fo? tbe leflTer fiDe of a diame^ 
tralle nomber,\3)'^tttvintii 3j XuoulD tjaue tou to aflfignc 
tbe otbec fiDe,anD tbe diameter, %u 

^cbolar* 31 follolue pour p^jeeepte , anD muItipHe 
9.bi? it felf,U)bereof commetlj, 8 1. tobofe balfe is be^- 
ttDene.4 o»anOt4 L s:{)erfo?e muff* 4 o, be tbe otbec 
CDemnO 4I.tl|e diameter, <3iV(ti bere tbe dtamttralknmf 
^frt0»?6o. 

fatten p^oue tbe Ufec i tobere ♦ i y ♦ 10 tbe Icflfcc 
nomber. 

^cbolar* i^multiplfeo fquare mafeetb«22j:U)bore 
ntgl)eftbalfe0ace«in«anD«i i ?«ofUi!)tcbetbefirfti0 
tbe feconoefiDe, anu t\)t later tjj tbe diameter : ano tbe 

diametralle nomher i0» 1 6 8 o« 

gaffer* tDbat fl^aU be tbe otber nombcr0 : tobere 
2 1J0 tbe leflTer fide:' 

^cbolan 2 l telDefb <it fquare* 441. tobofe po;^ 
ti'ons nfgbettbi0 balfe,are*2 2 o«anD.2 2 ir^nD fo ap^ 
peretb tbeir Offtce0,anO tbe diametralle nmher t0 4620 

matt. ^0 mm voa faie tbat bnto,27* being tbe 
lelTer (iuz; t\)t greater fioe (^all be* 3 6 4. ano tbe dis^ 

ur 



of Jrlthmetih, 
fer, 5 6 f .bicaufe tljc fquare otiyAn^y 1 9, 3nt) tljc </i^' 

wetralle nomher is. 9 8 2 8» 

^cljolar. ^omuftttbcbp pour rule. 

Smaller, ^ot ondp tbc rule Dotb tearijc rou tbat 
<t is fo , but a'lfo t\)t nature auD figure of foc!jc//«//c 

40* I I 2. 








2 2 0. 


21 


"^\ 


^ 




462 ©♦ 






115 ut to t^e (ntcnte f ou maie tfte better tinDerff ami 
t\)t nature of tbefe nomfaers:3i toil fet fojtlje tjcre tfje 
UfecfiDes iuiti) otOcr nombcrs : tDijcrefappoumaie 
fertotue, tl^at one toe male fcruc to utucrfe dUmetralk 



The feconde parte 



I2« 




3y- 



20. 







2)% 




300 






"^^^-^..^^ 








^^\ 




2 8, 







21 


\ 




T88* N. 



of j{rtthmetth. 



16. 



/ -' 




\ 






^ 






971. 




\ 




fqrfamc-4 n6mljcrs.9. 

i^.2i4nnD,2 7.rcta3tl)c 

IcflTcc fiDcs : :anD tlKic 

greater Goes are forbc 

nsDifagree fro tftefo;^27 

nTerrulc,:anDiiM)-.2i. 

aitD. 27. gfcctluoota' 

ricties,tjnlibe to tl)e fo;j 

nrec crample* )15utfcc^ 

rng tf)c fiocs Doe oifa^ 

gree,3 Doe not maruel tftat tl;c Mametr4lle nomhers are 

Diuerfe from tMc former, 

£pa(tcr» erammc tljercnombcrs^tuljctljcrtfjcf 
be true. 

^cftolar. 3? mud mumplic ccljc fiDe br it felf , anD 
tfjeit aDDe the together: aitD tftfici make as mocfic xw- 
ftlr^astbc^/i^wf^fr facing multtplieDfquare,t!)en are 
tl)cttrucnomfacr0. ^03; ree,tl)at.9.malietlj,8 i.ano 
12 Doetl) pelDe 144 M)\t\\t botlie aDDcD Doe maUe.225'. 
^nDfomoc^cDotl) lymakcljemgmnltiplieD fquare. 

3inictDaie0;fo2tl)efeconD figure ly.b^rngdb fo.2tt) 



e.tj. 



22), 



The feconde pane 

22^ and* 2o* siattl). 400 A\)at is fapaODitiom^^ c. 
Uioictje fomine Doetb amounts nUoyVohm^i^As nml^ 
tiplicD fquare. 

irt)c tbiroe figure l)atl) . 1 5- » alfo fo; tl)t one fiDc, 
lui)orc fquarc 13.22^ ano fo; tt)^ otber fioe- ;6- iul)icl)c 
ttia^etl) in fquarc* 1296. ano tljet faotI)c togctljcr giuc 
i)2KanD fo manr commctf) of 39 multiplicD bv it fclf 
In fquarc* 

again fo?tI)efourtl)efigure,2LmaUctl).44Lantj 
2 S.tioctl) pelDe*7 8 4* U)I)icl)e botljc bcpng aDDcD, Doc 
amonntc Mnto. 1 2 2 y, ano fo moct)c Docttj t!)crc arifc 
i)V' 5 ^multipIicD mto it fclf, 

%l)t fiftc figure fjatfj * 2 k alfo , ano bis fquarc 10 

44i.anDtl)efeconDcfiDcbcpng.72.makc«) in fquarc 
5 1 8 4. ^0 tbat botbc tbofc fquarcs Ooc mafec. s 6 2 y. 
ant) ti)e like nombcr is maoc bp , 7 y * multiptico in 
fquarc fo?mc* 

S^ohiin t\]t m figure 27 bcj^ng multipltco fquarc 
bi^ngctb fo;itbc.7 2 9» auD* ^ 6, imcUiaics multiplico 
joctb make, 1296. ano tbat luttb tbcotljcr U)in malic 
t)V aoottion , 2 o 2 ^ lufjicbc fomme (as is lucll fccn) 
tioetb come of tbe multipliration on4y,bp tt fclf 

3i« tbe fcucntl) figure. 27. multrplieo fquarc, Oocth 
gfue,7 2 9: ano tbe otOcr flocf iubicbc is, i - a ooctl) 
mvng fo^tbe, 14400. sCbcfc botbc io^nco togctbec 
tot make, i n 2 9- ano tbe like fommc is gatfjcrcD bv 
t\)t multiplication of. n^.fquarelp. 

^0 tbat all tbofe figures Doc appcre true. 

15ut boll) tbei male agree iuitbrourfo.jmerrule, 

Rafter. %ljat rule Did 31 make fo^ nobers tjucom^ 
pounDe. i^o^ nombers compounoe baue not onclp in 
tbcir otune name,tbe i3fe of tbat mlcMt alfo tbci fo^^ 
lolue tbc fo^me of tbofe nombers , of Uibicbe tbci bee 

^o.9,berng compounDe of. 3. foiotuetl) tbc fo;mc 

of 



of Arithmetike. 

Dfo-3nD tl)crfo:c a3.?.l)at!).4.fo; to maUc tljc fcroitD 
CDC luitl) i)i'm,ro. 9,(bcfvng tb.nfc.;. ' l^all Ijauc.i :. 
(U)l)icl)c 15 t():irc.4.)fo2 a matcljc fiDc iuitl) tivm. 

iliUctuaics.i^". belong compomtDc of. V anD.;.(l^all 
l)auc their fo;mcs in tt)c maUpng of tl)c dUmetralUno', 
hers. j^o;as.^.t)atli, 4. fo.iy.'bccpngftuctpmcs.;.) 
ft)all boijc. 2 o. (lubicljc is Mt tt^mcs . 4 .) fo.: tljc fc- 
foiiDcGDr. 

0gain,a3.)J)atf).i:.rofljall.i).;bcri'ngtl):cctjv< 
mrsrs-. nauc. ] 6, .tijat is tlj :cc tpmcs. 1 2 J fo: l)ts fc^- 
f onDr fiDc. 

}.iUcmaic5.2i.l)rintgcompounDcof.?.anD.%lljaU 
l)aucbotbcti)ciffo:nirs. 

^iHD. 2 7. lul)icl)c 15 compoimDc of. ;. anD. 9. Hjall 
ijaiic all ti)c tjaric tics of tbcr fo:mr5. 

Scholar. 3; fee itiscticu fo,anDtl)atin thcdiam: 

terras U)Cll as lit tl)C fCCOnDC flOe. 15ut t\)tdUmetrallt 

nomhcr Doetl) Dai ic luocbc lit tl)cm. 

a^aftcf. 13ct Doc tfiofc nomUers agicc in a mar 
neiloufe gooD p:opo:tton. j^o; if rou Doc confiocr rljc 
p:opo:tic^nofbotl)c tbcfiDcs m one figure, to botbc 
tbe fiDes m an otbcr figure:anD aDDc tt)ofc tiuoo p.20 
poztions togetljcr , the aDDitton of thcim Doeth maUc 
the nomber that rcp2crentcth the p:opo;tio bctluenc 

their tlUpO diametralle nomkrs.V:0.l)ldjC thvngc 5 lUlll 

notu onelr touchc,as biic^^ as mate bee, to guic ron 
Dccafion to niarke it better hereafter: 3>ith this place 
Doeth not fillip feme font. .ls.>. anD.4. bccpngthc 
tUjDO fiDcsofa'i^M^«f^>'<<//fW('>«^f^i0oenial?e.i2. e>oif 
9,ano 12 be the fiDrs cfa^/^^n-'f^/^f '^(""'•'^'■jthat nonr^ 
bermuitbe.9.ti7mes.i2.thatis.i S.J-o:.9.i6tripIc 
to.^:anD.i2. istripleto. 4. nnDbicanfc the aDDitton 
of p:opo:tions, is like the ninltipliration of frartios, 
^ mud mtilriplie. ;. bp. 5. o: els ; bv ] > luhichc is all 
]onc,anD that luill inaie,9. 
jLlil5eluaics,!f ^> anD. 4. be taltcn fo: the fiDcs of the 

cm. kflfcc 



Thefeconde parte 

IcCTf r nombcr iUmetralk, nno, i^»anD»56» fc? tfjc fiDCjJ 
of tl)e greater nombcr: as tl)e IcflTcr nonrbcr C^all bee 
i2»fo t5e greater muftbc,^4o»tl)at 16,4 vtfmes.i2» 
#0^ 1 5» tjnto,?. is tn a quintuple p^opo^tion,anti is 
lu^jitten tl)U3,|:ano« 36. tjnto 4 ts a nomnple p^opo^ti^ 
on,antJ is U);itten tijusf ♦ jpiotu if i'ou nuiltiplie tl)efe 
nombcrs together, tl)ci bill uialie 4) : toliiclje Dec la^ 

rct!)tl)Cp^cp02tionS Oftlje tlaOO diametralU nomhm^ 

jant) fo of all tljc reae,as ^m ntaic eafilt' confiDcr, 
^cljolar. 3! p?a«e vou,lct me evamtne one 02 tlooo 

Cf tbe,Ul COmpariTon to tX^atUxiXt diametrallmmber.il* 

3 feetljatiybepagtljclefrcrfiDe, anDzo.ttiegrea^ 
tcr nOe,DCC niahc, 500. as tOetr diamePrallcnomhehmti 
ti}at.5oo.ts.2vti7jnesromoeljeas«i2. is. 2:i)crfc;2c 
fcp reur fatvng tlje pjopoatton of i^tOo^anD cf.2c.t0 
4, inuft malic. 2^ 0nt! fo !tDoctl).#ojec!)CDft1]cm is 
a ^uintuph p^opojtton. ^jkxn it is quitUl^ gcflTcD, ttjat 
y.multtpltcD bv♦^Doctb mahe.is". 

J^O? fartber p;oofe, 31 taUe tbe diametralU mmher 

i68o.tubofcfiDes are.i^-.aiiD.in.j^irftBJ fcctbat.iy. 
io.j.bearetbajttJ«f«/'/ep;opo?ticn:anD.i i2.to. 4. is 
aB.28.to.i.2Dberfo?e3multtplic.28.bv.y.anDitma^ 
lictb* 1 4 o. sCben if 3 multiplie tbat nombcr bf. 12. it 
iuiUmaUc,i6So. 
%\M iB a (amcicntc triallc fo; tbcfc nombers* 

Of men fides li&ixt of focbc dtametralle nombers^as Ijauc CUClt ncm 

bers fo; t^eir lelTrr fiDc, you bane giucn no rule, no^ 
tl)ereramplcs,fauconcij?of.8.tDbcrfo;e3'p;aicvou 

tell me, bolu ftjall i! finDCOUttbe diametralUmmher^ 

HatbbiSDtber fiOc, auD tJic dimeter \\\ forbe euen 
nomberSt 

a^aiter. 10011 fijaU mafee it fquare , as pou l«iO in 
tl]t otber ncmbcr«>ti}at lucr 0DDe:anD of tbat fquare 
l^ou {!iaU tane tiuco quarters, tubicbe ^ou d^all alter 
m foclic fo;rc,tbat yim fijall abate, i.fro tbe one t\mu 
tcr, ano put it to tbe nt\m quarter 0nt! fo baue ton 

ttuoo 



of Jrlthmctike. 

tU)oo nombcrs, tJtffcrpng onclp by ♦: . auD botbi' be- 
pnof oDtJC. S^i)e IciTcr of tl)cm ttuoo^is X\)t greater ficc 

of tljtdiametralle nombcr : ailD tbc Otljcr 1'3 tbC diameter 

to It. 05. S. bepng I'our IcflTer fiDc, tl)c fqiiarc of it is 
6 4.1ul}ore fjiiartvi- is. 1 6. from luljicbr j: abate. i.auD 
tl)crc reUctl). i ). anD tbat 10 tl)c fcconoc fioc » aifo J 

aDDC I. to 16. anD it maUetl). I7:U)l)lcb: is tilC diameter, 

^cbolar. 2Ct)t6 is no tb^ng barDr. as by rraniulc 
3 iDiU p:ouc. Tf. I 2. bee tbc kScv fiDc : bis fqnarc is 
144. a5tDtacr|~i5artcrofitis.;6. ^^sn aJjatpng. i. 
3: fee tbcrc luiil bee. ; f. fo.: tbc otbcr fiDc of tbe diame.- 
trallcnomber. HnD aDDrng.i.to.;6.it matictb. v -to be 
t\}z diameter, 2\i^ it 'S nuiltiplic.^y.bv.i 2.it b:pngctb 

fo:tbC. 4 2 o. lublf be is tbc dUmetralU nombsr, 

0oU) fo: p:oofe of tbcfc nombers,^ multiplic. 1 2. 
bpit felf>anD it uiakctb. 144- SDben j multiplic tbc 
ctbcr fiDctbat is. ] y. bp it felf, anD it pelDetb. 1 2 2 v 
^bofe botbe togetber doc make. 1 5 6 9.anD fevng ^7 
inultiplicD bp u felfe , Doctb maUc tbcfamc nombcr. 
^berro:c are tbci all true nombcrs. 

an otbcr crample, i o. bcvng fct fo.i tbc IctTcr fiDc, 
3;Doemultipiteitfqnarc[p : aiiDtbere rifetb. 100, 
iubofe quarter is. 2)-. i-^- lubJcbe 3 take ( aspou 
tauo:btnir;.2 4.anD.2 6. anDfotbcU)bole^M"if'>i//f 
nomler 15,240. j^o^ p;cofe of tbc otbcr nombcrs, .X 
take. I o o. tubicbe comnietb of. I \nuutiplicD fquaiv, 
anDtoit3eaDDc.v7 6.lubtcbcisti)crquarcto.24.ano 
tbci botbe Doc make. 67 6. anD fc mucbeamountctl) 
bp tbe multiplication of. 2 6. fnuarelp. 

' 0^vifter. S^bts male fuff ice fo: tbts p:efcnie : if 
|50U marke tbat tbe cur nomuers banc not onclp one 
gcnerallc fo:mc , iDljicbe ;< did erp:eITc in tl}r former 
culc,but alfo focbe as be compounDe of anp otbcr no ^ 
bcrs,c«cn 0: odd:: i^aue tbc like nombers \\\ p:opo;^' 
tion, fo; tbc greater fiDc, anD fo: tbnr diameter as tbc 
nombcrisbauc, of UJbtcbctbcibeccompcunDc. ant) 

bicaufc 



The feconde parte 
bicaufc 91 bill not if ate to long on tW matt er,3f tjoiil 

J)ere (ctfO^thcnmt^Z twrteties of Smetrallnmbers^ 

luljercbf poumaiegatlber notonelp tftetruetnDec^ 
flanopng of tlje former rulejs : )I5ut alfo m tbcint pou 
juaie fee otljer notable cocluftonfi jano ffraunge too?? 
bc£s of tl)e natures of nomberg. 

£©arije iuell tbis table fo^me, loltii) tbe titles ouec 
tftlubicbe Declare tbe true meaning of it* 

:9lB0 tubcrc ^ou fee one nomber in tbe firfie fo^^ 
lumpne againft tUjoo,tb?ee,oj foiucr tn tbe otber r o-- 
lumpnes , rou Iball iJuDerHanDe tbat tbat nomber is 
t\)t fioe to fo manp feueralle nombers dimttnllt. 



Thetahle ofdtame* 
trallenomhers. 






12. 




'f-.ll 



The feconde parte 

SDljis table mate poii crtcnoc infinitely 0nD tWt 
tl)mge5 mate vou fe^as tbmges of greate aomf ratio^ 

!♦ SGtjere 10 no diametrdle nomber,bat it mm bc DlUl? 

tJcD fai\ 1 2. <Ml)erfo;e t\m bz all eucn nombcr0 cuen^ 
Ip auD oDOdp» 

^» 00ain,tl)cre tS no diametrdile nomher , but it enOet^l 

tn,o.in,2.o?tn.8» 
I. SDI)lcDcIp , tl)ere is no diametralU nomher , tljat can 

tjanc anp mo;e diameters ttjcn one* 

4» .^et male one nomber bee tbe dimeter tQ uiuecfe 
otljeCt 

aspou re2fj0tl)e<//4»;f/frto.i6cS»anDalfoto»3oo. 
S)0»6 5-. 10 tbz diameter to. I o o S.anO alfo tO. I y o o. 

iliheUjaieSt 1 4 y. 10 tbe dimeter tQ . 2 4 4 8. ano to 

y* i^tftelt? : i^o fquare nomber can bee a diamthdUt 
nomher^ 

^cboIar» i:befcp;opertie0 be notable* 
To kne-^es But Ijotu (l^aU 3 bnoloe , Uibcn a nomber (0 p;o/ 

didmetralle i^mfQM\ittbttitbtVLdimetrallenomher^tilX\Qti 

nomher, ^i^atter. BftttbattbpngBJfinocateDioufetrauell, 

bp anp rule0,in tbofe tbat U).:ite of lt*)l5ut 3 toil cafe 
jon of mocbe ^ainz tbcreim 

iPtrlte remember tbe p2opertie0 of tbofe nombecs. 

anD If pou banc anp otber figure in tbe firil place, 

tben.o*2»0^,8.lt 10 no diametralU nomher, 

^econDarilp , if it maie not bee DimoeD bp» 1 2. al^ 
tbougb itenDe m one of tbofe* 3» figure0, it 10 no dU: 

metrall nomher, 

Wberfo^c (fit bane botbe tbofe tluoo p;jopertte0 
(iubicbean infinite multttuDe of nomber0 Doe toant) 
ano be no fquare nomber (a0 none be tbat enoe tm2» 
0^8. 0^ toitb ooDe cppbers ) tben fette out all tbe par^ 
te0 of it, in focbe fo^te,tbat tbe lelTer parte Doc Uanoe 
Dirertlp ouer tbofe greater parte0,tDbicb btvtiQ tnuU 
ttpUeo to0etber,umi ma^ tt)z iobole nomber* 



of yfrithmetlke. 

anD tbctr craminc tWt partes , tuljicbc feme to 
\)mt ani> liUdil)oD:acco?Drng to tl)c fo;incc Oortrmc, 

00 fo: cramplc: tf.7 2. be p^oponcD to be cramutcD 
in tOat fo;tc,3i fcttc tjis partes m o;j3cr tljus, 

16, 24. 18. 12. 9. 

HotDbrtt % net)ct) not to fet Dounc.2. itot!)ei\4,fo; 
IcCTcr partcoj notUcr tl)ofe otber greater partes, tl)at 
aunfluere to ttjcmtf o^as 31 faiD befo:e,tbei ran not 
bee tl)c leffer fioe in anv' dtdmetralU nemhr. xiilIl\fCtto;c 
t\)c\ neoe no erammation. 

3^artl)crmo;c, fo^ tl)em tbat fou (l^all neoe to era^ 
mine , if tbc leflTer nonibcr bee an oDoe nomber , tljc 
fqiiare of it muft eontam Double to tbat greater nom« 
berrtbat is coupleD luitb tt)ano one ino.:e. 

jann if tbe leflTer be an cutn nomber Cof tbem tluoo 
tbatpoutooulD erammenben mufttbe fquarcofit 
containetbe greater nomber (tbatHanOetb bpit).4, 
tpmes,anD» 4. mo;ie» ano tbis.is not onclv a i\)o;tcv 
loaie,tben 31 fee to be taugbte bp otber artes menne: 
but it is alfo mo;c certaine, fo: all nombers not com^ 
^OWntitH of Othttdidtnetrallenomhers. 

^cbolar, Wv tbis Doctrine it appearetb quicfeelp, 

tbat»72.iS no dUmetr all nomber, 

jf 0? altbougb it Doetb enDc in.2» anD maie ht Diui^ 
DeD bp ♦ 1 2.pct no couple of nombers berc bauc tbofc 
properties tbat is requireD* 

i^o;jt)nDer*^is.24*tubicbe is to greate:aitD tjnDec 
6«tbere in, 1 2. tobicbe is to greatc alfo. 

jSuttnDen 8. ItanDctl)* 9 : tob icbe is to litle, bp g 
greatc Deale* 

fatter* 3Dben pjoue in tbis otber nomber. 1 3 2, 

^bolan ^is partes iDiUltanDetbus. 



The feconde pane 



5* 
44. 



6. 

22. 



12. 



timhttt 31 fc0 quicfeelp tl)at it cm not bee n diame= 
trallenomher, j^ojtljenomberstjnijcr* 3» and, 6. lie to 
gceate: fitl) no nombec t^at ftjoulD bcc fcttc tJUDcr. 3. 
matebeaboue.4. 

j^otbct* l)nDer.6*mafc anp nombcr bee fet greater 
tben. 8 ♦ 0s it ooctb fufficicntl^ appeare b? tbat tbat 
ijjtaugbtebefo?^. 

0no tanoer. 1 1 ♦ tbere can bee no leflfc nomber plaf 
ceo tben.6o: anl»tberfo?e. uab to fmalle, 

anti bcrem 31 percetuc greate Ijelpc bg tb^s table> 
tubicbe ^ou bane fet fojtbe* 

Rafter. itistucUmackeDofvou. But ret trie 
tibia otbec crample. 6 07 2. 

^cbolar. 31 fetooune biis partes in o;ocr,tbii0. 



23» |24» 
264- I 2^- 



^nti bere 3i fee a greate fo;:tc of nombers , tubicbe 
tan not feme to mi> purpofe,bicaure tbofe tbat bee c^ 
uen,anO are lelfe tben. 4 4* maUe to litle a fquare,to 
be 4,ttme0 fo mocbe as tbe nomber tjnoer an? of tbe. 

ano. 4 4, mafeetb to greate a fquare : tubcrfo^e it 
can be none of tbe euen nombers. 

again,tbofe tbat be oDoe t)nDer.25.Doe mafee to Uf 
tie a fquare, to bee Double to tbe greater nomber tjn^ 
tier it.anD tbofe tbat bee onne aboae.2 ^. Doe mahc to 
greate a fquare. ^0 tbat.2 ^.ooetb remain to bee tbe 
true nobcr fo^tbe leffer fiDe:anD 264 tbe greater Qoe. 

fatter* aiSicaufeeFercifc is tbe belle inilrwment 

in 



2024* 


6. 

1012. 


8. 
7^9* 


II. 


12. 
^c6. 


22. 
276. 


3^ 
184. 


44» 

138. 


46. 
152. 


66. 


69* 

88. 





ofArlthmetlh. 



iitleantpnor : tbcrfo.jc Ujilljpjopountic tofouoite 

XMim faie POU Of» ^ 46 o.-'js Ct a dUmetralU nomher 
0; no f 
^cl)oIar» 3f Uiill tdc it, fag fcttprtg oouiic fjis par^ 

tCB tijUS. 



IS20. 


1092. 


6. 
910. 


7* 
78o» 


10. 
^46. 


12. 

47y. 


420. 


14. 

J 590. 


If. 

564- 


20. 1 
27> 1 


2U 
26 0. 


2S. 

19). 


5o. f 
182. 




42. 




60. 
91. 


70. 
7S. 



^rtt) !)CL-c jB Tc Diucrfc atiD manp nombcw, tuijiclje 
at toe firfte figbtcapperc notftpng mctc fo; tbis pm^- 
pxjfe. i^o^ 2 o. 13 to fmalle a nombci:,a0 3^ mate fonc 
tuDgc : anD tI)crfo;e all otijec nombers tjnocr it,uiua 
neDes be to fmalle, of fo.:ce. 

%ame,3I f^c tl)at. 3 o. is to grcatc a nomber,3rtD 
tl)erfoje, of ncccflTitie , all otber nombcrs aboueit, 
mua ncDc0 be to grcate. ^0 tl)at;2 Lotber.2 S.mua 
be tf)t true nomber,o^ els none. 

u£r*)crfo;e 3 cvamineftrll.2i.tuI)ofefqnarci5 441 
Mncljz fi)oulD bee one mo;e tbcn Double, to tlic nom^ 
ber tjnDcr it,t{)at is to fciie,itfljoulo bcc.pi.ano fo it 
is not: K^^crtoit 3 rcfufe it,ano cramme. 2 S. tubofe 
fquare is . 78 4 . Znn tbatfljaulD bee foluer tpmcs fo 
mocbe as. 19)'. (tuljicbe ts tl)c nomber Uno:r it; ano 
4 . mo;e. :Cbecfo:c 3 Ooc quadriple . 1 9 y, nnD it ma^ 
feetl).7 8 o. ano tljen 31 fee ttjat it U)antctl),biit foturi: 
of tbc otbcr fqaace:luI)erfoie 3 take tbofe tluoo noni* 
bers,3 mra(Te.2S.anD. i9^fo2 t^ie true fiaes of.)" 463* 

lObiCbe 3 finDC to be a diametrallmomber, 

jailer. i5p ti)e U)aie,rcmebei: tljat \mi coulD ca^ 
filp perceiuctbat all nobcrs bnDcr.2 o.Ujcrc to fmall 
foj gone purpofe.'anDcontcarptuaics, all abour. 5 o, 

f.iti. to 



The feconJe parte 

jfjhdrte to be to grcatc* ^o t!)at t?ou neDeo not to fette Doune 
meme in fo maitp partcs of j?our firttc nombcr, 
n^cr^/w^. ^ !)crfo;jc if ^our nombcr bee focbe a one, as batft 
nian^ partcs^rou mate cljofe one bp gcflfctDljicb ?ou 
tbtnUc tuiU go niglb to Tccue pur purpofe:anD tf pou 
finoc tt to fjnallctbcn fct tljeim Dounc onelp tbat bee 
greater ti)cn tt, til ^ou finoe one otbcr luttctanD tben 
bauc von voiw purpofc, £)? if you finoe ant? to great, 
after ttjat iut)icl)e iuas to fmaUe, ano bettucne tbetm 
none luHe.tljen is notyour nomber a diAmetrdllnUer, 

^ut ano if tlje parte tubieberou tooke bp geflre,be 
to great,vou fljall refufeall partes aboue it,anD taUc 
oneli? leffer partes,til pu finoe a iufte parte fo;vouc 
purpore:o;t els one tbat is to title* 

anDtfinDefcenDrnge o?Oerlv,t?ou finoe no (uffc 
parte , before you come to one tbat is lo litle , tben is 
tour nomber no diametralle nmUr, 

^cbolar* %\m i^ a greate eafe in fl^oatenynge of 
Iuo?fee:iDbicbe 3 luill p?oue in tbis nomber.9 7 s 6. 

Rafter* 3|f you rememb?eo luell your former ru^ 
les , you iuoulD not abmitte tbis to be eramineb fo? a 
diametralle Mcw^rtbicaufe itcnOetb in none of tbe tb?e 
peculiare terminations: tbat is«o»2»o^8« 

^f bolan 31 eofeOe my faulte. ano tberfo^e 3| tafee 
tbis nomber«978o«tubore*2o»parteis.4S9» But fe^ 
yng»2o» ooetb mafee in fquare but4oo» tberfoje is it 
tjerymocbe to litle* 

SDben 31 take tbe* 3o»pacte of it,tubicbe (s*326*an0 
finbe it alfo to litle* 

STbiroely , 31 tafec tl>e * 4 o * parte of it, iobicbe is 
2 4 4i:anD feyng*4o*mafeetb in fquare* 1 6 o 0*31 fee 
tbat it is almolf e* 7 ♦ tymes fo mocbe as* 244-^: ano 
tbcrfo^e is it to greate. 

^0 mua tbe true nomber be betloene* 3o*ano*4o: 
0? els tbere is none at all* 

s:becfo;efirile31ta&e* 5r* tebicbefsitliemfbbelle 

nomber. 



of y{rithmetike. 

nomber(a0 the mottc aptc fo,: a coniccture;anD it rcl* 
Dctl).279|. anDtbcfquaceofo)MS*i22)-. U)!)ifl)ci3 
fane mo;2C tl;cn tbe Double of. 279;. 

2Dt)erfo;c,again 35 p^oue ioitb* 5 ^^ tubirl^c giiictl) 
5oy{-,^nDfevngtt)cfquarcof.s2. is.i ^24. it 10 not 
4.tpmcs fo niocljc ajs. 3 o )'|.fo.j tljat is. i 2 2 2 ,-. 

Wii}cttQ;c2i taUe a greater nombrr . bctlufite it 
ano. 5^ anDfirftBl take. ^^. Uibidic b^uigctl) fo;tbc 
2 96tV . tul)crbp 3 male fee tl)at. ^ vis to grcatcHnD 
fcrnof tbere is no nomber Icftc betluenc. ; 2. ano. ; -\ 
tl)crfo:cl luDgctljatfirae nomber, 97 So. to bee no 

diametr die nomber. 

a^aftcr. Crammetbis nomber. 4 ; 2 o o. 

^cbolar. iSicaufe 3 fee it to be a greate nomber, 
31 luiU begin luittj a greate parte of it. i^ntj tberfo^e, 
3! take, i o o. lubtclje pelDet^. 4 ; 2. ann conOocrpng 
tbat tl)e fquare of. i o o. ts. i o o o. tobiclje is farrc to 
greate,^: mull feUe a leflTer nomber. 

a^aftcr. 3 iDiUeafepouofpourpaines in tljaf. 
J^oibicaufebcreis mojetobeeconfioereD. ^^oure* 
member tliat 3 toloe vou before, in makpng of Mame; 
trallenfimhers,\)o\}) tbatfome nombers tjoe foUotue tbc 
rules ofotbcr,ofU)bicbetbei be compounDe.anDfar^ 

tbermoze, tbatfocbecompounOe diaynetralU mmhrs, 
DID bcarc p2opo.:tion to tbe leffer , as tbe p;opo?tioii 
iuas of botbe tbeir fiDes aDOeD togctber. 

^cbolar. aatistrue. 

SpaUcr. £Df like reafon all focbe dlametralU mm* 
hers , muft bee eccluDeD from tbefe rules , tubicbe bee 
maDe peculiarly fo? nombers tbat baue tbeir oUine 
p;joper fo;imcs,anD DepenDc not of otber. 

<anD pet fome common rule muft bee giuen , tljat 
maie ertenDe as luell to tbem,as to an^^ otber. 

mi)erfo;e let tl)is be it. 

SDbat tbe tUlOO (iDCS of all ikmetrdlUmmherSj fjaue 

focbc a p^opo;tion togetber^as berc ^ou fee erpjeflPeD 

in 



Thefecondt parte 

in fome one of ttiefe formed $ iftW htt cotttttmco aus 
Iberctljetbcbegon* 

CC&efii1teo;Dcr» 

4 * II ♦ 14 * 49 * to* 14 * ili * 144 * I8» ♦ »»• ♦ 

sj ♦ ir ♦ 17 ♦ 19 ♦ ?i ♦ M» 1? ♦ 17 ♦ !♦ ^« 

i64 ♦ 5TI ♦ US ♦ 455 • ?¥5 ♦ m ♦ TTi ♦ r«4 ♦ 3^* l*-* 

C2n!)cfecon0eo;0cr. 

5 , ij » i« ♦ JO « _J4 ♦ J« ♦ ;5 • ?« ♦ 4» ♦ J4 ♦ 

1< ♦ If * 6} ♦ 9S ♦ I4J ♦ Tsj ♦ JSS ♦ Til ♦ Vss ♦ 4»J ♦ 

^ece ^aue 3| fctt^ t^e Ictfer fioe as tbe numcrat o;:, 
anu tl^e greater floe as tlje ocitomtnato; , m. I;erebp 
^on mate percef ue t^e caufe of ttjetr otHtnmon. 

jFo; t\)t firft o^Oer t0,toto tlje leffer fioe, o; nom^- 
httjis oooe* 

%ijt feconoe o;oer ts , tu^en t^at UflTer ftoe i& an 
cuen notnber. 

StiM'm ooetl; fet tl&etnTojtIjat tfte numerator l!an* 
Det^i fo? tfje fecoitoe , oa greater fioe:ano t^e oenomt^ 
hatoj fo? tl)e firfte nomber^o; lelTer fioe. 0no fo;j ttie 
tno^ Oelectai^le contemplatton,to bebolo tMtit fo;me 
of pjogreirton,!)e fetteti) ooune a0 mani? l»l)ole nom^ 
l)ers,ad t^e fraction lutU giue* 

j^notf)t0t£(bt0fo?me* 

C2C!)efir!Ieo;Der» 

i\x 24: 3|: 4-I-: 5-77* 6^: 7fs* ff» 
dCIje feconoe o;oen 

t7* -1 Li* 2li» A"* Cii* />iZ« "T*" (*r 

It* -^ti* ^rs* 4>i» 5 34* Oi«» /ji* ?u 



of^rithmetike, 

ZZX l)cre tit tl)c ftvft o jt>cr,rou fc botlje in tht loljole 
nombrr6,anD?.lfo in tbcnumcrato:g of tijc fraction, 
ttic natiu-allc c;tcv of nombcrs ♦ xinn in tljc Dcnonu- 
natc:s,tl)c natnrallc p^ogrcffion of oDDc nontbcrs. 

13utint!icrcconDco;jOcr , foufcc tbattljclubota 
noiitbcrs go m tijcir naturallc o^jDcr , anD tbc numc- 
ratoic anD DenonT!nato:s,ferpc an Jrithmcticulle p:o^ 
^rcflTton, by cquallc Dittauncc of . 4 . fane that m tl/C 
mimcrato:5,aIl the nombcrs bcc otiDc:anD m t\)t Dc^ 
nominato:5,tl)ci ht all encn. 

j">olu bp tbis gcncrallc rulr.tf vou fiiiDc anj? tiuoo 
partes of anp nombcr. In one of tbcfc fo.:mcr p;opo; ■ 

tions^l'OU ntaie bee furc tbat fttJJ a diametralUnomher, 

13 ut fo; ti)c mo:e apte conference of tlje partes , pou 
fljall Doe bcfte to rcDucc tbem to their leatt nombers: 
as pou baue learneo m tbe firfte parte zt Artthmeti\e. 
^0 m f onr lad nomber,U)btc6c luas 4520 o.j^ou 
ftiall finDe bis. 1 8 o.parte,to bee.2 4 o» lobicbe bcpng 
reDuceD to tbeir fmallell nombcrs, iuill bee. ^^tiDbcr-- 

fo;e 3 ant aflrurcD,tbat it is a diametralUnomher, 

?aet one tbpng mo.:c l^all pou marlic. 

Jfanp nombercnDe in Cipbcrs, abate eucit CU 
pbcrs,as often as von can(Bl meanc.2.4.o.2.6.«,ami 
tf tbe rettC be a diametrallemmher,io Uias tbe firft» 0no 
tberfo;re \n tbis lafte craniple. 4 5 2. is a dUmetralleno'. 
ify^asiueUas. 45200. 

aifoifanpnombcrbcepng DiniDeobpanr fquare 

nombcr,DOC mahC a diametralle nomher in tbc quotiente, 
tbcn loas tbe firfte nomber a diametralle nombcr alfo.. 

0nD tbis,fo; tbis t'^raz , ll^aU fnfficc fo; diametralle 
nomherSt 

j^otu tDill3I fpeafec fomelDbat hm^iff t}fli{eflathsi Ofliks 

anD tben p?OCCDC to Ottftvfiguralle nombers, fiattes. 

^cbolar. 31 remember pou DcfineD tbcm bcfo;ic,to 
bee focbe flatte nombers,a0 baD one fo;mc of p^opo;;' 
tiott bettuene tbeir (tDes* 




The feconde parte 

0st)etC27. atiDi2«be 
lilieflattti : bicaufe t'mt 
fiDc5faetii oncp;opo.2t(^ 
on.i^o.ia0«9a5to,5.fo6 ^ 
ts to» 2, lotbc bectmgin 
triple p;opo;tton. 

£Da(ler :iPou faie luell . 6» 

0uotl)ati3tbefaurcUjl)ptbci j 

be calleD [lUe: fo^ t!)c UUcneffe 2 1 

tutoc pzopouiooftbcirfiDcg. • ^^* 

0ltbougl) fomc mmm Delite 1 _* 

SjttdrelikS ,^^5 j^ f q f^U tljem fquareliks figures : fatcaufc tljci t)aue 
figms, fomc p2opcitie0 agrcabte Uiittj fquarc nombers ( fo; 
as Nuclide ^mi\) xn bts. 8« booke,anD. r S.p^opofition: 
Buery f^oo nombers, beeyng likeflattes, haue 
one meane nomber betlfiene theim inproportU 
on . ^{nd the one ftatte nomber bcareth ynto 
the other flatte dmblt that proportion ^ that 
their fides doe, 

fo} Declaration of lubicfje p?opofition, marfec tbe 
tlxjoo flatte nombcriS befo;e : 3! meant .27. ano. 1 2. 
lubofe nuts arc in p;opo;tron Sefquiaker : {?no tbe flat 

nomber0tbemfelfc0 be a0|.o;.9,to,4:tbatts Double 
SefrjuijUArte. j^olu Ooc vou Double tl)c pjopo^tion Sefi 
^walper,mti it U)iU mafee douhU Sefquiquartc, 

^Ijolar, 2DI)usDoe3jrettctbemmo2Der. ': -'- 
0nD3fnTultipUetl)e numerators togetlicr , arrDt^e 
ftenomtnato^s alfo* (i^o?3J remember , pou tolDe me 
before 5 tbat p;opo;tion£f are aDDeD , as framons arc 
inulttplieD)anD tben Imll it ht.^xtixtn as pou farcD. 

fatter, agam ^«f//'/^fattl)mtbetUientethpM^ 
jolitton of tlbefame booSe. 

Jfany nomber Jlaride as a middle nomber m 

proportion^ 



afJAthmehh. 
proportion) hetlipene other ft^oo nombers,thofe 

tt^oo are likeflattes, 

2L!)at is to faic : if an^ tluoo nombcr£5,bcr'ng ntiil* 
jilicD together, fiocmafee a fquare nombcr ( fo^ none 
but focbe can baur a miDDlc nombcr bctUicnc tljcim; 
tbcn arc tbct hke flatten. 

;Hs. ?. anD ♦ 1 2» uniltiplicD togctljcr uoc maUc. 1 6. 
tul)tcl)c ts a fqiiarc nomber : anD.6.t!)crbv appc nrct^ 
to htt tbcntiDDcll nombcr bctiucnc tl)cim, 0nD tbcr- 
fo2carco.anD.i 2, Uksflattei 

i^iljclDaics.vnno.2 7.fo^tbcimaije. Si. iubicbcis 
a fqiiarrtano tbcir miDOlc nombcr 15,9. 

^nDfoarc.2.anD.8: 2»nnD.ib: 2.anD.^o, 2.1?. 72 
vantJ.4S: ^.anDjf: 4.anD,9, 4. anD,i6t 4.anD 
2f. c, anD. 20. 5'.anD.4^": 6.anti.24: 6.anD,5'4. 

:anDfoofmfinttcot!)cr. 

SiCbis crpoOtioit is confirmcD h^tMt firffcanofc- 
fonoc p;opo(ition of tbc ntnctl) boUc nttudide,]ii\)txt 
Ijcfaictbtbufi* 

Jft^oo nomhrsheyng likeflattesyhee muU 

ttplied together , the nomher that thei make, 
(hall be a fquare nomher, 

j{7idifz.nombersbeyngmuhlpliedtogetherj 

do make a fquare noher^then are thei Ukejiattes. 
i5p lubtcbc rulc0 it tioctt) appcrc,tbat ^ou ca banc 
no piogrcliio Gf(nM(r^m4//e,butitmuftbcmaOccitbcc 
of fqiiarc nombcrs, o;el0 QtUhjflattes, lubcrbp tbcrc 
appcarctb a grcate agreablcncs, bctlucnc Hksflanesy 
anD fquare nombcrs. Sno tbcrfojc faietb EucUdc aU 
(0 m tb:.2 6.p;opcIition of tbccigbt boobc. 

T^ombers that bee like flattes , haue foche 
proportion together ^as one fquare nomher bea^ 

G,y, reth 



The feconde parte 

nth to an other, 

s:i)is mate von P^ouc fap a»p of tfje former cram* 

pies. i^oM2.to«3»i0mliljcp;opo^tion,as.i6. to.4» 
o^56.to.9. 

aifo»27»to.5»I)at^)Ufecp^opo;jtionaj3o6. to.4: 02 
i44,to.i6»otfter.8Lto.9, 

ano farther , if pou DeujDe tl)e one of t!)ctm bp tlje 
otl)er,tl)e quotiente U)ill be a fquare nomber. 

^cbotar, SDIjat cioetli appeare euiDentelp at the 
firftctjetuc. 

i^o^» 1 2,DiuiDeD bp. ^.tioetft malte.4. :^nD.7 f .tjiui^ 
DcOb^5'0tuetb«2^, 

^o,M»l'^6»maljetI).9»anD.72*bp.2.pel0etft,56. 
anu fo 31 fee (it t^e refte, t^at all tbe qmtUntes Ujill be 
, , , - fquare nombers* 

Them.de . ^"^ ^ ^^^l^ *"°i^^ ^^ fenoU)e,l)otu tbofe nombers 
us ue made, j,^ p joOuceD, fo^ tbat 3 bnolue not vet 

Rafter. Cafec ani' tluoo fqtiare nombers, iul>at 
fo euer thti bee,ano multiplie tljem bp anp one nom^ 
ber,tljat rou littet anD thci lotU maUe.2./iitf^««. 

^o»4»anD.9.multtplieo bv.2.t)oemafee.8.anD 1 8: 
tD^icbe bee lihsftattes, 

jagautjifpoumumplietbcmb^f, tljeimaUe, -^ o, 
ano»4 s*, Ujbtebe be alfo Hieptui, 

^cbolar. 31 am perfett uiouglj in tl)is,tf tbat be al. 

fatter* an otl)er iuate pou mate make tbem al- 
fo : 3!f ?ou take anv ttuoo fquare nombcrs , tbat tuill 
aDmttteoitc Diutfo;,anD DiuiDe tl)tm botbe bp it. 

00 fb^eriimple.^epng 9»anD. 3 6.UnU be botbe nU 
ninth b^*3. 3f 000 fo otuiDe tbeun : ano tftetr qmtientes 

are, 5*anO* 1 2. tufltCfte are diametralle nomhers, 

^0 m ltkemaner,if 3f oiutoe 1 9 6 ano 4 9 (lubiclje 
botbe are fquare nomber0)bp,7« tlje iUQtientes MM be 

28.an0.7. 

%am, 1 6. antit i o o* hzvm botfje fquare nomber& 

ano 



of Jrlthmetike, 

atiD DiutDcD b^.4.Doe maUc«4. auD.: ^ a)3 tbc^r .yj^o//*^ 
#«/f ,antJ tbei be Uksflattes, 

©cbolan ;3nD tit tbcfc 3f Tee an otbcr fnaungc 
U)o:fec:tt)atiftl)ofctluoo//V/'<^mfacctnultipUcDto; 
gctbcr : tl)ti U)Ul maUc tbc greater fquare,of lubicfjc 
tl)ei came. 

i^o^5.tpmc0.i2.maUctb.?6:anD.7.n'n^c3.2S.gl' 
uctb. 1 9 6: ano fo.4.tpm?0.2 y.b^piigctb fo:tbe. loo. 

£Bafter. Jt uoetb fo bappen often tunes: but it is 
not altuates fo. 

J^onfrouDiuiDC.Id.anD. I oo.br.2. t\)t^uottentcs 
h)\li be.H.antf. y o. tubicbettooo nombers niultiplicD 
togetber, Doe make. 4 o o. farre Differing froni. 100. 
^0. ] 6. auD. 1 9 6. btpng botbe fquarc nombers.auD 
DiuiDcD br.2.uoc maUe. 1 8.anti.9 8.U)bicbc be likepi^ 
tesimn tbofc liksfiattes multiplieD togetber,ooc I'elDc 
1764. Ujbicbe is a fquarc nombcr,but tt is. 9.trmes 
fo greate as IS. 196. 

^cbolar. pet one Doubte 3 baue : iubetber all 
fquare nombers be likeflattes , ano fo bee not oitttnrtc 
fromtbent;' 

fo;. altbougb in tbe tiimiiow of figuralle nombers 
Vou DID Diamtte tbem, ret m tbe eramples oUiksfat^ 
tes^i^ou put certain fquare nombers cmongctt otber. 

Rafter 011 fquare nombers are likjpttes, bcvn$ 
tomparcDtogetber:anDclsnot.i^o;asanp.2.fquarc 
nombers maie be compareD togetbert fo male tbei be 
refcrrcDtotbetr rootcs, tuitbout comparifontoge^ 
tlitt-m els tbei maie be compareD to otber nombers 
tbat bee not fquarc* 

2Dberfo;te marbe tbefe tlno rules tuell. tbat no one 
nomber can bee calleD a likjptte : but m compar ifon 
to fome otber. i^o;j.2.br binnfelf is not callcD a Irkf 
j?4^/r,ei;ccpte be bee compareD to.8.0? to. 18. otber to 
3 2.o;?.5' o.oj fome otbecfocbe. 

feo lifeeUiaies«4«tubtcbe fa? nature is a fquarc no^ 

eag. ber^ 



The feconJe parte 

htu antj altDa(es d^all bee fot^ct is it not accepteD ta 
a likeflattey onlcB It be$ referred to fome ot\)tt fquare 
tiombcu 

^cljolar* m%atirithecompatti>i3)it\}^n.\i)Wf) 
rou named &efo?e to be a likjflattei 

95sfter» ^ou remeinbec : one of i^ucUde ij^g rules 
{ix3Wht 3 repeater faefo;c) is , tljat^l^^^/w beepng 
nmlttpltcD together, tutll maljc a fquare nbbet* ^no 
foDoetb notu»bepngmultipUeo b^4» 

^djolar, jiioluBHioe tjnoeraanDei?ourluoD;Df5 
better, ^o» ^ aiiD. 8, tompareo to0etl)er,bcc not Ukj 
flattes : pet er l)e of tbein compareD to otbcr nombers, 
male be like flattes, z&* 3»f ompareD to, 1 2, o; to.27;anD 
8«comparcD to. 1 8,0^0,5- o* 
C/mted {©ader* iSoU)UjtUUjelcttetbefeHf/4^^« alone 
uombers, fo.: a tpme: ano Intreate mo?e of rootco nobcrs . SlnD 
flrUBituill tellpou fomeiubat of tbe names anona^ 
tures of focbe nombers as baue rootcs: aDbcn fccon^ 
Darllp 3i tutU teaebe ^ou tbe O20er to eptratt tbeir roo* 
tes: ^noafteriuacoe iuiU^Kljetpefome parte of tbe 
tjfe of tbelm. 

Mberfo;je to begm, tubere toe lefte a Ittic befo;je, 
«/fm/r. tbeerplteatioof rootes:3i fale, tbattberooteof nomi» 
ber,is a nomber airo:ano is of focbe fo;te,tbat bp fons 
tj;jfemuIttpluattons of it, bpttfelf,o?bptbe nomber 
refultpng tbereof,tt ooetb produce tbat n6ber,U)bofe 
rooe tt !s.0nD acco^opng to tbe nomber of times tbat 
It 10 multiplico,tbe nomber tbat refultetb tbeteof,ta^ 
feetbbisnamet 

^0 tbat one multiplication mafeetb afquart nmUr 
^no ttooo multiplications boe make a Cubikevowhr, 

llifeeluaies. ^. multiplications, Ooe %x\xtafqMreef 
/pares, anti.4.mult!plications Doe ^eloe afurfoUde. 

^no fo infinitel^ 

fo} as multiplication batb no enbe , fo tbe nom^ 
bers amounting of tljembe innumerable, and tbeic 

rootes 



of Jrlthmttikf. 

ro9fc,3 ai iitfinite. iSut thzii mmz^ tljei talic certadi 
l)?,of tl)c n3)n Jcrs tbat tl)ct 032 ma^c. 

^3 tbe roo tc of a fquacc nombcr, t0 callcD a Square ^rjuire 
mtcmii tbe rootc of Cubikc nombccis nantcD a Cw. rooted 
hiks roote \ 3Jn Xi'^Z fo Jte tl)at rootc is calico a S^juarcd j-cuiiU 
/^fwreroofe.luijicbemikctbarquarcofrqiiarcsnina^ roote. 
bzv. aaD that cootc is a SurfoUd; roots, tljat Vcio:tl) a jfrqunred 
s«/y5//^e w£)m^fr:iiT lu!)icl)e fojtc of multipUcatioit,vou /;.„4^^ ^p^f^^ 
mate pjoccoc in&nitclp,a3 31 faico. JfrHrfodde ' 

j;iottuitl)Ilanoi>ng foz I'our cafe, J fjauc fct foojtbc ^^^^^ 
Ijcre lit a tabic, certain of tbc moftc notable liinocs of 
rootco nombcrjj. 

^no to tbc mtcntc von maic partlp conccine tl)c 
r^afon of tbclr names, 3| tuill after tbc tablcfct fo;rtb 
a bncf crplication of tbcir namea , tuitb tbc pjotrac^- 
turc of tl)e figures , that t}:iti Doe rcfcmble m multl^ 
plications GeomeHcalte : tuljcre poinrtes, lines, plattc 
fojmes , oj rounbfo<:mes bee multiplteo : ano b^pnge 
foo^itbc other formes agreabic to foche multiplica'- 
ttons. 
ai5ut firff marUc the table tuell: ano it tnill gtire 
pou greate lightc , ano aptncDTe to tinoer^i 
ftanoe all that folotoeth , mochc the 
better. 

jFo;cramplesarethc 
ligljte of tca^ 
ch^ng. 

She 




oo 
a <^ 




"37 



Co 






» 



ii 



C/J 



f 




^-« Co 


Pt^ 


■^ft 


***<.§ 


f-s 


5^ 


S-*^ 


<^ 


5^ 


s 


S^ 


'^ 






M 
*• 



^D^ 
tk 



Co 









£ 



•-Si 

ft 



^ 



o 
o 

o 



^>* 



3 & 



of Jrkhmetike. 

\i}txt rou fee Diucrfe tt\33t& of nombcrs , and ti« 
gatii^ cuecp roluc ttuoo names lu;ittcn : one on t^e 
rt0bt banDe^ano tl)e otl)er on tbe Icftc Ijanoctububc 
feruc tQi all tljc nombcrs tn tl)at rcUie. 

%\\t names on ttje lefte banoe lice tbofe nameg, 
lul)tcl)e bee eommonli^ tfeD , ano attrtbuteO to tl^ofc 
nomber0. 

Clje names on t\it rtgljte bante, arc names of mi? 
aODition , tDl;ube Doe aptlp erp;:eflrc tbe \itx\> natures 
of tbe nombcrs,\3nto luljicbc tbei bee aflfigncD : as a- 
none 3 U)iU Declare. 

j^nD nolu roncernvng tbe nombtrs , row fee firlfc 
in tbe beDDc of tbe tablc,a retue of nombers fct tn o;? 
Dcr,as tbci follolue in common nomb^f ng, from one 
fo;U)aro. ^nD tbei bee callcD rootes,fo2 tbat tbe mul^ 
ttpltcationofecbeoftbem,bvtbeimrelfes,o;bptbat, 
tbattbcreofamountetb,b;rnffctbfo;itbealltbotl)er, 
tbatbcefctbncer tbcm. £Dftbclobicbe,tbefeconDe 

retue is calleD Square nomberi:h\i({\x(z tbat tbeir lengtb Square 

anotbetr b;eDtb(U)btcbc3; tmoerftano butbc.z.nom-' nombers, 
faers of tbeir multiplication) is cquallc, 

0s.2.trmes.2.Doctb maUe".4.U)bicbe IS a 
fquare nomber,anD maie bee figurcO tbus* 

3lffeetuatcs.5. trmes.^. maUetb.9. Uibicbc 
iB a fquare nomber, ano is repjefenteD tbus. 

anD berei^ou fe,tbatlf pou Jimr}Ct\}C Square nomber 
fcp biB roote, tbe quotient luill be tbefame nobcr alfo* 

^cbolar. WatmuftneDesbefo* 

^Ratter. SCben m tbe tbirDe retue arc placed C« Cubike 
Uhs nombers : tubicbearcp^oDuccO bp triple multiple nombert. 

cation, ai^.i. tpmcs.2. tunfcmaUetb.S. i^no.^tp* 
mc0.5.tbJtfe,peldctb.27. ^o.4.tpmes.4. folucrt^^ 
ntes,giuetb» 6 4. Cbefc nombers can not be erp;ief^ 
fed aptlp in flattc, but p^ofpectiuclp, as ^kc maie be 
madeinpMi^acture* 



♦ ♦ 



♦ ♦ ♦ 



The feconde parte 
jSlnD tljefe are tWt fo^meg* 




ij ■ 

\o o o o^ 
OXp COO" 




3ln tlje firffe figure pou Tee ♦ 2 . crpjeffieD f n Icngtfte 
ii;ieDtl)e,ani) d eptbc* ano tit tbe fccono fojme.^to re^^ 
p^efeitteo in all tl)ofe,3,Dunenaon;3. 3|it t^e,4,figure 
4» iiet tbe roote,anD 10 D;aU)cn agrcablp to tljat f o;me. 
^cl)olar» %W w manifeae inougl) to figljtc* 
Rafter, jaet reafoit ougftt to iuatg^ it m o?e cp 
a(tlp,tben figftt can compjelbenoe ft fn aa tfjetr tri<» 
pie multiplication ooetft refeble tlje nature of founoe 
I)ooie0 , fo it migbt appeare mo^e inVtt erp^ffvng of 
tfteir figures , agreablp as founoe boDies ougbt : in 
tobicbe euerp parte can not appeare to figbte,fitb rsv 
tierfe of tbem lofee inluaroip* as b]? tbefe. ^ lattc tigu= 




^S^ 



«u« 


r~" 


— 




— 





\ 



\ 



tespon mate partelp contexture. £)ftuf)icbeattbl0 
t^xttz and in tftis place, fome tnen toill tbinbe it an 0^ 
uerfigbtetofpeafee, ano mocbe mo^e uuerfigbteto 
^atte of tbem anp tbgng largely ^au^tbat toe maic 
Dfe m^ fo? t Je apter ejcpiicatton of t|>at triple mtxU 

plication^ 



of jirithmetike. 

t(plff attoit,tu1)erbp tbct be nraDe. 

^0 tbat as it is multiplieD tt)?ifc, fotbe nomtcc 
tl)atDoctb amountc thereof, Ijat^ gotten* 5. Dimcnfi^- 
ones,tt!l)icl)e p<ioperIp bclongct!) to a boDicoj founo 

fo;jmf .anO tl)Crfc;c is it calico a We,^i tuhike uomler 

tCl liiclic nontbcr if pou oimoc bp tl)c rootc, tt)c qnotv. 
mt lull be tl)c fquarc of tbefamc rootcHsJ faiD afo:r. 
15utto p;occDe,ifrou Ooc niultiplic tliat Gi^ik? 
nomber bv fcts roote, tl)c nombcrtl)atrifctl)ofit, 13 
rallcD a s<]uare offquam commonly : bicaufe tbat not s^w^w •/ 

onelp it IB a Squnrs mmler, but tbc rootC of it alfo is Squdns, 

nSquaye nomher, ils ^ou maicpcrcciucbveramtna' 
tion,of all tbofc nombcrs tbat be in tbe fourtb relue, 
lubicbe nombers 3 Doc call/<»»^^ Cw^m: bicaufe tbei longQuhes, 
niaUc a line of Cubes ♦ 0no t)atb m Icngtbe fo man^ 
Cubes, as tbe ficfte roote Ooetb containc tjnitics. 

illjis line of Cubes , altl)ougb it baue foj bis 
b;eQtbe,anD Deptbe alfo,tbc tbicttcnelTe of one Cube, 
tct bicaufe itbatbno nomber of Cubes, in b;jcDtbc, 
no; in Deptbe(03 generally no nomber of tbat tbpng, 
lubereof it is calleD a lint) tberfo;ie maic it tollerablp 
fceare tbe fimilituoc anonamc of a line. anofoDoc 
lue commonlp call lines, tbofe fmalle cojDes^tubicbc 
are onelp longhand baue litlc b;ieDtt)c to tbeir lengtfj. 
IBut vet are tljei not Uiitbout all b^jeotbe. 

S>cbolar. J^no tbereof (31 tbinUe men call a line of 
^.2icUcs,anD a line of ^ffljclers ffones , tuben man? 
bee laieD m a roUje,in lcngtbe:ano but one (0; fetoe) 
in b;jcDtbe* 

a^aften i^ouratetrutbe* 0nti tbat name Doetfi 
continue If ill,emongeft all our countrie menne:faue 
tbatmotte menneooenot call it ftjarplpaline, but 
nto;ie b:oDer(after tboloe C uglifije language)a laine 
^no fo men tjfe to faie , a laine of ioinc buttcs, and a 
laine of b;oDe clotbestano focbe otbcr libe. 

Znh W batb fo largely applico tbis name ? tbat it 

l^Aj, mm 



The feconde parte 

matfe feme no greate abfurDitie, to name anp tfirnge 
a line 0.: lame, tbat t)at& mocfte mo;e ImstUthm 
bzeotlje : ano is maoe lip often aODition , o; multipli* 
cation of anp one quantitie» But pet fo;i auoiopng of 
ertoure, it ougbt to bee limiteo, luljereof tijat lute 13 
nameo. ^3 in our mater to (ait.aline ofvmties-.a line of 

Cubes ; a line ofCubi^e Cuba : anO a line ofCuhike Cubes Cm 

H^/)i-anDfofojtI)e» 

31n libeluaies mua Uie iuoge of platte fo2me0,tl)at 
tljei baue no Deptbe o,: tljtctjenerre , c^lfjcn one nom* 
bet 10 multiplieo bp an otber , onelp tluife: tbat is to 
faie, in b^eotbe ano \mQt\ic onelp : ano is not multu 
plieo tbe tbiroe time bp anp nomber,to mafee tt beare 
tieptbe, 

0nD tin's rnuH be confiDereo generallp.tbougf) tbe 
nombcr fo muUiplicD bee a Cube, 02 anp otl)er foimoe 
nober. j^o; m focbe cafctbat Cube, 0; founDc nomber, 
U)l}atfa cuer itbe,ttanDetb but as an tjuitie. 

^cbolar, ^ii:,^' Doe tjerp lurll unOcraanDctbe 
meanpng,anD rcafonablenefire of tbofc names, Itne, 
anO fqi^iarc.m anp totngroBut ^ bncVoe not tbofe ters 
mt&oCubtke Cuhes^m'^ Qubil^e Cubes Cubi^ely I^ltl)OUg^ 

3! fe t\itnx kt in tlje tabic, Uibicbe pou bauc giurn me. 
Rafter* #0 mo^e t\)m Doe pou ijnoerltanue Di^ 
nerfe otber names tbere , lobiebe jj lotU thcrfo;ie De* 
Clare Imto pou* 

Jf^ou agree to tbetjfe of tbe name, ofa line and a 

rqimre, in tbatfo;te tbatpoubaue tmknur^ tjnlo: 

m\\ If J multiplie a €\xhikt nomber by bts roote. 

gs to faie.8. b^2. 0^2 y.bp.^. otber.d 4* bp. 4* tbeii 

ItiallB! bauea Ime of Lubes . lubicbe j ooc therfo:c 

^.a^r.^ nf ?'^ "^' ^'!?'' • ^"^ ^ o«^»tonf P tOei hzz calleO 5^«4rf ^ 

5f«dm(,/ Squares, OlSc^u^res of Squares :mtimmKmtnt\)tlUt 

jquares. namep Zf«;</;^e»;^%;, 30 fqua^e nombers are calleD 

/!!^ir* !f ^«^J^ "a»«^ altbougft in founoc booies, 

tt tiato no tjfe , pet m practice of founoc nombers , it 

mate 



of Jrithmmke. 

tnaieant) Doetb erp2effe fomc p.2opcctieB aptlp. aa 
nam;:lp tbat all tbofc nombcrs,U)l)ict)e rife of 4 miib 
tiplication3,mciie be as toell maoe bp tuioo multipU^ 
cati60. 3i5uttl)en tbc roote of tbat multiplication fl^al 
be a fquare nomber alfo. 

S)Cl)olar. ^0 31 t)nDcrftanDc tbat, 16. tsanombcc 
of tf)at ro:tc,U)l)icb bcrc is callcD square off^tures.^nJi 
^etmale it bee calleo a fquare nomber: aiiDisfoiii 
DccDe,incomparifouto.4» ^nDtl)erfo.ie.jsperceiue, 
ft is fct tluife ux tbe table:oncs emongell fquare noni 
bers,l)nDer 4 lobube tben 10 bis fquare roote: 0tiD a^ 
gain it is fctcmongcft/fw^m o/y^u-^ro.tnDer 2 lubicf) 
in tbat place ttanDctb as bis fquarcD fquare roote. 

3li{jcluates,6 4.1s tluife fct intbefametable,one0 
rmongetty^W'jra ,t3nDcr 8»tubicbe is bis fquare roote: 
^nu again emongcltcw/'z^e mmbers, Xmnct,^. tobicbc 
is bis d^ubike roote. 

iT^aftcr. pou fate trutbe.0ltbougb tbe laffe tvh 
pie be not to iour purpofe, concerning square dfjuarer 

QlZen^^n^kei. ^nD Iff OU DiD note It OUCIL', fo 2bi«' 

caufe It 10 tlmfe fet in tbe table : tbcn mate pou fee it 
f\:\i\^t fette m tbefame table, fo^ it is in tbe firte retuc 
t)nDer.2. 

^cbolac &o 31 fee,tubcrfo;e ^ mtgbt ratber baue 
ta!je,8 Llubicbe is a2e«;<f <5»;<fX^»ow^''^anD fo batb 
fo: bis roote . ] : 0nD alfo it is a fquare nomber , ana 
I)atb.9.fo;bi0 roote. 

O^aHcr. i^artbertopzoccDe, iflmulttplictbofc 
fquArsi offquares bp tbeir roote , tbei luill maUe Sur/o^ SmfoUdeu 

itde nomber s. 

S>cbolar. 31 perceiue bp tbc nombcrg in tbe table, 
tljatt'oumeane tbe leafte roote oftbetluoo : bicaufe 
tnDer* 1 6. 3! fee. 5 2. m tbe cctue of SurfoHJes, 

n5aacr. K?afon male Ci^iue pou to tbinke fo.j^oj 
t!)c nomber ano bis roote , mufte bearc altuaics one 
name. ^0 tbat if J name. 1 6. as a fquare nomber-, 31 

^A\i, mull 



The ficonde parte 
nruff referre ft to ftis fquare roote . 0m) ff 31 wzmt it 

as a Zen:s:i;^n;^iie nemher : tt ntutte btt rcfcrrc D to ftt5 

Ze;^:^n^J^e roote* jSno ill lifee fo;t of al otljer names, 
as Ujften 31 call. 6 4. a fquare nombcr,f Demaunuc 
]ul)at is W roote : vou muHe neoes aunfluere bv bis 
square roote, iu!)icl)e is, 8» 2i5ut if 3 name . 6 4. as a 
Ctt^f, ano Doe tbtn fcfee fo;: Ins roote: ton mull t3noeri» 
ftanDe l)is Cuhtke roote,m\t) tl)at ig. 4. Wut if 3 name 

tt to bee a Spare of Cubes , 0^ ^n^cuhe : tl^cn IS. 2. ftlS 

roof (?. j^s ^ou maie bp tbe table perceiue. 0nD alfo bp 
t^e ojDerlp multipltr atton of euerp retoe , 0; ojucr of 
nombers bp tbeir roote. i^o^tberbp amountett) tbc 
ncrte reloe. 

0no fo mate f ou increafe tfje nombers of tbofe re* 
lues,o;! D^tuers , acco^Dpng to tfte tpmes of pour muU 
tipliratio,as mocbc as ^ou lift, ^inn euerp o;Der fi)all 
fceare foc^e names , as agrectlj to tbe nature of tbeic 
tootcs* 

Wibttfo7t tbti appeare to bee ouerfcne , tbat call 
t^ofe formers nombers SurdefolUes ^(cin(^ tbci are not 
anp tuatcs Smde nomhersy but tjaue tbetr rootes. 0nD 
tet, to confeCTe tbe trutbe, 31 cannot Uiell tell pou tbc 
true etjtnohite of tbctr namercrcept tbet be fo nameo, 
as it ijocvefoltde ^pon/olUe, ^nu tbat interpretation 
tuere to ftreiglitlp racfeeo. But tbe name bepng re^' 
reiueD ano tDCll fenotuen, tuee mate moje eafilp tuitft 
Ufcertietfc tt,ti)cn iuitb fcrupulofitie,curiouflp fca it. 

%bt(t nombers are Cmple nombers intbeir feinD. 
jf onbei rife of. 5-. multiplications. anD if tbeir roote 
bee a Digite nomber, tben is it tbefame nomber^tbat 
{fanoetb tn tbeir firfte place. Znn if ttjeir roote be an 
article, tben batb tbutSurfoUde,^, tpmes fo manp €v* 
pbers togetber m tbe firfte places, as bis roote batb: 
ano tbe nerte figure after tbofc C vpbers, is tbe firfte 
figure fignificatiue of bis roote* 

^cbolar. | fee it fo in all tbcfe nombers, tfjat bee 

in 



of Arithmctlke. 

(ntbe table* 

;^aftcc, i^iiD fo fljall pou finoc tt in all otljcrtf. 

0nD fartbec if tlje roate bee a ttombcc inirte , tljcrt 
tbefictte nombcc of tl)z fur/olide, IS tbc ficft nombcr of 
tl)c roote, ^iiD tbis 31 tioe tell pou fo.: fome Ijelpc , m 
gelTpng at tijetr rootes. 

JDbiB name tbcrfo^c of tl)c(m , 3i mcanc Surfolldes^ 
in Jrithmetiks > male fcrue to aDmonifljc poii of tl)ctL* 
cootc* 15nt m Geometric, o; m compofition of founoe 
boDic0,tt feructl) to no tjfctanD tl)crro:c 9! Doc call tljc 
agreable to tljclt figure , Squares of Cubes : fatcaufe tl)Cl Squares of 
malic a fquare fo^mc: but fo tljat eucrp ijnitic of tbat Qubes^ 
fquare , 10 ni it fclf a Cuhe : as bp tbe figures tljat fol# 
loluc,pou mate locll coniecture. 

anDalfotljciacemaDcbp multiplication of a C"- 

hil{enomher,ti\\XidiSquareno\nbertQ^Ci\)Zt -, bot!)C I)a^ 

upng one rootc : ana tl)c SurfoUde ^aupng tbcfamc 
roote* (22ai)crfo,:e reafon luitb tbe nature of tljcic 
founoe figure, info;cetb me to call tifcfyuares of cubes. 

19 et otljer menne attcnDpng mo:c to tl)c nature of 
tbeir rootes , tbcn to tfteir olune formes anD nature. 
Doe giue tbat name to tbe nerte reluc of nombcrs, fai^ 
faufe t\)ti mate be maDe of multiplication, of any Ca* 
hiksnomber bp It felfjtbat IS to faic fqaarclp. 

^Cbolar* 3|t is fo. i^0?.8, lubicbc is a Cnbi\e nomler 

multiplteD fquarclp maUctli,6 4. ^nD tl)at.6 4. ts fet 
emongeftC tbC Squares of Cubes. 

smaller, j^notbts commoDiticcommctb bptbat 
nameubatitputtetl) menne in remcmb.:aunce of tl?c 
fpeDie anD ealie extraction of tbcic rootc: ^s pou fljatl 
learne bereaften 

l!5ut3j confiDerpngtbcir otune nature anD ma- 
hpnge, as founDe nombers 0; boDics : Doe call tbcim 
Cuies 9fCubes,oi O^bih^ Cubes. 

after tbefe nombers m tbe feuentb rctoe, tbcre Do 
foUoVue tbofe nombers, iDbicbecommonlp are callcD 

hfurfilideSj, 



The feconde parte 

Secentte tfur/olt^es^oi tiJfmfoUdes, i\\Vi.t iS-,feconJefurfiliJes,ti} ilout 

furfolUes i'^'firfolideF. B Ut 3 male call tX^tmficondef^Mres tf cut 

•' * ^«,aUutirngattl)0famename,!^oU)bettif3iloobeto 

ttjcir fo?me aiiD nature,^ i^all 6c info^ceo to call t^e, 

Ung<i cubes 9fcubes,Qi Unge cuhtkt cubes. 

ano fo bv Ufec reafon,Doe 3 cal tbe ncrtc nombcra 

fiudrj cubes of cubes, Qif^Udre cubi^e cubes: tul)IC^e otl}CC 

jLarea redfiuares. ^ ^ ^ ^ ^ , „ ^ 

Cubes'ef Cwi/Jt' G/'W^O? C^i« of Cubes' btcaufe tl)C Cw^i^f rootCS 

Cm^w. of tfjofc nomfacra aic Cubike nombcrs affo. But ^ af- 
tertl)ctr true nature, coc call tl)cni Cubes of Cubes Cubi. 
hely.Ql Cubes otGibi\e Cubes, 

Squam of SDfte tcnt^ rctue of nombcrs f0 namcti tulgarclp, 

slrfolidL 5jK4/« offurfolides , btcaufe t^et l)aue a Square rootc, 

* lul)tcl)e 13 or it felf afnrfolide nomber. 0iiD fo? t!)eir ft? 

gure Crometricalle, 3[ name t!)P /ow^ fu^f5 ofcubi{e cubes. 

^0 ttjat i confioer^ng tftetr nature,t!)at tl)ci be ft* 
gurallc uomber0,am conftratneu to nante tljctm, ac^ 
co?Dt?ng to tijetr figure,33 meane in tbts place, U)l)crc 
3It!oemafeee)cpltcatton oftbeir natures auD names* 
l^ut otbecmen fo^aioe of tooo;Ue,tn ertraction of 
roote0,l)aue gtueu tbeim focbe names, as mate befte 
put menne tn rememb^aunte of reDp too^fee tfierem. 
Wl)icDe names 3 totU tjfe alfo hereafter, in mv y33iu 
ti?nges, btcaufe 31 lutU not bee an auctl)o;j of tmneoe-' 
full fingularttte* ano vet btcaufe trutlie tn nature ts 
as luell to be regarDeli,as eafe m Ujoojk^ng, ano ra^ 
tber mo?e,3l coulD not omttte tn tbts place,tbe oecla^^ 
ration of tbetr true nature ano tjerp fo;jmes» 

0nD fo botbe of ijs baupng gooo reafons, fo; tbofe 

names , nettbcr mate contempne otber, neitber con^ 

tenDe togetber» 

j'generalie 2inn altbougb tbe names tljat 3? Doe glue -, male 

N^onfor n4 feme to fomc menne c M)f)it¥ are fcarfe apte tuDges> 

mo^ 



0f ^rithmetike. 

wo:c otitoufe , fo? tbc nctwe inuention ( as tW matt m« «/fi^f/( 
tl)inUt;tl)en ncDcfuUto tfjcpjamfc of tliartc:fetfl)al „()ff;^ff>i 
vou fee in tbcim a naturall fequcl^ , ano o;ti£x\v p;o^ 
pagattoiT. 

fo} all tljofc nombcM are confiDercU, in one of.:. 
fo;mes firftc.3:tjat is to faie, other tljti Lcc taUen as 
nomUersabfolute^tuitljoutant'ecfiDcraticnofniul* 
ttpluation.^no fo tbei inaic be nameo iiomfarrfi one* 
lr,\uitl)out name of relation. iD: els tljri bee confiDc- 
reD as nombers multtpUcD, anD tbat ran be but uu > 
tartetiec . 

3lf tljei be multtplied but oneSjtben Doe tbei niabc 
a line of nombers,o; a linlarie nomber. 0nD tfjat no- 
bcr l)atf) onelv lengtbe , luitftout b^ebtbe, o; Depthe: 
^no tl)erfo;e male be tbe roote to a Sjuare,jO^ a C«^<?. 
15 ut is of It felf J mtbatconftocration, not!jcr5^<wr* 
no; Cti^e. 

&cconDar(lp,(t mate bee multlplietj tluiTctbe one 
nomber ftaOf ng fo; tbe lengtbe.anD tbe other fo; the 
b;eDtbe : ano fo i& it a S^ttarenml>er,dinti tl)erfo;c aflat 
nomber. 

Sri)trblp,tt male bee multiplieO tb;ife, anD tberbp 
0ettclengtbe,b;eDtbe,anDDeptl)c:Ujbcrbr it is maDe 
^ftunde number. SuD btcaufctl)c fiDcs bceequalle,it IS 

fpeciall^ a Cube o; Cnbii^e nomber. 

^oU) can there be no foluerth iuate,that anp mul s 
tipltf atton mate increafc:fo; there are no mo;e Dime* 
ttons m nature. 

aSut If anp manne Doe multiple the fourthe trme, 
then mull he accoumpte that he maketh a Um of Cuba- 
anD the fifth multipliration maUeth a square, m itht^ 
Che euerp tinttie is a Gbe: feo the Crtc multipliration 
mafeeth a Ghe of Ctt^M^^ictoumptrng cueip Icffer Cube 
tQl m tjnttte, i^nD there is a ftaie again. 

wa herfo;eif anr man multiplte the feucnth time, 
\t retourncth againe to the firlte nature of nombers 

3i,|, multiplicD, 



The feconde parte 

tnulttplietJ,tul)(cl)eace/mMWffno»»^w:amitl)e8»mut« 
tipUcattoit, iDOo^Uctb as tbe fcconoe DiD,anti maker!) 
Jiatte nomhers. %\iz ntiietb muttiplicatioit agrcablp 
iuitu tbc tt)icD0,ooetl) maUe Giles. 

aito fo manitclp tbefe* 5» luoo^fecs maie bee reite- 
rate, but a fourtbe fo^me can wmtt be DeutfcD, 

0110 tf)erefo;e Doe 3, as reafon Doetb c ompcU me, 
reduce all uombers to tbofe* 5» formes, as tljeir becie 
o^tgmalle rp;ji?nges ano fountatnes* 

Buttotbe iiitente tbatpou maictbe mo;ie aptl^ 
tuDge of tbetm , ano tijcic natures! , 3J baue ftere fctti: 
foojtbe tbe fo;jmcs, Uibicbe t\izi make in figures Geo^ 
metrical le,<ii founoc quantities. 0Dmoni(livug pou to 
remember ti)is toell * SCbat after anp nomber is be- 
come a fountje nomber,it is agamll rearon,to reDuce 
i)tm to an abfolute flatte nomber again, anD moffc of 
all br multiplicatioH. But noUj marUc tbefe figures^, 

llootes,o,:3LiiTes» 

4» ?» 2. 

6. 5". 



1.1* 



?»3. 



^ EEB 



of Mtthmzuh* 

S^Mres, 

V 



4 



■1 ■ . ■ m 

_j 

■— -— — — > 



5-0-. 



16. 



^n — i — m 

-rr-f- 

' ' iii M ii l ■ i l l . 



2^ 




lon^f Oiler. 

2,2.2,2. 5.5.5.5. 



^rvxx\ 





-w, P-. 


■1 ■■>-' 


- 


- 


• 


Tr», ' 


1—1— 


1 1 



81. 



S{u4Tn 



squares o/Cuhs. 





2»2.2.2.2 


» 


V \' 


X 


.X 


,\ 


N 




1 




\, ^ 


- 


1 


. \ 


... j 




N 


' 


— 1— — ■ 



3«3*3oo* 



C*iih Gttes, 





\ 



2»2»2.2»2«2» 
■r 



3»3o»3o'^ 



^ 





^cvt 



of Artthmetiks* 

^ere,a5 t'ow ^^^'3! ftauc fct firft certamc imcfl^conr 
taiiipng focljc partes as tl)ei bee maoe of bp multipli- 
tationttbat IS to fair,:. v4.o^5', aiiO tbefc bee p^oDii^ 
£CD bp tlic ftrttc inultipltcation , tuljcce an taitic of a* 
n? tbpng is multipli:D b^^ a nomber, 

jano fo an tncbc multipUeD bp.^malietb^^incbes: 
0n^ a footc multipUcD bp.6. mabcti). 6. footcianD To 
of otl)cr mcafurcs anD quantities , m liUc fo:te, au 
tobicljc multiplications, Doc maUc onelp longc lines, 
o; mcafurcs m Icngtljc onclp, Ujitbout b^cDtb o; tb tc> 
kenefTe. 

anD in tbis multiplication , notbcrtbe nomber, 
notber pet tbe bnitie, is accourapteD o; callcD a roote, 
But tbe line tbat is maoc tberbp : maic bee a roote to 
anp of all tbe otbcr nmoe of nombers bcfo;u^ reberfeD, 
anD fcttc fo;tbc in tbe table. jFo: if pon multiplie tb<J 
fame line, bptbcnomber tbat bis lengtbc Doetbin; 
duoe, tben tbcrc loill be maoc tbereof, br tbis fccoiiDc 
multiplication, a fquanc figure, containi?ng a fquarc 
nombcr in it : as pou fee emongett tbofe figures, tbe 
firilc foluer to be,U)bicbc are marUeD luitb tbcfe noms 
bers.4.9.i6.anD.2 5', 

S^cbolar. s perceiue iueil in ecbc oftbf , tbat tbe if 
lengtbe is agrcablc U)itb tbeir b;cDtbc , ano fo rbri 
make fquare figures, butj kjiolue not lubat tbofc 
nombers Doe meane, ttjat be fct oucr tbtir bcDDrs. 

£03tler. SSbe quantitic of tbe nomber , Doetb bi*^ 
tob£n tbe talelue of tbeir roote. HnD tbe multituD? of 

tbcramenomfarrrepcteD,Doetb Declare tbe nombcr of 
multiplications, fo;jcf be figure. 

0nDtbercfo;eti}e lines, lubicbe arc maDebponc 
multiplication, bauc ecbc of tbeim tbeir nombcr Dm. 
plpfet,onesonclp. 

%\)t fquarcs bauc tbeir nombers Double: m toUrn 

tbat tbcibaue. 2. multiplications, sibat IS , one in 
lcnigtbe,anD an otbcr in b;eDtbc. 

3?»<ti, Cbc 



The feconde parte 

«e t!)<rO fojntcsf, tul)tcijc be C^hes^ ant> are mane 
Df» 3. multipttcattoniSjtiauet^etr roflter0peteDtD?lif0. 

^iiD tfje Itfeenotnbers Dtd 3 fette , tn tbc fioe of tlje 
former tabie,agamfttl)e lifee ^mtitm.m l)u!)eCt>aU 
Ijelpe ^ou fomeluftat tn tlje crtraction of rootcs, 

^(Dolar. ^ototJocBipmctuctiotond? tbeirna^ 
nres^antj mulnpUcations^inocfte better t!)cn 3: did be^ 
fo?e: but alfo3! tjiiDerftanDe better t!)e Difference of 
^our name!S,anD tl)etr rcafong.^oj bi> tbofe filgure0, 
iDl)icl)e ^o« Ijaue ret in tlje foluertb place,anD Doe call 
tbetm hnge Cubes , 31 fee ttjcir fo;mc Doetb agree to 
tbatname* i^onbei are longer, tbcnttjei are otber 
b?oDe 0} Depe* anD faue foj tbeir Deptbe y 3i migbt lU 
^entljtim to/o«£eS^«4mtn Ceomttrie.^^oixjbcit-, Of 

tber men neglea^ng ttjeir fojtnc , ano looking onel? 
to tljeir rootes,Doe call t\itm,Squarc^ fquares. 

IButtf^ouluiU perntitteme , to fpeaUe m tbe Dc^ 
fence of tijeim, as a fimple fcljolar mate fpeaUc fo; af? 
fe(tion,«t tfte Defence of l)ts niafter^it apperetb to me^ 
t]battl)et mate tuell beecalleD fquareD fquares ; ano 
mtgl)tbefigureDt!)u0. 



2«2*2»2« 



■» •» 7 ■» 





j 











: 





16. 





i 




'■ 










\ 
















■■— 1 - 




I 





— 






1 












■— ■ 




1 ^ 












^ 







SI. 



Zimhttt tbe fmalleft fquares, tuljtclje becontatneo 
luitljin tlje p^icUeD lines? bc^ng taken as rootes,anD 

muUtpltco 



of Ar'ithmetike* 

multipiicrj bp tbcfamo noinbcr ngaut.tuljkljr x\)cx Do 
contamc; ( otl)ci* cl6 tluifc faj? tfteic rootes ) UiiU malic 
tl): U)l)ole greater fquarcs. 
ano Up ti)t3 figuring of tl)eim,tf)cr?DoctI)appcre 

no mconuciticiue no: abfurDiticm tbcir tjulgarc na= 
m:s : but ratbcr a mac crp^cCTpng of tbcir naturallc 
fo^m:s» 

jf 02 hi tbe fird figurc.z.ttanDpng as tl): fiDc of toe 
IcCfer fquarcaiio miiltiplieo bp it fclf^occtb malic. 4. 
lul)icl)ci3tlicquainiticoftbc Icffcr fquaix. C^bcnif 
3 multiply tbat IcflTcr rquarc.4:bp l)is olunc nomber, 
it ma«;tii 1 6.U)'itclic is tbc grcatc anD lubolc fquare: 

anO IS a Sqiare offjuares. 

fe)o in tJ)c fcconoc figure, ^.aan&ctb fo; tbe roote of 
tbc IcflTcr fqtiare, contamcD luitbin tl)e p:ickeD lines, 
anO If It bee miiltipliro bp it aif,tt maUctb.9' tubicbc 
istbequanttticof tlicfamc Icller fquare. SSlicnifjB 
multiplie tbat.9.bp it rclf,it luill make.8 i.lubicbe is 
tbequantiticoftbegreatc Square, m^i^ a square of 

fquires. 

Scatter. 3: commenDe pou luell : not onclp fo; fo 
DtUgcntccrcurpng of tbcim, Uibicbc fo: tbcir boncttc 
trauclU DrferuemocbctbtlnUcs, butalfofoUbatpou 
feUetob;pngmanifcftrearon,anDfomcfi)clue,attbc 
leaft,of Unearic Dcmonaration fo: pour purpofc. S>o 
tbatpou tuiUnotfcmcto fpcaHc,U)ttboutrome gooD 

grouuDe. 

But as in DceDc , pour figure uoetb truclp crp.zefTc 
a fquare of rquarcs,ro it Doctb fuppofc tbc otbcr nom-- 
ber,tubtcbc hy^ o:Dcr of multiplication, Doctb go ncrt 
bcfo;jc it, to be a flattc nombcr alfo. f 0; it is not pof^ 
fible tbat a founDe nombcr (as a Ctihe is altuaics ) be^ 
jngmultiplicD bpanp otber nombcr, mate Icfctbe 
nature of a founDe nombcr : 1i5\xt fljall continue a 
founDe nomber ttill. anD tberfo^e fcepng tbe nerte 
nombcr, bcfo:c a SGmre offyums Ujas a Ct^^ot is not 

poflfible 



Thefeconde parte 

poflTtble tl^iHitLSqusre offqMrn can ht a mtttflatttnmc 
hr^ ]M)u Ibaue D^tuett it 

tlionlo oc(u)ite t^e .4 .place,t4en l^oulD tbct bane fet 
(omtpUtfimt tn tt)e tbtrb place alfo* ?l2Ilbi(be mtgbt 
laue been mabe in tbts ro;te. 
0110 tben lotU tt be a Un^e 5^«urf,anb not a Git, 



3o.3. 



■I -• ; ■.. . " .... 

—-J ' ' 1 ^ 1 



IBtit fn as moebe am tbe( boe not atimittt this longt 
suture ( tobicbe \jv tbat name batb no roote ) tbf rfo;e 
tnaie not tbe nomber tbat folotuetb tt , face anp otbrr 
tben afitbide nomber. fQ% euecp Cuhik^ fojmcbf Cf ng 
ntulttpiteb b^ bus roote, boetbmabea s^wire ptlUr, 
mWt lengtb bearetbbnto W b;rcbtb tbtfante p;o^ 
po;tton,tbat b<0 roote boetb bnto an bnme« 

fecbolar. 3f am berp toell fattffieb noU):c6ccrnfng 
tbe name0 ant) fo;me0 of tbofe nomberg, ano hv tbt0 
tbat^oubauefatco, 3 toe fartberprrcetue, tbaty. 
multiplications Doetb mate tbeyju^r o/c«i«,U)l)icbc 
be fct in tbe fefte place,emongeac tbe former figures. 
sUiB alfo 31 tnberltanbe b? tbe former table, tbat tbei 
hecallt\iSurfolnies. 

iliUetsaics 3 fee in tbe tirteplace of tbe fo;ifafeb il^ 
grure0X«^'t* c«J«,ma5e bi\6. multipltrattong^liSut 
commonly tbcnombcrs of tboft quantities^be nameD 
Squares ofCuhts, fee tbat fo; tbeir names^tbns farre ^ 
amperfecteinougb. 

Cbe 




The extratlion 

of'J{dotes. 

£)lj)e totU3 a^Ctoe mUne extract 
l)otu f 0" ^'^l crtract tfjc rootc tm o/rtttes 
outofanvfof^cnombcr. 

^iiD firtt 3 ittuft amitonifl)e 
pou,tbat rou i^al altuatcs tjii 
DcrftanDc,focl)C a roDtcJas tl)c 
noinbcrDoctl) aDmtt. ^o tbat 
in a fquare nomfacr, pou fljall 

__^ fcbc a 5jtt4re roo/tf oncl\f , auD 

110 C«^'t<^»'oo/?,notl)cr an^ ottjcclstnoe, 

i^ibcluates a Cuhikf nomher Ijatl) no otfjcr roote> 
but a Cuhike note, crccptc t\)t namcbrc conipounDc, 
as ;<ew;5^cw^/)^f,o? Squdrtd Cube, foi ix\ focljc t\)cvt are 
2,fo;tc$ofrootcs,acco;jDpngtot!)C2.nantrstl)attl)ct 

beare. SCljat is botljc 5f U4rr ano Cw^/^.^ footc : as 3 
iDlU anon (i^cUic vou. 115ut firttc 3; i«tll bcgmne tuiti) 
Square nombcrs,antitbctrrootcs. ^j m r 
anotbtsgcncrallc o;jDcr mullerou J-/^^ *^«^**^ *!/ 
obferue , bcfojc all otljcr i patron /f«^''f '•''«'">"' 
^all fjauc bv I)artc, in rcaotc memo- compounds 
tic all focbe nombcrs , iut)ofe rootcs rj^ootes. iSguares 
arc Dtgttcs. f o;t as tt ts fupcrfluous 
to kU rules fo? tl)cim , fo mufttljct 
l)clpc in all greater nombcrs, tul)ofc 
rootes arc abouc 9. ano fo J roureafc 
In rcmcmb2auncc , 31 Ijauc l)erc fette 
foo^tbea table fo: fquare nombcrs. 



I. 



I. 



y 



4. I 16. 



Xm Ijerc in t\)t firftc columpncro" ^^ 
tbe rootes fct, anD in tbe feconoc ptl 
Icr, rtgbt agatnft ccbc rootc, tberc is 
fet bis fquare . Couebpng Uibicbc 31 
ncDc to fate no mojc, buttbat^ou be 
mt in unv tmcettatntic of tbem^tubf 



6« I 36« 
A I 49- 



<S. I 64* 



%}^ 



von 



The extraSlion 

pou a)aU nc DC tlmv ateo, lu!)tf l)c Ojall he contutuallp 
in Ijfc of fcarci)VMtg fo; other greater rootcjs* 

jj^oU) fo; greater nomberc,tl){6 is tl)c oMt, 

!♦ i^irftfct Doimetbcnoniberaisitts. s^bcnfcttca 

p;icUe tnDcr eiierp oDDe place, 3 meane tlje firOe^tbe 

tl)irDe,tt)c ftftC)tl)C feuentb.anD fo fo,JtI)c:anD fo fljaU 

cucrp p;icUe baue.z* nombcrs,e]cceptc tl)c Iaifc,U)l)i^ 

c\)t fomc ti'mes t)atl) but one* 

2, ^eeonnarilv, marhe tbc nombers tbat belong tn^ 

to tlje laftc p;jtcke, totuaro tbe Icftc bandc : ano lo!)c^ 

tl)erl3C?)auc belonging to (tone noinber,o;:tU)oo, 

looUc tubat tl)t roote maie be of tbat nomber,if it bee 

fquarc* ^iiD tbat roote fette bp a croUcD tme, aa pou 

place tbe qnoPienu in Diuirionri cumdl all tbat fquarc 

nombcr,belongvng to tbat p;ic|je« 

5» Butanuiftbcnombcr belongtmg totbatp;icUc, 

htt not a Square noniber , tben take tbe roote of tbe 

greatefle fquarc, iubtcbe 10 ccntameD in it,anD place 

tbe roote as 3! fatcD before* 3nD tbe fquarc of it lljall 

vou abate from tl)c nomber , tbat belontrctbtotbat 

laftc p acRC, anil let tbe reft be fct ouer tboYc nontberB 

ianceUcD,a0 poii Doe in Dtuifion. anD fo banc pou cn^ 

DeD pour Uio^Uc fo; tbat p^ic^e, 

^cbolar. SlCijiB mocljc i$ calte inougb,tf 3; tiiDcr^ 
Canoe pourigbtlp, 

Rafter* SDbcn pjoue it in a nomber , 02 ttuoo* 
anb firft tuo?Uc tuitb tbis nomber. ^ i >- 2 9 o o. 

^cbolan ^muftemartic eucrp oDDcplaccluitba 
p?icUe,tbu3» 

anDberegipercetue tbatbnto tbcfirfi 

5* U 2 9 o o, p;tclie,tl)ere belongetb 2 Cppbers one* 

' *• * * lp?anDtoccl)eoftbeotber»2.p;icttes 
folotDrng,tbere are appointteD, 2 » figures* B ut tf)C 
fourtbc p^icUe batb but one nomber,anD tbat is.^, 

0oh3 accojDing to tl)z fecouD rule,3i fekc tbe roote 
cf )% (fo^ bicaufc tbere bclongetb no mo^e nombers to 

that 



The extraSlton 

tliatpj(f!jc)an03( fcc-,itiisno fquarcnombcr^ttlljcr^ 

fo;c acco^Dpiig to tl)c tl)troc rule, 3 tahc tbc grcatcac 

fquarc in it, U)l)ic!)c 10 . 4, anD tl)c rootc of . 4 , ts, :♦ 

S^tjcito^c iJ Doc fubftractc.4,outof.5-. i 

ano caiucfl ttjat. v anD tbe»L tliat rc^ v i fz 9 o o (:♦ 

mauictl),3i fct oucc. vrts Ijcrc vou fee. 

anD tl)c rootc* :. 3 fcttc bcljntDc tl)c quotiente Itiicas 

VOti tang!)tmc,anD tljcit t):\i: nobrrs ttaiiD, as rou k. 

£l)allcr» IP on banc Docn Uicl. p;ouc again m tl;i3 
nombcr, iS 76 6 224. 

^d)olar, J^irll^rcttfjctmDounc 118766224. 
anD p;if Uc tbcinr, as l)crcDoct!)ap' | • • • - 
pcarc. ano nolu 3 fiCtbat tbc laftc p;zicbc batO tluoo 
nombcis belonging to it, tbnt is. 1 8. U)itb lubicbc 3I 
niiift begin. i^nD fe^ng it 10 no fquarc nomber,3l finD 
1 6.to be tbegrcateft fquarc m it:lul)erfo;c3 fubtract 
i6.outof,i8. anDfet2.oucrtbc.8. 2 
janDtbc rootc of. 16. U)bicbcis.4. j 8766 2 24(4* 
3ifcttcbcl)inoctl)Cf«o//V«/#lutC7a0 • ♦ • • 
berets fee n. 

05aacr. %\m mafc fuflficc fo: tMt fird Inoo;I{c. 

jjiola to p:oceDc, p" fl)aU Double vour rootc, anD 
put tbat Double tnDer tbe ncrtcfpacc , totuarDctbc 
rigbt banD,tbat is bebniDc tbc nertc p;tckc. jailDaic^ 
fo^fcrng, tbat if tbc Double Doc contain mo;:c figures 
tbcn one, tbat tbc firft ^a\\ be fcttc tjuDcr tbat place, 
anD tbc feconDc bnDcr tbc nertc figure , totuarDc tbe 
IcftcbanDe. 

Sl^bcn fetjc a juotiente, as pou Doc in Diuifion, tobi^^ 
tbc(l)alia)ciuc boU) often tbat Double nombcr maic 
be founD in tbat,tbat is oucr it,appcrtainpng to tbat 
place : U)btcbcf«ff^/Vwff ,pou Iballfct before tljcfirftc 
rootc, U)itbin tbe quotiente imc. 

HBut tbtsregarDc amine ^ou banc bcrc fpcciallv, 
tbat von maic Icaue oucr tbc nertc pjicUc, toluarD tbc 
rigbt baiiDc j as mocbe as t\)c fquarc of tbat j«w#/f«/#, 

11. n. UntI) 



I 

nT2 9oo (2. 



The extraSllon 

tuttl) tu!)ic!) ^ou luo.:Uc,fo;i out of tljat rcff,tl)c fquarc 
of tbat qmtiente muftc bcc abatcD . ano tijen niaUe 
botbc fubtractions^attD note tbc rcmamcr,(f anp be, 
ano place I'our iU9tia\t, ano tben baue f ou Docn Uiitb 
tbat p?tckc alfo* 

J^o; tbc mo^c plaiitc0,3 luiU giuc i^ou an crample 
in Jouc firftc nombcr, tutjtcbc ttooDc t{)U3,aftcr j^ouc 
U)o?Ue iuas enocD* 

Xt^cre 3 'ce oucc tljc laffc pjicfec 
fau'c one* 1 1 y. tiiDer tbe miDDcU fi^- 
gure of lubicbe 31 mull fct tbe Dou^ 
bic of tbc fo;mcc roote* 2. tbat t0« 4, ano tbcn }[ feke 
bolu oftem 4.10 to bee fouiioe in. 1 1. 0nD 3 finoc tbat 
5 matebauc ittluoo tpmc0,arttj.?. remain j?ng.CQlbt; 
cbe* 5, tDttb»^oucr tbe nertc pjtclse,tioc maUe»3 v nnu 
tbat ijjmojc tben tbc fquarc of m^^Mo^«»^r. 2» %\^zx* 
fo^c am 31 bolDeto fctteDoune tbat 
^mtimte\':^m acco;Dp"0 fo ^^jto a^' IJ J 
batetU)tfe»4.(iubtcbcf)3.80 out of 
I LanDtberereftetb»3.^berfo?c3} 
cancelLi Lant5fette»3»on^t^it snben0oc9?muWpiie 
tbelaffe jwo^i^w^^fquarelp: anDitmafectb. 4» iubicbc 
4*3} fubtracte out of tbe nombcr oucr tbe p;icUc, tbat 
I0» 3 5'* iDberc, 5'» nraie fuffice fo; tbis nombrr» S^ber^ 
fo^e3(abate.4.outof»^ anDcanfcUtbat.j-, anofet.i. 
tiibtcbe remainetb 5 oucrtbe* )♦ 
^no tben tuiU tbe U)bole nombcr 
ItanDetbug* 

3Cbw tuoo?lie,tubtcbe3?^auc 
lD?ougbt notu , muft be rcpetcD as often as tbcre bee 
anp p jicUcSjOa pricked nombers remaining. W. \)tv 
bp ^ou male eafilp geflTe , tbat it mutt bee tlulfc mojc 
vepeatcD in tbis erample, bicanfe tbecc rcttctb ret* 2. 
p^icbest)ntoucbet>, 

^cbolar, Tiitbougb 31 tblnfee,3l coulotjoe , 3031 
banc markeO routo tJOC:, pet foa mo;c ccrtaintie3! 

paate 



li 

fM129oo (22, 
4 



MI 

^ i f 2 9 o o (2 2« 

• . • * « 
4 



of^otes, 

^i:kic pou 1uo:1jc out tfjis cvainple* 

q^attcr. Sinijcn marke 1 1 UicU» 
3;(^all bcgm agaiiic luitl) Doublpitg of all, tbat is 
tuitljm the quotitnte \\m, aiiD tljat Double IS 4 4.U)l)i.' 
tbc j! mult fct UitDer. ? 1 2. tOat remamctlj of tijc lalic 
luooikc. i^nDtl)cnU)iUtf)cnonp I 
bcrsaauDc,a3l)crepourcc» 'cxc-^oon r^. 

SLbcnjIoUcboUjoftcittpmcs l-v;_r. • ^"*^* 
mate J ftuDe, 4 4» in. 512. ano 3} j "^"^ 
fee It UjiU be abatcD 7 tunes, anD 4 rcmamttubicbc 4 
Uutl) tl)c. 9. ucr tbc uert p;jicUc Doetb maUc. 4 9. 0nD 
tbat luUl fufficc to crtracc tbc fquarc of mp qtiotknte, 
7.j^o:.7.ti'mcs.7-inaUctbtuftc.49. S^buafcpng j| 
ntaic taUCw.fo; \\\v qtiotiente^j U)oo.2kc luitb It, as tbc 
rule teacbctb:abatuig ftrft.7.tuncs.4 4. (tbat is.^oS) 
outofo 1 2. auDtb:i'erclIctb.4» ouertbcfpaccljcfoic 
tbenci-tcp:icUc. Gabicbe.4. U)itb.9.oucctbcp;icKc 
DoemaUc.49.outofUjbicbe3}a- I - 
bafc tbc fquarc of mp^«9^/e»^^.7, U^^'?^^^ .^^^ 
(tbatis.49.)anDroreactbno^' r-'^Iy^. ? ^^^^ 
tbpug,but.2.<i:rpbers» 0nDtbc | ^^' 
iTomberftanDctbtbus* 

aito fcrng tberc remainctb one pjfcbc tntoucIjcD, 
31 n)oulo repeatctbcfamco^Dcr of tooo.jUeagatno, bj? 
Uoublpugalltbe quotiente, UibicbcUjoulD bee. 45-4. 
auD fcttj-ntg it fo tbat. 4. iufticbc i^ in tbc ftrfte place, 
(l)oulD be fette tjuDer tbc CDppbcr,tbat is U)itbout tbc 
pHckcanD tbc otbcr figures iit oiDcr,toluarD tbc left 
IjauD. I5ut all tbis luo:kc lucre m tjairtcfci^ng tbcrc 
is itotbpitg leftcto fcrue fo j tbc fubtraction. 

mi bicaufc tbcrc is Icfte one pzickcD place tjittou - 
cI)co,3? mutt fet fo; it a Cppbcr in tbc qmtiente. 

fo; tbis rule is gcncralle:tbat boU) manp p;icUcs 
fo eucr vour fquarc nombcr Doctb rontame, I'ouc 
^uitiente , 0: roote fl^all bauc fo manp nombcrs. 
OTbcrfo:c tbis rootc muft be maoc tjp tbus. 2270. 




The extraStlon 

rhcproofe. 2nt! fo it appcarcti) tijat rour nombci\5' r f 2 9 o o« 

<0 a tulle fquarc nombcr, C^ I^ictjc pou male p;ouc bp 

tUc ojDcilp pjoofc of crtrarticn of rooteg. 2Lljat 15 to 

multiplictl)at quctienu,o;toott ( tul)ifbe?ou!)auc 

rounDe)bi? it felf. antj if it cioc maUc tbe firtt nouibcir 

f ractlp,tl)cn tjauc rou lu;:ougl}t IdcU, 

^cl)o!ar. 2El)at p;oofc is as rcr^^ 

tamcasran be. anDtl)crfo;c33 tuiU 

p;iouc, tul)ctf)cnt lotU agree toitl) 

ttji0 ViioM* mi!)ecfo;cmuItipIifng 

2 2 7 o, bp It fclf, 3C fee tbat it pclDeti) 

tlje firftc fomme.0fii bete it ooctb ap^ 

peare. ^0 (stI)i$icoD;ljeapp;ouco 

gooD. )^r-9oo 

0nO!ioUj tutU3l attcinptc tbc lifecU)oo;Ue tittbc 
feconoc erample*m!)tcbc luas. 18766224. 

5i?ut after tbe firaclDOo^be tuaseuDco , anotbc 
greateft fquarc Cubtractct out of 
1 8»it DID re main in tbis fojnie. 

jjioluto fDnt4nuc tbc tuoo^Uc 
as pou UtD, aiiD as tt)c rule Doetl) tcacbe, 31 mutt Don * 
ble, 4. U)btcl)c is tbc roote, anD flanoetl) bv tl)c ^uoti- 
r»^dine:and mull fct It tJiiDcr. 7' tbatlTanDctb m tijc 
rpafe,betlucne ti)c lafic p^tckc ( Uibofe Uioo^Ue is cn>- 
DeD)3nDtbenertep?iclie toioaroe 
tberigbtbaiiue. janDtbentutUit 
fiaime tbus as pou fee. 

2Lbat Doen,3 mutt fcUc a ^uoti< 
enpe -, tfjat male Declare bolu often 
8. male bee fubtraitco out of. 27. a'nD tbat qmtUnte-]^ 
fuiDcto beo:bicaufe tbatafterj baue taUen.^tvmes 
8. (tbat ts.2 4 , out of. 2 7. tbrrc luill remain. 5. lubicbc 
V Ujiti).6.tbat 0anDetb oucr tbe p^irUe, Doc mabe.^ 6. 
0uD3ifectbatnombcrtDbee greateinougb i fo;tbe 
abatemtnte of tbe fquarc of \x\^ (imt'mtt \ lubicbe \B 
biit.3«tpmefi.5»tbatis,9« 

tCIbcrfo^c 



i>^ 766224 (4, 



i;S766 224(4. 
•8* ' ' 



iS:7r>6224(4^ 
' S ■ ' ' 



of^otes. 
CCll)crfo:e J fcttcoounco. 

fO.J m\> (juttientc, bcfO^C . 4 . UT 
t'OC quotiente luiC. ^IID milltl < 

plipng S.bv' tbat :;.tl)crc nfctl) 

2 4. U)!)irl)c 3; Doc fubtiatt out 

of.2 7. tljiU 15 oiicr.S.aiiD tijcrc luiU icniam. > E^ljat 

?»U)ttl).6.ouci-tl)CiJ2icU:,ma!jctl). 56» outof luljicljc 

3 mult iibatc.9:Ujl)icl)c is tbc rquarc ot'niv quotient, ;» 

anDfo U)iIltl):rcrcUc.2 7.oiHTtl)atp;iciu\ 

0nD ti)ii3 banc jf cnDcD. 2.p;:icUc3,anD vcl'.:.Hio,:c 
DOC remain: in tubtfbc botljc ^ imift icpcatc tbcfamc 
fo;mcofU)oo:Uc. 

2lDbcrfo:c 5 Double tbc lubolc qmtknu , anD it ina- 
lictl).S6: lubicbc > frt bnOf r.2 7 6. 

niiQ tben j3 fckr tbe qtiotienteMd^x\>\\^ bolu inanv 
tpmes. SY). maicbcabatcDoutof. 276. lubicbcmatc 
bc.vtvnTC5.;ilnDfo;tbatcaurc3J fct.j, \\M\)t qn^tiente 
Jbefo2ctbe.4^ 

Cbcn Doc J firttcmult^ 
plicS 6. bp tbat. 5. faipng. ], 
t}nncfi.s.nTaUctb.24,U)bicl) 
3! abate out of.2 7.anD tbcrc 
rcltctb. V ianD again 31 faie, 
^.trnTe0.6. 13. iS. lubifbejf 
abate out of. ] 6. anD tbeix Doetb remain. \ S. 

iTbat Doen , J take tbc fquarc of m)^ quotiente, tbat 
(0.9. lubicbejCDoefubtractoutof. 12. aoUbc.2.oucr 
tbcpnclic muft bo:otuc. i. of. s.; anD tben luill tbcrc 
remain ouer tbat p:ifUe. 1 7 ;. 

jHnD tbus IS tbat p:ichc cnDeD. 

j^oU), fo2tbeIallep;iclie m Uioo^Ue, tbougb be Oc 
firtte in place. Cbc Double of mp q»otientew,'S66* 
lubtcbe jj muftc fettc t)nDf r 
1 7 52.;Hsbere is Docit.lubcrc 
3IIcaucout manp canccIleD 
figures, as fupcrnuoufetn 



I 



I 7^ 
i;y:7r^,6^2 4(4H» 
* ■ 866 ' 



tbiji 






The extraSlioH 

tljis place. 

<ano tt}c:t fclipiTg fo;r a nciuc quotient e, 3| ftnDc it to 
be.2.1ijfttct)c 31 fct luttl) tlje otijer nombcra m tijc f tt#^ 
f/w^r.^no bp it 3! multiplic anti fubtract tt)c 8 6 6,fai^ 
j7ng;2.tvnneis«8»ts. 1 6. loljic^e 3i abate out of» i y.ano 
tftcre r0ftct!ji.0gauT.2.ti' 
nteie; 6 is* 1 2 tl)at 3 fubtratt 
out of* 1 5. ano tbcrc mnai^ 
itetft.L Wirolv, 31 faic.2. 
time0.6»gtuctl).i 2.tubtc!)c 
31 abate from. 1 2.ano tbcre 
lis left iTotl)t?ng.^aue tbat oucr tf)c p;if be tbere (Ian- 
Uetl) 4 tuljicljc tj3 equall toitlj tfte fquare of mp ^uotiet. 

tiail)erfo;c abating tbe fquare of mv fUftKnteout 
of it,tl)ere rcftetb not^png at all, 

ano tljcr bv 3i fee tl)at. 1 8 7 6 6 2 2 4. is a iutic fquare 
nemher. ^nO W toott i0.4 3 5 2. 
^e^mfe, Watttv* 0ltbougb 31 bnolue it to bee fo , ^ct fo? 
t?our better erercife , ano full pcrftuafion : 3! luoulD 
laue ^ou trie it,bt? fquare multiplication, 

^cftolar. Cftat maie 3 fonc Doe. 

janD fo 3! finoe it to be true, 

foh 45 52. multiplieo bv it felf, 
tocth maUe . 18766224, j^b tl)i« 
tooo?fee l)erc fet,ooetl) lljeUje, 

$©after, ptt bicaufe fomc otljer 
fmall ljoubte0,maie Ijappcn tn luo^- 
l?ing,tf)at maie trouble a ^ong pjac.- 
tifer,3 tuill pjopounDe to rou one oj 
tlwoo cranrples mo;ie, tI2l6crein poii (^all fmDe fomc 
tjarietie, as tucll m tlje nomber p?opounDcD . as alfo 

in tl)e piotientt, 

^m ftrfte to begin, 31 tuill i^ou to ertract tije roote 
of tl)i0 nombcr, 22071204, 

^cl)o(ai\ 3; muft fet oouiie tbe nomber , ano note 
itluitbp?ifto incucrvoBOe plTiceti^oj tbatrule3i 

pcrceiuc 



4352, 

866^4, 
12996, 
12996, 

17^2 S> 
KS766224, 



6 

zzo7i2o4{4* 



The extraBion 

percciuc ncucrfatlctl)» 

fl9aftcr, ,<iomojcDoctl)aiTf of 22071 2 o4( 
tijc ottjcc , a*ltl)oiigl) tt)c Ujoo^Uc 
maicljancinfoinc fmallcpomctcB: luljidjc pet maic 
be gixatc inougl) to trouble a voung learnci. 

^cl)olar. 'Cm acco^Drngto tbcfirftc rule, 3: fcUc 
out tl)c greateft fquarc in. 2 2. (fo^ 3i fee it 10 no fquarc 
iiomber it fclf)aiiD it appcretb to be 1 6. ilno bis rootc 
4,U)bcrfo^e 3 Doc fette ooiiiie, 4. in tbe ^uotimtcj nuD 
tben J Doe abate. 1 6. out of, 2 2, 
am tbr remamer 15. 6. Uibicbe 3 
fette ouer tbe p<jiclic , aiiD caiuell 
tl)e, 2 2.ai3berei£ifeeir. 

.4i>oU) gopng on iuitb tbe nertc p;riche, 3J (ball Dou^ 
ble tbe former roote in tbe^«»^/>«f^anD fette itDnDcc 
tbeCupbcr,betU)enetbe.2.p;icke)3. 

%\)m Do 33 feUe botu oftc tbat 8(U)blcbe is tbe Dou- 
ble of tbe *iu9tiente) mate be fouuD in 6 o ano 3! finbe it 
to be 7 times, anD 4 remaining to be fet ouer tbe Cp = 
pber. ^0 tbat fo;r tbe p;jlcbe tbere remainetb. 4 7.out 
of lubtcbcB! fi)oulD abate tbe fquarc of mr^tt»f/?M!6ut 
feing tbat.49(lubicbc 10 t1be fquare of 7)can not be ta^ 
feen out of.47.tbcre 10 a ik\33C quotient e to be fougbt, 

SDberfo^e J taUe 6. ^nJi (cc tbat it luill feruc.^o 3( 
fet.6. In tbe quotUtite : anD bp it3I 
multiplie 8lubereofcommctb48 
2Dbat .48. abatcD out of.6 <%lea' 
uetb. 1 2. s:bcrefo;e 3; rancell tbe 
6o.anDfet,i2.ouerit. 

SIDbcnDoe3i multiplie tbe qmtiente , 6, bp itfelfc: 
tubereofrlfetb. ^6. i^nDtbatabateDoutof. 1 27. lea^ 
netb.9 i.0nD fo baue3: euDcDtbe feronDe iooo^kc, 

^oto fo; tbe tbirDe UJOo;ifec , 3 Double . 4 6 . anD it 
DoetbfclDc.92.to bcefettetjnDer.9 1 1, as 3 banc put 
itbcre. 

;ClnD tbcn febtng fo,2 a imitnt\j fe tbat3; mate talie 

\.u 9. 



-19 

^^;..••'7■I2o4(46. 



The extraSlion 



7 

zz^^7jizo4{4t69* 
9^- 



p.t^abcrfoje 31 fct tijat 9 tn tMt 

piotiente U)ltl)» 46. auD bp It^j 

muUiplp 9 2 and fubtract tIjat, 
tl)atnfetl),iiitbii5fo?me» 

jjimetpmc)5.9.maketl) ,8 1. 
iul)iebe3 abatcoutof.9 u ano 
tbere refltctb i o^SDften 9 ti^mes 2 gtiietb 1 8. luf)icl)e 3f 
mutt abate out of* 1 o^anD tbere lutU miiam.8 v ■ 

antinoU) mutte 3 multiple tbatlafte^«tf/»«»^e»9, 
fquai:el|',U)l)erbp iutllamounte* 8 u tbatfljall 31 fub^ 
tract out of , 8 5 2. ano tbere luiU rcmam » 7 5- 1, ano fo 
ti;at p^icfee luitb bi0 U)oo.:ke ib cudcd* 

2Lberefo;jc p;joceDpng to tbe fourtbe p?iclie,3i Dou- 
ble all tl)e qmtknte , U)l)icbc tuiU 

HDben Doe 3! ft^fee a ncU)e qmtt',f ? ^ ^i ^-f 2 ^ ^'^^^ 
e»^e,tul)tcl)e31 finoeto bee.8.i^o;| ^ ^ ^ 

8*ttme0.9.0tuetb» 7~» U)btcl)e3!abateoutof»75', ana 
tl)eceremametb.?»00am,8.tpme0»5»ts,2 4»anotJ)at 
31 Deoucte out of* ? i. auD fo re(tetb»7.2Dben fate 3!»8*ti^ 
meg* 8* ts. 6 4 . iul)f ci)e bee^ng fubtracteo from* 7 o* 
Doctl) leaue*6* 0nD tl)at.6. tuitb tlje 4.ouer tbe p;jtclie 
maketl)* 64. outoftDHirlje 31 muffe iuitbDiatuc tbe 
fquare of*8.t{jat 13 mp j «o^;V»^, -mD it hty^xx^ 6 4.tl)erc 
rcttetb notbing* ano tije lobole VooM ttanoetl) ti)u0* 

tRlSI)erfo?e 3! fats tl)at tbe firtt nober 



22 07 1 2 04. 16 a fquare noberiauD 
Tiepmfe, ijat!)foMnsroote.4698 as^mate 
p;joouealfo, bp fquare innltiplicati? 
on* 5^o?,as in tins erample ^ou fee: 
4698* multiplied b^tt felf, ooetb 
fe^png fo;tl^et 2 2 o 7 n o 4* 



469S 
469S 



57T84 
422 8 2 
28188 
I8792 



22071204* 



9oi74^'^4iG 



of^otes, 

^atttr. ^ti one crampic mo^cOjaUrcu p;ouc; Mother 
aiiD tljat i£5 tW>9 o 1 7 4 c S 4 ^^ cyAw^u. 

&cl)olar. ^ atitDDunf,anDp;tcl:cttacco>Hingtu 
tl)c rule: aiiD tl)cn J fee oner tl}c 
lailc p;icUc , one cnclp ncir.brr, 
tl)at IS. 9. lcl)icl)c l;atl).^. fo: l;is 
fquarcrootc. ^^t^nt^.^f fctUjitI)tntlje'?«o//fKfdinc, 
anDt!)erfo;c3cancell,9. 

^ftertt)is3! ll)oult) p;:oceatJC Uittl) toublrngetbc 
roote. V anD ttiat Double 11)0 u'D I frt m tl)c iiert fpace, 
ouertul)ict)eremainet!)nonomber7fo;.9.bcimgfan^ 
cellcD^tbe Cvpber is notbvng. HnD fo am ^ at a aaie. 

05a(len &epng tbat ^ou can not fet tl)e Double of 
^our ^uof-iente Doune tijere, tubere no nomber is (0; if 
tt fo f l)aunce,as fomc times it Doetb.tbat tbe nombec 
cucr it,is leCTer tbcn tbe Double) tben fet a Crpber in 
tbe qtiotieKtc,mD to bauc vou Doen luitb tbat p;i(Ue. 
fo; m foebc cafe tberc ncDetb no multiplication, no; 
fubtractlom 

S)Cbolar, 2::benam3Ifnttruc^' 
tcD fuUp foj tbat pointfe : E^^bc 
luo;jhe is fo eafic. 35 mnft tberfo;c 
fet mpnomberstb us. 

£©a(ler. anD Doefounot fee, tbat tbe Double of 
tl)C(iuotientey i& greater tben tbe nomber ouer it:* 

^cbolar. 3 luas fo miuDfull of tbe one balfe of tbe 
rule,tbat3fo,:gatetbeotbcrbalfe. 

^>ut notu % fee,5 muft fet an otber Cppber ret in 
tbe quetient.^wh tbcu Ht^all J kt ti)C Double of all tbat. 
In tbe tbirDe fpace,aftcr tbis fo:te. 

aiiDnotuepjoccaDpngeto I 
fearcbe fo? anelue qmuente, j 9 :z 1 7 4 o 8 4 1 (5 o o. 
feetbat. 2.fl)allfcrueme. ' *6oo* 

t:2.lbcrfo;eXfette.2. mtbc 

quotiente Imc , iL'ltb. 5 o o. 2nD hv ft 11)311 J multipllC 

tfte Double afo?efaieD;famig,2,trmes-6.mahctb. I :♦ 

JL.U. to 



9;:^i74o84l()0 
. , • • • 
60 



The extraSlion 



S 4 
' '600' • 



9>^l74o,?4l0oo2^ 
60 04 



to bee abatcD out of. i ?♦ aiiD the ttmaincx toiU htc*s* 

^l)t\\ lljall 31 ouerpalTe 
tljetuioo (Ippbew^btcaufe 
tftet maUc nothing fap mub 
ttpltcatton: anD fo coming 
to tfje pacbe , 31 bate ttje 
fquarc of mp ^«o^V»^e:U)bicI)e to 4 out of.S.airo tbcre 
veftetl)»4»»0rfo?c3lcancclU8.anDfctDounc.4.ano 
fo IjaueSl cnDeo tbat ^mkc, ^no bauc but one luo^Uc 
nro;e beftinDc 

^bccfojc 3! fct Doune tin mn\hct$Mt\) tbc oou^^^ 

hlC of al thtfuotiente,t\}UB. 

^ntj tbcit 3j lofee fo;j a netu 
^tto^in^/e^lubicbc 31 fiiiDe to 
bc,9, bi> tttbcrfo;e3! mul^ 
tiplie, fird 6 ano it mahetb 
5 4» t!)at Ooeti) abate tlje ^ 4. ouer it. Cbcit omit | tbc 
2 C^pl)ci:3,anu multiplic 4. bv 9 lubcreof tbcrc conv 
met!). 56. Uibicljejlabatcoutof. 44. bcpngouccit^ 
ano tljcrc rcmautetb. 8. SEbat. 8 . Uiitb . i . oucc tbe 
pjicUemaketb.8i.out ofl«btcbe3imuac abate tbc 
fquarc of. 9. bcj^ng alfo. 8 1. ano fo 10 notbpng Icfte, 
tDberbp it appcaretb.tbat. 9 o 1 7 4 o S 4 1 . ts a fquarc 
iTomber , ano bis rootc (5.50029. 2Cbc p;oofc of it 
Ooetb fonfirmetbefamc.iPo;j 50029 
muItiplieDbp it felf, ooetb b;pitge 
fo?tbe.9oi74o84i. 

thtnighejie falter. »iis ftjall fuffice foj 

note ofi^nf focbe nombetc as bee fuUp fquare. 

f^mre mmf ^tber nombers tberc bee infinite, 

Iters, tDbicbe be not fquare , ano tberfo:e 

baue tbet no fquare rootc0» ^et of^ 
tm tpmes it bappcnetb , tbat lue fljall bee ocfafione& 
to fcarcbe fonbc nigbefte nomber , tbatmaie refem^ 
bletbeirrootes. 

mberfo?e in focbe tt{%tW fl^all pou ooe. i^irfte 

en'trartc 



50029^ 
^0029. 



27026 I. 
6 00 VS. 

9 o o S 7 ' * 



90174084U 



of^otes, 

ectract tljc toote, as if it tuer a (qimtc nolcr, ^nu u,^t 
rootc tutl feruc fo;j titt grcatcft fqiiarctljat is in vouc 
fo;jmcr nombcr: anD tbcrc UjiU be a rcinaincc facfiDc. 
^Df U)I)icbcrcmainccU)itl) tbc^«of/>»/,^ou ftjalmalic 
afra(tion,intbisfo;tc» 

^ct tl)c ccmaincr ouer tijc linc.fo; tl)c nomerato;:, 
anD tl)c Double of tbe roote (tbat pou Ijaue founDe)fct 
Ijudcc tbe Itncfo; tbe Dcnommatoj* anD this n)aU be 
a fufficientc p^ccifcncirc in greatc nombcrs , fo; anp 
tommon iuoo:Ue. 

^cbolar. 3C Unll bp an cranrplctalicn bp cbauncc^ 
p;ioiic tl)is rule. J^o; it femetb to bauc no Ditftciiltie. 
^'Clbcrfozc jt taUe. 296882* 

0nD ti)i3, J am affiircD, can be no fquarc nomfaer. 
5^o:,3; ccmcbcc pou tolD me bcfo:e,t!)at no focljc nom 
bcr might be a fquace^lubicl) haD 2 fo; bis firft figure. 

^l)m to fcarcbc bis nigbefte roote, 5 place it,anD 
packeittbus. 

anDtjnDec. 29. Jfinoc the grca^ 
tefterootctobee.^»Ujbicbc3iretintbe 
quotient e lme,anD cancell 2 9 fcttpng 4 
ouer It. after that 3; Double it, anD there cometb i o. t 
that Double J fet m the nerte fpace tnDer 4 6. SDhen 
fmoc 3! a ncluc f «of/f»fe, luhichc is 4 anD bp it 3: m\x\ ■■ 
tiplte. I o.luhercof amounteth 4o.to be abatcD out of 
46. anD fo rcmaineth.6» 0gam 3} 
multiplie.4.bpttrelfrquarclp,anD 
there rifeth* 1 6» tohiche 31 abate fro 
1 8. (fcpng.8.is to fmalDanD the re- 
mainer luill be.2. S>o ftanDeth the 
luhole nombcr:,as pou fe» <2ahcrfo;e 3? Double the qm- 
//>nff jtuhiche is.y 4* anD it pelDeth. i o s«that mutt be 



4 

2 9 6S82(f. 



.4 > 2 
Zc/,6',S'82(^4 



fet bnDcr s 2 8 303! hauc here Doen. 

SDhen 31 loobe fo; a quotiente, 

holtj often 3 maie abate, i o 8. out 
olS 2 8. ano I fee it toill be but.4. 



4)2 

,^,9 6' ,8^8 2 () 4* 
I 08 



^.iii. 



trracE 



prooff. 



-J 9 4 
^ 2; j5 6 



The extraSlion 

t^me£J.Ml)erfo?c 31 fct.4an tl)e ^Mo^Vwf^toitb tbe 0? 
ttjcc nomljcw, ano tbcn doc 3: l«oo;be iDitl) it:i^(rffe 
niuItipUung»4«anD»K togctt)er,lDljereof coiitctt) but 
4*lubicftc 3! abate out of,^. 0nD tbcrc rcmainctl)»i. 

agatn3!inulttpUc,^;bp,4»tD!)ereofrommctl).32» 
tl)at uoc 31 fubtratt out ot 1 2 8. ano tljerc luiU rcntam 
9 6» d)cn ftiall 3 take t!)e fquare of tnp qmtiente . 4. 
tul)tc!)e 10 1 6. 0no tl)at muft 3 abate out of 9 6 z.jauD 
fo rcmainett) ♦946. of Hljtcije 
itombct fet as; tbe numcrato;:, 
tuitl) tlje Double of tl)e rootcfct 
fo5 tbe Denominator , 3 ftjall 
mafee a fraction tn tW forte* 
^ . tubicbe is almoae, {- ♦ 

Rafter* ^ouftaueDoeniuel.0nDfot?oupemiue 
t^at tie nigleCe roote of rout fo;imec nonibet (0 
5:44^. i^o?t!)ofefracttons are all one* 

0nD berebp alfo vou mate tjuDerftanDe, tbat if tbc 
rematncr ouer pour nomber bee eucn,vou maie taUe 
^alfe of it fo? tl}e numerator , anD tl)e iubole ^uotitnte 
fortl)eDenommato^ 

^0 maie ^m take tlie quarter of tbe rematner(if it 
tuiU fo beeparteD)for ti}t numerator, anD tbe balfe of 
tfte roote for tbe Denominator* 

0nD m like maner generallv' , if tbe rrmainer nnD 

tl)e roote in tl)e quotknte -, bee nombers ccmmumcantejXiU 

uiDe tijem ro,tt)at tl)c Diuifor of tlie rematner^be euec 
Double to tl)e Diuifor of tbe ^uoticnte reote.^nn fo maie 
^ou eafil^ reDuce tbat frartion>to bis leaft terme0, 
rbe/i /it ^"^ "^^ ^^^ proofe of tbig U}oorke,tbere be tUioo 
j-tj prj t tuaiesube one is certain, anD tbe otbtr but m a nere^ 
neire.f=or as tbe reote of focbe nombers, i& not a pre^ 
cife roote : S)0 if ^ou multiplie tbat roote b^ a felf , \t 
Ituill make a nomber, tc rp nigbe to tbat former nom^ 
ber,butnotera(tlvtberame ♦ 
^bicbc faultc feme men tbinke ta reDrelTe^bp aD^ 

Drng 



5'44- 

2 1 76* 
2I76» 

2720. 



opitof of* I. to the Dcnomtuato;! : aiiD ^ct t!)at amcuDc^ 
mcwte fomctpmcs mcicafctl) tUe erroiirc. 

I3ut bicaufc voii rt) tU not luante a fure p200fc,D3C 
tl)u5 ♦ 3?ultiplie tl)c^«a//f«^<r,o;5^oo^cof lul)olcuom'^f ^"^(""^^ 
bcrsbpttfclf, aintjiuotbenomb^c tbatamountctl)/"'*"?/^- 
tbcceof^aODc tijc lonolc i*emamcr.$JnD if tljcn it make 
vouc firftc iiontbcr, ^oiir iuoo^kc Ujas Ui :U Docn : cl3 
baucpoumiffeD. 

^cI)olar. S^f)atmatc3lp;ioucI)cixquicUlT'. 2^be 
^uoHente m Uj!)ole nomb2r0 luas, ^ 4 4. lubicbc be png 
miiltiplicD rquai*elp,Dort{) pclDc.2 9)9^6. ijnto Uibi^ 
cbc nombcr,if J Doe aDDc.9 4 6. t\)M did 
ixmam , it iuill amouiitc to , 2 9 6 8 8 2» 
anD tbat luas tbc nombcc p;joponcDto 
me:lubcrfo.2citapperctbtbattf)cU)o;Ue 
UiasVufUbocn. 

!t0aftcc. VQW ft)all ncaQc no mo;e 
examples, fo; tbis fo:mc of looo^kc. -^ o r n ^ 

But one otbcr loaic lull jj (Ijcluc vou, - ^ ) y 5 o» 
bo\u pou fl)all gcflfc bene nigbc tjnto tbc rootc, ^itD M other 
pou fl).all go as mgbc as pou luiUDcfirc , m anp pjac^^ y^aietofinde 
tilie ioojU:. 3f pou Defire to geflTc loitbut IclTc tben ,'.. themghejle 
Dfonj7ti)cnfcibcfo;cpoarnomber.2,Cppber5. HiiD r<;5ff. 
(f poll Ujoiilo not crcc , ': . tben fet Datnte 4.Cppbers: 
But aiiO If pou litle to fcttc Dounc.6. Cppbers before 
pour nombcc , pou Hjall not miflfc 7:7; of an bnitic fio 
tbc true rootc. ^no ifpou liU to go anp bigbcr m p;rc^ 
circneflreofpartcg,aOQcftilleucnCppbcrs. 

©cbolar. 3lluoul3faincp:3ucri)i5fo.:mc , intbe 
famecramplc, lubtcbeB! U):ougbtclaO:c: Bicaufc Jj 
ipoiilD fe tbc ag rcmcntc betluene tbc botbc U)o:Uc3, 

ipailer. cDo tOvPour ronftD^^ration 13 rcafonable 
0nb btcaufetbe partes mate tbc better agree , fcttc 
tiO!me.6.Cppbcr0» 3nD tben fl^all pour rootc erpejcffc 
tboufanDc partes oftbe lubolc nombcr. 

^cbolac^ 3ir$ttcDounctbe nombcr. anD p^icUc it 

tbufi^ 



The extraBion 

tfjus* Wi\)tttlfvdi perceiue 

tW 3 ft)all baiie tSefame o?-- 296882oooooo( 

uerofkjoo^fee, anu t!)e felfe 

fame nombcrs ttjatSiljaD before, tiUt|)at3|romcto 
tl)e Cppbcrs anD tbcir pjickes, 

spatter. 2Crut!)£iti0, aiitJtf)crfo;emaiepoum 
foflje a cafe fette Douiie onelp tbe rcmaincr , tuitb tbc 
Crpl)er0. <© J els canccll all tbe nombew,faue ttje re*' 
niamer , ano tlje Cppfters : anu fet tbe former Uiftole 
roote,tDttboutti)e ftmiotxyin tl^e ^mt'mtt, 

^cbolar. ^ben tDtU I 
UttanDctt)u0» I 946 

j^otu acco?Dpng to tfje I Z;9')5'^ j>z o o o o o o (^44. 

rule 3 tuill pjoceaoc^ag if ( 

tbislu^olenoberUjectljefirllnombcrp^opoticDtjnto 
tne« anD tljerfoje 31 Doe Double al tfte ^uoti€ntc^\iiWMt 
maketb^i 0S8. anDtbatDoe3!fettjnDcr.946 c, jaiiD 
tben lljall 31 feke a quoth 
tntc , tW male Declare j 

l)olD often t^mes , tbat (J 

uoubletscotameomtlje f''^;^^/: 
nomberouent.anD^fe ^^Xlv r.^.r^ 

itUimbee.8.Ujberfofe3I ^Pi^|^^|oo 00 00 (h 4 S 
fetDoune.8. mtbefwo^^- '^ ^^ ^ 
*»>c,anD b^ it 31 multipltetbc Double, anD fubtracteit, 
int!)i6fo?te:fai^ng8,tt'me0,LoutDf9»lcauctl)w.rc^ 
itTalnj?ng* 0gatm8»ttmes«8. (tbat is.6 4. )out of 1 4 6 
iutllleaue.82. »enfartl)er3iabate/8«tt'mes,S.out 
of.82 o.auDtbcre rcaetb»75'6» atiD laftof all,3 tafee 
tbe fquare of tljc quotiente, tulbtcbe ts alfo ♦64* out of 
7^6 o, atiD tljere tutll remain 7496* 0nD fo !jaue3l 
Doen luitb tfte firfte p^irije of tbe Cppbersf, 
Anouhie ^ (Rafter. <i:onfiDerttotutl)atbptbofe.2, (T^pberg 
e^nfideratio. ^ou baue gotten 8 into tf)t quotient mo?e tben rou baD 
before. 5lnD all rour former nomber of tbe roote,re^ 
moucD bj? It into one place bigber^tbcn it ttias befojus 

^0 



of^otes. 

^0 tbat, tuljcre bp tftc flrft luo?fec, I'our rootc tuas 
c 4 4.anD almoftc -'; : bp tt)is toojfec t?ou Ijauc founDc 
tt to bcc ^4' , anD ^^t ■ of ^^ : tublcbe is tjerte nigbe tbc 
fame nomber^tbat vou IjaD bcfo;c» 

^ct)olar. jnDceocjif 3 rctiucetl)cfracttoiT0,itiuil 
bcc . > 4 4 ♦ 1^ anD ^5^; of ~ * ioljicbc is in one fraction, 
^*^'abouc«5'44. 

' '^"Patter. q^arUc tbt0 triall. 0nD ^jfe tbe like after 
eucc^j ttuoo Cpptjcrg arc cnDcD : anDrouH^aUreca 
gooDlr aorrcmcntc of tbc tDoo;jUc!3 together. 

pcljolar. jn tbe meanc trme, to p^occDe luitb ttjc 
former iuojtie, 3 fct 7405 

Dounc t5)c nombcr .:.nyy^s,S 20 00000 (H4S. 
luitbtbc remamcr, -V^^>:-' ' og^ • . unn 
anDttjcDobleoftbe 1 ^^yo 

^uotiettte,n& t)cre appcarcti)* 

0nD fearcljpng fo; a nctoc quot'mte , 3; finDe tbat tt 
ii)iUbc.6. 

2::i)erfo?e3! fettcDounc,6. fntbcf«o^/e«*^luttI) tbc 
otbcr nombers. anD bp tbat » 6 ♦ 3? doc multiplic tbe 
Double of tbe tobole ^uotientt , anD fubtract It o?Dcrlp, 
raipng:6.time0.KbC'' - 

pngabatcDoutof,7» gig 

Icuctb. I. jpizo 

^lUcluaics:. 6 . tp^' ^^^ ;, ^ , 

H,!)(cbc3 fljall abate ^-^^^^^Vsn'T'^ 
out of. 4 9- anD fore- '1>pv." 

ftcti). KXbcn 6.t(mes.9, (tol^tcbc t0»5^4vmutt be fub^ 
tratteD out of ♦ i o 1 6 ♦ anD tbere tuill remamc . 9 6 2» 
iJganie 31 (Ijall abate. 6* tpme6.6» (tfjat is. :; 6.) ut of 
9 6 2 o . anD tbcre is lefte .9^84* '^^^^^ taUc 31 tbe 
fquare of mt? f «<?fie»^f,tDl)ic!)e ts alfo 6 tunes 6,o;j 3 6. 
anD tuatBi mull abate out of.4 o.anD tbere reaetl),4. 
0nD tbus is tbc fee onoe pjicfee of t!)e Cvpbcrs enDeD» 
j^nD no^ 31 finDc in tlft iuotmtt not . ^: as S ^i^ <» 



The extraSiion 

tl)e lade liioo;tije before tbts. y£>\M 3 finoe -^i \ii\)kht 
goetl) mo^ae itfgljc to ^ ♦ i^o.2 ^^ luoulo be ^': aiiD ^- 13 
equaiie luitl) ~ ♦ anti 3i niaie caQIp ^t, tl»at ~'^ t£? riJo^ 
ingl)crtOT^:tl)cnto^: facfiDetljeiemamcri iuhichc 
U)i» maUe t^ of ^ . 0? tU ,M;^, of one. 

Rafter* 3 fec,a Uiell loiilvng mi'iiDe can marlte 
t)tligcntlr,anD Icariie fpeDU^: luberfoae go fo;iuarDe 
U}itbpourU)oo?ke. 

^cljolar. 3 muile fettc Doune t\it Double of all mv 
qmtiente^ \3ii[}Xt\^Z lutU be. I o 8 9 7 2. ^IIO It luiU ftanOe 
tl)U0. 

Mljerfo;^e3 Doe 9 5" 8 02 

feljcfo^aneU)e^«o; ^.9j5'^ b'zoooooo(s-44S6 
//>«/<?, ano 3! finDe it 108972 

tobc.8. U)I)tcI)e.8.3; 

fct xn i\)t quQthnte, imtt) tl)e otljec nombers, and br it 
3i U)o?ke after mv vuie, faipiig:8.tpme. his,8»Xubicl)e 
3! abate from, 9, anotbererelfetb,!. 2i:bcntahe3',8* 
tpmes.S. abatis?>640outof. ijg, aitD tbe remamec 
iDiUfaee.94» 

again 3 fubtrart . 8.trmes.9. (bee^ng. 7 2.) from 
♦94 o.anD tljere is lefte.S 6 S.i^aitbermo^e 31 tatte,8. 
ttmeiei.7*(iubti:beii5.5:6) ( ^ . 

outof.82»anDtbere.rc^ ^.^^4 

fietb ♦ 2 6. SCben Doe 3(- i^^f^^ ^' 

lDitbD;^atue.8.tpme)8.2. L^^o o^^'^''^^^ . 
OM6\outof.6o. ano ;^^^.SvS/rQooooo^5-44S6S 
t^ere remainetb* 4 4. I ^>^-S -97-2 

liaft of al 31 take 8 times 8 o? 6 ^{mUht 10 tlje fquare 
of m? laft j«o^/>»^e)outof 8 6 2 4 4 o ans tbc remamer 
tuill be.^ 6 2 3 7 6* ano fo Ijaue 3| enoeo all mp luo?be* 

anD noto 3i ftaue fo? tlje rootc ^^ tljat is . r 4 4. 
^^^pMmwM ofT^ o;j m l^ec t^mesi^^ of 
T^^ tjjat »s ,,,if22|z of one: ^ftiebe hm^ reonccD into 
one fraction Uiitft tbe ~~ iuiU ma&e ^^^i^i^ , 

Rafter* ii^oulbaiieDoenipeat """"" 

janD 



of%ootes, 

QnU Mttt pou fee, tt)at rou 0;aUic nigger t nigljcr 
flia,to t\)t tjcrv rootcif It mtgl)t Ijauc anr-j^o: -"^ is 
a iHgl)cr nombcr to ^, , tl)cii ts t'x ais tbat tuas jngf)Ci: 

tljcn./. 

j^iio tf poll luoulD ir)o;jUc tuitl) nio;:c Crptict*5,rDu 
ll)OiilD pcrcciuc fttll, tljat it tooulD O^ali r nigljcr anO 
nigt)cr,5i5uttI)isniat*cfuffifCfojcramplcsfaUr, 

^cl)0lar» SElKitBSpjaic ^outcUmc, iDliatistljc 
rljicftjfc of tl)t0 rule: aito fo? U)!)at maters it fc ruetl)» 

vBaUcr, £?itc y>ixt M\\ not fufficcto crpjcDTc t{)e 
f omnioDittos of It. 3:t fcructl) fo inanp tuaics^m buU^ 
tjinann pioiccrioii of plattei5,fo;mcaruring of grcuD 
Sluubcr, o; aonc: anD alfo In luarrcfo; franipng of 
battailcs, fo;maUrng of Diucrfccngmce, anD gcitc^ 
rallV' fo: all tuoo;bC3 of Gfomf/r/VanU JJlronomie. y5ut 
fonofattffici'oupartip, ^iKillfcttc fo^tlic tluooo? 
tt);tt queftions,t!)at tepcnoc of tfjts iuo^Uc of crtrac^^ 
ttonoffquarcrootcs. 

anti ftrac of a battatle : btcaufeit femctb to fcru^ 
Icaacfo?tl)atpurpoff. X,f«fy//o>» 

a capitatnc gcncrall Ijanj^ng tl);icc grcatc armies, V' «» '«^^<"- 
tuoulD catle tljcim into tl).:cc fquare battatlcs , but be 
Iniotuctl) not botu manp men, be l^all fct in tf)t frotc 
of ccbc battailc* 

^\)t nombcrjg of tbe tb.Jee armieMrc foj tbc firffc 
562r»#o;Jtbefcfonti92i6:anDfonbetl)troiri29 

^cbolar. 31 Dooc pcrceiue eafilv^. tl)at fo? ec^jcof 
tbcfe nombcr0,3i muftc fcarcbc out tbc fquare roofe, 
ano tbcn baue 3! tbe fronte, 0; flankc.^ttb botbc arc 
cqualle in a fquare battaile* 

vm bcrfo2c 31 fet Doune tbe ftrt! nomber tbu0, tuitb 
IfiB p;if Ucc anD tben tmuer tbe firtt pjtcfee ( 
totoarDe tbe lefte banDc, 3! fi»5e tbe grca^ U62S( 
tefte roote to bee , 7 . feerng tbe grcatelfe | 
fquare is. 4 9. %l)at rootc Doe 3 fet tuitbm tbe ^uoth 
me line: anD bw fqtmre Doe 3f abate from, s 6* anD fo 

^aj» remainctb 









i,S 



The extraSiion 

rematuetlj* 7* 

JCben Doe 3 Double tbat rootcanD (cttc tlje Double 
tJitDccy 2«aiiD fee tbat tlje neloc quotient 
null bee,^', 0iiD tbcre tnill remainc. i ), 
iDfticbe is tbe iufte fquare of tUe latt ^m; 
tiente. 

Wihtthi^ it IS catDente,ti)at hi^ firff arinic contat^- 
neD a fquare itomber,anD tl)e t:oote,o; fiDe of it is 7 y, 
Sinn fo manp menne (^all be in tl)c frontcof tbc firtte 
battaile,anD as luanp m tlje flanbe, 

^oU) fo? tbe feconDe battatlc , 3 fche tbe fquare of 
9^i(^»anDfinDeittobec»96. ^5 in 
ti)is eramplt 3 l?aue lu?ougbt it* 

fo:, the firfte nombcr ig»9» ferirg it 
tg tOe greatefte fquare roote , tbat can 
bee founDe 111*9 2* ^tiD fo ts tbc Double 
of it. 1 8. anD tbe quottente fo? it»6.a0 it appearetb nia 
nifeftlp inougb* 

Mbcrfoje 3i fate tbat tlje fecoiiD bmMlt flial bauc 
(n eueri' rankc.9 6. mciu 

anD nolo fonbetljirDebattaile, 3 fette Do unctbc 
iiomber,acco?D^ngrto tljis rule: anD 3 fiuDe tbc firttc 
rjODtetobe^Lbtcaufe.i.t^mes^Lttta^ 
feetbti* anD biiS Double 10. 2. lubicbe 
31 abate ttoife from tbc nomber oucr 
tt:anD after Double tbofe botbe nom^ 
lieri8,tjD]^!cbemakc»2 4. anDfiuDc 
tfiatto be abateD.^.tpmes. 

anD fo baue 3 gatbercD tbat tbe nomber is fquart 
anD tbe roote 1 2 j\ acco^Dlng to iubicbe nomber^tbat 
tbtrDe battaile mull be marfljalleD. 

50aften a>epng you are fo reDp f n tf)is potnrte fo 
fone» Cell me boUi manp menne,l^aU be fette in tbe 
fronte,if all tbefe* 3. armies be ioineD into one fquare 
battaile. 

^c!)olar» i^irUe3ImuaaDDeall,3* nomberdtogc.' 

tljer. 



17J 
'2*4* 






of^ootes, 

t\)tt, 0nD fo tutU tljti mabc. 2 9 9 6 o. as ! 

tjcrc bp crampic Doctb appcrc. \ f^- T 

')!3ut tl)i3 nombcr can bcc no fquarc 9 > 1 6 
nombcr^bicaufc it bati) one oDOc Cppbcr ^ n - 9 . 
(ntbcfirftr place: fo;3i remember I'onc 29960/ 
faipng, ttjat fquare nombcrs ran not bc^ 
gin iDiti) oDDc C)'pbers.^;£ll)crfo;c tbis; nomber Unll 
not nuljc a fquarc battatle* 

;pct lull ( p.:onir,U)bat mate be tbe frot of tbe grea» 
telle fquarc battailr,tl)at maic be maDc of tbat nobcr. 

ano fo; tljat pnrpofc 3 p;uUc tbe nombera , ano 
finDctl)egrcatellerootein.2.tobe.i ; 
ano tbefantc nobcr to bee tbe fquare ^ ^ ^ 
alfo. iEbcn Double I tbatroote^anD 
place 1)15 Double t)nDcr,9. tljat 15 Xin^ 
p;tcljeD:anDfcrcl)pngfo;ia^<wfif»ff, 
36 finDc It to be.7- U)itl) tubicbe j luooiiUe by tbe rule, 
auD fo Doetl) remain fo: tbe nertc p;icUe. i o. 

S!Ll)a\ Doe J Double tbat. 1 7. lubercbp rommetb ; 4 
tobtcbe Ji fet bnDer, i o 6. anD fo; it s finoe. ;. to be tbc 
ixtctcitc quotient e : luitl) Uibicbc if j! U)oo.2heacco:Dvn- 
glistberclmllrcmainc. ^j.astbcerccire abonctbc 
grcatcfte fquare, 

Wi berbr It appearetb tbat 2 9 9 2 9.13 a fquare n6» 
bcr:anD batb* 1 7 vfo; bis roote» ano tbat fljoulD bee 
tbe fronte of tbis greate battaile» 

£&a(tct. J^.oh) Unll 3; p;ouc rem iDitb an otber 
quellionoflihcfo;te» 

ap;mcebatbanarmlet)cncgrcate. ZZWthhjbiThefcconcle 
cbe be paflTetb in a clallte , fo tbat m marcbvngc tbe ^ue/hon of 
fronte can be but. 1 8.menne. ano bp tbat mcanes tbc an amis. 
fiancee containetb. 4 4 9 ^ 5- 2» 

after tbat tbc armie is paffcD tbat tjalic, tbe Ujntg 
nnnDpng to occupie all tbe bette grounoctuillctb tbc 
battaile to be fet fquare. Ii)o Id tuoulD pou Doe it:" 

^cbolart fira 3i multiple tbe flancke>bp tbe front. 

£15.11). Xtno 



The extraStton 






nni)fo3ifinDctl)eiuljolcnomijfrtobe«8o88336. 

sni)at nomfacr Doe 3 p^icfee 
ns mv rwlc tcaciiet!) me , ano 
3finDetl)efirllrootetobe»2» o, 00,-^/0^^ 
aiiD!)ij3fqunre.4»lDl)ici)efirll >^ /^^/^- ^^^^^* 
3;fulJtract0outof.8*anDfore« j '"^^^^p 
ftct!),4* aenDoeSJDoufelc i -^ 
fijat j«o^/ra/f,anDfinDetl3at Double* 8* tpmeitmt^c 
fommcoueitt. 

anD fo Doe 3 p;o«Dc tiU 31 ftaue founDr ut all t\}c 
4. figures, aao?D|?ng to t(je.4,p;icfecs tjuDec tljat no^ 
her. nnD tficn tbc I'oote appearettj to be* 2 8 4 4* 
Thethrde W^ittu .i^etonequeftton itiDje, fo;toerfercire 
^ueponof poui:pcnne,lom3lp;opounDeofaIiUemater» 
an amk. gcueralle Ijattj tt);ee armt'ea , to tbc nomUer of 
2828 9.men : anD none of tbofe tMttt armies is apte 
to mafec a fqaare faattatle, ^et Ije Is appotnacD bp ijis 
foueraigne,to fettc tbelm in t\^it:t fquare battailes* 

3i:!)efebetti0,^»nomfacrsoftbe.5« armtcs. 3ntl)e 
firtte tlrcre are. 10296. men:3^n tf)e fcconDe. 9493: 
anD (n tlje tljirD.S 5-00. jpioiu let me fee tioUj rou can 
rati i\itm tnto tt)?ee fquare battailes* 

^cbclar. 3 tbinke it reafonable, to tafee tOe grea^^ 
tefle ^t^u^ixts of tbe firff anD feconD nombers, auD tljc 
frceireoftt)embot!)e,toputtotbett)irDcnomber. 

^5a(lrr. ^oarepounot furetl)attbetl)trDnom^' 
bcr,tt)ill be a true fquare. 

^djolar. s:tjcn UnoiucjfnotljotD to Doe it. 

spatter. ^Tabe t\)z grcatelie fquare in t^t tftirDc 
nomfacr alfo. auD ittite tbofe tf)jeecrce(res,anD ttjeir 
rootesalfo* 

Sben put one to cuerji roote, anDmarhe tlje fqua^ 
res ti)at U)tll rife of tijem. 

srbtrDlr/ubtraft tfjefirtte ^nombersjoutoftftofe 
;.nciue fquarcs^anD notetbe Difference of ccfjeof tbe 
f3rjJenombcr0,fromtl)ofefquares:anDfo!]aui^pou.5 

nombers 



of^ootes. 

tiomberjsiofctccOrcanD.^.otbcrofluantc. 

j^oU) compare tbofccrfclTcs ano luaittcu lurU to.- 
getl)cr:anD poii fljall cafil^ fee from UiOicbc rou fljall 
take anp nombcr,aitD to U)f)icl)c vou fljall aDOe an)'. 

^cbolar. 5n tl)c firftc nobcr tl)e greateft fqiiarc \s 
lozoi.aitD t[)crbp tbe erceUe 10.9 vaiiD tlje rootc 11. 

3111 tl)e fecono nomber tbe grcatctt fquare 10. 94c 9 
anDl)t0 roote 97. S>o 10 tl)c ercelTe.S 4. 

niiD in tlje tl)irDe nombci* , tl)c ofiratcac fquare 13 
S 4 6 4: ano the rootc of tt. 9 2. taijcrfoic tl)c ercclTc 
appcaretl) to be. ?6. 

0nD ti)U0 bane J founoc tbe. verceflfcs. 

jl^oUj fo.: to finoe tbe , Defaultes 0; luantes.J aooc 
one to ecbc roote, auD multiplietbeun fquare : ano fo 
of. I 02 . 5 ftnOe tbc fquare to bee. 10404. ano if X 
fubtrarte tbc firftc nonTbcr,lul)icbe is. i o 2 9 6.out of 
itjtbcrc U)Ul remain, i o 8. fo; tbe firftc luante. 

S:bcnfo,:tbcfcconDcroote.97. X take. 98. lubofe 
fquare loill bee. 9 6 o 4. out of tubicbe J abcUe tbc it^ 
fcnDcnombcr,U)bJcbeis.9 49;.anDtbcrci0lefti i i 
a0 tbe luante of tbc feconbe nomber. 

s:birDlp,3; taUc 9 ] fo: tbe nciue roote, nertabouc 
9 2 . ano 35 finOe bi0 fquare to bee. S 6 4 9. from Uib(' 
rbe tuben tbe tbiroe nomber.S s o o. is abatcD, tbe De 
faulte appcarctb to htt, 1 4 9. ano tbus baue T tbe. ,\ 
Dcfaulte0 0; tuantcs,anD alfo tbc^crccflTcs. sabicbc 
fo^ cafe of comparpng,3! fct m o;t)cr tbus. 





jf. 


®. 


c. 


^•^.anijr.bcto-- 


Ixcejfes, 


9S' 


84. 


^6. 


Ucn tbe o:Der of 


tfiantes. 


Io8» 


I II. 


149. 


tbc^ftrftnobcrc. 



0nD bcre 31 compare tbc crcefTes luitb tbc tuantcs, 
^0 fee If anp.2.erceirc0 tuill ma&e tip tbe otbcrs Uiant 
3nD 3( fee bp a Itgbte p;oofe,it ImU not feme. 

2it fo^ tbc tuantC0,31 Doe not compare t\)tim to tbc 

crccftes, 



The extraBioH 

txttfCt&jtoi^ fe tWtutt^ one toant,i« greater ttew 
mv one eiccelTe* 0nD tberefoje^i^tuantes are farre to 
grcate aboue an^ one ercelTc, <anD fo am 3 at a ftaie» 

spatter. S^ljerfo^c altbougl) tijat rule bee gene* 
raUe,pt tol)ere tt fatletl),tl)ts l^all ^ou Doe. 

iafeet^e.2. tuantes,ofanp.2. nombers^anoaODe 
thtim firfte togetlber , ano then abate tbeim from tbe 
t\)itu nomber: ano if ttje remaincr be a fquare nom^ 
ber,tftcn baue ^ou gotten pour purpofe. 

^cbolar. Wat toill 3! p?oue bere.antj firtt 3: tabe 
t\)t lDantes,of tbe . 2 . firttc nombc rs , tul)icbe mahe 
2 1 9. 0UO tbat ooc 3 abate from tlje tbiroc nomber 
8 5' 00. anotbereremametlj. S28 1 . Iul)ul)eas3: fee, 
mate be a fquare nomber. anDtberfo;re3i p?oucit,m 
mp tables,ano 31 finoe tt fo to bee. ^no. 9 i.to oec tijc 
roote of it. 

Mberfoje 3 fate to tl)e quettion^tbat tbefe fijall be 
tbe nombcrg of tfje ] battaile0, as bere 31 baue fct tbr. 

SCbefirttebattaUe.i 0404. anoljisfrontci 02. 
2Dt)e fecono battatle.9 6 o 4. ano W fronte. 9 8. 
2CbetbtrDbattat!e. 8281. ano bis fronte. 91. 
SCbefommeofall^ -,5050 
tbe.>battaUes. 5-2^-^^* 

ant) btraufe tbefe noberg are not onelp fquare,but 

alfo tbeir lubole fomme ooetb agree,tuttb tbe fommc 

of tbe 5 feuerall armtc0,pou male be furc tbat tbct are 

luellpartcti,acco;DrngtotI)etntentcoftbequeaion. 

TBnt btraufe for be quettions, baue mo;je oiflfif ultic 

tbcn commoDttte,to tbem tbat are not mete.to be tra^ 

uelleD in focbe marft)allaffatres,3 Mi leaue tbatma* 

ter to marfljall men , ano Uiill come to lotucr maters 

tn tuarrc. 

jfqutflion acitieD^oulO bee fcaleo, feeing Double DtcbcD.0no 

offcaljng. t\)t inner Dtcbe . 3 2. foote b;oaDe . 0nD tbe tualle.2 u 

f05te !)igb* %Mt cnpitatn contmaunDetb laODers to be 

maoe 




of^otes. 

iiiaDc of tl)at luttc Icngtbc , tljat mak t:cl)c from the 
Utter bioii) cf tl7C inner Difljc,to tbc ic^jpe of tlje U;dl* 
a^m tl)is figure C 
ts partlt^ crp;cf. 
fcD. 

U)!)rretl)elinc 
^^.ftanDctbfo^ 
tijeb.JvDth oftbc^j 
m])C ♦ !:^nDtl)cr 
Imc.s. c. fonlie 
licightc of tl)c 
U?alir. j;ioluc3J 
DcmaunDe/oiliat 
Iljallfcctticlrgtb !B 
oftljclme.y^.C:U)l)ts 5:. jf. 

cl}t Ijerc ijoetb rcp:crente tijc laDOi r:' 

^ebolar. :ilt)is figure Dotbocrafio me to remrbec 
tl)c ^ vtl)eo;iemc of t^e patljeUjaie^lulncbc faitl) tljiis. 

hi all r'tghte an^uled triaugles^the fquare of 
that fidcjl^hicheltcth aTauift thcrio-hte anzlc. 
is eqnallc to the tli^oo jquares oj hothe the O" 
ther /td^s. 

u3. lierUp 3; tnDerffanD.tljat j muU nmlUplv tOofc 
tU)oo fiDcs rquarelp,tl)atis,ccl)e of ti)eim b\! itfelfe. 
^nD ti)cn aDDpngtbofc. 2; fquarcs together, 3; muttc 
crtrart tbc rootc of tljat luijolc nomijcr: lubtcbe rootc 
fljall be t'i]c true lengtbe of tljc dope Inv:, 
C2n)crefo;e,firflc 3! nTuIttpic. ; 2 . br it 
felf,anD there rifetb of it 1 c 2 .4. 
agatuc,^ multiplie.2 i.bv it 
relf,anDttpclDetb.44 1. Cbcfe 
j faotl)c fommes, be^mg atJUcD to -■ 
igctl)cr,DocmaUe.i 46 5'»lBbicl)c 
i nomber mate bee fquare :, btcaufc it begins 

j^.j. ncth 



21 

21 


21 

42 



441 



■i^ 



6 4 

I024 






The extraSiio?t 

ttctb tDtti)^ s* 

f39aftcr, 3t 10 no fquarc nontbcr, us it appearetlj 
at the firOe figI)tc.#o? altUougi) tt)c firftc nombcr be 
S-^zt tn focl)c nomiicrs it is rcqniftte^tbat tl)c fcconDc 
figure n)oulD be. i, els can it not be fquare:anD tjcrc, 
^ou fce,t!)at tt)c feconDe figure ij3.6. fo ttjat it can not 
fee a fquarc nonibcr. 

^ tjercfo.se ?ou fliaU feUc tlje ntg!)cfte roote , tbat 
t?ou can finoe tn it,anD take tl)at fo: vour purpofe^ 

S>f{)olar» ^ere is mp tuoo^hc fct 
fo;tbe» 

anD fo it appearetb luell tbat tbe 
nigbette roote 10 . 5 8. ^{t 7 lubicbc is 
Icffe tbcn a quarter of a footc.abouc 
5 S.foote auD tbat mull be tbe lengtbe of tbe laODer. 

cpaftcr. :ii3etonequcfttonmo;c tuiUJp.zopounO 
agreable to tbe firlle fo;me» 
J" qaeftio of ^ capitate generalle baupnge tb?ee armies , tti 
mami>yng. tb^ce feucralle battatle3,irt tbe firllc»4 9 o o»mcnne, 
tn tbe feconoc ,2401, ^m in tbe tbiroe , 2 5- o o. (fo 
tbat tbe greatcfte armie , is as mocbc as botbe tbe 0^ 
tber,ercepte one nianne) is Infojceo to iome all tb:ce 
battatles in one, l!5ut is in tioubte, lubctber be mate 
baue gooD ano conucnicntc grounDc to cncainpe tbf^ 
in battaile fo?me.(2aberefo;e confioerpng.tbat all. ^. 
battatles togetber, are but Double to tbe greatcac of 
tbe^.alone. SCbecapttainCUcfir^ng a mete grounoe 
fo? bis armie, fo iotneo in one fquare battaile , is m 
Doubte, tobat fquare ofgrounseluUlferuebispur^ 
pofe ♦ iBut fure be is,tbat it muHe bee double to tbe 
grounDe>tbattbe greatclle armie of tbe s.biD occupie 
nno tbat luas fquare euerptDaies.2 1 oJoote.tlSrlber*' 
fo;e bis oemaunse is , boUi man? foote fquare, Iball 
tbe fibe of tbat grounoe bee, tbat is Double to tbe fo^^ 
mer fquare platte, u^bofeltDe toas, 2 1 o. foote cuerp 
tuate." 

^(bolac, 



14 
K8 2oo(:96^;: 



^d)Olar, i^iiUcBi muHmultiplir. 2 1 -» bp ttfclf, 
ano fo baue Be tt)c luft plattc of gcoun&c, of, 4 4 1 o c 
footctbat mul! 3 DouulcaiiD tt tuill bcS 8 : o . lIuo 
o«toftI)isnojnbcr,ft)aU3J fclictbc mgljcScrquarc 
rootc.#02a lullc rquai*e,3 rc,tt ts not:bp ixafon tbat 
after tbccuenCvpbcrs, there foloiDcti). 2, iu!)ict)cis 
one of tbofc figurea^ttjat ran not begtnnc ang fquarc 
nombcr. 

tzat)ercfo2c, fefepngfoUbc 
nigbeftc roote,3I ftnoc it to bee 
296. u, ttjat 10 almotte, 2 9 7* 
footc euer^ luaies fquare. 0iiD 
fo mocbe niufte tlje fquare fiDc 
oftbat groiuiDc bee , lubicbe 
lljoiilD fcrue fo? tljat Inbole ar- 
mie, 

j^nD berebp 31 Doc percctue,tl)c oucrfigbte of man? 
incn:tut)icl)e being reqnireo to Double a fquarc plattc 
DO Double the fiDe of it.tbinUmg tbe uiatcr caff Ip Docn 

^utiftbci marfee tt iucll , tbei 
mate percetue,tbattt)et Doe maUe, 
bp tbat meanes,a fquare foiucr tu 
UKS fo bigge as tbcir firiie fquare 
tua0. 0s'bu tbis figure , mv man 
mate fee. 

J'O? If 2. be tbe fiDe of tbe fquare 
then ts tbe fquare 4*liPut if 3 Dou^^ 
blc tbc fiDe, anD ntaljc tt4. tbe fquare tbereof iutll be 
i6.U)bicbe ts.4.tj7me3.4.anD not onelp Double. 

^0 tbat tbe loote of tbe Double platte , (boultJ bee 
tbc roote of. 8. tubif be ts fomeiobat leDTe tben . 5, anD 
tberfo;e mocbe lelfe tben. 4. 

Rafter. iPoumaie pcrceiuctbcfame , iuttbtbe 
rcafon of tt, b>7 tbc 1 8. p^opofitton of tbe. 8 . bookc of 
K«c//Vf ,a$ It t0 before allegeD. 

liSiUnoUi fo^tpH^ietue tbe larger Ijfe of tbis rule. 





: ' 


! 



The extraSllon 

feoirapbical ST^^^rc be.2.t0nncs, aS Chicbe/ier ann Tbry^? iuI)lcI)C 
ipc ^outljc aiiD ijio^tljcanD bctlucnc tijiin.i 2 o.mu 
Ic3. a tbirtje tounc as ExcejlerMti) plaint' tSil c(lc fro 

Chiihejler, I 2 o, milcS. 3 Dcfirc tO UuolUC t\it luHc 01^' 
flauncc of Toke from ^xcejltr, 

g>cl)olar» gmuttfetttjore, ^ touncs.mfojincofa 
&ianglc, tuitl)^ 
tl)cir DtaaiuKcs: 
i!s Ijere is rtp^c? 
fentcD» mfjrrc 
^.aaOctf)fo;rx-;M 

/^fr,|.Cfo;ro% 
:^ttQt!)cnacco; 

Dvngtotbcrulc, , 

3!multiplic,i2o» !B, 220. c. 

fquarclP:anD it maUctlj. 1 4 4 o o » Liiicluafrs 3; Dooe 
nuiItipi!c,22o.aitDitvdoct!|.48 400. 

^{Kfebotf)enomlier6 3Efi)alliornc inonr,nnt)ro 
tjanc 3;,6 2800. lo^ljofc roote is tjerr nigl),2 s c.niitcs 
anD 5'ofamiir. 

0nD tl)at IS tbetruc Ditacmcc 

aiTorkstimExce/ltr, 

liSp tljis crampic 3? gatfjr r, 
tljat tf)is rule Doctlj !)clpc to Gf 0. ^ 

^'''&'':J^^ ^^ ^^^^^ ^^^ ^^^^ P^'"^^«^ of anf f ctintric. 

seaam 5f^Il)oulDaanDrtrtp;2opounDrngcra^ 
pics of tins ruletjuto pou , ur^ng but one fo;cuerp 
arte auD fcicnccano fo; eucrp Different kinDc of conu 
moDioufe p^actiTcnf luoulD nialte a greate boolte. 

4uti tbercfo:c omittpng tfjat, till occafion feme • 
tDerb:>aies,3i UjUI p;oceaOc 1 tbc eFtram on of r«^;le 



• . • » -^ ^ 

4 



.€>f 



of^ootes. 
OfCubike rootcs. 




J^. iKnanp Cuhikenomhsr is p:opoiI!t' 

tJI i^ftcrtfjc nombf r is lu;tttcn Domic 

'^-'^ o:Drrlr:vou fijall fit a p;iclu- tnDcr 

I tljffirftc figure :anDl)no:rtl)r. ^. 

anD fc tjiiDcr nmy tt)irD figure , o^ 

^jnitttynqriliU.z. t'tgurcsljiipacUcD. 

nnDlcoUc t)olu uiany p;icUc0, vom' nonibcr Ijatfi, 
fo manp figurciJ iljall tl)c rootc of ^oi^r uobcr contain 

Clicn to begin tbc fearcbe , fo; tijr firfte figure of 
tl)e rooti \n t\m o;t)er' rou tijall loouc lubat mate be 
tl)c rootc of ti}c namber, belonging to tbc lafl p;iftic 
tolunrD tl)c Icfte l)anD:. an5 tbat rootc l^aU rou fate 
bp a quQticnte lui?,as vo" O'D m fquarc rootcs. 

anD If tbc lubolc noniber oucr tt)atp;icUe,bc a c« 
Inks nomher,vm tl)ali canccll It all. But If It bee no C« 
hiks no.nber,K\\n\ fubtractc out of It, tljc greatettc G^^c 
m It, anD cancel! tbc lutjole nomber , ano fet tbc relic 
oucr itras pou did m fquarc rootcjs. 

i3ut conrtD:rpng,tbatpou ougbtto fiauc in reaDp 
rememb:aunre,alltbofcC«i'/^eroo/-fj,\ubicbcbcD!gi* 
tzB^ luttl) tl)c Cubes tbat tbn ma[;r: fo; U)itbout tljrtm 
vou can not pjcccDc \\\ tbie U}oo:lie. % tbinUc it gooD 
to fet fo:t!)r fierem a tableau thofc rootcs Umiubcir 
c«/'«, tbat tbcrbp i^ou male be tl}c nio;e j 
aDfurcD m tjmic of pour U)o:Ue. J-o: els 
a litle miftaUpng,migbt be tlje occafton 
ofagreatccrroure. 

anD nolu fo: this firH rule X fate, as 

3; faiCD QtSfme rootes.XiMQ ftjall bc CUCr 

mo:e the firfte luoo:Ue,anDn)aU net be 
repctcD in anp one Cuhike nUer. \M bcrc 
as all tbe ot^rr rules fololupng,ft)al be 
fo cftcn repcatcr,as tbere arc parUrs m' 



I. 


It 


1 


8, 


^» 


2 7. 


4- 


6 4» 


f« 


MS. 


6. 


2 16. 


/ • 


H^ 


S. 


^2. 


(\' 7 2 9. i 



The extraSlion 

?outnombci% 

2, anu of tbetm tW is ttjc firfte: tftat vou fljall triple 
tljc firfte roote. ano ttjat triple l^all rou fet fcnnr r tbc 
nerte nomfaer, tcluaru ttjc rtgbte l)anDc , before tftat 
p;icbe,tul)Ul)c rou Dtti lafte enue. 

3, Cljen ttiultiplie tbat triple , fap tbcfamc quotients 
ano fet it Doune tsnrjf r tl)e firtt triple : anti tbat notm 
faer (Ijall be calleD ^our Diuifo;» 

4 SCl)trDlF,lokc out a ^tto/i«#,tl)at mate Declare Ijolu 
often tlje ouitfo; is in ttie nomber ouer it* 

^n tuljicfte Dopng,f ou mull taue tijis regarD,tljat 
bettuene tbat p^icke tbat is enDcfi, anD tbe nerte tbat 
llanoetb totuaro tbe rigbt banDc , j?ou muft fubtratte 
2. otber nombecs, SDbat is to faie , tbe fquare of tbe 
lafte f «of/V»ftf,multiplieD bp tbe fo:mer triple, i o, 1^;= 
mes: anu tbe Cube of tbefanie quotiente, 

^cbolar, SDbts rule is tjerp obfcure in tuoo^Dcs. 

5^after, ^benioill 3! termc it tbus, 
j,f.5» 2^afeetbefquareofpourU)bolej«e*jf«/f, 500, fp.' 

4, mestauD tbat tljal be i^our Diuifo;, 2Cbcn feUe a ncluc 
^«o^iV»/e,Deciarpng boto often tbat oiuifo; , mate bee 
founoe in tbe nomber, tbat Doetb belong to tbe ncrtc 
pjtcke. ^utfo tbat tbe fquareof tbat nclur ^nfticnte, 
multiplieD bp tbe laUfwo^/Vw/f, 5 o.tv mes; anD alfo tbe 
Cube of tbat neiue ^uotiente,iovncii ail in one fomme, 
mate be taljcn out of tbefame nomber^ant) If pou fciu 
tierftanDe tbts,tbcre rctletb no nTD;je otflTicultte. 

jS)cbolar» JtruftbpevaplejtotnDerttanoitbetter 
gaffer* 2Lben tafee pou tbis eraple. 2646^^92 
tubicbcS; (ball fet ooune ano p;irUe, as 3 taugbt pou 
befo?c:ano as pou mate bere fee. XM bere tbe-3,p;ic=; 
fees Declare ibnto me, tbat tbe roote 
^ijillbaueo.ftgures, 2646; ^9 2( 

anD tbcn tiiDcr tbe pjicfec tbat * ' 
ts mvtc tbe leftc banDe, tubofe nomber is.2 6.3? ftnoc 
tbe greateUe Cnhih^enmkrtQ beet8»anD bis rootc,2. 



j?o:.2 7.iul)icbe 10 tl)e ncrtc Cuhe,i$ to grc^itf .^ 

png.S.gi DOC abate out of 2 6.anD fo rfmaiiictl). 1 8- 

Sl^bat. i8..^0ocfctteoucr ,2 6. t 
tut)icl)c iJ niuic caiiccll : anD then i i S 
ftaiiDctb tl)c nombct , oxs Ijcrc jjou i 2 6 46 S ) 9 2 '2 
Ooc fee, 

iTbts 13 ti)at firtle iDOo;hc,lut)tc{ic is not rcpcteD. 

'C\)tn to p:oceDe fo:U)arD,3: Doc triple tl)c quotunte 
2, anD fo banc j.6.U)l)icijc I l^all fct tinDcr. 4. bcrng 
tbc ncrtc nombcc , on tbe rigl)te tjanDc of tbc p.2iclic 
tljat 13 euDcD. 

anD tbat triple mua^J multipUcbp tlic firft quoti- 
f tt^f,iul):rbp amoutetl) tbat nomber, tbat muft be tlK 
Diuifo::anD it is m tl)i3 luo^he 1 2. 1 . <^> 
\uljicbcmuCbefetl3nDcrtbefame | ^^ .^^^o-^^- 
triplciasbcreJbaucplaccDit, -^ -^O,)^^;.. 

SSbcn iljall I ii:\kt fo: a ncluc | ^ ^ 
^Mof/V?;ff,Dcclar|^ngbolu often tp' f 
nies. I 2.maic be founoc m ttje nomber oner it,tl)at \& 
1 8 4. ilnD J fee it male be in appearaunce. i r. tpmcs, 
but mo2C tl)cn.9. pou (^afl ricuer talic fo2 a quotiente : 
lubcrefo.;c It appearctl) , tbat jmaiefaolDlvtake. 9* 
U)l)tcbe 3 il;aU fctte tn t\\tqmtientt tuitb toc ftrftc . 2. 
2nDtbcn iV^llJEinultiplie. i2.lobicbetstbeDiuifo;, 
bp. 9 . auD tocrcof commetb. i o8» to bee fcttc ijnDcr 
184- benctbe tbe line, lubicbe 
(ljalUuermo;c bcD.:aluen\3nDer 
tbcDuufo:. 

j^otu muae 3: taftc tbe fquarc 
of mp lade quotiente.g, (lubicbe 13 
8 1 JanD multiplic it bp tbe triple 
of tbe fo jmcr ^«o//f Mf f ( tbat is bp 
6.)anDfo baue ^^.486,to be fettc 
onctplacemo;c totparotbcngbt 
^anDc« 



18 07 4 

2: ,^^•4.6' 179^(29 
'6 • ' 
1 2 

486 
729 

16 389 



The extraSiion 

Uaffc of all;. 31 Ojall ntwlttplic tbe laffc quotkntt Cn^ 
hihh\mxi tljat mafeetl)*? 2 p.iubicbc muft be frt , vet 
one place mo?c totoacD tlje ngtjt tjanD,ttjat is to faic, 
tnDer tljc ncrte p^uUe. ^tiD tften ftjall 3; aODe tbofe ] 
fommes tnto oite:Ujl)ccbp tuill rife* 1 6 5 8 9«to be fub^ 
traitco out of* 1 8 4 6 5. ano fo luiU remame oucr tbat 
p?tche.2o74. 

0nD tJje luooikc of tljat p^icfee (0 Doem 

STbi^s o^Der of iuo?lje,if t^ou ntarfee tuell^^ou baue 
learnctj tbe lubole arte of emattion of Cubif^erootes. 

fti2 boiu greate fo cn€t pour noinber be: pou Ojall 
not bauc ain> rteloc feinoe of luoo^Uc. 

li^utpetbiraufe^I Dioteacbe rou before, tbcfame 
iMQoM in otber luoo?oes , BI iiJiH iuoo^Ue tbcfamc e^ 
irample agaiir, arco^Dpng to tf)c(c luoo<iDes. 

aiiD firtte, after tljat tbe nombcr is fet Donne, nuD 
tl)efirft Gihiks roo^f taUen,anD tbe Cube abatcD. Kt)cn 
take tbe fquare of tbat rootc ♦300, tt^mes, ttjat is 111 
tbts erample,4.ti)me«.? o o,U)l)icl)e mafeetb* 1200. 
ano tbat l^all be ^our oiuiTo?» d)ts nomber, ano all 
otl)ermtl)tstuoo:fee, ftiallvcu fctoouncfo , tbattljc 
firfte ncntber,(]^all be ijuoer t!)e nertc p;:tcUc,toU3aro 
tberigbtebanoe* 

%i)cn ftbe^outfmfienteyijjith tbe former rautele, 
anD It Uiiil bet, 9» xm b£rcfo;:c 
ntultlpltpng. 1200. bp. p^tbere 
tutu amounte logo o,to be fet 
tjnpert'oeltne. 

aftertbis, 3IltjalltaUetbe 
fquare of, 9. (Uibtcbe is tbe nelu 
^mtknte) anb ttiulttplie ttbp.2* 
(IcbKbeluas tbelaHe quHtente 
befoie;, o.tpmcs. ^0 mufteSi 

multiplte8 Lbp,6 o^anD it Uiill malie,4S6o*"U)bicbo 
3 place oiDcrl^. 

%\)tn fet 31 JDounetbeGt^f of tbef«<r^«»»^ff, Inbicbe 

tnafeetb 



18 

2:/^^46t)-92(2 9 
1 200 

1 o Soo 
4860 

729 

16589. 



of^otes, 

tttafectl)»7 2 9. ^iiD f arc tbc ], nomUcrg pUccb,anli 
agree luitt) tbe former luoo^Ue,tn alltbuigcs/auc m 
-^.pomctcs. JFoj here tljc triple of tl}c^"</^/rtj/f,i0 not 
fct Doune^but fecptc tn mcnio^ie. ano again, Ijcrc arc 
Diucrfc crpt)er0,lut)icl)c arc not ui tljc former luo;fec. 

fecbolar. &tr, i pcrcctuc , tbat tljc Cvpbcrs Dooe 
notbvng els, but fct tl)c nombcrs m tljcir Diic places. 
>1nD tl)c\nplc of t[i£quoti€nte,\si fupplicD m \iio;\xC bp 
:.nTulttpltcations.f irft bp. 5 o o^anD tf)cn bi'o 0,^0 
tl)atiti0alloncmcffcac. , , . 

anD bv the one luo;Uc.3e bnDcrlianDc tbc otljcr tbc 
better: U)t)cn 3 compare tbcun botbc togetber. lout 
vet J p;aicvou,cnDctbeU)Oo;Uctbatrou began. 

i%aflcr, 2:0 contuuie tbat Uiooihc, firllcjj mutt 
fct Doune tl)c nombcrg^ao tbci fljoulD reniamc, aftcc 
1 6 5 8 9. is abatcD out of. i S 4 6 5« 
ano tbcn luiU tbei ttanoe tbus. 

Si:f)enft)aU3I repcatctbc fc;^ 
nterluojbcbpfcttpngDounetbe 
triple of all tbc quotient c^ijDW^t luill be. 8 ?♦ anO tljat 
muabeplaccDtJnDer.4^ 

4]5crtc tbat J (^all multtpUe.tbat. 8 % bp . 2 9. anO 
tbcre tuill comc.2 s 2 ]. lubicbe mutt be ti)c Diuifo;. 

;i23!)erfo;e33 fckc fo;i a nclu 
fuetiente, ttjat mate fljeUie me 
t)otu oftcn.2 s 2 ]A5 contatneD 
in. 2 07 4 ^ 0nD It luill bec.8« 

%\)at 8 D0e| fct m t\)C quotient 

anDbp ttl multipUc . 2 y 2 5. 
ant) It giuctl).2 o 1 8 4 lobicbc 
3 fette Donne^as bere ^ou fee. 
SDbcn Doc 31 multtpUc tbat 
quotient fquarclp^anO tbat lull 
be 6 4. tllbicbc J ftjall multiplie bp tbe triple, tbat is 
S 7, anD tberc luill amountc. y y 6 8. to be fet one place 
mo;z totuarD tbc rigbte banoe 



2074 



2074 
'2:/5'4j6'^5'9^-(29S« 

87 • 
2 y 2 ] 

20 I 84 
2074^92, 



iD,^ 



3ia(E 



The extraBion 

llaftofalljBlmuIltafectbcCttkof.g^tljafig^n^, 
auDitr^allbec fvttepet outplace mo;c totuaroctbe 
I'lgbte 6an0c» 

:anD tben bp aODitto, 31 fljall b;png tl)c all into one 
nombcc:anO it UitU bee,2 074^92, lul)icl)c is equall 
luttl) tbe tul)olc nomber aboue,tl)nt is tJucanccUeo, 
auD t\)ttfo;t If 31 abate tbe one out of tbe otl)ci-,tl)ecc 
iDill remain notbi?ng» 

timi)ei:efo;c 31 fee, tijat the firttc nomber, is a iafic 
Cuhil^e nombcr»0nD l)is rootc is, 2 98* 

^cftolar. 3 banc marUcD ^ou fo tucll, tbat 3; triitt 
to Doc tbe IlUc,U)itljout errourc, 

116ut3|p;ateiJOuU)oo;kctl)islafte parte alfo , bp 
Voarfeconoerule,aspouDiDU)oo,2Uctl)eotl)cr:tl)at3l 
itraie fee tl)e Due agrcmcntc of ttjrlm botbe : anD alfo 
pcrcciuc tlje rigl)te tjfe of t^is tuoo;lic , tlje better bp 
tbatotberfojme* 
Thefeconde cpatter* 31 innff in tt)at cafe fetteDounetl)e nom^- 
'»oor/{e, btth , as thti tuere fet in tt^c 0^ 
ti)tt \j)OtiM* HuD tbcn 3 fljall 
multiplie al t\)t ^mtiet, tubiclje 
is»29» bp it felffquarclv^auD it 
Xuill makers 4 1» Uiljicbe mull be multiplieD bp» } o o^ 
^nD fo tbcre amountetf),2 y 2 3 o o. to be fette Doune, 
asftercpoufee* 

Cljen3! lljall fefee out a ^m-. 
tienti^ Declarpnge 1)0 tu Often 
2 5'25oo,maiebee fouuDein 

2074^92. ?in,D tbat quotiente 

iuill bee,8:\ul)tcbc 3i fet in tbe 
5«o^/V»^roome,lyiti) tbe otber 
nombers* 

0nDtben3lDooe multiplie 
tl)e Diuifo;^ bp tbe qmttente,tim tbereofrifetb 2018400 
iubiclje 3! fet tJuDer a line, as pou male fee, 

^ejcte tW,J Doe multiplie tfje neiue imtimt, bp it 

felf. 



2074 



2074 

2^2 j!oo 



2 o I 8400 

n68o 

5'12 

2074592" 



of^otes. 

fdffquardri luiicrrof commctlj. 6 4« anti 
U)at fquarc ott\)c lall ^uotient,^ fljall muL 
tiplu' l\\ 87 o. Ii3i)icl)c IS. 1 o. tunes t\)c tn* 
pic of ti)c fo:nirr qmtiente, 2 9 :anD tbcrc of 
conimctl) . > ) 6S o. U)!)icl)c3; fct Domic o,b 

fo o;Dcrlv » 
Lattc of all. ?: multiplic. S. ( tijat ij3 tl)c 



64 

_ ^''''^ 
"4480' 

,_ ., ^_ . 112 

\T[.^^ quotievte\Cuhil;jly ,iim\imVi\\tt\i,S \'^' ^ ^- 6 8 C 

iubicDe alfo J fct Dounc \\\ coucntrt o:Dcr. i 
0nD tl)cn ft)ail 3 aODc tbcim all togctlirr . 0nD fo 

Ijauc J tlicfamc fommctljat J baD before in tljc otljci: 

fojiiKU* U)Oo;Uc,anD It 1S.2 07 4 ^^ 9 2. 
s>cl)Dlar. jncaDc no nio:cinftrurtton foulufi: 3| 

tljuiUc mv fcfrro cunnrng, bv occasion of rourcram-' 

pics, luljicbc vou Ijauc U);ougl)tc fo in Double fo;mc. 
il^aHcr, E'tjatmaicvou p;:ouc, b^tbisnombec 

47<S52I47» 

fecliolar. J^irffc % fljall p;tchc it , ass po" taugljte ^» of^^^y 
mconnttrngfiUL:. nontbcrs. exambh, 

anDtl)cnoutoftlK nombccoucrtlic laftrpiiffec, 
3 ll)al fckc out tl)c Ctihike rooff ,anD abate the Cube tljcr* 
of,outoftl)cfamcnombcr:ranDfcttbcrcmaincrouor 
iC^canrcUpngtbcrcftc. 

<anD fo ui tilts nombcr , 3; finDe 2 o 
tn.47.tl}c grcatcftc C«^f to bcc-2 7- 4 7^8 32 1 4 7 (^ 
anDtticiootcofit5.CCll)crfo.:c3a>- * ' 
bntc.2 7.outof.47.anDftnDctl}ercItctobc.2o,tI)cr^ 
fo;c 3 canccll,4 7.anD fct.2 o.oucr it. i3nD tijc. ^ \m\\U 
t\)t (^ tJic rooted fct m ttjc quotient, ano fo 10 tljc firll 
tuoo:UccanDcD. 

XClKn DOC ;< triple tl)iit quotiente , anD itmafectl).9. 
tubicljcj fctDounetjnDcc.S. 

again 3 muitiplie tl)at.9. b^^. anD it rclDetb. 27* 
tul)icl)e 3 fet tinDcc tbc triple, anD taUc it fo;t mr Diui^ 
fo?» 

Mt)crcfo?e 31 D^all noto fcltc a qnotknte, tljat mate 

£>.tK Declare 



The extraSiion 



20 

4^/832147(57. 
•9 * • 

27 



1 89 
441 
___J4i 

256n 



Beclate bolo often.2 7» i0 tn.2 o 8 

ano 3 fee, it luiU bee ♦ 7 ♦ tpmes. 

SI^berfo;5e33 fetteooune.7* in tbe 

^uotiente-.mh h^ It 3 multipUe 2 7 

aiiD itinaUetl).! 89*tiJbic*)C 3 T^t 

tjiiDcr tl)e line : ano tbcit 3 uooe 

inultipUe ♦ 7 • bp »t felf , \ul)icl)e 

niafectl).4 9 ♦! tbat fquarc Doe 3 

multiplie bp tbe triple of t!)c fo?^ 

tner j«o^/>ff^^tl)ati!5,bp,9.ano it pelDctb. 441. \Xil)U 

t\)t 3 fet one place mo?e toUjarD tbe ng^te bauDe. 

3laft of all, 31 taUe tbe Cube of»7» Ujbtc be is« 3 4 ?♦ anD 
tbat Doe 3 fette Doune, pet one place mo;e toluaro tbc 
rigbte banoe. 

ibefe, ?. fommes bepng aobcb togetbcr, boe mafee 

256n» 

Spatter. SCbat tuill be barwlp abateb out of a lef« 
ferfomme* 

^cbolar. 31 fee notu ntp errour. 3 mull take a IcCTc 
j«oficn*:tubicbetbtng31 ntigbtbauE perceiueo bp tbc 
feconb^ number. i*^o:tbeitU)oo tuerto grcate,befo^ 
tbe tbirbe iuajei adoeo. 

^0 tbat3l IboulD bauetahcrt but.6.fo.2 tbe quotient 
0nD tben tuoulD tbe ftrfte nombcr baue been but 1 6 2 
ano iXyt feconbe. 324. ano tbe 
tbirbe . 2 1 6 . but tbat tbeir pla* 
cpng luoulD mafee tbeim to te of 
otber tjalues, faue tbe laft of tb&. 

2^berfo;ie, 3 feteuerptjiteiu 
bis Due roome : ano abbe tbeim 
togetber, anbtbereamountetb 
1 9 6 5" 6 . to bee fubtratteb out of 
20852. anbtbe remainer totll 
be 1 1 7 6. ^nb tbus tjj tbat p^icfee 
tuttb bis lDOo;fbecanbeb. 

%1im fo^ tbe ney te p^icfee, 3f t^eate tbefame berp 

fojme 



I 

Z9i\ 16 
^71^-iZl 
'9 

27 


47 06. 


162 




324 

2 16 




I 96y6 





1 I 76 

' lo8 • 
^8 8S 

I I 664 
972 

iz 

I I76I47 



of^^otes. 

ftintt of iDojUc again, j^icft fcttt?ng Dounc tbe triple 
oftl)c\j)holcjuotienfe,\jitnc\:\ciii. 1 08. fo tljatitfljaU 
ftaiiDctnDcr.i i76K0?t)nDcr.76i. aaoumpt^ng &;? 
gurcfoi figure. 

Cbat triple mull 3: multipltc againc bp tbe Uibole 
^ttof/>n/f.56.anDitU)illnTaUe. 5888. tuljuljenomber 
3 muttc take fo; mp Duiifo?» 

t2It)crfo^c3;fcUc l)olu ma- 
np timc6, 3E mate ftiioc tbat nu 
uifo^m. 1 176 f.ano 3 fee, it 
imlH)cco.tpmc0. Si:i)crfo;:c3! 

fct.^.aS mp quotienteAn 1)10 DUC 

place: aiiD bp tbat qmtimt 31 do 
miiltiplie, 5 8 S 8.anD fo Ijaue 3 
fo; mp firfte nomber. 1 1 6 6 4. 

£!gamc3!Doe multiplictbe 
laltc <imtiente, ^. fquarclv,anD fo banc 3?. 9 . lubicbc J 
l^all multiplie bp tbe triple of tbc former ^«cn>w/>anD 
it pelDctb.9 7 2. tbat lljall be fct mo;e mgbcr tbe rigbt 
banDe,bp one place. 

s:i)irDlp,3! tafee tbe Cuhe of. 5 . lubicbe is. 2 7 • anD 
tbatDoeSifetpctone place mo;c. to UiarDc tbcrigbtc 
banDe. 

2Dben Doe J abOe tbofc 5 fommes into one.ano tbei 
make. 1 1 7 6 1 4 7. tubicbe is equallc fomme, iuitb all 
tbe nombers oner it,tbat be tjncanrcllcti. 

t£lberfo:c l faie tbat.47 8 , 2 1 47.1s a Cuhik« norm 
ifj-^anD tbe Culfihe roote of It is. ] 6 ;. 

£l9after. j^oU) Doetb tbe ojDcr of teacbrngc re- Thenlgbefie 
qMire,tbat l! l^aulD inftruftc pou, boU) to ertratte tbe roote in am* 
\\\^\)<:^z Cube roote ^ outof aiif nombci', tbaticnota ^fmo^ (.«' 
true Gihe, 00 tbis nomber fo; erampic maic fcrue. likj, 

694r829n. 

^berefirftej muftc ertracte tbe nigbellc roote, 
as 3! tangbte tou,fo; tbe nigbeffe Square'rootes, nt no^ 
bers tljat are not fquare: anb tben ft^all 5 note tbc re^ 

£>,itt. mamirr; 



I 8 2 



The extraSiion 

maiiTcnlu!)icl)c3; C^all fetfo? t!)c itumerato;j.0n6 his 
dtnominato: l^all be foimDcaie; 3 Ujill tell pou ancn. 
j5ut ftrac DOC T'ou iDO^fee tbe.ejcamplcto ftis nigljcfie 
rootc in iul)Dlc;nombf r0. 

^cljolar* 3 fct tt Dounc, and 
pikkt it, ano ftiiDetbe grcatcfte 
Qihe oucrtbe laftc p?ichc to bee 
S I z.antf tije roote of it t6,8» 

Wtl)ttfo^t3 fet^Joune♦8♦mt^e^«o*/>»f^ 0nii3! a^ 
bate.j- 1 2»out Qt6 9 4.ano fo rett«tb 1 8 2»ano tlje fo^ 
mer.6 94»canrelleO» 

%iim to p,:oc0de,3| muff triple tbat roote, 8, antj it 
mafeetb»24,U)l)ifbe.24.3fftt!nDer.i 82^ anotbcu 
3! Ooe multiplie tbat again, bp tbc quotiente oi roote.8 
ano it ntafeetb 1 92,tobefet\3nDertbefaieDtriple,2 4: 
as tbe muifo?» jFoj trsbtcbc 3 fcfec a ncU?r quotient yunn 
it tDill be 8. SLbat.S.i fet ut tbe quotients i.lacc,anD bp 
it 3 multiplie tbe Diaifo?, 1 9 2, ano tbcre rifetb* \)]6* 
to be fet l3m)er tbe Ime^m conur niente o;jDer, 

i^erte 3 multiplie tbc quotkntc fquarelp : iobiclje 
^clDetb. 4 6, anB tbat fquare 3 m ulttplic again bf tbc 
triple, anD fo baue %\y^ 6.alfo. :515ut tbi0 muil ftano 
mo;e fo;ituarD!p bp cue place* 

2laft of all 3 take tbc Lube of tbc ^«o^/Vw^S♦ano tbat 
tei^f 1 2, lubicbe 3! fct tjnocr tbe 
otber tluoo fommes , ano tbat 
b? one place mojc fo^iuarDlj?* 

jjiotu gatbcri^ng all tbefe, ?♦ 
f6me0 into onctbei i»illmalie 
1 6 9 4 7 2 lubicbe 3 ftjall abate 
out of, 1 8 2 y 8 2 ♦ anD fo remain 
mtb tbere .15110. ^nD tbat 
p^tcke iDitb bis iDo^feeeanDcD. 

^I^berfoje bau^ng one otber 
fpacato Uio^fecBi muH repeate 
ibefame o^Der of lno^fee again. br tripling tbc Xubolc 

t^uttientt 



I 3 
if^z I I o 

(^9^>-^-2:9n(88t 

24 ' ' 
I 9 2 



169472 



I 42 s 
J'.5'i i'^i^ 26 

' 264 * 

25252 


I I 6 I 60 
6 600 
I 25- 


I I 68 2 I 2 5" 



qMttente, 8 8 . aitD tbat luf U bee. 264* ^tiD agaf ne 3( 
mud muUtpltc tl)at trtpleoe nomber, bp tbcfatcD f m^^ 
tiente^m'^ It luill niaUc ♦ 2 5 2 5 2 ♦ iobtcbc l^all bee tlje 
Diuifo:. 

i22at)ccfo:c 31 fckc a nctuc 
^Ho^/e«fe,U)t)icl)c 13 caQlp pec 
fciiieD to \ic.s> S^bat vDoc 3 

fCt in tbc quotiente , ailD b^^ It 

31 Dooc mnltipUc tbc Dcuifo^ 
2525 2.anD tocrc amountctb 
1 1 6 1 6 o . ns tbc firftc nom- 
bcr^tobccfcttJiiDcrtljcltne. 

ilgatnc 3^ fljaU multipUc 
tl)CfMof/>»tfquarclp, lubicbc 
giuctb .27. anD tljat fquare 
Sjall J multipltc by t()ctnplc.2 6 4»anD fo tuilltbere 
nfc,6 6oo, tobccfcttc, a^tbcfccoiiDcnombcrtjit^ 
Dec tbc line : aiiD one place mo;e fo^tuacDlp, toluacDc 
tberigbtcbanDe. 

Ilaftofall, jy fl)allfettet)nDcctbcmbotbe,anDoiic 
place mo;c toluacDe tbccigbtc baiiDc, V^cCuheQty 
iubicbet$,i2y, 

0nD tbi^n il^iill I aDtJe all tbofe. 5.rommes togetber 
of lubicbe commetl), 1 1 6 8 2 1 2 y» to bcc abatcD out of 
1 5 1 1 o 9 )- KaiiD fo tbe ccmaiitcr luiU bee. 1 4 2 8 8 2 6. 
<:^ti)cccbp 31 fecjtbat tbc 6cttc nombn* tbat Uia^ pzo- 
poncD, 31 meaitc 6 9 4 > 8 2 9 f i is no G^^ik? nomherMxt 
tbc gceatcttc G^he in it 10*6 9 5 1 )' 4 1 2 ^anD bis cootc 

!B» 8 8 y 

anD ro,3 fee,all otbcc nombccs of liUr UinDe mud 
beeU):ougbte. 

^ut nolufoj tbecematncc , boto ll)all3;Doocto 
b^pngc It tjnto a fcaition, tbat maic aptlp crp;cnrc tbc 
r jbede coote in tbat fojte.-' 

(paftcc, SCbcrcbccaamanptuaieg, astbcccbee 
tujitccict almodc ? fo: euccp mmm Dcuifctb ? boto f 

b?r"5 



G^rdanc^ 



Scheuhill, 



The extraSiion 
b^^ngc it mo0c nigWe to a true roote , ifmv focfjc 

yjOCVt:h)l)ttCOfGriiane Ijlg rulc 10 t\)i6. 

Multiplie the roote fquarely^ and againe hy 
^.and that nombcr jhailhe thediui/i)r yntotbe 
rcmamYy 

tCl berc be nng!)t Ijaiie ijreD mo;e plameflTe in U?o;« 
rje0,if l)e ^ao faicD:anD tl)at nombcr fl;al be tfje Deno^ 
mmato; , to tl)e remaiiter. tICl berefoae aj3 bcre four 
roote 10.8 8 >■ fo ts t!)e fquare of tt? 8 yiis aiiD ttjc txu 
Pl5oftbati0.2 34967y- S)0 It oulDtbat fraction bcc 

' 'feut fjolu ntgb tljt0 Doet!) go to tbe trutf)e,3; Icauc 
tt till an other tf me. 

Scbeuheliui Doetl) allege an otfjer reafon, ano infcr>- 
retb an otber o;Oer,Diuerre fro tljis, ano foclje as n\v 
Ijngneth tl)i0,raif ng: 

Triple the roote , and the fquare of it alfo, 
and adde bothe thofe nombers together , andi. 
more:j{ndfo haueyou a denominator for your 
numeratour, 
%i)e numcrato: euenno^c (0 bnoerttaD to be tbc re- 
mamcr. Il5v tobicbe meane0 tbe frattio m tf)i0 U>o;Ue 
luoulD bee •^^ : lubicbe 15 a lefTer fraction bp a gooD 
tjeale,tben 10 tbe former frattio, after Cardans fojme. 
i3ut bfcaufe at tbi0 pjefcnte, 3 mate not fpenoe fo 
mocbettme, to fcan tbeir reueralleoptnion0, tobere^' 
in ecbe of tbeim, pleafetb b^mfelf luell : tbe one alle^- 
gmgoemonftration (Uibicbeftarfelt^ feruetl))anDtbe 
otber nampnge it a fecrete , 30 it 10 luo;tbte to bec:3! 
tuill p:oceoe to a tbirue tDate,mo;u; certain tben etber 
of tbcfe botbe. 0nD tbat is b^ aODition of certain Cr^ 
pber0,to tbe remamer, in focfje fo<ite,tbat tbci muHz 
aUlDaiesbeeternariegjag.^d^p-o^^iz.fc. ^notben 

fearcbc 



of^otes. 

fcarclbe fo^toarD tuttti tbc Itbe o;Dcr of iuo;lie, a^ tou 
\3fcJ3 before. 

5n tois maner of p;amfc,looIxc ftolu manv p;icbr3 
Vcur cipt)cr0 hatli co; els IjoUj maiiv tr rnartcg of Ci^ 
pl)f rs, tijcrc bf c fct to pour uombi r) fo nianp figures 
ttjall tVx numerator of pour framon contaim TinD tl}c 
Dcnoimnato;ift)allcutrmo;cTContaini.nio;i\t:^l)fr^ 
of tijc laftc onclp fljall hzt an tJintic, ano all tijc ot^cr 
lljallbccCppbcrs, 

2:t)at IS to falctbat (f 3; aODc but 5.CipI)cr£j to tbc 
iiombcr,tl)c fraaion fljall contain certain, i o. partes 
SnD If i? nJ3Dc.6. C:vpbcr£!,itfl^aIIerp;eire.i o o. par^- 
tes. ^0.9. (Ivpbcrsmaiictljtbe Denominator to be 
iooo.parte6:0nDi z.Cppbersgcuetl) loooopart:^. 

j^o: crample. 3; tuill aooc to our lalte no:nber tl)at 
remamcD.i 2*Cppbers,J?.nD t!)cn U;iU tl)c ncniber be 
1428 82 6. 000 000000000. Ijuto Uji)ui)e 3 f^t 
no more piicUcs^ tijen fcruetb fc^ tlje cipbcrs^bicaufc 
3 bane )^cii\f:\i all tbc otbcr p;ickcs , tn n:p fo;niec 
luoo;'dc. 

xir.D noti) to rontfrtiiemp tuoc;Uc,3: (Ijall tnj:lc all 
t!:t fo.imer f Mo/;«jrf,ano tt luill be 2 6 y y. lubicbe nom« 

lv:i- J a-a:! place , a3 j . . ^ ^ . 6 ooooocoooooo( 
Ijnvvcuf.eit fct^ano , ^ %. •_ ... .^ 
tbcnfijalljmultiplic \ ^ .,q?^1 
thattnpIr,bptl)efo;i^ - 5 4^0,) 
jncr '/<.'(?//Vtt/?.S 8 y.lubtcbe luiU pelDe.2 5 4 96 7 y^to be 
fct tMDc r tijefaieo triple : as 3f bane fette it berc alfo. 
j^nD tbis nomber (I)all be tbe Diuifo;. 

Sbc n fball J fceUe fo; a ^uotiente , lubtcbc can bee 
none otljer tncn.6:Ujberefo;c3; fttte, 6. tn a ^uottente 
line,anD bp tbat 6. 3 Dooe multiplic tbeftiicD Diuifo; 
2 ^ 4 y 6 7 y. anD it giuctb. 1 4 o 9 8 o y o. to be tbc ftrUc 
nomber bnDer tbe line. 

j^fter tbat , 3; taUe tbc fquare of tbcfaieb quohcnte^ 
VoW^t is, ^ 6. ann bp tt J multiplic tbc triple. 2 6 y r* 

p.;, lubcrbp 



The extraSiioH 

tul)erb^<j{ matte I 18064984 

9 ^ ^ 8 o.to be tlje ' M^^^^^P-W^^^ooooooCS 8^6' 

feconoe nombcr ) 26SS ' 

254967^ 



tjnDcrtt)cliiic:f 
fct, a0itougbt, 
0ue place nio;e 
toujaco § rtgbte 
))anDe* 

ilaftofallfo? 
tbctfttrocnombt 



140980^0 
9^)80 

216 



14 I076I0I6 

r 3J tafee ti)e C«^* of tbefaCcu qattiente 
ljDl)tcbe t0*2 1 6.anD place tt 30 pou fee, loitb m firlte 
figure t)nt>crtljep?tcfee. 

^l)en Doe S ^^oe tl^ofc* 3. nomberd tnto onc,U)l)t' 
tl)cmabetl)«i 4 1 076 1 o 1 6. ^nutbatbepngfubtra- 
cteO out ofi 42 8826 ooo.tioetblcaue 1 8 06 49 84. 
0nD fo t0 ti)e Utoo^be of tl)e ftrfte p;tcke eatioeo* 

WX fjcrebp it appeacctb , tbat t^e fraction is fouic^ 
toW mo?e tijen ^ 0^ | : aa it l^all bee trieti better, bp 
tl^e U)oo;to0 tbat O^all enfue. 

2Dberefo?e 3 pjoceoe to tbe nerte pjicfee* flno fir0c 
3 triple tl^at toftole quotientCyixiW^t pcloctl).2 6^68* 
to bee ret,a0 iti0 often befo^te repeateo,anD tberefo;e 
neoetb not Ijereafter to bee teoiouOp rebearfco. 

%hat triple D^all 3| multtpUe again, bp tbe to^ole 

|tat'f^«mN°^4984ooooooooo(88,6o 

greater anti notraOlp io;jougl)te bp 
mcmo?ie) ano it ooe 31 fct m bi0 Due 
place,a0pouree» 

But t\)tn fcepngtbat Diuifojia 
greater tben all tbe nomber ouer tt, 
3 r^all ret a Cppbet in tbe ^uoticntet 
in tofecn tbat tbe Diuifo;:, can not be 
23^2 bi 6208 abated one0 out of tbe nomber oucr 

it. 



265*68, 
8 8 r 6 i 

1 5*9408 

I U840 

212^44 
2 125-44 



cf^potes. 

It. Hud fo (6 t\)c iiiooakc of ttjat p,jlcfee canocD, lultD- 
outanpmojctrauclL 

tatjcrfou; to go fo;tnart!,3? triple all tliat quotttnte 
anD fet it Douncas tfjc rule iooulDji as ijcrc is fccn, 

I ^948 I 9^S'64 W 

i^^;^^Pi^M'P>^2f^pyooo (88^607. 

2 6 T 6 8 o 
2^28620800 



I6470054T600 
I 3 o I 8 5 2 o 

; 4 



26 $"6 8 o 
49 



I 647 o 16474 3 M ? 

Si:!)cnDooe3!mult(pltct!)at triple, b? tl^e lubole 
jwo^Vn^^tuljercof comcti), 2 3 f 2 S 6 2 08 o o. ano tijat 
tt;all bee tljc Diuifo;. 3nO tl)f juotieme fo;j tt lutU be.?* 

§5>o ttjen if 3 muUiplic ttjat Diutfo^ bp.7.tt)crc tutU 
amountc. 1 6 4 7 o o 5 4 5*6 o o. foa ttjc ferft nombcr to 
bcfcttuDertbelme. 

atiD fo2 tlje ncrte toooiiSe, 3 t3(jall multiplie . 4 9» 
(U)!)ic!)e is tbe fquarc of tlje myst^uotiente ) tuitl? tbc 
triple of t!)c former ^«tff/o«^? ^ atiDtt 
tumb2pngfo;tbe. i3oi8 52o.Uj!)t' 
ct)el]^allbeett)creconDcnomber, to 
bee fettJiiOertbe line. 

2Cl)c t^irDc nombcr f^all beetlje 
C«*tfof.7.U)bicl)ei0.M^ iH o i 8 U o 

ano tbofe . ? . fommes aUOcD togetljer , Uiili mabe 

16470164743^4^^^*^^^^'^^°^^*^^^^^^^°"^°^ 
1 8 o 6 4 9 8 4 o o o o o o» ano tbcn d^all tbere remain 

if948i92y64T7'3n5fo t)aue 35 eanOeo. ^ p;itcfees 
of t!)e Cppljers. anD ttiercb^ maie faie^tbat tbe frac^ 
tion is j^'j; anD fomelubat mo;c : %W is fomclubat 
mojetbenl. 
^ctjolar. 35 fee bp tbc fraction,tbat it is | ani» t^ 

p.ij, bcfiDe 



2591120 
106 2 72 



The extraction 

beCuc tlje quantf t(c of tlje rcmamer, But 3! p^ai'c rou 

canDe tl)c tuoo;Uc of ttjat otl)cr p;ufee,U;'t;icot Dooetli 

remainc. 
Smaller. 3! miiftc triple all t!)c ^mHcnte: lubcrcbp 

tutu rife, 2 6 ) 6 S 2 h U)l)tti)c murtc bcv n:ultip!icO I)p 
tl)cfaiiDf«o//V/i/f : an:) thereof 
luill p:joceoc tt)c Oiuifo;, bepng 

25T:i'^99:i7n47 ♦ ^»0 tits 
jao^/fw/f iuiUbcc. 6. 

^ l)crefo;c 6rfte 3 fct, 6. in 
tiT ^uotiente imc, tcltt) tl)C otfjec 
iicmtjcr0: ano tijcn Doc 3e nnil> 
tiplict&c Diurfo;j bptljat inoti^ 
tnte , aiiD It b;j?ngct6 foo;ithc 

141175956^2082. s^Qim 



26)6821; 

88)6 o 7I 

T8T97747 
159409260 

15284105" 
21254^68 
2 12)4)68 



25)289927T54' 

flrOe nombcr to ht fctte tjnoer tbc line* 



185078734793024 

'^^A^i^Z.^^^^-7f^'ri^ (8^5-6076. 
26T682 I 

23) 289927n47 

il4 I I 7 3 9f6 52082 
9y645'H6 

2 I 6 

[141 I 74052 166 3976 

0nt) again tbe fquarc of 6. bepng multiplied b|? tl)e 

265682 1 
36 



15940926 

7970463 



trlple^lorll ^elDe. 9^645556: lutji- 
tbe0jailbcetbefeconDe nombertjn^ 
dertbeltne* 

3Ct)c ttjirde nomber fl[jaU be.2 1 6* 
bit aufe it is t\^t Cubt of, 6. 0nD tfjofc 
3 ♦ nombcrs bee^ng aODeo togetl)er, 9 5^ 6 4 5 5 5 6» 
uoemabe.i 4 1 174 052 1 66 3976. to beabatco out 
of. 1 5 94 8 1 92 56 457 000. ano fo Do«t^ tbcre re^ 
main?* i83o78734793o24» 

^Ijcrefo^e 



of^otes. 

mhmfQicittiticth appcarc,tbatbe0Detl)c firff,? 
nombcci3ortI)croote, tbatu. 88). tl)e rcac(tl)at 10 
6o760ttAUDeti)fonbcnumeiato^ofafra(tion, ano 
tl)c Denommato: linto tt is. i o o o o. 

^0 tUat the nigl)cftc roote ts . 8 S ^ t';!! ♦ bcfiDc tl)c 
fiamon tl)at Doctl) fc!nainc:U)l;(icl)c luoulo maUc but 

^:i)olar. ^^ijj ts a fufftcicntc p;jccifcnc2. 0nD fe 
$ luD^c tt fumciciitlp taugbtc. 

(ZfOcrcfo^c J p:aic pou pjopounDc fomc qucGios, 
tbat Doc rtquirc tl)is artc,fo; tJ)cir folution. 

iB;iftcr. 3 am coutcntc. ^nD let this br tl}C tutic, 

JLi)c Crcciaibi oMim to iDle baniicttmg.auD fochr Aqncflion 
like UjanujnucitViDiD p<:ocurc t'oercbp focl)c moUallc ofdonblyng 
frckciirffL^iJitliattbc quiche U)crcrcarfct)al)Ic to buric dC«/^^. 
tl): DcoDc. CilljcLXiQje confultpngc Uiitti tl)cir C'0o= 
D:0,fa; rrO;circ thereof, thci rccciiicQ auufiucre^ihat 
lubcn thet luonlt) Double the llltare , luhiche i:;a3 of 
C«^% fo,::nc,tlKi fl)oulD bcc DcUuf reD fioni t!jat pta. 
guc. iBeanrngc t'^at learning 10 a Due iiuancto Dc - 
Hucc rcalincs ficm plagues aiiD eno:nnt;e£;» liSiit to 
thcque!l!on,U)i)atraicvou:' JfthcfiDc oca iubeh^.i, 
footc(a0 that altarr might htt) holu manv footc lljaU 
the frDe be of that Cwl'f^Ujhtchf muft be Double tjnto (t. 

Scholar. SI^hiB 3 cofiocr. 2Dhat ftrdc 3; muft finDc 
XMt quantitic of thc'c«/'^that is pzoponcD. anD then 
fljall I Double that quantttic. 2:hirDlv> . 3 Jna2:: cr^- 
tractc the C«^% ros^if^ef that Double uombcr. 

S>o m this queltion, the fiDc of the Unolncn Ghe is 
3,aiiD therfo;c the luholc Luhe (0,2 7. Uihofc Double is 

5" 4. anD the C^hlks roote 13. 5. attD t; hv Grdancs rulc: 

SDhat 10. 4, Uihichc is plauilp fa[fe,fo2. 4. is the I'ootc 
of. 6 4.anD not of.v 4. iSut bp SihsuhsUm rulc. 11 luU 
ije.;{jthatis.SialmoSIc:luhtchci3iiioihcnigh!;rthe 
truthe. i'o^vvmultrpUcDCt/Hf/)*, DocthmaUr. si, 
J ; . tuhtc'^ is to title bp a gooD Dcalc , that is bp . U,- . 

p.tu, iuhcrcae 



The extraSlton 

Idljercag.? ^ ooet!) mafec a leflfcr fomme:tftat (« to fap 
but 5 1 '"".ano fo Ujantetlj.2Vr6a-3nD altljougl) botlje 
t^cfc fommw goe nigbcc to tht trutftc, tben CarJdnes 
ralctufjicljeimflrettj.i o. U)]bolr»?^^tttTaieitOcea(ilp 
fecn,tbat5c^««^e//M rule ts not To gooD , as tje luoulD 
it Ujcre, ano tftc Uiojfe l)ere,foj tlje aoo^nge of tijat 
one mo;tc, 

S0aacr. ^ ou are lepte tjcrte fooenlp from a fcfto^ 
lar,to a c optroUer. ano pet 3 can not but p;aife i^out 
OiUgente obfcrupng of focbe tbpnges. 

p^oue noU) bp tbe Cppbers,boU) it tuill frame. 

^cbolar. 3 fctte oounc tbe nombcr luttb ♦ 6 . €p 
p1)f ri3,ano paicfee tbem tbus. 

^ben Dooegi tafee tbe greatellc 

Cuhil^ nomher m. <; 4, lubicbc IS.2 7 

ano tbat J boe abate from s 4- anD 

fo rcfietb.2 y.tbc roote of tbe Gbe i0» ^tububc 3! fette 

in tbe quotiente line, 

0nbtbcn 3 triple.?. tDbicben^aUetb.9* tbatmulfe 
be multipUeb b? tbe qmtiente againr,anD fo commctb 
2 7.to be tbe btutfoa. ^no bis ^uotimte fcmetb to be.9. 



27 

^-4000-000(3 



^bcrfo^e tuoo^fe^ng luitb it, 
tbe firUe nomber is. 2 4 ^anb tljt 
feconDeis.729. tbatis.81. mul* 
tiplieb bya 9. lubtcbe is tbe triple. 

0gaine,tbe Cube of. 9.10.7 2 9. 
0nb all tbei togetber^Dooe mafee 
^2319 iubicbe feme is to greate, 
anb tbcrfo?e 3 muH taUe a lefljer 
^ttttientf, .^s 3[ migbtc baue per^ 

27 

^4ooooooC38. 
9 • • 
27 



2 16 

n6 



27 

f4ooooo<^(39 
9 

27 



243 
729 
729 



32319 

ceiutti hjell inougb bp tbe feront) 
nDber,if 31 bab marfeeo it in time. 
y&ut nolo amenbpng mp ouec 
Cigbte,3 tafee. 8. fo; tbe qmHmtt^ 
anb U)oo;5fepng toifb it 3 fce^tbe 
tttU notitber bnber tbe line,Unll 



of%potes, 

btc. 2 1 6. anb tht fecotiDc. s 7 6. anD fjere all rcaa? J 
efpiempoucrfiigbteagam* ^ , ^ 

^Tbct:fo;jc 31 ta&e,?. to betfjcf «w#/«i#^ 0nDbp itj 
wultiplte tbe omifo; , ano fo Hut 3 
3j, 1 8 9*fo? tbe ficllc nombcr. 

aaofo;itl)cfeconsenomber, 31 
5oe U)o:kc U)trii.49*tol)Jcbe is tbe 
fquarc of tl)e juotientg , multiplies 
bp.9, tf)at t0 tbe triple : ano n vth 

snijirDlp , 31 tafee tijc CmI^ of. 7* 
iul)ict)e 13. M 5. ano tt)cn aaopnge 
al.^.nombcr0togctl)er,3SitDctl)e , ^ , ^ 
fommc to bee . 2 ^ 6 n» lubicbe is to bcc abatcD out of 
27ooo.anDforeftctb3547*^berbpirer,t!)at.s,„ 

luitl) fometubat moie is tbc roote tl)at J n)oulD ftnnr- 
i3ut fo? farther triall, 31 triple all tl)c fiohtnu^tim 
finoc tbereb?. 1 1 l lutiicbe 3 mul- 
tiplte bp tbefame quotknu again, 
anD fo commetfj 4 1 o y.to bee tl)c 
Uiuiro;j. anD bts f «of ^>»f ? toil! bee- 
8.as it femetb: ano fo tijc firtt no^- 
berU)mbe0.32 856« anotbefe- 



-2.7547 


^^p>^ooo(?7' 


9 


27 


18 9 


441 


M^ 



2 5 6)5 



i^ 5 47ooo(57S« 
I I I 
4 l<^7 



]2 



8^6 
^^^^ ^^^ ^^ __ _ 7 104 

wnoe '(^airbVcIf / o 4. b^^^^ 

<smantfefteallrea5ie. ^ i. ,. , 

mberfoje 3I tafee 7 fo? tl)C qmtmu, anu br it mul 

fiplipng tbe muifoa,tl)ere rifcti) 

2S749' 
aiiD fo;^ tbe feconne fomme? 

tbereisfounDf.y4^9. 

anD fo J tbe tbiroe lome. 54 5» 

ail lubtcbe. 5« fommes iotneD 
in one, oooemaUe. 2929655* 
anotbat bcc^ng abateo out of 
tbe bigber fomme .5547000, 
fioctblraue.417567' 



417 567 

;'3'4.?/^'p>'(577' 

III 
4i Q7 , 

\ 87 49 
^459 

54^ 



1292:96 5 5 

Wberefa^e 



The extuBion 

^Ijercfo^e 3 mate bolDlr Taie, that t%c Umicn m 
^ aiiD nte;je, bp tbe poaticrt of tbc re mamcr , U5!)ic^e 

:anD It ts four fccn tW^ are equalfe to -y. toj^ere.- 
fo2e^ fljall be tno;te ttjf n |. 

^iiOfo Dooetlj ScbcubeljfAstulttttcmoie , tijcnj 
tijougljt before, 

a>o tjs t'our qucfltoit aunfUicrco,tl)fit tl)e fioc of t{)0 
Double Wffji^all be.?»foete anc ^ll ano | of-^ . 
Ofthmtes £Bnft£t* fu}thttoott!ioffrmm\Sf^Sv,klt\ctic 
if/radons, to fate 110 mo;e but tfet0:tbat if tljc nuiiTrratoiano nct 
nominator bothtbt^^uares^o^Cuhes^u-tlKn maic t?ou 
finDe in tl^at frattio tbe lifee rootc.lSut if anv of botbc 
Doe flcarue from tljat name , tbcritjatljtljatfraatcn 
no foclbe roote. 

^B'^\$ ttotfter Cuht^emiS^uareMcatitc i)is partc0 
tiooc not agree tn Sy «4rf name jid; in CuhiJ^e nanicaL- 
t{)ougb tljc numerator bee a s^«4r^ , ano tlje Dmenu;^ 

t\c[tO;aCuhe, 

^cl)olar. SDljattJoetl^appearcrcarcnablc, ntt'oc 
tl)eftrfteftg!)te. 

a3aaer, %^m fectng vm arc fo rcaDte in Icar. 

n^gr:aanf&7erc me to tbis quc0tcn. 

Jpiejim of ^<0onne of fiice incbes Dtanuter in tijc tnomi^c, 

sComf. fioetbi^otteabolletof tltJfntkpot'jiD iDctgf)te:U)i)at 

liietgbte {^alUbat boUette vaiit, that ftcuctbfo;a 

gonne of, 1 4* t ncbes in the rno u i li r i 

But to ftelpe^Du fn tim qmiium , anD tn aUfotljc 
Ii6e,pu (^all marfec iurll BucUde 1,13 fai; iig,tn tije i^ 
jp;jopoCtion of b^s^i 2»boot:e, hjthu he ts tlM, 

Ml Globes hire together triple thatpropor^ 
tlon^that their diameters doe 

^0 in tw erample, tbe p^Dpo.ation of tbe dtmam 
be?ng as. 1 4.to,6* £D? 30-7, to, ^» 3 n)aU triple it,anD 
ttjen baue3J tbe p^opo^tton of tbetr c^lobes. 






WMtKtlts%t 31 fettc t^e»^fractron« tfjus, ^ 4 -;^ anu 
tftet make *^ ♦ ftat te. 1 2. i|» 3itO fo i% tfje p;!opo;tton 
of tfic (IDlobc5,a0 toell m toefglitcaig m btgneffe. 

Si;SI)crfo;c 3 mull muUipUe* 2 o.t^at iu tbe tucigljt 
of tl)e Icffer boUette, b^ tl)e numerator of tfje p;topo;* 
tion^atiD Dtuioe it bv tlje Denomtnato^ 

ano fo fl)aU 3! l)aue ♦ 2 H A fo? tlje iocigbte of tbe 
greater bollete* 

j^oU) p^ooueroutbelibe 
iDoo^be. i^emembjpngt^at ^ 43 

CuU^ alfo , 35 Uiell as dDlo- 20 

be0, Doe bearc triple p?opo?' 6860 

tton , In compartfon of tijeir 

(iDes. 0s fou learneD befo^ b^ tbe.i 9^ p;opofition, 

of tl)c.8.boobe of Nuclide, 

a CttJf of is^aflfe of.4. tncfjes ^quare,^)oetb lueigbe jfqueflm 
7.pounDctueigbte,tubat(|>aUaC#^af3iB^ireof, 9, ofi/cuba, 
ijK!)esfquare,tuaie;* 

^cbolar* kMt p?opo;rtion of tl)e Ct)e0 it ad | iu^i 
tbe 3 mull fet ooune tb?lfe, ano multtptte tijem toge^ 
tber,a0 fractions I^oulD bee* <^o fo tutU tt bee tt)u0« 
ii|.tl)atmafeetft.'i|, 

tSIbcrefo^e 31 multipltetlje toelgftte oftbelelTer 
C«^^bepng.7,bp.72 9.ano itmabetb*^' i Q3»anotl)at 
Hoe 3! DiuiDc b^.6 4.attD fo fiitDe 31.7 9,i2 , toberebp 31 
male fenoioc , tbat tfje Uidgljte of tlje greater Cuhtj\% 
79.poanoetuelgbte,ainjt3erpnlgl)e|* 

^allt r, 2Dbefe»2. quellions Dooc teactje pou, r«^ 
tber tbe p;opoatlon of («^«, tijen tlje tjfe of tbe rale: 
iu^erfo:e to mabe tbe quelllo0 mo^e agreable to t|yi5 
tule73lp;opontidttbemtbn0,ln bacher o^ec* 

a boUette of p?on of.7« Incbes didmeter, uoetb toaie 
27.pounDc toelgbtetiDbatl^allbetbei/tfmf/^yto tbat 
bolitttt tbat l^all toale* 1 2 ^»pounoe toelgbte:' 

^cbolar. 31 pjaie ?ou aunfiuer to it ^our felf,tbat 
Bl male fee tbe apte fo^me of appli^ng focbe queHions 

2^.4* to 



The extraction 

spatter. ^&- tl)c CuUs are in triple p;opo?t(on to 
tfte fioes , fo are»tbe proportions of tlje fioes , to bee 
founoe bp triple Oiuifion:tt)atisto faie.bg fefeingtlje 
Cw^i^tf mtcs,ti$t\)z 2»termcg dftbe proportion. 

mbcrcforc 31 Doefirtte ret mjiine tlje termcs of tl)0 
proportion of tl)c botlettc0,tiju0:"^^ . ^m j; fee , tbat 
tljeC«H^rooffof. I2 5^«i0»5', anotbclikerooteof.27. 
<0. $.Uj!)icl)e nombers 31 HjaU fet in tfje roomc of tl)e.2 
others , tftus : I ano tijei declare tfte proportion , \}tf 
ttDenetbe^/iiwe/woftbe.2. boUetteg. Mbercofone 
t\iSit iB tlje leirer,i0 fenotocu tabe. 7. SDljerforc j mul« 
tiplie tbat.7.bp.^U)bereof eottrmetb.^ ^. ano tijat. ?w 
Doe 31 OiuiDe bp. 3. lubicbe giuett). 1 14-, 

mberfore 3! faie, tbat if.7. mcbck bee tbe ii^wrf*/ 
toabolletteof.27.pounDeiDeigbte, tbem 1 1. mfijra 
aim I fl^all be t^c <i/tf?»f/<f to tbe bolletc of. 1 2 f » pouDe 
ioeigbte. 

^cbelar. Me proofe oftljis ban neDe bee certain^ 

fce|?rtg tbeluoorUe is obfcure, to tbe common iuDac^ 
mmtt, * 

rhtpmfc. Saffer. ^oufaieloell. 0niJti)isis tbeterpor;* 
5cr of proofe for it. a^ultiplie botbe tbcfe rootes C«^ 
hikely, antr if t!)eir C«^« be in focbe proportio as tbeic 
Joaigbtes bec(tbat is to faie in tW eraple as '-Otbcii 
to tde tooarfee goootels not. '' 

^cbolar. Cbatmutt neaties bee fo.0nJ3 tbercTorc 
totU 3( prone ii Co in tfjefe nombcrs. 

anofajtbateanoe, fiirtte3(multiplie.7. Cullhh, 
anoitgiuetf^.M^ Cben^rniritiplie.! i4.C«^Mfiy, 
ano it maUetb ^'% . )i5«t naln fepng t\it one nomber 
isafract<oiT,3(U)iH foreafe tournet^e otber into a 
fraction of tbefame Denomination: ano it tuiU bee "^ 
<n tobicbe.2.frartions,tbe proportion mixVtz coQft be^ 
tUjene t^e numeratonrs. fi>o tbat tbei botbc hztvna 
WnioeD bg one common nomber j nwtte come to tbis 

fratttoK 



of%potes, 

fradon'g. 

auD fo 3! fee it ttjf II be t fo? tbc leflTer bepng tluttjeo 
\jy>, 5 4 5. tutU f clDc 2 7. 0m) ttjc greaser oiuiDeD bp tbe 
fame o 4 5 . U)iU giuc . 1 2 f» ^0 tbat bp tnall , tbat 
U)Oo;iic ts appjoucD gooD. 

SaUcr, ^ iuiU nolu paouc pour r uniipiigc , m a 
nctoe qucat o« , lubicbe )!5;aGcrs often tvnies , bauc 
Offafiontotjfe:a0tbu5. 

3? ^aue a Dtre of 3i3?aire of, 6 4 ♦ ^"ff s of SCrore j-^ne/Hon 
iccig!)tc , luftofc Coe t0 ♦ ? . Inches anD I- ano luouID ofmigbus. 
fjaue an otber Due of tIjefamcfnettaU of . 1 8 . pounce 
tucigljtc. 

q^p Dcmaunue fstlubat l^all be tbe ODe of tbc Dice:' 

^cftolar, Cbts queftion mutt firac bee reDuceD to 
one hinDc of Denoimnatton (n tbc U)eigbtc0,anD tben 
UjiU It be nio;e apte to be aunfUicrcO. 

m bcrcfo;e 3! Iljall tourne. 1 8.pounDc into tjnceg, 
multipavng it bp» 1 2.anD it Unll be. 2 1 6. 

0nD tOcn 31 confiDer tbe p;opo;tion , tijattsbc* 
tiuciic tljofc ♦ 2 . nombcrs of luctgbte. 6 4» a"t). 2 1 6» 
anD It is ccrtainlp. ^ -^ ,0?-^ out of lubic!)c p;opo?tion, 
3! uruil crtrncte tbc Cuhtie mte^ns 3; mate eafilp Dooe, 
fcritg bctbc tijc nmncrato;i anD tlje Denominato;,arc 

CuhiJ^e nomhers. 

$!no fo 10 tljefr rooteitlobtcbe (0 tl)ep;opo;tion of 
tl)c fiDcs of ibc ttuoo Dice. 

0nD fi'png t\}t fiDc of tl)e lelTcr Dff,f0 fenotoen to be 
5.tnc!)cs ano \, t\)t ot!)cr l)t0 fiDe ntuU be in SefquUlter 
p;opo?tion to lt,t!)ati0.^^ : lubiclie 10 injougl)tealfo 
t!)U0.3;multtpHe♦M'b^vanDttmaket^J,I o; lubicbe 

31 tl^aUDiuiDcb|>,2.anD there commetb.fT* 

Si^aflcr. ^ct one qneCion mo;:e3 totU pjopounDc 
f giue vnu or cafion , to tjnDerifanDe tl)c apte confer 
renccofmaire0, ofDiuerfcduffe. 
* anDfc;tf]atpurpore,5 rupporetl)t0 p2opo?tton(« 
ipctgbte,to bee betUjene maflee of one faiggeneffe. 

Q,ti. %ut 



Q^OODC, 


60, 


|i 


&tone» 


100 


irrn 


p^on* 


150 


It It 1 1 




2 00 


ho 2 14 M 


280 


h4lt4'j)^ 7 



TT^e extfaBion 

Bxamph o/nnt it I «w^Pare|^,[iSS7-pf-;^ 
rates for MoWK an^ftoiteofene ^ ' ^ ' 

yttill^is. quanttttetogetfjcr, tt)e ' 
ftowel^all tuciglie mo^ 
tUcn t^c UjodDc bp y, 

Jlitetuaiegp^ontob^ 
fjeutertbenaoneb^^. 

janD B^aCTe to bee be-- 
tttcrtfjett^aonb^T* 
ilcDiK to be t)euiec tben ^jaflte bp y, 
au \x)W\it rate0,altbougl) tbei be tafeen foj eram.- 
plc0, anO not of trutl)e,?et tberebp mate pou learne, 
^oU) to tDOO^^e U)ttb true rate^, fet (n a like table. 
0nO notD fojtbe tjfe of tbls table,tafee tbis qucftio. 
jf^Htftion 31 UioulQ baue ♦ s » iiietgl^tes of G^^t^ fo?me, maoc 
^f-r^eightc, oftbefe.^iluffes* 

%%t tuetgiTte of tbe looDOe i3[)all be^2 8. pouuDe. 

Mefione*$'6*pounoe. 

2nbeF?onai2.pouniie. 

^^e )15;uiire.2 2 4.pounDe. 

janotl)elleDUe»448. pounDe. 

£Df all tbefe 3I ftaue but t^e ,^;ott tueigfrtc ; iubofc 

fiOe,0? Gihiks roote is* 1 2* llTCftCiS y. 

0nQ mp oetire t0 to fenolue , of Ujbat quantitle t^e 
SDe0 of all ttre otber tnetgbtes (l^aU bee. 

^cbolac. JCljc qucftioir 10 pleafaunt:anD ret fome 
tobat baroer tbctt tbe otijer. 

Softer. 2Dbe table iuUl be^e pou fullp , fo tbat 
fOH c6ferre It tuelUtuitb tbat ??ou baue learneo before 

iSutbicaufeB! ^aue litle leilJer, tofpcntie mocbc 
tpme tottb pu ( fauetbat jealeto ^our furtberaunee 
5oetl)mafee me partlp to fo^gette m^ otonc bufinelTe; 
t^erefo^e \xi\ii ^ leaue t\^\^ queHton to ^our felf,to be 
autiftoereb at^our latfure. 

j^no fo tn all tbe rell,3l muftpolte it oner:attD gtuo^ 
anl^etofocl^e maters ? tbat toucbeme mo^enigbe: 



of^!(ooteu 




and tocig^e mo;e IjeuUp , tben all foci^e UitisWHyh^ 

XM\)tvfo;ty touctjpng all tlje rootee of compounoe 
nombcrs,pou fl)aU atmp fjanD noti^baue no p^tuate 
DcclaratioiMiSiit focljc as ^ou Ijauc IcarncU all rct)ie. 

Ofcompounde rootes. 

f tije nomber bee r om - 

pounOe, Otber of Square nem-. 
bers^OlQiCuhli^nomhtn , tl)cn 

acco;Dpnglp astbccopofiuori 
ts^fo l^alpou D;alu tt)c rootc* 
anD iDitljout one of tbrfc tluo 
tbere can bee no compofitinn. 
^l)crcfo;e to begin l?jitl) 
tfae fmalUtt compounoc ncm- 

heC i\\ tbat fO^te , tufticbe 10 a Square of /quarts , ^ou 'S^WrfW < 

l^all firfte ertracte tbe fquare coote,as i^ou bauc lear- Squares. 
neo befo jc. ano out of tbat roote(U)bicbe nuitt neoes 
bee a Square nmher)'^Q\x (^all ettracte bis fquare roote 
alfo: anotbat roote Is tbe ^:^:^en^{e roote, of tH 

fitUe Square offquares o; ^n:K!!K5"^'if nomber. 

jFo? erample take . 1 4 6 4 ' ♦ yxiWt Square route ifi 
1 2 i.ano tbat fame roote ts it felf, 
^Squdrenomber : anO batb fo; btS 

roote»iK 

taa berfo;jc ? nia(e faie,tbat. 1 1. 
Is tbe S^udred/quare roote fO;: tbe ;<f»;^';^e»;^'(e roo*^ of 

1464I* 
again Sso]o^6'isaSqu4re 

iffqu&res,n.m tberfoje a s<ytt4rtf 

nmher. HnO blS Square roote tS 

4 2916* tubicbetsa 

20 1 6(J4 ^jwdrf norr^ber alfo, 

i-o* anD batb^H* f^^ 



•224* 



^^^'/^^;^^ 6 (2 9 1 6. 



Th e extraction 

Ijfjjrootc. 

^0 t\iat.^4,mak tucU bcc callcD t\)c iK^^^'KS^t^^^'^ 

roote of. S 5" o :; o ^6* 

^.wu fo (l^all pou luoo?l:c,U)ttl) all of tijat name. 
Zenri^enf ^But aiiu ittht nomber he compoimDc, ofo-;<f«;^v 

■^^ ,'w>^fw;^'c>e«;^";^f(U)btct)cfomcmcnfo:ft)o;tncnc,caU 

zens^^n^n^ks), %\}m fljall fOU DjaUrC fuUc il)C 
Square roete^mJi thin tlje S^«4rf roote of tljat rootcailO 

ti)irDlv tfjc s^«4rf roo^f of tt)at laftc roote. 
51s fo2 rvample . 6 f 6 1 • is a Sqture of 

fjuaredfpiares.^nlil)Wfl\:liiCl'OOtCiS.>s I. ^ en CS I 
lul)ict)ri6airo a square mmber, nut) \)i\tl) j ,. • 

9.fo;j!)t0iDotc. 2CI)at.9. liUrUjaicsisja , " 

Square ncmbe),mti tjatt). ^.fo; 1)10 lOOtC. 

^0 tl)attl)C ;:;fM;<|;<er;^';>;;M;:v/^f roote of,6 f 6 I.fS.?* 

auD fo.2t1)fft formes or nonibcrs, 3; l^aU not ncDe 
to ftasc fo? any mo;c r rplication,o^ cvaniple0;fecvng 
tl)r mater fsplainc. 

jjlOlU fo; COmpounOC Cubik^e nomhers , ^0" fijall tU? 

ticrtlanoc t\)t liUe fo<:mc. 
fuhes of 3if t^c nomber bee a Cw^tr o/c«^f5,vofi fl;all firCte cr* 

c«J«« ^'''^^'^ ^'^^ Cubiks roote. ^nD btcaufi tfjat I octc 10 a O*:" 

^i^f »ojttk/-airo,tljcrfo;ic Vi)a\{ yen frUe Uje Cubtke roote 

of It. ^inD t[)6:t fcfcnDc locte iljaii bvC Uje Cubkubil^e 

roflfroftfjc fi rite nomber. 
0s fo^erample.s" 1 2. !s3 a (,«/ / v' ^-^"'^ ^. c? a itiheoj 

Gibes. £\\\tit,iQ Cuhike roote ij%.s„ u}>i:i;;:.8. agautcisa 

Cubike nomber ar.D !}atb.2.fD; I^IS rcoic. 
^Ctt)at.2.l5tf)C iuhicuh'i\e lootc.? v ' 2. 

i^ikeluaicG. I o 077696. IS t' LH'^-'f?;'-/-!' .vo,;fr,nnU 
Ijis firfvC G/'/^e roo/f IS . 2 1 6 . ::■:< yvn ipr.ic eaf:!;^' per? 
cfiuf tv tbcfi UjooikrstUibn;: i f;a?a ::tic »^^o;u)c tljc 
o?Der of crtraiiton of hl0 Ctihikeroote^ li)iiicl?c is. 2 1 6. 
;2nD tljat . 2 1 6 ♦ ts 3 CuhiJ^ nomber , f cu ;uaDe not to 



<Si 6 

6 ] ' 
M 2 5 



Doubtc , fo.: tl)at it ts one of 

ti)f,U)l)lcl)rou 

iiatic , J Dare 



795S 

2 268 

2 16 



faicmpcrfcctr 
in;mo;ic: ioi 
raufcljiurooti 
i5aDigitc,antJ 
rl)atis.6. 



z8i6 
{■■'/;;.7696'::i 



6 
]2() \ 



, 12 () ! 

i3r tl}i3 rou iiMif itiDgcof ("^'i"^« 
C«^f;o/'<:«'.'/V«/^f/. that 111 tl)*,un (-idi^ely. 



S 1 () 6 9 6 

VOU Iljall fuftC fcUc tlinr Cuhi{er90tc: ilntl tl)i M tijc C« 
i/;^e j'Oi/rc of that fOOtC. HllD tlJItDlp tljC ( ''^/(,f '"^o/t- of 

thatrootc ag;ainc. ilnD fo Ijauc^ou the Cul-'^n^t>^iii'\f 
rootc of that uilic noinbcr. 

Che thiiDc luaic off oinpofition (r> , luh':n s.jH.trei The thi.de 
anD (w^^j be fotnpoiniDr together: as ^en:^cubcs,Ze}i tomt'u^ition 

ncth Oiucrfelr, 

jnallthcfc rou fiiall as often abate the /::n::^^j 
YootCj ai3 that name \3 \n the contpofition, ano fu lut.- 
UiaiesoftheCn/'/lf'oo/'f. 

©0 that in a /c7;^/i;</';^f,von fl)ali crtraae firite the zi 

Square roote:,-\\\\i out of that SijHaicoote , j'ow lij.ltltr 

tra(tcthcO'/'/l'^''0''^^« 

05.6 4. IS a 2'f«;>^V«/^/(.f wow^f r,ly hofr i'r/rt.jri' r(j(?/f Jj3 

S. anO that . 8 . 15 a Cw/"!*- wow/^frjano hath. 2 . fo<i im 
rootc. 

$i>0.y ) I 4 4 I. »J5 a Z?^"<f«:cf'f«^'- ^'10<*<^ firft Square ^ j^ 



V 



:ClihiKt, 



Yootexi, 729. luhiehenoniberiB 
a Zcn:^cnbey f hath fo,2 hlfi S^Mjrf 

roo^r.27. HnOthatno^ 
bcri3a(l«^f,anDhath 
foihisroote.vUihcrc- 
fo:C:? mate lulllr fate, 

t\)?,Xru\^X\)i: Zen:>^X!^XtCHbtks roott QtS 5 1 4 4 T* 

113 ut 30 31 faicD before y that J lufght not llaic long 



.?4 

.'/^?(2 7 
/4 



i4,4 

5 Vr4'4l(7 2 9 
;<4/l4 
4 



V 



liube. 



The extraSiion 

at tW pjefente, fo tftetjfe of tftefe greate nombtu is 
tare m p^actife: ano ttjerefoje 3 him ouerpaflTe tijcm, 
fo^tl)i0ttmie. 

<aiiti r^t fo^ pour ateo in t^c mcane fcafott , 3; Ijaue 
Ijere o^atoett a table^lpftlcfje mate bee calleo tbe tabic 
ofeafe: lit lubicbe pou baue greate plcntie of tbcfc 
nomfaersjtxittlj tfjetr rootes m Diuecfc Utnoes. 

%\jt table it felf i& fo man(fette,tl)at it ncaoctb no 
Declarattonuf pon tiaue not fo^gotten^tubat pou leap 
net! before, 

!^txt} ff ?ou Kite to enlarge tW table,?ou maie ca^ 
filp Doe lt,multipllt?ng tbe nombers fttll bt? tbetr roo^' 
tes^lubtebe bee fet ouer tbetm^, in tbe beooe of tbe ta^ 
ble ♦ j9no fo mate pen make it to ertenoe infinitely: 
tubicbe D^all eafe pou toonoerfullp , itx tbe crtrattion 
of anp bmoe of rooter* fo; tubicb at fome otber time 
if mp leifure feme me better, toitb quietneflfe, 3 Uiill 
Siiut pou mo?e fpecialle rules. 

j^nD alfo 3i f ouncell pou , tuell to cramine tbi0 ta^^ 

ble^ano truft not to mp cattpnge. fo; balle ano 

otber troubles, male often times caufc 

erroure in fupputation* 



Thefmtefulle table;^hkht mm he called the table o/eafe. 



l\%ootes. i = 1 3 


4 


J 


6 j 7 j 8 j 9 j .0 j II 1 11 j IJ 14 II j 16 j 


^ 

•7 


IK 


19 10 1 Zl 1 33 


2J j 24 


2 Squares. \ ^ 
I Cubes. 1 « j 


9 

17 


64 


2f 
I3f 


;)6 j 49 j 04 1 «« j »oo j 


»1» j >4-> 1 2'9 j »96 217 j 276 1 


2f9 


324 


j J«« 1 4CO j 44« j 4*4 


1 Jl» j 576 


'■•« 1 543 j 


7 <2 

4096 I 

}276« 

10JI44 1 


729 loco j 

(JCI loooo j 

79049 1 100000 1 

551441 j looocoo 1 

47S1969 1 lOOOCOOO 


1)31 1 1718 j 2197 j 2744 j 


5575 1 -to96 j 
50627 1 M53« j 
759375 1 1048576 I 


4913 1 5832 


6879 «ooo j 9261 1 10648 


21167 j IJ824 


^iSquares o/f^uares. | '« 


316 
jo:4 


*3t 

3nr 


1 :)6 140' 1 
7776 j i6Sc7 j 


14641 10736 j 2f76« I 38416 


J351I j 104976 
] 1419877 I 1889768 


•3C31I j 110000 j 194481 1 23415* 


279841 j ,,1776 


1 ySurfolideu | j^ 




16IC71 1 248831 


371193 j 737814 1 
2826809 j 77:973s 1 


24-6099 I 360C000 1 4084101 j 7173632 j 


6436343 j 7962614 


C}^Squares of Cubes. j " 


409c 


1561T 


46676 1 JI76-19 j 


'77'56l j 3987984 


11390617 j 16777116 j 


14137769 I 34012114 j 47047J81 j 64oo«ooo j 87766121 j 115379904 


148037889 1 I9UC2976 


-J, Secmde Surf elides. •-« 


21S7 


• OS4 


7*i3S 


2799}G 1 *25543 j 


167772I6 j 
134217718 


I9497«7> j 35831808 j 61748J17 j ioC4'35"t 1 »7°«59375 j 168437476 j 


4iD;;!'673 1 611120031 | 893871739 | ilSoooooco 1 1X0I088741 11494377888 


3404817447 14^86471414 


:^Sjtt4res0ffqu4redf^rts^ -56 j 
^ (jibes of Cubes. | J'^ | 


(J6» 1 


6JT56 j 


;9o«:s 1 1679616 1 wf-4*o' 1 


430467H 1 ICOOOOCOO 1 


114468881 j 41998«696 1 8l775072« 1 14777*9056 j 274; ti2o627 1 42 949<^71S< 1 


(.9;7^744' Ilioi99«o776|i6983763o4ll j j 
111 ' 1 ' I 


1 


.9fii} 


26114-1. 1 


I9!}i2f 1007769s 1 403n«07 j 


387420489 1 IO30OOOC00 1 


2379I5769' 


7179780371 106044993731 1 j 1 


{ i III 1 


1 1 


loSqudrtSifSurfolidis. ( '"4 j 


79049 1 


I 048 J 76 ^ j 


S7Ci6ii j 60466176 j 181475249 j 


I073741824 
8789934591 


3486784401 j j 


1 1 1 1 1 1 




1 1 


UiCSurfoUes, | "48 | 


•77^47 


4194304 


4i!8l8ii; j 361777076 jl977}2«743 


1 1 


1 1 1 1 1 


! 1 1 1 1 1 


i 


I2JS^«4rf 5 of:t^:i^cubes. 4° 9« 


yji44« 


IC7772'« 


144140(27 1 217666136 1 1 


1 i 


1 1 1 1 1 


1 i i i i 


1 i 


V^p.Surfolides. \ ""^ 


1T94J2} 


I £7108864 


1220703117 


1 1 3079974026 






1 1 1 1 1 


1 i 1 1 i 


1 1 


l4\S(fUdresofBfur/oUdes. | "jh 


478i9«9 


2684314J6 


6Io)717627| j 


1 


1 1 1 1 1 


1 1 1 1 1 1 


1 


\^\OtbesofSurfolides. \ i^^'■* 


1 >4}4''9o7 


lio7374'»24 






1 1 


II 111 


1 1 1 i 1 




)Jb\zi^^^x^<i^<p:ii^^\ ''''' 


43046721 


1 42949f'729£ 






1 1 1 1 1 1 


1 1 1 i 1 ! 




ll^furfoUdes, \ »J'*7» 


1 IJ914016} 


1 






11 1 1 1 1 


13 


1 i i t 1 




l55lSjtt4rMo/Ctt*if«*M. | »"'44 


387413489 




I 1 U [^otes, 1 


2J J »S 


»J 1 »« j 29 1 30 


»• 1 M 


J 14 J If j }< j 17 j »» 


}« J 40 


Mffurfolides. »^4a«8 


1 iiciisi4e7 




1 1 2, 1 


Squares, 
Cubes. 
Squares of 
)urJolides\ 




[ «i; 67« 


I 7» J 7«4 j «W j 900 j 9C« j »oi4 


£089 


1 II76 
1 J9J04 


j i2ir J i29« 1 'JO 1 '444 


1 iSlt 1 l«oo 


lo^en^Xf^^Hrfolides. \ '°'**57« 
2\^esof$Jurfolides» '097.52 


348£7i'44o« 




I 13-1 
1 1 4- 1 


'Squares, 


17627 1 I757« 

390617 j 456976 


196JI 1 »I»71 j 24389 j 27000 
1 7)1^1 1 £14676 1 707:81 1 810000 


2979« j 327(;» 


K937 


1 42«7t j 46676 1 7062 J J y4«74 1 isji» 1 «4oc» 






923711 j 1048776 


1185921 1 1336336 1 1700617 1 lC79i^26 | l874«C« | io«7lJ6 j 2}I}44t j affeeoo 


22^qu4resofCfurfolides. | 4«94jc4 








5^ i 


1 9767627 II88IJ76 


X1448907 17110368 j 20711149 24300000 28619171 }37r44»» J J»'3r39f 45435424 j 51721871 j Cc466i7« j 69J4J977 | 7923Ji6« 


9«134I99 j 10340000* 


2]^furf9lides. | »'''8«>°» 






j 6» [Zeniijcubes. 244'4o«if j''89>777<s j 


336U0489 j 481890304 1 794813311 729000000 88770)681 j 1073741814 


I519I467969 jl 74480441 6 


J 1838267717 j».7«7«2„6 2,67726009 | ,0,09,6,84 |r.»7417«i 4oS»«o«oeo. 


24^w»;^i^«»;<;^?;^«V«^«l if777ii<s 






1 1 7* ^furfolidcs. 


6IC37I762) 1 ?03ltloi76 


9075173103 


«)49»9287ii 


17149876309 11870090000 17711614111! 


1 


1 


1 1 i 1 . 




ofCof^ike nomhcYs. 
Ofnombers denominate. 

li^us Ijaucjligljtlpoucr x\m^)tmQiitKmlen 
common femocs cf nombcrc Mpaih, iontraffe, 
auD noU) reftctf) tl)c trcattcc of itom.- 

bctSiContraSiejO;<De»omiHate. £DftuI)lfl)C 

UmDc tfjcrc bee fome calleD nomhers Jcm^ 
mindte vulgarelyi ano ottjer bee calleD nomberc </r«ow/; 
nateiofitl{cly, aiiDattjlrDe fo;tetl)erciflof nombers 

radUalU, Ujl)tcljC fommoillp bce calkXimmbersirratio, 

w«//«:bicaufcmanpoftt)eimarcfoctie,asrannotbec 
crpzcfTcD t bp common nombcrs jljlraHe , notljec bp 
anp certain rationallc nomber, ;®tljer men call t^em 
mo:e aptlp Surdcnomhers, 

jano altbougt) manp mennc tooulD not accoumpt^ 
tbcm,Uiitl) nombecB denominate, pet ^ mate llifilp Doc 
It, fo: ttiat tbei require a rcDtirtion to one Drnomina 
tion,iftt)eil)auefcuerall£fi0nesofqiiatmc0,a3poii 
ttiallbcarcljereafter. ianDtbofenombcrciuucrgoc 
alone, Uiitbout fomc ot\icx fignc, ano name of rootcD 
quan title. anncrcD to tljcim. 

iDf tl)e fan UmDc of nombcru tjenominatduljfcljc 
arc tulgarelp Denom;nate,as, I o.n^iUinges. I o, men 

2 o.fijippcs, I o o.l^epe. I o o o.pcrejs, aiiD focljc liUc, 

3 luiU fpeakc notftpng in tW treatice* i3ut of tl)e ou- 
tlier tiijoo UinDes 31 tulll fomeUiljat Ic^ite , fo: poure 
learnpng anD contentation ♦ 

^cl)olar. S>fr,3JammocI)ebouKr)eljntopoii:anD 
t!)crefo;e remit all to pour olnne Difcretion ano gooD 
luill. STruHpnge fo to appUc mp ff uDicanD emploic 
mp fenol3lcgc,tl)at it fl}all neucr repents pou of pour 
furteficintl)i3belialfe. 

fatten Hften marbe tneH mp tuojDrs , anD pou 
Itjall pcrcetue , tl:atj irillfefeaB mocbe plameire, aa 
Jmaie^tn teactipng: ?.nD t?icrfo;c UiiU begmne luitb 
O/^^nomber^fira. 




The jfrte 
OfCo^ike nomhcrs^ 

€>mber;5 0/;l^ate^ocDe 

as bee rcntiaac l;itto a ncnot 
minatfon cf fotiir Cofii{e fignc 
as i.HomlJcrjaooU;j.fquarc 

IBut as fouopcnmoufncflFe 
mttictifcoftbcim , tljcrcbcc 
certain figures fct fo; to fignu 
I fictl)em;fo J tbmbc it gooD tfl 
cvpicITc ^nto foil ttiofc figures, bcfo:c luee enter anp 
farther, to tljmtentc lue maie p:oceDe alUiates in ecr^ 
tentic , anD hnoujc tl)c thvngcfi tf)at toce intermcDlc 
iBitballtfo.: tbei are tl)e fignes of all tbc arte, tljat fo> 
lolueth t)ere to be taugbt. 

0nD alti)ougf) tbere be manv lunties of irrattonall 
nombers, vet tbofe figures tbat feme m Gj^ike nobm, 
bee tbe figures alfoofall Irrrtionalle nonibers,anD 
tt)erfo;c being ones lueUHnotuen,tl)eiferue in botbc 
places eonmiotJiounp* 

^bcfe tberfoze be tbeir figncs , anb fignifirations 
b;ieflp toucbcD:fo; tbcir nature is partlp oeclareb be- 
fo,:c» 
f ♦ Bctohcneti) nomber abfolutc-as if it bao no 

figne* 
^. feignifietb tbe roote of anp nombcn 
^. Itepjefentetbafquarenomben 
ct* erp:cffetb a CubiUc nomber. 
5" ^* 5s tbe fignc of a fquareof fquareSjO? :ztx\\U 

J^* ^tanbetb fo? a ^urfoliDe* 

^ c£. SDoetb fignific a ^enjtcubike, o? a fquare of 

Cubes. 
V%^* SDoetb betofeen a feconbe feurfoUie. 
y ?r 5"»^octb rep^cfent a fquare of fquaccs fijuartB 



ofCopih 



mhet 



,e nontDers. 

CC cf . ^tgniftctl) a Cubs of Cubes, ^ 

^^ \f^ ', Q rp^CffCtl) a Squars cf Surfolidsh 

c/^', iDCtoUcnctb a tliiiDc Surjolide. 

^ -k- CC. iACp;cfcntCtt) a Square QiSjuaredO^hci : oj 

2>/5 % ^tailDCtU to; a fOUrtl)C Sur/oltde, 

r^Sb Z>. gs rtje fignc Of^O, fjuan Cf fCCDIlDC Surfolidci 

cf/5 '. ^igmflCtt) a C«^^ otSurfclidrt. 

5" 5^ 5^ 5'* BetoUcnctl) a 5^«4^f 0r/j'«4w, fquareoig 
fqiiarcD. 

^/^T** 3S tbC firttC Surfolide, 

>'CZCZ* Grp;ctrct!) a fqua. C of Cuhil^e Cubes, 

^fl^' 3S tljC firtC SurJiUde 

5 * 5 '^5'* &octl) rcp;cfcntc a fquarc of fquarcD fur^ 

foUDC0. 

C ^ /">;^*. ^tanDctb fo; a Cube of fr ronoc SmfiUdes. 
^ ' ^J-^^* Js ^ fquarc of tljtroc Sur/ofidts, 
gf^ «! Docti] bctoUcn the fcucntlic SmfoUde. 

5'5-5^'Ctl,»)tgnifictIjafquarcoffqiiarcs , offqiuv 
rcD Cubes. 

HnD tbougl) 31 mate p^ocraDc tnfinttdr iti t\0 
ro,:tCi Kt jC tbitiUc it ft)all be a rare f!)auncc, tfjat vou 
(l)aU neDc tbis mocljc : ano tberfc;c this uiaie fuffirr. 
jp)otU)it{iftanDrngc , 3 loiU anon tell pou, boU) ^ou 
niaief6tmuctl)cfcnombcrs,bpp,:ogrctrton,asfarrc 
aBfouliftc. 

itnD fartbcc pon Hjal bnticrftanDe ,tbnt manf mert 
tjoc ruer mo:c call fquarc nombers ^enxikf^pd,)^ a fl^oi 
ter anD aptc r name , otbe.r men call tfiofc fqnarc0 tbc 

firpe quantities , auD tbc f «^f 1 tbet idSS-ftcondt quantities; 

fquares of fquarcs tbci call thndeqtiantitics.anD furfo 
liDcs/o«r//'<' '/ttd«//7/>;, :lnD To nampiig tbcrn all quan^ 
titles ccrfcpte uombcrs anD rootes ; tbei Dooc aDDc to 
tbcm fo; a DitTcrencc.-an o:omall name of nomber,a$ 
tl)ci DOC (Toc m o:occ fucfcflTiuclp. 



The Jrtc 

3s ^erc folotDctlj tn example 
5-. jTirac. -V 0nDfofo;jt!)c,ofa5 

ct* ^cconDc. / manp a0 male Occ 

^^%\ "C^iixx^t* I rcckcncD. 

/5'. f ouctl^c. / 13ut altljougtic 

5"ai* Jiftc. I fanTcntcnacfompte 

5"§"S*» ^cuctitl)C,A*^ *lDaic:btcaufctI)co' 
ct.cC' Ctgljtfi* j tl)er names be corns 

5^/t>, ^incjtlji* \ licroufc,pcttbofco- 

ff^d, %tx\t\)t, \ tljcr names before, 

^5"cC»^lc"f"tbc \ tiocrp?eirctl)cqua^ 

dfz>, SLUjclftlK. imcoftfjcnombcr, 

better tijen tbcfe later names fioe. 

^cbolar. g^tljanbcro" double , fitftfouarecon^' 
tentc to teac^bc me Double names : to% fo ftall 31 be ac? 
quainteu tuitb botbc fojme;s , as % fl^all tXimnzt on 
tbem In otber mennes boofees. 

S^berfo^e noU) pou male p;oc£aDe to numeration: 
to^tfljeBltblniicttnerte* 

Smaller. %Mtxz be otber.2»Cgnes In often \ikj of 
Ipbicbc tie firOe ts mauc tbus — ~j— ano bctoUenctb 
mu}tx tbe otbec is tbus maOe ano bctoUenctl) 

ano tobecc tbel tame in, an? nombec C»ple . b? 0^ 
tljer,t!)at nomber is calleu a compounoe nomber, bt 
caufe it confiftetb of,2,nombcrs. flno lubcre ncltber 
of tbeim is , tbt nomber is calleo tmcompounuc , al^ 
tbougbtf^efign^ be compounoe* JfojtbccompDunDc 
(tgne t mabetb not a compounoe number* ^no nolu 
31 tDiUp;o<eoe to numeration. 



^f 



ofK' 




ofCoJIike nomhcrs. 
umeration hi nombers 

n)aacr. 

embers O/^bnccmpounDc, !)aiie no XnmrMien 
Difttcuiticintt)cirnuiHci'atiDn:fo;ciici- 
n!o;r tl)C nobcr rcp.2cfcntctt),fo mnnr of 
tliat Co^/A? Drnommati6(bcti)cin6bcr5, 
iootcs,fquarc0,Cubc0,fqiiarcs of fqiia 
res ,o<^ anv otl)cr likc)a3 thcr ht tnitics m tljat nobcc 
&>o. 6.f . !0.6. nombers r^^no. 6. :::^ .10.6. rootcs: 
2o.^,^.io.2o.rqiiarc0oo.cf..bctokciucl).^o.CnbC0. 
^ci)oiar. 3 fee it lucU. J^o: bv tbi0 nobcr.zo.r^ . 10 
not appninaro ani' nobcr abfoiutc, of one r crtatntu% 
but onclp fo manp quutitm of tbat buiDcrlubicbciiiaic 
bce.So.if.4» bconcfquarc.0nDif.9. btconcfqiiarc, 
tbcn zo.fquarcs make 1 8 o.0nD if«2 y.be one ottljofc 
fquai:c0 ti)crcbj'rcp;efcntcD,tbrn»2 c.fquarconialic 
s" o o. 0nD a0 fo: tbc figncg^^ou taugbt inc ti)f bifo;c. 

Ofjddition, 

JDaUcr. 
1)10 numeration (0 fo platnctbat iucc MMtlonof 
mate paffcfrom ttDnto aDDition: lubl- lil^eftines 
cbc 10 a0 cafic alfo, ff tbc quantmc0 be 

of oneDcnominatton.i^onbcn ncDctb 
no mo;jc , bat to aDDc the nombers to- 
'gcthcr , anD to put tbat fame common 
Cofiik.^ Dcnommation,to t\}c totaU tbcrcof. 

^cbolar. 3 take it tl)us,2 o. xo . aoocD to. 5 o.::^ . 
tclU mafic. <ro,t£, 0nD.i2.^.au^cDto.i6.5'.b;)ti 
gctbfo;tbe.2H 5'. 

evader, ils'^pou Dae cafilp fee al tt}c mater of tl)i0 
at)tntion,ro maic i^ou as caliip conceiue^all tbe \do;Uc Suhtratiion 
of fubtraetto./o^ it 15 iD^ougbtas m \3ulgarc n5bcr0 t>/likffii»ti 

^.19. ^cbolar. 




The Jrte 

^djolarv Sl)cn if 3: abate *6>c^* out of. i o, c£» 
tlKrclx.iUrfftC4.cc*^"0fo.9.5^5^N0utof.2^.5»5» 
Uoctl)Icauew6.^*§^. 

^eattcr. %\m is all foj nomlicrs of liUc figncc 

^cl)oIar. ^I)attl)cntf3] ioottlDaDDc. i o. ::^. to 

6.§-:'U)bcre tlje ligncs bcc \jnliUcrmaic it \it Doc nrfc- 

f ng tF)ci ht not of one Dcnominatio, no? figne Cofiiks* 

Mditionof 05aftcr. as tuell a0 fi)iUrnffcs niaic bee aonco 

vnlikeftpncs tctt!) pounDeie;,o;penie0:anti m Itfee fo;ine. 

fo; tbet fl)all ftanD ftiU as tbei U)er,lmtl) tl)c ftgnc 
of aoDitton,iul)iclje is tijts. — I — 4 betohcnetl) mo;e. 

^0 tl)at. I o.x^.put to.6.§^* maHctl).6. ^. — 1 — 
I o.5:o^.tbntis,6^.ino;je.i c.x^.o^d^g^.ano.io,::^ 

^:t)olar, anDiob^not.i o.xp — ^~6^^^f 

CBalter, :ii5icaiifc it is moftc c^oerip :, to fette tf)C 
grcatctle figne O/J'^fo^nroUe in cjDrr. 

as tow faie,2 o.fi)iil^)nges,anD.6.pennies: ratljcc 
tl)en.6.pcnmes aiiD.i o.l^iU^nges. 

^cljoiar. 2n}cn 3 fejf» I vc^.-l'f*^^^*^^ ^0,18.5^5^ 
it tuill make. 1 8.^ ^» — I — .ivc^» ^nfo- 1-/§^*» 
lorneD Ujitl).2 o.§- cf..UDoe maUe.2 o. §^> cc* — t — 

Of SuhtraBlon. 

spatter. 
Vhtrattkn fs as eafic: fo;j it tioetl) oepe no 
onelp of tlje figne of abateme nte,U)i)ic^ 

Is ttjis. .ani) fignifietl) UDTe , oj a^ 

bating, ano tfecrcfo:? if 3 lioulo abate 
6.x^. out of. I o.§^. 3; niufi: fette ittljus 

I o,^\ — .6.2^:tbat is to faie.i o.§-. leffe.6.2^» 

o^abattng.6.2g^. 
^c!)oIai:. 3Ltjen if 31 bauc 3 o cf;.anD tuoulti abate 

out of tbe. 1 2.9.3: mull fet it tbus.? o.cf . . i2f . 

t!)3tis.3 o.c«^«faue.i i.nombcrs.anoifmultiplicai 

tiott 



SuhtraWioof 




ofCoJIike nomlers. 

tion anu ntutfion , bet as eafic , tW fl)aU wane no 
grcatc Uuoic, 

Of \iult'tpHcation. 

Rafter. 

'^i^'£ '0^N^»^K'"^«^ outtl)c ncUic figncs.as j; toil 
E^t^fc^^^l^i^''^^^^" ^"°" • 15utfo,: fiuDvngr of 
L^^^gx^^^;;i)cnoml)cr0, the connnonniuiti 
M0S ^B^>|^Mi ^I'f^^t'O" '^"^ Diuifion Doctt) knic. 
\m&f5x^: A^\ 5)otl)atlul}cn. n. ^>. ismuitu 
plirD bi'.r).:io .it inaUctl),7 2.ct» iHnDif.z^.LX* '^^'^ 
mnltiplicD l)y.v5'-tl)cic cifctb.i 2 o./^>. 

s>r!)oliir, £:Ui5paflrctt)nircunnrngc, fdnticfii!^ 
Dpng of tfic m\jot fignc . alttjouffii t^c niuiiiplicatiaii 
uniit mmbttsM as cafic ass ran he, 

ayadcx. jfyou DID lucllrcnicberjiDljnt roil Ijauc 
learncD before: tljc mater luoulD not feme fo 'oarDc, 

S:)ocnotvouknolue , ttjatarootcniultipluobra 
rootcDoctI) mahc a fquarcr' auD a fquarc niultipiicD 
bp t^is rootcDcctl) bzpng fo.:tf)c a aihc^ 

^djolar. Ii:0at5linolucngIjtUulI:anDtljrrfo:c 
a S^adre if Squares multipUcO bt? blS rootc , Itlll rclOe 
a Surfol'ide, 

^paftcr, sri)cn br Ufee rcafon , a Cube muUiplicD 

b? a Squire j^dXi maUe a Surfolide. 

^diolar. ^n DccDc tt 10 all one, to nmltiplic a cuhe 
bp a Square J anD a Sfuare 9f Sutures bp a rootC, 

^t^aftcr. SCbni fo: a gcncrallc rule , 3 tutU fcttc 
fojttje bcrc a picfiucntcfo^rou : luljcrebppou male 
inoiDC tl)e nctuc Ognc,tn all nmlttplication 0; Diuifi;: 
8n:not onelp bp figbt t3crpfpcDilp,but tljat pou maie 
alfe commit it aptlp to m«mo;ie. 

tSUlicrfoje marhc loci tt)is tabic folobfng: tubcrc 
i?oufeemtl)el)igt)ecrotue, almeofnombcrs, fetm 

natnrall 



The Jrtt 
naturaUpjogrririoti : anotinocr tl)cmvcufttt!}cCi 

The table ofCoJ^ike fignes, 

and their ^eculiertiombtrs. 



o. U 1 2. 


1 ^» 


4* 


1 ^' \6. 


. f* i ^. 1 y^ 


1 ce» 


\h^h'^ 


1 /S^ Is^cc 










7. 1 8, ) 9. 


1 lo. 


1 1 1- 


1 12. 1 I :. 


^-^5^♦!^^§^&^^xce« 


1 h'^^'' 


1 ^z-^- 


'%'h^^.^fy' 



SCijts tabic 10 largely fctfouljc, m the title of p;o> 
grcirion,Uil)cr£tmto ^ou male tiauc rccourfc,tf ^'Ouc 
nomber be to grcate fo? tl)i5 tabic. 

13^ thjts tabic mate pon cafiip I^nolue , tbcfignc 
tbat ftjall fcruc fo^ i^our ncUjc fommc, m multiplica^ 
tion, 

asfo;j cramplc, if | uooc multiple fquatcj? bf roo^ 
tes : 3 looUc tn ttje table , tuljat nombcrs Qanoc oucr 
tljrmfeotbc,anDputtinigtt)ore,2.nDmbcrj;togetbcr, 
^ fcke tlje tQtall in tW(imc Imc, ano l;nDcr it 3 fitiDe 
tt)e neiue Dcnommation <:o/?;^<',U)l:ii:tic iT fljoulD fjauc 

^cijolar. 3!pcrcciueoucr.Jf .thenomberfff i.anD 
oucr ♦ ^ ♦ tfje nomber . i . tut)ul)c botbc aDOco togc^ 
tbermake. h ^nDbtcaufetntJcro.itftnotbe figure 
01 flfgne of.c£. 3i muStt take tljat to; tljc nciuc Ocno** 
mination, 

Rafter. ^ouf^ietrut!i0. 

^tl)olar« s:i)cii if 3! multtplie. i z. z^ ct • ^V'S. c6* 
tljcfommc luill fae,96. cf c^. i^o^oucr. cc* JiSnoc 
^.ar.Doiicn§^ci^,aanDct^j.6. tuliu!)e botfje togctfjcc 
J)oemafec.9,anDt)nDcr.9.3fcc.c£cf»tiil)ifljc3itaUc 
fo^tftctJenominato?. 

0no if tftf fame rule bee gcncrall, 3; am cunnrnge 



ofCopkeuombcrs- 



bi^aftcr. 3iisgcncrall,fo,:nTulttpltcationlntbt« 



bmDc* 



Of'Diu'tfion. 




at fo; DiuiOon , ro" nruffc abate tbe one '^'^uijt&t^ 
ucnibcr out of tl)c otljer , to hiiDc a nctec 
:DcnonHnation. 
2rii£rfo;c If ro« tooiill' l'i"i^^96.Gcoc 

' bp.8.cf^.tl)C quetieute UJtU bc I 2.^ 'CC • bl« 

faufc tijat oucr tl)c figne of tour DuutifnDc, uanoctb 
9. nnD oucr tbc Duiifo;0 fignc 15 fct > til l)f rfo^c aba« 
tutigo.from.9. tbc»-c rcftctt). 6 . tnDrr lubicbc ig tbc 
(ignc. Y c^.tbat it muft takcto put to mv quotients 

^^c^olar. ESbcn fo; an otljcr triall , if 3 luould Di> 
uiDc.2 6 o.c/^'.bp.s-y^'tbcfwo^/T/ UjiU be y :.§-5-cc 
/^o; bicaufc tfjatouer.r/^. 3 fuiDe.i 7,anD oucr. /p. 
llanDct!).^ti)cnfubtraapng.r.fro,i7.tljcrcrcfl^rl)W2 
tjnDcr U)!)fc!je in tbc table 31 finDc. ^- j- oi. 

$)ODtuiDvn0.2o.cc.by.4.f. tbcf«o/'>»^'tuiUbce 
y.cf.:anDfoofot!)cr. 

£l9aftcr. ^ut anD if p ou tooulD \imt^c» 1 2.cf.^ bp 
j-.^^. tbat muft be fct in fo;me of framon.tbus. \'f, 

^.IS.^^b^7.^.malJctb.y|-a^0 6,5*•b^2.cC. 
reloctb* f^. of tubicDc fractions , luce \m\\\ fpcaijee^ 
mongcftc tl)C fractions of G7?%uompoiiOc*i^o; tljei 
Degenerate out oCt!)ij«5 kmoc. 

taaijerefoje ttjis maie fuffire b;icfl|> , fo; tijc cufto* 

Wable lU00;kCJ3( of tofjolc Cepksnembers. 

Of FraUlons In Co/like nombers, 

0ti as fo? frartions.tlje tDoo;ktng i6 like Offumm 
m tuerv polnttc, tnto tbe U)o:ke of nom-^ in uomhn 
bcrs^i^/?r4/ifr:rettiemb;ing onclr that as c#/j%. 
tl)ore broken nombers , liaue aCo/?/^/ De^ 
nomination annejreo tottb tbcm, fo mutt 
%,u tbat 




Tk /Yte 

tijat Denomination follolue t\)t rules , nolo laUc De-^ 
clareu. 

xm^tu^Qit^^ysiW notn:DctoDocanpmo:c, but 
to fet ro^tl)c oncli? cmam cpamplt's,of cucrp liinocof 
U)oo;jbiJint?)cm. 

Bxampks ofl>{uimrat'ton, 

-^i£^» ^igm6ct!)|ofa%^f. 
45^-» BctofecnctI) *ofaS^«4/'^ 
"oS* ^Amzkx\tzt\)\^QtaCube, 

0nD fo of all otbcr fo^nrcs ofCoJ^i{e Ogncs: luljcrc 
ijp tsintcnocD, tbattl)c Cefit^epiaHttth^m Diuioco tn- 
tofo many partes, as tbc Dtnoninato,: contamet!). 
anD tbere is l)cre rep;jefrntcD oncly fo nianp of tl)cm> 
as tbe numerator Doetb impouf. 

&cDolar» ^ercfap 3J Dooe pcrceruc, tfjatafiaiiion 
Lopieymm fignific a nomber, ano not onclp a parte 
of an i)nttie,as it did in mmhtvBM/lrafh, 

po; mtn 3! faic -| 5% if tljat square be, 9. tt)en that 
fraction Ognifictl). 6. 115 ut tf ttje square bc.4.tljen tbat 
fraction Doetf) rep;efcnte«24^» 

iliSetoaies jcf . if tfte Gu be. 8. tbm tljat fraction 

Doet^0gnifie,6.l!5Mtift|)er«/'<rbe.27,tf)cntl)atfrac^ 
ttonlscquallcto.zo,^, 

fatter* ^ou Doe conRDer it luell. 
Ofjdditlon. 
AUtt'm, ^^^ fo? aDDit!on,taUe tWt eramples* 

T§^aDDeD to I5-.D0C mabc||§-,o?. i j>^» 

4<e toineD luitb icf. Doc mate ?| cf, 0; i, eg i|, 
0nD In t}nlike fignes. 
Igr'^aDDeDto^cf^Doemaftclc^* — 1 — i^-oaels 



tijus bp one common Denomi* 
nato2. 



20^ 



ofCopke nonthers. 

.^f luljicbe s tuill fpeabcmo^t (n tljc ^inomialles , mn 
lijrvffDZf U;ill cir.ittc tt,tiUlDCComc to rl)cm. 

^cljolac. 0sfonl)crcac , ^fccitu^cU : /o;tl)c 
tuoo:Uc ID all one luttl) fracttono Jhftrane, 

^nD licrc tl)c Dcnoniinatio QiCoji\e fignc is not t.v 
r{cD,altl)ougb l)crc be tjfcD Diucrfc niulttplicationg. 

{Rafter. iHnD gooD rcafon:fo2 tlje tuliolc qufittente 
m\c\ic 10 rcpjcfcntcD bf tljat Co/?/^f figncts not n\i\U 
tipltcD,butccrtainepartc0ofit:anDtl)ircto;coug'otc 
t!}at Copie fignc,to CanD ijnaltcrcD, as tt)e qnantitic 
i*cp;eftnteD bn it, is not multiplicD no; altcreo. 

Exawpks ofSubtraiii m, 

.tc^.abatcDontof^cf ♦ UocIeauc7\<£* 
^ ^,oiit of -^ ^-.tbere rettctt) ^1 5^. 
i535.,fubtraacofc6 :^5^Docleaue;:y5',o; ,^ 5 y 

0nD in tjnlifeefigncs. 

-i^ct a^atco fro t't <:^^ foe Icue ;,ccc^ ' S ~C. 

■^5^ taUcn out of ,'^ ct .tbc rcfte is let i 5 ^» 

iliUeluaics as in at>DitiD,ro in tbis fo:tr of fubfr.r 
tion,tberc mate bean otberkinDe of Uioo,:'.ic,Ui?)tcI;c 
3 UjiU remit to tl)e tceatice of !Bmo?»/rf//f5. 

Examples ofMultlplicdtiou, 

•{ 5^ muIttpltcD b^ -l^?j-, dot make I ^' ^\ 
4ir£^.multiplicD bp Jj^.bj^ngctl) fo;tl)c.^T 5"* 
,>^ ^. multiplicb hv -f 5"- ^ot reltjc ^^ 5^^ 5 ',0? ;, S» a'* 
^cre the fictnes uoe altcr,a0tn t^e multiplication 

of Iw^ole OJ^'ks ncmbcrs* 

2C»tj» ^cbolac 



The Jrte 

^cliolar. %W ^octl) fometu^at trouble mt\ tijat 
t!jf Cofiih flsncs Il|oul0 c!)aungc !)cre,rat!)ec t[)cn m 
atmtion.ozfubtraaioii : ^epngrtberc luns as mocbe 
inultiplicatioii.in anp of tbem botbe,as tbtrc is Ijcre. 

i^aO^r, ipachettjc niatcr iuell,anDpou ftjall bee 
foncfatiffrc0. 

i'o^m aoottion ano fubtrartion, tbc mulrtpliratio 
feruetDonelvro^tl)ereDud:ion.oft^.2»fraaion3,tjrr' 
to one ocnominatton: 3iiD tbercfoic m tl)em,pou nj> 
ner multiplte tt)c numerato;j3 togctbertbut pou mul* 
tiplie croflTc luaiesf , tbc numcrato: of tbc one , bp tbc 
Ocitommato; of tbe otber,lDberc as at nmltiplicatio, 
pou tjfcno ceDuctton.bwt Doe mabe aplainc multipli^ 
eattem 

0nO fo libcU)aie0 in Diutfi6,tbere (0 tjfeo no meaitc 
of reDudiontano tbecefo;c m it ttje fignes ntiitt alter, 
iijs b$fo;c (0 Declared* 

'Examples ofViulfion. 
T S" 3r" ♦ ^M'^t^ bp ^ 5" ♦ Doc make (n tbe <iH9ttente 

^' i ce/DiuiDct^bp -'^ ct* Doetb pelDe '|[ f » 0^ elj? l{, 

i^o?re?ttg3i(l^allDtutDe.cC.b^ce-3fmutttberr:^ 
fo?e abate.5»from.3.anD fo reHetb notbmg,lubube hJ 
(igntfieD b^ tbw Cipber, o. ano tbat aanoetb ouer tbe 
figne of nomber : tberefo^e tbe framon^tbat I5 as tbe 
f «o//rwff,muft be Wscn as a itomber ^^/-4<f?f. 

ilikeUjaies | gr^ 5-,DtuiDeD bp 4 §- §-. ooetb mafet 
Sf tbat ts to fate.^ ^uD fo ^ ce cc^DiutDeD hp-tct 
noetb b>pKgfo;jtbe 4° §- ce«o? ^ S^cf ♦ 

^cbolar. «is ts fuflftdentefo^^DiutfioiT. ^otu 
tf pou tbinfee goOD to fpeafee of p;:ogremort,3 can not 
tjut remember ?o« of ^aur p;jomtre« 




ofCoJ^ike ndmhers, 
Of^eduSiion. 

spatter, 
iltbougb %tdunion lljould go (n o;2Dcr be T{cdHHkn, 

|fo:c Tr»irepon,^tt fCCVng tt)l6 %eduHion, 

jconfiftctt) m tbc onclv nonibcrs, anD not 
\\\\ tbe figncstano therefore agrccttj luitb 

juulgarc rcDuttion of framons ( as here 

pou mate fi^c before ut Diuerfc crantples^ thi rfo;c U>il 
tue omittc it,anD go m banoe Ujitb froirejlioit:\iyW\)t 
i5mo;cftraungc. 

Scholar. 3 p<:aic fou fotj^o: 3f fee tbis reDurttori, 
ifi but to riOuce the greater frartion, to a leffer in no- 
ber:aj5 3 leavnco long a gone bp yoar otbcr booUc . 

OfTro^reJTton in CoJIihfi^nes, 

Rafter* 
y^irepiort 15 tbus i»?oug!)te : i^irde fcttc 
'ooune as manp tiulgarc nobriB , m tbeir 
naturall p^ogrelTion, as pou IiHc to bntjc 
gCV^'A'fig^fJ^, tbat bp tticmpou mate the 
J better fenolctlje triie places of tbe Cofsikj 
figtitstfo tbat pou fet in tbe firfte place a CTipber.anD 
tjnoer it.-y. 3nutl>en tinker, i. fet. i:^.t)norc.2»put. 5^ 
anD t)nDer.5. to^ite.cif^. 30 pou fee in tljc table fololu- 
vng. ant) bp tijrfe l^allpou fet,a0 manp as pou Itfte. 
Joj all tf)ct3algarcnonibcr£f,U)l)icbcpou banc fet 
(n tbe btgber reUie, be otber compouiiDe nambers ,0:2 
ebst bnco)npounDe:anaif tbe place, lubere pou tuoulu 
fet anp L4'iK^ figne, be noteo luttb a nomber tnconr^ 
pounDiNtbcn muft tbere be fet one of t'oe SurfoUrles. 

i^o: tnDcr t'oe ftrft nobec ijncompounDr, pou mult 
fct'tbe fiirfte SurfoUde, and tbe fccont) t^noer t^c fccoaD 
womber tjucopounoetano t^c tl}irDe tjnDcr tbc tbiroc. 




The Jrte 

iTJoc nombcrs tncompsilDcnrc iiOcfc in t! cii pja- 
grciiioii. 

f. 7. ih n. 17. 19* 2^. 29, )i. ;7. 41* 
4> 47« )^ T9» 6i» 67.IC. 

nni3crnctl)c.)Mnuavou fct./v.anDtnocr. 7,^^^ 
t}nBcTti i.t/gvauDtjiiDciM I, dfy, anDrofoo:t!)c, hi 
i;nuconicro.6 7.t3nDcrtul)ii:i)Cvoumuaftt.rA'. ano 
ViSiDcr 7 1 vQu mutt kt sy^>, aitD fo 20 farrc as pou liC-. 

liiMit fo; anp otljcr piacc, iJicaufc tljc tulgarc ncm - 
Ijcr IS (ompounDc , tijnt 15 (tt{as> tijc pcujUarc rem • 
bet , m tl)c !)igl)cr rcluc ) tI)cvcfo:c tlK Coy^//^? fi jnc 
iiiuilncDc»;bccompounDc,ott)ccof.2.o;of.^o.:cl0 cf 
l)Dt'oc.3nQifitbcf6poimucof.2.tbcnrctDounc.^-.ro 
Drtcntvniics,as.2.i£5mt{)cccpcfiticnDrt!)atiicm&cr. 

00 fo2 crampic: 1 6.10 fopouiiDc or.2. ioUicr ii^imf J5 
(rot h"^ aDDiticn,but bp mult!pi:caticn,a0 m faigng, 
tU)ifc.2.t\v.oo t)iinc0,tlutre» 

^cl}D!an 3 perceuictUnre.2. to bcc. 4. anD tluiTc 
tijatto br.S.an^ tlutfe tl)at to make. 1 6. 

^aHfr. feo maic vou lyo^iic bachctuartjf ,111 (ai< 
piig. 1 6.tJmiti£D fcp.2, niafeet^.K.tOat is oncfi:tf)en.8» 
£p,2.pctDct!)»4.tt)at!0tU)tfc. ^gain.4.bv.2.iiTa!ict!) 
2.t!)at t0tD;tfc:auiJ,2,fo;l;unfcfo0 tl)c foiatl):Uji)cc* 
fo;c tjnocr, 1 6.3P '""ft fet 5ounc, ^" 5 ' 5 ' 5 '♦ 

5!nofot2ntcn 52,3mu!tcfcttc.v5-^tn one tI)U3» 

ano isnDEC . 6 4 ♦ 31 tljall fcttc it. 6. tvmtn , tl^us* 
S' ^ 5" ?r* P §"♦ X5uaufe,6 4.10 ma:)ccf.6.multipU5 
tation0bp»2. 

^cbolan li^erebpBI fee,t!)att3nDcr,S»3l mtiffe put 
?.t^me0 tljat OgnctaitD tjnDcr,4.tUiift tljcfame. 

fl9aCer» ^0 mutt f ou tn Decor. 

£no nob fc^ ot!)ei:pIa(C0,!f tljwr nomOeris bee cfiv 

(lounoe 



ofCapih nomhcrs. 

|jounDeof.5.onel^> t^cn muttpou fet Do mtc the figne 
otCuhf , as oftentimes as ♦ 3 ♦ is muUipIieo. to make 
tl;atnombcr. 

^as fo; example, 2 7 ♦ ts compoutiDc onelp of, ^ano 
not of.2.(fo; of all otber compounoe nombcrs bercm 
tljcn of rocI)c as be copounDc of.z.o;.?. Vot take no rc^- 
gatDc^ano. ], multtpUcDt!)are,Ooett)make, 27. ni 
faipng.^. tpmcso. tb^ifcanDtbcrcfo^emiDcr. 27.3; 
fljall fet tW fignc of,cc»tli:cc times, tUis.ct ctd . 
lul)icl)e betohcnetl) a CubeofCubgs Cuhiksh. 

But anD if tl)e nombec bee compounoc.botbe cf. :> 
an'D, 5 , tncn fo; euer? tpmc tljat. 2. is miiltipUcD , a 
tbatcompormon,3lf5aUrettc.5^.anDfouucrptvnic. 
tt)at.5.ismultiplieO,3il^aUfetai.remcmb:vngCU5 
torct.5\bcfo;e.a^.anDnotaftert)fm. 

as fo; crample. ainOei:,2 4/J fl)all fc t. 5 -j^, -^ ht 
btcauretbnt.2.2.2. 5.tl)attstoraie.2. tvmcs.zaluuc 
tl)jire,Ooctb maUe.2 4. m bp rcfolution,tiiiis.2 4^t:» 
uiDeDb^2,gluetb.i2,f=o;itbatfirfte.2.fet.5r'.^gain 
i2.DtuiDcDbp,2.)?elD:tl),<$: foni)tsreconDr,2, fct.5^\ 
alfo. aen Diinoe.6. bp.2.anD it maUetl). ^.j^oj tlje.2. 
3! mud ^et.5^ anD fo^?. 3 muft put-cC^anD fo all to 
gett)ermaUctb.pV5>ceantbe,24-Plncc. ^ 

3liUeU)aies tmoec* 5 6* 3 muft fclte. ^-^ ct cc- bi., 
caufe tbat. 2.2.5. ^. Doet!) make it,tl)at is. 2. tpmcs,2.. 
tb:tre,tl):ire. anDb^rerolution,tt)us.56.DiuiDcDtp 
2,giuetl).iS.fo2tbat.2.3ret.§-. againe.iS.DtuiDcD 

br.2.maUctb.9.f o;tl)at.2.93fftf^^o""^^«''H"«^'5(^ 
3:t)trDli',fo;bicaure,9.cannotfaecbunoeD Op. 2. but 
bp.5,^tpmes:tl3erefo;e 3i mutte fette Doime, fo2 tbofc 
ttuoo. ^.ttotfc.GL.'i fo tbe U)hole figne is. g^-^^^OLct ♦ 

0o\s) if the nomber of tl)c place,o;j pcculiare nom< 
ber^bce compounoe of one of tticim txuoo, luitb fomc 
otber nomber t)nc6poimOe, tben mufc lye tovnc tt^eir 
fignesto«Tctl)er, 

as. I o?i5 comuounDc of. 2. anD, s. tbcrefo:e mnH X 



The ^4arte 

fet tiiDcr. I o. tl)f fignc tijatfg fn tljc fiftlj plare,ibl)t^ 
cl?c i0/5-.anD before it 3 mulle fettle figneof.y.fo? 
2»&o mufttljatfigne be. g^/g^', 

lltkeU)ate0> btraufe.i )% 10 contpoutiDe ofo» sitD.f. 
31 fl^all iomc togetUer t^e fignc of cc» aiiD of./§>. ano 
tnafeeit.cc/5^« 

^cbolar. ^0 3; tuoerffatiDc ft notu^tbat ^ cannot 
nttffc it. $s>auc tljat fo? lartte of tjfcanD tb^ougbe foy^ 
getfuinclTe, lubenjf beatetbc name of rontpofttion 
in nomberB,3i ooe mt0abe it rometimes (0; aoDttion, 
H& bere can be no etroure. ^n toben 31 boc confiocr, 
tbat2o.iscompounDeof.2.2.y.tbati0ttuife.2.anD.f 
(ritb.2.t?meg.2. mabetb.4: ano.^ti?me0.4. makctb 
2 0O3 maie fone conriOer,to ret.?r*. ttutfe facfo^c.A-. 
anb tbcn it tuill be.g* j/g^.to be put tn tbe,2o.pIace. 

ILifeetcaieg in tbc.2 i.place,3i fctcc^T^-* fepng 2 1 
10 fompounDe of.^^ anb.y. ano.cf ♦ i0 tbe fignc to tbc 
tbiroe placc,as V^.feructf) fpi tbe.y.place. 

Rafter. M bat (l[)all ton fct tn tbe. 8 4. place:' 

fbcbolar. 8 4 ♦ i0 compounue of. 2.2, 3.7. tberc fo;c 
|i0 figne mirit be.§> j* cfe Vg^* 

i!l3attcr. i^oU)3I fee,^ouarecunnrngincugbin 
fbi^^anb tberefo;jetake berc tbis table^fo; a patronc; 
anb tben IdiU lue p20cm to tbe tooj^e of GfsiijnQmf 
^ncompounoe* 



T/^e table for prop'efiori CoJ?iie, 

>hi(hemdie mnafe it felf infinitely y 
"Pitbtuinny M/fimltit, 






5 ^5a-€ i^/s-'- ly'kfS^fir 



16» 1 17 * I i8« |I Q . I 2 0. I 



^7. i ^S. I ?9. _l _4o. I 41. 1 42* J 4^. I 44- I 



21. I 22. I 2;, I 2Z I 2T- I 26. I 27. I 2X. ! 



29.i vD- I U-l :;2- I jji- h4. I il- ^ ^6. I 



■ 4v J 46. I 47j_4S' _ f 49. _| So^ I p- ( 



g.^j)/^, 0/^ i^-c ec^cei/ rT-^/5-uV5-^'^-/5"i'x//5- i5--^/5"l 



? 9-1 6o. I 6l » I 62. I 6^ I 64' I 65'» | 



_6 6^ I 6 7* I 6S> |6 9»l 7 o» I 7 f' 



72^ _ 1 7 y 

Ws^c£ceir;p 



7v I 76. \ 11. FtS^ l79- I 



74. 



So. 



gWK7a/&^l&'S ^4^^ 



3;n tbts table. y.cZ ♦ anoy^. arc tlje grountcs: 

of all tljc rette aboue tbfm.j^o^oftbcfc 

tt)?e, all tljofc otl)cr bee maoe. 



£>f 




The Jnt 

OfCop'ih nomhers compounde, 

V^^^W^'r^W-r Ofiks nemhrs compounDe,at*c inaDc 
-.-.-.. *^ "^ ^^ Jbp aODttion of* 2.0? mo^c fimplc Cofs 
i [tks nombers togctljer : 

1 2,ct-~^ — 4-^ ♦ —- f— vf ano 
fo fco;tl)c m Diucrfc fo;jmfs,tul)irt)0 
be infinite* ll^otubett fo.2 b?tcfncffe, 
iuc mate comp;iebcnDe,bnDer tbefamc name (bicaufc 

of tbe like lu0^i5e)aU Otbcr rejidualles Cofstl{e,\xi\)it\)C be 

tnaDe bp fubtraction : as. 5. rP* — ♦4.§^^.3nDaU 

fotbofetbat bee niaDebp aDDition anD fubtrartion, 

botl)eto5etl)er:a£;.9*^5".— 4— .4»^* 6.^. 

3J« tobofe numeration 10 no ijaroneCTe* 
^cbolar. %\)tr\ ^our rules maie be tbc lljo^itcr. 

OfNumeratiojh 
£paaer» 

■■^i& KumerathniB cafilp tn&crftaDC 
bp atiDition of fimple Gfstiis, foi 
tljiB 10 tbe fo;5me.6. ^ -H — lo.f • 




tbati0.6.5-/tt4y«,mo?e.io,n6ber0. 
liltlicluaies. S.cf*^ — |— .1 L^rp^.is 
j;8.C«^fjanD.iL2^. 

^ j^olD fo? refidudlUs , tafee tbefe 

eramples«9*§^ ^.•— — .1 2,c£,lul)icbe ts.p.SfWrfm 

of Sj«4r«, faue. I iXu^es. 0lfo, 4»y^. ♦ I y. §^, 

tljat 10. 4;yttr/»//V«, abating* 1 ^/^uam, 
Znn fo; botbe together, tbis is tbe fo;mc» 

io*^5> — h-.6*<£. 3o.5:p,tDbicl)efi0nifietft 

10 . Squares ef Squares, nnt.6* Cuhes^uhut^nQ*^o.r90fes, 

fecbolar* s:!)is is plaine . J^ojfo maie 3 tinoer^ 

0anDe of all otber as.9, x^<t* . ^5^» — \— B»f 

tW ia.^.S^Harcs 9fCHhfSjicftt ^.S^uarsSymo^tXnUers. 

$^allcr* 



ofCoJsike nomiers. 

ot mo :c ano IcHTc in o^tscr^tbcn to follotoe tbc o,2Dcr of 
ti)r fo/li/^tf agncH.'bicaufe tl)at aomtion.ts ojDcvli' pla^ 
ccD bf fo;c fubtr.»moiT. &o Uierc it better to fct tljcim 

tbus 9.^^ce»— [ — S.f o.5-a^)obcitin dccdc 

allt0onciutl)efcbtnDcofnombcrs,but notfo mo* 

t\)tt Surde nombers, ijoljCtC tht QM^ foloUiCtI) of HCCCf? 

fitic,a6 fljall be DcclarcD tn t!)cir place mo^c largely* 



Ofjddltkn, 




|<i aDDttioit,vou mud liauc confiDei 
ration of tt]c 0/"^^ Ogufs : fo^ noc 
oti)cr ncmbcr,maie bcc aoncD into 
one,tt}cii rocl)c as appertain to one 
figne Cofsiks^ 

I 2s m tjulgarc tjcnonunations, 
'»v^>8-Trvi;2c:-^ Vcu Coe not a*DDe tl)c nobers of fljtl^ 
trngesTto t!)e nombcrs of pcnnics-.but rou tonic 11)1^ 
pngcstofbUltnges^ano pennies to pcnmcsttpoun.^ 

DCS to ponnDcs,ro m Cofiiks nombcr^,C^^^" mutte bee 
tolncu to Gti«, S^fwm to 5^«4m, ano gencrallr. lilxC 

to U^ie« 

Scholar, 3Jf «)i3*c al,3l ra marixc it inell mougrj. 

Q^aller. 2i:bcretsromcliil)at wo;tcto bcconfiDc^ 
reD , tliatiftljcrebee anr fignc in tDc cnc nombcr, 
tuDtctje IS not in tlie otljcr , ttiat reuernllc fignc iuit^ 
Ills nomfacr, mutte bee fctte twunc U)itl) W ngurc of 
_-|Il_,o:» .as itftanocti) tljerc. 

0nD fartlier, touct)r"g t!)orc tluoo fignes.— f — ♦ 
—-- .lulncbe bee tlic figures of moje anu lerrc, pou 
mutt giuc reccarBctuOetber tl)ei bee liUe o; t)nltkc,iit 
tftofe nombcrs tUat mutt be abOcDt f o; if tt)et be Ukc 
fn nombers,of one Dcnominatton, then mutte tbet fo 
t^mainastbei be. ISuttf tDei bctnliUe,cuermo?ca. 
fcatc tl)c fmaller nombcr of tl}cmT, tDat folloU}c it^orc 

a.t;. tjnUUc 



TheJm 

tjr.ltUe %ne0,out of tbe greater: and fctte iroune tl)e 
rcae,lMitl) tljcfigne of tl)c greater nomber, 

^ct)olar» Bi> craniplexs , 3 ajall better concetuc 
tljofe rules* 

a^after* CaUe tftcfe eramples* 



14-5^ 



4» §-♦— +— 8-f 









.20,f 1 145-- 



20 f. 



I45>* -.2o.f I i4»§-»— 1— -20.^ 



4.5-. 



4*5-* 



I4»5>» 



► I2«f« 
♦ 8»f 



►4»f 



4*^» 



-.I2.f 
■,8»f 



I4»^» ' -♦4 ♦ f ♦ 



io,5>»— 4 — .8.f» 
4.^. -12.?, 



lo.g^* 



►8*f 



4*5-»-+— ♦n.f* 



M«5*» ♦4»f 1 i4»5^»-+~*4«f 



^ete]^aue3| tjarteo one eirample Dtuerflr^to tlie fi 
tente pou mate utarfee tije tfe of pour rules in tljetm* 
;ano fo? tibe reafon of t^ofe rules ? latx D^allmarhe 



fo Cofike nomher 



tijofe examples UjcU. 

i^o?lut)crc in tlje firtte example , boti)e figncfi arc 
— 4--jit mutt ncDes be,tt)at after tbe aDpttion of tt)e 
Srttc nomfacrs, tbe fcconoe muftc bee aUDcD iuitlj tl)e 
fignc.--| — . . 

jn tbe fcronDc eramplf,lu!jerebDtlic tl}e figncs bc 
— -— .bicaufctOcrdutintetl). 2i.f.oftl)cfirft.i c.^ 
S^t)crfo;c 10 ti rcafoiLtl)at botbc tljofe iuantes fljcuiD 

be fcttc Dounc luitl) tl)c figiie of. : auD fo in th^ 

tIjirDc auD fciirtbc cramples. 

3:n ti)c fifti) ciamulc^tbe fcconDc fonimc tc notful^ 
i^.4. V buttiicrc luantctb of it.8. f . aiiD t{)riTfo;ctf 
l)ouputtJDnnrtS)C.4.5-.fullp^pounma abate. 8. out 
of tbe. 1 2.f . in tbe bigbcc fomme : ano fo of tbe otbrr 
crampics. 

lout fo: mo:e p.2artire,anrj better Declaration of tbc 
^fcoftbcm,bercareotbereraplcS7 0f"io;cv=anetic. 

2o.s>ce 

3>.s-ce-H— .14-5-*— .104.:^ 



12. §-^5- —-] — .12.^. ♦ip'f- 




3n tbe firffc erample of fbefe.2. fou fee ♦ 1 2 o. t^. 
irttb tbe ftgne of leffe , to bee aubeD toitb. 1 6. i:p . 
iultb tl)^ figne of mo^: anb tberefo:e, feepngYbe 
ftcrnesofoneQ/5%Denommatfo!t oifagree, ^cuooc 
fubtrarte tbe leffcr,out of tbe greater:ano tbat. i o 4. 
iubtcbe rcmainetb ? 3! 00c fet Doune luitb tbe fignc of 

Uaij* Icffc 



TkJrte 

lcffe,bicaure tfje remstnec tsoftljat mmhtt^iW 
baretftatfigne* 

M^ in tibe feconoc eicaplejtfje placing of tfjc figne 

-H — before ttiabetl) nomberB to bee fettc bc^ 

fo;e fquarestanD fo tbe like Dcnommationij,Dooc not 
ftanoe one ouer an otber«i^et is tbe tuoo:fee Dooen as 
\tt\)t\ Dio danOeecbe ouer: bis liUe. 

^ctjolar. 3? P^aie roulette me trie mvcunnrngc, 
iuitb an example o^ tUjoo. 

I6 5^c£- — i « I 2» §^» .6.X^. 

— +— 12»§>, ♦g^J:^* 

3! fettbeerample, as nombcrs came to m^ m^nDe: 
but I baD almoltefctntpfelf on grciiuiDc: fauetbatBI 
calico to temembjaunce, t!)c comyaiifbn tbat ion 
matje,tot3ulgaretJcnominatton6ofpount)cs,(l)ilUn« 
gefi,anD pennies: ano fo toas infiructeo to placeeuc^ 
rp feueraUe Denoininatton (tuerailp. ^no to fette tbe 
greatede Of nominatio firff, $ tt^t otbcr in tis o;Dcr. 

^oiu Jotll 3 p;oue an otbcr eramplc, o; tkcc 

3 ♦Tg- ♦ —^ — ' 4 ♦ cc» ♦ 2 o.f. 

2o.c^, ~— , 8 , ^, < 1 6.f . 

? ♦/§"♦ — 1 — .24«cC« ♦ 8. ^» .56-f. 



1 3 • §^^* — h^ — tS^cC* ♦4»^» 

7 * ^<^4 — — .&»(£♦ -7^^ 

2 o.§-c£»-~l— tZtcC. ♦4»i^.» ♦7.? 



of CoJIike nomhcrs, 

4. ^,'-\ — ♦ 1 7 . f ♦ r-*7.5:5» 



10.- 






.9.?. 




4.5»ct:» — ' — .S.cC. »?'5"»- — — .4'^* 

£Daaer. liionliaucDocnlucU : artDfo;p:Dofcof 
Vonv \\)o;\ic,\mi mate in tbis arte not onclp p;one u, 
bp tl)c rontrarp kimocas ro" t»iO i" ncbcrs Jhftrach, 
but alfo bp tt)c rcfolutton of all ti)ofe G/?% nombt rs 
tnto nobcrs ^^y?r<f/f ,tafern5 a"!' nombcr fo;i a coctc 
anD tben tt)eS^/<4m ano Cw/'fj^caccoititnglp as t)cre 
tn tl)ts table^vou mate b^tcflp fee,but nio;c largclp \n 
tbc table at tl)e cancc of nombers figuralic 

Juhlefortmlk by Yefolution^ 

of any t»oorks in this arte. 



•^ 


?^ 


^T h-fr 


f^ 


kct 


V5i_ 


2 


4 


8 16 


32 


64 


128 

2IS7 


1 I 9 


27 1 81 


24^ 


729 


4 i 16 


64 25-6 


1024 


,4096 


46384 






2S 


?[!5' 


1 ctct 


u ^^/s^ 




'S 


1 121 


r" 1024 




6T6I 


19683 ^9049 




6yT56 


1 262144 1-048 v6 





janD if tbtjj table in anu parte , feme to (l^o^tc 02 to 
* Ittle: 




The Jrte 

iitlct^ou mafe Wtje recourfc to tbe table, attl)ccnDc 
of Sgurallc nombers ^tobtcbe tberfo;e 10 mafic largi- 
an^ gineraUe:fo tijat ft mate iocU be calicu tbe frute 
full table,oj table of cafe. 

iSutnoUj fo; triall of tbe lafic crample: 
firtte tbcre UJ.4.5^ ai:foj tuljofe roote 31 taUe 
2.an0 tl)ccefo?e toofe, 4. ^ cf.* ntafee ♦ 2 f 6» 
lul)icbe 3 fette ooune in nombec>^y?r<r??, 
j^erte is»^rquare0,tDl)!cbe acco;Dpng to tbat 
roibtcmuff neoc0 be, 2 o.ano tbat. 2 o, 3 let re 
ooune alfo x anD tben*6*rootcs,tubtci)c maUe 
1 2. i^no all tbei pelDe*2 8 8. ano tbat ic all tbc 
firlte fomme. 

i;bcn fo; tbc feconbe fomme,3l fee ficffe.8. 
C«i'«,U)btc»)e maUc,6 4. to bee aODco. %\)t\\ 5 2. 
foloUj2tb.8.fquares leire,tbat 10.5 2.to htt a- 2o« 
bateb,antialfo.6.roote5leffe,tbatisi*2o,airo r^, 
to bee abatebt^o mut 31 abate.^ 2 > (fo^ tbeim 
hiit\\z ) out of. 6 4. anb tijen tbere reaetb hnt ^ 4* 

1 2. lubtcbe aDDco tjnto 2 8 8. sf tbc firft fomme __lli_ 
Hoe^elDc^oo. 12. 

j|>9Ui!ftbctotall agree tui'tli tb^0jtbent0 ^n^ 
t\^z luoo^ke gooD. *■ ^^ 

i?^oatctaIllubereof,3irerolue.4.5>c£.0^ — ii* 
to nomber ^i^/^r^fff ,ano tbei tutll niajic.2 5-6. 3 o o 
tben.8.a^«ntafeetb.6 4tlDl)icbe botbereloe 2r6 
320, SEbenfoloUjetbmtbefamefomuteo.^ ^4 
ano.4.2^.tobeabateD.SDbe.5.5f\mahc. 12. -7—^ 
anotbe4.roote0 velbe. S.tobtcbetogetbcrDo ^^ °* 
amourite to.2 o. anb tbat muU bee abate D fco 
tb^fatu fotttmeof52 oanDtbentbere rcmainctb one^^ 
Ip 3 o o.agreableto thz former fonmie aboue tbc line. 

^cbolar. 2:bt0 p;}oofe 31 Itbe tuell: 0nD 3, "^ttttim 
tbat tf 3i luoulD tuooalse tbc lifee > tabrng foj tbe roote 
3,0^ anp otbec nomber.tbe p;oofe iutll fueceDe a Itfee. 

i^affar, ^oitj to mafee an eanbe of ;^bbition , W 

taufc 



ofCoJsike nomhers, 

caufe ?ou tl^all tl)e better remember tl)e rules ofit,^ 
iaill 0tue ^on tbcm in tbis b;jife fo;}me» 

Ittf^reatencffe liJ^e andjlflies alfo, 
jfdde like to like there nedes no mo: 
jfndyi'herethe greateneffe difagree, 
iPUce eche by other fctier ally. 

tfiith figne ofeche^as doetb require, 
'But iftheftgnes Vnliks ap^err 
Ihenfrom the more abate the lejjfe, 
Ibe greater his fts!ne "^ith the exccjje, 
tvill make the/omme. 
Of that addition. 
The proof e is by refoluyng, 
tche notnber into lis rei.kenyng. 

%\){^ IcflTon Doetb containc tbe fo;:mer rules oitelp 
(n b;:tetanD tberefo;ic itcabetb "o Dcclarationrbut tbc 
greatcnelTc Doetb betofeen tbc OA/lfDenomtnatioit, 

anD fi:^nes betoken fpecialli', — \ — anD ♦ . tbc 

fignes of mo?e anD le(re,anD no otbcr figne5» 

^cbolan %W b;fef ^^CTon luiU bclpc ntemo;ic 
jttocbe;anD ft)all fufficc fo^ tbe ruleiS of aDDitifln* 

OfSuhtraSiton. 
falter, 

X'lcn fo? fubtrattton, tbts Hjall rou 
markc in efpecialUtbat lubcn rcur 
nombcrs arc fcttc Dciinc, after tbc 
comon mancr,firlle tbe totall, auD i.i^/f. 
tbeu tbc DcDucticn:pou (ball confix 
Derluell, tubetber tbc fignos bee 

..=^-_^ c ,.-^.. -- I ■ 02 . foi in tbe Dc^- 

Duction,tf rou baue — j — tben muH tbat be fubtrac* 
tfO from tbe like aboue* 
anDtftbatfommctntbe Debuctfon, tbattjatb tbe 2. ?(«/?. 

%.}. figne 




Thi Jrte 

ftgfne —I — .bee greater tfjcn x\it nomber of tfjc like 
quantittc ouer brm ,>tt!) tbe like figne — t— . tljcti 
abate tf)cf)ig(jcr out of tl)c lougbcc , anu U);<tc tljc 
relic U)ttl) itjifi figne — — » 

3.\tt/^ iButif tt)e Ufee quantittc fn tbc total!, bane i\t 

figne ,tl)cn vmt faotbc nombers togctbcr anD 

fet tbcm bnDcr tbe line tottb tbat Ogne . 

4»^«/f. ano if tbe fccontjc fomme ( tt)at is tbc Oel>u(tton oj 

abatement) iDitl) anp nomber^baue tbw fignc of leflTe 

,it mud be accoumpteu fo.j mo2e, anD nuift be 

aot^eD to tl)c IiUc nombcr ouer it,crcepte t\^t oucr no^ 
bcr |)aue tbe figne of IcflTe alfo : jFo? tben mull pou a^ 
bate tbe lcirer,out of tbe grrater,ano fcttc Uoune tlje 
refte , Iwitb X\st Dgne of tbe greater nomber : tubicbc 
tbet bauc at tbi0 conferecerj meane to regarue tubat 
tbe figne of ttjc feconDe fomme is bp elliraation, ano 
not bp tD^itpng,fo.jtbei are contrary. 

^cbolar. % izz gooD rcafon \x\. tbis : f^i iw anp a? 
batementc, tbemoje is abateo , tbe leCTe bp fo mocbc 
fi)aU remain t anD tbe lelfe 10 abateo ? tbe mo^e Doetb 
remain bpfomocbe* 

^,^t, fatter, pet one tbpng mo;e fjs to bee marUcD, 

tbat if tbere be fome DcnomaTatmns,in tbe one fome 
tbat are not xn tbe otber , pon (ball marbe m lobicbe 
fomme tbci bee^i^o; tf tbei bee m tbe fir(le,tben Iball 
tbei feepe ftill tbeir otonc figne* auD if tbci bee in tbe 
feconoe fomme , tubicbe is tbe Deomtion , t\^tx{ ^^all 
tbei cbaunge tbeir figne to tbe contrarp: liBut Ujberc 
foeuer t^i be, tbci mull be fet m tbe remaincr. 

^cbolar. 3! ean better tmoerftanoe pou , tbenre^ 
memb^etbofe rules. 

£©ailer. Cbcn tafee tbis b.:icf le«ron,3pter to bee 
rememb^ctj, tben to bee bnoerllanue , but bp tbe let^ 
ter before, ano ^i^^ tbe eramples folotupng. iSnt mc^ 
ino;ie It&etb tueU focbc aioe« 



ofCofiikeuomhers. 
jfbriefruk ofSuhtraSlm. 

1 , "^fhenftgnti andgrcdteneffe hothe dirtt. 
Tour T)t0or\e prtcedetbfonhe commonly , 

2. ^ut if thahatemtntegrcdttr hce, 
jbexceJfeJhdllchdUHgf bU ftffte tberhj, 

^, jfnd'P/ieretbtfignesd$etlijfagree, 

The bighcrfigne mu/i reft duely: 
jfnd though the hatemente he the greater. 
The rtftefttllioynetb hitbefonmes together, 

^ If qudntitiei d$e dijfagrtf, 

^lace thtm 'PrithfignesJlfeuirdUr 
The tot all \efetb tbefigne be bad. 
The hatementeftiU,to (baunge is glad, 

^cljolar* i^otD fome erample0,t)Dtll Itglitcn tWt 
tulcB toelt* 

^aOtr. 31 totll p;iopounDe tl^e ltkc,a0 3! DfO (n aD^ 
ttttbn,to t\)t tntete ^ou mate (uDge ti)e It^nclD^ano 



10. J.. 



4-f 



♦ 






6.$-» 



4.f 






.I2.f. 






.8.f 

>12»f. 



-~f-~»4« 



10,5. 



The jfrte 









6»5-»— H — ♦lo^f* 



j^S-> .I2.f 









6. J>» 



►2 0,f . 



10,5.. .8.^. 

4.§^»-+-.i2,f 



6,5^ 



► 2 0,f. 



Clje firffe aittj tbf rOe epampl^s be berp plafnetanD 
iw tfte feconoe lubece » 1 2 ♦ i^oulD bee abateD ont of.8» 
tbere f0.4»to feUjc:aitD t^ecefo^c 31 abate tbc biffbcr, 
out oftbeloug]ber,ani?3l fetooune. 4. tuttb tbe Cgne 
of iDarTtviTg,o^abaf entente. 

3ln tftefourtbc example tbicaufe tijeljtgber: nom^ 
bertst^clelfer,3l Doefubtracte btmoutoftbe nctljer. 
ant> fette toirne t^e tefte» 4 ♦ Iwitlj a contrary figne of 



But tn tlje. 4. later emmples, lubere tlje fig neis Do 
Wfagree , tbe nobcw tbat foUoiue tbe fignes,ace not 
fubtraeteo one from an otl)er,but are aDDcD together: 
anD lAiti ta^e tttll tfte btgber figne. UBicaufe m baluc, 
tbe figne of abatemente is contrary, to tbat it appear- 
cetbto^ee. 

arno fo;j pour crercife, to make pou full p;omptc in 
tl)i0 arte J baue fet fo^tbc mo;je eramplcs* 






.I2o,f. 
*4Q^f 



89»?-— 



-89*f 






ofCoJiike nomhers. 



18. X 






6 ♦§-.-}— .6.::^. 









6 ♦ X^* — 1 — ♦6»^. 



^5"* 



18.2^' 



-lo.^. 



I2.2^«— I — .8.f. 



>§'♦ 



.6.i^, 



I8.f. 



4/5-— f- 









4/5--f-io.cc. — J-5-'5-.— 18.5-'. — ?f 

^crc in tijc firffc rramplc,tuf)crc 3J ti/oiilD abate 9 
cc.outof.6.c^»3I»taiccariIppcrcciuc,tl)attl)crcarc 
o.cc.tofcluc. anDtl)crcfo;jcDoc3;fcttcDounc,^.aL. 
tuttl)tl)ts figite — ,U)l)icl)cfi0nifietbUjanteo;a- 
batcmcnte: airo tbe. 2, nombcrs tijat folloluc tbe tjn^ 
like figiTcs,3J fet Doune botbc aDDeD into one:anD put 
tljerto tl)c fignc of tlje totall o^ oucrmotte fomm % 

3in tbe feconoe cramplc , there 10 tbe Uhc tuoo;fee: 
ifonnaljatpng.9. outof.8.3lfinDe.KtofcUje:tbat.i. 
Doe 3! fet Doune luitl) Ins Denomination of. ^ cc :anD 
tljefigne. . 

0nD the nombrr 8 9 tbat folotoctl) tbe figne 

(n tlje feconcc fomme , ftauDetb in fo^ceas— ^ — ^,fo; 
tbe leflTe 10 abatcD,tljc mo;c mull remain: tberfo:e m 
tbe remaincr , 3 fet not tbc fignc of mo;c, befo;e tljat 
nomberof.89. but^Sputttin tbefirttc place oftye 
fomme;U)l)tcbeplacebfitfetf,figmfictl)ftiUmo^c. 

3Eai|, anD 



The J^am 

0nti bicaufc ouec tbat nonibcr 8 9,tl)ere aix r.c x\h 
lcr0 m t^e totall, ttjcrefo^ g muGr putte Ccunc tijat 
fonime 80 it is, tuitijout aDCirng to It, 0? abat^ifg fro 

It, tit it felf» 

^cbolac %Wt*i> crampIc0mtgl)tl)efettt)U0,a0 
^ t^infee,bicaufe tl)c places doc fo require. 

6.CC* — I — .I20,f» 

9.ct. .40.^. 






Rafter, l^cnrcwbcr pour fclf iuell , ano mariic 
t^e rcmamer IjoU) it is luaittcn. 

^c^olar. 3i fee mp otunc ouer0gl)te: j^o; no nom^ 
ber mate bcgiit,toit6 figne of leflTe: ano tf)erfo;c mutt 
tt)eir plaff cfl be altereo of neeetTiticano fct m o^oer a5 
t\iti luere befo^. 

{patter. jCbf n fo^ all tbe rette of tl^e eraniple^.o; 
anp otber lifee, 3J d^all not neatje to glue pou ant> far;* 
tijer inftructioiT j fit!) tbat bt tl^efc fo;mer, pou maie 
iuogeofallotl^rr. 
fmfe. 5inD fo? tbe erantinatlon of pour lBo;*r,tbe triallc 

bp refolution Ooetb feruc bere, as tucll as t\& lubere: 
rcmemb^png onelp ( as tl^c o?Der of fubtraaion maie 
aomonillje pou)t!jatti)e fomme of tl)c totalle,lDbicl)C 
XB tbc firtte fomme.mutt counteruaile tbe otljcr bot^c 
fommes:tl)atis of tfje DeOu(tion,ano of tlje remainer. 

^0 to trie tbe firtte erample, tafepng. ?.fo; a roote: 
d.c^.mafec. 1 62.Ujftiefte 3J put to. 1 2 o. aiiD it pelDetlj 
282. d)eit in tbe fee onoe fomme . 9 . c£ . are. 2 4 ^ 
lubrreof. 4 o. mutt bee abateo fo? tbc ttgne ^fo 



cfCof^ih nomhers, 

f0tljatfommc.2 05. again in tbcremaincr.^^cf .are 
8 1. lulju!)c mua bcc abatcD out of. 1 6 o.anD fo rcttctft 
7 9.U}tiicl)c U)itb.2 o j» DoemaUc.282. agrcablc luttb 
tl^c firtte fommc. 

^:l)olar, s:i)t3 Doc 3; toell buDerffanoc, ano p^aic 
^ou to p;iorcDc to multipluation. 

Of Multiplication. 

j'i tnultipltfatlon-, tljcrc r3 no oiff^ 
culne, fo tOat vou Dooc IccH marhf 

ibeOgncjj — | — anO- , luljb 

clje bepng l)otl)C like, tuiU baue tbc 
fignc — 1 — fcttc in tbc totallc.anD 
l)f vng tnliUc, tbet imtl Oauc in t^e 

totallc tbc (ignr . 

0notilicU)aic0 tn OiuiCon— ^ — .DduDco bp 

0? f otrarp Ujaicc bp — j — tuill alioaics bawc 

m tf)c totallc : but — f— OtuiocD bj' — | — ,0: 

bp — — ,lu{ll make alluaic — | — . 

CCl^icbc rule fo; rcaDf rcmcmb^aunc^ 3 banc qu 
ucn von bcre in mrtcr. 

"Hfho that tfillmulti^lie, 
Oryttd'midetrulie: 
J/^all liks/lill t9 hdue more, 
yfndmip\e Icffe injlore, 
Thtir quantities doe h^pefoche rate, 
Tbdt.M.doetb adde:and.<D.dlf4te, 

^cbolar. ^omcanepou , tbatUbeCgnesmnftt* 
pl(el> together, Doe make moje,o; —\ — : 0nD fenlibc 
fines multipIicD togetl)cr,Doe yeltie IcflTe, oj . 

^ader. ^0 is tbe rule. But to go foiUiarD notu: 
of tbe nerteDtffifultie^astoncbpng G/?»^<f quantities 
that ci)aunge thetr oenomination, here ts no mo :e to 

bee 




The Jrte 

bcc ratcD,ti)cn Iras taugljtm multtplfratfoit of noin« 
bcrs Co/?%tjiuomvounDc, ano tn iiyc tatic fet fo^tlje 
fo?tbect)aunge of tljcir names. 

^d)olar« 3 DnDccflanoc, that tn nTultipliration 
Ctbat i6.^.)tl)cir figures ntua face aoncD.^riD tn-SD. 
(0? Dtui(ion)tl)et mufte face abateo* SEftcrefo^c a fcUic 
examples H^all fufficc fo^ tbe reffe* 

spaften 51^abe tljefe fo; a p;iefiDentc , of all tbat 
iuob^fee:fa^ tul)icl)e ^ou mate iuDgc of all otljei- lihc* 

lo. cC* — ! — ♦9*5^»' — |— ♦20.^. 



8 o ♦ <x. ♦ ♦? 2.§-. . 1 6 c.i£^» 

tq/^—- 1—- ♦455-S— +-I00 ^* 

T^/5— h- Iiy^S— +-8?ct~f-6S5- 16^ 






— "—♦7 ^5'^* — ^~ <^ ^♦S^ 



2lo.^5>§— i— 30.^/^ 7 y 5-^ce 168 ^s- 

^cfjolar, 31 pei-cefue.tbat tbefe tuo?I«0 toe apperc 
moic IjarQjtijen tijet bee in Deeoe, anD tfjatbtcaufe of 
tbeic ftiaunge formes t but bp tjfe 31 trulle to bee ac^ 
quatnteO Icttl) ttiem Uiell inouglj.and tfjer fc»;ie 3 ltitl>. 
begin U)itb mo?e eafic ^ampl^s. as ibefe bee , tbat 

foloU2C 



Collotoe^ere* 



ofCofAl^ nomiers. 

'r-^e* ^-4.^ 



-7 2*5'. 



.8o,f. 



270.C6.. — I — 5oo::g. 
27o.ct. — h~Soo.2^.~ 72. J-. .8o.f» 



S.ct. *7.f 



112.: 



i28.y^.-4— ii2^5>. 



-9^.2^. 



ri87^- 



-112. 



^5^ J J 2.5^- 



-98.:g^, 



rente mo;:e Dtfflculte to looke on, t^cn tbct hz \\\ p;^cs 
ttfe^ tf a ntanne gtuc gooo lieDeta tl^c figncB, ano t^c 
quantUtc0. 

£0aacr» Before Uie go anpfatt^er,3l toiH ll^ctuc 
^ou fomelubat of tlje reafon,U)ftp tlje fignes ougbt to 
cbaungc. £1 JTD tl)at bp tUioo plainc tooo^UeSjin nom^ 
bcrs Jh/iraUc'^s tjere folotoetft^ 

QCJljcrc pou fcctliat loben 
31 IjaD multipUco.i 6—-] — 1 2 
bp 2 o it maoe. 3 2 o — | — 240 
tbat Is in all.y 60. 



5i5ut bicaufc tftc muUipf j< 
are ougl)t not to be fo mocfjt 
bp 4 therfo;jc it is reafon,tbar 
31 fijall multiplic tt)c bigtjer fomme bp , 4 ♦ ano abate 
tbatontoft^efoamectotaU, 



16.- 

20- 



.12. 

'4. 



-64- 



^o- 



48. 
240 



^60 — ' 121 

tbatfg. 448. 



p,U 



XMljiclit 




44'^ 



The JyH 

m\)\<^t tl)^naf ro" f« !)cre Docn bp » . 6 4» 

.48. iubicftebotbemafee, 1 1 2, to bcc DrDuacD 

out of) 6 o.anD fo cemainetb 4 4 8» %\^t luftc fommc 
tbat rominctb of tbat multiplicattoiT, 

,^cl)olaf. SDf)i0 3i tinDetftanDc UjcII; ano 
maic pjouc it iw tljts fo;te ♦id* — i — ♦ 1 2. 

iimucti)»28:anD« 20. .4JS, 16. 

2Dl)cn If 3 muUipic,2 S.bp. 1 6.tt imii ^>clDc 
4 48* as tbe Ujoo<iUe bcre Dcclarctb* 

0nD fjcrebp mate 3 luDgc, t^tcojiiks noJit« 
bcr0 libctuaies!. 

£|?allcr» ^ct one crample mo;jc MIX 3; p:opoimD 
btraiifc 3 UjouId put ^ou out of all Doubte.mijcifo^c 
luache tl)t3 fojme of luoojke. 

Ipftiz ^ou mate fee , tftat if 2 4. . 5» 

tbeficfte fommc of 24 3 i ^ . — — .2» 

iDcc multiplteo bp i y it iuoulu a 8. P^6~ 

inafee.560. ♦45'.t!)atij5 55o* Vr 

3 1 V.)i5ut It ougbt not to htt fo - ^ -^ av 

mocl)c,butleirebp.2»t)?me» ^^, ., 9> 

2 4 3.tbat ts»48 6: ^^^*^ *^*^ ^ ^* 

bicaufe t\it multtpUcr uoet!) lDantc.2»of, i s> 

^iiD fo abatpng»42, 0;, 48* — .^6» outof»3 1 5-. 

tbf re rcltctO.2 7 ^ tobicbc iis tbe tuffe totaIl,lubcn.2 1 
ijs multiplico b^» 1 5* tobcrbp tbc multiplication i3 oc^ 
clarcDtobcegooo* 

2lnD fo^ btcaufe tfjat— multiplietr iuitfj — 

tjoetl) maUe — H — : marfee l)ere,tba't pou maie not a^ 
bate fullp,48. but 4 8* 6. 

SDben f»::epng in abatemente, tbe Cgnes in figure 
are contrary' to tbcir otune catmatton and fo;jce:tt)cr* 
fo;e tbat* 4»8«muft be mane -~ — ♦ano tfte— — hti 
fo?e.6.tourneointo — [— ♦ 

^cbolan 3! fecit tuc«, it matt ntnt^ be fo. 

i^o^iftfjettoerefetto beefubtractcD, tbenfl^oulu 

tlti aanoe fo.4 8* ^:ipl|tc^e Dcclaretb tbat 4 2 

n^QulD 



ofCofikenomhers. 

3i3ut luben tl)cfamc nombcrs^arc fcl c mongcftc o^ 
tbcr to be aODcDtag tt ta berc in luoitjpitg of muUtpli- 
fatio!T,tben muft tbei bclo^ittcn tlms.- "4> - 6 
Ucclarpng tbat if pou abate, 4 8, ro" J""ff "^ aDDc.6. a- 
0am,btraufe vou abatcD,6.nto;c then vcu ougljt. 

Scatter. ^iJou tjnDeraano it h)dl 10. t)crfo:c bcrc 
tuiU luec maUc an canoe of niultiplicatiDn : ful) tbcic 
refletb notbr"? but tbc p;cofe of it: b)l)icl)c tnaie bee ^i,^ py^fg of 

U);ougl)t bv refolution, of all tbe C^M^ »o^"^f ^^^'"S,»to//r4. 
to nombcrs JbJiraHeM m otbet kinoes bcfoic. iDnc^ ^ -g^,^ -^ 
IpconfiOcrrngtbatttjcrefolutionsoftUcfirllanDrC' 
f onoc rcnniie0,muft be aooeo togctben 

^no tbcrfo;e if vou lifte to p?ouc tbe fii'ITe erampic 
takrng.2.fo: tbe tootc, fou Hjall 6n0c tl)c ftrftc louic 
J o, ■ I - -j6--h--4 o.tbat la. I ) 6. ^no tbc feconoc 
fomnTet»,2o. — (—♦ M- — -.8.tbatis.2 6.SLl)C 

tbtroc fomme ta. 1 6 o o. — I — . 1 8 4 o* '6 6 4* 

-+-272. .32o.lubicbcmaUctt).4oj6.ano 

fo Uoctl). 1 5: 6. multiplteO b^2 6» 

Scholar. %W maie 9! p^ouc at an^ trme: fo tbat 
pou (^all not ncDc to ttatc aboute it. 

OfViuifton. 

fatter* 

Fjuifion tfl nertc in o^ocr.anD agrr ^ 
'able m tbe gcnerall rulestano batl) 
noe moje fpcciall^tbi n tbe tjcir "a^- 
iturcoftbc luoo;UcOooetl) require. 
rio2 as concetnynge tbc fitncsof 

\ | — anD ,ti:cfameo;Ocris 

^ ^.,_. ^ ^_^^,^ ^ihcre, as is m nrultipUcatipn. 0no 

Tou'cbrnl'tbe Oph fiS"*^^ ' ^^ '^ ^^"^ °"^ ^^^'^ ^^^^ ^ 
fatcD in oiuiQon of liombers OM^ ^ncontpounoe. 
Scholar. %^zn a feUjc eramplcs maie fuppUc fbe 

13. U. Declaration 




The jtrte 

neclacaHon oftlietireoft^c x\xU&,Mt\i t^ep^tbs 

^uSizu 2Ca&etil)erefo;tourpurpore. 
jan eirample of tlje fIrKe too^ke. 

60. 

:El)e remoni^ng of tbe Dtaifo;t, 
fo; tlie feconoe looo^* 

i^. 5*» — I — •s^^g^* 

SC^ep;oofe in nomberif AhJlnUcf 
aaoafmptHn0*2»fo; rootc* 

ZA* — I — i^* 



4S0 

^5(^_| — ^^iz^^-H — 32o» (8— f— 20. 
24, — j — 15, 

Cl^efame tuo?ke in $ere 3; Idaae not onel^ partek 
tjulgare fojme* tbe U)oo;Ite, fo; ^our eaife m tin^ 
nedtanDtng.'bnt Bl bauc alfo }^\xt 
A err. f '^Q agamll(t,t!)e Declaration of tl^c 
1^0 ^^^^ ' ^^ refolurng the Gfsike 

^^ n6ber0,tnto nombcw JhfiraUt, 

^no flnall? > 3| iiaue putte on$ example of tbefamt 

nomberj(^ 



ofCofSih nomhm, 

lumbeiMft^r tbetjulgarefoamcaU lol)tcl)C«3 agree 
t^tbenanD bQttct)^ imi m atber. 

falter* !^eieiisanetber. 

C[2ni)c firfte crtractioii 
oftbeDtuifo;. 

^;2^.S<€.-+— >t^jS'.^5>— [— 2oce— i~2 4.ie* ^^.§-5-, 

^ ^. — ^ — ^.f 

CSbcremoupngc fo;- 
tDacDoftbcDiuifo;. 

CCftc comp^jobatfon of tbefame bp refolu? 
tion,accoumptHng (t<ll,2.fo^ a rootc. 

m9i Tf^^—^i 6 0-4—4 s» 

^^_l„^^ (I2S. 

CXbe fcttrttg fo;ilDaro of tfje Dtuffo;* 

^cbolar. i^etones agiitn, 3| p^aie^ou tDo;]^e tbe 

iPo;t altbougb 31 pcrfUiaoein^ lelf, tbat 3f percef ue 
tbe iDOo;ke:^ct IdouId 3ij fee mo^ie con6rmation of tt, 
bcfo;je 3 tooulo be to conttante in mp perfualion* 

Rafter, ciooD aomfemete is eucr furcjbut if pou 
5oubtc,pour couiTceJloure is not farce abfcntc. 

^cbolar* 31 maie iuftlp r^oice tbcreof : "Bixt to^ c^ 
tttrv mater to require aieo , ano neuer to traueil mp 
otcnelDitte,itmigbtfemetiiereoaaarDlincirc. ^no 

jp.itj, fo 



Thej(m 

fo tuetc it plame babtd^eneO^^to cottet eoertmo^iU 
to be cljatueo bero;!ebant>e, ano pat into tn^motttl^ 
eBaftcv. %\)tn tafee tb<0 otliei: erample , (n one 
platte complete : ISut luitb a caueat , to beloare of to 
jtToclic confiDcnce, \of)ik ^ou feme t» flee uoubtefwlle 
talteroltncire* 



^♦c£« — t — *^*^» 



-1% — ^j^ — ^ce (7ce-+-'8J^ — 35* 

-z.t^. zee— H-^^e 



^cbolan i^oiotoeBf^tbatlloolteOfo^ 
falter, ^ofte, lette t)s trie tbis tooo^be, u toee 
baue Doen tbe otber:bcfo;ie tue goe from iu 
^cbolar« HI p^te^oa let me ooe it* 
gaffer. MitbagootitmU* 

^cbolar. 3iJcpettmtbeo!Drc0tg 

2. ICben is tbe* ^c€ ♦64* tobicbe be^ 

ing multiples b^,i 4. mafeetb* 896* 

ano fo.5 o.^^»iJoe?clDe.4So,anli 

16. fquare»ma*e»64»3Utbeitoge«' 

,,, jtl)erpeU)e«i44o« 

»e reffe of tbe nombcr0,muft be abated, §05 

bicanfe of tbe figncs . . ano tbei malje 4 g o 



64 
14 



2y6 
64 



896 



16 
50 



480 



32 
6 
192. 



j6^ 
48 



„ [2 4o*-fo?0uerr-y§^'W-?2.anD ^4 

^ tben,6*ttmcgtbat,tf)atmafee!j ■■ 

i92.tDbereunto3lput48*foi ^^-^^ 

6. CuUs t ano f0 baue B[. 2 4 o«to be aba* 

teD out of. 1 4 4 o. airO tben remainetb- 1 2 o o« fo; tbe 

WutoenDe. SDbe tmiifo? i« bitt.2 o.fit!).2« c^* r ^ ^ q 

are.i 6.anD.2.rootes;mafee.4, ' ^2o 

3Df3DimDenolu.i2oo«bp.2a -£±r 

00^^^ tl)eftt#//>Ti/*l»iUbe»6o«agreabIp 1200 

to tbe foamcrf w^>»#f.if o;7,c£«mafet.y 6 

ano 



ofCoJ?ih nomhers, 

0nD.8.t:oote0 r^lDe. 1 6» tljat i». 7 2» fx^m luljfcbe K 
muft abate. 5. g-tbat i0» 1 2. and tljen it us iufte. 6 o. 

fatter* 2Dl)ts 10 iocil ooen, 

§^cbolac. ^ea h\tt,3, am pecfccte inouglb ? in tijis 
fcatc of uiuiCoiij 3i trotue. 

05aacr» ^aouDoctuelUoOoubt 

^c^olar* 3ltI)ttiUe mpfelf fure iuitbout ooubtet 
00 bp one 02 tiuoo Ci:amplc0,3 totU Declare. 

ano firfi .^ tafec ti)i0 nobec 3 2 2 ^y^~ 1 — iif §-cC 

42.0^— +—69.5^.— I — ^oxo.tobeOtuiDel) 

bp. 1 4.^. — ! — ).%p ♦ luf)erefo?e 3 fettc tbcm Doune 

522^/5— f-ilj' 5-ce 42 cf ~-l-~69S— f -3 o»^ (23/5-— M ^ 






145^-i- -r*^* 



artDfinoe tbe fidJe qmtiente to bee . ihf^* bp 
Mtcbe Bl multtptte tbe omtfoa , ano it ta^eti) aluate 
all tbe nomber0 ouer it : ^berefo^e 3? fct tlic muifoj 
fo^toaro, f finse 5 2^.fo;j tbe imt'mtey tubicbe 3 mul^ 
tiplie into tbe Diutfo^, f it maketb 4 2 o£"-i — i T 5^» 
tDbcrb? 31 am at a Hate, fm altbotigb 3! H^^ i« tbe Di^ 

uiuenoejtbe lifee nomber0,^et tbe figne of oe^ 

daretb^bat tt 10 not poCTible, to abate tbt0 netue no^ 
bee tben0:fe?ng 4 2*cf.. 10 lelTe tben naugbte. 

£©after* 2I^b^rcfo?e confioer it , in cboft?ng ^ouc 
flMtienmmti giue t?our fuotimte tbe Itfee (tgne. 

&cbolar. ll5uttben rifetban otber ooubte. fm 

tbere ioill be ^.i ^§^.lDbicl)e Dtfagreetb in Cgne 

from tbe nomber ouer tt. 

SOaftcr. pzt male ?ou fubtracte it tuell i!io«igbe. 
If pou bauenbtfo^jgotten^^our rulc0 of fubtrartiom 

^f bolar. j^olo 3i oooe better remember mp felf: 
tbat bp gooo rearon,3l mull leaue as a rematner, not 
•nelp tbe ijubole nomber ouer it , tubicbe i0 . 6 9 ♦ J** 

fee 



The Jrte 
but I mutt aODe ttittto* i s*^*tnoit* 

^0 l^all 3i cancell tlje* 6 9* anD fet ouer ^,8 4. fl!it5 
tljen Doe a remoue tljc otuifo^ fo;Xiiart>.fetttiig 1 4 5> 
tjnDcr. 8 4* S"* a»tj tbe refte m omty toljercbpjf »er^ 
cc(ue,tftat tyc netoe ^wtiente Juill be»— -} — ♦6* f ♦ 

84. 



■f4*^»~i— .^^^e 



:^^/5--i~--fi?^ce 



-Met — —i^^*~ 



Mlb^cljc quoitente^ rwe tnuWpIte into tfte Dfutfo?, 
anD It Doett) maUe«84* §"♦ — { — *3 o.s^. agreaWeto 
tljc fomme ouer tt ano fo tftere rcmaitetft nothing. 

Rafter* 30 ou tjauc Oooen tocU. HPut in cftoff nge 
Ijou i: DtulDcnoe, and tl)e Dtutfoj, pur lucfee UiaB bet* 
tcr tt)cn pur cunnpng. 

^cftolar. %\^t (^aU 3f p^oue againe , bp an otfier 
trample, tafepn alfo at all aouenture^. 

B! tuoulo Dimoetljis fomme. 

l6.S^»ce— -f— ♦2c.^.— -^— .12«S^» ♦S.fbi? 

4. 5^.— t— 2,j^.anDtf)erfoae3 ftttftctm&ounem 
D3Oert!ju0. 



4.^. } 2.^« 



'(4.5f^. 



0nJi firffe 31 fee,t!)at4» r^ contafneo tm 1 6. foluer 
t^imes : ano fo maie3i finDe. 2. in anpotljer nontbcr^ 
tljere. 4.tpmes. miXitmit 31 fet.4.tn tijr quotimte. 

^nDbicaufetl)e.4.mtl)eoiu!fo?are.5- anDtlje 16 
to ftee titutDeD,are»§> ce.acco;Dpng to t*»e former ru« 
tvB,! finoe tije neUie Denommatton Cofii\e to fac.^^ 



ofCopike nomiers^ 

Vo¥tf)t 3 fet in t^e quotUnt Mtf) 4 aiiD to is it 4, 5^^ 

2^l)en fate 31.4.^ ^♦multiplieo fap.4.^-«Do niaUc 
1 6 §>cC«anO tfter&E dcaretl) ano confumctft al ttjat 
fome ouer it. snijcn fartOec faic 3; 4. 5- g-. multiplicD 
bp»2. ip . tJoct»dD0.8/p: 115ut3 fee noc focljc Dcno? 
ntination in tfte Diniucnoc. 

sealler* %l)cn mate rou pcrceiuctlbat rou f;aue 
tnilffD. 

&cboIar. WSHhv fir*3i tfjinfee 31 ougbt to Doe as vow 
6(0 : tbat is to multipltc tijc qnotiente into euer y parte 
oftftemutfo;* . 

fatten Cljatis truetbut^ luil Dctcrte tlje fautc 
tjnto pou. and tbat is t!)is» 

S^ljatall nombcrs Ctfiih^ts compoiinoe,fan not face 
5iutt)c0 o;0erlp , fapttuifo;sccwpounDe. j^nutljofc 
tibat can bee tiiuiDcti , tuill not uttint anj? otber: Diu(# 
foMf tt)efamefeinOe,faut one of,2.nombcrs,br mill? 
tipiication of lJDl)icfte>tt tuas maDetano fo tbc otbec or 
t^ofe.i.fljall be tlje quotientei <as it came to paflfe in all 
tbofe.3.eFamples,lDt)icb 3! ftt fo^tbe, anotberefo^e it 
U toUe laboure, to goe aboute to oiuioe tbetm tn tbat 
fo^te* 

^cbolar. Cben are tbere but fetue nombers of 
Ofiikss compounDc,tbat maie be OiuiOct). 

5©aften &o manp men faie* )!5ut 3 faie tbereto, 
tbat tbougb manp of tbem can not be DiuiDeD, bp Uiie 
nombers GM" copounDe, ret are tbere manp tbouj» 
fanoes«tbat maie be fo DiuiDcD. 

ano again 3! faie^tbat all fojtcs of tbeim,maie bee 
OiutoeOjbp an jfi^r/jfif nomber. ano alfo anp of tbem 
maie be oiuioeD^bp conuerfion into a fraction :0nD fo 
jnaie pour eramplc be fet tl)us. 

l6.5-ce>-+ - *2o-^%-+-I2.S:g^>-- .8-f 



The /frte 

antJ ixi aU ctljer cafe0 U^e.attc tbc Uf uitjcnsc oucr 
a line , ano tl)eDiuifo2 \jnDcr tbirfauic I:iu > ano To w 
^our otutaon i-anocD : ano ti}i3 13 tbc rcDDuttc luaie, 
aiiD tbc moftc tuDiffercnre,!!! aii for l)c nombera. 

«»cl)olar. Cijat is fone learns D. am ti^crro^v nca« 
2)et^ no moarc Etautpics* 

iltis UUetn noniberSi^/y?«//f , tubrn tbcgrratcc 
nomber,Doetb Oiuioe tljc Uffec. as.d.omiOvO bp. 1 1. 
nta&et^ ^ . 

Smaller, g^omclubat lifee it is. li^otubcit Ijcrc is a 
iDooafee moare lihp tbcreijntcns iuben lur l^oulo ot» 
moe toe leffer C#/i^# nomber,bp tbc greater, fo2 ttjt ii 
loe mutt fct mzm m tbat fo;me. &o,6. 5-. DiutDcD bp 
7,cC.ft)aU be fet tbus : 4|. 0nD . 2 a(^ .oiuiOtO b^ 
y.y^^^.muft fianDe in tbis maner; f-^. 

^cbolan mi\iy^ii o.maie be niuioeti bp. r. 

$©atter. But rP. ran not beomibeo bf A*. $ljx5 
fn Co/?/^(f nombers,tbe ctjief regaro is to be bao,to tbc 
W't« figttcs* 

^cljolar. Cben, as fb; anp otber fo;me , of rcca* 
larcDiutfion,bereisnone. 

39after. i^oe,ercepte pour Diuifo;, htt a nombcc 
AifiraBe: iS);attbeIeaae, ifitbaucone ornXvCo^ih 
figne,an0 be UncsmpounDe, tbat fi gne mutt be otbcc 
rquaae,o; leflfer tbcn tije leafte Co//(f figne,in tbe Di^ 
uiOenoe. 

i^oa fo.6 o.g. a: .-+-4 8 ce«--f— 1 8. ^.maie 
bee DimDf p bp anp nomber, baupng one of tbefe. ?.fiv 
gncs i^jitke, ^. j^, ^, ^ 

fecbolar^ 3 tinDcrttanD it fof iLi^o;. 2^, fs tbe laffc 
%ne m tbetJiuioenoe: gno.sp^.anD. q. are not onclp 
lelTe tbcn it , but alfo . f . teauetb tbe nomber,as if it 
tuere a nomber ^/-y^r^f/r. 

^oifai U)onlD Oiuioe pour nomber, alTiffneD bp 
4o.§-.tbef«o;imeU)ouIobectbus. 



ofCof^ike nomhers. 

40.5-. 40-&*- 4o-5"» 

99a(lcr. 15cfo?c U?e ranuc tills tr oiUc of Dlnificn, 
^ t}ill amiionift)C rou, of one cafic aicCin tl)c Dtuilio 
ofDiucifenombcrs.3uDtl)attP,tPconnDcr,U;l)ctl)ci: 
rourDiuiDenOc, DOC omit anr CoMf oc»on^it^atirns, 
bctU)cnctDcm,U}l)tcDcitl)att).j^o;ifitDoc,iouniua 
pet fupvlic tbcir roomc , U)itl) figncs anD C ipl)crs» 
00 bp craniplc^ou U,all tnocrftanoc. 

3 require to l)auc tljis nomber.S'.cC* — }— -6 4»T» 

muiDcObp.i::^^.— 1 — ♦4''y« .,, yr >>r. . 

$>et}olar. :£i:oatU)m goocquicUelp. iPo;3irc^4» 
IciU be tbe firCe ^mticate ano ^10 Dcnonnnatton Usill 
be. ^. fitb.c€ . DiuioeD bp.^. ooe malic. ^. 

Wixt firtte 3; fctte Doune ttjc nombcrs o;DcrI^, ano 

tbenjmulttplie tbeDiui* o^ 1 — ^.^x)^ (,r,^ 

ro;bvtl)cf«ofir»i/e,ttberc ^;_T1 ^ A ^ ^ 

Matter. S>tanOe you o ^ , i^^TT" 
noU)amafcD,fo; all tone ^^ ' ^ 
greate conftDencei-tiou fee tbat pan ran not finDc nnp 
^.mtbemutDcnDe. %\)ctfo^t fet Doune tbcnomber 
as 31 tolD pou befoacjin ttjls fo;te» 




z.5^— I — ..4»?« 



ano tben J take t!jefamef«<»//V»^ tfjatpou D(D,anD 

31 finoc the remamer to be. . i 6. ^. 221 bcrcfo^ 

S O'j' attain fctte fo;iluarDtbe Dinifo; : anotinDctljc 
^uotieute to bec— -8t2p .bp U)l)tct)e 31 multiple t\:ic 

^,y. Dimfo?, 



Thejrtt 

t»(uifo2,attD it maketl). 1 6^.- 
bating tljc. 1 6. ^* tlje reflte, t] 



'*l2,%p fottjata^ 



fl)aU be tbc remainec tuitlj tbc agne— 4 — bp tbc mz 
offubtraatom 



.s ce— h-j^5>— K^2g^--+— 64f (45- — S:se.'~*+^i6f 

2C!ien tnDer tbat remadtcr, 3J remoue tbe D(mfo;, 
anD finoe tbc ncUie qu9tiente to bee — }— 1 6.^, 0ntr 
fo IS tbe tiomber cUcelp confumcD. 

6>cbolar. 3|f3} fo^getteanp parte oftbf0,3l am Dc^ 
cetiieo to foule. 

£©aftcr, SCbcn baue pou learneD tbts parte, locll 
aiougb,fo; tbis tpme. aito tberfo;e IdiU lue go fo;tb 
tjittofractiona, lubicbe partly tucre omittco before, 
ano partly are componnoe of tbcm fclf* 

Of fraSiionSyand their numeration. 

Umions of tbrs feiitDe appere fim* 
ple:anD pet arcfraitte fo to bee luo^ 
gco : aB-||bctoUenctb4.^. to bet 
DiutOeDbp.^c€» liibetuaiectbis 
frarttoVi^Doctb mipo;ttbat i ih* 
mufit b'ee"DiuiOeDbpo''^cf.. Ii3ut 

lli^bctobenetlMo.^^.tobcepartea 

Into. 1 9. portions, 

ano bcrc fljall pou note, tbe DoubtfuU fo;me, tbat 
manp menne in tbts arte bfe, tubicb^ lu;ite tbat laOe 
fraction tbus.^i 5^. tobere as tbi0 fractio Doetb rep;:e;' 
fent II of a fquare:anD not i o. ^ to be DiuioeD bp. 1 9. 

^bolar, Bicaufe pou fa(c,tl}at fome Doe fo h(c it, 

aitu 




ofCopih nomhers, 

snDBItooulDglaDlpmure aIlgooDtD;ttcr0 : ^mate 
fatefo;tl)cm tl)atafimt3ulgarenomfaers,U)l)cn.i o. 
i^oulo be DiuiDcD bp 1 9. ano 10 fettbus j° it Doctb tm- 
po;te botbe tbat. i o. is diuidcd into. 1 9. ano alfo tbat 
cucrp po;tion of tbofe. 1 9.1s ~ ef an bnittc : fo tbat if 
1 o,t ft)oulD be pacteD emongett. 1 9.mcH, eucrp man 
H^oulDbaacMof.i.r. 

fatter* pout Ujo;Dc0 bauc fo mocbc appcrafice 
tbat tbci mate pcrfuaDc bpm, tbat is not tjerp p;ccifc 
in termes, cfpcctallv fcpng tbere ta no otber ^uotiente 
tbere , biittbefamc nombcr. )i5uta& tbc fommcof 
I o.e. bcpng diuidcd bp. 1 9. iB farre mo;c tben ^ of an 
tjnitte: ^0. i o.^^.f bee oluiocD bp. 1 9 . Differ mocbc 
from 1} of a fquare.j^o; tbe one ts 1 9.tpme0 fo mocbe 
a0 tbe otber. ano tberfo^ ougbtc to baue a Dittmrte 
fojmc in to;itpng. 

^cbolar. 'SDben pon toonlD baue me to tu;(tctbe 
fo,tbat ~ of a S>quarc , l^onlD baue t])t Cgne againtt 
tbe ltne,a0 bcrc 10 fet '^5- : ano tuben 3 tooulD rep^e^ 
fent. I o.§> . oiuiDeD bp. 1 9 . 3 (ball lo^ite (t tbu0.7;^. 
toitbtbe figneabouetbeliuc. 

£paftcr. pou maic fretbciragremente,anD tbefr 
fttfftrence bp rcfolution, in tbi0 manec f^^: ujta inafee 
?5 accoumptpnge.2. fo^a roote,anD ;|^.maUctb r^ of 
4,o?v^of.i. 

again,accoumptpng,5.fo;jtberoote, tben ^l^v^U 
Detb Ti : ano ;:§- maUetb ?: of an bnttierfo tbci appere 
to bee cquall m baletue b^ reouction. 

l!5ut noU) maie pou fectbat tbe one boetb betoken 
tbc firftc n6ber,tDbtcbe 10 to be muirjeD:anD tbe otber 
Doctb fignifie tbe^«o^*«»**of tbe Dmt(ion : anD fo arc 
tbet Dtftinrte in office ano nature. 115 ut bicaufe bp re? 
folutiOjtbc one tournetb into tbc otber, tbcrfo«r ma? 
np men arcoumpttbem 30 one. ill)oU)beit,lDe ftano t<x 
longe aboutc tbi3 , conaoerpg tbc crroure,i0 not al* 
\uatc3 oaungcrouB, 

^♦itj. But 



Thejrte 

itnfplacc tl)e 0gne, lut}cn tt ftjoulo bee fette tJtiDer t^e 
line: 30 a greate clcrbc Doetl) (except 31 fijall fo; bts ep 
(ure,tnipute tl)e faultr to tbe p;itnter)fo; ^e ntcantng 
to OiuiDe.5»bt». 7» §^ 5-. tu^ttctl) it tt)U0.^ §> j^.luljerc 
fte C^oulD \3ii\Xt It tt)u0. j^:ano againcmpnDpng ta 
DiuiDc»7» bp.^'g-g^^ betu;itttl)itt!}U0^^ j>,lubece 
liel^cuU)U)?tte,|g^» 

^cl)ofar* Cl)ts faultet0ntantrcfie,anoocte(tet^ 
tlbc firlle negligcnce:/o J ^i*^' ^°^^^ "^^^^ *" "°"^* 
ber , after tl)e fo;jmei: rcfolution . -^ ano . -j^Dooct^ 
mabe* |g , 

Rafter. tKHcUifcpngtoupcrceiuet!)cfauIte,lue 
iutli 0anoc no longer aboute tt. ^l)crfo;r to pi^oeeoc 
tiiftinrtlp ano ccrta!nlp,U)l)etl)cr tbat frartion be f om 
pounoc , o^fimple , U)bcre tlie numerate; \s a G^tkf 
nomber , ano tlje Denominator , a nombcr abfolute, 
jet male i>cu bolDlp tijinUc , tl)at fraction to htt tom^ 
poftnDe,lubore numcrato: ts a nombcr Cojaks and tfje 
Denominato? an otljec Copks of ijnlikc (ignc : as . -?g 
ano^l^. 

pzt as m nombers abffrarfe , it mate feme moffe 
apftp to bee raUeo a fraction, luben tbe numerate 2, is 
leffer , tben ttie Denominator , fo in nombers Mike* 
tnolfeaptlptljc eigne of tl)c Denominator, I^oulDbec 
tbe greater, ^ct botfje formes come m t3fe. 

0no forbuaufe cafineflTe in u orbrng, Doetb often^ 
times bring certaintie iuui) it before toe talie m baoe 
ttjeaDDition of fractions, g tbmlie it gooo tofpcafee 
fometobatof i^eDuftion , to anctbetDcnomination, 
^0 tbat^ou forgette not,tbatanp.2.nombers (.ofaks 
rompounDe,U)ttlj a line fcettDcnetbcm,maie be eaUcD 

aftattion. astbus. '^'=^^^Ii:I^~^'=^^\tWiSs 

^c£. —^ — ♦ S . sg^ ♦ ♦ 6 ♦ ^v to bee DiuiDco fajj 




Eyami>Ies ofl^umerathn, 
;,5>»— 1 — '. 1 2»f anD fo of otljcr like* 

Of ^^cdaclion offraclims, 

(rW; £a^^ ^j^:-i^a(ttons GM*. "ot onclp fn tljcftr 

'"" nombas , uut aliomttjiirDgncs 

maic be rk^DuccDto otbtr tjalutiong. 

ianDnaiitclpto tbeirUattc tcimc0, 

ano pet continue fliU in one paopo;» 

^jtior, betU)£ ne t^jc numcrato; , ano 

gJjDcnominato^ 

l&o ^,^^maic htt rcOuceO to ^^ :fo; fo ftfglj as. c£ 

^ aboue . f . tbat is in tbc tbtroc place from tt: ^o td 

g^^.in tbetbtro: place aboucr^. 

agamc . \]^ . bp rcDurtion Ooctb mafee ]^ : 3nD fo 
.,gc^ ■ t ., ^ — — -^5^ ir III brr bp rcOuaion. 

0nD fo tn all otbcr fracttonj?, lubcre tbe nombers bcc 
commcnfurable. 

^utifanponenombcr»bcefncomcnfarabletoit(i 
tlje otoer , tben ran tbcrc be m>=iDc no reDuaion in tb8 
nomocrs. B?et in tbe figncs C»/i;^r, tbere mate be a re* 
Ouition, otbcr to greater, o; to imallcrCgneu J /o; 
tijofcfign:3 be eucr commcnfurable* 

anD ti^cre t5 no erceptton, but ttjci maie bee rcDu^ 
ceDto fmaller quantities , ercepte anponequantitte 
of tbcim b:e.f. tbat ts a nomber./ o; tbat can bee no 
fmaller. anD tberfo:e none otber maie be altereO,(it^ 
euerp one muCt be abatcD alihe. 

^m looUe bolo mocbe , t!)c fmallcffc quantitic of 
tbatfra(tton,isabouca number, fo mocbe maie tbci 
all bee abated : fo: tbct are ncuer rcDuceO to tbe fmaU 
leftctill one of tbem be a nomber. 

^cbolar. 3nD lobp mate not tbis reouition, feruc 
ioi iDbolc Cofsike nombtTS:' 

falter, iSicaufe tt;c toljolc nombcr,t)oetli not co 



The Jrte 
fill of a p;opo;ttton,a5 m ttmion Doet^, and fo mate 
bee crp;circD in Diuerfc umtB i but it tmpo;tetb one 
fommccertaine, U)t>fcbe mate notljer bee increafeo, 
no; DccccafcD,but it }iiiU cbaunge W t)aleiue>anD ab 
tci* bis office. 

ano If g faic: a foote 10 ^ of a rarDe,3[ maie faie a^ 
tvuclf jincrrafrng botbc nombcrfi,iii tbe like p;opo;« 
tton,a foote is fj of a parDe:o; m leffer termes:a foote 
is-fofavarDc. 

15 ut lubcn 3i faie in tobole nombec , a pacoe is . ^ 
footco; a foote is, 1 2. Fntbcfi,31 faie trnelp : ano if 3| 
oor inacaCc 0; abate anp of tbofe nombers, mp Ujo;^ 
Dffilumbcfalfe. 

^0 altI)ougb in tbis nomber. 8./§-. — h~. 6. cf. 

1 o. S-.bp reafon of botbc nombcw ano Cgncs. 

tbcrcmtgbt bee a reouctton, pet btcaufc it is a tobole 
nobcr,it fl^oalD tberbp bee abatcO mocbctas bcrc rou 
n^aic fcc.4,ce.— 4— 5. V .— — ,^,a ^^irfjc bv re^ 
folution into tnlgarc nombew, 2. bepng fettc as tbe 

roote,Doetbmabe.n-4— 6. .j.tljat is. 3 ^ano 

tbe otbcr nomber befoje, ooetb peloe bp tbe like tttof 

iutton.2j6— I— 48»-- ♦4o.tbatis.2 64.anDi0 

S'tvmcs fo niocbe as the otber. 

^cbolar. 3 perceine nolo gooD reafon,tDbp rcbu* 
rtion fcruetb fojftactions oncl^ano if tbcre bee noe 
mo^ Dime nftie in it , tben pou baue Occlareo ♦ 3 can 
too^beiteafilp. 
^tdumm'm i^o; otber tbe reonction cofilfetb in tbe fignes 0/5 
fgnti mtlj fike onelp, as fr^tobere tbe nombers bee bncommen* 
furable , ano tberfo;e tan not bee altcreo to anp IctTec 
tcrmes. ^ut t\it fignes Cofsiks maie bee abateo hv* \ 
benominations:fepng tht fmallelle of tbem,is fo ma^ 
np in 02Oec aboue.f ♦ ano tbcrfoje it mate be rcoucco 

%tduH'm in £)tbec cls fcconoacilp, tbc i?e0uftf on confi0etb in 
ffbers tndj tlft mmhcts onelp^tuben tbe nombers bccommuni> 

cante* 



ofCo/sike nomlers, 

Wttte. flnb tMz fignc6Co/}% htt all rcDp at eftc Icaac: 
as tuhcn one of tbcun is. ^ , ^o ;:f IciU tec rcDuccD 

to . '^. 
*** * J.J' ♦ 

<ID; cl0 tDirolp, tbc rcDucticn mate htc lu;oucr!)tc, (jf.Juttion ;« 
botOc m figncB, anoalfo iii tiombc re. «EClbni alltlje Zms/nd 
figncB be aboue.f aiiD t()c nombcrs be comniunicaiu^^J^/,^,^^, 

^0 \\^ mate be rcDuccD lueU Unto. ',4. 

£aaacr. ;iietonefo;meofrcDuct(onmo;e,3:li)iH Mother 
fijolue rou.lubere not onclp tbelibc Ujoo;Uc nraic br, rrdumon. 
h\xt alfo m nombcc mate be b^ougbte from bis c om.- 
pontion,toamo;jc fimplccitie , bp abating fomcof 
btc partes. 

^5 tbis nomber :^E±^^=ri^^, mate bee reDuceO, 
firHc bp l)t0 nombers to > <— +r-^ ^' , 

*- 46C-0 1— '-..6 

^er onDartIr,bp W Ggncs it mate be altere D t\)\\». 

I H — -) — ^ f 
Vto's—- 1 — '^ce 

£:f)iroclr,bpabafrn0ctbcnombcr0,tbatfoHotue 
fignc of compofitaabat ts— I — ) tt maic be b:Dugbt 
to . ,:!^^, . 0; . i'^ . Uibicbc fratttons, feepe tbc felf fame 
p;opb Jtion , that tlje firftc fractton Dto» 

jLiUcUa!csliit:;tl)cligncof .nombersrcC^ 

Ciuallcs.maic bee rcDuceo.as.^g'^IIZl!' ^\ hitUbec 
r,Di:.TD,a5t:;cotberiDastoi^-. ^^' 

&i:::olar. Si^btststntomeamaructleufe mater, 
tV;:ii tl)ofc. :, ccntrarr nombers,aioulD Ic leDureo to 
one fraction. 

fl^attcr. _^^ litxe Ijappenctfj tn tiilgarc nom> 
bcr«5.i^o?. r^iZ i^g . lutU bee rcDuccD to ^ . i^o: firtte 
tt nrahetb ;: ana t\)tn | . ^0 ItUctoates r'EIZri UiiU 
makcfirftev'anDtben }♦ 

ianD tbc reafon of it,Doetb tiepcnOc of tbc. 1 9. p;o^ 
pofittottjof ttjc fiftb boobe of f«f//</f,t«bcre it 10 lo^tt^ 
tentljus* 

Ja.j. If 



The Jrte 

If the proportion of the abatemente ynto a* 
hatemente be^a^ t he iphole is in proportion to the 
HoholeXhenjhallthe reftduebte in iikepropor* 
tionto the reftdue^dsthe'^hole is to the i^hole. 

SCtjat <0 In tt)e latte erample. as, 1 8. i& tinto.i 4. fo 
ts 6 tjnta 8»»erfo?t l^all 1 2 be to 1 6»a5 1 Ub to 24, 

0nD fo? to ereccife pou ttje better, loc, !)ere are one 
oa ttooo eramples mo;ie,of tfje liUe reouctiotr, 

l]5ut"tl)is mutte pou farther marfee , tUt in Ofsike 
nombers, not onelp tbe nombers , butalfo tbe ckh 
flgneg muft bee,arco;opng to Ewhdes p:opofit(on, 

^cijolar, 2D!)atiioe3lfee» 

^ojtnt^elaffeejrample; <a5.f tsto. ?>, fo«io .(5 
to»cc* ^ ^-' 

ano mtlbenerteerample before: Sls,cfAfitor^, 
^liketuaies in tlje otber eramples^as cP fg to kk^ 

j3l tbtjEs (0 good and reafonable. 

i^affer* iptoto Doe pou fee,botbe tbemaner of re^ 
iiuetton,anD alfo fome reafon fo? iu Cberfo^c 3 ijnili 
p?ofetie,to declare tbe feoojke of Addition, 

Of Addition and SubtraSlion, 

i^ildditton tfjeretgnotlipnffemoare, 
tfjen pou ^aue learned before : J^o? a0 
fojtbe multiplirattons of tbe denomt^ 
nato?0 together, and thm croOe tuaie0 
toitb tfte numerator of t!)ot^er,i0 lude 
agreable \3Mt}) tlje reductton0 of ab^ 
tJractc fractions , to b^png tftcim to one common de^ 

nominator 




ofCofiktnomhers. 



nomtitato;. 

^m ^tn tljc iiumcrafo;js aUDcD together , uaoe 
tnafee tlje netuc nnmcrato^ tn aODition* 

aiiD UUctuaic0 the Icffcr numerato^,ftibtractcD fro 
tbe otbcr, Doetl) make tijc nutnerato^ m fubtraaion: 
Uiftcrfo^c afcU)C eramplca mate fuffic^ 



Examples of addition. 



H-^»-i— «28.ct» 


40.^-. [ »42.i:^, 


T-s-** to A. ce. 


v'5^» to -}.se- 


6^ 

SDljat (0 (n fmal^ 
lertcrmc0» 


48. 

2 4* 



l^ercpou fee fjblD tl)e.2.frad(ons be fcttc bctluciic 
2, Imeg : aiiD tinder tl)e netbcrniofte line , ts fette tl]c 
iietuc Denontmato? : ano ouer tlje l)igl)cr ltne,are fet 
t!)c.2.nel»e numcrato;20 topncD m one. 

2St)e firlle of tl)cm,can not be reDuccD to m\v tivaU 
Icr tcrme0,b tea life tl/e nomber0 be not all. ?. comme ^ 
furablc:f tljeDcnonnnato?,alfot0anombcr^/y?rrf^. 

Stie fcconDe batb alfo a nomber.^rd(5?ff fo;bi0 
Oenominato5,ano tl)crfo;je tl)erc can be noc rcDuctton 
tn fignc0: battue nomber0 all.5.betingfommcnfuva= 
ble,^ Duufiblc bi'.2.mait be rcDucecas tf)cre pou fee. 

More examples of Addition, 

^_^ i6/p«— ^ — ♦4«ci^* 

1 2/5-.— h- .9'Ct« to 4/p» -^cf. 



2o.?;-ct» 



2o.^c^ 



aa.ti 



Mat 



The Me 

2CIj3f fd fit fmal^ 4'5^»—- 1 — ♦ i»f ♦ 
UttKxmts* r^^^ " 

^ete (g noe multfpKcation to^ougljte , bicaufe t\t 
t)cnommato?sareltUc* 

^;i otherExampkofMJition, 



^ere<0 noe muWpKcation , no^cDurtfon to one 
tommon Oenominato^tatb thti bet one all rcauptno^ 
tbec can tfte nombers be ceOuceD, to an? otber IclTer: 
but tf)c quantities onelp be reOiiceD aa i'ou fee. 

§>cljoIar« 3[ p^ate pou let me p^oue. 

j{notherEx4?npk. 

8 o> V§— +-9 o S-ce— f-6 o 5^ ;o/^. 

S,ct— +-9'^» " 6«ce ^ *T^ ' 



lo.O^, lo,§^^* 



Waller* ^arlie tour tuo;fee tugll, before ^ou re-- 
tiuce it 

^cbolar* g fee mp faulte:^ baue fette.2. nombers 
feuerallp, tuitb one figne C&fike : bp reafon 3: Dio not 
fojefee,t^at,c£»inultiplieoU}itl),c£»o<jeti)maUet!)c 



of Co/Tike nomhers. 

like quattt(rtc,a3»^ ^. multiplied bp»5-» Xhtttt9;t 
ft(][)oulobet^u0* 

Wfiicbe mate bee ceouceo , bp mw?ane of tbe nom^' 
kr0:,tott)t£(rommc. 



$lnt) ttotD confiDcri'ng tl)c Gy5/if Qgnes, anD Ido;? 
fepng as 3B Ijauc markcD pou to Dooe: 2^l)at is to abate 
tbc Icafte fignc, out of tbcim all : bicaufc./p. is bcre 
tt)c lcaac,3I abate it out of. ^"5-- anD tbcrc reftetb.5^* 
anD fo Doing loitb tbe otbcr fignc, ^c^.tbere remain 
ttetb*::^ I tlmxf^ out of/5- Doetb leaue.f . 0; nober: 
^0 tuili f be fraction bee tbus: ^^jrrfciLre.-rizrrii b^ 
reDuctton in figncs anD nombers alfo* 

spatter, ^cpng pou banc fo hscll mavUttj tf)c xzo 
Duction of tbc fignes ( tobicbc follotoctb tbe fo;me, 
taugbt befo;je in Dtuifion ) 3 tbinke it not neDefull,to 
ftaie anp longer aboute tbi3. 

C(aberfo:e lue toill goe fo^jtoarD to rubtraction,af;j 
tcr tbat 3! bauc aDmoniD^cD pou of fractions, m appce 
raunce fimplc, tubic be m DceDe bp aDDition,bcc come 
(ompounoe.astbis^ct.aDDeDto {§>» maicfiraebc 
aDDcD by> tbe common figne of aoDition , tbus. 
T'X*—^ — .^g^'.lobicbe bp reDuction,t)nto oneocno* 
mination,tml be tbus lo^itten. lc^~j— js_ 

liSut as tbis IS eafie inoug!) to UnDcrffanD^fo maie 
it bclpe often times, fo; fpcole luozUe^as luell in aDDi- 
tio^as in fubtractio,bp tbe onclp aDD)^'ng of tbe figne. 

^s if Jl UjomId fubtracte tbi,«3 fraction : ^^ x,^,o\\t of Suhiranion 



The Jrtc 

iV^ cC ♦ 3f mate to?tte it tijus* ^ 5- c£» xy, y,^ 

ano fo IS tije a>ubtractto!t to^ougljte* '^ 

|9et mate rou reduce tI)ctm,to one oetiomdiatioit, 
if ^ou iuiUjdftcr tbefame fo;ime, as pou Dto in aonitt* 
cm anDttjenluillit \itt,'^^^^^=^==^i:^^\x)\i\t\^tcm, 
not becreDuccD to anpfmallec termea, btraufe tbe 
nomfaersare not commcnfurable: anoone oftbeim 
(tijat is to faie,tl)e OEnominato;j)ts a nomber yfhkr^Si 

^cbolan 31 fee in tbis, tljere is no Difference from 

0omtton,butintftea0ne0.— I — ano. ~.U)bec^ 

fo;c 31 tDiU p^oue an otber erample, bp pour leaue. 

3UjoutofubtractcV5-Dutof. is^^.anDitloiU 
^f /i!5i?^|^ ^ 5^ S^* l/5-.ino> rcouction 

$iaaffer» ^our tuoojfee is tueU Doen,acco;opng to 
rourfirtte meanpng: Butast^c numcrato;ioft!)i« 
!aae rcouction ooctft Declare,it can not bee tucU,tbat 
1 ^yp^maie bee abateo out of. 1 6« v ^*S^o^ tlje grca* 
ter abfolutelp, can not toeU be abatto out of tije Iefi» 
fer:anD tlierfo?e pou migljt rather ftaue abatco ''Z^^ 
outotlTI-* '^o 

^cbolar* 31 fee it tuell nottj:fo^ tbeA- ♦ i& alioaies 
Rouble 0.2 triple , 0^ pet mo;e t^rnc^ greater, mn tbe 
5^e!r/)i5icauretl)e./^. contmett) bp multiplication of 
tOep^bpljisfirtterootc* 

fatten ptt bere in is Difcretion to be tjfeD fai in 
fractions, fometpme tbe nombcr of tbe greater 0gne 
male be tlje leffer* 0s fo? erample ifx^ is leflTcr tbew 
i y jj", as bp refolution pou maie p?oue,accomptina 
iJoube common roote* 

^cbolar* 2.bepngtberoote,32*istbe.A>»ant>br0 
^ mafeetl|.6.tbett. | ^ ^, beepng. 1 2. Dooetij appere 
Double to ittauD tberefo^je greater bp moctie. 

3f|Doebptl)elibe refolutio,p;?ouetl)eotberfrac/ 
tionsbcfo^Ci/p.tuilibee. 2 4:anDf ?>^, iiixWhtt 
i2i;U)^ici?eisfeirermocl^e» ^^ 



ofCopike nomhers. 

^0,91 percetue t\)t grcatncCTc anD fmalneffc of tftc 
ftaction3,muft he con&oereD^as IdcU m tf)e nombcrs 
afl ttt tlK Co/?'!^ Hgrtcs. 0no farther, tf tbcir fractions 
ht nigl) of one gccatcnctrco^ tljc fcactton of tbe Icffcc 
ftgitctbc greater , tljencan not tl)efubtraction,ap^ 
pcarc reafonablc. 

Rafter, E^ljat 10 true,tftbofc,2. fractions ftanoc 
alonc:el3 bepng partes of otbcr nombcr0,it mate ap? 
peare reafonablc mougb* ^b in tl)t0 erample of com^ 
pounoc fractions. ;\ ct* — 1 — i g^* niaic bee abatcD 
out I c€, ♦ — I — } ^ • anlJ pet m tbe abatemcntc af« 
tec— I — notonclp tbe nombcr its greater, tben ^ 
tn tbe otbcr, but alfo , tbc Cofiiie figne ♦gr^^ts greater 
tben tbe otbcr Ci/^'l^ ngne.i^. 

^cbolar. 3 confiocr it to oc ro:anD pet ] cf.«tioetb 
fo mocbe crceoc -t c^.tbat it fuppltetb fufficicntip tbc 
otber Dcfaultetels could it not be luell Docn» 

2i3utfo;tbtsU)oo^fee,3muacrauepourbcIpe:bi^ 
caufe 3 bauc not fccn tbe like. 

Scatter. .19 ou mate Doe in tbts, as 3 faieD befo;j^, 
generally foiall fubtractions* 

^et Doune botbe nombers m Due o;Dcr,ro tbat tbc 
abatemcntc oooc fololuc in o;Dcr:anD putte bcttoene 
tbem tbc fignc of fubtractton:as tbus« 

l^otobeit, tf pou tDiU firac ccDuceeuerp c6poant>c 
ftactton , into one fraction , it toill feme mo^e apte. 
^B tbus»f c^. —- f— ♦ ! tp .bepng rcDuceD bp aODitio 
tDtU mafee -^^-^-=pr=^'& . ann bp fartber reouction of 
ncmbcrs.-i^^=7^^^ . itifeetoatcs f-. . cC -+— 1 5-- 
toill make bp tht firffe aODition . ^'^:=-^^'£ . and op 
fartber rcDuctton '^—-J^'-^^ , 

j^otu topne tbctm togetbcr, luitb tbc fignc of fub;^ 
ttaction,anD tbei lutll ttanoe tbus. 

^cbolar. 



The Jrte 

J&>c!)olar. %W Doetl) appeare tcrfe l!raangc tit* 

to me: but bv tjfe 31 (l)aU ftnoe it nio;^^ familiaret^cc^ 

tng 3! fee tljc rcafon of tftts U)o;be, to agree luitft tbe 

rj^ . c luo?Ue Of coimnon fractious* 

ibcpmfe. iButfo^pjoofe of it, 3 ioill refoluecrftc U3o;jbe, in^ 

to nombers abfolute, accoumpt?n0.2»fc^ a rootc. 

fatter. ^0 ftjaU von finoe it true : 3but fo^ cafic 
tuoo?Ue,tafee ratljer, i o, fo? tlje roote» 

^cljolar* 31 ttjanfee pou fo? ^our aloe, 

%fitn If. I o.be the roote, tbe fquare luiU bt. i o o. 
ano tbe Cuh, i o o o» ^oto | cZ * tljat ig -^ of. 1 o o o. 
tfli.6 o o.0nij I of. I o.ttJljicbe 15 tlje roote,tDta hce,6. 
Hjbic!)e botbe put togetljer, ooc mabe.6 o 6. aiiD tijat 
10 the greater noniber. 

%\)cn (0; m leffer ^ c€. are in tl)is erample.4 o o 
i^o; tl)e Cube bee^ng. i o o o. bis -^ is, 1 o o. ;againc 
tbe fquare be^ug. i o o.|r-. mutt neoes bee 
7 s, micU bee^ttg put Unto. 400. ooocttj /- n < 

Wenooef abate.475',outof.6o6.antj ^^— 
tljeretoiUreHe.ni. li^oiunoUj., ^ ^ ^» 

^aaer. 3 perccluc pcu llaic , as bee^ng nHoniC^ 
mtty bif aufe m tbe former luo^fee, tijere is not lefte a 
rematncr : Hdut tbe.2. fir0e fommc s cnelp altered bp 
reouction.ano lorneO togetI)er.ti>itf) tbc figne of fub^ 
traetion ; iuljercin if rcu tao rcntinucD ^^our UJo;Ue^ 
Vou d^oulD I]aue founoe tl)crame ncmbcrs. 

ifo?.^c6.niuttncDcsbee.5ooo.frrng.i.<:€.fsa 
iooo.anDalfo.5J:p , are. 30: irbicbebotbeaooeDto 
getl}er,mafee.^ o ^ c^riuice tijcnt fcr-r* (as tbe oeno^ 
ntinato; luoulD) ano it tuiU be.6 o 6. as tf)e taleiue of 
tbefirflte fraction. 

,J^t^.uTo ^?i^^ ^^^'^^ nomber:anp tou ntaie fone 
tftmfee tbat. 8. ct * are.80 o o. ^no. i ^ Squares are 
If 00. apDe tbfim together , ano tbei toill mafec 
9 )- o o. mu\)t mm bee oiuioct br* 2 o, (as tbe ocno^ 

ntitiatoj 



ofCofHh nomlers. 

ntfnatojtmpo^tctl)) anD tbcrc tolU a^ 
twountc. 4 7 T« tlje tjalcluc of tljc IcflTcr i -r 
fraaton:Ujl)ic!)c nombcrB appcaretbc 9 ^ o o (47 f 
fame , tbat lucre before :anDtbercbp 2^2:0 
tbe luoo;Ue ts gooD. 

^ut If f ou luiU b;jr"Sf tt to a remamer,Doc tbus. 
i^coucetbcfe.2. netue fraatona, tiito oiieocnotntna^ 
ttoit. 

^cbolar. SHbat can 31 Doe,bi? miiUfpIipitg tljc nu^ 
merato;stogctbcr:tbatis.2o.bp.^.anDtbercofconi^ 
metb.i 00. lobubcfljaU be tbc common numerator: 
tbcn mull ^ multipUc in ccoffe Uiaic0,tbc numerator 
of tl)c ftrac , bv tbe ocnommato; of tbc fcconue , ano 
contrai'ilp. 

feo fo;: tbc firftc numerator ] . cC» — I — ♦ 3 ♦ ^^ 
^ljjoo:Uctbiis. 3nDtbcrcbp 20. ^ 

0ooctbamountc(ai3poufcc) ^^ ^ — 1 — 7-— — 
6o.ce-4-6o:%^.anDfo; 6aceH-6ai:^. 

tbc feconbc numcrato;i,3I multtjUe* S,ct — | — i >- ^ 
Ip.^anD tbcrc ooctb rife. 4 o.cC — | — *7 f. §-♦ ccbc 
of tbctm baurng one common numerator, i o q, 

tZ\ bevfozcfcpng botbe nombers , bane one Dcno^- 
m(nato?,3 fljall abate tbc leffer numcrato:,oiit of tbc 
grcatcr,as bcre m crample is fct fo;tbc:anD tbcn ti)e 

6o.cc« — ! — .6'o,S^» 

4 Oct' — I— 'Tyg-'* _^ 



rcmafncr twtll bee (as t'ow Tee). 2 o.<x . — h-6 o,xp . 

7 5"» 5^« tnto.U)btcbe ? mufte aoDc tbe common 

bcnomtnato;* i o o, anD it ibtU be tbu5« 

2 o. cf ♦ — ! — • 6 o, to ♦ .7 J", 5>. 

15b.|, /^otti 



1l7e Jrte 

i^otu p^ouc Uil)etfter tW cemaliter, doe not aaree to 

tbotbcr rcmainer befo^Jn pour trial: luljtcb loas nr 

^c()olar» 2 oc£ uo make* 2oooo,i6oiD .pcloe 

6 o oitbofe 2 fommcs 3! mua aDOe togetl)0r,t)Saufe of 
ttje 0gne»-— f— ♦and it Una fae.2 o6 o o.t\im* 7 ^^. 
are,7 f o o, inbtclje 3 mud atiate from tfte '^ 
former fomme of»2 0600. ano tbcrc tuill 20600. 
remaine, 1 5 1 o o. fo? tlje numerato;r, ano 7roo 

I o o.fo? tbe Dcnomtnato^,tbu0.^(f^ . nToo" 
gaffer, ano tubatoo^poutlnnfeeof tt:' 
^cfjolar. I15p tbat 3! learnco in ttjc tjulgare fractl^ 

tion^, 3? Unoinc tftat it is lulfe, i , i. ano fo Docth it a- 

grcc pjcnfelp,lnit!) t^e former p^oofc. 
^aO^r. Mell pet fo^moareeratfneflrern this 

iuo^fee,3i tomfartl)errconcetl)atfraai6,bp oiulDincr 

tfte numerator tp tbc Dcnominato;i:U)lierfo;e.2 o cP 

oiuioeo fap.i oo.Doetftpeloe.^ce. ^nD,6 0.20 . oiut. 

Deo bp. I o o. Doetb mahe 1 2p ; :ano Iamp.77r?^, of, 

?n r'.SnF;J ^^^ ^^ll?' ^ * ^° ^^ tbefame fradtoii 

foreoacen-f ce-j-jf xp . aj., anonotntrie 

iuf)attl)ati0,bptbefo?nierp?ODfe. ^ 

< »?'^'^1?^^5: |^"«^^fcneprrceiue,tftat^ce.^0.2oo. 
Uitjcn tOc Cube 10,1 ooo:anDfo;2io.ig.6*U)hifhe^ 
mwH atiDe togetber,anD it luill be.2 o6.2i:bcn -^ V 5 

7 y ♦ \siWMt if J oooe abate from . 2 o 6.tbcre l«m re* 
mam. 1 5 uagrcabtp ag before. ;ano fo r^ i\)\^ U)oo:fec 
fuUperamuicD. ^^uimu^kc 

a^attcr. ptt Intll J p?opounDci)ne 0; tino cram^ 
pics mo;e,partlp to p^actife pour memo^ie, ano part^ 
Iptoaomomn^c pou,ifpoubappen to fee anp foche 
mirre l»:ougl)te,in fame otbcr bo to(asjj baue oocn) 
50UJ pou mate ameitDc tl)c erroure,anD not (fate at it. 

#trae tafee tW eramplc.31 Ujoulo fubtraae. 

— rr-^- rr- outof—L-I 

12.2^. 5.5^. 7,^. 

^cl^olar. 



ofCopikenomhers. 

^c^olan 3( mult firHmttlttpKe t{)e Detiomfnato;0 
togetl)er> ano fo (t ioill make , as ^ere \& fette foo^tt^ 

Stben 31 multtplic tbe nu« ^ "* "C * 5* er'* 
mcrato J of tfte firttc , bp t!je 7*S;"« 

foo;tl)e. 5 3 6. p : tobtclje w tfje numerate; 

fo?tl)cabatcmente. 
aftecluacD^ muUipUc tljc numcrato; 

of tbc fcconDe, . « ^ , ^ 

faptbcDenom(nato;oftbe ^ J'p ^er* 
firtJe , atiD tt lotll mafee 48»y» 




^oU) if 3! fubtracte tfiat ^ g 

536.^.outof,|76,xe. -,^^^^_„ 7T4:^-: 

I44,^atajillbee >'^'^*x-' ^^^'er* 

S76>yp 48o.5^Jo^tbcabatemctctbat(l)oulD 

be fubtrattcD notu,t0 fettc after tbe figne tottl) 

tbe former fommc of.i 4 4, 

i^f nallp , to mafee t\^t rema(ner complete , as tbat 
latte nomber 10 tbe numerato;i7fo titito it 31 mull aODe 

tbe common DcnonTtnato;.84. c£. .2 1» J^^* 

nno it lutU bee, y,;.^ rg g . tbat Is In IclTer termes 

Rafter, ijioitj p;oue rour cunning In tnis fome, 
.T^l!_,^'fubtraarng it out of/i-^^IE^P^^. 

^cboian j^irtte 31 rtiwVt reduce tbelm, to one com- 
mon Denominate;? bpmultlpUpng botbe oenomtna^- 

84,5^. — .2I.c6* 



I o o 8.CC* ♦- ^ ^'^ S^' 

-^^ ,^b,y. to;s 



Thejfrte 
f o^s togetlber, 9ttD fo iutl tt be. 6 5/§- — i — t o o 8 ct 

JCtjen t)oc 31 multipKc tlje irumerato; of tijc totail, 
b? tlje uenominato; of tfje abatements as t)erc alfo 3 
Ibaue perttcularip fet fo;tl)e tn Uioo^ke , fo; mp oUinc 
eare,ano auoiornu of ecroure: ano fo j finoe it to be 

l^all beetle numerator oftftetotalle. 

232.i:^. — i— .5'76.f. 



2784^. — j — ♦6 9l2.2p. 
696*ce* 1728.?;^ 



1 0^6.5-.—- i— .69 1 2.X0 . — .6 96.ce. 

%\itn Doe 3! multfplie tfje numerato.? of tbe abater 
riete^bp tfte DenomOtato; of tbe totaUe(U)bicf}c tbmg 
is eafilp Dooen, btcaufe tbe one nomber,t6 a nombcc 
Jhpaae)^m fo l)aue 31 foj t^e numerate; of t^e aba^ 
tementc.4o^2.g-- 1 oo8.cti. 

0no fepng tbefe tluo nombew, baue one common 
Denomrnato;, 3? l^ali abate tbe leOTer numcrato;,out 

ro5'6.5'.— +-69l2.Sg^.— .696.ce. 

40)2,^-- • . ■ >-, I o o 8.ce. 

69T2.r^-H— . 312: ce7=:2 9 7~6:^ 

of tlje greater, f fo txi« tljere be left fo; tbe numerato^^ 
oftl)eremamer69I2::p^-~^ — l\i<:t—i976^ 
tjnto tobici)e,3i l^all aODe tfje common Oenominatol, 
antJtijentomitbe* 

6912^^.-4— >n2.ce. — -.2976.^. 

^h/^^ — i — .loo8.cC' .504.5^^, 

Cbat 



fo Cof^tke nomhers> 
SDbat ic in ItUct tctmzB* 

2 3o4,f-4-- io4.§-, .992.:z^. 



jpaOcr* pou Ijmz lo^ougljt it toelL anfi Ijcrebp 
3 coniertucc, tljat pou arc crpcrte Inougb In fubtrac^ 
tron. wa\itt(o;c noU) tue UjiU got m l)ano,luit!) mul^ 
tipUcottonattODtutriom 

OfMultlplkatlon. 

X^SD firffc, concerning niulttpUca? MuhipU; 
tion , bcrc ts no mo;ic to bre fateo, f 4//#«. 
t!)cn batb been taugbtc before. 

jpo^tbe nombcrsftjallbccmul^ 
tipUeo 7 as common fractions are 
U)onte to bee: tbat 10 to faic, numc^ 
rato? , bp numerato;, ano Dcnomt^ 

nato2,bp ticnomtnato?. 
$!nofo: tfjc cbaunge oftbctr DcnomfnattonsCff/^ 

/J^tf,tbc rules gtuen before l^all futficc: fo tbat a fetuc 

cramples O^all fuffidentlp mltruct pou, tn tfje Ujo^bc 

of it» as tbis fo; tbe firfte. 













"H-nj^e* 



1 2 o./j--— f~i 1 4. §-. §. 6 o. ^- 

^bere J oooc multtpUe. 

2o,g^, — (-— 19.^:^. b^ 6. cc. — 

0nliffe({e3( *^an muUipU'c , numerato; bp numc;' 

Bb.iij. rato;; 



>^f 



The Arte 

1 2 o J^ ♦ as tlje former table of multiplication , fo; 
cl)aungc of GM* ngnes Doctlj Declare, 0nD fo in all 
tlje re(le,t!jere i0 no Dtfftniltie,if ?ou remember t^at* 
tijat ^ou fjauelearneu befo;ie, 

^c!)olar. 3! i^tmiut it Uiell* 0nO fo tbe tuljole 
neUie namerato;t toill bee.i 2 o/§^,— f —i 1 4,5^ §^« 

.6 o, x^. — 1 — ♦ 5'7*5:p ♦ ^notbeoenomtnatoa 

iuillbe.i2 4./^, 

^0 luill tl)e luibole fraction bee« 

I2 4*/5-* 

Cfiat i0 not to bee retiuceo to fnraller termer of nom^ 
bcrfijbicaufe tbei be t)ncamme«furable,but tn Ofiih 
fignegjit mig^te bee baougbte to one leflPer^aB, 

l2o.5>§>-~f— Il4«ce 6 a5^»— 1—5-7. f, 

~24*5-S^* 
ji^otaMl 3lP?oue anotbcrnomber, as fortune 
Doett? offer it to mpnoe.SCftat ig ^'^ -:-:r:^>' /5- . to bee 
multiplteo \i^,i2^ -^l^ X '"^ 

j(u Ahfmdt £paaen 3rt appearetb tbat ?ou take tbeim,at all 
momher txf aDuenturc0, fn po«r firffe nomber, femetb to be an 
^rtSftih lejfe Mfurde nombcr, ^e^ng l)is numerato^j, is IcflTe i\izn 
tbtitMuibt naugbte , in appcaraunce. 0nD tbenmaie it not bee 
Btutueo b^ anp nomber : ano mocbe leflTe bp fo greate 
a Denominator 

a>cl)olar, atiseafietofer, nototbati am aomo* 
nfl^eo tbereonj^oj it is not poiriblr,tbat anp SurfoliJg 
nomber, can bee lelTe t\i^ folwei^fpmeg fo mocbe, a0 
tbe Cuhe of tljefame nature . ^eepng euerp SurfoiiJt tg 
maDe,b^ multiplipng tbe Cuhc bp tbe/f «4rr of tbe li&e 
^otiyhut leflTe tben, 4, is tbere no Sjudrr ano tbere^ 
fo;je euerp SurfiUde^mttXi erceoe W 0»^t^lwr ttotetf 
attbeleatfe* 

^0 



ofCoJIih nombers, 

^0 t!)at.^ 2.cf..^ S/5>» lucre itotl)pn0,anD fo 

(0 an Abfurde nobcr» ano tljccfo^c. 3 2.0: 2 8./^*. 

I0 mocbe IcITe tben notfjpng, ano ^5 tliecbp an Alfurdt 
nomber alfo* 

spatter, ^et mate pour ewmplcfcruc, toteaclje 
anD p^acttfc muIttpUcatton bp^an lucU as anp ot^er. 

0nD fartbcrmo.:c, 3 Intll tell pou bp tbis occaCon, 
t^at 31 fpabs to pou , mo;e after tbe optnton of tbe i'Q> 
mon nombsr of actcs men , tben after mp olunc luO^' 
mente* 

^c^olar. BSmigbt tb(nliefo,bptcrmpngcofpour 
ferttencctbut pet luas pour faipng true. 

Rafter, ^ti male that fraction llano tucU, if pou 
take a b.:oUe nomber AhjlrdHe foz tbe roote. 0Ubouglj 
in Ui^olc nombcrs,it bee an AhfurdenQmhtx, 

^cbolar. E^bat luiU 3 p:oue,bp fcttpng ♦ ^ . fo; a 

;|^bc^«,^-^n^le ^"^^^/,f;^\opy^^^^^^ bprefo. 
^:i ^beS<o//^f. j^j^j^^^ Dotumpnomb^lMm 
rire,3ita!jc.u.cf.tbatl0.-:t^o2M.T»U)l)icbe3?notc 
as tbe firftc fominc. lEDbcn 31 take IiUeluaie0.^ '^•f^- 
tDbtcbepcloetb f,:;] , tbat is.6 \7s • ^"^ nolu 31 fee tftat 
31 mate abate it tjerp lueU,out of. 1 5 f , 

£^after. S>o mate pou fee, that as in tubole nom^ 
ber0,eucr moare tl]c greater GM^ figncs, luf II baue 
tbe grcatcfte nombers : S>o m fractions refolueD bp 
C»fu^e figne0,tbe greateft fra(tion,aunf tueretb to ttje 
Icafte fignctanD tbe If atte frartio,agreetb to tbe grea« 
tettc fignc. 

SDbe rcafon of it ts tbis. S^bat tbe moare anp frac^ 
tion IS multiplieo bp a fraition , tbe leffcr it luaretb* 
j^o; as lubole nombers bp multiplication , maie in^ 
treafe m&nitelp; fo fractions bp multiplication, mate 

Dccreafe 



The J'rte 
liecrcare(nfin(telp. 

315 ut befo^ luce palTe from multiplUaticn y ^ iDiU 
p;touc^ou luttboneerample moarc* B(U.oulo^aue 

fecljolar. % am troubled tutt!) tbv multiplier i^o^ 
3 fenotue not toljatto mafee of it:* 

spatter, p ou tJ0ubte(3B tbinbe) of tljc numeratlo 
of It, bteaufcrou bao nottbe Ufee erampie before : fo; 
ft 15 a mtrtc nomber of a fraction , ano a Uiijole nom^ 
ber. t5ut fepng tbe figne of abatemcnte 10 fet agamtt 
tbp lubolc fraaion, ano notbcr agatntte tbe numera? 
to;,no J Dcnommato;5,tberfo?e mutt tbat4.2p.be tn- 
t>ei'ttanDr,to be abateo out of tbe full fraaiom 

^cbolar. j^otu 31 pcrceiue tbe mater, j^o^ tbere 
migbt be, 3.biuerfe fo?mes,to place tbat abatements. 
3s bcre 3 baue fet tbem.i^-^==^i^ $ T5-==r^. 

0nDasitUiasfetbppou,-\|.-^ — 4.2(f^.U)bicbc 
3 luiU refolue into abfolute nonibcrs,tc fee tbcir Dif^ 
fercnce tbe better. Zm fo,taUing $.fo; tbe roote,tbefc 
tDiUbetbeir.^.fo^mes. 
ThMe. ^^^ ^^^ ^^^ ^-^^iF^"-- ♦ ^l els % tbat ts Mv » 
•^ ^ i^oa t\it fecouDe ^, ''' ., 0; els ^f| tbat is ^^ . 

BiiD fo^ tbe tbiroe nomber, tubube Is our fpcciallc 

nomber . ^f, i2.tbat is.8. . 1 2.anD is an 

Mftii'de nombcr.#o? it betoUenetb IcflTc tben naugbt 
b^.4. 

ipatter. 3f pou tnoulD baue ft no Mfurde nom^ 
bcr, pou mnVt mcreafe tbe p;opo?tton of tbe fraatcn, 
bp augmenting tbe numerato?,o? abating tbe bcno^ 
mmatojjo; els tbirDlp,bg abating tbe nomber,after 

tbefigncof abatcmente. Hs-,<| 4. 2^:0? els 

feccnoarilp , tbUs . ^^ ♦ « 4, X3 . 0; tbiroelp 

l^otubett foj eramplcs fafee , i?ou mate ti}003fee,as 
tucll Ujit^ Ahfurde nombars^as UJitb an^ otijcr. 



ofCo/sike nomhm, 

*B\\t fo2 vo" cafe , 3? Uiill f^ctoc pou t\it tooo;&e of 
tlMs cramplfjn tluoo formes. 

i^irft,rou (Ijall muUipIic tlje firUc luftolc nombcr, 
hv tl)c fraition of the fcconoc nombcr, tbat is, 

irr^*'P-.f ♦ anD it IdiH htt. 



'9r£ ! — ; 



4)-6-^cf>— |-7:>S"y' .12 ace. 



00 \ittt in tuoo^Ue rou mate fee it platne* 

1 9'Cf .—- 1 — 1,'x^ .— .y.f . 
2 4.cr. 

" 4 )■ 6V5-"cS-f-72.§>^» -.izo.ct. 

9. ^. 



6^5^^ 54-^- 



E:i)atf0 fn Itflrertcrme0,l)otl)cofnombcr0, anDof 

U 2.y $-♦—{— .2 4-^> .4 o»^> 

~ zT.^. ,1^, 

jant) tl)(0 (3 tfje firftc parte of rour fomme. 

2Cf)cn fo2 tljc nrrtr parte, multiplic pour firlfeno^ 
faer,tbat ig i!£^;r±ii!!ii^;=:=3l! b^tlje abatement of 

tl)e feconoe nombtr,tbat 10 bp . . 4, zo . ano (t 

Ujillbe* ^^ 

Ct,U ^ 



The ^rte 
9i^ bp m^^ tuoo^Ue ?ou ma(c fee. 



1 9»ct:.-+-.5-ie.*- — ♦^f 



tulbtcfte being reDuccD to tbe Dcnominatton of tbe fo^ 
mer nomber,UjUlbe triplcD(Gtb tbat Denomirtatoj ts 
triple to tbis)ano fo Imll it be "'^~'^Zf'^^~-''^^— 
^oU) aODetbofettuonomberstogetbec, bpputt??ng 
tljeic botbe numerators ivi oite,ano it UjiII be. 

2L§-. .I8.f. 

j^gs bere appearetb in iuoo^fee* 

i5'2«^5-«— 4 — ♦24.5-. .40.^. 



2 0.2£^.^- ♦76.5-^* .12.5-. 

tubtcDe tutU not bee reouceo to anp fmaller fraction, 
bicaufe tbe nombers be incommenfurable.ano one of 
tbe CopiefiQncB ia.f .ano fo 10 tbat tbe fomme of tbe 
multtplication. 

0notberUjaiepou maieiooo^ljeit , anoallfocbe 
lifee,bp rcDucpnge tbe multiplier, into one t)nifo:me 

fracticm asberein.^^ 4,^^ .poulballmuL- 

piplie — — 42^«bP»9»5**UJbicbefstbefo;merOC'' 
nominator ano it ioill be — — .36.cC»2bben putte 
tbat to.2 4.cf. ♦ oucr the line , ano fet tbe common oe^ 
nominator. 9. g-.bnuer tbe line,anD it Uiill bee m one 
fraction rcouced ^^==^^'^. 

§>cbolar. laereBimaiefeeattbefirHebetDe, tbat 
t))iB fraction is an Mfurde nomber:for tbe abatement 

after tbe fignc .is greater tben tbe nomber be^ 

fo;e 



ofCofikenomhers. 



fo;c tt. 

spatter. SHfjat tuas cofcffcD before* 15 lit pet mate 
von tuo2bc tl)e cramplc bp it. 

^cl)olar» 2C!)ati0 truc:anDfo tuilltljc numcra-' 
to;i0 , bcepng muItiplieD togetfjcr , niaUcerartrlp, 

6o.cc» 228. ^-cT. o6.5-g-N :ast)crc 

m example of iuDo.:kc, J Uaue ^tt iUtQi m^ oUinc cafe 
anoccrtcntic. 

1 9.ce» -- 1— o.^» — ♦y.f 

2 4.cfl. 06. c^. 



684-^c:>-- 1085-5^—1— .iSocf^ 

6 o Tct .— — .2 2 8. 5- ct . — — ♦! 6 § '5^. 
0nD tl3at IS tlje nclue numcrato;. 

HnD tl)cn fo: tbc fcronDc nonibcr,tf tljc firHc Dcito-- 

ntinato;.? ^ 6.<y.bcmn(tipIifDbri"?irfrconDc 

acnoimnato;i,9.^. ttiscafilrfccn -, tljattbciUnll 

maUc.6 ? 5- §- y 4 5- U)t)icljc fijnll be tl;c ncluc 

UcnonaMnto:, 

^iiD fo t\)z tntcrr frartjon fijnd bcc. 

6o. ce.— -- ♦2 28.f,MX> >^6 .^5->, 

sniiat ts m the fnTuUcrie nombcrs aiiD figures Cojiiks* 
^^^^^=^~7^Z: !^y7,P^^^^^: lulnctic fommcDoDctb m all 
th^ngrs fulli? agree, luttl) tl)efo;jmer nomber tljat 
^ou lu;oiigl}t. 

iBatten p;;oiie tfjcim.botlje bp refolutlon : and 
tljen n)all pou Unolucttic rcafon of tijetr agremcnte. 

^fbolar. gfeetliattbc U)oo;ibeoftt)e Denomma^ 
to js,Doctb agree. CCI I)erfo?e 3 Uitll take. 5. fo;: a rootc 
to p:oue hoU) tbe lco;jfee of tbe numerators tuil agree 

^nDfofoM9,ct»3C^ain)aue.nD'^"^fo;,^x^. 

Cc.t;. 31 



The Jrte 

311!)aUf)aue»9,fobeat5DeDto,5'n»^nt)roI;aue3[,5'2 2 
out of iufticlje fomme j, \m\^ abate . ^ ^ . 5 

:anD tljcn cematiiet!) . n 7 ♦ to bee mwMU %'fi 

plieo bp,2 4-cC«tt)at ts bp.6 4 8. anD tljc ^-^-^ 

totallc Ujill bee Ca0 bere in tuo^ke appear 4 J 5 6 
retb).35roi6* lubtcbefommemuabea^ ^48 
bated to a fmaller nomber, in liUc rate aa 3240 
tbe otbec Uias ccDuceo,rtrae bp partition 33^015 
into.^ a no tbeit U)tU it be. 1 1 1 6 7 2* 0nD 
again,ft muft bee oiuiDeo bp, 9 . foj tbat !■$ tbe quaiti' 
title ofafquare,bptubicbetbe former reDu£tion,luafl 
IuaougI)te fo; tbe Gliike fignes : ano tben iuiU it bee» 
1240 8» 0110 tbat \^ tbe firtte parte of tbe firtt luo^fee. 
SCbenfoj tbe feconbe parte of tbat Uioo;jfee,3i (^aU 
multiplie tbe firtte nomfaer0,tbat is 5- 1 7 bp tbe abate* 

mente of tbe fraction, tbat is bp 4 ^^ ,0? 1 2* 

Clitb* 3a0 tbe roote)ano tbereof iuiU com^ — 6 204. 
iubicbe fomme 3| muft triple,asa oio bis equallcCtbat 

is.2o,2g^» 76*5- §-» —1 2,5^0 auDfo toil 

it bee 186 1 1. ^otu (l^all 3 aooe tbis fomme, 

tyitb tbe firfte parte,Uibicbe Uias j 240 8»ano it luill 

bce.i2 4o8, iS6i2,tbatis.62o4aeffetbeii 

notbvn0:anD is m numerator of tbe firfte tooo;He» 

Mlberfo^je 3J p^oeebe to t\)t feeonoc luoo^fee.iDbere 
tbe numerator of tbe frarttoit , ht^^n^ rcDuceo to tbe 

common Denommato^is»2 4.cC» ~]6.ctM\\U 

t\)t is 1 2»c€ ♦ano m nombers refolutecueping 

^ftiUasaroote)itts 52 4-bpiDbicbeif3lmuU 

tiplie* n 7- it loiU peloe ♦ 1 6 7 5" o 8» ^no tbat fomme 
be^ng abated, bp Diuifton into, 5. anb.9 ♦ as i\)t otber 

iDas,o?eIsDiuiDeDb?,27«Uibtcbeisanone,itgiuetb 
6 2 o 4. as tbe former ly oojjfee oib, 

Rafter, sCbus 31 fee , ^ou areerperte inouffbe in 

muItipHcation:^berfo;e3iU)iU(hcU3erounoH3,tbe 
o;DeranDfo;meofoiuifiom ^ «^ 




cfCopih nomhers. 
OjDmfion. 

^ttt iB noe fpccf all rule to be g(ucn, fo^ 
itW tuoojfeeof 2Dtuifion,otbcctI)en focftc 
m arc all rcaDp ta!igl)te tn otljer loo^zbes 
of Diutfio before. Mfterfoje 3! tuiil hv one 
o; 2.eramplei3,ll^etue ^cu tlje iuo^Ue of it. 

Thefirjle example ofDluifion* 

i4>ce-H-j9>S-- tobetJfulDeufay ^-^•~+~^^' 

i)oeffjpclDe.li:M!ClzlZf^ tljatls tna Icf^ 
fer fractfombt? botbe reouctlons of nombers f Cgitea . 

^;z of &er example. 

3^"miM!: Diuloeobp 19- S-- --^-f 

2»ce» .^^e* '^•S"* *^f 

tjoet]^ ma^e* 

iDljofe nombcris bee litcommenfurablcanli tijerefoje 
mate not bee reouceOjbut bt? abating one Oenomina* 
tion Cofii\e, 0no fo tuill tt be* 



48«S"C^— I — ^Q*^S" 64*ct ^o* 

Cc^lij* ^cljolat 



Thejrte 

&c1)olar» 31 ^tt tftat pou multiplie croOTe iuafe0(a« 
f n tjulgare frarttong ) t^e numerator of tbe one nom^ 
berjbp tlie Denominator of tfte otber^ano fo xt^ oiuifi* 
on of noebtmcultie,to Ib^m tijat rememU;eti tt^e fo> 
mer ruled* 

Of the golden rule, 

fatter* 

l^e golden rule , tftat tK tfte rule of 
!paopo;tton,l^oulDfoIo&)enoiu,b? 
tftetommo o?oer, HButfepngtljere 
10 no otmcultte tn it, notfier an? o^ 
tljer forme of Uioorfee , tfjentsin 
lijulgare nomber0,3i ioiU notffaie 

anptpmeaboutett* |&>aue tfjat fo j 

^onr pleafure , 31 Ijaue fet Ijere rertaine eramplc0,a5 
iuel in to^ole nombers a/%, as in bjoUcn* 







2rfC^^ 






Mce — A 



-5^ 



4^f 



KS.- 



dms 7 



M»^*H-^«4»f 



^ ' ' ice. — I — sS' 



5!6»e 



.3ce — i— ss- 

S)cl)olar« %Wt fctueerample0,t)ooerumctentlp 
teac&e tfte forme of tlje tobole rule* ^0 tljat bere nea^ 
Oetft noe farther erplication, 

Wl)erfore,if in tfjis arte,tbere be anp forme of er^ 
trartion of roote0,3i praie pu to proeeoe tberto. 




ofCoJ^ike nomhers, 
OfextraSiion ofrootes. 
gaffer* 

^ (n nombers Ah/lraHe, txxtt^ nombcr (3 
notarootcDnombcr , butfomecertatnc 
|onelp entongcft tftclm , fo in itombers 
|G/?/feaIi nombcra bauc not rootes: hnt 

ber£( are rooteD , Uiijofc nombcr batb a roote , agrca^^ 
ble to tbe figure of bts Denomination. 

^otbat* 16. c^«isnota^qiiarenomber, notfjcr 
Ijatft anp roote. i^o; altbougb. 1 6. bee a fquare nom^ 
ber,anD l)atb. 4.fo^ \)\b roote , t^et tbe Dcnominattoii 
(Uj^ictje i0.c£Obatl) noe fquare roote: but ♦ 1 6.^»i0 
a fquare nombcr:anD batb.4.:::£^,fo^ bis roote* 

3liUeiuate0.8.c£.i5 a Cuhike nomber,anD bis roote 
(tf.2.i^:but.8.5-.i)atb noe roote.^o^ bicaufe.8.batf) 
no fquare rootc,agreable to tbe figne. ^». notber i& it 
a Cuhike nombcr, altbouglj it bauc a Cubik^e roote, hi^ 
caufe tbe roote is difagrcablc from tbe Ggne, §-. 

^cbolar. 3! pcrceiuc t%^t m tbcfc nombers^as tuel 
as in all otbcr, tbe roote bccyng multiplied bp it felf, 
luill make tbe nombcr, tobofc roote it is. 0no tbere> 
fo^^e can no nombcr be callcD fquare, o^ Cuhihe.n anp 
iuaieselsarootcD nombcr, crcepte tbe roote of tbe 
nomber agree luitbbtsfignc: Mbcrebplperceiue 
tueU,tbat. ? 2.y^.is a rootcD nomber, fo;bicaufe tbat 
1 2 ♦ batb a SurfoHde roote , agreablc to tbe figne. S>o 
liUetoates. 1 2 v.c€.^^s a rootcD nombcr, fcpng 5^. is tbe 
Cuhths roote of. 1 2 s* But.2 7^^ As no rootcD nober. 

5©after. Cbus rou tnDcrftanDe fufficicntly, tbe 
iuDgemente of rootcD nombcr0,anDtbcirUnolulege, 
in fimplc Ofiih nobcrs, tbat be tjtterlp tjncopounDe. 

?I2!tberfo;e,fo?crtraction of tbctr rooteSjtaUe tbis 

cDrtractc 



The Arte 

cl;]ctcacte tl)erooteof rour nomfacr^asfffttocre 
abfolute^anu put to tt.2^, fo^ toe ocnominatiom 

;anD,49»p» l)atl).7«2^. fo;l)is xtioit, 

;affam,tt)crootcof.2 ft.<:t:.is.6.2£. 

fecliolar. 2Cfti£f3i perceiuc. ano bplifee rrafow, 
tlje rootc of. 24 ^♦/g^. is* 5. 2^. IBut tuftp ocoe j^ou 
name noberg 0>/?%\)tterl^t)iuompouiiDc r'i^o^asg 
Ijnoerttanoe , tjat tftere bee noinbers compounDe, tti 
tfteir fignesjfo 31 fee tbat tbei male baue rootes alfo. 

^5. 1 6» §- 5^ ♦ batb foa bts rootc. 2.2^ . ano iibe# 
Juaic0.64»5"C^«barf).2. ^ . fo; bis route. 

#a0cr. i^no Dooe ^ouliot fee , tbat tbofecom;* 
poimoc nombers, mate bane moare rootes tben one/ 
$>itb. 1 6.§^ 5^.batb fo? bis fquare roote.4.5^.as U>el 
as it batb.2.^ , fo? bts js^n^i^n^if^e roote. 

&o.4.§- 5^" batbfo?ms Square roote.2.^. 0nD 
9atb no ^Kf^^^en^iet^ngrcable to bis tuboie figne. 

iltRe\Daies.9.^ ct* tjatb no i^en^cuhifie roote^ac* 
coinitiQ to bis twbole figne; but it batb a fquare roote 
agreable to parte oftbe figne,anti tbat is. ? c^. 

^rbolar. 31 fee tbat alfo. anofo batb.S.^-cf. noe 
ji^en^cubike roote, but a Ctilih roote: tobtcbc is. 2. ^, 

iPaffeA SCberfo^e in r opouoe fignes, if tbe figne 
male baue forbe a roote , as t\^t nomber y)i\\\ pelDe-,it 
Is a rooteo nomber,eIs not. 
^ Mberebr pou mate pcrceiue , tbat if anp nomber 
fopounoe in ligne,baue a roote agreableto bis tobolc 
jigne,tbcn mate it baue aIfo,as manp rootes^as tbcr 
be partes iw tbat e ompounoe Ogne. 

$>o 4c965-5^«c£batb not onclp a ^n>i^n>jcuhili 
roote, tobicoe is.2.io tbutit batb a 5y«4rr roote tbat 
is.6 4*5^ cf ♦ ano aifo it batb a CuUlie rcotP-,tbat iB^ 
I ^I'StS^'i^^^^^** y^mtM batb a ;<f«^/;^f«;?-/if roote. 

^I'^^lllf:"^' ^"bfourtblr , itbatb i;^»W'l' 
rootejtbatts.4.j*, ^ ^ 



ofCofiikt nomhtrs. 

;3nD To (l)atl ?ou tuDgcof all otljer Wu, 

$ixl)oIar« SL|t5 fl^alt Oiffta, 00 3f UitU p^acttfe t^g 
mater , at moare letfcr. 15ut ano tf tbe noml^ers bee 
(ompounde, lottl) figne^ ofaODtttou, tistbere t^ena<' 
ti? fpertall o;Der fo: tbeic rootesf :H0 tit tW erample* 
8i.5>^. — |— -27.^^♦UJ^re3!^)auemaDeecbepaltc 
to be a rooteo nombcr* 

bailee* 3;n oeeoe. S i ^ ^. batb botbe a Square 
roote,anD alfo a :^:KS^KJkf roote.^ut.27 cc 'ftatb 
none of tbofe tUioo roores , altbougb it baue a Gt^Hd 
ro0te,U)btcbctbeotI)crnomber toantetb* ^notbeP 
fo;e 10 not tbat Uibole nomber,a rooteo nomber. 

115 ut to tl)c tntcnte, tbat fou mate Hthc mo;e cer^ 
tetn of rooteD nomber0,3l iDtll tell p ou eerteln notes* 
boU) tt utaie bee knoluen, Uibetber^onc nomber be a 
rooted nomber. 

i^trffe,tf tbe nomber annereb to tbe greateff (Igtte 
oftbatcompounoe c*/j/X; nomber, bee not a rooteO 
nomber, tbe tubole nomber can not he a rooteo nobec 

^econDartlt^.tf tbe nomber tbat td iopneo tuttb tbe 
leafte Gj^ikf ngne,be not a rooteb nomber,tbe tobole 
nomber can not be a rooteD nomber. 

flnb ecbe of tbefe botbe roote0(<f tbct baueanp)are 
partes of tbe tubole roote , fo; tbe compounoe Gpkf 
nomber. 

SHbtrblF, tf tbe nomber be a rooteb nomber, euer? 
parte of it, tbat (0 not a rooteb nomber, Isameane 
nomber, bctluene tbe greateUe ano tlit leade. 

i^ourtblp,if.5:f^. bee anp Denomination in ityt^m 
i0.f .an otber Denomination m it alfo. 

jPiftlp,anD generally, all rootcD n6bcr0,otberare 
fpectallp rrameD,bp o;iDerlp multipUcation,o? el0 are 
nombers cqoalle to fome one rootcD nomber Ahftrati, 

i^otu fpcciallp frameDare rocbe,a0are maoebp 
multipltcatto of one nomber bp it rrlf,anD Ittle 0; no^ 
tbrng altered from tbat berp fo;me. 

STiD.;. C^rampie 



The Jrte 

Offittirt C-rapIeof. r29 ^ce-H— 184 ^5r*~4— 165^ 

r9Qtes. iufijicfte IS a Square itomber , maD0 bp multiplicatton 

of.2 ?,a£ — f— 4.2p • bp it fein SCftlg nomber mai'^ 

^29*5^ce-- h-l84S^5— f-l6S-(23ce+4S^ 
2 3 46*c£* 

$11 16^ firfie nomber,3l fiitoe tfte 5^«rfre roof e to bee 
13, ana fo? I)f0 t>0nommatton, 3 tafec ftalfc tfte GM*? 
ftffne ^ ce »ano t^at i^^cC-^o^ as^ce^multiplieD hv 
ct^rmti^msXit.^ct, &o in Diutfion b^z.ano mcr^ 
traction cf Sy«4y^ roote«, 3 fl^all tafee tfte . ce ♦ fo^ tbe 
Ijatfe of §- cc ano tbe Denommatioit of bw rootemno 

fo fi^t it Ooune in tbe qmtiente, 

Hubert 31 (balloouble tbe nomber if^y^y^^* oftbat 
iuotUnte (fecppng bis OM^ Cgne iJitaltereo) ano tijat 
Double fl^ail 3 fet euermo^e tJtioer tbe ntxtt nombrr, 
totoaro tb^ uqW banoe* as bece,^ou fee, 31 baue fet 
4 6(tobtrbe is tMt tiotO^lc of 2 5)Xoitb bis (lgrTrc€»bn^ 
oer tb^ feconoe itomb^r* ano tbere 3f percciue,^ maie 
baue it4,tpmes,if 3; ooe Unxnt cas 3f ougbt) 1 84, bp 
4 6» anti tbat*4,3| fette tn tbe f ^o^/w*(?, tuitb tbe 0anc 
— {— ,anotbeoenomiitatiort,2p :fepiTg» ^^^» oi«ij» 
ocob^cc«Doetbi?eloe»2^» ^ ^ ^ 

Eaae of aa,3! moOe rmiltiplie tbat parte of tbe f «o. 
tUnte, 4.2g^.bp it felf^^no it toill ^elDe* 1 6»5-.U)bicbe 
bepng fubtra£teaalfo(asttl^oulo) ieauetb notbpttg 
remampng of tbe fquare rtombcr. 

SDbtso^Dcrmttftpottftcpe in all fquare nombers, 
boU) create fo euer t^ti be* as in tbis feconoe eraple* 
■ — 905^^. 

2> ^ce-+-8 o/p 2 6 5^5- i44ce-+--8 1 ^(j ce-f 8 5.« 9 X: 

).c£. ioce--f-645-S-» 

—- }— locC — ! — 16 V-. aso . 

S^be 



ofCoflikenomhers. 
Cfte roote of t^e firft noinbcc id.'; ctMit\)t 3f fet 

(it a qustiente* 

%i)t n Doe 3 touble tftat,^ , and tt mahttli, i o, to be 
fttte tmjer.8«U)itft ijiss CenommatioibUJtliflJ^tiS^cS* 
lifec to tt)e roote* 

SDftat I o*cC ♦iwat^ befoutiDefrr. 8 o»y^,8»ttme0,f 
t\)ttfo;t 31 fet, 8, tn tbe quothnte, toitb tbc ftgiic— 4 — 
ano tl)0 tjeitomtnation^g-, and tlwrtt Dooe 3 muUipIie 
tl)at,8.5^.fquareDlp,lDl)tcl)e giuctft,— H — 64-^^ §^» 
to be fubtracteti out of —2 6.g- 5-,ano fo itmatr 

netl) 9oǤ^S^* 

after tW^toublcant^tiuottenteagainy Mjtvtf 
cf commetl) — +— i o.ct*—^— 1 6< §-♦ 0nD bicaufe 
tl)cre is a rcmatner^ouer tlje nomber tbatj iu^ougljt 
laffe,3 mutt fet, i o,af ♦ tjiiDer tbe rcntatncr, anti the 
otber nomber tn o^Der, as ?ou fee tt fet. 

%\)cn fekeB! bote often t^mes mate, i o,ct,ti(utije 
9 o,^ ?>, anD 31 finoe tbe quotknte to be — — 9, 2£^» 
anDlifeluates— -h-i 6 §-.muUipiteo b?^ — —9 x^ 

Doetb mafee 1 4 4'Ct* cquallc to tbe fomme 0^ 

uer tt: anO fo fubtractetb tt cleane,^ berfo?e to enoc 
tbat tjjoo^Ue , 33 mwltiplte tbe lafte qmtiente.h^ it fe!f 
fquarc,anDttpelDctb . — I — 8 1,5-, tubtcbe t0 to bee 
fubtracreD out of tbe Uhe fomme, in tbe fquare nom^- 
ber : anD fo reftctb notbftig » Wi\itxtiQit 3! iuaii' af;^ 
firme-,tbatt!]e firffc nomber t$ a fquare nomber, ano 
batb fo;jbis rooteo%ce*— +--8,p 9-5^- 

^cbolar, SCbat mate | fone p?oue , tf 31 multtplte 

5 c€ ♦— f-8^,' , 9'^» 

' 2 y ^ce-t-4 0/^-^-4 J §::^ 

_i_.„4o7^--|.-64^^§- 

8 1, 5- ♦ 7 2,aL— 4 5 5"^ 

72,cc- 



2r^ce-+-8o./3.— 2<S5^p--i44ce+8,5^. 



Thejrte 

tW toott bp it felf,as "hm 3 ftaue Doen it Wl\)tth^ 
3 ^aue not onelp confirmed tt te be a fqnare nomber: 
hut alfo ^ l)aue efpteo , tbat pou bfcD tfte nomber not 
fo plainly fet Ooune , as tbe pacttcularc multiplicati- 
on 9to make tt : but ratber as a reafonable reouctton 
iuoulo crp^eflTe it* 35 meane in tbc ♦ 5- c» ♦ iubere tbe 
parttculare multiplication batb — i — 6 4»§- §>,ano 

9 o §- J»*i^o? iubicbc ttooo nombers pou fettc 

one,tbat rcfuitetb of tbe botbe,tbat ijs 2 6 5^v» 

Rafter. Il5ut if pou tuoulb take tbc ncber m tbat 
ro^te,tbe tuoo;jfee IdouID be notonelp all one:but alfo 
fometubat plainer to bee perceiueD of a learner, ^no 
tberefojefo: vourpleafure,3f luillfctfo^tbe bere,tbc 
example of tbat tuooaHe* ;^nD loe^bere tt i$. 

2 ^ 5- ce-f 8 0./5.+6 4 ^5—9 o^^~l 4 4 ^ f S i^ (Sct+S^~9^ 

^cbolar. l!5p comparpnge tbefe botbe fo;me0of 
^oo;Ue to|fetber,3 bcoe better bnDeraanoe,tbe rea^ 
fon of tbe ftrUe luoo^Ue. 

Rafter. £)nceramplcmoareoftbts hinueofer^ 
traction of rootei5,Ujill 3i fet Ooune, tbat maie be a tr-** 
ncralle patrone, fo^ all t\)t barteties , m tbis foitc of 
rooteDnombcrs. ano tfpou eramine it Diligentlv, 
ano marfee it iuell , pou (ball neabe feUie otber eram' 
pies, fonbis ^mae of fquare nombers. 



Xf)c square nomber, loitb tbe 

liioo;keofertra(tion 

of bis rootefo^ 

toU?etb 

be re. 

E6fe 



Thejrte 

tW toott bp it felf,as "hm 3 ftaue Doen it Wl\)tth^ 
3 ^aue not onelp confirmed tt te be a fqnare nomber: 
hut alfo ^ l)aue efpteo , tbat pou bfcD tfte nomber not 
fo plainly fet Ooune , as tbe pacttcularc multiplicati- 
on 9to make tt : but ratber as a reafonable reouctton 
iuoulo crp^eflTe it* 35 meane in tbc ♦ 5- c» ♦ iubere tbe 
parttculare multiplication batb — i — 6 4»§- §>,ano 

9 o §- J»*i^o? iubicbc ttooo nombers pou fettc 

one,tbat rcfuitetb of tbe botbe,tbat ijs 2 6 5^v» 

Rafter. Il5ut if pou tuoulb take tbc ncber m tbat 
ro^te,tbe tuoo;jfee IdouID be notonelp all one:but alfo 
fometubat plainer to bee perceiueD of a learner, ^no 
tberefojefo: vourpleafure,3f luillfctfo^tbe bere,tbc 
example of tbat tuooaHe* ;^nD loe^bere tt i$. 

2 ^ 5- ce-f 8 0./5.+6 4 ^5—9 o^^~l 4 4 ^ f S i^ (Sct+S^~9^ 

^cbolar. l!5p comparpnge tbefe botbe fo;me0of 
^oo;Ue to|fetber,3 bcoe better bnDeraanoe,tbe rea^ 
fon of tbe ftrUe luoo^Ue. 

Rafter. £)nceramplcmoareoftbts hinueofer^ 
traction of rootei5,Ujill 3i fet Ooune, tbat maie be a tr-** 
ncralle patrone, fo^ all t\)t barteties , m tbis foitc of 
rooteDnombcrs. ano tfpou eramine it Diligentlv, 
ano marfee it iuell , pou (ball neabe feUie otber eram' 
pies, fonbis ^mae of fquare nombers. 



Xf)c square nomber, loitb tbe 

liioo;keofertra(tion 

of bis rootefo^ 

toU?etb 

be re. 

E6fe 



Thefquft noniier,'^'uh the TUfOorh ofextraSiion of his roote. 



■A^ 



-'s^ 






-1 w 
















-If 



*4^V«— f-lo^-g- .8*ce-4— 65-* 4Sg^-+-I»f 






»Lf. 



Theproofe ty b^iitiplkatkn. 






36^/5—4—3 ocfce -245"^§^ 



-.2 ♦ is .- 






=r8?/5 



so.cece- 






-I s^c€ ^lo/^— Hi §'5'« 

-I6^ce 12/^>-+-8 p 5*- 

-1 5" $" ce — i 2/5—4—9 5" S"- 

-1 2 ^ lc/§— I— 8 p ^ 



6/§>* 



55^^- 



-♦4»c€» 
-4*c€»- 






-22e-+-i' 



^ftolar* 3(t mate appeare eaClr^tl^at tlife erample feruetb fo^ manp otfte:, it ijoett contain fo man^ tarfeties of Cgneg C(»y?%, multfpUeD to 

ano in tftis nombec alfo,a0 tuell as m tlje otber,3i fee tftat manp nombe« be omftteft , op reoutfion:namelp in tbe tfttrDe, fourtbe,fifte,anti 
fijftbe ojDec5 of nombers«#o? in tbc,2.firfte o?Derg,anD tn tbe.TJattctbere fe no tjariette of tbe fignci^ — | — ano . 

Mberfo^e to fee tbe t)anette of U)oo;{ke,3i tutU fette Doune tbe nomberg^as tbct rife in particulare multtpltcatton -, ano in it tutll 3 malto an 
erpertmente of m? canning* asberefololwetb. 






-h-/^i2^cece-+-zi'5^5-^- 






--^^5^^^ 









iZ./^—i — loar^ .8.CC* 



-^/4&'ce*-4— -J^/^ 



Z0 



ID 

2i^ 



-^^;^- 






E)D. ig. 



~^^/^- 

I2./p, +- 

^btrc 



9^-^ 



8 6 



-42^ — 1 — 1^^ 






--8»(^,- 



»o«5' 



"4*5:^* 



.l.f 



cfCoJ^ike nomhers. 

Whttt foj mpne otone eafejattD a(eO otmtmo;ic, 
f \)mt fet tjitoec cuer^ Doubling of tftc ^imtientex ^nn 
m fomme tftat amountetfj , bp tbe multiplication of 
tftefame, into tlie rrctoe ^mttente, tuitlj tbe Square of 
tiefame netoe <iu9fmte, 

^berebp 3 perceiue tbat tbe nomber£i,Doe not go 
in rocl)e o^Dcr , tbat euerr oi»i»c place,maUetb a neiue 
roote,a0 it ooctb in nombcw AhflrA^e,)5\xt fometime 
31 mnft take* 2. places nerte togetfter^ano at an otbec 
t^xmy 3 (]^aUfcippe»2»o?,5«placej5. 

5©affec. ^ou marbe it tnell. 0nD pet tbatiffa 
gooo anD trne rule , tbat fome menne teacbe t tbat in 
tbefe Cofiiks nomber6,afli iuell as in otber jibp^^e no 
ters, poufl^allmarbecuervoDDe place, anDtjnoec 
igcbe of tbem to 8n&t a Square roote. 5!5ut tbat is to 
be Dn^erftanDejtoben tbe nombers ate fette, in tbeic 
b^efefte ano eicattefte o^oer. 

SDbefe fetue eramples mate fuffice^fo^i a oeclaratio 
of ertratfpng tbe roote of Square nombers, maoe bp Themtes of 
multiplication, 0nD notu toucbpng tbofe nombers, nUers equd 
tbat bee equalle to fome rooteo nomber , anD namelg to hejtiurts, 
focbe as be equalle to a fquare nomber, 3 tuill teacbe 
pou boU) tbetr roote mate be eit:tra(teO» 

mm firfte pou (ball marbe, tbat a Square beepng 
compareMs equalle to rootes anD nombers,tbe roo^ 
tes mate bee couple^ toitb tbe nombers onelp , in, ^ 
fo^mes.2Dbatis,tp .— 4 — .<y(tubicbe i& all one luitb 

f ♦--+--• 2^) 0; els tbus.f , .^<, ^i tbtrol??, 

^_J — sj, (3nD fo; ecbe of tbefe, ^To^tes , tbere is 
fome tarietie , in tbe ertrartion of tbe roote* ^uD in 
tbem all mocbe agremente. 

i^o^ tbe firft fo^me, iubere 2^ — [ — f is equalle to Jhefirlle 
§> take tl)efe eraples i ^ is etjuall t0.4.2^ — \ — 21 ^form, 
oi. I p.is equalle to 5 j,*^.— 4^-2,2^ ♦ liketuaies i §* 
isei|ualleto,io2^, — 1 — 7 f.f^oM'g^'-ts equalle to 
lo),^,— -j — ♦8.?p* 

DD.titI, iw 



cfCoJ^ike nomhers. 

Whttt foj mpne otone eafejattD a(eO otmtmo;ic, 
f \)mt fet tjitoec cuer^ Doubling of tftc ^imtientex ^nn 
m fomme tftat amountetfj , bp tbe multiplication of 
tftefame, into tlie rrctoe ^mttente, tuitlj tbe Square of 
tiefame netoe <iu9fmte, 

^berebp 3 perceiue tbat tbe nomber£i,Doe not go 
in rocl)e o^Dcr , tbat euerr oi»i»c place,maUetb a neiue 
roote,a0 it ooctb in nombcw AhflrA^e,)5\xt fometime 
31 mnft take* 2. places nerte togetfter^ano at an otbec 
t^xmy 3 (]^aUfcippe»2»o?,5«placej5. 

5©affec. ^ou marbe it tnell. 0nD pet tbatiffa 
gooo anD trne rule , tbat fome menne teacbe t tbat in 
tbefe Cofiiks nomber6,afli iuell as in otber jibp^^e no 
ters, poufl^allmarbecuervoDDe place, anDtjnoec 
igcbe of tbem to 8n&t a Square roote. 5!5ut tbat is to 
be Dn^erftanDejtoben tbe nombers ate fette, in tbeic 
b^efefte ano eicattefte o^oer. 

SDbefe fetue eramples mate fuffice^fo^i a oeclaratio 
of ertratfpng tbe roote of Square nombers, maoe bp Themtes of 
multiplication, 0nD notu toucbpng tbofe nombers, nUers equd 
tbat bee equalle to fome rooteo nomber , anD namelg to hejtiurts, 
focbe as be equalle to a fquare nomber, 3 tuill teacbe 
pou boU) tbetr roote mate be eit:tra(teO» 

mm firfte pou (ball marbe, tbat a Square beepng 
compareMs equalle to rootes anD nombers,tbe roo^ 
tes mate bee couple^ toitb tbe nombers onelp , in, ^ 
fo^mes.2Dbatis,tp .— 4 — .<y(tubicbe i& all one luitb 

f ♦--+--• 2^) 0; els tbus.f , .^<, ^i tbtrol??, 

^_J — sj, (3nD fo; ecbe of tbefe, ^To^tes , tbere is 
fome tarietie , in tbe ertrartion of tbe roote* ^uD in 
tbem all mocbe agremente. 

i^o^ tbe firft fo^me, iubere 2^ — [ — f is equalle to Jhefirlle 
§> take tl)efe eraples i ^ is etjuall t0.4.2^ — \ — 21 ^form, 
oi. I p.is equalle to 5 j,*^.— 4^-2,2^ ♦ liketuaies i §* 
isei|ualleto,io2^, — 1 — 7 f.f^oM'g^'-ts equalle to 
lo),^,— -j — ♦8.?p* 

DD.titI, iw 



The j{rte 

git all tl)cfc eraplcs,ano otljec foe lie ime,pou mud 
firft confiDec tl)e nomber annereo \xiit^ tbc fignc. io ♦ 
CU)bicl)e istlje miDDcU quantitie)anDtt)cftalfeorit 
fl)aU pou notr^foj U,itl) it l^al pou luojUe tUiifciPirft 

rou(t)aUnusItiplic!)alfcoftljatnombtrbpttfelf.ano 
tl)i5 w tlje firfit U3o;Ue, ann to it ft^all rou aODc tbc o^ 
tbec lul)olc nonil)cr,tI)at is iopnco luitb.f . SnD tljci 
iuill cuer n:o;:c mafee a fquare noniber: out of lubicftc 
rquacc i?ou ftjall cmatte tlje roote. ani) to tijat roote 
lljaUrcuaODe ftalfc tlic nomber, tbat Ujas annerco 
Jwitl) tljc fignc of.2p ♦ (luljiclje luas tbc nombct tftata 
baoe pou to inarftc/raiio tljis 15 t^e feconoc tDoo;be» 
SDlje totall tliat r ommctfj of tbts aooition,t« ttjc roote 
Of tljecompounoe Gfiikemm\)tt. 

jtn example CErample of tl)c firftc,4.5:p .-If — ,2 r.^. jjalfe tlje 
nomfaer anncwo toitil^.^^.tir 2. txjftoft Square 10.4. 
tftat Ojall 3 put to. 2 K ano tl)crc rifetl). 2 ^ becpng a 
fquare nomb0r,aiiD Jjaupngr.^fo; ftts roote. SCo tljat 
S S iopne ftalfc t!)c iromber annerco luitfj. j:p . ano it 
ntafeetft. 7 ♦ toljklje in m ttomber ttjat 3 fcbe fo;:ano 
(0tl)e roote to.4.s^.— I— 2 i.f. 

riepreofe, i^o?triaUU)l)ereoftaftei4.roote)g,tljatf)5. 28. ano 
i^uttz to it. 2 1, ano thereof rommetr). 4 9. tofjicljc 10 a 
fquare noniber,aiio ftatb.y.fo; fjw roote. 

^» o^kr ^cljolar. SCben tan 31 Doe tfte Iifee Untb tlje fecont» 

txaxnph, ^ >^amp!.3 f.f— +— .2. V .^no 0r0e tlje !}alfc of.2.10 
Lanotfte^quareofitis.i.tDfticlje^putto.^j-.anoit 
mafeetl).^6.a Square nomfaer:U3f)pfe rocte 10.6. SCo 
t^at.6.ifBI aODe. i.tfjat twas tbe tjalfc bcfo2c rcferuet, 
it Ujill mafee. 7. Uibic be i0 tlje roote tbat i Doc fcfee. 

Tbepmfe, S:bep^oofei0tbi0:2. roote0mafectb.i4rano.u. au 
uetb*49«lJJbofcrootcis.7* 

The thirde ^ife^tcate0 fo;j tbe tbirDe erample i o xo — !— 77 ^ 

txampU, 3 U?oo?fee tbu0.l(^alfe. i o.i0,5-,anD %\b ^.quarei0.2 /. 
tbat booe 3 aWe to . 7 f.anb tbere rifetb. i o o. tubofe 
roote 10. 1 o, to bbtcbe rooted aoo. j. ano tbererom^ 

metli 



ofCofSih nomhers, 

metlb.i f*tl)at to t^e roote Uibtcije ^ tDotriD bnur. 

anD tbat 31 mate p;oue b^) trtaU tn t^ts fo^tc . i o. 
rootcjf gitte.i j ctjnto toUicbe if ^ aooe,? ^ tbcre UiiU 
amounte.z 2 f*U)l)tcl)c t5 a Square nombcr:anD i)ati) 
I f*fo;t bts roote. 

SDijc fourtlje example is. I o),^—^ — s.if> . lul)crc Tbefourthe 
31 take firfte tbc talfc of.8. tbat w.4.anD it tiT^quaic ex4mple. 
giactb. 1 6. tobicbc 31 aODc to* i o ^, ano ti)crc amoun- 
tetb*i2hljepnga^quarenoml;er,an:itI)crootcofit 
] i*V)nto lubicbe | il^aU anDe. 4 * fo; balfc tbe nombi r 
of rootes : ano fo tbcre nfctb. 1 5- . as the rootc tljat 31 
feUefo^aiiD to app;joue it3( take.S.tuncg. i j. lyljici)? Thepmfe, 
is* 1 2 o.anD aDDc it tjnto. i o 7. anD fo commctb. 2 2 j. 
/o; tbc fquaccano tbc rootc of it is. i s* 

aaaUcr, Cbe like o;^r of Uio;kc fljall pou t)fr,iu Other/or^ 
all nombcrsCo/z^fCompouiiDc, lube any. 2. nomUcrs »»"»»/% 
luitb immediate Dcnoininatios GJ^tke > arc cqiiallc to forte, 
one of tbe nertc Dcnommation, m o;Dcr abouc tbcm. 

HB.i.c^.tfl equallc to.^.gp*.— f— i o.ip^. 

anDagain.i.y5>.eniiallcto.6. 5^^ §— 1—4 o,cZ* 

ilikeUiaie0. i.g^c£ .equallc to. j./i — ! — 2 8. ^ ^> 
But in al tbcfc tbe roote fl^al bearc name of tbe grca» 
terquantte. 

^cbolar. llBp tbe fo;mcr o;Der of tuo;hc,35 Hjall in Thefi>iU 
tbe firOe of tbcfe.5.erampleB, take balfc.:^, (btcaufe tt example 
i9 tbe nomber of tbe miODell quantitc), ano tbat 13 ^ . 
ano tbat ttjall 31 multtpUc fquarclp, anD fo Icill tbere 
rife i,t)nto Uibicbc 31 ftall aoDc i o 0; f . ano tbat ina^ 
betb^'.lDbicbc igafquare nomber,anD bi0 roote is ^. 
tjnto Uibtcbc 31 muft put tbe firftc balfe,tbat is i, ano 
tben bill it be t>o;i els.^tobicbc is tbe Cuhke roote of 
tbatnombcr.^.p— ] — i o. ^.bcpng equallc to ic€ 

foi p^oofc Ujbereof,3i muittplie.7. Cuhil^elj, ano it ihproofe. 
maketb. 1 2 y. SCbcn ooe 3; raultiplie it fquarclp,anD it 
Urtllbe.25'.j]ioUj.^j^.is.7 y.ano.i o.ts> . maUctbo' o 
tobicbc botbc aooeo to0etbcr,gmc. 1 2 ^ 

in 



TbfficMJe 
txumfU, 



Thejrte 



Theproofe, 



7%9thiri« 
example. 



3;n t\it feconuc erampte, tol^cre. lA*. ifi tqfinlh to 
^*?r?r—'^~*4 o» c€ . 31 ftiaU tabe ljalfe,6. (Ui^ufte 
IS iDe nomber of tf)c miDOeU quantitie ) and tftat ».?• 
anu tl)e fquare of It f0.9.U)Wclie3( mua aDOe tjtito 4 o 
anD tijcwof commetl),4 9» Uibtci^e 10 a (quatt nombcr 
auo batt,7.fo;{l)is roote,l)itto U)l)tclje3| aODe 6,anDfo 
^aue 31 1 o fonbe Sur/o/ii/f roote, of 6 ?^?^-— |---4ccP 

ano foj p;joofc 3 faie,if. i o. bee tl)e roote , tljc n li 

1 o o^tfje fquare,!. i o o o.tl^e W*,tljc ^5^ 15 loooo* 
0nD tlje Stirfdide, I o o o o o» tKH fterfo;e;6;^ x^ mate 
6 o o o o. anD.4 o,cC.^elOc«4 o o o ©♦ aiiD botfte tijei 
togctfjcr Hoc maljc» i o o o o ©♦ lul)Ul)e 10 tlic qiiant<^ 

tic of tl)C5ttr/o//Vf. 

3n tic tft troe eramplc* i. ^ cf ♦ fg cqualle to. ^A-♦ 

ir."^\^ ^ ♦ 5" J* • ^ft'Jff ^<ri:^cuhHe roote,3I ftlie wi 
tqis ro;^tc* 

i^irde g tabe lialfe.?(a0 tlic nombcr of tftc miODcU 

quantitte)tl)at is i,f tbat maftctlj in fquare i.toW^i 

31 aUD: tjnto 2 8(tl5at mabctb -*) t it pcloctb^i UibicN 

10 a fqHare nombcr,anD bi0 roote 10 v. Ijnto toljicljc 3| 

aODc ^-,ano it Voill bs ^,0^7. tofticlie 10 m ^n^ijcubike 

roote tjnto tbe fo^cfaio nombcr. ?/5— -^— 2 8^?^ ?^ 

tlfpmfe. fo} p^oofc tobcrcof 31 multipltc. 7 . s^tnxicuhihh, 

anD It mabf tb 1 1 7 6 4 9» SLbcn muft tbcA>. be 16807 

anD.3.y^.jo42i. agarntbc^j^.i0.2 4ohanDrb 

2 8»5>^«ll)aUbee.672 2S.janDtl)ore botbe together 
telDe^ii7649. 

fatter, j^et one otber fo;me ^0 tbere, tbat fn ail 
tfc(ngc0. faue (n one pointfe oncIp:fo«otDctb tbcfame 
rule, fl nD tbat (0 lube tbe 3 Dtnominat(on0 Doe not go 
ImmcDiatl? togetbcr,butret arc cquallp DIffante. 00 
S'S'* Sr. anD.f.UjbcretbeDiflaunfewoneonclp 
quantitle. 3LikcUjaic0.^ cC.ce. anD.f. UibicbeDif^ 
fcrbp.2.quantittc0. ;an^inUbero^te,(t (Z-f^mu 
^r^.are Dittante br.^quantitic0. :3nD fo of otbef,boto 
manpfocuerbec omittcD, fotbat tbe Difference bee 

equalle 



Mhirdc 



ofCo/?ikenomhers. 

cquallc* 3!n all iotjicftc rou fljall luo;bc,aig pou d(d In 
tl)c fo.imer rule , till pou baiie caiiDcD all tijat too^Uc. 
13ut tbcn bauc vou t)crc,onc t!)ing mo;ie to bet contv 
rcD.j^ojtbc laftcnombcr.tuljtfljcv'ou IjaucfouiiDcio 
not tljc roote,but a rootcD quantitic: 3no ftifi rootc 10 
tt)C rootc t^at vou fcUe fo;, 

&cl)olar. "Oiit vou nicanc tftc fquarc rootc of tljat 
quantttic,o;ifomcotljcr:' 

C?7aftcr. 3it mate be anp UinDc of rootcfn oiuerfc 
numbers, but not in one noniber.^Clbcrfo^c fo;pour 
ccrtctnttc marUc tl)is rule. 

3;f tbe Dcnominaticns of vour nombcrs,Difier onc^^ 
ip br onctljen is ita fquarc nobcr,tl)atpou Doe fintJt 
bp tt)c pzartifc of tl?? lalle rule. ^nD tl)erfo?c fljall f ou 
ud{c l)is fquave coote,fo;i tl)c roote of pour nomber. 

13 ut If tl)cocnomination Differ bp. 2 . quantities, 
tlicn fi}aU pou crtrartc a Cubil^e roote, outof pour lattc 
nomber. 0nD if tl)e Dittauncc bee. 3. quantities, tl)c 
rootc muft beca ;<?n^';<fw;<?^f rootetanD fo^4»quan^ 
titles Dittantc,aS«r/o//</? roote, anD fo fo;tl)e. 

iasfo;eramplc.i.^5^,isequallcto.8 0.^ — | — , Mexm^U 
200 oA r>oU) fo; to flnDc tbc rootc of.8 o.^ — f— . 
200 o.f .*3P U)o;Ue tl)us.i?irftc g taUc tbe balfe of 8 o. 
(bicaufr it istbe nomber of ttjcmiDDlc quantitie)anD 
tljat balfc is. 4 o . lubicbc 3i multtplic ^quaiT,anD it 
maket!) . 1 6 o o . to it 35 aDDe .2000. and it luill bee 
?6 o o.U)bicl)c \^ a fquarc nomber,f .6 o.is Ijis roote: 
to tbat .60.3 fljall aDDe tbe fo;efaieD. 4 o. anD t\)tn 
iuill It bee. i o c. U)l)icl)c nomber in the firde rulcljaD 
been tl)c true roote. 15ut ftere conCDcrpng tbc Dittacc 
is of one quantitic,3! in»llc crtrartc bis fquarc rootc, 
lubicbe is . i o . anD tbat is tbc ^n:i^:^n:^ks rootc, 
tbat mp nomber e ontainctb. 

3n otbcr cramplc. i. ^^ a: .is cquallc to.4 o o.c^. Thefecmde 
— j — )- 7 5 4 4.f .1 tahc 2 o o.fo; tbc balfe cf tbe miD« example, 
DcIl quantities nomber, anD multiplipngr it fquarc, 31 

Cc.i. fiitc 



The Me 

fiiiBc. 40000. hW^z 31 put to. y 7 5 4 4» mxn tfjcn 3 
!)au0.97 5 4 4» iufticbe is a g>quarc nombcr , aito \m 
rootc ts 5 1 2 tjiito Ujbiclje | (^all aODc tfte I)alfc of 400 
ano fo tuill it bet* s 1 2. ISut itotu mull 3! ta'ac tbc 0<' 
^/^« roote of tbifi nombccCtljat 10.8X0^ mp roote^tbat 
31 oedrc: Btcaufe tbe Denommations in t^e nombcr. 
Differ bp, 2. quantities. 

^cjjolar. 3ireet)erp toell tljeo;«icr of tbis iuo^ke: 
0nD tbe pjoofe is in like fo;te,tDl)icbc 3i maic pjattifc 
b? mp felf atanp tpme. ^berfo?e 3J p;aie pou, p^o- 
ceDe fojtbe to otber rules. 

The feconde ^^^^^ ♦ 2Dft^3 j^ fumciertte fo;j tbc firac fo;te. 

fort of Jl I ^^^ ^^^ ^^^ feconoe ro;te , in nombcrs tiiminute 0? n^ 

^Cr, J*'^^^^'^^^^^'?r*^^ equalleto.f ^^^.tbc fo;me 

of Uio;fee is libe tnto tbe otber, m all pomctes faue in 
one» jFoa in tttue of tbe laae aooftion, pou l^all life in 

SxampU. tft^f^ rtombers , Subtraction, as bere fo; erample, 

iuben3i faie. I. g^. is equalle to. 60.^. .4.2^ .to 

finoe tbe roote, firae 31 tafee tbe bactfeof.4. (bicaufe 
it is tbe nomber of tbe miDDell figne ) ano tbat balfe 
b9:^nq. 2.t)octb make in rquare.4. tubicbe 3J put to 60 
anD fo is it.6 4* a fquare nomber, ano batb,8. fo; bis 
roote.i^rom iDhicDt rootc(fap t\ic o;Dcr of tbis rule;3| 
muii abate. 2 . tbat is ttit balfe of tbe firlte nomber of 
rootes. 0nD tbcn tuill tbere rcmaine. 6. fo; tbe tjeric 
roote of.6o.<5>. — — 4.2^.bcpng equalle to. i. ^. 

'ike^mfe, ^cbolar. a^bat is fone p;oueo. i^o;.6. hcti^uQ tbe 
roote,tbcn.4.r3 .mafeetb»2 4. tubicbe bcpng abatett 
out of. 6 o. leauetb ? 6 ano tbat is tbe iuttc fquare tjn^ 
to. 6. as tbe equation fatetb* 

Thfeeonde Rafter. 0n Otber erample istbi5.!.A>.isequall 

example. to. 1 6 2.ct* ♦ 9* 5* ^* 

fecbolar. SCbatcan 31 Ujoo;ke,tlms:i^irffc 3P fahe 
tbe balfe of. 9. (bicaufe it is tbe nomber of tbe miDOell 
figne)anD it iB |, lubiclje 3 multtplie fquarelp,ano it 
toill be ^,tbatmua bee aD:)2D toi 6 2.o;*-\' , ano tbcn 

tuill 



ofCoJ^ih nomhers, 

U»tll tl}crc amouute '-^ . tuljfcfcc is a ^tjuarc nctnber» 
aiiD !)atf) fo; W rootc ',- out of tutjubc, tr tbis rulr,H 
tnutt abate -}» ano tbfn rifctfj -/ , tbat \b>9, li-I^ubc is 

tlje tjcrt roote to 1 6 2,cc 9'&' y* bci "0 cquaU 

to.i/§-. 

janD fo; tljC p;oofc,3: multiplic. 9 . Cuhilsjlyy ano if Thetmfi, 
giuctb.72 9.fotUt.i6 2.cf..DociiTaUe.i ihopS.out 
of tubicie 3 muft abatc.9.^ 5-. tljat is.s 9049. (Ip 
tbcfamc rootcfitl). \^i\ ifi. 6y6 1). i3not!)cn ItiU 
tt)crc rcmainc. f 9 o 4 9. Uibicbc is tl?c luftc quaiuttic 
of.i./?^. 

tBafier. mi one crainplc mo;c fl^all I'ou Ijauc cf Tt^f /^Wr 
a tl)irDc ro;te. ex4mp/e. 

mijcn.i^criftcqualleto.27f45'6.f 16 ct 

3 DcmaunDe of t'ou,U)bat is tbe tjaleUjc or. i.^^:* 

^cl)olar. 3i fearcbc it thus . SSljc nombci of tlje 
miooell figne ts.2 6.tDljorel)aIfc 3; muft taUcajiD fa a 
nmltipltc It fquarr lp,anD tbcic lutU rtfc 1 6 9. Uibtcbc 
3aDDeto,27y4^6.anDithjiUbcc, 27^6 2 5^.UjI)1c!)c 
iB a fquare nombcr,anD tjatl) fo; bis rootc. s 2 7. from 
tubicbe nomber 31 miiC abate talfc tbc noi«bcr,of tl)c 
jiTiDDcII Cgnc -, tbat is . 1 5. cxn fo tbc re luill rcmaiitc 
f 1 2.trl)of£ Cubi\e roctc 3: jrr.ll crtiartjbiranfc t\]c Dc* 
noininaticns Differ tr. 2 . cuai-tittts, ano tljnt rootc 
totll bc.SMt>i;!fl)C totlir Ct^i/^ffrcotcto.f 1 2.biU to tbc 
iionTbcrp;oFci?nt!tt!,itistl;c;^f»;^/f«('/^f rcotr. 

SnCcr. s:i)is ts tncugbc io;tI)c luo.zUc of tbc fc^ 
foncc fo:tc. jj^olu ft\: tqc tbuDc fo^te of equation, Thethirde 

tuljerc. ;-.ts cqiiallf 10 ::^. ,f .3 luill gme vouforteofe^ual 

a bjtcf aomonttfon cnelp,[l50iigb itDifferfrom botbc mmhcn, 
tbectber. :. rulcs,tnfo;nieofiuuoo;iUc. ifD;actl)cc? 
qualitie maic be in Diiicrfc fortes , fo fontr ipmcs rou 
ntaic Ijfc tbc Uioo:be cf tbc firtlc fo;tc, bp flDtiition of 
balfc tbe nomber cf tbc miCDle Ggnc: anD feme times 
fou G^tdl U)o;be hv fubtractton. Wi berciu thw is the 
Otffercnecfrom tlje fceonDc rulc.SCbattbtrc rcu doc 

Ccc^^, fubtrattc 



The Jrte 

fubtiacte Ijalfe tlje nomUcr of tbe miDOeH fc'ijii:^, from 
tl)crootcUjl3ict)et?oufonDc. 0nDm tbistotrocriilc, 
^ou fljaU fubtrarte t\)t roote from tiyt l)alfe , ano not 
tbe Ijalfe from tlje rootc ♦ jf o^bicaufe tl;at tijat roote, 
is euei- Icffer tben ti)at l)aUc. 

0nD in tW tule , tbis ts fpcciallp to bcc obfcrueo: 
tbat tiie Square of ftalfc i\)t nombti* , of iht miDDcU 
figne,tmll euer moae bee greater, t\m\ tijc nombcr of 
tbe leCTcr figne : 0nD tberfo^e H^all tbc nomb£r of t?)c 
leffcr Cgne, bee abateo out of tbat fqiiarc. anD tbe re 
mamer UjUI bee a Square nombcr, tuirb M)\t\)t v*ou 
fijall tooo;ke,as 3 l)aiic taugljt von bcfo:c. 

ano faitber in tbts rule, it is commonly fecn, tljat 
cuerv foclje equalle nomber, batb. 2 . tjaluation-j fo j 
I)isroote,3!meanetl)atani?oftl)ofe.2.nombcrs,U)iU 
bee as tbe roote in tbis equation* j^oj otberluaies no 
ncmbef ran baue.i.rootescf one denomination. 

^cbolar. BitjnoerCanoei'outbus* Cbatncnom^ 
bercan i)aue.2,fquarerootes,o.:.2, c«i/t^rootes,anD 
fo f02tbe: €'ls onenomber maie baue.3.02.4. rootcs. 
^s,6 4.batb«8.fo? bis s>quarc roote:4» fo; bis Cuhiks 
roote:ano.2.fo; bis ^n^^cuhil^e roote* 

^r^aflcr, pm tatje ittueK* ^no fartbcr fo: tbc 
eafteiinotuIr5geoftbofe»2.nombcrs,o^rootcs:s:bet 
mult bee foebc,as beerng aDOcD togctbcr, tuiU n!ake 
tbe nober of tbe mtODrll fignctamj beeimg mtiltiplteD 
togetber;,lDilmaUetbe nombtr of tbe Icaft ftgne. :3nD 
fo mate lou finoe tbeim Uiitbout fartber multiplica? 

rhefir/Ie tion,o;icrtraction of rootes. 

example. i^o^ej-ampIe,31 fettcfirfte.i.g-. equalle to. 1 6>Xf . 

^ .6 3.f .tobere Bi maie efpie qui£Ucl^,tbat,6 ^ca 

bauenomo;epartestobi3compo(itton,but.?.7.9'2i 
anoifjtafee* 3. ano. 2 1. tbcntbctraODttton toiUbee 
greater tbfiT» 1 6. but 7. anD9.malJctb tulle i6.br aD' 
£ttton,anD.6 ^.fcymuIttplicaticiT, ^tUOtbcrefo^ctbet 
foallbetbe.2.rootcs. 

^rbolar* 



of Coph nomhers, 

^cbolar. 3; U)illpjouetbatbperam(iTatioit,tl)U0. 
jf.7,bc tl)c rootctbcn (6.4 9.tl)c fquarc. aiiD. 1 6.::£^ 
maKc, 1 1 2. out of iul)icl)c3i muaaUatc.6 ?. anDtbcrc 
rcffctl).4 9,cp.iuiU2 Uiitb tl)e ^quare:fo istljatatruc 
rootc. IC!)f n fo;.9:l)i3 fqunrc Ij5.8 i. 0nD, 1 6.::^. t»oc 
pclDc 1 4 4 fro U)l)icl)e ij %\\ abate 6 3« ^"0 tljc remain 
ncr luill bc.S i.cquailc to tlje fquarc»anO fo w tbat al 
fo a true root?. 

q^aacr. j>olu U)o^Ucttbptl)cotbcrrulcj5,tbat31 
taugl)tvou. 

&cl)Dlnr. i^u'ftciitabc. 8 . as Ijalfc tbc nombcr of 
tijc miDDfllfi«nc,anDtt)atinultipUcti Square, Doetl) 
giuc6 4from lul)icl)C I C}allabatc6 ;anDil)cn Doctl) 
tberc vematn but. i.U)Bici)c is coumptcD ar. a ^jquarc 
nombcr,anD bts lootc to bc:« u alfo,U)l)kl)c if 31 aDDc 
to. 8. It luiU maUc.9.tI)at is one of tbcroctestaiiD if a 
abate it from . 8 . it tuill leaue. 7 ♦ U)t:if Ijc ts tgc otbcr 
rootc. 0nD ti)U0 31 fee one luo;Ue r66rmetl) tbe otber, 

£l9after. 2:aUe tbis fo? tljc fcconoe craple. i p cC Thefeconde 
10 equalle to.S./5-.-— — ♦ ^ ^'5^ ^.lubat 13 tV,t roote f;,^^„^/<.. 
faiei'ouf 

^cbolar. "Eo fiiiDc it,firac 3: loljc fo; tbe partes of 
12. anDtbcibe.2.^.4.6.ofH)l)rcl);^.2.anD.6.Docrcruc 
m^ purpoff/o: ti>cir aonmon maUctb»8.anD fo DoetO 
itot.5.anD.4. cai)erefo;e3:faie,tbat.2»maicbectbe 
roote, anufo mate. 6. l5utfo; fartbertrialleofit:31 
luooiUe It b\> tbe otber rule, fait'ng balfe. 8.i0.4'anD 
bis fquare 10. 1 6.#rom lubicbe 3 abate. 1 2.anD tberc 
remati!ct!).4.U)bofc roote i3.2.tbat 31 maieaDDcto.4 
m^ fo baue ^.6.fo;; one roote: 0; cl0 abating it from 
4.31 (ball baiie.2.fonbc Dtber roote. 

SLho ojoofe 13 manifcae fo:. 6. bce^mg a roote,tbe ^^^^ .^^^c^^ 
:^e»;^'f«^ei3.466S'6.2Dbe5«i:yo/*WtftG.7776.i!nOtbc ^ ^ * 
^n:^:^n:^{e 13 1 2 9 6. &0 tbat 8.^^.t)0C malje622oS 
anD.i2.y^.are.i)'^^2.1ubicbcbemgabatcDoutof 
6 2 2 o 8 00 kmz 4 6 6 J 6. tbe true quantitie Qt\^<t 

QtA\: ana 



The Mtt 

^ototf.2»beftttefo;arcote;tl;cniBt^jt.?>j>.i6. 
tbc/^*5 2,anD tbe.^ c^ 6 4.^ntJ fo aa\8./£.,?quaU 
to. 2^6. 0nD. 1 2, ^ §-♦ pclDc, 1 9 2. m bcr fo;e at ating 
1 9 2.out of.2 f 6.iijue teftetl),6 4:» tlje m0e t;uaiititie 
cf« ^S^c£* ;anD fo 10 tljat Icoo^be alfc ficoD,ano.2.« 
true coote* 
rhethirJc i^after. i]iotu pjoue tljijj tljirfie erainple, libcre 

txamfle, l^h/yiB CQUalle t0»2 o o o*^ §3 4 7 o o 1 6 1^ » 

|»c!)oIar. f^alfctfjenomueroftljemiUDcUOgnuj 
looo.aiiDtljefquareofitw. 1 000000. j^rctiiluW^ 
cfeej fljall abate .470016. aiiD tibere lutU retwame 
j29984'iuibofe fquare rootc bp trtalle of ertramoit, 
3 ftute to bc7 2 S.iufttcftc 3 mate otbcr aODe to.i o o o 
ano fo tibf re rifctb. 1728. lubicbe 3 finoe to bee ( 00 it 
ougl)t)a W/'^e uomber.anD Ut5 roote to bc» 1 2. 

iSutano if 31 abate7 2 8.from i o o c,tf)erc Juill re^ 
tiTatiT.2 7 2.iDbicfte ijs no C«t//tf noanbcr. 

spatter. ^0 tbat ftere femetb to be but one roote. 
jano?ettWe.2.notttber0.i72 8.ano.272.feepefocI)e 
a rate , tl)at bcc^ng multipltco together , tftei nrabe 
47 o o 1 6. tDi)ir6e t£( one of tbe nomb£rB,anD bcttng 
aotictj together, tfjei mabe 200 o. luljif fjc is tfte otber 
nombcr of tljefame Co^ke reftdtulU, 

asut nofe p^oue in otfter libe n6berg,XDl)icfte Ijaue 

fome mdaunf e, betlucne tbeir ficnominattonj5,U)be;f 

The fturtbt tl^cr it Vaill fo Ijappcn fiiU. as namelp in tljis, Uifjcre 

ex4mpU, I , Vp. i£i equalle to. 1 2. §^ §^. ; 2. Xo . 

^cl)oIar. li^alfe.i 2.t0.6.ano bis Square. :6.from 
tubictje abating. 5 2.tljere is Iefte,4.lDbofe rcctete.2 
0no if 31 aDDc tt?at 2. t8.6. it mafectb. 8. Uj b tr be t0 a Cm 
hkfmmhtr^mn batb.2.fc? bi0 roote. 3iE wi if ^ abate 
2.from.6.tbcre rcniainctb.4. tubicbe is no c«H<f no^ 
ber,ano tberfo:e batb no focbe roote, 0nD ret tbefe.2. 
nomber0.4,anD.8.bratimttcn,nTabetbemiDDfUn6^ 
ber,anO hv multiprica(ion,tbei maUc tbe la0e ncfaer. 

©atter. 



ofCofiih Honibers. 

fatter* ^^;oue pet ones againc in a nombcr, Tbeji/te 
tD^rc one quatrtttte onelp is otmtteD, 0s iubcn i/^> txam^lc, 
tee«iHallcto,2 4.cf.» 1 5T*^» 

^^olar. I z.maUetb in Tquarc* 1 4 4. from U)b(cbe 
Bfi^alloeDuctcHf* anDtlt)C^nre(tetb.9.U)l)orcrquar0 
ro8tetd.^U)l)tcbetf3j aDDe to.i 2. it tptUbee. i ^ attD. 
^at^ no fquare roote,a0 })ere fs rcqufreo* ^ut if Bl a^ 
bate.^from i z.tben rematnetb 9 Uibofc fquare roote 
w. 3» ano feructi) to tl)e nomber , as 3l Ijaue berc p;o^ 
ucD in mp tables. :3nD. 9* ano. i ^ bepe tbe cudoma^ 
ble rate, jFo;i bp aODitlon tbci make. 2 4. ^m bp mul* 
tiplication,tbci pcloc. i ] s* 

jl5ut in all tbcfe eramples^tobcre tbe Dcnomfnati^ 
ons be are a oiSaunce, 5 can fiinDe but one roote,ano 
not.2. as it tuas in tbe otber eraples of tbefame rule. 

0n3 in fome of tbcim , tbe greater nomber contain 
netb tbe roote : but in otber , t\)c leflTer nomber batb 
tbe roote* 

ipaHer. Bicaufeglcannotttaicnotu, about tbis 
tarietie,3J toill remitteittill an otber tpme-Buttbis 
bp tbe t3Daie,3! mu\l aomonilbe pou^tbat 31 ooe folotue 
bere, tbe common fo^mc of ioiitcts , in calling tbefe 
rootcs,tbat rife in equatio, lobere as tbei are not tbe 
rootes of tbofe nombers,but are tbe balue of a roote, 
if 02 of a OA'^' nomber , tbe roote mull neaoes bee a 
6//t* nomber alfo. anDfocbe asbp multiplication 
totU make tbe rooteb nomber: But fo can not tbofe 
KombersDoe* 

0nb bere toill 31 mabe an eanDe,of tbe tDo;ifec^ 

QtCoffiie ombers. anD noU) toill 31 ap^ 

plie tbem to pjaaife in tbe rule 

titejuMti(m,tl)nt iB Com> 

monlpcaUeD^/' 
fibers rule* 

CCbe 



The rule of 
equation. 




The Jrte 
The rule ofequation,common* 

Ij called Mgehrsiule, 

rcrtf rtD Ijauc 31 taugljtc ^ou , tijc 
I rommoit formes of luo;lic,m nom^ 
Jjbcrs ^emmmate, tI2l tjifbc rules arc 
pfeu alfo tn nobcrs MJlrstte, q iifee* 
ijUjaics in Surde nonitjcrs. ^^Itfjougfj 
pe fo;:mes of tijrfc Ujo^Ut s be fcue? 

.>_J|rane,tiTcfl)elimDcofiiombcr.il5ut 

no^ lutil 3f teaclje ^ou tbat rnlctljat is tljc p^mclpall 
til Lojitke iuoo;Ucs : ano fo^ iufticfjc all tljc ctftc r oooc 
fiJrue, 

SEliis^ulc IS rallcDtftc WuXtttJlgeler, after tf)c 
name of ttje muentoure, as fomc men rljinUe: o: hv a 
namcofrmgularcrcellencie,asot!}criuDgc, l3ntof 
l)is life it IS rigbtlp ealleD,tljc rule tit equation: bieaufe 
tbat b^ f5/tttf^/o» of nombers , it Doc t!) DiOfol we ooubte* 

full quertionstann tnfoloe imricatc rioles* j^no this 
is tfte o?uer of it. 

Tbefmme of the rule ofe^uatm: 

Hen any ^ue/lion hpropouded^ 
appertemyng to this rule ^you 
/hallimagm a name for the nom* 
ber^that is to heejou^hte^asyou 
.-.===:,^.,^.... remember , that you learned in 
the rule offaljepofttion^jndlptth that nomber 
frallyouprocede,accordyn^tothequeJlion^Vntil 
youfinde aCoJitke nomber^equalletothat nom^ 
ber,thattheque/lionexpreflethMicheyou/hal 

reduce 




BfCoJiikt nomhers, 
reduce euermore to the Jeajle nomhers, ^/{?uUhen 
Jiuidethenotnberof the leffer denomiuationfy 
the nomher of the greatejlt denominatiojij and 
the quotient doeth aunfipere to the quejlton Ex* 
cept thegreaterderiominatiojdoe bear e the figyie 
of fome rooted no her. For thenmujl youextraSt 
the roote of that quotiente , accordyng to that 
figneof denomination, 

s^cljolai'. BCt fcmctl) tljattljis rule is all one, tuttb 
tl)c rule of falfe pofition: ano tljerefoje migbte fo bee 
raIlcD:rei?ngittaUetl) afalfe n6bcr,to luo^bcUjitbal. 

£9aftcr» Cljts rule Doctb farre erceU tbat otfter* 
0nb Duoett) not taUe a falfe nomfaer, but a true nomi^ 
berfo^ W pofition , as it l^all bee DeclareD anon. 
Mberbp tt ntaic bee tljougfjte, to bee a rule of iuon^ 
terfull inuention, tljat teacbet!) a mannc at tbe firfte 
luo^De,to name a true nomber,befoje be Unotuctb rcs 
folutelp,tDbat be batb nameD. 

lirut bicaufe tbat name is common to manp nom-- 
bers (altbougb not in one que(lion;anD tbcrcfojc tbe 
name is obffure,till tbe U)o;be Doe Detettt it, 3 tbinbc 
tbis rule migbt Icell bee c alleD,tbe rule of Darke pofi^ 
tton,o; of ftraunge pofition: but not of falfe pofition. 

0nD tQi tbe mo;c eafie anD apte tuo^be in tbis arte 
tuee Dooc c ommonlp name tbat Darfee pofition. i. 2^. 
0nD luitb It Doe Uie U^oiUe^as tbequeftion intenDctb, 

till toe come to the equation. 

Cbis rule ote^Mthn^ DiuiDeD b^ fome mtn^ into rbe barta 
Diuerfc partes, as namclp Scbeuhelius oooetb mabe. ], ^ffy^. mU. 
rulesofit.0nDintbefeconDerule,beputtctb.^feiie« 
ralle cannos. &ome otber men mabe a greater nobec 
of Diftinrtios in tbts rule . 115ut 3} intcnDc(as | tbinbe 
htUt fo? tbis treaties , tobicbe maie feme as farre 

ftu as 



The Jrte 
a0 t\m ixioMa fioe tvttntic ) to minac it oirelp Into 

tUjOO parted* ^^ereof tbc firfte is, y»hen onenomberis 
equalleVnto one other, 3nD tlje feconDC is,T>i>r» one nom^ 
htris compared as equalle lfnto,2,othernombers, 

aiUiai0£t iuiUpng pou to tcmeber, tbat pou reDucc 
tournombew , totbetcleafte Oenominatioiw , anD 
fmallefte fo?mes,bcfo?e )?ou p^oceoe ani? farther, 

ano agairt,if ?our equation be focbc, tbat tbe grea^ 
telle Denomtnattoit G/?%, be ioineo to anp parte of a 
compouiioe nomber , pou fl^all tourne tt fo , tijat tbc 
nomberoftbegreatette Cgne alone, maieaanoeaa 
equalletotberede* 

ano tbts 10 all tbat neaoetb to be taugbte , concer^ 
nfingtbtstuoo^be* 

!!)otDbeit,fo? eafie alteratio of equatiom.j lutll p;o^ 
pounoe a fetuc eraples, btcaufe tbe ertrattion of tbeir 
roote0,mate tbe mo^c aptli? bee U);ougt)te, 0nU to a* 
uoiDetbeteOtoufe repetition of tbefe iuoo;jDes : ise^ 
qualle to : 31 luill fette as 3 noe often in luoo;Ue t3re,a 
paire of paraUele0,o^ d^emotue lines of one lengtbc, 
tbu0:=— =,bicaufe noe.2. tbpnges,can be moare 
equalle, 0no noUi mar^ tbefe nombers. 

3» 26, J— h-I oJ^=-=9.5.. 1 oiLo^~4— 2 1 )\f 

4» 1 9»2^~l — 1 9 2,f ==i o 5-~f— lo8f 1 9 2g^ 

5. 1 8*2£^-+-2 4.f ^^-=^ 8.5^.--{— 2.^. 

6. 345, i22g^-.=4o2g^— f— 480? — 9,5. 

!♦ 3}n tfie firffe tbere appearetlj* 2 ♦ nombers , tbat is 

1 4,i^ ♦ 



ofCofsih nombers. 

1 4,::^. — ( — I j-.f . cqualle to one nomfaer,tu!)icl)c ia 
7 1 .«^7i5ut if rou marUe tljem tocU, pou maie fee one 
tjenominatio, on botlje fiocs of t!)C rf«4^/ow,lu!)ic!) nc' 
uer ougljtto aanD.tSHljerfoje abating tl)e Ieffer,tl)at 
t3j ^'J^.out of botI}e tbe nombers,tftere \joiil remain, 
1 4.2^ — =5-6.^ .tl)at t0,b^ reourtion,! t;0^— 4.f ♦ 
^cjolar, 3 fee, pou abate. i f .f .from tt)tm botbe. 
i3nD tben are tljci eqiialle ftiU,fepng tbci Ujer equalle 
bcfo;e. acco^Dpng to tl^e tbiroc common fentence, in 
tbepattbcluaic: 

Ijfyou abate euen portions ,from thynges that het equalle, 
the partes that remain fhall be equall aljo, 

cpafter. ^ou Doe tuell reme ber, tbe firffe groun^ 
t)C6 of this arte, jpo; all fpjingetb of tbofe p2inciplc0 
Geometricalle.'m berfo^e call to pour mtnbe Ufeetuates 
tbe feccnOe common fci.tcnccin tbefamc booUe, anD 
tbcn baue pou anotber reafon,Ujbicbe iuiU belpe pou 
not onclp,in tljt otber fo^imcs of U)oo;jUe bcre,but a^ 
fo ticrp often m tbe jp^attifc of tbis arte. 

^cbolar. S^batistbis. 

Jfyouadde equalle portions Jo thynges that hee equalle j 
tfihatfo amounteth ofthem.fhallbe e(juaUe, 

99after. :2!:befe tluoo ftntencefi Doe (nffrurte pou 
tbat U)ben p on fee on botbc tbe fiDe0 of tbe equation^^t 
nv one Dcnommatiu Copks,'^m l^U marUe tbe fignc 
tbat is anncreD to tbe IcITer of tbem botbeianD if it be 
t\)t ftgnc of aDDtiion.— \ — .tben (ball pou abate tbat 
IcfTcrnoiuiicr, front botbc tbe partes of tbe equation, 
jasjctuuubisfirffierample. 5I3utiftbe Dgnebeof 

a batcmc n te — ,tbcn (ball pou aDDe tbat leflTer no^ 

bcr,to botbe partes. anD fo (ball pou Doe,till tbcrc be 
noe one Denomination on botbe partes , butDinerfe 
anD DilTinftc. 

^0 tbe feconDc nomber inill be.2 0.2^— =-i 2 o^ 
anD m tbeUattetermes.i.^ -— =.6.f . 

^cbolar. 3i fee tbat ^ou abbe. 1 8. f . to botbe par^ 

i^f.y. tcs 



The jfrte 

tc0 of tlje equattom IBixt h^ tbat rcafon , 3; Doubtc in 
tl)e tbiroe fomme,b!caufe. i o, xp , is tn bot^e partes 
of tlje equation: in tbe firtte parte tuttfj —f— ,anD in 

tbe feconoe parte iDitb , toljctljci: 3 l^ail aotie 

1 0.2^,0; abate ttjem* 

$0aaer» Jn focfte a cafe , pou ma(c Dooe citbcr of 
botl)e,at rour Ubecttc:anD all Uiill be to one eanDe. 

^cljolac* 3lf3iaODe«i o.2p. tbeiTU)illitbe,26.v» 
— f— 2 asg^^zzz:. 9 5-*-4— 2 1 3.f "^ 

S0aftcr, 0no Doe pou not fee. 5-. on botljc fiocs of 
tbe equations 

^cbolar. 3| Dto lokc h\xt foj one alteration onelp. 

spatter. 31f tbcre luere tiuentie libe Ocnominati^ 
ons,i?ou (^oulD alter tbem alt ft>i tbat 10 tbe p;inci* 
palle ano pcculiare ceDurtton,tbat belongetb to cqua- 
tton0» 

^cbolar* %\itn mull 3l abate. 9.5^. on botbe par^ 
te3,anoro tuilUbere remaine^i y.^. - | , 2 o.'zp ♦ 
z=:,2r5.f. "^ ^ 

Smaller* jjiotD reuucc it bp abating, i o. 2p . 

^cbolar. ^0 it luill bee ♦ 1 7 » ^ . ,21 3 . «?, 

>— .2 0.2g^. 

ano notoll remeber , tbat tbi0 fe tbe better fo^me 
ofreOumomBuaufe tbe greater Denomtnation,tbat 
i0,§-, is alone toitb Ins nombcr on tbe one fine of tbe 
equation, ano tbe* 2* IcITer Denominations , on tbe 0^^ 
tberfiDe* 

£©affer. !^oUj Doe pou reoucc tbe otber equatios, 
to tbeir rmallcftefo;jmes:' 

^cbolar* 3intbefourtbeerample,tbereis noeoe** 
nominatio, before tbe figne of equation, o;j in tbe firft 
parte , but tbe lifee i& in tbe feconoe parte alfo , after 
tbe figne of equation. Mberfo;ie firae,bicaufe ^ fee 
1 9«^» on botbe Goes, 3 luill abate it on botbe fitjes. 
janPtb enluilli tbetbus. 
1 92»f zzzzzi a§-n-'+-i o8«f— -.38.^:^ ♦ 



ofCo/?ike nombers. 

iSut bicaufe 31 fee f ♦ pet remaining on botbe pactcs, 
31 abate tlje IcCTer, tbat 10 . i o 8 ')\from botlje partes, 

anD it UjiII be.8 4*f*. i ^'S""^ * ' ^'^ * 

fatter. 5I^t)ts equation luouiDIiee better , if tbc 

greater Denomtnattoii , Dto Saitoe as one parte of tOe 

equation alone. Wi btcfje tbpng pou maie cafilp Doc, 

bp aDDpng. 5 8. 2^ . to botbe partes : bicaufe fo muf)t 

folotucti) -Ton tbc one parte. 

0nDeucrmo.:e tubenoccafionfcruetb.totranflate TrMjlation 

nombcrscompounDe, on tbe one fioe is equalle of timbers, 

to — ^— on tbc otijcr fiDc. 
^cI)oIar* Cben it tuill be tbus. 

8 4f -+- 1 S.Xo^.=. I 0.5% 
spatter. 3!t iucrc bcttct tl^usi. 

I o.§-N. ♦ 3 S*::^. -+-'S 4'^- 

0nD ni fmaller tcrmes. 

y.j-.zzrz.! 9.^5.—- h— 4^'f 
^ut nolu p^occDe luitl) tljc eramplcs. 

§>c()olar. SCOcfiftbc is eafilprcDuccUjbp abating j. 
2.tg^. on botbe fioesti^o; fo UjiU it bee. 

8.§-.=i=:z:»i6.2o.— ^ — »24.f. 

Cbe firtbe equation IdiU be,bp aDUpng. 1 2.if .on ^^ 
botbe fiucs. 5 4«p~^~^) 2.r^~-h^4 8 o.f — ^95^. 
But pet 31 muft reDuce it fartoer,bp aODpng.p.^.on 
botbe fiDcs. ^nD tben it tuill ftantic tbus. 

45.5-.i:zi=.S'2.i:£.— -i — .48o.f 

«0a!ler. jjiotu tuill 7i fbeloe pou tbe tjartettetf of r^r/rt/M */ 
equatio3,taugbt bp Scheuhliui,bimi(c pou maie per* equations: 
ceiue,boU) tbei bee conteineD in tbofe. 2. fo;mcs, na^ 
meD bp me. as fo;i tbe manpfoloe t)arieties,tbat fome 
otber Doe teacbe,3l accoumpte it but an iDlc bablpng, 
o;t(to fpeaHe moare fanourablp of tbem)an tjnnelTarp 

^.iij. mftmc* 



TheJrte 
Oifimctton. 
tbejir/ie g^^g firtt tqmtio utttt Scbeuhelitu, f after mp mea> 
9fU4tto». npng airo,ts,lufteiT one nombcr la equaU to ait otfjer* 
meaning tftat t^et bottje muft be fimple nombers Cof, 
fiksj ano tjncompounoe. 0s. 6« Xp ♦ equalle to. 1 8»f : 

4»5'*=:=:»i2.2£^. 1 4.^,zr=:.7o.5.: 

1 )/5-»=rz:»9 o.^ ^: 2 o.g-.ce=r:i 8 o./^; 

2 6.5V^»i:=. 1 1 7«c€ cC* 

3n all tftefe examples, as pou fee hnt one nomber, 
compared to an otber ; fo to finoe tbe quantitie of one 
roote, roufl^alloiuioe tbcnomberoftljelcfferCba^ 
ractcr,bv tbe nombcr of tbe greater CIjaraaer,anD fo 
fi;sll tl}C ciu9tUnt€ b?png fo?tbe tbe quantitie of. 1 . xd . 

^cbolar. 3t femetb at tbe firfte teUie^tbat it ib^ 
gainfl rearon,to Diuioe tbe nomber of tbe lelTer fiane 
bptljenomber of tbe greater. ji5ut UibensifonfiSer' 
tbat tf 31 compare a nomber of crounes,o; anv liUe Dc' 
nomination , to a nombcr of tljillrngcs in equalme 
tlje nomber of croune0,oj otfter foclje like, muft nra' 
ties be leflrer,tben tbe nober of Ojillinges. ano fo r^im» 
tJing tbe nober of tbe OnUinges (0; otber leffcr name) 
bp tlje nomber of crounes(o;j otber greater name)thc 
quottente iciU (l)cUie, ftoUi manp fljUlfnges make a 
croune : ano generally , boUi man^ of tbe leflTer, oooe 
mafee one of tbe greater. 

as if.2 o. crounes bee equalle to. i o o. mivrnts. 
tben. 5". D^tllimges Dooetb mafee a croune. &o Men 
6. te.-beeequ^Uto. 1 8.f tben.^.f Doctb mafee.i.i . 

anD.4.^. . 1 2. tp . Dooct!) f aufe tljat. ?. «9.mutt 

bearootc. ^ ? 7 "•««, 

». S^^f' M i?^ ?°"^ eramplarie p?cfc Is gooD,fo rc^ 
tmmon luill be a fufficicnte p;onfc in tbis 
^cbolar. 3 fee It manifcmF.i^o? if. i ^cf . bee e. 

qualletc.70.js t!;cn.i.rc.ts equalle to,y,g-.bptbat 

rcDuction 



ofCoj^ih nomhers. 

rctittctton in nomhtts. 0nD again bp reDuction in Uf 
gne0. i.2£^.is equalle to.^f . 

iliUeUjaics. i s-f^* bepngequallcto.9o,5>§>«re^ 
t)uctton bv figncs ano nombcrs alfo, tutU mabe 1*2^ 
«=^,.j>.^o ^aU.2 o.^cC»~.i 8 o.y§^,bereDucco 
to.i.2^.-=.9.f . ano.26.^y^»===.i o4.c£ <:£♦ 

UiiUmafec»i»::^.=-==.4' ?• 

^©aftcc. 0no fo gencrallp,lDl)en tberc w noc Oc^ 
nomination omtttetj, betlucnetljofe.z.tbat bcecom^ 
pareD in equalttie, ttill the DiutQon of tbe nombcr , of 
tlje leffcr Denomination,bt the nombcr of tbc greater 
Denommation , iuill b;pngfo;tl)e in thtiuotientcjtlyt 
quantiticof.hzp . 

But If tbcre bee anp Denominations omitt:D,bC' jhtfecondt 
ttoene tbofcz.lutjiclje be compareo togetber in t^nsLt forme of the 
litieUoiJC boUj manp Denominations areomiitcD,anD m^mo^/i'S 
fo manp in o;oer is tbe rooteD quantitie, tubofe rootc 
pou muft ertrartjfo? tbe aunfUiere to tbe queftio.i^o; 
(n focbe a cafe , eucr mo?e rou (ball ertracte tbe roote 
ofpourlaftenomber. 

as fo?crample,lubcn.6.c€«be equalle to.2 4.2£^* 
bp tbe former rulcpou fljall finDe. 4. m tbe ^uetiente, 
Wnt bere tbat. 4 ♦ is not tbe quantitie of a rootc, but 
(s a rooteD nombcr, tobofe rootc 3 (ball crtractc. anD 
fepng bettoene. c^. anD,2^. tbcre is no quantitie 0* 
mtttcD , but one, tbat is . §^ . SCbcrefo^c 3 (Ijall ac? 
coumpte.4«tbe firfte quantitie,tbat is to raie,a square 
nombcr, anD fo take W square roote, bei?ng. 2»fo;tbe 
quantitie of a roote. 

again if. 7.^. be equalle to. s 6 7»^ • tbe quotimte 
tutu bc.8 1. anD oeclarctb a ^ri:^:^n:^e nombcr,bi* 
caufc tbcre are omittcD bettucne. f^, anD. ::^. tb?ce 
nombcrs : anD ;^e»;<»<«»^'^f is tbe tbirDe quantitie: 
as von DiD Icarnc m tbe beginnpng of tbi3 trcatice,of 
nombcrs Denominate. 

^c^olar* 31 perceiue iU anD tberfo;e 31 mull taUe 

tl>e 



Thijfrte 

tlje T^n^i^^^^hs rootc of, 8 k tutiic^c in. ^. ano tbat if^ 
XMt true roote, Uil)ere,7./p. be cquaUe to. f 6 % 2p • 

Rafter. 0111) if tljore.7/§-.U)cre accopteo cquaUe 
to.y 6.5^ ♦ tfte quotiente \uiii be. 8. fliiD bif aufc tbcre 
arc omittco.2.quantitiej6,tbat iB.c^.ano.^ ^-.tber^ 
fo?e ^ou ll)all arcompte tbat.8.to be i ct*o} aieconoe 
quantttte. ano bts roote C«H' Js. 2. Uibicbe ftanoetli 
as tbe talctue of a roote,in tbe former equation. 

0nD ttt0 iiotpo(ftble tbatanv otber nonibcr,mate 
be placeo as a roote ^ in tbat equation :o;i tn m\\! otbcr 
fo^nie of tbis firffe feinoe. i^otubcit in one ro;te of e? 
quation,of tbe rceonoe fetnoe^tbere mate be.z.biuerfe 
rootesjluben one nontber batb.z.rootes in taletuc* 
00 31 taugbt vou before in tbe ertraction of rootcs. 
Thtfmnde %iit fccoD feinoc Of equatio, after Scheubelius minoc 
kjttde of anD m^ne alfo,ts,toben one fimple nomber OM^ is 
tiuatim. coinparco as equalle to.z. otber fintple nobera Cofiihs, 
of feueralle Denominations, ano Itbe oittaunce. 

0nD in focbe equation, bet ng rcDnreo as i& taugbt 

befo;e,tbc roote of tbofe.2. nombers rompounDeD,as 

(n one(o? ratber tbe tialelue tbereof)(bal be crtratfebt 

0s 3 baue before taugbte alfo* 0nD tbat roote ooctb 

aunftoere to tbe queHtom 

Thefeconde l^olubecit,bereistbelifeeobreruation, as luasin 

forme ofthet\)t(econtit fo;mt Of tbe firttekinoe. fo} iftbofe.?. 

Jccondkjnde Denominations be not immeofatc, but Doe omit fome 

otbcr bettuene tbcm,tben l^all pou ertrattc tbe roote 

of tbat latte nombcr,in alt poinctes , as ^ou did in tbe 

fir(tc equation. 

eramples of tbe firfic fojtr. 

tubtebe betng reDuccD,toiU bee: 

i.§>.— =.i.2£^. — |-— ♦ i.f ♦0nD tbe roote luil be.2 

0nD.6/5-.— .1 2.5- 5-.- [— .1 8.ce» 

SCbat 



ofCojiihnmhtrs. 

i.g>..«=-=.,y.2g^-- — .6»f.lBlioferootete»3,oi,:. 

itiUclualw. 2,^ 1 .i 2o«f«-. — 8.5g^ 

d); bpre<)U(rt6, i. g ■ . »■■■ . ^m^ oJ^- — — ^4,3^ ♦toljofc 

Cramplesof t<)c re<ottde(io;tc« 

E^t)dt ttidbct^ bp ccoucttoit. 

^tiD tlje fquare rootr*4, 

3liUcUjatc0.8^cC.==.4o.c€- — } — ♦JozoS.f. 
^; bp tlje o;Dcrlp reDuctloiu 

j.^cC=^===j.c£«— h-3776.ftDbofeO<*/t* 
tootc tl3* 4. 

jagainlnrefiDualfcisr, 

&^at maketb bp reDuctiom 

J&o.9.*/^«:=«=9 0*53 ju, — -"J 4 4,*^, 

^; br retjuctlom I.St. c€===i o^c£.— — 1 6.?. 
tDljofcroote 10.8.01.2. 

l!5«t nolo Wcaufe ScleultVm tjdoctft mafte, 2. t^pj 
raneequattonsof rtjefe.2« foiimeg : aifO0taeti).^oi^ 
tietfc rule s,o^ canon0 fo^ ccbe of tfjcm^j tbtlliKClats 
l)(0.6. ratToti5 to be altcoiitafnet) tn t^0 fcconitelitn) 
jfcQuattoti* 



^t mntttl) 1)10 Dtntfion t^u«* "mHn « i « nombct 
t0 compareD as «{tmilrto.2^tbfr,ott)eir that one no^ 
bee is of tl^fXmalLeOe deaomtnation* 3nD t^cn 10 tt of 
tiit firilc Canfitt.a35»i.§>— H — 8»3 ^-— ^ — = 6f,f«c; 
rl0 tfjfft ot!einwAer,w of tl)e grcirfufe ftenontmatro : 
-as. ^.2^. — I — 4*f=*=«»-='i.^.0aottKni»itoftt)e 
feconue Canon; £);d0tf)tri)elp, tl}? alone nombccm 
of tl)? mtt)Ole oenomtnat{6:anD tben (0 tt of tbe t^tcoc 
CanomSUi.f.g*— ^ — «J2tf> -S ^sak 

Clie UI)efo,unel)et)fet^ > CD;t()£ npmbf cdof oena^ 
mmattoif0OtSaunte* 

mtittbiS poum^ic pcrcciue, that m mp rule tijcrc 
(0 noe fo;me of nomter04tke tt)f of tbe firttc Cancn, 
notbec pet of tUe tbtcDerbut onel^ of tbe fcconoc, liSot 
tl)cn again tnn!trnte,tl^ere are.2.fo;ite0 of erampUs 
iubtcbe be batb not. an)>ifi?8u compare tbem toeU to 
getbcr, pou (i^l peff ^Hie^tbat tbet bw agreobte to^ 
getbcr. 

jas fioi; erAi>(e^ BIQlUiSf fi);lle canon,tbt0 isi tbe fo;me 
i,^— I— 6.2^.-===-=*27* ^. lubi(b«4^uationm 
sir i^i|l^)^? traaatton,i0 epp^elTeo tbu0t 

i.^-=-=2 7.f 6.2g^.bicaure 3 coe Uill fct 

tbe greatefte Denomination alone. 

agavtini)i5 tbirDe Caoort , tbt0 is anr eranrplc. 

K§>-H — i?.f-==-=*^ sg^anbtbatnoinberooe 
3[ tcanOate into tms fo;nre i V.-= 8,2^ 1 j f . 

j^otir W(>erea0be giuttbreneratlernlee/o^euerp 
Cmtntrtr,^ (^tefo? tbem atttcrtwcte tbc roote of tbat 
eompt^fvle itimber* fxii all im rule^Dof teacbe no^ 

^cbolac. ^boe bnoerllanbe tbe ottieditfe ^ and a^ 

gttmtntt off ear Hite 0«tft bts. But fo; mp ^rereife, 

BI booeicouttte (omeapte qaedt^m^ ap)»^tatnpn0 to 

ik^ntititltm^ 

jpiepion !<: apaftfr* . %nkttw &|tbefirffe^t«fftoit» 

^-"[g"* ;aie]ranber bepng afHeo bol» oloe be iuas? 3^am,2* 



ofCofiih nomhers, 

Sio* anDtnt fattier loas as olDcas Ut£ bdtij0,ano.4« 
prrtd moare. ^nb nt^ fatber i)a:>tn9 all t^of^ ]^ rc0» 
fatcD auranoer, tDajs.9 6»i'erei; of agpe. 3i oonaun^ 
nolo of ^0tt,l)otD olDe Inosccbeof tbeur. 

^ct)olar. 31 p;^u: rou auitfUiax tlyt tpxdtitn roue 
fclMo teacbe mc tt)c fojatc. 

fatten ^tDtUbcjctnlbtt^tijc^oiTgcttcmamifg 
agc,an5t!)at*toUl3cairM-^.UjljuDti5t^fciiTmon i t^-fsthe 
fuppofitiott tn ail focljc qumtoiiiS, C^jm ts aicraii* commonfup-. 
DccB age. 2. pews tnoarcttjal w. Lr^ — 1 — 2.f . HiiJ /<>/<//««. 
ttjofc botlje together Oooc niabc* 2. i^ . — I — . 2. f . 
lubcrcunto if pon put.4.ino^c,tbni t^aut sou tbc age 
of C-pljcttio l)is father, tIjatUiiU bc.2.::^.— I — .6.?. 
anD all tbcfr pot togctljcr, tbat ib. 4* ^-H — ♦S.y. 
totll make 9 6 U)l)tc6e t&tbe equation t^ O^all open 
tt)c quc0ron. 

tHai)etfo;ie 3J fct Douitc tbc equaftoit tl)u3. 
4. xp . — f— 8»f — -=9 6. f ano btcanfc 3f fee on 
botbe fiDe0,one Dcnomtnatton of*f . 3> ^<3C abate.8*f . 
fro botlje liDeBii tljc n tljere rematnctb.4t;3^=— 8 8 f 
ano br renuctfon o? Dintfion,K^.==— = 2 2.f . 

^c!)olar. s:i)cnmaie3}cafil^fatc, tljatcpbcftfoxif^ro^, 
tua0.2 2. pcres olDc, fcpngpou did putte. i.^^. fo? bw 
age:anD nolu.i 2£^*ts foundeto be.2-^no tbcrbpnll 
tbc otbcr perc« be mantfcfte. i^o/aicranDrr facpng.r 
pcrcs clDcr,muft bc.2 4. ano (Epbcftio Ijw fa^ 
tber l)aD tn agc.2 2*anD.2 4:anD. 4«»io;e,tbat 
t0.f o.^eres. ailtubicbemabe. 96*ji»ot0tl)at 
qiteftlon full? aunfuitfreQ* 

a^aacr, an otber queftfon (0 1%\9> 31 ban t 96 
a fotmne of moncp ototngtjnto mcitobe rcof Jf tfo re^ ^^»ffli*n 
cetue at one t^c ^ anD aftertoarD 31 rer ciucu ^ of tbat 0/ 'iebtc* 
reHmie, tubicberematncDtjnpatro* anoforcmatncD 
tbe relte of tbe Debte 2 7*r.3i tuouUi knoU^ tubat loajs 
tbe grUDei^f ^fiUt tucr tbe*2.fi^tieraIlpa!ntTentc0 

C.t|. feictolac 



22 



The ^rte 

^ciiolar* Ci)ti» anx&e ^ obferue Htll, tamnwtt^ 
firfte UDobtfuU tbpng* Ki^o .ioljercfeje 3 fate tljat the 
§ir(teDcbte tuao uxp ^lubcccof 31 recctucD ^ 0nD fo Diir 
tf)ei-^r0main.|.2g^.ai toljlrijc rcaB>agatn^3 recetaeo 
',tl)at (0 /-.of t^etu^le fomine,o; /^r^^* ^o tliat be^ 
i^nq abatet) alfo,tben DiD tljcrc rcniatnc n 2^^. U}l)(cbe 
pou namco to be.i7.U*K^m if ^j . i^^. bet tquallc to 
2 7,tt,Dimoe.27*rt»b? ,1^ iW tbc qnotiente UiiU bceJ^'g, 
tl)at (6.6 o.lut)tcI)e loas tm sbole Dcbte:0tio tben 10 

ttpIamc,tl)atsJ*of itte.i ^.aiiDf . oftbetcfioiie W.I 8f 
ti^khj?.maiietb.5 >flttBtbcn rcmametb.27. 

^afttT. SD^rc isltotbpngf better tbcii ererctfe, 

m attainpnoranpi^nDe of bnoUjlege:^iiD tberfo^e 3| 

)utlip;»}uc^uluttbiituerfe quclltons, totnaUe^ou 

tbc moare erperte tn tbi0 cule. ^iiD tbiB is one. 

jtaaefiton of SDbere tBafloo^cp.aueoiDitb^quarc )i3;ubc,tbr 

tauw£ lengtbcof tiiataooae bcpng longer tbenxbe b;c^Dtb* 

^ •' ^* b\? 7 , anti tbe tubole pauemente xontainetb ♦5^84. 

b;icUc0: 3 require to knotue tbe b^eotbe aito Icngtbe* 

^c^lar. :S^be Iclfer quatttte^lubtcbe 10 tiie batmh 

3! Ooc name, i, 2^* Slno tben tl)e Icugtbe luiU bce,b? 

pourmopo^tton. i^*^* iJioiumuftBimuitiplietbe 

b;eDtbe bp tbe lengtbe { fo^ tbat la tbc ozDerip iuajfee 

m all flatte fo;me3,to fiuoc out tbe tubolc plattc)tbat 

i& bere.i. 2^.ip . |. xg . anDtbere iDill aniountc tbe 

lobolc platie.J. 5^. tuljicbe bp pour fuppoGtion 10 e^' 

qualleto. ;5'84. 

Mberfoje acco?Dpng to pour rule,3l DiuiDe. K 8 4. 
bp 5. anutbt f wHra/tf luiu be. 3 1 56. tobtcbe (0 a 5^««re 
nomber, btcauretberet0one Denomtnatlonomtttrd 
in tbi3 egaatio. fm iiettocne ^ ano f. tbere rs omit^ 
teo.to .anotberfo;icmua3f ertrsttetbefquare roote 
of. I IT6. ano it tuiU bee tht guantitie of.i. ::p . tbat i 
looo^itie in tnp tablr0>and ftnoe tt.^ 6* iubicbemult be 
tbe b;cotbe:fo? tbat g namcD. i.tp . Cben tbe Icngtb 
mnSt be moare bF?*of tttottb fo (bail ^t be.6 4. 



ofCopih notnhers. 

j^ato foa to coiifirmenip Ujooxbe , 31 niaU!pUe.j6. 
bp. 6 4 ariD It luill make. 5 J S 4. Uiljicfte ifi ii)c mmbcx 
t^at pou 010 name. 

95aller» JCtjat quettion tg UicU auiiriueccD : anO ^» "/irfr 
Ifppu t)aD put.uxp^.fo^tbelengtbejaBpoumiafttoo. y^oorkjof 
tbeit tbe b^eotfjc uTul be ^.x^ . anD m fquace ^ ^.anu tbat^ueftith 
fo.i.ip . luoulDbee. 64.as^ou mate pjouc atletfcc: 
but inttje meane time, toljat faie ^oji to tbis qucttio:' 

Sphere ij3 a capitaln,lpt)icbc batb a grcatc ai:mir,t J'^ue/ttan 
iDOulD glaolv ^©aclbaU tbe, intaa fquare battatlcas •/<» ^'w/V. 
large as migbtc bee. tXIl.bcccfo;e in Ins &iac p«;oofe of 
fquare fo:mc, be bao rcmainfng.z 8 4«to mmv, ^no 
p;oupngagam bp putting, i. moarcuUbcfronte,be 
founDe luaute of.2 y.mcn. l^olu manp fouloiarg bao 
be,aspougcirc.- 

^cbolar. J caUtbefii'tte fronte. i .2p . ano tbcn 
niulttpUi?ng it aquarclp : 3 n^all bauc fo; ibe lobolc 
battaile. i. ^ anofobppoucfaipng, tbcrelDaslcftc 
28 4' men, Uibccefo;e tbe lubolc nomber of men, load 

l.V— ^ — a84»f> 

jfioU) fo^ tbe fcconue p^oofe,, tuben tbe fronte toas 
increafeo bp.i.man:^ Hjall fcitbe fo;merrronte,anD 
I. maitnc moare , tbat w , ^^ _i_ , x> 
h2£^-f-i.f,anomu^ ^'[[f^- IZ I* 
tfpupnge- tbat nomber, Ki^» t^W^y^ 
fquarelpttbercUjiUarife i.r>,— j — .i,tp » 
fo; tbe tubole armie* ^ , ^_j r ^ 

I'?^-r-22p-'f-if ^ 7 ' 

out ofU)b!cbc""3lmuftea^ i.5r*— 1— 2.^. — j—i.f* 

tateirtbat^poufatejtJiD 

toattte, to mabe tip tbat fquare battatle. ^nD tben tt 

tDtUbe.i.^.—- 1 — 2*iP' -2'4.f 

^Q\x) mit 31 one nomber of menncerpjcflTeo bp.2 
Qflike nombera: €)f neteflfitie tbercfo;^; mutt tbefe»2. 
nomberi^be equAlletfetng tbet re^jjefente one armie. 

m'if^^t^l fet tbem tbu0« 



aiiD jinDritfir* I. §-. on botbc parte0 of tbe cquaf ron,^ 

toe auate it,t ttjcn ftanoctlj.z 8 4^-«2 xo 2 4^^ 

^0t ajjain 3 fcc.f .on botfte iiow of t^je tquation,anD 
tl)trfa;e,f0iiig rlje Iffferof tlicm f;atlj tbc figneof ftib. 
traction, 3: 00c aD0e»2 4;to botije nombera , ant tbcn 



If 4. 
1^4. 



61 6» 

770 
15*4 



2 37 16. 
284. 



luill tijcre bc.5 o 8.— =.2.5:^.tbat w. 1 5- 4 
^otljatfcmgw2p^toai5 fcffonbefirtl 
fcontc:tbrCamc ftorif muff be» i y 4. lutjofc 
^qnarc <0. 2 >7 16. tjnto tobicbc ^ muffe 
aDDe tftr.2 8 4.tbat did abounDcano tben 
luiUtbcluboIc nombcrbc.24 000. 

i^c J fartbcr trtalle iDbercof,3f tafec tbe 
fcconDc fronte to be. i <; j-^tbat («, i. moare 
tbcntbe firffc; anobw^uare lutlf bee 
2 4 o 2 ^ ano fo ts tbcre* 2 5-. moare tben 
tbc tuffe nomber of tbe armte^as tliz mxt^ 
ffton fuppofeo. 

jfn other OJaffcc. 2Cbat itjuclhon mafe betojougbt alfo bp 

xiKKirks^f "anting tbe feconDefronte. us^.an&tbentumbU 

that^mjlit fquarebeew.g^.butfetngtbcre ujantctb.25'.mcnne, 

to malic tlnA Square battaUe, tbe nomber ftaU bee 

h-^.- 2f.f 

%\itx\ fo; tbc 0rffe front, ^on muff ftM.man We, 
a« tbc queftton (mpo;tctb,« tbat U)tU be. i.:^ — u^ 
lubofe fquarc luiU be r.g- — |— Lf . 2:2^^. 



24000* 






I. 



J.f 






.l,f 



i*^*— h-«Kf 



i.r« 



bntoiubfrbesf nmftsbDcfbe.284^ mwinertbattrDft^ 
bounDctDbc tbat battaUe iuw nmtt,mt tbcn toill 

tbe 



ofCofihnomlers. 

fijcnombetbca.t> — +---.28^f •— -.2.^- a«0 

rtmuflbeeequalleto.T.^. r^y§,m^^ao^to 

rcfiace tbat etjuatiotijflme 3 at«)c w botije QQw ^ Hr 

f tbeiT rettctfr. i.?^.f qudHt to. i.^~-H-?io 2,^ 

ICtjcn Bf aODt.2.%.btcaufe 31 ftMllftauc mt — -^nr 

abate, i .v^Wbotbe fifes of tbe equation tanotben 

reinantct^>2.::p --=.? i o.f tbatw.i.ip .-«iT>f' 
^bcrbp tt appcaretb tbat t^ fecon D^ f rontc^tuas. I r r 

ann tl>c firfte frontc. i s 4» t fo fo^tbc^as pott to^ougbJ* 

tt before. 

anotl)crquraiont)5tb!3. 

XbtTc 19 a tipng Uittb a greatc armir: ^nD m ab- jf^ue/tim 
nerfaric cojrtiptetb one of bi0 bcraultcjs U)itb gtftcs, ofanarmic, 
aao nuUetb bpm f lucre , tbat be U)tU tell Jjpm , boUi 
manpE>ukc0,€rie!j anD otbrrfoulDiarfitbcrearcin 
tbat atmre. -S^bc l^eraulte lotbe to Irafe tbofe gifted, 
anD as lotbc to bee* bntrue to bw prince, omifctb bw 
aunfujerclabicbe toaa tr«e,but pet not fo plain,tbat 
tbe aDuecfaric toulo tl?erbp DnDecttanDe tbat, tobicb^ 
beoefirco.anbtbatatinruierelDastbw- ^ ^ ^ 

iLoo^c boil) manp aDufees tbere are,anD fo; ecbe or 
tbcm,tbere arc tlutfe fo man? C rtes. ano bnDer ene^ 
TV eric , tbere are fotoer tpmea fo manp folotarB, a0 
tbere be Dukes m tbe adoe. ano lubcn tbemuftet of 
tbe foloiars teas taben,tbe.2 o o.parte of tbem,toa» 
atrmesfo manpastbenemberoftbeSDufees. 

SDbtfi w atrueDeclarattd of ecbe notnber, quootbe 
^raote: ano 3 baueirtfcbatgetiiiit»otl)e. j^oUjgefife 
v'ouboUrmanpofecbcro?tetl)etctDa3. 

^bolar. aitboagb tbe quettion feme barDe, 31 m 
tnarrr tpmco , tbat Diligence mafectb barbe tb^nges 
wQcanS tberfo?e3 toillattemptetbetoojteeof it. 

jgrnb firffe fo; tbe nombet of SDuUes, 38 f^te. u 2^. 
tbcnujtfttbenombcc0f^rIe0Jce.2.^.tbati«.i.ip 



Thtjfrte 

bv.i»'¥.»'"lftp«e5tiutfe:)anot!^eiiomberoffoto(aril 
are.g.rP.tftatiB, i.j-.ntuUipltcDbM.?^. fohiert?/ 
me0,bat bicaufe tl^c* 200, parte of tbefottuaw itf^o* 
tpmw fo moclje as t^c nowbcr of tfte fl)ufee0,tlicrfo» 
uiuft tt)e.2 o cpam of. 8»ccbe equaUe to.ax^ » and 
f0Con^equcml^8.cf-=-^I 800.20 anDi\fe«=, 

i^o; (f 3i fet^ly aiiD. 9. awqualle together, f IdoqU) 
tjp tftcarte of fj:acticii«,b;tngctl)cfantcp;opo2tion 
ttt Uibole nombcw , 3 ll^aU ftaue fo; . 9 * t^is fraction 
\^. aitlj ferng tlje D£iiommato;0,bc all one m -^ and 
in -,^T ttjc p;opo?nott confittetlj betiurne tbe numeral 

SDlKit to pjoceDc,ff.2 2 y.bc cquaUc to. i. t-.t (hall 
taUc tbe fquare roote of.2 2 ^fo2. 1.20 • aitD tOat 10. i c 
\mt\it mua be tbe nombcr of Dukes. 

0nD fo baue 31 tbe firfte noinber, U)bercfo;e tbe fc- 
(onoe nombcr, tbat 10 tbenomber of carles, niuftbec 
I y.tptnto. I sAMitx tljat 10.4 f o . i^no tlje nombcr of 
rotoiar0ll^allbe44,tvme0,iy.mult(plieD 
bp. 4 y o . tl)at 10 ♦ 2 7 o o o. aufi fo; tuttc 
trialle of tbis tuoo^fee , 3 take tbe . 2 o o. 
parte of tbe foloiar0 tbat 10. 1 3 ^ o. ano 31 2700 o 
fiitOeitto bee.9. tpme0.i 5^. tf)att0.9.tpme0 fo moche 
as tbe nomber of tbe SDuto. ano fo 10 tbat queflion 
foluet),anDtncD. i-^^utvii 

Mother m^tu 2i:bt0 10 an otber qucuion. JTberefsa 

^uifiim of ffronnoe mclofeo kirtb.4.t»ane0, be vng like lambes 

W/«. mn of one belgtbe. Cbe longe0.2. iKaU?0 are in oTo^^ 

fif ?L°" ? *l^, l^ojteae,a0.Mo. 5 ^nD tinto tlje befabt 

geffe br be ftonefle, ano tbat totaUe bp tbe be gbtr, 
Ibtre m\ rife. ? 99 5 afoote.3 bcmaunDe tbenXbat 
i0 tbe lengtbe ano tbe betgl}t0of ec^ tualle!-' 

&fl)olar. s:tjeieaftq«Stitiei0tbebeiffbtf,lu6fc!ic 
mil u^^ 5«J»tjntoit^b(?iongj^0^jj,2tei0OoMbte 

Sef^nUUtrx 



4yo. 

60 



cfCoJ^ike nombers, 
SeffuUheritt^at is,2 ■ ♦ i^. il^otu tHt (ante longeftc is 

in p;opO)tlonSuferbifMnienU^uint4S,tO tbe (^O^tcftt 

ioaUc.^o mull tljc (^o?tcr luall be i ; Jg^, SCbcn muft 
3 OTultipUealltl)ofeo-noberBtogct!)cr,tt)atis 1.2^. 
bp« 1 1 if . iDljcreof Doctb come. 1 §>. tfje n l^all 3; mul*' 
tiplie teat totallr,bp : ^.unn it fiill be "^ a! . oH ^ ctf, 
lobtcbe mull be equalle^bt^ tt)e tooojOes of tbe queHl^ 
oit,to.39 9?o. 

fe>o bp reDucpng tbem to one Denomination. -^ cz, . 
fljail be cqualle to "^^ tbat is. 1 5".c^=i f 9 7 2 o.^. 
ano. i.cc*"^^^* 10648. luberfo^ 3( fljall ertratte tbe 
Cubil^f rootc out of. I o 6 4 8 . ano tbat i$ tbe quantitic 
of. i.%p^. 0; tbe betgbte of tbe lualle. 

jn mv Sables 3 U)oo;jbc tbat crtrartton of Cuhil^e 
lootc, anD finbe It to be.2 2. 5lnD tbcrfo^ muft tbe 16^ 
gcftclDallc bee Double Sf/jf«;W^fr to tt, tbat is. jj-. SnD 
tl:)C fl)o;tcftc lualle iuill be. 5 ]. 

f^o^p;ioofciubcrcof3JDoocmult(pl(e.2 2.tDttb.y5'. Thepmfe, 
anD it iitaljetb .1210. lubtcbe nombcr 3: I^all multi^- 
pile bv. ^ 5-anD it tuill be. 3 9 9 5 o, acco;!Din0 to tbe fup^ 
poQtionoftbeque0ton. 

a^aller. j^ ou Doe f boCc dill tbe leadc nombcr, to 
be cqualle to. I x^.as tbe caficftc fo;mc. IfJotobeitpou 
maic put. i.i:^) .to; tbe Icngtbc of anp of tbe luallcs. 

3nD If i?ouTcttc it fo; tht longeftc toalle , tben tbe Jn other 
fl)o;tcftc tuallc toill be ] z^.anD tbe belgbtc % t^mn forme of 
all tbofc. 5. nombers tuill mabc, bv multiplication to? y^torfie. 
gctbcr ,* cf^. cqualle to. 5 9 9 5 o. anD fo U)ill.6.ci^. be 
cqualleto.99S2 5'o. f.anD. I. cii.=-=". \66^75*9* 
tubcrcof tbe CubiXe roote is y j.anD aunftuerctb to tbe 
quantitic of. I X3 ♦ 

i5ut if. I. x^^c fet fo; tbe mcafure of tbe D^o;teffe The thhdc 
toalle, tben tbe longeftc lualle luill bee \. tp . anD tbe forme of 
beigbtCv.i^.anDfoall.?. nombers multiplied toge? y^orh. 
tber Iuill maUe v c^»-=. 3 9 9 3 o» ^0 fball. i o.c^. 
beeqHallto.^5'937o.;^nD»itc£»*==*U9?7«tuber0^ 

l?b.;» oc 



The yfrte 

of tlje Gy^k? toott is. 5 ^ ano is tl)t t^uc of. i.tp^» (n 
tbis pofition. 

^cbolac. 2r^is iianctie of tuoo;jUe, isnotonelp 
plcaraunte,butttmafeetl)tl)creafoiioftl)clyoo:ljeto 
appcarc moare plainly, &o tljat J coulo neucc fae Ujc? 
tie to bcare focbc queftjons. 

Rafter:. SDbeu ioill 3 p;opounDc one o; z.moare 

before tucpaflTc from tljisUmoe of equation. Ml Ijcre.^ 

of one fljall be fomctaljat lifee tijat laa. ana ttm it is. 

Jjueflioa ^ 113.:ickelciar ^aD a pile of )i5;ic!ie, MjuUt Ijc folo 

ofhricke ^VtljefarDc. Cbclengtbeofittua^lto tl)c bjcDtljc, 

^ ^ * tljat is TripUfefymalters. anD t!)^ tjcigOte bag auc tv* 

mc3 fo mocbc as ttje legtlje. Htjis pile t\)t oU)ncr foID 

fo;.98 o.ccounes.l3p foc^e ratctljat be t)aD fa; eucry 

l^arnefo man? Crouncs , asttjepilebasr paroesui 

b,jc?Jti)r. jpiotu is tl)cqucftion,lubat tuas ti)c kngtbe, 

b:iDtbr,anD beigbte of tblspile.-' 

^cbolar. 3! fuppofe tbe bxt^tht to bee. 1. 1^ . t!)e« 
tuas t^e lengtl) ? ^i^^.ano tbe bcigl^tc. 1 7\ '^^\\t{i 
I . fommcfsDooe^ multiplie together, anotOcimafec 
'-^ cf . iobicbe ttanDctb as cqualle to all tbe parlies m 
tlje U)!)ole pile. But pet ioliat tijat is, 3f fenoUie not. 

tKafbecfo.:eto p.2oceDc farther, J conaDertbateue=' 
rp parDe code as manp croauirs^as tbe bjeotbc contai« 
neopacoes. jjioU) tbe bjcDtbc being i.ro 5 muttfair, 
tbat euetp p'aroe oio coftc. i.r^.of croufes. ano tbeii 
bp tbe (i5olDcn rule: if. i.paroc 
cclle.i. ^. ofCErounes, Uibat 

^oo^kpng bp tbe rule, 9? 
finoc tbat it ITjall coft ^^ ^- ^. anD tfte queffion Doetl^ 
fupporetbatitcolle.98o.crouncs.^berfo;eB(mutt 
faie,tbat.9 8 o.c rouncs, are equalle to '-^\ ^ ^, ano 
conrequentlp.2 4 T* ^ 5"- -=• 1 9 2 o.f . iDbcrfo;ie nU 
uiDpnge tl)t wontber of tbe leflfer name, bp tbe otber, 
tbe ^uotiente VoiW be 1 6. Ujl>0re :K^n;^^ns^ke roote IB 2 

ano 



^IcZ ' ''' 






ofCopike nomhers, 

2nD t1jattIjcifo;c mutt be t\it Wut of a rcote,anu tljc 
baeotfte otm pik. ^o Ojall tije Icngttjc Ijc.7-rar0c5, 
anDtlicljdgfjte. ^T-Pai^^^J^. 

f 0? triallc ont,5niut:ipUc ti)c Icngtlje, hv the j^^ .^^j^ 
fajeDtbc^anD tljat totaile bv tl)c t)cigt;tc,anD fo Ijnuc^; 
4 9 o.fo;i all tl)e rarDts of 3L^;tclie. E:t)cn confiDcirng 
tl)at cuerr varoe coac.2.cronnr0,bicaurf .z.rarDcs is 
tl)c l):f Dtbc cf tbc pilculjc nombcr of crouncs muft be 
tluirc.4 9 o.tl)at 10.9 8 o. ^nD fo is the iuoojUc gooO. 

spatter. j^.o\Xi luoo^bcttjatqucflion, bpfettimgc m other 

I.X:3.f0^tbeIcnQ[tt)C. forme of 

^ctjolan 3ftl)clcngtl)cbc.i. x^, tbc b^cDtbc muft y^oorkj^ 

bcCy« X2 .tbatlS Subtrt^la/efquialtera to A. 'X^.^\\ht\:iC 

f)C!g{)tuituft tcc.y. ::^. aillul)Ul)cfomiiu0maUebp 

multtplicaiion Vcc. 

STben fartljcr, if urarDe cottc I i^. y cc.fljaU cottc 
^^ . 5' J- , luljicbe is cqualle to — -^ i . ^ 

98o.anDfot5.2o.§y.cquaI * / '^* 

to . 48020. anD bv Duufion yC^. ZL 4.'5"5^» 

Ki^'^^kf rootc 15 . 7 ♦ ^no ttjat i$ tht Icngtbc of tijc 
lualle,anD is t!ie tjalue of. i.i^. 
ILbc rcfte of tbts U d;Uc, id lil^e as bcfo;c. 
maticv, ^n paour ilje t^,trDe hiaic. j-fhirJe 

^cbolar. ^\)t bcigbte becrng. i . ^ tljc tr itgtbc ^^,.^^ . 
ntna be tbe fiift part of it,tbat is ; x^. ano tbe b;JeDtb<^ ,/ 
;,i^ . m tbcfe mal$c bp niulttpUcation „; cC» S:!)en ^ * 
fontiep5ice,!f.i.rarrjcoftCv:io —7 > w, 

U)bata3aU,,;cf.coac-15rtlje '' ^ /^'-» 
ColDcn IJuletl^eic is fount?, -,?'c€. Z„ KT.^S^g^ 
r, at • ^^ ^'^ Ujbicbc is cqualle to 
98o.anDrDfl)all.4. 5-5-. becnur;Hcto.6oo2yoo. 
anD. 1.5 ?;' ---=1 S 0062 S". toll 0fC5^f»;^\<n;^'^er00tC 

is. 5^ ant)tl)atist(;e\jalueof.i.%^, anotbebcigbte 
oft^cpilc. 
fatter* £Mic qacCticn moarc UiiH j pjopounDc, 

I'b.i'- aiiD 



Thejrte 

anD fo eanDe tnftb tW cquatiom 
Aqtitfttm of ^ poo?e man DicD, luljicbe l)aD fotocr cf)i!&;en,ana 
»T€flament. all !ji5 gooDcs immt to.7 2.crounc0: lubicbc t)c luoun) 
Ijauc partcD fo, tbat tbc fee onoc $ tbiroc cf)iIoe fl)Oulo 
!)auc.7.timcs fe moclic as tbe firllc. $liiD tfjat tt)c po;- 
nonc of tljc tljiroe aiiD fourtfjc rl)ilDc fljonlD becy-tt'* 
me0 fo mocljc astl)c fccoiiDes pattctaiiD tl^attljc firlt 
anDtt)t foartOc, l^oulDljauc tluifcas niorfjc as tbc 
tljirDc.Bif pou U)o?Uc tbc folution lucl, vou mate fcjnc 
luo^tlvp to be matter of tbofc luarDcs. 

$>rbolar. 3! trull to obtatncmoarc bcnefitcbptbc 
qiicaion^tbcn faptbatomcc. mbcrcfo;c j tuiU aiuc 
gooD bcDc t)nto xu ano fo^ tbc firftc nebcr^ fet. l^ 
then muaetbcfcfonDeanotbtrDe po;tionsmahctir 
0ctbcr.7-2^. ^.nD tbc fourtbc muft bee all tbc rcttc of 

tbc, 7 2 . tbat is.7 2 8 x^. jiioU) tbc tbiroc mutt 

be balfc tbc firfte f tbc fourtbc , tbat is.? 6 — ] " xp . 
:aiTDtbctbtrDtfourtbe>i£f.^trmcstbefcfonD.ti)bu:^ 

fo>c tbc fcfonoc D^all be tbc^part of. i o 8 1 1 ' x^ 

tbat ts.2 1 ) — 2 AtX^, lubicbc noniber J fl;all fet 
in o;i)er iDttb Ilettcr0,asbcrc3i baucDoocn fonnu 
oU)necafc,anOaiDcofmc' U .^ 

mo;p.9nDtbcnfijal3aDOcU|5 2t»^- -> fv^* 

tbemalltogctbcr.Mbcrc.' Lr -/^' __.^^.'P* 

oftbcrccommetb. ^ --,7 _'V:^' 

129! i2^%e,^bkbe p-:^i: ±.1^_ 

<£(cquaUeto72,j>irfttber^| ^^9-' ^'^1^* 

fo;jc3J 00 aooe all tbat folotoctb — to botbc partes of 
tbecquatto.0nu fo bauej 1 2 9 ^=12? ip —4^— 70. 
"m^At blcaufc tberc are nombers abfolutcon botbc ft* 
l»cs,3I l^all abate t):it lelTcr fomme , tbat is . 7 2 from 
fcotbc partes , anD tben luill tbcre bee lcfte,y 7 . ] = 
1 2 ^ ::g^ . tbat is. 2 8 8. « . 6 4- 5:p • ^HD bp mmaon 

rtcpmfe. ^ol^all tbc firtlcmanncs portion bee 4 ^ anD tbe 
feconoe ano tblroe mawnw po^tion.y.ttmes fo mocbc 

tbat 



ofCopike nomhers, 

tW the fourt!)£ mannc , l^all tjaue %5 n ^ 7 , » . 
tl)creftcof72,tt)atts.5 6. CD 2o\y^ '* 

S^benfccpngttictljicDemannc, D 56. 

Ijatli balfc fo ufocbe as tl)c firft ano ir^ 

t\)t foiirtbe, t)i3 po2ti6 l^all be 2 o ',. ^ ^* 

2nD tijcn bv Diucrfc rcafons, tl)c fccoiiDc ntancs part 
fljall bcc.i I v^"0 'itl tbcfc partes aoocD togctbcr,ooc 
maUc mac 7 2. cat)crfo;c tl)c U)oo;Uc is gooD- 

£Raftcr. 110U bane U);ougbteit lucU. ^iiO pet Ju other 
mate pou U)ob:Ue it tbiis. j^irfte fette nounc.i.-z^Ao; forme of 
tt)e ftrfte mannrs parte, ano tben fo: tbe rtLonOv ano •^oor^e. 
tbtroe lojmtlr.T.xp . fo l^all tbe foiirtbc niamic bauc 

y 2,s. ,s. xo . iluD bicatifc tbe feconDc manncs 

parte IS ;. of tbe tljirDc ano fourtbc niaitnrs portion, 
If pou lopnc all tbcir . u partes togctbcr , tbe fcroncs 
mannes po:tton luiU be ; of tbat totalle. put tberfo^c 
7.X0 , lubicbc IS tbe partes of tbe fcrouD anfi tbe tbiro 

Ijnf^. y 2 8 ^ , tubicbc is tbe fourtbe mannes 

parte,anD tbe totalle ujill be.7 2.f 1.^^ .lubofc 

firtc parte is i i.f. 1^-, fo;tbc fecouoeman> 

nes I^arc. Ulbicoe fomme if pou abate out of.7 . :^* 
tbcrc lull remain fo: tbe tbtrDc ^ 

mannes parte 7',^ 1 2.f ^ i.x).3i_.v^ 

ano fo bane poueuerp man* Z ^77 '^ 

nes porno allotted to bpm Duc^ ^ /^-^i ^'i 

Ip. as 3 bauebcrcfetitfonbc SD 72 » -S-^- 

fo;p0u. anDalUbctaODcDto? 1:7; ' 

gctber,ooemaUe.72. - "* ,. 

^cbolar. usutbcreisnaccquatiopet, tbougotbt 
partes be DtuiDcDiuftlp. 

Rafter, jjiolul^allpourectt. 

STbc qiieOton fatetb,tbat tbe tljrroc maimes po;t(^ 
en is balfc tbe portions of tbe firttc anu fourtbe man. 
toberefflje fepng tbt flrfte ano fouttbc mannes po;tf ^ 

ins oocmabe.72^ 7'2p .tbetbtrocmannespo^ 

^ l^b-tii- tion 



jffjutfiion 
tfftlkes. 



The Jrti 

tioit bfctng5oublcO,(l^aUmabca«ntotfee. Buttfje 

Doubrcoftl)etI)irDcmane0partc,is 14 'xp 244^. 

mio t«)crf0;jc 3; faictfjat tt)cfc»2,iiombf rsbe equalte^ 



fXt^t aDDC.7. Xs 

7 2.<y.=2IlXp^ 



to ccl)c parte , a no it Uiill bee 
—2 4-^. £:ii£n aDDc.2 4. f .ctl 



botl)efiDcs,anb if)crelu(Ube.96.f. -^—^ 1 ^ v .tfjat 
is b^ rcDurtton, 288,-=. 64. 2p. as rou ni^Oc it» 
:anDtbcnailagrcctb. 

3MUcUjaiesfonbccquatton,rouma(cfcttbctbirD 
inanncs portion, U)it!) tijc balfc of tbc firttc t fourtbe 
incunrc partes* ^nD fo Uiill. 7 ' , ::p . . ...y ^.i^.be 

cquallcto » 56 .f. 5t^\ SiD fcr reDuaiott, 

J0 7^.=— ♦48.f.s:f)atistnotbcrtermesoflut}OIg 
iiombtr.?2j£^.=-,i44.afit)bpt!iuificnitiDUIbee 
i.:lo ,=-=.4 ^ . aiiD tbus iMill tuc eanuc tbeeramplfs 
of tue firftc equation, fo; tbls tpme. 2im \a\{\ (bcUje 
rou fome qucflions of tbc feconoe equation. 

Examples ofthefeconde equation, 

ly quejlims propounded. 

l^^ereare tUjo mrntbatbawefilfee 
to fell . SEbe one batb. 4 o. elnes, 
anDtbeotbfr.90. ^nDtbcfiiCe 
manbisfilfeeis not fo fine as tbc 
j fee onDe man bts filhc . ^0 tbat be 
feiletb in euerp angcll, p;ice nio;e 

- _ .JbrTOfanelne^tbcntbcfefonDetna 

Doetb. anD at tbe eanDe,botbe tbcir mcneis niaoe but 
4 2. angelles. ^oU) 3 DeniaunDe of t'ou, botu mccbc 
ccbe man folDe fo; an angeli:' 

^f bolar. 31 tuiH fololue nip olDc fo:mf^it\ putting 

i.::^. fo;tbcUaffequantitte, U^biftjetstbcfefonDe 

irannes fomme ^ ano tbcn (Ijall tbc fitfie maHncs 

fommebe.ils^. 

q^aHcr, 'pQii are DcreiucD all reabie . f 0; vou fet 




ofCo^iXt nonthers. 



I. V .ffl; ait cine, ^cpng pou name-; of an clncto be 
f.i^,0nDroU)crctl)cpotttionrTcaDclcirc, anDlibC'- 
U)*miiaUtI)cluoo.:kc. 

S>cbolar. ^ fee mp faultc:but 3; Unotuc not I)oUj to 
anuMiDi* It. J^o: il>at. f.^ .maic bcc a parte 0; partes 
ofandncranDfomaic it bcmoarctl)cn.i.o.:.2.clncs 
fo tljat J Dugijt not to banc fet ; Clutjicbc is caratiil)> 
rcfcrt cD,in tl)i5 nucllion, to an cine) a<3 the parte of a 
I)oubt\"uuqnantitie,butratl)era3tf)epaitcofaquan5 
tiucrcrtainc, c^TOcrcas.i.x^.iocucrputfo^anom 
benjnkuoiuen. 

i:i?.xacr. 2Do l)clpc you l)ercin,3[ bill fct tl)c firHc 
nonibcr3,a'3 pou began t^am. rD:)e rceouDc man l)i6 
no;nb:r3ofclni:i5,n)iUbec.i.x^.aji.n3u!DtDname it, 
anDt!)cn U)-illti)e firrte iua^-'^j ^ , « 

nc5pa:tionfcr;;a3mocbc,anD; j "^ x. Jxs* 
ofan tine moaic: lul)icl)e ; i i '*^' 

maic boftc call ; >? . 0nD fo ll)alt it bee D(ftaunte from 

I.::^.clcrelp in al[\JO0Q;)it^ritb)neticall, 

iJut nolu to p;oceaD:, 31 iljall DiniDe ecij: mannesi 
nombcr of clnes, Uj!)icf)e be IjaD, bv tbc nombcr of Ci> 
nc3 , lul)icbe be foloc fo; an angelic, ano tbe fuotiente 
ioill orclare botu manu angelica rcbc uxan baD rccei^' 
ncD. »>o tbat tbe firttc mannes nombcr of cines, bcc- 
png. 4 o . fijall bee tbe numctcifo.:, ano tb: fommc of 
meafurc , lubicbc be foloc fo.j an Angelic , tljatis 
I zd - I — ; y. fljall bcc tbe Denommato;, Znn fo 13 
tbeTsiutfion canDeo. ^tnD tbat frac^' 

tioni^ tbe quotient, 

Scholar, jjioto 31 pcrcciuc tbe 
iuoojkc. Zn^ bp lifjc rcafon: tbt fe- 
conoe manncs fommc of cines bcc* 
pn J ♦ 9 1 . fijall bcc tbe numerato:, 



40j 

90, 



ano.i.ip. b:pnjtbefdmmeofm:afure,folr)efo;oup 
i^ngelle^il^aU be tb- Den3minato:,tbat 13 \n one frac* 
tio:i ;,^ : acco;:)irtglp as 31 Ijauc fcttc bot'oc nombera 



Thejfrtt 

fatter. 3t lucre moarc cafe fo; rou (n too^fepntr. 
If pou DID tournc tijat frartio of^, into an tntcre trnttc 

^cljolar, Cl^at luii eaaip be Docn, bp multipUpnd 
tucrp nombcr, of tbat Uj^oIc framon bp. 5. anDtbcw 
toiU It be ^^^7-- 't'— r^, lubkljc 15 all enc m talne tottb 

40* ianDtl)(s3 conODcr fartbcr , tbatw 

i«^— +— -ff* tfjcfc. 2, fra(tions,fcucraIIpDooecir* 
P?circ tlit fonimcs of angcllc0,tt)at crljc of tljcm ttttU 
ucD,fotopnrtIpbotbc togetljcr , Dooc Declare tbefull 
fonimcof all tbe(r angclles. m. f)erefo;ie a iau aDDe 
tbci m faotb e togctbcr. janD tbct Ujill make* 
" "^ ^— "* 35 fjcrc in ltJOo;lje 31 f)auc crp^tlTcD. 



^2o^ 90. 

>^- — I — *i«f 'iT^ 



3.^,. 1 .I.5£^. 

flnp bp pour fuppofitton , tbctc botl;c fommcs tt^w 
gcUe5maDe,42. ft>o tbat tbofc . 2 . fommcs are c> 
quallc : anD tberefo;eam 31 come to an equation, in 
lubicbe 3: fee a nomber abfolute, cqualle to a frmtton 
CoA%conipounDe. 

iJ^aCf r. ta ben fo euer tbat , 0; tMt lifee Dooctb 
cbaume. pou I^all rcDucc tbe Uibole n6ber,to tbe lifec 
denomination: anD tljentljcir numeratoifliuiubec 
rquaUe* 

^bfllar. s:ben l|>a« 3f multiplie.4 2. bp the Deno* 
mtnato; ? ^—f-^i V f it toil be 1 2 6 t^jJ-a-^Z 
tobicbemuaebee cqualle to. 3 90,:^ . ^Ll _i n^^r 
2r:bati6tnleirerterme0. ^ ^ ^ ^ 

tbcrc remainct^ t^cn.2 1 j-^^^.j 8,i£,.~+--j^.f 

i5ut 



ofCo/sike nomhers, 

)£a\t tiotD 3! rcmf ber ^our aDmcnittctl^at hU:.iiic 
f})cnombcrannercDtotl)cgrcatcftc firrne, tsmonrc 
focn. I. 3i l^all DiuiDc all tl]c nombcrs bv it, ano fcttc 
tl)etr ^uotientes lit tljeir IteDf,U)itb tbcic Ogncs. anD fo 
luill tbc nombcr of tbc gfcatcttc fignc.cncrmo;c be k 
^nDtf)i5cquationU)iUbei.^N— =lv^. — r-'.!^- 
cabcrct nuiacrtraitc tbc fquarc roott of tl)c laicc 
pait.acf02DmgtovourDominr,anoitU)illbc.vaoit 

appcrctb in tbts U)o;{ic folotuing,tubtcbc 3; m frame 
in mp tables, 

' r in fquarc Dactb mahc !;' , tnto lubicbc 3f mute 
at)De "'% Uibicbc 13.aU one luitb :sbp reduction to one 
Dcnouunatio.&o is tbc full aDDitio ^It- Ujbofe fqtiare 
root:^ 13 ! NDnto labicbf 35 Hjall aDDc ;: , anD tt UjiU bee 

'"' 9^aacr.' %\)iii is tucll Dccn.^i^.oU) lt)0.:Iie tl)efamc 
que(tt6,as (tluasp.:oponcD,anD you ftjallfaOivfinDc 
all tbc otbcr nombcrs to bcc trucanD agrcallc to tbe 
qurftion. . 

^fbolar. ^fimgtbcfcconticmanncfolDr.:.clnc5 Tbepmfc. 
fo:anangcll,tbcfivftemanncDiDfrll. v elncsanD4, 
^c.4 o;ii)bicbc ic tbc fommc of cIjics cf tbc fira man 
1)13 niUe) DiuiDcD h\\ ^ ; . Doctb rclc:, 1 z.riHD Hjcluctb 
bolu nianv angcUcs tljiM man rcrciucD. 

i'vgain fo: tbe fcconDcman,^Dlr«cbc bao.9 '^•clncs, 
DtulDc tbai.9 o.bpo.anD fo fijnll vou fmDc. 5 ^.fo: tbe 
nombcr of bis 3ngcllc3 . Slv.n tbato o.ant;. i :♦ Dooc 
maUc. 4 2, It ncaoctb not to be y:oueD. 

O.iailcr. ,Oolu agamc fo;j I'our crcccifc , fuppofc ^n other 
tijc Rifle maifnes fommc to be.i.t^ . forme of 

&!>rbolar. :ir ben mufte tbc fcconDC mannc fellfo; y^oorliif, 

fin angelic. 1 .t^ i'f • ^"0 tbcir nombers of ct- 

nc3 , DUitbcD^tbofcnomberfi lu'U make .';!-. ano 
-.^^ — -^ , lubicbc botbeaPDcD togctbcr , Will bcc 
^^^^^ —^ ^cquallc (0,42.^,^31 IS fcp rcDudion. 
59o,::p» — ,4of,-=, I26,j^. — ,42.X^. 



The Jrte 

ano bp aooWon of* 4 i.ii^.iin botljc patter. 
i^ ^}:^~ "^ o.f -==. i 2 6.p. anD bp Diuiffon it 

^0 tbat noU) 3i mutt ertracte tl)c rootc of tbaf com* 

pounDeO/%fra(tton,tl)uj3. Tfquardp, Doocmafee 
^K out of tobicbe j fljaU abate ^^. ano tbcrfo jcfirftc of 
all 31 Doe reoucc ttjc to one Dcnoimnattou,i tljci make 
nif ♦ aitD ^. lubercfo^e if 3; abate tbe leffcr out of tbc 
greater : tljcre luill remaine l^^ . tbat is m icflfcr tc r - 
me0-^i ariD 15 a fquare nomber,luI)ofc roote 13 '* tn? 
to ivbicbe If 3: aODe V tbat 13 \^ . it tuiil niaUe % o<j i% 
tl)atist(jct3aleUjcof, i.xo. anoistbcfirftcmannes 
nombn- of clnes , agreabiy as 5 tricD it befo:c . ^no 
lo Ooe botbc luo;bc5 agree. 

13ut noUi connnctlj to mr re mnnb^zaure, tljat tbis 
iiombcr, Ujbofe roote 31 Dio ertract.fn this Unt luo:hc 

IS of tbat fo;te,lubere.2£. <y. 15 equalle to. v. 

,9.nD tl)erfo;e bati) i:t it.^.rootes : tbone br aDDttion , 
as tbis, lubicbc 3 nolu fomiDc : 0nD the otbcr bp fub^ 
tiacticn,tDbicl)ein 11)15 eramplcbp abating '^ out of 
H. U)iU bee ^, But boU) 3 maie frame tbat roote,to a- 
gree to tbi0 quefiioiT,^ Doe not fee* 

spader, £:bat uarietic of rooter Dooetb Declare, 
tbat one equation in nomber, maic ferue fo;. ->♦ fcue^ 
rallequeaions. 15uttbefo;meoftbequcaion, male 
eafilpinaru(tpou,Ujbicbeoftbore.2.rootes,i7oufl)all 
take fo; four purpofc. li^olubeit fonietpmes pou (ball 
take botbc, 00 fo^ crampic again , marke tbis quc> 
ftion, 
Jqutflian , 3 gentilman , toiUrng to p;oue tbe cunnpng,of a 
ofmonep hxnsQi[>nQ^rithmetictan , faicD tbus : 3 baue m botbe 
mp banoes. 8. crounes : ^^utanD if 3; accoumpte tbe 
fomme of ecbe banoc bv it felf feuerallp^anD put tber^ 
to tbe fqnares auD tbe G'^rs of tbe botbe, it tuill make 
in nomber. 194. /^olu tell mc(quoD be ) lubat is ui 
ecbe ban5?;anD 3! UjiU giue ^ou all fo; pour laboure. 

^cbolar* 



ofCoJ?ike nombers, 

Scholar, ^oc^e tncoragentcntc0»tJDOulD make me 
fftiDictjarDc, anotrauelltjcrv lutUpnglj' tnlcarncD 
ercrrifcs : tfeougt) learning bet motte to be IoucD,fo; 
){noU)IeDge0 fake. )i5ut fo^ to finDc tbt true aunfluerc 
t\)U6 3 Doe p;ofraDc. 

fivftc 3! fuppofe the one nonibcr in one l)anD,to be 

i.io , 3nD tljen muft tbc otl)er ncDefi be ?,.^ . 1.2^ 

dicrt Doc 31 maUc tticint bothe Squares. 0nD fo; tl)e 
firfte 3; l)auc. i g^.ano fo; tbe fcconoe. i.p — | — 6 4 f 

^i 6.5^.iCt)UDclp 3! multiplif tfjem botjjc Cultf 

ksh''^^^ fo liaue 3; fo2 tbc firfte. i.c6 anD fo; tljc otbcc 

2 4. ^'. -4— .^ 1 2, f 1. <t, 1 9 2. x^. 

jEbcn muft 3; aODc botbe tbc nbbctHyhyitf) tbctc fqua^ 
rej5,ano tbctr C«^a,tnto one fomme. 00 bere (n too;!: 

I . ^» i ♦ I • cti« 



8.f . 'l*^- 

1 , 5^ .— f— 6 4-f -^ — ^ ♦16.2^. 

2 4,^ — I — .T 1 2 "^ 1 ct 1 9 2-^ « 

2 6. §-* — [ — ^ S 4* f ♦ - — '-*-♦ 2 o S. ::o . 

fi IS fetfo:tbc . mbcre fo;eafe 3; baucfct. i.::,© ,1. J^. 
atiD. \,ct (lubtebe ts tbc Hootc , tbc Equate ano tbc 
Cubcofonc nomber)aU in one line : and tbc otbcc 
Uoote,&quare,nnD C ubc, 3! baucfct fcucrallp. <^nti 

fo all tbet ooe maUe. 2 6 ^—^ — s 8 4 ?— 2 o S ::o 

iubtcbc tjs cqualle to . 1 9 4 . bf tbc mtentc of tbc que* 
(tton.(:cabccefo;ic 3! aODc fitflc. 2 o8.j^,to botbc paiv 
tc0,anDtbercrcmalnetb- 

26.5-.— +-584-?— --=2o8.t^— -f— i94-f 
SDbcn 3 abate. 1 9 4. from botbc fiDe6,anD fo reftctbc 

tbeequatt6tbu5.26.^— ^ — 59o.f-=«2o8t^ 
jXbat is bi? Of utfion. 1 . ^-. — ]~. i y.f . =— =s.i^. 
ano fap tranflatton of. i vf . tofette.i.^.alone , it lull 

be.i.^:^==r=:c.8::o 1 ^.f .0nD noU) bauc 3 tbc 

crartc ano completecquation , luberc 3; mutt feke to^ 

^t.y. tbc 



TheJ^rte 

tbctjalue of.i.sg^.bp ertrattpng thi tootc, 2Cf)crcfo:e 
firftcBI taUe ^dlfcof.8 autJtoatts. 4. u^Oofefquarc 
t0. 1 6. oat of tuMcbc 3J abate. 1 5-. anD the remaincr 13 
Lluftictje 3 male etttjcr aDDc to. 4 . aitD fo bauc J. ^. 0^ 
tl)cr,3 male abate itfrom 4anD i'o Uaue jt ^.^bicbc 
nombers alfo aaojDing to tbefamc rule, bc^tiQ aODeo 
to0etl)cc oooe mak^*S* ti}at is tt)e nomber of tbe iniD' 
Dell Dcnominatton* Sinn bc^ng multtpitco together, 
tijei Oooe maUc* i s* tbat is tbe otber pane of tl;efamc 
compounDc Co/i^ notnbcr. 

cpaftcr. 0nt) if poa baD marbeD tbat ftrCc , poii 

tirtgbt cafili' bauc fownD botbe tbofe nombers, bf tbc 

parteisoftis-.Uibicbccan be noaic other, but. ^.ano v 

:anD farther, fcpng tbci. 2.DoemaUe. 8. ano.S. IS t!)c 

nomber (uameD in tbeciueflio) that tbei fijoulD make, 

therfoje pou (hall taUethem bothcHnD name Inhir he 

of them t?ou liftc to be. i.::p . anD the other fljall be of 

ncceflTUfctherefteof.S. 

Tbepmfe. scholar. %o eramme thctm , bp the o;Drr of the 

qucfuon^Sl Doe p;oceaDc thus. 5. luith i)iB ^yquare. 9. 

anDhisC«i*,27»DooethmaUc. 39. ano.^. Unthhis 

rQuare2 5anDlnsC«i^ iz^.DoepelDt i^y. ^nDbothc 

thet together Doc b?rng fo^the. r 9 4.acfo;Drng to the 

rating of the queStontanD therfo;c it is certain, that 

the luoo;ke ts gooD. 

Another $©after» IBefo^^'ou palTe anp farther, ^ iDtlUD^ 

tfoor^tfor Jttonifhc ^ou Of onc Uiaie, Uihiche 3 ofte tjfe m rcDuc^ 

!^u4tions, ttonof foche e^ations,as this 10, iuhen there 15 nos 

Denomtnatton on the onc (iDe,bnt the Uhe is on the c 

thcx fioe, loith a greater nomber annereD to tt. £:hcn 

mate pou abate all the leflPer nobers^out of thctr grea* 

ter£i,anD the relte l^all bee acroumpteD equalle to no;* 

thFJ^g. iSEhlche cl)auncecan neuerhappen:ercepte 

there bee fome nomberfi on the greater fiDe ;, bitth the 

ft gne of abatemente.— . 

a^hcreponhaD* 

2 6^. 



o/CofTtke nomhers, 

istcaufc on tDe one QOc,tl)crc is noc nobcr but 1 9 4 «y 
ano on tljc otbrr Gdc, tberc ts.T 8 4^f ♦ bcepng a grca^ 
ter nombcr,anDof tbcfanic ^cnommation : tbcrcfo^e 
maic ^ou abate. 194. fioni botbc a*)C3, tint) tl)in re- 

mamctb. 26 v~i"59 o*? -2 oSi^-=— o 

Ua berfo^c vou maic Uiell conODrr^tnai tot nonibcrs 
tobicbc be loirtf D luicb — [ — .arc cqualic to tbc nam- 

bei's tbat bee fct U)ttb — ♦ 3nD tljrrro:c tijc one a^ 

bating tbc otbcr luftlp , Dooe rcmaine togrtbcr as c^ 
quatlctonotbi'itg. 

^iza bccefo:c it is rcafmiaWc , tbat feci ng tbc nom^ 

bcrsluitb beccquallcto tbc nombersluUb 

— ] — tbat 3; niaic ttanflate tbc nombers tuitb 

from tbat fiDe of tbc equation, aitD fct tbem on tbc c6^ 
trarp uDcluitb tbc Qgne of— f—. ane fo m tbis cra^ 

ptcit U)ill bee. 2 6 ;?^'.—^ — o9o'y.— 2oS.::g^» 

ilnD tbis fo;nic fl;all cafe rou mocbc, in reDucvmjc of 
equations. 

a>cbolar. 3; tbanKc rou mocbe. anD 3; ijjUl not fo;:i 
get to tfc it,as oaabo iI)aU bappcn. 15ut J p^aic ^on 
p:opounDe vet fomcmoarc qiiclJions,tbat 3; maic fee 
tbcirDiuerfcbariettcs. 

$l3aaer. Sbere lucre tluoo feucrallc mcn,iubicb jfquejlm 
bao ccrtainc foinmcs of angcllcs , in focbc rate, tbat cfmsney. 
tbc fcconoe mannc bis fommc , ^joastriple/ef^^uiquaru 
to tbc ftrftc : ano if tbeir. 2. fommcs lucre multiplieD 
togetbcr,anD to tbat totall tbe 2 ficftc fommcs aDDeu, 
tberclDoulD amountc. 1421. KDcmaunDcofpou, 
tDbat luas ecbc of tbcir fommcs m angellcs:' 

^cbolar. E^bcfirftemanncs fommc 3 call. i.2£^, 
gnD tbc fcconoe manncs fomc fl)aU be. 5 t ^* tobub 
2. fommcs bccpng multiplicD togetbct, Dooe make 
^^^> tjnto tubicbc 31 mwC aODe botbc tbc firftc nom< 
5)ccs,tbat IS 4 T ^« 3nD it luill be ] -1^— |— 4 ^^ 
equallcto. 142 ;. auiubtcb^ nombers, 3: t^^*^lb;ing 

3;t,iij, into 



The^rte 
into mole nowfaewjlf a multtpUe tiicim hv. 4. amj 

fo Ml it be, I ?.5^.-f^i 7,2^==^.^ ^ 7 o.ano bp 
reOacptig tftc g ir^tclle Denomination ufiiie.to an tj- 

nitie. i.5-.-+-,:2 V .— *4 5 K. ano latte of a!l,bp 
tranflatpngtbcnombcrof, xo . tofct» i.^., aloncon 
one CDC of tfje equation, it toiTl be. i ♦ ^=£==4: n & 

• It ♦ ^- ii! berc 31 muft emaac tbe Ualue of tbe 

rootctbufl.l^ fquarelpDooe waUci,^?. bnto lubicbej 
f^all aooc tbe.4 5 ^; (it beepng firfte multiplieD hv, s 2. 
to b;v"g it to tbe Denomtnation of. 676. ano fo ma^ 

fe?ng^S^)anoitU)iUbei^'Ujbicbei0afquarenom^ 
bcr(as 31 banc p^oueb in m^ Cables) anb bts roote i0 
-'^1 . from U)bif be roote 31 mwfX abate ^ , ano tbcn tuil 
tbereremain'l*,tbatw.6. 

anbtbat.6. istbebalueof. i. Xp ,anDffanbetbfoj 
tbe firOe manneg nomber. &o tbat tbe fee onbe man^ 
He5 nober muft be ag.^ to it: tbat i& tripUfefauituMru. 
attbfol^allitbe.i9i. r jjj 1 

Thepmfe. mtttt. i^oU)p;ouctborcnomfaers,acco;binffto 
tbequcfiion. 

^cbolar. 1 9 I multipUeu bp, 6. Doctb maUe. 1 1 7. 
tmto iobtcbe 3 (ball aDbc.2 y v.amounti?ng of tbcir,2 
aDDttios.anb all Irnll be. 1 4 2 i^aceo;Dtng to tbe pur^ 
po;te of tbe quelf ion* 

Jnther 5paffer. ^0 is rour iuoojfee goob. ^tt luoo^Ue it 

•Bfor^e of tbe again,bp cbaungrng tbe pofition. 

fimtfuiftio. ^ebolar. 3 maieput.i.sp . to bctcfeen tbefcconbe 
manne bis fomme. anb tbcn (ball tbe firttc mannes 
fomme bee^ . 2^ . lubtcbe botbe multi|jlieD togetber 
boe make ^ §-. itno tben abopng tbe.2.firftc fommes 
to it,it Uiil bee ^ ^ — f— i a 5;-, . ^ino tbat is cqualle 
to . 1 42 T . ail lubicbe nombers iuill bee rebuceb to 
lubolenombcrs,b^ multiplication eonueniente.anb 
fo Uitll it be.8.5-— ) — 3 4.2^. equalle to. 5 7 o j: tbat 
IB bp reburtioii, i. ^,~~\—^ ^2^ .=.=4 6 3 -^ 4>, 
a«b bp tranOation of tbe termes. 



ofCo/Ttke nomben. 
number 3 mi txttd tDe tjaluc of tDe rootc , tn m 

^^%zfiz 3 faic '^ multiplied Square, uocti) niaUcJ^, 
1>mo lubicV nombcc J mutt aODc. 46 ] ; , ff "J,^f4^ 
it o«gJ)t,anD it U,iU bcc in all ^'^'^^Y^^^^' 

nrift abate ' . ^itD tl)cn luill tl):rc rcmaui -, , t. at is 
g ? fo>tl" lvalue of.i. xp . ano fo confequcntlv fo^ 

^^4itlcr tdbat faie pou tbcn to tbis qucaionf ^f^J^i^ 

2:be artte oaic be muft goe 1 1 milcano cucr)' Da>e af- 
tf r tbe firacbe mua mil augcmcntc bis ^omv^^lr 
of a milr.^o tbat bi3 io;mcv njall p^occDc bp an ^r'J^« 
mff/c.//. pzogrcffion. anobe batbto traucllfojbw 
iubole iomci'.2 9 S ^inii^s. 3 DemaunD: in lugat no* 
b;rofDa!C5,n)all be canoe bis lomep:' 
^cbolar. 3i Unoluc not boU) to p;occaoc m tbi3 

^"^ato. Doc von not beafc me name it,ait Anth 
me'ticillc pjogceffiom'aibecbp pou migbt be aDfuceb, 
tbat It Doetb appcrtame to tbat rule . ano acco jopng 
to tbe canons of tbat rulcmuft fou Ujoo^ije tbts que* 
mon. I5ut fo;i rour better mOruction , J U)ill belpc 
pouintbt3tuoo2Uc. , ^ 

f irftcaunfluerc to tbe qucfiion , bp tbe common 
ijoQtion:ano faic tbat tbe tvmc of bis io2ncp is. i.t£^» 
of oaies. ano tbcn (ball all tbe fxc^i ( lubicbc maie 

alfobecaUcDtben»m/>fro/"^k/^4Cf;)bc.i.i,f^— 1? 

5E:t)r.fOJttJ»o»fxre/tfiuasfupporeotobec. x- of a mile. 
anU tbcrcfoJC t^fommt of all the txcelfcs mufte bd 
"n "^tbat 13 to faie, tbe nombcv of all tbe excip 

fti multiplicb bv {^ , tbat is bere,tbc ftptc parte of tbe 
*• nomlict 



The ^rte 
nombcroftl)f rxfc/M» 

^f iWommc of the cxcejf.s , anD fo t)auc 3; i\)z Ufit 

IS /*t ;^< nmltT of tlje pjogrcirfort. 
JSOU) vou remember, tijat in ^%Q%xtmn At-time, 

tm\) amonntc the fomme totallc of tbat p;osrcS 
2nO tbrrfo;e m tD.s erai-lr.if gou aoor. iKtobul^ 
(« tl)e firttc iiober in tlje pjogreirion) bnto ^^&=+=iy 
amo!m/X'^5|i!?!''''f f "f «<« piogrelTion) tftere m 
amounte ^s==t=iJ, u,i„tbe ^^j,, mulHpilico bp 
tUe nomber of balfe tftc places, tbat .3 ■- ^ [Mi 

— -t'^ =',tol)tcl)e (stlie fotalle fomme of all the 

milMtanO tbcrfojc le equalle to.-' o c T 

noui. sicpng ■ ■ — • '« equa c to. 2 f c. ? iniii 

firttc b;png tlje fobole nombtc to tbc likr ocnm-.M- , 
non,fcitl)tljefcamon,anoitu>iilbre -S-=?^" 4^ 

bptranitition L5.=>i=Z7^,y r-i'ir'i'I'''' 
tobofctoofembalucj njallfinoe out tlnis tii^r' 
ttplit •-} fquarelv,anO (t fcill be i','. tmto luhkhc ^Zr 
aODc^5 f 4 6 o. } It torn mafec ^-4 M^xXTi^yl^t 
miKber, anb batl, foj Dts rootr «; fro Lt f' 'c t m.ii 
abate V',anO tbcn rema(netl) h» , tlwt Is s o i.-Hrhl 

.r-f . r ^'*''"5'"'«t<'«1ucliloiircDu(rctl). 
Tir/rM/r. 9?affcr. ^bc pjooff in tbis , anD the liHe nucff f- 
ons,!3, to fetfoo^f&elbep^offrcirion ImtbaKtcr: 



ofCofiikB nomhers, 

mt&, c!;rffptc^oiitDtllfo:(^o^tiTCtrc, fcttcDounctfic 
firftc tcrmc, tubicljcinthiscramplcis.i ;:anDtl)cit 
h\> tf)c nombcr of tbc encejjes^o; tudaumcs ( lul;icl)c 15 
ciier one Icffc tl)cn tljc noucc of places ) muItipUc tl)c 
quantitic of one exccjjfe : anD put to it tbe firftc termc: 
ariD fo banc pou tt)c laftc tcrme . SXben baurng tbc 
firfte tcimr aiiD tl)c lattc , luitl) tl)c nombcr ctexccjfes 
pou knotur bolo to ftnoc tbe totallc 

!^3 in tbiG crampic , tbe nomber ofexcejffes bcc^rngf 
179. OhD tbe quantitic of one rjfcf/tf be^^ng ;^. tbctc 
mnlt(plicatton giuetb - "I-Unto lubicbc if I'ou aDDc tbe 
firttc nombcr,tbat is i ;-, it UuII be -*. ^un tbat 10 tbe 
laac nombcr of tbat p;ogrcflrion.2niicn to trie tbe f o> 
tallc fommc of tbe miles, aUDc tbe firtte nombcr. 1 4-to 
tbe lattc, anD tbci luill make ^l-, tbat p ou fi)all multt^ 
plie b^ balfe tbe nomber of tbe places , tubiebc m out 
erampic are. 9 o (Gtl) tbe lubolc nombcr is. 1 8 o^and 
tbf re UjiU amounte. 2 9 y > . acco;bpng as tbe queftion 
faietb . 

fecbelar. S^bwisfuffuicntfo^tbisquettlon. ant) 
at fomc lOle time, 3! luill not fticfec to trie it out,bp fct 
tfngtbe p.:ogreirionfoo;tbcatlargre. Kntbemcanc 
tpme 3 P^aie poufo: better crercife,, giucmc fomc 
moarequcftions. 

a^aftcr. SLbere is a nomber , tobicbc J bane fo:' yfn other 
gotten : anoitUDluiDcDinto. 2. partes, Uibereof tbe quejlion, 
one J baue forgotten alfo,but tbe otber loas* 4. anD 
Fcttbis J rcmcmber,tbat If tbe parte, tubicbej banc 
forgotten, be multipUeD bp itfelf , anD tben alfo U)(tb 
4. tbofe. 2. fommcstmllmake.i 17. ,^oU)UjoulDi 
knoluetubat leas tbe tubole nombcr , 'anD alfo U)bat 
IS tbe parte, lubicbe J baue foigotten. 

g>cbolar. 3 fuppofe tbe tubolc nomber to be. i v . 
anD biraufe. 4. is bis one parte,tbe otbcr parte mutt 

ncaDes bee. i.io . 4.2vbcn Doe 3E acco;Dpng to 

tbic quellion,multipl(e. i .i^«-~ — 4.flrae bp tt felf, 

lih.), anD 



The Jfit 

Darilv.S oocmultipltcit^tljatuj^i.sp ,-JL.4)j,„ . 
0nO ttgtuetij,4*z^. ^ 1 6* ^'' «^ ^ 

2C^en al>oe ^ botlje tbofc nflmfaer0 togetfter,a»i» it 

MM be. 1 5- 4.s^,h}tHcbe bp tijxj qucttton iball 

beequaUcto.117* -, m, » 

!♦ ^^ — j — A 6* f* .8.5:g^» 

igu^t^^en muftjl Wietbe atcuaomco tranlTatron, 
to bjpng tt)^ gr^atcHe quantitic in ocnommatioH , to 
ftanoe alone* aijD fo toiU it bce» 
Kg^. 7=^=* 4 ♦ ^♦--f— . 1 1 7-«?» 

«^i:53lmuaai:cbefo;(ti)etjalif ofaroote. anD 
tberfo^e 3.m«ltiplie,2. bg ii felf (^tiacely, anofo ijauc 
S . 4 . ^.n^to tobicijeat aOD^ » 1 1 7. ^no.itmalt£t&. 1 2 u 
luljofe rootc is. 1 1» tjnto totjicbe 3; mufte aODe . 2, ano 
tbmcommeti),! 3.aatb^l)aittcof.i2p ani^tbequan 
title oftbcluliolenombcr. ^ 

Thepm/e, i^; p?Qofe of ttjis ti?o^k^,3! abate.4.out of. i .\anD 
mm tmm*9^nB m oti)mmtie:.%t)en ooe 3! nmiti. 
plie.9.bp it felf, ano tbccof rifcttj.g i. aifo 3 ooe imb 
tiplie*9»bs.4* anD it maHeil). 56. ipbic^cbotbetogc^- 
tftcr, JiacmaUe. \i7,u$ ti)c qncftion.iDoulD. 
Mother i^aUer. $s>eti.2^.foni)clinbnoU)enparte.anD 
itoor^e, mn tooa^fee it, to jfee tbe twuerfitie of tbe tooo;Ucs. 
^xbolflir* 3fi4.faee one part^, anoj.ji?. , tbe otljer 
p^arte,tbpn ioiU tt)c taljoteinom^cr bt.},&\ — 1 — 4^* 
HeaijewfOieftrfteB^multipliM.;^ , bp tnclf^anou 

pdoetb.L5-.2i:benDooe3ftimltiplieu»2p.bp»4.anD 
<t^actf),4,ii9 .lubicbe botiie fommes togetber, Oooc 
mate.K5-.~^4~.4.2^» ltibicbeiseq«alletowi7 

lUiDbgtcanflatio. i.§-.— =-==.1 1 y,^ 4^^^ ^ 

«3a?berero;e3 boemultiplip^ 3, fcuacclbano hqu 

urtb 



ofCofih mmhers. 

llct^.i, lu!)tc!)c aOOcDto. 1 1 7. maUrtb . i : t. anutlic 
rocte of that is. 1 1. from Ittitdic i HjaU nban. 2. ano 
tt)-. re totU rtftc . 9 . as tl)c otlicr parte of tbt nouibcr, 
£:!)ts 15 tcric plain, f the p;ofc of it as it U»as before* 

99aacr. s:hcnaunfU}erctotbi£5nitrmon. 

S:f)crf are ? nSbers m p:oponion Gnmetriali, aufi ^m^'o** 
one of tbe crtrcmes 13. 20^. tl)c ottjer crtrcntc, tuitt) ffi>roportr 
tlK Oouble of tl)c miODcll tcrmcDoitU maUc 2 2.i!^olu 
luoulo ^ hnolue of ^ou , tubat tborc.2. nombcra bee.-' 

fe>ct)o(ar. f onriailcKname tl)eot(;erertreine, 
I . xo . aiiD licaufc It , Ujitl) tl)c Double of tbc miODle 
tcnitc Dooctl) mafec . 2 2 . tljc ntiODeU tcrmc (^all bee 

1 1 . . ; . xp .fo; 1)10 OoubU is. 2 2. 1 x^- 

mtchc \i)itlh I. ip . ooctlj maUe. 22. ^ 

2:t)cn to p:oceaac , 3! bnotwc the p2opcrttc of tljofe 
nombtTs in p:opo:tton CtemrtricMll to bee foclic , tbat 

the ynultiplicatton tfbotUt the ejctranes , is cqudUe to tire 
/^udre of the middellttrme, U)bcrefo:c J multlplte the. 2. 

crtremc0togetl)cr,anDtf)crcU)tIlrirc. ';2^. ESbcii 
tjooc J multiplic. 1 1 — — -\ 7^. br It fclf m Square, 

anu it tuill bee, 1 2 i.f. — | i ^ • • ' ' ^^ 

lubicbe nuia bee cquallc to V ^. 0:. 2 o ^ t^. SZinn 
to rcDiKc it,3i aDDe. 1 1. a^ . on uotbc fiDcs,anD (t U)ill 
be.5 1 -\ ^^ .— — =} ^j^'^-^— • ' - ' f ''^"^ ^^ f*"^^"' 
Cation. r5> -^^-= 5 1 -I ^--r--' ^ - ^ •?♦ ^^^^ '^ 



i.p.._^^ I z^.xe . -4M.f 



^^oU) reftett) nothpng , butto fe arc f)c the balue of 
i.V . mucrfo:: Ttakc-;anOniulttplieit&>quare, 
anD fo banc ^ ^-I . from tufiicbe 31 miitt abate .4^4- 
tbat 10 -':• inD tbcre luill remain ^-'^^I tobofc roote 13 
* ': , tobttbc 3; (Ijall abate i'rom '^1 , ano tbcrc U)(ll re - 
t«atn *,tbat t0.4,fo: tbe otbcr ertrcme. 

K^htn f02 tbe miDoell terme,tbus (ball ^ Doe.$l9u^ Thepnofc, 
tiplie. 4. anD. 20^ toge tbcr, ano tbcre tuill rife .81. 
tobofe roote f3.9.artD is tbe miDDcll nombcr. Cbat 9 
ooublcb loUl ntaUc. 1 8.anD 4.ioinrt) tberto.giuetlj 2 2 



^k. i;, ^0 



The Jrte 

00 aretbofe. vtcrmes in pjogrc ffton Ceometriull.ac^ 
to;Hvm to tbe conoitions Iimiteo in tbe qucaion. 

muet, ]aaouc tlje Uio^Ue nolu.bolu it UjU frame 
if» i.x^.bc fct fo; tljc mmtll nom&cr. J^o; it lucr foL- 
lie, tocricUJbetljectbisquefiion, U;otilo aDmittcaD- 
mtton of ti)i?.2,laae nomfaers, illtbougl) tbc rule Doe 
Declare tijat m foclje fo^te of efluation0 , ttjcre is doh^ 
ble taluatton to ecl)e roote. 

^cbolar. ,iPet 3 befcbe ^ou, let mc eraniine it a li- 

tle,torectl)ecaure,U)lip3n^aienotaD0ctl)cin,anDro 
take tl)e roote* 

£©aaer. 3 "tuft faere U)itl) ^eu fo mor 15c. l5paD. 
Ditioit pon ree,tbere tuiil rife '-^l^hat is 1 2 1. Hno rljrn 
tl)c miDDell nombcr Uiill be.4 9 t- ^"D To m p;opo^ 

tlOU is ^.tbat is ^u^lsfuptrquadri^artiem mnas, ZU i}CVC 

asintbcothcc.^.nomtiers, 4.9.2 o;« t!3ejp;ojj option 

IButmtljequemon i^onecoDition, tbat fee luDctb 
tljc roote,tl)at rifetl) bp aDDitio. fo} the Double of tbc 
miDDell tcrmc, iuitbtljeotljertjnUnoUjetiertreme, 
ll)oulDmakc.22.0stn.4.anD.9.itDoetl).Butin49;- 
anD 1 2 Lit luoulD be 2 2 o.tbat 13 1 o.tpincs fo niocbet 

^cbolar. anD if rou baD faicD in the qucdioiubat 
tbe Double of tbe miODcll nombcr, luitbtbeotbcrep 
treme,UjoulD baue maDe. 220. tben 3 ft)oulD baue ta* 
ben tbis later roote bp aooitiMiiD not tbe fittte rootc 
bpfufatrattlon. 

janD fo 3: perceiue tbe tarictie of conDitions in t\)t 

queftionDooetblimite,tubicbeoftbe,2.roote33:fljaU 

of necelTitie take,anD leaue tbc otbcr. 

Mother But noU) to ijarietbat lu0?be,?U)illfct.T.5:p .fo;r 

mrjffryte. tbe mtDDelt terme , ^no tbcn tbe Double of tt /^itb 

tbe otbec terme,luill mafee. 2 2. ICbe Double of. 1.20 . 

is.2.i^.0o mull the otber terme be 2 2 f 2.:;^ . 

SDbcn tafeke out an equation, 3 multipUe tbeTf. 
mumcs to0etber,tbat is ,2 2,f — — 2 2,0 Jbp 2 o ; . 

:^nD 



ofCoJ^ike nomhers, 

0nD tberc nfctb . 4 4 7 1 ♦ 4 o ; . t^. 0m> tlje 

fquareof.i.Xuo .bci'ngtl)cmiDOcIUcrmc, is folic pcc^ 
ceiucD to be. iV§-. aitD fo tl)c ftrftc equatton is, 

2aUcrcfo:c ^ tafec Oalfc .40;, tljat 10 . ; , lubofe 
Square is ''':\ . HnD tinto it 3: puttc. 44^;. U)i)crebv 
tOcrccommctb" ^% lul)ofc rootcia ";.from U)ljicl)c 
rootc 3; inua abarc ';, anD fo rcmamctl) ':; . tOat 13.9. 
^3 tl)r tialuc of. i.xp^. 3nD fo2 tf)c niiDDlc nombcr. 

SCben fo,2tl)c p:oofc:if.9. bee tbc miDDdl noinbcr, Thepm/e. 
t\)t fquare of it, \u\)icl)c 10.8 1, fljall bcc cqiiallc to tijc 
inultipluationsoftbcertrcmcfi. naijcvctoicit^W 
iiiDc.8 i.bpzo },tbi;/«o^/>«/f bcr«3:« 4. Dcdarctb tlje 
otbcc crtrcmc. 

OBaHcr. t^ou feme crpcrtc inougl) m tl)!:; fnr.ne 
of luoo:Uf . LSl)crfo:c .^ AuiU p;occDc to otbcv iiucfiioB, 
tl;at Differ fomcUiliat from thcfr. 

2^})crcarc.:.incnnc talUj,n'0togctijcr oftljrfrmo- ^^(,ttj/f 
n(c5, anD notl)rr of tbriin U)Utimgtocrp:rucplainlp queflion. 
l)i0fomiiTC,biitintl)t3fo;tc.i::i)cnombcrofangeIlcs ^ 
in mp piirfc, faicti) tl)c ftrftc mannc, maic bcc partco 
Into focbc 2Jioinb:r0,lubtcl)c ht'^^w^ mnltiplico togce 
tber, Unll maUc.2 4. ^nD tbcir Cuha bci png aDOcD to^- 
gctl)cr,\t)tll make. 2 S o. TLi\tv\-, quoD tbe otijer man. 
SnDtlje like maic3J fate ofmpmonci^, fnuetbattbe 
C«^f5oftl)c.2. partc3,lutllmakc.n9' iiioUj^Dcfire 
to knoluclubatmontcccbe oftl)cm IjaD. 

^cl)olar. 2^1)c ftrftc manncs fomc,3 fctto be vtp 
1dI)icI)c J muft parte into tujoofocbc partes, tfjattbei 
botbc multiplicD togctbcr,maie make. 2 4. 

£©aftcr. ;l3ou crrc \)crtr mor be. j^o; it is not pof- 
Cblctbat tl)e partes of ani^ Cofiike nombcr multipItcD 
togctlirr, can nuke an abfoUite nombcr. ^1 bercfojc 
in foc!)c cafcs/ojficrc pou percctue tbat tbcrc is rcqut^ 
rcD, after toe ftrftc pofitiou, anp multiplication to 
make an abfoiute nombcr, ^eu ftjall call tbc ftrftc no* 

l\k.iij. bcrs 



The^rtf 

bees , b^ fomc oiber name of plcafure . 00 bcrc von 
maic call tbe firftc manncs fommc.X ^iiD tbc fc ciio 
manncfi fommcS.ano tOen m ttjcir paciition,ia ttjc 
namcof.i.x^, 

jaiiD 30 tlHi arc tUJoo qacfltonc tn one,fc fiall rou 
make fcucrallc luoo;fec0 fo; tt)cm, 

^cl)olar. SLbcn (I,al 3f faie, tUt ti]: fir0c tmimtti 
fommc 16.^. ano it is DimocD as t)e Dcciareo.^ bire? 
fo^e fo; one ncntbcr of that tiiuifion,^ fet. i.xp . gmn 
tl)en tbc otUcr ttjall be rj, fo; as tbc one nom&er mul- 
tipUeO bp tbe other, Hoi it) mal;c.2 4. ^0.2 4.f .mu(^ 
DeD bp tbe one of tfjcm , muC ncaoes b;pna fo;t^c the 
otber . 

SDader. SCfiatislL'dlrcmcmbaeUofpou.i^o^as 

4»antJ.j.brniultiplicatiDn,Doemalje.2o. ^o,2o.Dii' 

moeU b^y,balngctl) foatt)e4,ano DiuiocD bp.4atrc^ 
Detb.^ 

^cbolar. ^0 V is hut 4, auD ^^ is.f. 

Rafter. dDo fo<jtb tbcn iDttb tOe reft of tbe ixoiie. 

fecbolan CbeWf of. i. xp .is.i, ct«anDtbeaA# 
of ^^ ts^rj Ujbtcbe.2, uombcrs ? maic not aDOe to^ 
getber , tmtiU 3 banc reDurcD tbcimbntooneoeno^ 
«unation:lubicbetbpng3!flja«0oe,tirrcftrng.i.ct^» 
as a fraiTiontbusi^. HnDtben luoo^ferng after tbe 
rate of fractios, in ttje firlfc rebucrion tbci luiU (lanDc 

[|^l^~ir "'^^i^* ^^^ ^^ ^'^"^^^'^ aODition tins* 

anD bctberto tbe Ujoo?!tc of botbe tbefc. 2. mrnres 
rommes^aremmffercnteanDagretnge. feo thattbia 
one tuooabe fernetb fo;^ tbeim botbe. l^ut nc li» tbci 
iBill Differ, fonn tbe filrae mannr 5 Uioc;Des,anD fg 
tn tbe U)02kcfo? bim '^ ^--^i'^il ts equalle to 280: 
but in tbe feconoc mannes U)oo;he, it mutt be ac com^ 
ptcbetiuallcto.np* 

Butfirffeto gocfoati^arba-ftbtbcfiraeman. fec^ 
r»ig-^^~5^^^t0cquaaeto, 2Sc. ^bmfo^ebp 

reoucton 



ofCof^tke nomlers. 



re&nrtionto one Denomination, ^^^^~te^"-"^ 10 c^ 
qualleto -,|^. ^no renioupng tijc comuron Denomt? 
nato:,tfjc numerators fljal kcpe tbefamc proportion: 
an& tt)crf0;e, 1 ^ c£ — ^ — 1 ] 8 74.^ .l^all be cquaile 
to.2 8 o of . 3no b? tranflation,to Uauc tfjc grcatcHf 

U.nominatton alone, i ^'cf.— =2 8 o oi 1 5 H 2 4*5* 

^bere J ajall it^t ttjc tjaliic of. i . i^ . tulnclic i^all 
not h? here accoumptcD the fquarc roorctiut tljc j^^n- 
^cuhiJ^ roote, or tl)e C«^/\5 roote of tt)c fquarc rootc, 
acc<)rDvng to tbe greattfte Denomination. 

ClCffierfore. i 4 o.in fquarc, malictlj 19600. from 
li)l)tcbe J mufl- abate 1^824. anD ttiecc aoct<| remain 
S776 Ujt)ofe fquarc roote is.7 6. Uibictjc bepng aDDcD 
tnto.i 4o.Dooctf)giuc. 2 i6.ano bcfngnbatcOfrDin 
It, !t leanctf). 6 4. of luljiclje botljc 3 mufli' cutraftc ttjc 
Culfi^e rootc, bicaufe m ti)c equation tDcrc arc.2.qua' 
titles omittcD. ^0 that of. 2 1 6»tbc Ctilu{&taotc is 6, 
J^iiO of. 6 4 . the C»ii>i^e rootc 10. 4 . l^ccc $ fee bothe 
rootes fcrue fo mppurpoft^tUat 3?rt)Mltaitc thcbotlj* 

!?©after, HuD gooD reafon.i^or as m fcttpng i to 
for vour pofition , vou coulD not tcU U)l)ctl)er it luxive 
t])c greater parte , or the leflpDc , fo mate pou not nolo 
applic It to either of theim boHjc, but taUe bothc roo- 
tes for the. 2.partesoft>ournomber. 

$>chotar. ^0 Docth the ftrfte mannes nomber np-- Theproofc. 
peare to be. i o. fei^ng the partes bce.4, ano.6. lohiche 
35 mate eramme thus. ilhat the! maUe. 2 4. br nTiilti- 
plication, it IS cafilj' fecn, 0nD thattheir C«^« aoDcD 
together, Doc make. 2 8 o.ts fone perceiucD: fcpng the 
c«^of.4.i6 6 4:anDtheC«^^of.6.ts.2 16. luhiche.2. 
nomfaers bp aDDition,Dae malie.2 S o. 

;3paffcr. j^oU) prouettiefeconoe mannes toorbe. The'tfor^e 

Scholar, jn his tuoorUe '^^^^==^1^;=^^^^ tcequalle cfthefccond 
to 559. CinDbprcDuction to one Denomination, it is f'»'^f- 
cqnallcto'-';^. ^othnt. i. §^cC — ^ — 15824.^. is 
equaUc to.^ 5 9.'X.anD bp tranflatton. 



The^rte 

T^n^cuhiks roote 3 fcfee^tbus:^^ ootl) mabe in fquaw 
^^^^^i, from lotjictje 3 m«a abatt '^'|,anD tbcn rcniau 
net!) -;:T.iuI)ofe roote is ^^I bnto lufjicbc 3 maie aooe 
1^1 . anD tfjcn bill it bee i^^^ttjat is. s 1 2. Uj^ofe C«^»)f c 
root^is.8. ^nD 10 one parte oftbc feconDe manner 
nomber. 3nD for tljc otljcr parte, 3 l^all abate ^' out 
of i^,%and tbere rcntainctft ^, tbat ts,27.tul)ofe Cuhikt 
roote IS. 5. ano Is tbe otber parte of tbe feconoc man^ 
Tbeproofe, IKS nombcr. Zb It male fone be ttico tbus. {^o; ^ tp* 
mcs.8» mabetb* 2 4. ano. 2 7. lubtr be is tbe Cuh ti.l 
aODeD iwttbo" 1 2.tDbicbe is ttic Cuht to. 8, uooet^^ mabc 
J 3 9,as tbe quellion mtenDetb. 
jfquejlion spader. £>ne otber queflion 3 Icill p;opount)e, 
9f Mamie. of.2.armtes bepng botbefquare,anD of likcnomber. 
flnD if rou abate. 4. from tbe one armie,anD aDDe. i o. 
to m otljer armie,anD tben multiplie ti^tm botbc to* 
getbcr, tberc toill amounte.9 8^272. 3 oemaunoc 
of rou, iDbat IS tlje fronte of tbofe fquare battailes. 

^cbolar. 31 call tbe fronte ixp^. ano tben mutt tbc 
battailebee. i. 5-. jpoUi afaatpng. 4 .from tbe one, it 

Ijjill Off i;^.- .4*f SCben aDDpng. i o.to ttjeo^ 

tber,itU)ilmafee. i.^.— f_,, o. f anDifpoumul. 
tipltctt)orc.2.nomber0 together, tberc luillamountc 

bpit.i.§-§>--|--6.^. ^4o.f Uibicbefomme 

mutt be equallc to. 9 8 y 5 2 7 2. 

i« 5*« — h— .lo,^. 
K^. .4.f 



4* ^« . 4 o« f ♦ 

f .«5?^!f.?f ? ^^^^^"^ ''*?'^'' *^^^^^^ P«^^<^« '^f^e c«l«a^ 
tton. It UjUI be. f j> ^^-^6 S^.equallc to.p 8 n ? 1 2 



ofCoJ^ike nomhers. 

out of lutjicbc laftc equation, 3s Hjall fcarcl;cfo:tl)c 
ljaIucof.i::^.brninItipIipngfira.vfnuarclr,U;l)crc^ 
offonimrti).9.anDtbcnaODpngitto. 9Sfi5 1 2. ilno 
fo commctt) .98)5521. \dWc rootc 10 . 3 ^ ^ 9 • ^^"^^'^ 
U>l)if t)c 3 muft abate . 5 . anD tljcn icnwmctl). ; 1 5 6. 
U)l)icl3Cij3tl)c fullnomfacr anD Square of tbc one ar. 
nnc 2un IjatI) fo; btis rootc. y 6. 

i^o; ai3 ijcre to one onclr quantittc omittc D , fo the 
firfte nonibcr, lubicbc ni otbcr qucfiios of immcDiate 
equations, tuas tbc tcnc rootc, m tljcfc tntcrrupte c^ 
quations 13 a rootcD nomber,anD is bcrea fquare no^ 
ber:U)l)ore rootc tbcrfo;c,3; banc D;aU)en acco;Dpng^ 
Ip. HnD fo; triall of t!)is U)oo;ke.^ 6. tn fquarc mabctb Thepmfe. 
? 1 5 6 . from tubicbc if pou abate . 4 . tberc U)iU relic 
3152. i^gain If pou aDDe . i o. tbcrc luill rife o 1 46. 
:3nDtbofe.2.nombcrjsmuItiplieDtogctbcr,DocmaUc 
9 8 T fl 2 7 2,as the queftion intenoetb* 

cpafter. S^bispoufectubattfe icintbcfeequa^ 
ttons, vet are tbcrc manp otbcr cquati6s,lubtcbc bcrc 
ht not fpohen of: hut bcre after pou fljall baue moarc 
largelvDcclareD, ifpou fljcluc pourfelf Diligentciii 
tbis parte. 

<anDoncqucIIion31ump;opounDc,f aflToplc tuftb J'queftion 
out tuoojUc fo? b;efcneirc , tbat pou maic fee tbcrc is of/lraungt 
moarc bebmoc. 2Sbere tsa nombcclubofc Square c^uatitn, 
abatcD bp . 1 6 . anD tbc firfte nombcr augementcD bp 
8. anD tbcn botbc tbct muIttpltcD togetbcr,tuUl b;png 
fo;tbc.2^6o. 

s>cboIar. 33 Uiill p?ouc tbc U)oo;Ue of it.^no tbcrc* 
fo;e fuppofe tbc firfte nombcr to be. 1 2^»E^bcn is bis 

fquarc i gp'tcbtcbcabatcDbp i d.Icuctb.i ^ 1 6f. 

anD tbc nober augcmctcD bp.S.pelDctb 1 ^ — ! — S^- 
S!^bcfe.2.nomber3muItiplicD togctbcc, Ujillmakc 

I cc—-\ — .S.^, 1 6.x^» 1 2 8.f .bepng 

equaHcto.2 5^6o, 



L5>, -A 6,^* 

I.cC. — h-^S,^-*— A 6.%^, . I2 8.<y. 

;9nD aDO^ng 1 2 8,f ♦on faotbe fioea of tljc cqimtton, 

(ttotllbejci— H-8p 16:^- ^268 sy 

againc aDDpngc. 1 6. x^ .on botljc QDc6 , it lutU bcc 
l.cC~|— 8^ -^=:=.Y6 xo^ —4— 2 6 8 8 f 

S©aller» CMbcreatftaicpounoUi:* 

^cljolar. 3i fee no ll)ifte7t)ut otbcr to Icauc tt^ac tt 
(0,2.nomberseqnalleto,2:otl)erclstonTabc.i.nom' 
berequallcto. ^ HnDalltbatifsabouempcunni^ng. 
i^o;t l)etl)crto 3; Ijaue learneo noe rule fo;: anp of tl)cni 
tiotbe.&o tbat 3i can not geflfe^lubat ttje firftc nomlcc 
jn<g!)tljcc» 

Rafter. SL^enomber(g, 12. i^nD ftis Square is 
144. from tubictje if poii abate. 1 6» it loill bee. 1 2 8. 
anD If pou abDe.8.to. 1 2. it luill pelDc.2 o. SIbcn mul- 
tipUpng. 1 2 8. bp,2 o.tl;e fommc U)iU be, 2 > 6 c. as tbc 
qnefttott oerlareo. 
Of other )[5ut to put pou out of Ooubte, tljis etjuation is but 

equations, a f tiflc , to otfter tbat bee bntourbeti . 0nD pet ^ toill 
tourne tljis equation a Utle,to giuc pou fome ligbt in 
iU ant! otftcr foclje. 0s ftere. 

l.C^.-~-=«l6.2o. — \ — .2688.f ^8^. 

tul)crepouree.i.cC«eqnalIeto.3.otbernombers.0nD 
IS It not certainc to pou. tbat t5)is equation is ttutf 

^^cbolar, peSjS amaDfureotbereof. 

50aftcr. ^0nu pet to auoioe Doubtfulnes tbc mo;e 
trie it bp refoluticn, accoumptpng. 1 2.foM.::o . 

^cbolar. ^bcrc.i 2.is.i.::^,tbcrc.i,5^.icVi 44. 
anO.i.cf^.i0.i72S.luf)iclje.!728,mullbceequaUcto 



ofCofiike nomhcYs. 

I6.XC tl)ntts.i92)anDto.26S8-rauctl?atvoamua 
ai3atcrs.?r>,*bat is i h" 2. aoU) if K atiDc 1 9 2.to 2688 
(tU)iUnTakc.28So.outofluDictieatatrngc. i \si* 
tijcrc Ujtll rcmamc, 1728. U)!)crbii 3 fee tDc equation 

\5 mac. 

naaacr. dim vou fee tijat tbe equation 10 true. 
0aiD can pou Doubtc , tliat anp nonrbrr , ii-bicbe is? c^ 
iiualle to a Quhike nomber.tiatb m it a C^H^f roote:' 

^cbolar. 3t nuin ncaDro be a c«t;^f nonibeistbat 
13 equallc to aC«^'l<nombcr : anDit'rrtfo;c nujftc 
neaoeg i)aue a Cubiks roote : altbougb 3. luiolue not 
toll) to crtrattetbat roote. , 

a)aaer. LiUeluates, luben J fate: " [' ^^' 

c^^:,,^^,^^___.,2./r.. — ^_ I 2.S.<y. itisfcrtatne, ''^"^f '^•> 

notonclptbat. 1 2./v«- '—-i -^-f '^o"^"'"^"^^^ i" if 
asmorljeas.S. ]y>cc . butttiattbe. S. parteofttioa 
5-cf .nomber,anDtiatl)a^fw;<''f«/'/^ciootc. 

i^nD fartber it is manifettc , tuat as euerr • 5 - cf ♦ 
itomber,Dooetb cojitamc in itcertaiw./^ . nonibers 
cra(tlv,fo If anp nombcr be anncrcD luitl) tl)Dfe Surfo^ 
li(ies[iis bcre m tbis erample are fct 1 2 8, it is of nerer 
fitieltbat tbat. 1 2 S.niua contame m it ccrtauic Sur/o^ 
lilies crarrlr. 

S)0 if.S.^ ' ct . bee cquallr to The roote h 5 

io./5^-i>-2o.^-> >;.->-- ' -4oo.ce-4-;i2)-.f 
itnTiiilneaQrcbcibattlir.H. partcoftbisfonipounDc 
noinberfl)allbeea.5>rc. noinbcr. riiiDnlfotbattbe 
V V' ^^'^^ ^'^"^ ^'^^'^ iianibcri; fcloU'inig Dooetb con^- 
tain: a certain no "brrcf./5-.nombcri5.;lnDtl)e.c£. 
inltljef?:teinrlMD:tl)anouibcrof.^5:^-.n-)niber0. 
r!i*t) laac cf all.; 1 2 s -^.f . Doctl) comp;ebcnDe certain 
O.^f/(^cnoinbcr5craitlp. 

jn liiie fo:te,li5l)cn lue fate,tl)at.i./§>.ts equalle to The roote 
(5.^:^. — ! — '8.5-. — 1 — .9.<y. i^lltbiscompounDe herenr^ 
nomber 10 a Swfolide , and batb a ./^^ .roote . 3no 
N v« ~4 — * 9 ' *?♦ mcluDctb ceciflinc G^ba . HnD fo 

"^ Xl.U. Doetl] 



Thejrte 

'Bwt oftbefe ano manp otbcr tJcrieercdUntcanD 

tuondcrfullc tDOo;ke0 of equation, at an otber tpmejl 

tDtinitftructe^ou fartbec, ifj fee ?our Diligence ap* 

pltto toell In tbts , tljat 31 banc taugbte vou, 

janD tljercfo^e ftere tuiU 31 maUr an 

wnDe of Ofiikf nombrrs, 

fo^tbtstrme. 




OfSurde nomhtrsyin Jiuerfe fortes 

Mifirftf ofSurdc nombtrs 
yncem^ounde . 

^to t!)at pou !)aucfomc^ 

tuf)at IcarncD tbc arte of Co/^ 
ftkj uoniijcrG, Uiitl) tUc rule of 
equation, It fcmctl) gooD time 
anO aptc plarc , to tcaf l)c vou 
tbcartcofSMr</f nobera, UjI)1^ 
cbt aieoiucffc m namcacco^ 
opngas tljcrcare Diucrfcna^ 
tuic0 of rootcs , Ujf)uljc mate 

giuc tljtinmamc. , , , . 

jTo: gcuerallv ,a S«r</e nombcr ts notljrng cls,but jsurde 
focljc a nombcr fct fo; a rootc, as can not be ci;p;eireO f.ombcr. 
bvanpotbcrnombccabfolute. 

as t\izSquAr( rootc oM o,o: of.S.o: of anp nombcr, 
tbat 10 not fquarc. LiUcUiaics tl)c Uihike note of. 4. 02 
of. vo: of anp nombcr tbat i3 not (jihikc. ^0 tbc ^:rr^ 
^Ven-iVmffof.S. i2.o;.2o, o,:ofanpnombcrtl)at 
\&no^enxi:KS^^ke, ts callcD a s«r^f nombcr, anoiii 
like manccanp otbcr rootc of anp nombcr.tbat batb 
noe focbc rootcDoctb caufe tbat nombcr to be a Surdc 
nombcr. ^ . ^ , 

i^o; If ^ou fee tbofc Cgncs annercD luitb nombcrij, 
tbat batb focbe rootcs , tbofc nombcrs arc not Surde 
Mombcrs p:opcrlr.but fette ItUc Surdts, aa tbc Square 

rffo^<rof.4.o;of.9.o;.2vtC. %\)tCubike rootc ot, S. 2 7« 

0^ 1 2 y. f c. iDbicbc fomctpmcs isi tofcD fo; aptc luoitic, 
ad pou l^all fee bcre after. 



OfNjoneratton, 



/^^ ^ie numeration of tbfooctbconacic, (It l^ttotu.' 

5 4 lege of tbcir figurcs,tubicbc partly be Declared 
•^•fcefo;e.^uttbcir common anupcculiarcfigncs 

aro tbcfc,/* W. s/. aitbougb tbcrc maic be moarc 

Ll.u). Imnctiee 



The Jrte 

tjarictics: t^f t tl)cfc fo;tl)tfi t^mc mate fuffifc, 

s:i)efirac,tl)atis. v^ tj3 cuftomnfalp fct, tofigntfic 
tJiS^Mrerootc, 00tt)tsV.T. bCt0hrnctl)tl)C5j«drf ro»/# 
Of.f. ianO, v/.l 2. iSt!jcS^«<reroofeof. I 2. |))Otubcit 

man? tpmr 3 it liatlj tuitl) tt , fo; i\)t moarr c rrtcmtic 
tl)cCo/?/lfCgKc.5^.anDt6U)?ittcntl)U0.'/§%2o.tl)e 

S:!)cf£CcnDcfigncisai«nc)t;eD \M\tiiSurdeCnhes, to 
crp5cnrctticiriootc0. :a0tl)ig. w'.i6. teljicfiefigni^ 

flptl) the W/^f roo^r of. 1 6» ;311D.^^v^2 o. faetoUcilCt^ 

il)c Cul^iiero9te of.2 o. ;?(nt> fo fo;tl;c» HSutmanr t^^ 
nics It tiatt) ti)c G/?/t* fignc luitb it alfoiajt \u/.(nL 2 > 

tbc Cubil{e rooteoti y, ^nD. ^v\ .cf . 5 2»t!)C C«^/;^c roo^tf 
Of.52» 

IXbetbittJC figure Doetl) rcp;cfente a ^n:^^n:^illie 
roote, ^J3.\>/. 1 2.1s tl)C ;<f»;</;^f»:^% roo^r of. 1 2. illlD 
v/» 3^t0tt)C;<f«:</;<'»;<'^fyflo^ffOf.3v i^nDUUcUiaicg 
if it 6auc luttb it tiie CoJ^i^e fignc. ^>g-. 00 w^ >; >^2 4 
tl)C s^^i:^7:jkf route of. 2 4. aitD fo of ctljcr, ' 

^ctjolar. 3it lucre agatnfte reafcii , to ft Uc rrafon 
fo^tfjofe fignrS) toiiirlje be fettjoluntarilr tc figntfic 
ant' tl)r"0'altl)ongt) fome tfnrcB there bee a ccrtainc 
nptc ronfo;mit(e m for be tbimgec. ^iiiD in tbcfe figu^ 
re0,tl)e nombcr of tbeir ininome0,ri'rinicri) Difagrca^ 
faletotbeirciDrr. 

spatter. Jn ffiat tbcrc 10 fome renfati to bee fljc? 
lDCD:fo; a0.v^Dcclarttb tl c niu(ti^Ii:atfcn of a nom^ 
ber,onc0 b^ it felf: ^0, wv . re p;i fcnli tli t^>at multi^ 
pltcation G'H^r,in U)bifbctl)c rocteis rrp^efcnteD 
t'o;tfe,^nD.\v.CanDetl)fo:.v/.v'.tl)atfjE*2.ftgarc0of 
Squire nrulttplication : anD 15 net erp:urcD n^iti) . 4. 
mmcme0. j^ojto fljoulD a fcire to crp^clTc ir.oare 
tbcn.2.S5M4rfmu[tiplifat!on0. 13utoflJoIiintaiieft^ 
gne0, it 10 inougbe to feno tie tbat t\\\Q tlie i Doe figni^ 
ffe. nnt» if anp marine can DtuiTt Dtt)cr,moare rafie 0? 
apter iw t3fe:,tl}ci mate lueU hz rccc;ucD. 

15ut 



ofSurdt nomhers. 
15nt fonccrnfng tbc mimcration otSurdt nmhcti 
tW ftjal pou macUc:tl)at lubcn nn? compounuc Ogne 
tsputtc bcfo:c a nombcr,tul)iclicl)atl) anp roote,tDa^ 
maicbcccrp,2cffcDbi? parte oftl)at figne, tl)at nom-- 
bcr IS not abfolutclr fo to bcs crptcfTcD , onlclTc it bcc 
foKafcojaptncrrc mU)o;kc. a0.^v/ 5;5%56' ^y^^^ 
cbcbctoUciKtti t!)C<f«;;jvf»^'l'? roo^eof. ,6.S>crng 
itisUJcllluioU)cn,tt)ato6.t)att).6.fo;r,i3S^«4rfroc-ff, 

itlucrc moarc aptc erpjclTyngc tl)at nonibcr t^us. 
v'::>6.tl)atistbcrquarciootcof.6. 
'btlKi-luaics, If tbc nobcrtbatfololuct!) tbc ngnc, 

banc a rootc acrrcablc to tbatfignc : iti^; imSurdc 
nombcr. as..".^>. i6.is.4.anDisnocS«;v/fB»nii'a. 
S>c, Av'.^"-. IS- vaiiDncaDctb not to bcc Uj;ittcnin 
Surde ro:iuc,crccptc It bcc fo,2 aptnrlTc of U)oo:Uc. HnD 
bv tbis mate pou iiiDgc of all otbcoac tbci come m \Dfc 
>fl)olar. jiftbisbccall tbatisrcnuifitc tonuuic- 
ration , ^T p^alc ron p,:occD . to aociition . j^o; that is 

ncitcino^Dcr. <!.,-.,.» 

a^altcr. Cbr;tistbcconii:!on o,:na-. noiuucu in 

tulgarcfraciions^rciircincmbcitbatmnltipluatioit 

ano Dunfion , arc fct bcfo:c aDDiticn awa fubtraction: 
bicaufc of tbc cafic r fo:mrG of U)oo:Ur,tn mtiUipUca^' 
tionanODUiifio. 2n^[omi\)tii:Surd€nowbers,bicau^c 
tbc U)Oo;Ucs of multtplication^ano of D!iiir5on,bc not 
onclr moarc cafic, tbcn tbc tuoojkcs of aDDition, anO 
of rnbtraction^btit alfo br rcquifitc to tbcm, tbcrefe.JC 
U)ill j; begin luitb tbcm,anD fo come to tbc otbcr. 
Of hiulttplicatlon. 

>UUipllcatl5in Surdemmhcrs tncopoiinDC 

itiatbnocmfficulticiftbci be of one Drno* 
Jmination:clsmulitbcibercDatcDtoonc 
jDcnomtnation: anD tbat bp muUiplicati 
,.^s.il3n,acco,:Drng to tbctc figncs. 
"^utlu^'^rcnoc r^Diution ncaDct'o , von l^aUnmU 

xwAx 





Thejru 

tfplie i\)t nomljew togetlicc , anD fcttc tWt romrnoit 
figne before tbc nomfaer, t^at rcfuUctb of tfjat multi* 
plication. 

Examples of/quare SurJes. 

iif^?outDmmuItipHeV»^,K.bp./.^ 
|26atU)mmabeV.^.59o. ^ 

!l ^oV.^.32.mulIipUcD fapV.?>. 48. 

tJoctIjpelDeV,j>,ii76: ^^^^^i- 

l^otobeitfometpmesUtappcnet!), ttot^enom.- 
btt,i3)l)ic\)c IS maDe bp tbat multtpliration^ig a nom* 
oer abfolute, ano not nSurdenmhr, 

Examples offoche as make 

ntmhrs Mfolute, 

j^o* ^ -/. ^ 

i/* 3 6.t&at t0^« i/.i44.tl)atl!e;.i2~ 

^*4»-f> v^.7-' « 

v^»y 6,^ tbat is.7 1. -/.2 o y.,'^ tijat U5 1 4 1. 

^•56oo,tbatr06o. v'.42 25-.tlmi0.6^ 

0no generallp tobeu anp nomber is multiplfco bp 
an otbccif tbe p2opo;tton betUjene t!)ore.2,nombew 
bet rcp^ftnteo bp a Square nomber, as bp. 4. 9, 1 6. 

mU^plSn "^ ^^^^ "'^^ ^ ^^"^'^^ "^"'^^^ ^^ "^^^*^ 

Crample^ 



of Surde nomlers. 
Examples of Cubikt rootes, 

Wv'.C^. 91. sw\'7"\* W. 1)6' 



^vv . C 



'Examples offoche as mah 

jlbfolute nombert ♦ 

^^v^ 7 4* ^^^'» 6S6» 

wv . 1 7 2 S.tljat 10.1 2. vw'.i 7 4 4-tl^at tg.i 4 

A\\\ 4S6» 
\vv . 9 6* 



Examples ofi^n\t:^cn:;lhrootes. 

M/'. ly. W» 204. vv/. J 6 2. 
vs/. 7. vv^» 2 6. \v/. ^2. 



^a/.Ic^. 


W.no4. 


iv'.n84-tt)atis.v^72. 


u- 7N 




\a/. 2 7. 

\/. 1 2. 



^v . y ;;.tt,aii^.. .2 ;. -v/, $2 4.tljatis.v/.i8* 
Examples of :^efj^i:^eti:iike rooUs 

that ma{e abjolute nombers. 

s\\ ?2. w^. I2S» 

H^. 8. va/. ^2» 



v/»2$6,ttiatl0.i6» iv/.4^96.tijatls.6 4. 



w"* 288. 

'W. 2o736.tljat 15. 1 2. 



115at bere ffi to face itotcD, tfjat if pou tooulD multi.- 
pltc artp 5«r^r nombcr,bp an abfolutc nombcr,o;j anp 
5«r</rnombei:ofoncDcnominatio , fapa5«r;/f nombci' 
of an otber Dcnomination:tou muft 6rac rcouce tljat 
abfolute nombcrtotfte like Denomination. anDfo 
mullpou reouce tf)c,2. ^wr^/r nom'berfi to one Denomi- 
nation. 

anobicaafctbattljijs tDoo.2bc Doetfj feruc often in 
rtts arte,anD tbat in Diucrfc U)oo^be0^3 ujill fct l)crc 
tbcarteofreouttion* 



Ofredu&ion inSurdes, 




|eoumonin5«r//«,isft!)cb;in5pn5 
, of funD;ic Denominatios tjnto one. 
;?ICl^tcl)e m abfolutc ncbcrs ijs t\m 
.Docn..i3oun)a[lmuItiplictl)cabfO' 
I lute nombf r,affo;Ding to x\)t fignc 
, of tbc Surde, auD tljcn ft rtc before it 

I'tbe like figne.fefl tbat if tou luoulo 

Double. v/.§-.8.tt}at IS to faic, if pou IdouId multiplic 
it bp.2.pou mutt firttc muItipUc t!jat.2. fquardp,anD 
tbcn multiplic tt)ofe nombers togetljer. SDbatiBto 

faic,poufi)aUmultipUc,/.^-,8.bp»/.v.4,anDfoi« 
it Doubled. ^ 

3likcU)aic3,to fnpU at\^ Equate Surde, i^ to multi^ 
pltc It by. 9. ano fo to iusdriple anp fquarc S«r</f ag to 
multiplic It bp. 1 6. ano fo foitljc. 

ISutif pou Double anp C«^?nombcr, pou fljall 
multiplic it bp. 8. t!)at id ti)t Cuhe of. 2. anD fo if pou 
hjoulD iripUa€ubi^e rootc , pou muOc multiplic it bp 
2 7. anD if pou tuoulD f «4^ry/;/* it,poii l^all multiplic 

4t 



OfSurde nomhers, 

rtbf.64. 0nDfoofot!)erlibcU)oo:fec3. 

ilgain,ifvouU)iU Double anr;<f«<'<f»»^'^'rootr, 
Xtix muft multtplic it \i\\ 1 6. i^nD if j'ou luul trti>le it, 
f ou fl)aU multiplic it bp.S 1. and fo if rou U il! quddr'n 
fit it.vou muft multiplie it tp 2 ^ 6. ^nD m liUc niancr 
cucr moarcfo: the nombcr abfolutc, rou ftjal^ J'ct bi3 
:^fw:vr^f»;^'^f noiiibcr. Like as m Squares , fo: an^ 
nombcr abfolutcpou C^all fct Ijis fquarc-^nD m Cuba 
poul^alltaUcbteCa^^ 

^ct)olar. %W 10 platnc inouglje: pctBJ p^atc rou 
put an rramplc 0; tU)oo,of ccbc kmoc* 

i^allcr. SDaUc tljcfe examples fo; fquarc rootc0» 

2. 6, 12. 



IJ2. '/.§-.46o8. 1/.469976. 

Examples in Cuhike rootes, 

v>v^. ^2. w/* 16 ). W. 4806. 

____± ^ §1 

V.V. 416. ^W\2ol7S* 5w/. 2460672. 

Examples in :^c}i:::i^en^tke nomlers, 

K'/. 69. ^v^. 2 p. w^. IjS^ 

2* 4» S* 



^".1104. \v/.642J6. v\/. 22^0625-. 

fe>cf)olar. 'CW S pcrcctuc U)cll.li5nt notn in SuUe 
nombcr6ofDtucrfeDcnominattons,tDl)attbco;Derof 
rcDurt'o is,S P-aic ^oti to fct fo;itl) U)itlj fomc cwplci 

SDaftcr. s:t)tfc cramplcs tout) tbetc Declaration, 
male fufficicntl^ fcruc foj a lt)cU)c, if Bl iDoulO multt? 
pUc.vW.u.bp./.^^.BimuftfirftcmuUiplicttjenom* 
bcr of ottc figne , a«o;Dijngc to ttic Ognc of tt)e ottjec 

Q^m,\^. nomber. 



The Jrte 

nonibcr,anD To alter to:m botljc. tabicfjc luoo^Ue 10 
lihc X\)t rctinaioii of fra ttcni , to on: ro-iimon bcno^- 
mniation. ^sljtccn niuacmt'I ipUc. ).(jihii{tly,^\\^ 
I z.muft bcmuUiplicD IquarclisniiD tl).n iljall jt aooc 
botljcfisncsinontjfonlicirro.nmonrig-nr. ^oOjall 
5 liauc fo;tt)cm tbcs- ccaootc of. 1 4 4- to be multi^' 
plicD bv'tbe ;5;5n^icw^/^erO0tCOf 
izy. aiiDfo luiii tDcre come of s'^ycZ* 144. 

that multiplication, tUr^f»X/f"' '^'♦5"^« '-v 
^ii^e I'BOtc of. 1 8 o o o. ^3 Ocrc bi» - ^ -^^^ — ^7~~~ 

3lil5cU3atc3 If 3 UjouIO nniltipUc. \/. >i" n\2 >' o. bp 
ann/» 54. g (^allfir(lx;multipUc, 2 > o. Cuhi\ely, ano it 
iL'ill btc. 1 > 6 2 y o o o. 0HD 3 4. mud 5 multtpUc ;^f«^ 
s^^en^{ely, anD it U)ill vclDc. I? 5 6 5 ^ 6. ca l)crcfo;e 
iHiiltipui-ng thcim together, anD aatipng tbcreto tbe 
ro']TmenDcnonnnation,itlmUbecti)e.5-^-ai.reoi$ 
of. 20880250000000* 

i::i)t3lDOo;Ueisaptlv rep.:crcnteDin figure, after 
tnisfoatc. ?vnO'tt)cn l^all 
l^oumultiplie croHcloaus ^^ ^eir'S"' \/ ^^^'Cf.. 
tbenomberoftjjc onc,bp /\^^ 

tbcfigncoftbeotlier, 0nD ^^ ^* '"+' 

fo mate vou Dooc in all otljcr like nombers, of Diuerfc 
Denonuiutions. 

Sbis reDuition Doetl) fcrue fo; anv otlnv U3002UC, 
ass tutll as fo; maltiplication. 

OfDiuiJion. 

s3;uifioni£iascafiea5muItiplicatiom/o;i 
m It tbere is nocregaro ba^ to tl)e fignes. 
13ut tbe one nombcr DtuiDeD br t\)c ctl)cr 
as If tbei locre nober£l abfolute. fjiD tben 

^^^^ 3.tlje firtte fignc aObcD to tbe qustiente, fo; 

tb5mo^Ugl)teaiiDcmatntic,3! Ijaucfet f)nc,eram> 

plesofecbefo^tCt 




ofSurdt nomhers, 

^ndfirjl examples offjuare rootes. 

Examples ofCubike rootes. 

'^''- ^^\f (W. 74,',- 
Examples of :^en;^:^en:^ke rootes. 

^nt) tbi3 mate fufficc fo; Diuffton. 2^f)c p;ofc offt 
(0 bp tbc f ontrarp lunDc. i^02 r»7nIttpIicatioii p;ouct^ 
t^iuifioitianD ZMuiDon trictl) O^ulnplication. 

^c^olar. 0UtI)isi3caricinoiig:bc to remember, 

Oj Jddttion. 
^Batter. 

-.4p*5^^pMttonf0notro caQcbuthatl) niwikThefirJle 
P J>j|o,t3aricttC0 of Ujo;kc,a5 anon lljall appcre. /ormr*/ 
fll^^^'Oi^ Ocrcof tbc Orftc is as cafic a0 can bee. Mdition. 
|r4^ ATlf o; it requirctl) onclv tl)c ftgne of aDSin^ 
fe^;i^^on. — ! — . 00 If 5 tooulD aDDe.v^ i :.to 



The Jrte 

i/.26» |C9alirctftrtju5V*2 6. — t — V.i2,a«0ft 
v/.zo^puttonto.v.M.ntabcttj. */.5'4« — (— -V.zo. 
%\)\s fo;mc fcructlj cbteflp fo; rootcs of Diocrfe na* 
inc0. :^s » iv^ . ^ ^%2 o. — -| — . vw ♦ c£ . ) o. Inhere 

Thejeconde j-fj^ feconDc f0;me <s noi fo taOeianti pet manj? tt* 
/orw#, mcs!tisinoarcccrtaiiie» ^nt)tt)isist1jco;Dcrofit, 
pou l^all fcttc oounc t?our . :. noinfacrs , ttjatp oa 
luculD aDDc togetljcr , fo;fepng t!)at tij^i be of one Oe^ 
nomsnation. £:t)rn (^allrou aDDempIame fo;mc, 
t\)t\t nombcrs together, puttriig thereto tt)c (ignc of 
tbc rootc. ^nDlicpc that as a parte oftljc aUDitiom 
3gain pou fljall multipUc tlje.z.firtJe nomberfi togc^ 
tber. :aMU ttjcir totalle pou l^all multiplu; bp.4. :anD 
before that l^aUpoufette tijcfigiieoftljeroote, ilno 
itl^allftanDcas tbc fcfonoc parte of tijat aDDition. 
^0 tbattfjofc. 2, partes, fljall be aDOeD iuitl) tbc fignc 
— 1 — ♦ iano ttjenistlje luoo^ke canDcD. eramplc 
tjcreof. 3J luoulD aoDc the . 2, firftg fominrs , tijat is, 
v/, 12. to. v/.2 6.Uil)crfo;e3| 

fet tljetn tbus. SlnD tljen 00c . ^'A^^H" — ^^ J ^ 

3? aDDctljf lJDtl)e plamlpto^- 
gettjer, anDtI)ctinaUe.^/.;8 v^oS* 
1d!)!cI)c 3 fet bp, as one part 
oftljeaDDition. E:i)entJoe3l 
ntulttplie /.:6. bp./ 1 2.anD 
t^ere rifetl). v'. 512. U)l)icbc 
3 mutt Double, o:tiTuIttpUc 
bp.2. anD tljerfo;e f£t?ng tlje 
tooojbe ts in fquare rootes,3! fet tbe fquare of 2.tD(tb 
tl)efigne of.v/. fo;j«2. anotften muItipU^ngtbcim to^ 
getl)cr,31 t)aue. -/. 1 2 4 8. txjbicbe is tbe feccuDe parte 
of ttje roote. ^ bcrfo^e aDO^ng tljofe. 2. partes togc* 
tljer, Ujttft tbe figne . — | — ♦ tbcre rommctf). -/■ 5 8* 
— f— V. 1 2 4 8. as tbc totallc of tbat aODition. 
^f l)olar, j^s me tljinbetf), tfje firtte fo;me otahtiU 

Hon, 



1/. 


2 6. 




5 1 2. 
4. 




ofSurde nomhers. 

tfoit fcruttb better fo; tbcfenomber^^tficn tW(t^ 
conoe fojin:. f o: it ts moare eafic to t)fe,(n anp binoc 
of UJoo:Uc,anD moarc fpcDil? Dofn:aiiO it fcinctl) tbat 
t^ifl lattc nombcr, i3 moare obfcure tljcn tbc firftc. 

:®ailcr» petiB tbis Ujoo^Uc gooD,anD tjcrp neceP 
fane, fo; in tbcfc nombcts > ana focbc otber Ube, it 
fcructb onclp(a3 appvCctl))to alter tbc ttatc of tbe no^ 
l«r3,U)bcrcbptbcimatcbcccoinmcnfurablc,U)itbo^ 

tb:r , tbcn tbei tuerc before tbat alteration. But in 
fomc nomb:rs,anD tbat tjerp manp,it reoitcetb tbem 
to one ample fojme of rootc. as bp tbe crampUg folo- 
luyngroul^allpcrceiHe. 

anerample, 

^l)eramc crample otber ^ tbirJe 

Ujaies lujougbtc, forme of 44* 

, diiitn. 



./ ^ 



\ 7. 






2 8. 

7. 



?V 



v/.i96.U)boferootei3i4 
14. 



v/o5^»-+-'2 8* 



V. 196. 

'/.78 4. 1 

v/.?^— i~v/.7 84. 

£>;V.?v-4~2 8. 

S^batisV* 6 5» 

tiailbcrefiraeSI bauefet fo^tbc. 2 , crainples of one 
atit)ition,tbat i?ou male fee tbe agremcnte of tor bott) 

anDfirtteBDiuoulDanDcV. 28.U)itb.v/.7' Ujbcre^ 
fo2c % 000c io?n2.2 8.anD.7. m one fommc, Ujbicbe 31 
fet a partcaa tbe ftrae po:tion of tbeaODitton. 2:ben 
^ uoc multiplic.2 8.bp.7. ^nD tbereof commetb. 1 9 6. 
iDbtcbe 13 a fquare nober,anD batb- 1 4* fo^ bis rootc. 
too tbat 3 mate ^fe nolD.2. U)o;ike3. f 0; otber J male 
tonttnue mv tooo jUcas J banc Dotn (agvcable to tbe 
flrfte eramplc) mmultipUpngtbat. y^i 96.br. vM« 
(tobichc is but ooiibling}anD fo tbere cometb. v^7 s 4 
^ ' \Dbiibe 



The Jrte 

tul)icl)C i6 a nomber afafolute: biraofc it fjatlj a rceffi 
acco;jDpiigtol3i5fignc, tuftuljeicote 10.28, anoirat^ 

j^oUi m tl)e feconDe iu0o;jbe,bicaufc tbc firffr muU 
fipluation of.2 8.U^ y^latt^ maUea fquare item be r, 
9 Doe tabe t!)c roote cf ttjat nombc r foi it : fr jng it ts 
all cue tl)??njj to faie^ -v/. 1 9 6, ano. 1 4. fo;, 1 4. is tlic 
roote of. 1 96 ♦ ;antJtl>^n baupng tl)c rootr^ 3 muUe 
twiiblc It , acco;!t)^ng to tljc rul^, o;i muIttpUe it bp.2» 
«HOti)ereofcomractIj .28^ t^birb^K ft>all aDDcVuit^ 
^'5", aiTD fo feaue 31.6 6* toljcfe rootc containetl) tl)e ao? 
mtion orV»2 S.anoV.y. 

|5cl)olar. 2:^15 luoo^be fcmctlj ftraunge : anu fai» 
tftcfte fcom comwcn tcafcn , of all otbcc U.oo;hfB ni 
tbwartc, 

£©aifer, 3^ migl)tc cafilp bp Demonff ration make 
foiuto perceiue as mocbr rcafon m tbis U;o;be, as ca 
be in anp:fo^ it Depcnoetl) of tbe. 3 8, 2Ebi:o;cmc of tbe 
pattbelDaie,115ut bafte of otber bufmeCTr, inaUc tb me 
to omittbeDemonllratton attbis t^mc, icticbc Ibo;;- 
tlpi'Ou fballbaue, foa alltbe equations , ano ottjec 
luoo^fees lilj£Ujaie5» 

5iB ut fo; tbis p;arcnfe fpme,it ajall be rumcir ntc to 
tuojUe an crample m rationall ncmbe li^^ns if tbci Uiec 
Surdc ncmhtXBiti^ut tlnibv i^cu nraie pcireuic tbc o;^ 
irccanD tl)c txmhz cf tbe luoo;:tie. 

^bfifo^eSJtaUetJ^gretluoo ncmbrrsV* ^6♦ anu 
s/»4 9. to bte aODeb togctber.Mbcre 3 Doc fciiic act/e 
tbe ttuoo nombersplaineip togetber: anotbci maUp 
8 5" . fo? tbe ferfte parte of tbe aODition . SZbcn ddcc 'i 
multiplie.49 bpo6.anDtbcrcnrctb, 1764. lybuijc 
16 a fquare nomber. :ano tbercfo;r niaic 3 W.i. \x:op 
lies,as vou fee. jn tf)c firi!c 3 multipde tbat fenuarc 
nombcr b^2.o.2 bpV.4,l«btcbe is ail one : aiiDtbcre 
toetb amonnte.y q 5 6. a Square nombcr alfo, lubofc 
rcotc IS* 8 4» 

SCbe 



OfSurJe nombers. 
%\it firtte fo;mc» JTbe Tec cnoc fb^mr . 



v^.^6.- 



-V.49-1 









49 
294 
1764 

4 



49. 
j_6. 

8'^ 



v/.56 — h— v^»49 





49. 
?6. 


2 94- 

147. 


|Ct)atts.42* 

2. 


v'.Sj:-- 


84. 
4-84- 



1/. 7056 

iD;V.8r.-H— '84 

SCftatf^V. 1 6 9. 

fti eftc feconDe tDao;1fee 3P tafec tbe rootc of ♦ 1 764* 
tDl)tcbc 10 4 2 ano Doubling 1x3 t)auc 8 4.a0rcab(e to 
f l)c ott)cr iuoo;^* Xbcn Doe 3E fettc tbore.2.nombcri$ 
Donne tottt) I ) ano putte to tfjcm tfjc figncV.fn 
toUcn tt)at 31 tnuHe tabc tbe roote of tbat (ompounDe 
nombcriano not of anp one parte of it, 

^cbolar. ICbat bauc 31 marbcD UjcII: j^o;j 8 y. Ijatb 
no rootcnotljet: 8 4. batb anp rootc, IIB ut 8 s — f — 8 4 
tbat 15. 1 6 9.^atl). i >fo;! biB roote, 

anD fo 3; fcr,tbat tbc rootc of. \ 6. hjbtcbe ffi.6. 0nD 
tbc rootc of. 4 7« that is, 7* bcepng botbe aDDcD toge^ 
tbcr luUl make, i ^ tbat id ttie roote of. 1 6 9. 

iT^aftcr. jnct one otber fo^ne of cafie tooojke , 3[ Ofnmhrs 
ItJtU t^clucroM» iubicbc 10 botbe picafaunte ano p;0' ^^i^^enfurA 
fitabic : 15ut 13 not gcneraUe, fo: it fcruetb onelp fo^ ^lf,*fomtht 
nomhm ccmmcn/urahh, 3 meanc forbc nombecs, as b^fi'^f* 
onccommonDiutfo:, mate bee b^cugbt into Square 
nombcrst ^ttb lubicbe nombers, ^ou f^ait U2oo;ke 
tbusu 



Thejrte 

^itttt hittint tbeim bp tbe commoit o(uifo;:ano fet 
fo? tbem tbcir rootcs. SDtjcn aooe tWci. rootes to^ 
getljccauD multtplte it fquarelp.anD tbat fi^uare be* 
trtgmttUiplictbp foe common Diuifo^, luiU b;^ngc 
fo^tbe tbe Square of botbe tl)c rootes. as fjcrc follow 
tuotl) tti erample. 

i/»584t)nt0iv/.ijoUjl)icb 77 — ^ -"^ 

nombers30pferamm,ta 6.) °7 ^^* 

3J mate fiiiDe tbeir commo, ^ ^ 

linir Icafte &iu ifo; , lubfcbe * ,^* 

l)erets,6. SCbcn DiutOtng — L> 

tbem b? tbat6» 3J baue fo; 16 9* 

384.afquarenomber.64* 6. 

anDfoM^o.jbaueana* ioi4» 

tl)erfquare,tbatis.2^iDf 
lubicbe botbe fquares 3! fet Doune tbe rootes: anti tbe 
rommon oiutfo;i alfo, Cben Doe ^ aDoe botbe rootes 
togetber,anotbereofcommetb. 1 5. tubofe $>quarc is 
I69.tl)at3IDoemultiplicb^ 6. iubicbe is tbe commo 
ti(u(fo;i,anD it tuill bee. i o 1 4.U)t)ore rootc uoetb com 
taiH botb^ tbe rootes hcfo}c named* 0s pou C^all fee 
it paoueD anon bp ^ubtramom 

^cbolar. Bin tbe nleane feafon 3 con0oer,tl)at one 
of tbefefo;jmes,maiecontirme tbe otber. ano tbere<f 
fo^e if 3IU)oo^Ue tbtslafte erample,bponeoftbeo.f 
tber f02mes,attij finoe tbefame totaU, tt muft neaocs 
be tbat tbe luoojke is gooo. iMbicbe 3 pjoue tbus. 

i^irac«fett^ng Dotine tbe nombers,in fo^me of tbe 
tafxtitt aoDition. aiio tben ao&png tbeim togetber, 
31 finbe.n 4« iobicbe 3f fttte a0oe,as one parte of tbe 
tiomber^tbat 3 boe febe foj. 

C^ettuoo03| multiplietbe. 2. nombers togetber, 
and t^ti mabe . f 7 6 o o , tubicbe 3 iwoe multiplie a* 
gain bv 4» 3rtt) t^ere nfttb.2 ^040 o^being a fquare 
nomber,anbbatb.48o. fo^btsroote. Mberefo;«:3 

fet 



of SufJe ubmhers, 

v^>^84 — 1 — v^.iyo* fet.n4'a"l'»4So»to^ 

gctlicr. iMitbtbcfignc 
584 of aoDinon, t!)twi. 

112 n4— +-48o»ani> 



584 
170 



19200 
584 



5'76oo 
4 



250400 



) $ 4 tfte rootc clifeat iwm-^ 
btVfiBttiUSLlie to bot^iC 
tlit dtttt tootca, y^nt 
confiDertng ttat botbc 

, , / , bee jQptico lafteofall 

v/.j,4— H-/.250400. ij,^jjj__|_^arenom' 
£»;♦ v^» 5 54 —I — ♦ 480. bcrs rationaii aiiD aiM 

jCbat is. /. I o 1 4. foIute,3l mate aooe tijt 

in one ^ f fo tbet make 
io'i4.agrcablp totl)eot!)crtiioo;Uc. mi)crcfo;e3| 
tnDgett)em botljeto be gooD 

Rafter, ^eu mtgbt baue U);ougbt thin ^oo;Ue 
Dt!)ectoa(e6 /bicaufe tbe firlfe nombcr , tbat rifcUj of 
tijc multiplication is ?i fquare nombcr. 

^!)olar. »cn3i pccccitie , 31 migbte baue taken 
tbe rooteofit,lul)ic!)c is. 2 40* anu Doubling it, 3I 
l^oulD banc. 4 8 o. as 31 ban in tbe otbcr hJo;be. j^no 
fo all Doe agree in one. 

HDUt mp cbicf DOHbtc nolu is , botu to fenotuc tbofe 
nombcrs tbat bee cowme«/«M^/ff; #o;tf3; Hjallftanoc 
long in fearcbrng fo? tbat,3! mtgbt fonrr Uioo^be tl)c 
otbcr ro2me of U)o;ftc,tbcn to make tbat trialleof«wj 

menfurahltnejlfe. 

fatter. &c eaCcfte Uiaie is,to DiuiDe tbe grea- Thefindyni 
tcr nomber , b^ tbe lelfcr , as if tbci lucre botbe noim ofnombers 

bcrs abfolutetf tbe qu9ticnte\\i\\{ Declare tbeir Squaret, commenfuf 

0s if roil Doubtc,U)bctber. 5 8 4.anD. i j o. bee cow- rable. 
we»y/<r4^/V,DiuiDv.5S4.bp. 1 5"o. anDtbCf«o//>«/f Urill 
bc2^i,tbatis.]% iCbcn DiuiDe lubtcbe of tbe. 2. firtte 
nombers pou liffe,bp bis like nombcr tn t\)t quotient e: 
0nD tbe common Dtuifo;t Unll amounte. ^0 if i>ou Di? 

jpin.v. HiDc 



Thejrte 

nfoe tf)e gw«itec nombcr. 5 8 4. bj> tfjr orrcate r nombcc 
in tbe ^uiHentt, U)bic!)c 13.6 45t'ou fljnll fincc tbc nctu 
jHetieH(e6' lubtcbe. 6. tstbccommcii nomber. £):tf 
Vou?)iiitoe.i7o.bp2).tI)ccommonnomber.6.U5iUbc 

tbt quotient f, 

ilt^ut ano if tbcy «w//>»^p br a tubolc nombcr,anD no 
fraaioii,ano be a Square nombcr^tbrn 13 it ti)c Icffcc 
fquarc. Mlbercfojc if v^ou muiDc tbc IcflTer noinbcr of 
tbe . 2 . bp tbe qutiiente, tbc common nombcr U)iU ap* 
pearcnttberefonDCftfo^/w^^ S!notben ifpouOiuiDc 
tbe greater of thczjiombcrs^bi'tbatcommonnom' 

bee, bis quothnte UliU (bciue POU tbc Otbcr S<iuare. 

inD ifTo bappen,tbat tbc quotUnn of tbc firftc Diui* 
fion be not a fquarc nombcr , tbcn arc tbofc nombcr^ 

mcsmaunfurabU, 

§S>0.\/, 5 2. anDV. I 28. htC ctmmen/urahle t anD the 

l«/)fi£»f*oftbeirD!uiCton is. 4» lubicbc t3tbe IciTcr 
fijiixuc* nuD. 8. appcarctb to be tbc common nombcr.. 
HnD tbc greatci Iquare is. 1 6. 

lioUjbcit bp tbifl nombcr it maie eafilp bet cfprcD, 
tbat fomc nombcrij mate bcreroluiU,into mo;c fqua* 
re0 tlycn one. ^s tbcrf;2.nombcr3, ucpng dhudcd bv.2 
D00cgiuM6.anD.64. anDbccimgDiuiocobi'.8,tbct 
b?"^ngfo:tbe.4.anD.i6. 

"ili5utfojtbcir aDDitiojT, Uibat 3»qnarc3 fo ciicrrou 
taUc,tbat rcDoimD: bp one common Diuifo^tb^ triall 
luUl be UHcnnD tbc roote one. 
^cbolar. 3 p :aic vou let mc pzoue tbar barictie. 
SBaftcv, SDben p;oue itm focbcnomberSjlubcrg 
l^ou maie fin&c moare Uariette . ^s tbefc bee. /. 2 8 8. 
anD.y/.i IT 2. 

fecbolar. ifflbiuiDci 15:2. bp.2 88.tbey««^/f»^# 
luiUbee.4.Ujbicbe5imnfttaUefo;tbeleaac'^riuare. 
KlKW br it 3; DtniDc. 288. anD tbc qn'itientt U)iU bc.7 2. 
asJtbe common Diutfo;. Bplubicocif^DiuiDci i^, 2. 
tberc luiU iifci 6.aitb? fcconoc fquare. Cbni fct ]3 

tije 



/2. 



2)-:. 



ofSurJe nomhers, 

tbftiofacrs in o^Dcc tbu3, s^Ai^i—V—^ ^-i^ 
aiiDUnbcr.i iT^.Bircttbc i6 4' 

one Square 16« ^notjn? 72) 4 -♦ 

Ucr.2 8S.3pnttctt)cotl)cr 6- 

rquare.4. anotJHDereclic 6. 

ot'tt):imlii3roote. Sben ^^^ 

ao:jc 3 tl)c Kootes togc^ 
t}:^ct , lubichc nufectl).6. 
luborc fquarei3.;6. 3nD 
tt)at bcvng muUiplicDbp ___________ 

72. tl)c common nombcr, ^, 2)92» 

Qo:tl)iTdDc. 2^92. lubofc 

tootc 03ct!3 containc botl)c tbe otbcr. 2. rootcs bp aD^ 

mtton. 

l^iitnoUiboUjjj ft)aUfinDc anj' other ariuarcstn 
tljofcnobcrcto make aup farther tnal,ji;knQU)c nor. 

ip.iilcc. DiuiDcalluaicsoncoftbenombcrs, b^ 
f jmc fquarc inber,ttjat Voiii parte it craitlr* U)itijoat 
anp rcmamcr. anDinarUe tlK^uotientc^o: bp it fl)al 
vou Dtmo: the other nober, auD if the quotiente m that 
la(t DiuiQoiT,be a fquarc nombcr, then hauc von i^our 
purpofc. CIS muac gou p;oue Ujith an other square 
nombcr. . ^ , ^ 

Scholar. 35 tjnserffanDe pou.nnD thcrfojc m thefc 
nombers,}! UnU make trialle U)iti).9. bp luhichc :< Di* 
uid:.2 S S. ilnD ftnDc the jHoiient.] 2. s:hcn bp thcfamc 
^ M! Duiio: 1 1 ) 2.anD t\)tqu9tiente i9o 6. &0 hane j} 9 
an:>.^6.fo;the.:.fnuares,anD.52.fo2thcc6monDini' 
fo:.:Hherfo:cJ fctthc nobers m 020cr aj thei ought. 
anD Uno:r them i place the. 2. fquare nombcr^ luuh 
their rootes. S^hcnaaopngcthe rootes together, 3| 
finDe.9-U)hichc X mnltiplie fquare, am it pelDeth.S i, 
that, s I. .^ DOC niultiplie bp the common nomber. 5 :« 
aifo therc'amonntcth. 2^9-- ^3 it om befo :e in the o^ 
ther UJOzUc.caherebp Jperceiuctnatthrrc tnooiUe^ 
DOC conSnnf one an other. 



81 



Thejrte 

\/a 1^2—- 1 — v^.288. )anD tfjcrefoje 3I foru 

22) 6 5 ties of tfti0 iuo;fee,3 mag 

o finli^mtbcfenombers* 

o 0nD fo; nip purpofe , 3( 

totll DtuiDc tbe leffer of 

tl)e*2. mmbtvBjh^ an 

manp Squares as 3i ran, 

162 fojtljatfcamctbtobetbc 

-4 1 rcaoicftc toaie. 0nD firdc 

v/ 2^92 3!P50uetua!),i6. ^nofo 

t}:iC quotient ISA ^,hvh)\iU 
Cbe.l 8.3 Dtuinci 1 5-2. ailD tftc ^uotiente 10.6 4. lDl)Ul)C 

is a f£;uarc nobcn ^0 tbat 33 l)auc tbat tanctic nto^e. 
S:i)cn again 3J p;oue U)itb.2 ^-But 3 fcc,tl)at iutll 
not frame, to bcrfo^e 3; alTaic luit^.^ 6. ano finDc t\)t 
^uotiente 8. bp iDfeu^e 3 DtuiDc tl]c greater rquare,anD 
t^e qmUtnte is. 1 4 4,a fqnart noniber alfo. ;anli tber^ 
fo;e 3 note tbat fo? an otljer tarietie, 

S:i)irl)lp,3! P20UC toitl}.4 p.but tfjat tiuil not agree, 
STben attempted ltJttl).64»^ni> t^at fcruetl) as cuiK 
ipertetliatjaflTate.Si. loo.anD. 121. but none of 
tl)cmlumciiuiDe.2 88. IuI)erefo;e3:pnirct3nto. 144. 
iul)tclie IS ttuifc fontamco In 2 8 8.bp tliat2. 3 DtuiDc 
11)2. anD finoe the quotiente, y 7 6. tubicljc IS a Square 
nombcr alfo. 3 no fo Ijaucs:.^ otficrfcantttcs befioe 
tl)c.2Jo3mer U)oo;jUes:U)l)Kl)e.5,t3artftics,fo.:m^ rc» 
mentb^aunce 31 fet Ooune,tl)u3. 



o/Surde nomhers. 



/.IIP- 


--!—/. 288 


64 

IS) S 


16 
4 

I 2 
I 2 


144 

I 8 




I1T2 

144 



\ . 



2^92 



/♦IIV2— ' — /»288 
144 ^6 

S) 12 6 

18 
18 



524 
8 



V. 2^92 



V/.1U2-I— /•288 



2) 



^76 

24 



144 

12 



smaller. £Dl)cn fo<i to 
gratifiepoii,^ iuill fette 
Dounc. 2 ♦ otbcr nombcrs 
U)Ul) 6t3arictif0. £Cr Ijicbe 
maic fcame to fuffice fo; 
tbis U)02Uc,U)Ul)out mo;ie 
eraplcs. 0nDbkaufevoii 

IjnoU) tbc o;Dcr to trie tl}C 

/. 2^92 35 U)ia fcttc tbcim Douue 

iuitljout ang crpUcatton, otljcc o^claratiott. 00 berc 

tou fee. 



36 
36. 

1296 

2 



»^,288qo— I — v^«7 2oo 
3600 
60 



14400 
2) 120 



180 
180 



32400 

2 

/♦ 64800 



/♦288o< 



i) 



3600 
60 



V,72oo 



i/» 



90 

90 



8100 
8^ 



900 

30 



64800 

/.28800 



The^rte 

»/«288oo — 1 — /«72oov/,288oo-^^^,72oo 



J 600 
18) 40 



400 

20 



60 
60 



900 

n) 30 



3600 

18 



28800 
?6 



V. 64800 



4S 

45- 



2 2^ 
IS 



2o2f 

32 



40J0 
6o7y 

^* 64800. 



v^.2880 



v^.7200 




^* 64800 



v". 64800 



^tmav* SCbw tiatietic ia pleafauiitc. 

J©aaer. 3! iDtUfatiffie pour Dclite better at mo:c 
lelfare, iSut ret one tbrng moare IdiU j faie, before 
IDC can^e tbis foztc of aoDitioittjat if rou UjouId aODc 
aiiFrootctottrelf.^B, v/.6. toV.6. o;V.io, to, 
j/. laic.pou r^aU onclp fuadri^Utlit nombec : ant) fo 
l)auepouDoen. 
^cMar, ^fcegooD r^afonm tbat:/o;aotjmoii 
of anpnombcr to it felf,is but Doublpng tf)at nombcr 
o^njuIttpUrattonbp.2.^nDtf)atniuabcDoenbrthat 

^S2/« .^;tf ''* i^°toU)Ul3ifet fo;tt)efomeeraniple0of 

c«H.ra.m aDDiionm6H^roote0, i^on^c JLo^kcisiifeeDnto 

tl)i0 Me fo;iiie in Square pcotw, fauetbattbemul. 

tipiications, 



OfSurde nomhers, 

tiplicatiDns,tDl)ic!jclucrt Square \nt\]ixt (DCo;Iir, 
ir.tia bt Cuhike in tl)i0 U)D>Uc. niiD tljat cnclv irtnoni ^ 

ljer£5^o;»»"»/"'^^''vi^3^nombcr0 incommenfurAhh h^t 

aODcDUiitl) tl)c(:gnc.— i — . U)itl)outmoaieUio:Uc. 

"^tdXi^tiiht CubikjXZQtiQ commenfurJde ^\33\\\t\^i br QihiJ^eitoi 

png DiuiUcD bp anp common nombcr , tuill maUc C«^ ^^^ comment 
ij\embtts m t\)CiT ^uttientc^s.wV >Z 4. anD.\u/.8 I furabU. 
lubictjc DiuiDcD bp.^.Doc makc.S.anD.: 7.bott)c bcvng 
C«^ii^nembcr6.&o«\\/o2o.ano.\W«ny.bcpngoi=' 
uitJcD bp.f.Doc maUc.27. ano. 6 4. botbc Cubil{e noni;- 
bcr0.LiUcU)atcs.uv^.;i744»anD.uv/.iooo. become 
vifn/r</4^/f,bicanretbnmabc.M?.anD.i25^.botbcC«' 
H'"o"'tjcr0:?ftbctbcDiuiDeDbt'.8» 
^cl)o. 31 p;aic^ou make pour cr5plcs toftfttbefc* 
i^9a(lf r. %l)ctc ncDctl) noc tuojDcs in tins Uio^bc 
It (0 fo like tl)c aoDttton of fquarc rootcs. ^wD tljcrc^ 
fo;je marUe tbcfc tramples luelK 

M\/. 8l.--r— Wv/,2 4. W^.o2 0« ■ {■■ UV^ny 



27 


1 


8 


3.; 3 


s 


2 




125* 






3 





W. 37y 



JO 



64 
4 



27 

5 



H3 



W. 1715 



ua/. 2744 — 1 — nv/.iooo 



8.) 



34? 
7 



12 
12_ 

1728 
8 

W. 13824 



127. 
5"* 



^o.U ^idolan 



Thejrte 

^cl^olar. ^tttisinoc Ofuerfittcfcomtbcfojmrr 
tD0jbe3,but in fettpng tbe Cm^/I* coote^fo; tijc fquarc 
roote* lanu in muUipUpitsr tfic aDDition of tbe.z.ioo- 

^aXtcx. srOat is alL aitD tl)cccfo.:c tuill 31 aauDc 

noe longer aboute it: But Ujtll p;occaDe to an otijer 

foame of aDOition , Ujfticlje feructb alfo fo; Cubik^e m* 

Jnother tes cmmertfurdble. CbS rulc iS tl)is. fi>CtDounctI)C 

forme ofad- Ca^i;^^ rootc3,tuitl)tljeir common oiuiro,2,anDtbeC«« 
^itign, hes that nfc t!)erbp,ano tbcic rootcs alfo. ail tbis pou 
DiD in tl)i3fo;utcr luo;fee, 51But noUi peculiarly m tl)is 
rulcf ou (ball fet Doune.^ otber nombers o^Derlp, tn-^ 
Dec tbofe.^fojmer nombers . Cbc firttc is tbc fquare 
of tbat lafte O^hiks rootc:t!)c fecoDe is tbc triple of tljat 
Square : anO tbe tbiroe isanomber rcfultpngoftbe 
multiplication of tbat triple bp tbc otber roote» 

'Ebcn taUe tbc.4 ertreme nombers, tbat is tbofe 2 
laftc nombcrs^ano tbe.2. C«^«,anO aODc tbcim togc^- 
tber into one fomme» ^uD tbat fommc bcpng multi^ 
plieD bp tbe common Oiuifo.2,U)ill make a Cubike nom- 
ber,tubofeC«^i^frootefl)allconta!nebotbetbcfirae 
rootes , tubicbe pou intcuDcD to aDDe. jjioU) marbe 
tbefc erampIes:aiiD coferre t^^mn UicU Ujitb tbe luo;^ 
liesoftberule» 

1 ^vv^'I y 9 7 2—1 — wnAz 5-92. 
W.384 — ! — N>v^»4S, 



64 



8 12) 



6) 



4 
16 

48 



21 

4' 
12 



II 
121 

^63 



IIS8 



48 



216 
6 



96^ 



uV« 1296 



49 i 



2 I6 _ 

6 

?6 
_lo8_ 

2178 



12 



\vV« 



9826 

49Ji 

J89T6 

\v/.y2 488» 



of SurJe nomhers, 

W.^y2488. h-♦'^^v^ 24696. 



S 8J 2._ 27 44^ 

9) ^" "is 1 4- 

524 196. 

97^ T S S. 

ioyS4 15608 

52768. 
9i_„ 



fecl)oIar. 3:ntl)crcframplc6 3; fcctbe tuoo^Des of 
rour rule obfcriicD-.f oi tjiiDer cctjc s«r</f C«i/^e rootc, 
tl)crc 10 fct a true Cul^'kf nombcr,U)I)icl)c ts founoc bp 
tlie common Diutfo: : tl)cn fololuetl) t!)c rootc of tljat 
true C«^<r : ano,befiDe itflanoetl) tbc common Diuifo;. 
S:i)en in tl)e fourtbc roomc is tt)c Square of tlic true 
Cwi^ roote. anD tjntjcr it l)is nomber triplcD(as.4 8 
tjnorr .16, ano .12. tjnDcr . 4 ) Ujbtclje triple be c^- 
png multiplicD by tljc rootc oftbe otberfiDcDooctb 
maUe tlie loUjcttc nomber In t!]at roluc, ^0.48. 
multiplicDbv. 2. maUctb.96. U)t)tcl)ci6rctt3nDertbe 
rootc.2.?inD.i2.mHltiplicDbr.4.rcl0ct^)-48.tubicl)e 
t0placcDtnDert!)at.4. 

X:i)cn tliofe. 4. ertreme nombers, 6 4- and. 4 8,8.f 
9 6. Doc maUc hv aODition 2 1 6.Uj!)tcl)e fommc ts mul* 
tiplcD br.6, tliatts the common Diuifo;, anDforifctl) 
1296. iuljofe Qii^'k? toote comp?el)cnDct!) botbc tlie 
ftrfferootcs. 

ii^after. ETlielihematcfou fuDgeoftijcotner.:. 
eramplcs. 15ut bicaufe rou ^^^^ tnoeraanDc t}jt 
certamtie of tbis luoo:Ue tl)c better,3! IJ^iut Ijerc fettc 
fo;tl)e. 2. eramplesoftrueC«^/lf rootcs,fo;!mctjUUe 

S/«r</rnombcrs, ^ 

^OAj' \vv .4096 



Thejrte 

Mv^«4o 9 6. I •-•. W. 1728. 

8, 

6 4' 
192, 



8) 



864. 11^2 

2744 

8 



2 16. 


6. 


^6. 


loX. 



w. 



2195:2 



wAl^6 8 ?. — ^ — « w/- 5 ? 7 y* 



27) 



729 

9 
81 

243 
67J 



i2r 



2y 

7f 



2744 
27 



I2IJ 



1920S 
f488 



v\v' 



^cftolar. 5 pep 
ceiuefapcraminati^ 
onofU)oo:Ucinmp 
2:ablc3 t}crc , tbat 
4096.1s a C«^^% 
nombcr, anD Ijatl) 
i6fo;rt)i3rootc.§>o 
i72St3aC«^/;^fno' 
becalfo,! l)t3roote 
is . 12. tl)ofc botf)e 
tootcs aDDcD togc-' 
74088 t!)fr,Doemaljc.2 8. 

rt-t . * . , '^nDtl)at.2 8. istijc 

Ca^/^f roote to.2 1 9 f 2. as the firttc crampic UjouId. 
, aiiD fo^ tf)t recoiiDc erample , 3 fee IiUcUiaics tijat 

i9683.l)at!).27.fo;l)isC«Herootc.anD.5;75^.i)atl) 

I r.ro2 l)t0 rootc. ano tijci botnc mabe. 4 1, lubic br 13 

t^e iubt\e rootc to, 7 4 o 8 8. acco^Dpng to t^c iuooiUe 

oftbcfcconoe crampic. 

Mditmef i^aftcr. ^pngpouarcconucnfcntlpmarurtcD, 

;<e«^;^«j. ,n t!)cfc Jiombcrs, nice loUl goc in banoe luitf) :^«^i. 

<;^r rw/«, :^tf»^'tr rootc0,anti tbcir aODitio: toftcrcin 10 nobittc 

re:cofU)o;Uc,batoncIi'fonbemultipIicatio,lubici)c 
mutt be agrcafalc to tbe nature of tbc nombers , zm^ 
<';<f»<.^W.^notbcrcOucrionbp tbc common oiiu.- 

fo^. 



ofSurde nomhers, 

fo:,fnIfte fo:ntc,!nto ^^?<^f«^»^» nombcvsMe tbe 
firftcnombers bcc commenjurdlfU, i5utif tbel bc««w5 
mcnfuTdyie, tiicti muft tt)e aDDttton be iD^oiigbt^ bt? t^e 
fi^nc. — I — ,iuitbout anp otOcr bufineffc. 

BxaffipUs of :^en:^i:^tn;^ikes 

httynf comrncnfurahle. 

W^.6 4 8— ] — iv ^.f OOP ' ^v^»I2So — - I — \v^. 6 480 

' 2)6 



8) 



w 



81 

3 

8 

8 


62y 


4096 

8 


^ 5276S 





)) 



4 



I 296 
6 



10 

10 



I 0000 
S 



w 



<j ODOO 



:conDccramplcstl)cnc^ 



16) 



240 I 

7 



409 6 
8 



IJ„^ 

' 5062 y 
16^ 

yo62 S 



K\/» 8 I O O O 3 



bcrfi arc Surdes , but m 
jtt)ctl)irOcci'ainplct!)ei 
arc rationall nombcrs, 
jfraincD like iJiito SurJcs 
to tl)c intcntc tbat poa 
migljtc tl)c better per* 
ceiiie tl)c fo;:me of tf;c 
U)o;fte» i^0M8 4i6.tfl 
a s:en^:^xjl{f uomber, 
f t)att),i4.fo;t)isrootc 



^0.6 T ) ] 6.t3 a xf^xi<^^^^' nonibcr, ariD bat!). 1 6. 
fo2l)isrootc.anDtlKfr.2.rDOtcsDomaUc.5o,lu!)!cl)e 
(stbc::c''«^'^<'«<,'A;rootcbnto.Si 0000. iI!nDtl)cres 
fo:e mate ir ore tru:lv faieD,tbat,\v^.8 1 o o o o Doetb 
contatnc tbc tluoo firltc rootcs. 
^rbolar. 3 p,:aic you p:occDc to ^ubtrarttott. jf 0; 

all tI)W J DOC UieU pjrccti!:. 

iDnJ'l. Of 




The Jrtt 
Of SuhtraSlion. 

Ubtraction toct}) tifttt from aDDit<on,tn 

litlc moarc tften t^e frgne .toljtcfic 

:fignc fcructlj generally , fo; all nombew 

imommenfurahle.^nliContlDet^ngthtXtUi 

Utlc Difficultte in ^ubtraaionrjf pou re^ 

mcnrtcr tucll tljcacteof auottton,3i toil ligbtl^palTe 
itoutt in tlicfameeramplesjtfjatg lmuc\xi;ouQhtin 
^DOittoiT,bicaufe it mate bee ap?oofe of tf)attuoo;be: 
aiiD tl)at iuoo;kc alfo a confirmation of tbis. 

^nt\v tW fl)aU pou obferue In this rule peculiar* 
lp:t!}at 35 m tl^t feconoe fo;}me of jaooition, pou muft 
aODc tbc rootcs togetber, before pou multiplic tbctm. 
^0 bcre pou (^all ^ubtrattc tbe kflTcr roote,from tbe 
greater, before pou Doe multiplie tbeim. 

(ticample of ^ubtra(tion,luitb . 

\/a 2.abateo out of \/.2 6,mafeetb./,2 6 /. i :♦ 

anofoofotber. 

Examples ofthefeconde 

forme o/SubtrdSiion 



v/.6?, 

63 

28 


V.28. 


— 


6] 
28 

91 


2nbereccnDcfo;mtc 
oftbatU)oo;I{e, 

i/»6v ,v^28 


yo4 

126 






6; 
28 


V'.I764 

v^. 4 

v'.7oj6 


— 84* 


y6 


1764 
luborerootei0.42. 

42 

2 


V9J 

£D;V.9i — 


1 


8 4 
v/»9i 84» 



'bati0V»7» 



/.I dp 



169 
?6 



ofSurde nomhers. 



V. 



1014 

6084 
4 



|I69 



anotbcrfo;mcof 

v'«I69«— — V o6. 
i 169 

! 36 



1/. 60S4 
tuljofcrootf t0,7S. 

78 
24356 ! 

V'.zo^ /.24356J iy6 

0;./. 2 o ^ v/« 1^6. I -/.Z o f 1 J 6. 

£Cl)att£5.v/.49. 

&cl)olar. 31 fee tn all tbcfc cramplcd'oii take tbc- 
fame itombci*5,tbat ^ou baD before irt aDDition. 0nD 
ficHc rou fct tbc totaUc , out of lubicbc pou abate one 
of tbc n5bcr3,tbat befo;c toerc aDOeD, j tbr ronaincc 
b:ntgctb fo:tbe tbc otbcr . j^o; m tbc firftc oftbcfcz. 
erarnplpB V.2 8. is abatcD out of. v^. 6 > aiiD tbcrc rc^ 



mauictb./.9 i. 



84.tbatt5.v'.7.fo;.S4.t'iUcii 



Dutof.9 f. lc7iuctb.7.^nDintbcrcconDccraplc.v/.59 
abatcDoutof.v/.i6 9.DoctblcaucremainpngV.49. 
" Q^ttcr. 2::bctbii:Defo:mcof^ubtraition,i5liUc 

.:ietbirDcfo:mccf aDDitton: fauctbatUicfct . 

fo^ — j — . auD bcrc Voce miiHc abate tbc lelter roote 
fro tbe grcater(a35 faiD) before lur Doc multiplte tbat 
nombcr bp it felf. ^s bp tbts craplcpou mate pcrceiue 
ix:ibcre Ti Dooc a>ubtraiteV. i o). outy/. i o 1 4, ano 
tbe remamer is. 1/. ^ 8 4. j^oU) marfee tbe U)oo;kc 



V/.IOI4 y/,\OS, 



169 


25- 


6) I 3 


5 




8 




8 




64 




6 



./. 



384 



lr}creFou fee all tbmgcs 
agree , irntb tl)2 fo;mc of 

^ODition, fane . fo; 

— i — . auD luben 3! begm 
togatb:rtbcnombcr,tbat 
ItADctb m tl;c miDDlclubi' 
cbc "C multrplic bjMt fclfc^ 
anD j^ Dcoc not uraUe tbat 
nombcr^ 



The^rte 

nonibcr,bp aDOrng botljc rootcc tcgctJjcr: j*^o; fo. i ?, 

anD.'>.UioulDnfaUc«iF,t)ut3!aO£itc.y.outoti3.anoro 
tbca Doctl) remain. 8. luitl) l\)l)Ui): 3 ^iQunt as g cto 
m apDttion. 2nD tl)en commctu fo;{t!)c t!jc rcmairrcr^ 
v/o 8 4» 

^cl)oiar. 3 tjnDcraauDc it berp luelL anD 3f p;ate 
rpu tfjat foj ap;oofc,3 maic banc tijc olbcr cjramplc^ 
of aODition.partlp fo; mp creriifc, aiiD partlp fo; era* 

minanooftljcfojmcraODitions, bftljcc6trari^|jfnD» 
fatter. XM itl) gooD UjiU. 

^cbolar. SCftcn itU 3 fct tbcm,aiTD luo^kc tbem. 
a0!)crcfololuctb* 

Beit firlle 3 Vuill tJcgimluttI) jtfje UjD;fee of tbia laft 
erantplr,after tbc feconDe fojmc of ^ubtratfiontfo: a 
(oablc CQii£irmationof it* 



v'.loi4 -/«iTo 



1104 
150 



$0700 
I014 



1014 
II64 



i^. 15*2 1 00 
v. 4 



\/.6o84oo 

V'.II64 v^«6o84oo 

;©;V.II64 780 



•E^notbcr fo^mcof 
tbefamc luo;ke. 



I 014 
15-0 



^0700 
I014 



lyzioo 

iiil)oferootc{jei.39o« 

590: 
2 



*/.ll64- 
2:bat(0Vo84» 



-780. 



ariDnoto fiercare t}^t tjarfatfonsoftlje otberep 
ample0* 

l/.2J92» 



i/.2T92- 



OfSurJe nomhers, 

-V.2S8. )v^.2^92- 



36 
7Z) 6 



4 
4 
16 

72 



32 

I 12 


/. nj2 

♦/.2^-92 
144 

18) 12 

8 
8 


i/,2 88. 

16. 

4* 


64 
IS 


n2 

64 


^\ liy2 
v^.2f92 


v/.2SR. 


1296 

2) 3 6 

24 
2 I 


144- 
I 2* 


W6 

2 





n) 



■v^8S. 



6 
6 



36 



72 
108 



^.25-92 — — V/.288. 



8) 324 
18 



36» 
6. 



12 

12 



144 

S 



V* II 52 



v/.2T92' 



V.nr2. 



v/. I I S 2 



1.296 S76. 

2) 36 24. 

12 
12 



v/. 



144 

2_ 

2 88" 



£Dtl)cr cramplcs tarteo , fo^ pjoofe of tlje like . 6. 
crampUain^ootttom 

pp»i. V/.64800 



Tk^Jrtc 



^",6 4 8 oo ■ — V >72oo 



32400 
2) 180 



120 

120 



14400 

2 



v^» 28800 



3600 

60 



v^.6 480 o>-^-/»7 200. 



8) 


8100 
90 
60 
60 


900, 
30 




3600 
8 





v/. 28800 



v^.6 4 80 o y/ .y 2 o o. !/,6 480 o v'^.7 2 o o. 

3600 400. 

1 8) 6 o 20 

40 
40 



1600 
18 



12800 
16 



4/* 2 i> 8 o o 



2o2) 


22 V 


^2) 4T 


I >• 


30 




30 




900 




32 





/♦ 28800 



i/.d 4 8 o o x^,-j 200. 



TO) 



1296 

36 



144- 
I 2 



24 
24 



J76 
TO 



v^«64 8oo v^7 200^ 

900 loo, 
72) 30 10, 
20 
20 

400 

72 



V. 28800 



I 1/. 28800 

SuhtYdUhn %sSltu %\}xz Difference \& there in §>ubtrartioti 

0/ C«ii^r of Cuhikt rcotes commenfmahle, 0nD tljerfo^e 31 ftt tl)e 

w(?/«. eramples onelp, lotttjout anp larger Declaration, 



-w 



/.^7r- 



V 



I2f 



wV* 24 



ofSurde nemiers. 

•W.8I. l'w/.I7IT- 



27. 
5 



T( 



54 5 

7 



4 
4 



64 



•VVV/.I55'. 



W» 520 






\\\ ' . 1 5 S 2 4— \wVlooo 
125- 
5 



8) 



17:S 
12 



345 



^v 



jn tl)c fcf onDe fo;mc of ^» (,/^f r 

aDDltlon Oi Surde Cuhes,^ou T^oor{e of 

remember tbat pou aODeD SuhtraHion 
4 tiomOers together. 15ut/£,r5«r^ff 
m^ubtrartion, i^oufljall Cubes, 
aDDe to eclje roote fcueral* 
He tfjat , tbat commetb of 
!)i5 oiune multiplication, 
U)itt) tlje otber triple , ^nn 



2744 
tl)cn fi)all vou ^ubtrarte tbc IcITer nombcr,diit of tbc 
greater. 0itD tl)e remaincr pou fl)aU miiltiplic bp tbc 
common Diulfo;. :3nD fo l^all pou bane tbe rootc tbat 
rcmainetboftbc^ubtraition. ^b inerample* 



WV/.I296— 


— \vv.48. 


216 


8 


6) 6 

56 
108 


-> 

4 
12 


72 


216 

64 
6 



w. 384 



uA89S6 \U'^ 1 5-972 



4915 

12) 17 

289 

867 

6171 
216 
12 



I55I 



II 
121 

565 



SS^7 



\W» 2592 
pp.U. >\v^2949l2 



W'«2949n- 



9) 



^2768 

1024 

^3072 



J832 



W. J24S8 



The Me 

•w^«2 4<^96 | ^cljolar^ 3;ttali 

^< tbeton&rmattcnof 
ti)efo;mcraDmtt6, 
0nD in tbffc laffc 
U]oo?to,tlji3 3jfcc 
pcculiarc from aD* 
Oiti6,tljattl;cC'«^^ 
ts aDDcO tultb tl)c 
lolDcCe nombcrui 



14 
196 

^^8_8 

43008 



7 2. anD maUctD. 2 S 8: aiiD, 8. is cDDcD Uiith -> i ^ thS 
PclD^tb. 2 2 40 ano turn 15 tDe leffer artcD> ouu^c 
grcatrr(aB.2 24.from2880 anotbcicmamcr (U)hi^ 
cDe tUcre 15 6 4 ) fct in tbe rniDolc DnDcr bot^c tl^^^^^ 
lues of nomfaerg, ^no tijcn 10 muUtplico br tberoni. 
monnombcr,tomabetJ)eremamcr. 

^0 in m firOc ej:ample,tfjc rciiiamer i3.^^^/. 2 8 4. 
iul)crc.AV.48.isabat£Doutof.vH/.i2Q6. ana?n 

outof.vw»y89T6»tbercmauier(0W.2fQ2, iLfkc. 
mits iix t\)t mm ewntple. xW. 2 46 9 6. is abateo 
ciitof, vvv 2949121 If auctO rcmampng. ^H/f 2 4 8 8 
^aaer» liSutnoUj in aODition mvctoiohjttV-^ 
otftcr eramplcs, lobicfje bp fabtrattion uiaie bcc pio 



W2i9f 2- 



8) 



2744 

14 
196 

y88 

2688 



MV 



:W4o96 1 W? 4 o 8 8- 
TI2! 



216 



1728 



8 

64 
192 

4704 



27) 



2744 

14 
196 

__T_88_ 

3402 



i2'r 

27 



W_[96_8^ 

72 9 

9 
81 

__24i 

J292 



W. 337J 



^c^olar. 



efSurJe nomhers. 

^rtjolar. 3! fccUn tijrfc eramplcfi of Subtraction: 
tl^t tbc fiffte ixombtt ts tl)c totallc^o; latte nombcr m 
aODttion. anotljcfcconDcnombcr, U)l)tc»)cfoloU)etl^ 

,10 tic nombcr to be abatcDMtiD tl)en lattc ano 

lolucac of all,i3 tljc rnnaincr,U)bicl)c luajs one of tl}e 
firfte fommcs in asoitton. 

3nD tl)ougl) tl)crc rtmatnc. ?. otber craplcg of ;<r»^ 
fK'Kf^Kl^' nonxhctSr'i fee no Difficulttc in tbcun , but 
tuat 3 can Ujoo.:Uc tljcnr.aa bcrc 3 Ijauc fct tUc fo;tlj. 



^v/n7(^s- 



1V/.648 



4096 

S) s 



s 

62) 

s 



\-/, jooo _ 



w^.f 0000 


-vv/1280 


I 0000 

10 

6 

6 


2y6 

4 



1296 

5" 



v/» 6480 



w^Slooon w^'6fV?6 



16) 



J06 2) 



4096 
8 



2401 
16 



a^aftcr. Sccpngt'ou 
arccrpcrtc inoiigbintl)c 

S . UjOO:hC0 of tlKfe Surdes 

Dncopounocj lull tcacljc 
tou tljc lifec luo;iJcs in co* 

^OUttHC Surdcs. 

^cljolar. 3:3tbcrctlj0o/re^«ff/fl« 
nocreDuction, notbcrcr^ andextraffh 
u/* 3 S 4 1 6 traaion of rootexj , to bee <,„ o/yp^^^; 

taugbtc in tl)cfctjncompounDc5«r//a.-' ^ * 

spatter* 30 fo; reOnction, 5 bauc taugbtc pou all 
rcaoic in muUtpltratton , a0 mocbc as 10 rcqutrcD m 
tbefcnombcrc 

anDfo;ertraction ofrootc0, roumaicfonct)nDer* 
danoctbat bcrc can be none, fo; tbtn toerc tbci not 
SHfde nomber0. 3no tberfo;c 31 faieo linto rou before, 

pp.ii;. tbat 



The Jrte 

tT>atV«^»^ o o.fu not a 5«r</rnombcr,aUftou0ft ft be 
U>?(ttcn UUe a Sutde nonrbcr,bif aufe it fjatft a Square 
roote,atco^t"0 to W$ figite:aito tbat i^.i o.3iibeiua« 
tea. /^i 5^ 6. is rto 5«rie nomtin::fo; tjrs Square roote 
isfenoU)eittobe,i6. 

^f l)olar. 3 migljt Ijauc cofi&creD as mocbc bp tbe 
Df fihition otSurde nomber:5, tbat tbcir rootw can not 
beaDTigneDmnombcrgabfoIute. anD ttjcrfo^eBifee 
t!)at» W.I 2^10 noc5«r^?nober, fitbhU5Ctt^i(tf roote 
is.5'.0nD.iv/.2 5-6.10 a nombcrr4^/«M4//f, anD mSurdt 
nombcr:fo; btS ^en^:^en:^i\e vtott 10.4. 

Rafter. 15ut.i^/.6 4. i0a5«r</fnombcr,anDpct 
batb . 6 4 ♦ a 5^«4rf rootc, anD a Cubikc toott aIfo,but 
not a;<5w;</;<fw^;^noote,acfo;Drng to bi0 fignc.0nD 
tbcrfo;c ongljt better to be iu^ittcn tl)U0. v^. 8. 

^cbolar. 3 pjaie^outo p;ioceoe to Surdemmbtta 
componnoe* 

OfSurde nomhers compounde, 

jailer. 

'Xrde noniber0 rcmpouDc, arc made 
' notonelf of. 2. o;.^. o;nioarc5«r^f 
!!nornbcr0 trcompouncif^but alfo of 

IrattOTjille 0; Abpa^e ncmbcr0 lov- 

rncoiuitb^wr^f nDinber0.0g. 1/.1 o 
j-i—v'. 1 2.anD.8.— f-v/.6.liiie^' 

l,itDaic0V.2c. 5.anD.w/',4o. 

~f— V/.14. .3. 

Compounde l!5utl)ere(l)aHpoumarfee, tbatSjcaU fompounoc 
Surdes, tTomber0,notonelv ftclje as bane tbe 0gne of.—^ — , 

but alfo focbe as baue tbe fignc of fo^ altbowgb 

in nature of tbe nomber ^. 1 o 1/, y»be not com^ 

po"unDe,but abateD, pet (n name be 10 rompouDe,anD 

awgemcnteo* i^o,n . Doctb aa UieU augemente 

tl^e 




ofSurdi Homhers. 

t!)enamc,as— '-— DoctI). 

^cljolar. Jt r:in:t!) rrafonablc. f-oi lubcn J faf:, 

■/. 1 2. — v^.7. tl): name is r o:ni30untir,an lucll as 

(fJfiaDfairD.v'.! :.-- l-s'.-:, alt')ou:jI)t!)cnuanti> 

tic bcc not fo grratc. j^o: Ooctl) frier abate tl)t 

quatitic Oi tl): nolj:r, t»)oug:I) it Do uurcafc tl)c name 

CBaftcr. j?ct to: a DKT-rcnci-/!):nontbcr5tl)at be 

COmpoUllDe iiltlj— ! —be WllcD ^imtdiaHeiUim U)3^C Bm':dUlles. 

tijat be compouiiDc luitl) M nameD ^f^dualln. T^fiduilies. 

^no if tt)e 'BimedialUs l)auc ail tljcir nombcrs aiiD par- 
tes of one Denominations, tbcu bee tbei calleDonclr 
bp tbelr gcneralle name BimedialUs, L'^ut if ibcir par< 
tes be of.:. Denominations, ti)cn arc tbet namcD ^ino^ 

ntialles p:operlp. ih)0U)beit,mani,i Ufc to call Binomiallcs 'Binomjjlles. 

all compounDe nombcrs tbat banc — 1 — . TiuD fo iu:l 
31 let the names paflTe. 

^Hclides Definitions Doe not tjcrj,> aptip agree to this 
place,as at iiw ottier tpmc 3: imll flKU)e vou,anD tocp 
fo;e i Doe omittc tbcm fo: hjis tvme. 

Btittoticbrngour p:incipalle intent.^ , tybtrhe is 
to Declare tbe p:aitit^e tuoojUc Qt^momialUs, anD 'i(f//.' 
dualUs ,i\)txc 13 litlc Difficultie,if poii mark: IdcII tnat 
Uibicbe 13 taiigbt befo:e. ,-ro; as ^inomialla anD \ft^ 
duplies, beemaoe of5ttr(/w,o;el6 ofrationalle nouibers 
toitb Surdes , fo tbc U)02Ue of tbe compounDe nombers 
DcpenDetb of tbe U)o:Ue of tbe fimplc nombirs.ano 13 
all one luitb tbem. 3nD concerning tbe fignes — | — 
anD . — . here is no moare to bee faieD, tben Uias 
taugbtc in CoJ^iks nombers compounDe. 

fecbolar. pet of euerp hinDe,it maic plcafe pou to 
fct fojtbe fome eramples. 

Rafter* ^ tbmke tbat mcte,luitbout m^n^ h)ov 
ttB els. 0ot fo jgettpng fap tbe luaie , tbat ^niuerfalU 
rootesyatc not accompteD emongcftc tbefe compounDe 
Surdes ', but arc refcrueD to tbcir pcculiarc tceaticcaa 
root^0 of compounDe5«r<^«, 

£Df 



Thejfrte 

OfTS(umeratlon. 

^umeratton 15 inoare plant , tbcn tf)at % ncaDe to 
ttanDc in oedartng it,otberU)aic0 tbcn bp ejcamiilcg; 

Examples afB'momialles, 

6»— I — V .S. 2CIjat ts 6 mo^c ttjt ^^K^rr rootc of 8« 

v^.2o— 1 — .5. gstljcS^wrf'^^roDtf of 2c.nioarc.5» 

Woo — 1 — v/.p.^ignidetl) tlje C«M^ rootc of. ^ o» 

mo;je tljC <f »;^/;^f n^/^ footc of.9» 

j^no fo of oti}cr» 

Examples of^efidualles, 



2 4» / 9 6. 2E:!)at I5'2 4-abattng tl)c roof e of 96 

-/.I JO, .9. 3stt)c5^«4reiootf of i)o.abatmg9 

i/j 2 o 8 /$ ) . 2Dl)C ^Kpi^K}^ ^Klkf r oote cf. f 2 o 8» 

fauc tije 5^«4rf coote of. 3 j. ;anO fa 

fo^tftc. 
^cbolar. ^0 31 fee an^ 5«r</« mafc hct compouiiDc 
tuitlj ct^cr:iano an^ noberg rationalle loiuco UjUIj t^e. 

Ofjddltion, 

gaffer. 0DDftfon fs as plafnc. i^oj as tfjc partes 
face, fo ftjaUtbe aoDition bee, arco^ovng as pou f)auc 
leariuD before. 

Examples of^inomialles. 



-/.JO — j — IC 

/.2.-4— .8. 



-w^. 48«- 
w^.24^. — 



'iH)3 



1^ — I — -/.I)-. 

[8-4— v/.6o, 



V-4T^ 



vv%Tg7J---f-V.8o. 



-v^W3y. 

vvv/.32 — 
wv/*4« — 



/♦I264— +— 8. 

2S-+-/516 



]6~- 

Vlo. 
Vi9» 



V2844 



wv 108 — j — v'29 I • ■ V760 

Crainplee 






v^.io8' 



OfSurJe nomhers. 
Examples of^cfiJualles. 



.414 
1. 



16 — 






2^0- 



^74- 



1^1 o 8, 



V/.44 76. 



/.:?). 






V/.72.- 



9. 



MV'2 4$- 



w^96!\v^. 32. 



W.6, 



-\Vl 6 2 



-v'.2 4. 



vv/.;2.— 



w^.JI^ 1/.29- 



V4>: 



Examples of^inomialles'^ith ^fiJualles. 



v^So.— h-6. 



, — . -) 



vM2)- 



12. 



-j — v'\2o; 761- 



-V^)!2 



42 — r— V.y. I 901— ^'1^68 



z\s>/ 1 6 o'\\v/. 520- 

V, 7 ,— [-^^wv" . 2^ uy . 4 o— 

v^, 1 1 2 W6 8 4 'n\n/ 1 6 8 cr- 



1/. 6 V- 



-^^y6. 



.24. 



-v^So— V)376 

^ct)olar. 3;fccti)ntpDnmaIiC rciicrnKc^DDitions 
in all tijcfe uombcrs. J-n;^ ]t^ij aDDc ttiU like nombcrs 
iDitb tbcic matcl^rs. ^0 tijar ijcrc is notlji'n^ Diucrfc 
from tfic U'.oo:U!:3 of fimp!: Smdes, jilUliougl) m cue* 
rp tbii'Dc cramplr , tljrr: 14'pcarcmoare Difficulttf, 
tl)cn there 15 in 5:^5:: tiTJ lien J conGDcr tl)e liUc trart- 
fporiricnin Co^i^lks nombcrs. j^o;itl)eluoo2Uc aDDetl) 
liUcnoniDcrs together. 

OfSubtraFlioft. 

fi^aftcr. Jn Subtraction tlierc Ifl as Iitlc Dutcrd^ 
tir. as tbcfc examples luill fuffictctlr Declare: tufjtcl^c 
bt fct a0 trtallcs of tt)c former aoofticnc;. 

Cq.j, Crantplcis 



The Jrte 

Examples of Binjmlalles, 



V/.72- 

1/. 2- 



■18 

-8. 



v^5' c>-*-| — I o 



36— f~/2S44 
vAi26 4— h— 8. 



28" 



v/ol6. 



33- 



V.I351 



18— h-v/.6o 



w^«l87T— +— /♦So. 



v/, 243- 



V.4^ 



W.I 08— 1 — /.2 9. — I — /.760. 

W. 4-r-*4-— V^.19. 



^vV« 32- 



•v.lo. 



Examples of^fidualles. 



/.1 08 i 

/. 3 1 



v^is- 



-4 



I74-- 
v/.44j 

2^0. 



V.27^ 



30. 
14. 



V*I2. 

V.3- 



I6»- 



'vw^.2 45' 



-iV'.l6 2. 



V'.27. Av/. 72 — vv^.96. 



iv^.fI2-^ v^.2 9.-"| — ♦/.480. 

K\^» 32— V'.^. 



iV^. 32- 



V.24. 



Examples ofhothe together. 



v^.I25'-+-4 

V. y 2 

v^, 8b— f— 6 



901- 

/.288- 

J6I- 



340 
/.J 1 2* 

42. 



of Surde nomhers. 

42— +— v^.7. 1/.112 \u'.64S« 

12' ■ V 'S* \^* 1 — ^ — ^u'.2 o. 

50— j — v/«2o, v'«65 -nv.l6o. 

W. I o 8 o, v^.S o. v''.n7 6. 

\\\^ 40. — f— .v/-24. ^ 

W* 52 o. -v' .) 6. 

^cljolar. Cbts « as r afic as acDition , fauc to;. 5. 
rramplcc^lubicftc 3; tnDcraanDc not. f^i altljouglj 
3; fee tbc lafte cramplc , of cclje of tbc fo;tes of nom^ 
bcrs, to bcc agrcabtc tuitb the Iibe tramples m aDDi^' 
tton , ^et 3 can not fo tuell pcrcciue, tl)c o:Der of tf)cir 
Subtraction, as 31 Doc hnotoc tbc mancr of tbcir ^D 
mtio. j^o; bp tl)cartcof (hnpU"S«r<^«,3 fee tbarV.i o 
anD.v^i9.Docmalic./.2 9— +'-V.76o.lrutlubcn 
v^.2 9.— I — ./.? 6 o.is fct as a totally ano. \\ \ 9. to 
be &iibtra(t£D out of it,l)oU) 3 fl;aU U)oo;Uc tbat,anD 
IcaucV. I o.fo;thcrcmatncr,3C fee not. 

^0 in tbC refttiuMlles , 3 UnoUjc toU) . v^ . f . auD 
V/.24. DocniakcV.29 — h— i/-4 8O'15"t3:fc«0Ujc 
not botu i/.y abatcD out ofV.2 9 — 1 — v''.4 ^ o.ooctl) 
niakcfo;tbcrcma!nir.'/.2 4« 

j3nD the tiUc Donbtr 15 in tl]c tljirDc fo:tc otSurties, 
tul)tcbc arc mtrtc nombcrso'^o^i ^^ci^c i. fee m 2!D3i 

non- — I — v/.2 4.aDDct)U.irl)~ ^\^>)6. ^notbc 

tDtallc to bcc. — — .v^So. v/o"576.^hncU.c 

tbc rcafon of tl)cUioD:Uc,fD2tlicfignes. — r~-nnD 

.Ip tl)at J IrarncD m Cofiiks ncmfacrs 1 ^no tljc 

rcalte is manifcHc tp aoDition of fimplc Surdci.fo; it 
tsUj:ouglUbvabatvngV.2 4.outof./.>6.15uttl:cn 
tn ^uJ;tra(tiDn,')olu — -| — v^2 4. bn ng subtractcD 

from v. ho /.9 576fi)aUlcaue — v/0'6 

i ran net tuDrrc. ^nD v^^t bp tbc figncs J geffr ( as 3B 
IrarncD m G/% nombcrs'tbat it is Docn bv ^DDitio, 
bicaufc tbc ngncs doc DifagrcCt 



Tie Jrte 

fl9affer. 3Jit tliat vou remember tfje former rules, 
to f onff rre t'ncm aptlv U)itl) tbefc later U)o<:be0,3i can 
pjaifc rou tucll. ^ut in tbat vou can not bnOerCanoe 
tbe rcafon of tbat , tuljicljc luas not vettaugljtc pou, 
3 can not grcatclp blajne ?ou. aitbougl) 31 can not 
p^aife rou,fo;^ tljat you tfimUc vour fc!f to be cunni'n* 
0cr t!)cn ^ou arc. foi m tijofe aDDiiions , tbat pou 
tbmUc v'our fclf to be crpertc inoug!),3!'oare fatc,tbat 
pou bee DircciucD, if vou tahe tljcm to bee nombers of 
anp focl)e,a0 betfjerto l^atb httn taugbtc t)nto pou. 
^Dcljolar. 31 take t^^tm fo; compounoe Surdes, 
fl^aSer. ESbet arc not fo: ji^otber i^ tbeir tuoo;!jc 

agreable,tu!tbtbcU)oo?beofcompounoe5«ri/«.ii3ut 
tbci aretbe rootes of compounDe5«r</M: ano tbcrfo;jc 
are calleo yniuerfalU rootes of Surdes. 0no acco;Dpng to 
tbeir proper nature, tbei ougbt to bee calico rootes of 
Surdes,anu tiotSurde rootes. :as 31 h)Hl tell pou anon. 
TM ben 3! toill alfo DifcuflTe pour Doubte. 
15utbefo;e3ifpeaheanp moareof tbcfm , 3: Ml 

canDetbetuoo;iUesoftbefecompounDe5ttr</«:UjbcrC' 
of.z.bmoes petremamc bebtnoe. 

Of Multiplication. 



|P^m^^;aitiplicotton of compounoe Surdes, 

^^^^^ H'^ "^^ ^"^^^^ ^^ ^^^ ^^^' ^"^ Differctb 
jin notbpng,fro tbe U)o;Ue of fimple 
Surdes. hDnclp tbi0 mutt pon marbe, 
ag reafon UjouId , tbat pou muttc 
muUipIie cuerp parte of tbe one no* 

bcr,bpfucrp parte of tbe otberno;^ 

facr:a3 pou remember tbe U)o;Ue of compounoe OMt 
nombers* 
©f!>olan 3? p;afc pou gf uc me fome cramplcs. 
Oi^aac r. SSbat ft)all pon bane, n no tbat mate fuf^ 
fiCc fo; tbis U)Oo;Kc. S9arbe tbem luell tberfo;je. 

Cramplejs 



efSurde nomhers. 
Examples ofBinomialles, 

6— [— -/.S* 



138— I /♦I20. 

— I— >/.Ho»— 1 — V/.4232, 

138^=4=^^^.4252— f—1/,5 40-4— /,I20, 



1^.120 — -f — \/.I2, 

v'. 12 — [ — v'.y. 



i/1440— j — ^.84. 



12. 



12—1 — V/.1440— -[~'/»84o^H--^«8"4~» 
Examples of^^e/tdualUs. 
5". ■/.lo» 



V.I 



o. 



2^— 



to. 
V.270- 



•2^0. 



3r 



V. I o o o. 



v^. 2 4' 



■V.20. 



— -lA 



v^.720— i — v^.480. 

; ^ ■ ■ 2 4 -^ ■v/'.d o^, 

" —24- 



v^.72o— t — ■/.48o- 



V.6oo« 



Examples of bothe together. 



32— H-v/,14. 

v^.124 —6. 



v^.126976— H — ^.17 36. 

__ . 192.- 

/'. 126 9 7 6--1— /♦ 1736- 



V. J o 4. _______ 

-192 — — /.5;o4 



1/. 



Thejrtt 



vM )02S — \ — 28y. 

T2 — 



V. IT 028. 



57- 

^cl)clar. £Bulttpltcation , ao 3 fee , ^b tljc cafif l!e 
tcDo^Uc of all tl)c otljcr.^o tbat 3 Dooc mathc tbc rc^ 
Cuaion , tn gatbcrpng tijc totallc : lubicljc ts caDci^ 
nougl) to VnDc i fiancfap tbat 3 tjauc If arneo m C»M* 
nombcrc, £nD n^iutliou ht no ftarocr, U ximt lone U 
IcanicD. 

'tUtSWll, 



OjVi 




fallen 

;3'uiffoit {jp oncftmpic nonibcr,i6 
I no moarc cifficiUte: ajjtbcfc cram^ 
' pics DOC Declare.^} \iixt t^c otutfo; 
ts a nombcctncontpounDc. 

V/.26 — h— i5^iHiOeDfcr.^5octl) 

k^.7p?^:«u::^>u:3j1 :againc.i/.y6 — I — i/.24.Dtu(^ 
ticDbr.V,6.0oet6P«lOc«'/.9T — r— 2. 

anDfov/.7S— ■/. 4^«C''iii^0tirV-5*tiooct^ 

t;vn{jfo;tl)e.T» 4,tl}atis.u 

]lUicU'aicsV.?2c. — h— i/tScbcrngpartcDfcp 
v/.^.coctl) mafee tbc juotitnte. 1 4. 

^cbolan g fee it fo, fo; at tlje firflc It 15 . v^6 4« 
— 1— i/o6.tt)at 10.8.— -f— 6.tubicl)c mahctb. 1 4. 

fll^aflcr. ISO ntaiTFOn iDCi»;fef all lifef DtuiCcns. 
l^ut lut)rn tl^c Diuifo; jo a ccmvonnDe number , tUcix 
jr lift pou W an otbcr nrcane : tbat is to rcDucc tbat 
compounDe nDbtr,to a drnplc ncmbcr: Ubicljc thing 
tou mate cafilp Docbv multiplirng anv ®/»ew/4//f,lp 
Ms %efidualU , c; ccnti ar^'lcatcsj ttjt '^fiduMt fcj^ jri^ 

2iss 




ofSurJe nomhtrs, 

336—] — /.io-multipUer»bp.6— 

mitQ. /.S v/.^.multipUcD br /.S-4 

Doctl) pclDc.S. ^ that 10. ;. 

S>cbolar. jC pcrcctuc a b;icf luatc in this muItipK' 
tationt fQi 3! ncaDc not m tl)c firftc crampIc,to miil 
ttpUc 6.i)p. v/. I o. fitb It luoulQ amountc to not hrng. 
^nfomocljeas atone miiltiplicatton , tt luoulD brc 

— i — ,anD at an orbcr. . Hno fo tljc one luoulD 

abate tt)c otbf r,ano icaiic notb^ng fo; t!)cm botl)c. 

(Batter. s:t)at ig locll marfecD. ano it is fo genc^ 
rallp. (Xll)crcfo2eCaspou^ce)tl)emul^o;bvtt)l3m^a' 
nes,malc ligbtlp be tourncD mto a Qmplc nombcr,o; 
a platne abfolutc nombcr. 

ano nolo to make tbe Diu(rjeDr,t« t^crarnr p:opo^ 
tion,to tins nelu: Diiurio:, tbat \x laas bnto tJjc olo Dt: 
iiiro: , voti 11)311 maltiplic it brttienfwncnombccbp 
iDbicbc ti)c Diuifo: loas multiplieD. f o; if anr noiii^ 
ber£5 bee mukiplieD , bp one common nomber , tl)f ir 
neluetotaUcokepcthefamcpjoponion/tbatUjasbe^ 
tioenctbcfiraenombers. 

^rbolar. Chat mutt ncancs be fo. f'QiZi^.U^fef 
qulAUerd unto.z.fo If pou muUipltc tbem br. vtbei luill 
mafte. i y.ano. i o. Vobicbc be myf/7««4/fera p:opo:tiGn 
ant) likcluaics U)ttl tbeic p,:8po;tion rcmant,br lubat 
fo cucr nomber tbei be multipIicD. CCIberfo^c it mitft 
neaues be reafonablct'oat if tl)c DiiiioeiiDc an^ tbc tJi^- 
utfo;,bc multiplieo bp anp one nomber , fimplc o: tb^ 
ponnoe, tbei (ball kepe tbc fame p;opo;tlon,tbat ttjci 
tjaD bcfo:c. 

Scatter. j^onno^ccerMfntmDcrttanDrngoft^ts 

rulctahetbefceramplec.SDbcfirac I / ^_ i ^^ , 

13, lubcrc /.6 8 -+- /.^ 4-t3 fette \y\^ __! ^'l 

tobecOiutOeDbpV.6 —- 1 — /• ^» 



tl^erc firftc i«multipIictb?Diuiro2 il, . ?' 
^ bis contraric , tbatls im "Binomi^ I ^^J'^* ^^^ v 



6 

V 

alie 



Thejrte 

dU, /.6 — -- j«3iio x\»tt rifetfi, 6 -— ^.ti&at fcs^i 
iubiclje 31 H^all feepe foj tibe netoe ttuifo;* 
%\im ooe 3 multiple tlic DiuiOeDe v^^S- 



v^ 6. i/?. 



^♦4o8-H — /•324* 
i/".2o4 — — >/>l 62«. ^____ 

■/.408— i-^v^*32 4- '/♦20 4 ■/♦idi^ 

0tio tijcre Dot^ amoute,a0ilbere tn tuo^be iA ejqp^eflTeo^ 

/.408 — 1 — /♦524 '/.204 '/.162. 

tp1)f cbe nottibtr il^aU be taken fo^ tbe netoe OfuiDr De: 
anD muft be utuioeo bp, 5. t^iat ts tbe netue muifo^Bf w 
iul)ofe ffeoe 3 fetV«9»fo? moare reDineffe in tuoo^fee. 
SDl)ct:fo;e Bi Tet tbe Doune tn o^oer^a^ bere fololoetb* 

^408— +-/«324 — *^»2o4 /.162 (/♦45't— 1— -6 — /22| — A Si 

/♦ 9 ^* 9» v^* 9 \^* 9 

0nU tben uoe 3 fefee botu often. /. 9. maf e bee founoe 
(it V*4 o 8*tobttbe mate bee. 4 y^ of tpmes. ^ bere^ 
fo?e 3 fet. /♦ 4 T T J" tbe ^U9tiente, 0nD tben uoe^ ref * 
teratc tbe D<uifo?,anD fette It bnoei. v'. 5 2 4. icbece 3( 
fmtje lt.?6.tpme0:ano tberefo^e fet3(.6.fo? it,btfaufe 
tlit^Mticntetls tuoulo bee. v^. 56. Xubtcbe is iuitlp.6» 
SD!)iri)Ip,Bi reniouetbetrtuifo?t)nDer/.2o4. lubere 
tt waie bee founoe .224 tpmes . fo^ tubicbe 3! fette 
/.2 2 4 in tbe quotievte, 0no tben fet 5 tbe oiuifo? laft 
of all bnoer. 1 6 2. U)bere ft Is foiinDe. 1 8, t^mes : ano 
fo;j tbat caufe 3 fet v". i S.ln tbe juotientei2inti fo 10 tbe 

Iw'OOle qUGtitnie /« 4 y ! — j — 6 v/.2 2t -/♦ I 8- 

fe?cl)olar. 2^bi3DtutfionlsnraungetOfretiite, al^ 
tbougb it be not Difftculte to luoojlie. 

a3aOcr. ^f vo« Doubte of it , i?ou male tjfc tbe ac< 
(utJomablc tnnllebv tbe contrary? femoe. 

^cbolar* 



of Surde nomhtrs. 

^c1)olat» feo mnft ii folotue , tljat tf 31 uooc muin 
pUet^t0 ^mtitnte fa^tlje firtte Otuifo j , tijc firftcDiut 
ticnne UjtU refulte hereof. 

ano fo? tbe p2oofc of t!)at,3 foo^ ntultipUe , 

v^.4 j4-» — +— ♦ 6»' V*224- V. 1 8. br 

^^5,_4_l_^,3,)lButfo?tt)emoarecafe3t«oetournc 

all tbe mtm nomberu into oncl^ fractions * anD tljcn 

ooc 3 multfplic tljem o^uerl^ 

'/♦IS* 




-/♦lo8» 



-1— -/.loS. 

/♦ 68 — i--^vr74. 

J^irfi 31 multtplte /. ^f br v'^d. anu tl)crc commct!) 
^/.iii tftat tg V,2 7 2. again 3 Doe multiplic.6. c: V .:6 

by /♦6.ann itntafeetl) v/.2 1 6»«en 3 ntultipiic v .t 
b^v^-6.litgiuet!)V/-;,lwt)icl)et0V.n6;j0t5itai^ 
v/. 1 8 . ntulttplict bp* \^. 6. Doottl) mafec / . i o s. ail 
hjhtclie 3! at Doune tDttb tljeir conuenicnte ^gncs* 

after t!)at^ multiplieV,^! b^. v *?. anD itpcloet^ 
v^, ^* tftat is. v^* n 6. t>l)icbc 3 fcttc Dounc luitl) bis 
^m^t—V-^'^M^^ v/o6br.^.«'a!ictl)/.l oS.airD. 
IpV.T bp / ^.tioctlj ijtuc V.6 8. anD lafi of all,/* 1 8. 
m'ultiplicD brV.^bjviigctli fojtbc. /.-> 4» 

®Sbcn aUtljcrcb0placcDtoniicm0ntli?,31 Doc con ^ 

fiDcrtbat— I— /.n6.anD /. no.matebcc 

fcothe cancellcD,bif aufe tbe one Dottb abate li)c otbcr. 

anDUI?cltiaies5,— f-/*i o8.anD Va o8»ecl)e 

abate o'tbertfo tbat VcMmwi^ botbc be reiettcD. 

Chen 33 fecttiat v/.68bcvng abateD out of ^.272 
there iDiUreniain.v^.68.anDinime,i/.H*bernga^ 

faatcD out ofV.2 1 6.Doctb leaue. v^.t 4; ^0 that the 
lobole wultiplicatio Dotb maUe ium\? v ^^-i^y 4 



Thejm 

iul)f c^e i^ tlie firSe of ntoenoet j^tiD fo i& tl^af DtuCiton 

app.:oueD gooo. 
^» o^i?jf r Rafter, ^et fo? pou ererrtfc^ou fl^all feaue fome 
tfx^m/'/r. erampleismoareofotutitom 

v'»4j6» ♦/♦72«fj3fettetobeeDfuioeD!jp 

^cbolar. CijatDfutfo^mullBf -/^is,—- 1 /.6 

multtplic b^ t)is fontrade^tuftictie ^^ { g] . ./ 5* 

fo , 33 pou mate fone perceiue, Vft^ft* t\ 

tbere luill rtfc. 1 8» 6, tUt 10 -^^^"^ *^* ^ -* 

1 2. lubicbe muft be kepte fo; tbe netoe Dfuifo^ 

£Di)cn ll^aU 3 multiphe tbe former DiutoenDe, tljat 
t0 /•4 ) 6 — /t? 2 bg tljcfame rtftdmlU /, 1 g — V6 

i/.4^6 '/♦72» 

-/> 18 -/♦ 6» 

^^.82 08 ^v^.1296* 

^,4; 2*' ■ /«2 7 3 6» 



v^»82o8— I— /.432 — — '/♦2756 v'.l2 96. 

0ITD tljere tutU rife of tl)at multiplication, as ftcre bp 
example apperetb /♦S 2 o 8—1 — 1/. 4 5 2 — ^,27 5 6 

1 2 9 6; tufticbe iiober 3! lljall Diutoe bp» 1 2» tbat 

luas founoe fo; tlje netur Diuifo;* 0uD tbcn UitU tbe 

^uoHentc bee«/.J7 — 1 — /.? ^/« 1 9.' -/♦9» 

0s l)ere in iuoo;&e Doetlb appeare* 

i^.82o8~-h-v'»4?2 i/.27l6 V.i2 96(/»r7+v/»5 1 

v^i44. '/144 1/.144 v/,i44» 

WiiitYt 31 baue fet./* 1 4 4.fo2. 1 2. fepirg tljct be all 
one: but tbat. v/. 1 4 4. is moarc apte fo; tbis tDoo;ke. 
anD J baue repeatco it as often t^mes, as tbe Diuifo; 
fljoulD be remoueo. 
ihepyoofe, ^ut Holu to trie tbis luooihe, iubetber it bee Uiell 
tu2oug:i)te,33 l^all multiplte tlits quotiente bp tl)e firftc 
oiuifc;,! tljrn oujbt'tbe ferHc Diut^^nDcto amountc* 



OfSurde nomhers. 
00ftereine>:ample, poMfcc lw;oug|)tc» 

sf. 18.-4— v^«6* , ,____ 

/.1 026-+— '/♦5' 4-- '/♦Mi i/«i62» 

v/» M2-— f— v/a8 v/.ll4 ^» y^* 

♦/,io26— I— Va8— =^v/*ll4- ~Va6i. 

tsafterc — +— v^. H ♦ Jw^t^ t^x^izW -/♦ y 4» 

aiiDtscameUefib^tt. 

^o__-^—^,542»am) — — V.M2.ercluDcone 
an ott)ec, aiiD tljerefoje muflt bee botlje reiecteo* ^no 
then rcmainetl) onel?, 

/*lo26— ^-'/'iS* '/♦II4 '/.I62* 

Xul}tct>e nombers 3! uooe toell eramtne: aiiD fiiiDc tbat 
/, 1 1 4* berng abated out of* i/* i o 2 6. tberc lutU re^ 
inatneV4y6. 0gatne<f-+'-v^.i8.bcfHbtra£tfO 

put of /, 1 5 2.tl)ere totU reffe— ^ — ^.1 2. Uno 

fo is tfeat iD^ole ntulttpltcatto onclp^ /. 4 S 6 — -/? 2 
agreable to tbe firffe oiufDenue. Mbcrbv it ts mant^ 
felle,tl)at tbe former DtuiGon iuas gcDD. 

^patten it^otu can rou luoojijc tbts evample;^ Thethirde 
Mberc»2 4«is fetto be Diuioeti fain^-H — v/.8. , 

^cljolar* 35 maaHtU obfcnictbe genecaUe rule* "^^* * 
ano multtplie botbe tbofe nombers, bp tbecontrarie 

of tbe Dtuifo?,tbat is.hv t^e refidualU.] v/*8.0nD 

of tbe firUe multiplication of it, 
tott^ t^e D!utDcnDe.2 4,tbere rt;^ 

fetb.72 '/♦46o8»£)ftbc 

feconDemultiplicas 5, — | — ^,8 



2 4» 

J - v/. 8 

72 1/.4608 



tton , iubcretiie ^inomialle is multiplieD ^. -v/^g 

bp tbe 'I{efic!ualle,tm i3 bis contcarp,tbe ^ g" 

totaUeti3tllbe.9 §♦ tbatisbuti* g-ugtis* u 

ann tberfoje fepng. KDoetb notber mul^ -^ ' 
ttplie no3i)iutoe,tbefo;nier nomber. 

Cbat is. 7 ^—V ♦ 4 6 o 8 .is tbe f tto*/f»^tf,tuben 
2 4as OiuiDeD bp3.---t— v/»8. 
^ i^r.t^. i^o;t 



tfjat t0 tfte fU9tienu,hi^, 3. — 1 — /.8» 0ntj tljere tifttft 

216— +— /♦41472 i/»4l472 v^»36864» 

U}ftereof.2» nombers Differpng but by — h- f • 

ntufie bot^e bee retecteD , as nomber^ fuperauoufe^ 



Tbefourthi 
example. 



72- 
3- 



V»46o8. 
V.8» 



216- 

v^.4l472- 



V»4I472 
V»5686 4 



216- 



,I92» 



SCtiatW.24. 
^I^affer* ^on il^all baue one example tnoare, and 



2Dben. 36864.1 s a^«4r(r 
nomberr,anobatb»i92 
to;\)is roote.^ berfo ^e 
tbe lobole nomber td, 

216 I92tl)at<3(a0 

tttsmantfette tnougb) 
2 4.0norof$tbetobole 
luoo^^e phoned 0000* 



tben Uitll 31 mabe an entie of dtutfion* 

M ^cn ^*6 5" 7 o.— 4 — ♦ ^.2 y 4. w pjopounoeo to 
bee nimntn bp /♦y 4 /♦6, 3 tooulo fenotoe tbe 

tiuotiente* 

^cbolar. 3( fee tbe netoe ntuifo; luttl be, 5: 4 — 6» 
tbatis,48» 
0ndtbcnfo? tofinoca tJfufUenDe comxtmente^ 3l 

D^ail multiplie tbe firfte 
OfutDenDe,bp tl)e contra* 



V. r4 



-I — /♦25'4» 
-f— /. 6. 



i^.5y478o- 



i/,39420— f-- 



V.13716. 
V» 1^4* 



rie of tbe firfte Diutfo;, 
tb3tti5bp/.j4— { — /6 



v/,?f478o- 
i/«2 3o4. 



^not^cre tDtllrtre^a^ 

pou fee . / .5J478o» 
—I — ^♦15716'H — /.5942o» — 1— i/.jf24» 
Sbbat Diutden&e mu(t be otutoeo by* 4 8* 0; moare ap^ 
tly by. /♦2 ; o 4. ano tbe qutthnte tntU bee* 

00 ^ere appearetl^ m luoo^Ue* 

V.157I6— h-/3942o+v/l5'24(v/lJ5^+v/'.j};if/l7.-||+v^^ 



'/.2304 v/«23o4 \/'23o4. 

0nl>t!)nttljt0 U)oo;lici0gcoO3 BjiusUpaoue ttb? 

ntnltfpluatton* 



ofSurJe nomhers. 



tnultf pUcaf ion* as tlje crample folotupng oooetl) oe^ 
dace. W3i\iztt bp tbc ficlle multipltcation tljereromp 
met^. 8* nombcrs, tftat 10 ♦ 4 . U)itb ♦ — ]— ♦ anD»4< 
toitb —♦ 

v/»f4. vA6» 



V» hi 1 V*-75i 1 ^V* — -iji [- V*-,yi» 

>^ ♦ 79J ^ • I»a ^* 191 V ♦ ,yi» 



^/ i?96fio__J / 1711 ./ '97'° ,. . . . ,./ 7" 

V* -qi 1 V*-T9i V*-~T9i V^Tgi* 

anD btcaufc tlje firffe nober loitlb ^»ts equaUc 

to tl)0 tl)icDc tuttb — i — , tlierfoac tl)eC bot!)e \n\xVi be 
reiectcD.agatnmasmocbeastbefeconDenobertDtt^ 

ij5 eqiialle to tbe fourtlje nombec luitlj — | — , 

tbet botbe i^iall hzt canccUeD. aiiD tftcn remaincttj.z. 
nombers Ujitl) —\ — ^ano otber»2*Vuitb ♦ 

^0 if t?ou abate tfje tfjiroe out of tbc firttc 

jt\}Cquotientc VoiW be.v^.6 ^ 7 o. 



i^ihcUjates If pou abate tlje fourtlje — — out of 

tbe fCCOUDc) — i — ,tlje ^uotlente UliU r«^lDC V.2 r 4» 

auD tW botbc Ujill make tlje firile DtuiocuDc . /♦ 
6 5" 7 o* Mberbp tbe former uiuifio is app.:oucD goou. 
fatter. Mis ft)aU futfice fo: tiuiliou. 

OfextraSiion ofrootes. 

\^t rterte toooake is cictcaction ofrootes; 
Uibicbe vou mate tjerp eafiUe luoo;!^e, b^ 
puttpngtbe figne oftljecoote, tbatvou 
oefire , befo;ie tbe iubole nombec. as if 
^_^'pou UjcuID tjauetbefquacerooteof /. lo 

/♦^.tbisisttvv'.io— I Z.^"* %i^tCuhikt 

rootcoftl)eramcnobecis.\\\^./.io— 1 — v^.^ano 
tbe ^^xixs^^^ toQtc of it 13 v/./* I o. — -| — v^.v 
)15ut If t'OU Uiil! ijmziUzS^tiarc uoote of* I o-+— v''^ 




Thejfrte 

it ti5 V. I c — \ — -/.^^SUfi W Cuhi^e rcotc iB.w/* I O 

— ^ — i/. y. Lifeetoaies !)ts ;<f«;</:^r»;^% rcote id 

— — 2. SSlicCtt^/^frootcis^wV.W.iS 2» 

^iiD tl)C :^fn^ixf":K!hf roote t0,u/» W. 2. 

&^^c^)ctar. ifjcrcbv 3 pcrcriuc tijat tftc tater parte of 
tl)cccFofit!cn,isnottsaricti atall, lJutoiiclpt!)eferfie 
parte tafectl) tjutotttt)efi5neoftl)crootc, 2nDtljat 
figne ts referred to tlje lubolc coinpounDc nomber. 

VninirfalU spatter. ^t)cferootfs tl}crcfo;cbeefaUco>»/«<r; 

rofl^«. /4//eroo/«,btcaufctl)ctaret!)erootcs, tiotoftbcfcue^ 
ralle partes of tl)c rompouncic nobcr,fcut of tlje lo^olc 
fompounoc nonibcr. 5lnotl)attstbeOtffcrcncc,bc;' 
tlucne tlit f omnton Surde nombcri5,ant) vnfuerfalU rt»f 
^«.j^o:if/.2 4 — [—\^A44Mkncfo;ncommtin 
Surde nombcr,t!)cn ooctb it betofectt:rt!)at 3; inull taUc 
2.rootcs,t!)at(s.'/«2 4.an0^.i44, antiiornet!)cim 
together, 115 ut if it ftanoe fo; an yniuerfalU roote,it re^ 
pjefcntet!) tbe roote of tbis lobole nomber.2 4 — 1 — 
^/, 1 4 4. U)btcbc is.6,fo? tbe lubolc Sqiure is. 5 5* 

Scholar, g percetue it luell. fD;,v\ 1 4 4,beet?ng 
i2,tl;at.i2.tuith«2 4.t)ooetbmafee.?6.^nDtbercfo;c 

mull tlie ^n'merfalle roote of, 2 4 — V— v^. I 4 4, bec.6. 
jaiiDrov^2 4 — f— /. I44.istuff.6, 

l=^nt If. v^i 4 — F^/. 1 4 4» l^oc dniicic foj a com? 
mon Surde ncmbcr f omponirtje : tbrn !s it maue of. 2. 
rootes^tbat is v/.2 4.lubicbe ts almcI^e.y.anD 1/. 1 44 
bcrng.i 2. i:inD fo tbe U bole rompcuntie roote^in tbat 
fo^telsalmoGe.i?, SiiDisnigbe. :;. trmesfomocbc 
as tbcfrme rombcr^bf vng an yniuer/allerocte. 

!tl9aficr. ISicnufe pen niaieprrnine It tbe better, 
Shilli^ut an ersmpIeinS^tt^/ercnilcrs^maDelikc 
Stf^</M.astbtS.v'.8I — F — -/06 1 mthcnnvniuerfallt 
m^Mben It is equalk to i c, j^o: ^ nuul tafec firfttbe 
loctc of tbr laUe ncmbcr;, Irljicfje rs, 1 9. jano aODc it 

luitlj 



ofSurJe nomhers, 

iDitM Ltoftecbpt^ereamountetfj*! oo.luljofcroote 
t0. io» y^utifitttSLnnaftct the common to;tt of SurJe 
nombf rs, H faetokencti) ttjc raote of. 8 i.anD tljc rootc 
of, 5 6 u (t^at IS. 9.anD. 1 9)to Ucc aODeo together. ano 
f0ttjrimaUc.2 8.Ujl)icl)c isfarreiabauci o. 

i5utfartbcr nolu, if it IlaitDc fo; a common 5tt>v/f 
nombcnano 31 luoulobauctbeSfM-erf ioct:ofif,tl)cii 
is tbat . v^v^ . 8 1 -^ — v/ o6 1 . aiio betoUcncrt) t'ljz 
SjUdre rootc of ti)C fquarc rootc of.8 1. ano tl)c square 
rootc of. 5 6 I. aOOcD togctbcr, tbat is tDe fquarc roo:c 
of. 2 8. 15 ut moitc generally ano moHcaptli.', It bcio ' 
bcnctbtljcrooteoftbc v»/« ?{/*//? roe^f of. 8 i.t v. 5 6 u 

^cl)olar. j;ioiu Jpcrcciuctbatm 0DDttion , ano 
^ubtraition of 5M/'</«,tbc;iaff nombcrs tbat DiD rcfult 
of tbat U)00;l;c,U)Crc ^ntaerfalle rootes, 

a)a [tlT. :1a oil faic trutbc. 13 ut barUc Uibat mca- 
nctl) tbat battle knocking at tbc Doo^c.-' 

^cbolar. jtisamclTcngcc. 

jailer, tubat is tbc mcflfagcr'tcl mc in mine care 

=19 ca fir is tbat tbc mater:' SCbcn istbcrc noc rcmc^ 
Die, but tbat :< muO: neglect all ttumcis, ano tcacbing, 
fo;to ImtbHanDctborcoanngcrs. rppfo.Jtuncisnot 
To gooD,to banc quictc tLmtc to tcacbc. 

^cbolar. I5ut mp fo:tune anomi^ fcllolucs , is 
mocbcU)o:rc,tbat^ourDnqutctncs,fobinDcrctbour 
IxnolDlcOgci p.:aic Cod amenoe it. 

£©aacr. 3! am infoicco to make an canrjc of tbis 
mater: I5ut i^ct Vutll jl p;jomifc you , tbat U)l)i:!)e pou 
lljallcbalcngc of mc, lube you fee mc at better laifcr: 
Subatj luiU tcacbc i^ou tbc lubolcartc ot^niuerfalU 
rootes, ^no tbc crtrattion of rootes in all square Surdes: 
luitbtbcDcmonllrationoftbcun , anDaUtbcfo;mcr 
U)oo;kcs. 

3;f X migbtc banc been quictlp pcrmittrD , to rctJ^ 
but a litic \ubilc logcr, 31 baD Dctcrmtn:D not to l)au: 
cearc5,t!U J oaD cn5c0 all tbcf:: t'jmgrs at Lr;af:.r3nt 



Thejfrte 

notD fareiuelL ^nt> applte ^our ftuhit tttfgpntip in 
tl)t5 tbat i^u l}au0 UacneD. 0nD if B mate gctte an? 
(iutetnelTe reafonable , Bj lutll not fo;iget to pcrfo^ttu; 
mi? p^omtfe tuttt) an augementatton* 

^c^oiar. £p^ tarte 10 fo opp^clTcO tsttib pefifcnct^ 

bp tbiB foDainctjnqiitctncffc , tl)at 3 can not crpjcffc 

mv grief. I5ut3 loiU p;aie, fcitb all tbeim tjat 

loue l}oneffe i^notulcoge ^ tbat Cod ot tjtg 

mercictDUl fone cnoc pour troubles, 

anD graunte pou focbe rcllc,a5 

pour traucll Doctlj mcrite. 

j^no al tbat louc Irar^ 

npng: fate tljer^ 

to.0meiT» 

cpafter, ^mtn, 

ant 0men* 




Otnp?mfc0at|lonDon. 

yfnnodomini. 1557.