ROOT

, in Arithmetic and Algebra, denotes a quantity which being multiplied by itself produces some higher power; or a quantity considered as the basis or foundation of a higher power, out of which this arises and grows, like as a plant from its Root.

In the involution of powers, from a given Root, the Root is also called the first power; when this is once multiplied by itself, it produces the square or second power; this multiplied by the Root again, makes the cube or 3d power; and so on. And hence the Roots also come to be denominated the square-Root, or cube-Root, or 2d Root, or 3d Root, &c, according as the given power or quantity is considered as the square, or cube, or 2d power, or 3d power, &c. Thus, 2 is the square-Root or 2d Root of 4, and the cube-Root or 3d Root of 8, and the 4th Root of 16, &c.

Hence, the square-Root is the mean proportional between 1 and the square or given power; and the cube-Root is the first of two mean proportionals between 1 and the given cube; and so on.

To Extract the Root of a given number or power. This is the same thing as to find a number or quantity, which being multiplied the proper number of times, will produce the given number or power. So, to find the cube Root of 8, is finding the number 2, which multiplied twice by itself produces the given number 8.

For the usual methods of extracting the Roots of Numbers, see the common treatises on Arithmetic.

A Root, of any power, that consists of two parts, is called a binomial Root; as 12 or 10 + 2. If it consist of three parts, it is a trinomial Root; as 126 or 100 + 20 + 6. And so on.

The extraction of the Roots of algebraic quantities, is also performed after the same manner as that of numbers; as may be seen in any treatise on algebra. See also the article Extraction of Roots.

A general method for all Roots, is also by Newton's binomial theorem. See Binomial Theorem.

Finite approximating rules for the extraction of Roots have also been given by several authors, as Raphson, De Lagney, Halley, &c. See the articles APPROXIMATION and Extraction. See also Newton's Universal Arith. the Appendix; Philos. Trans. numb. 210, or Abridg. vol. 1, pa. 81; Maclaurin's Alg. pa. 242; Simpson's Alg. pa. 155; or his Essays, pa. 82, or his Dissertations, pa. 102, or his Select Exerc. pa. 215: where various general theorems for approximating to the Roots of pure powers are given. See also Equation and Reduction of Equations, APPROXIMATION, and Converging.

But the most commodious and general rule of any, for such approximations, I believe, is that which has been invented by myself, and explained in my Tracts, vol. 1, pa. 49: which theorem is this; . That is, having to extract the nth Root of the given number N; take an the nearest rational power to that given quantity N, whether greater or less, its Root of the same kind being a; then the required Root, or √nN, will be as is expressed in this formula above; or the same expressed in a proportion will be thus: the Root sought very nearly. Which rule includes all the particular rational formulas of De Lagney, and Halley, which were separately investigated by them; and yet this general formula is perfectly simple and easy to apply, and more easily kept in mind than any one of the said particular formulas.

Ex. Suppose it be required to double the cube, or to extract the cube Root of the number 2.

Here N = 2, n = 3, the nearest Root a = 1, also a3 = 1; hence, for the cube Root the formula becomes .

But ; therefore as 4 : 5 :: 1 : 5/4 = 1.25 = the Root nearly by a first approximation.

Again, for a second approximation, take a = 5/4, and consequently ; therefore as 378 : 381, or as 126 : 127 :: 5/4 : 635/504 = 1.259921 &c, for the required cube Root of 2, which is true even in the last place of decimals.

Root of an Equation, denotes the value of the unknown quantity in an equation; which is such a | quantity, as being substituted instead of that unknown letter, into the equation, shall make all the terms to vanish, or both sides equal to each other. Thus, of the equation , the Root or value of x is 3, because substituting 3 for x, makes it become 9 + 5 = 14. And the Root of the equation is 4, because 2 X 42 = 32. Also the Root of the equation .

For the Nature of Roots, and for extracting the several Roots of equations, see Equation.

Every equation has as many Roots, or values of the unknown quantity, as are the dimensions or highest power in it. As a simple equation one Root, a quadratic two, a cubic three, and so on.

Roots are positive or negative, real or imaginary, rational or radical, &c. See Equation.

Cubic Root. This is threefold, even for a simple cubic. So the cube Root of a3, is either . And even the cube Root of 1 itself is either .

Real and Imaginary Roots. The odd Roots, as the 3d, 5th, 7th, &c Roots, of all real quantities, whether positive or negative, are real, and are respectively positive or negative. So the cube Root of a3 is a, and of - a3 is - a.

But the even Roots, as the 2d, 4th, 6th, &c, are only real when the quantity is positive; being imaginary or impossible when the quantity is negative. So the square Root of a2 is a, which is real; but the square Root of - a2, that is, √(- a2), is imaginary or impossible; because there is no quantity, neither + a nor - a, which by squaring will make the given negative square - a2.

Table of Roots, &c.

THE following Table of Roots, Squares, and Cubes, is very useful in many calculations, and will serve to find square-Roots and cube Roots, as well as square and cubic powers. The Table consists of three columns: in the first column are the series of common numbers, or Roots, 1, 2, 3, 4, 5, 6, &c; in the second column are the squares, and in the third column the cubes of the same. For example, to find the square or the cube of the number or Root 49. Finding this number 49 in the first column; upon the same line with it, stands its square 2401 in the second column, and its cube 117649 in the third column.

Again, to find the square Root of the number 700. Near the beginning of the Table, it appears that the next less and greater tabular squares are 676 and 729, whose Roots are 26 and 27, and therefore the square Root of 700 is between 26 and 27. But a little further on, viz, among the hundreds, it appears that the required Root lies between 26.4 and 26.5, the tabular squares of these being 696.96 and 702.25, cutting off the proper part of the figures for decimals. Take the difference between the less square 696.96 and the given number 700, which gives 3.04, and divide the half of it, viz 1.52, by the less given tabular Root, viz 26.4, and the quotient 575 gives as many more figures of the Root, to be joined to the first three, and thus making the Root equal to 26.4575, which is true in all its places.

Also to find the cube Root of the number 7000; near the beginning of the Table, among the tens, it appears that the cube Root of this number is between 19 and 20; but farther on, among the hundreds, it appears that it lies between 19.1 and 19.2, allowing for the proper number of integers. But if more figures are required; from the given number 7000 take the next less tabular one, or the cube of 19.1, viz 6967871, and there remains 32.129, the 3d part of which, or 10.730, divide by the square of 19.1, viz 364.81, found on the same line, and the quotient 293 is the next three figures of the Root, and therefore the whole cubic Root is 19.1293, which is true in all its figures.—The Table follows. |

Table of Square and Cubic Roots.
Root.Square.Cube.Root.Square.Cube.Root.Square.Cube.Root.Square.Cube.
111644096262144127161292048383190361006859000
248654225274625128163842097152191364816967871
3927664356287496129166412146689192368647077888
41664674489300763130169002197000193374297189057
525125684624314432131171612248091194376367301384
636216694761328509132174242299968195380257414875
749343704900343000133176892352637196384167529536
864512715041357911134179562406104197388097645373
981729725184373248135182252460375198392047762392
101001000735329389017136184962515456199396017880599
111211331745476405224137187692571353200400008000000
121441728755625421875138190442628072201404018120601
131692197765776438976139193212685619202408048242408
141962744775929456533140196002744000203412098365427
152253375786084474552141198812803221204416168489664
162564096796241493039142201642803288205420258615125
172894913806400512000143204492924207206424368741816
183245832816561531441144207362985984207428498869743
193616859826724551368145210253048625208432648998912
204008000836889571787146213163112136209436819123329
214419261847056592704147216093176523210441009261000
2248410648857225614125148219043241792211445219393931
2352912167867396636056149222013307949212449449528128
2457613824877569658503150225003375000213453699663597
2562515625887744681472151228013442951214457969800344
2667617576897921704969152231043511808215462259938375
27729196839081007290001532340935815772164665610077696
28784219529182817535711542371636522642174708910218313
29841243899284647786881552402537238752184752410360282
30900270009386498043571562433637964162194796110503459
31961297919488368305841572464938698932204840010648000
321024327689590258573751582496439443122214884110793861
331089359379692168847361592528140196792224928410941048
341156393049794099126731602560040960002234972911089567
351225428759896049411921612592141732812245017611239424
361296466569998019702991622624442515282255062511390625
371369506531001000010000001632656943307472265107611543176
381444548721011020110303011642689644109442275152911697083
391521593191021040410612081652722544921252285198411852352
401600640001031060910927271662755645742962295244112008989
411681689211041081611248641672788946574632305290012167000
421764740881051102511576251682822447416322315336112326391
431849795071061123611910161692856148268092325382412487168
441936851841071144912250431702890049130002335428912649337
452025911251081166412597121712924150002112345475612812904
462116973361091188112950291722958450884482355522512977875
4722091038231101210013310001732992951777172365569613144256
4823041105921111232113676311743027652680242375616913312053
4924011176491121254414049281753062553593752385664413481272
5025001250001131276914428971763097654517762395712113651919
5126011326511141299614815441773132955452332405760013824000
5227041406081151322515208751783168456397522415808113997521
5328091488771161345615608961793204157353392425856414172488
5429161574641171368916016131803240058320002435904914348907
5530251663751181392416430321813276159297412445953614526784
5631361756161191416116851591823312460285682456002514706125
5732491851931201440017280001833348961284872466051614886936
5833641951121211464117715611843385662295042476100915069223
5934812053791221488418158481853422563316252486150415252992
6036002160001231512918608671863459664348562496200115438249
6137212269811241537619066241873496965392032506250015625000
6238442383281251562519531251883534466446722516300115813251
6339692500471261587620003761893572167512692526350416003008
|
Table of Square and Cubic Roots.
Root.Square.Cube.Root.Square.Cube.Root.Square.Cube.Root.Square.Cube.
253640091619427731699856315544963791436415443993944219536486350888
2546451616387064317100489318550133801444005487200044319624986938307
2556502516581375318101124321574323811451615530634144419713687528384
2566553616777216319101761324617593821459245574296844519802588121125
2576604916974593320102400327680<*>03831466895618<*>88744619891688716536
2586656417173512321103041330761613841474565662310444719980989314623
2596708117373979322103684333862483851482255706662544820070489915392
2606760017576000323104329336982673861489965751245644920160190518849
2616812117779581324104976340122243871497695796060345020250091125000
2626864417984728325105625343281253881505445841107245120340191733851
2636916918191447326106276346459763891513215886386945220430492345408
2646969618399744327106929349657833901521005931900045320520992959677
2657022518609625328107584352875523911528815977647145420611693576664
2667075618821096329108241356112893921536646023628845520702594196375
2677128919034163330108900359370003931544496069845745620793694818816
2687182419248832331109561362646913941552366116298445720884995443993
2697236119465109332110224365943683951560256162987545820976496071912
2707290019683000333110889369260373961568166209913645921068196702579
2717344119902511334111556372597043971576096257077346021160097336000
2727398420123648335112225375953753981584046304479246121252197972181
2737452920346417336112896379330563991592016352119946221344498611128
2747507620570824337113569382727534001600006400000046321436999252847
2757562520796875338114244386144<*>24011608016448120146421529699897344
27676176210245763391149213895821940216160464964808465216225100544625
27776729212539333401156003930400040316240965450827466217156101194696
27877284214849523411162813965182140416321665939264467218089101847563
27977841217176393421169644000168840516402566430125468219024102503232
28078400219520003431176494035360740616483666923416469219961103161709
28178961221880413441183364070758440716564967419143470220900103823000
28279524224257683451190254106362540816646467911312471221841104487111
28380089226651873461197164142173640916728168417929472222784105154048
28480656229063043471204094178192341016810068921000473223729105823817
28581225231491253481211044214419241116892169426531474224676106496424
28681796233936563491218014250854941216974469934528475225625107171875
28782369236399033501225004287500041317056970444997476226576107850176
28882944238878723511232014324355141417139670957944477227529108531333
28983521241375693521239044361420841517222571473375478228484109215352
29084100243890003531246094398697741617305671991290479229441109902239
29184681246421713541253164436186441717388972511713480230400110592000
29285264248970883551260254473887541817472473034632481231361111284641
29385849251537573561267364511801641917556173560059482232324111980168
29486436254121843571274494549929342017640074088000483233289112678587
29587025256723753581281644588271242117724174618461484234256113379904
29687616259343363591288814626327942217808475151448485235225114084125
29788209261980733601296004665600042317892975686967486236196114791256
29888804264635923611303214704588142417977676225024487237169115501303
29989401267308993621310444743792842518062576765625488238144116214272
30090000270000003631317694783214742618147677308776489239121116930169
30190601272709013641324964822854442718232977854483490240100117649000
30291204275436083651332254862712542818318478402752491241081118370771
30391809278181273661339564902789642918404178953589492242064119095488
30492416280944643671346894943086343018490079507000493243049119823157
30593025283726253681354244983603243118576180062991494244036120553784
30693636286526163691361615024340943218662480621568495245025121287375
30794249289344433701369005065300043318748981182737496246016122023936
30894864292181123711376415106481143418835681746504497247009122763473
30995481295036293721383845147884843518922582312875498248004123505992
31096100297910003731391295189511743619009682881856499249001124251499
31196721300802313741398765231362443719096983453453500250000125000000
31297344303713283751406255273437543819184484027672501251001125751501
31397969306642973761413765315737643919272184604519502252004126506008
31498596309591443771421295358263344019360085184000503253009127263527
31599225312558753781428845401015244119448185766121504254016128024064
|
Table of Square and Cube Roots.
Root.Square.Cube.Root.Square.Cube.Root.Square.Cube.Root.Square.Cube.
505255025128787625568322624183250432631398161251239591694481636334255384
506256036129554216569323761184220009632399424252435968695483025335702375
507257049130323843570324900185193000633400689253636137696484416337153536
508258064131096512571326041186169411634401956254840104697485809338608873
509259081131872229572327184187149248635403225256047875698487204340068392
510260100132651000573328329188132517636404496257259456699488601341532099
511261121133432831574329476189119224637405769258474853700490000343000000
512262144134217728575330625190109375638407044259694072701491401344472101
513263169135005697576331776191102976639408321260917119702492804345948008
514264196135796744577332929192100033640409600262144000703494209347428927
515265225136590875578334084193100552641410881263374721704495616348913664
516266256137388096579335241194104539642412164264609288705497025350402625
517267289138188413580336400195112000643413449265847707706498436351895816
51826832413899183258133756119612294164441473626<*>089984707499849353393243
519269361139798359582338724197137368645416025268336125708501264354894912
520270400140608000583339889198155287646417316269586136709502681356400829
521271441141420761584341056199176704647418609270840023710504100357911000
522272484142236648585342225200201625648419904272097792711505521359425431
523273529143055667586343396201230056649421201273359449712506944360944128
524274576143877824587344569202262003650422500274625000713508369362467097
525275625144703125588345744203297472651423801275894451714509796363994344
526276676145531576589346921204336469652425104277167808715511225365525875
527277729146363183590348100205379000653426409278445077716512656367061696
528278784147197952591349281206425071654427716279726264717514089368601813
529279841148035889592350464207474688655429025281011375718515524370146232
530280900148877000593351649208527857656430336282300416719516961371694959
531281961149721291594352836209584584657431649283593393720518400373248000
532283024150568768595354025210644875658432964284890312721519841374805361
533284089151419437596355216211708736659434281286191179722521284376367048
534285156152273304597356409212776173660435600287496000723522729377933067
535286225153130375598357604213847192661436921288804781724524176379503424
536287296153990656599358801214921799662438244290117528725525625381078125
537288369154854153600360000216000000663439569291434247726527076382657176
538289444155720872601361201217081801664440896292754944727528529384240583
539290521156590819602362404218167208665442225294079625728529984385828352
540291600157464000603363609219256227666443556295408296729531441387420489
541292681158340421604364816220348864667444889296740963730532900389017000
542293764159220088605366025221445125668446224298077632731534361390617891
543294849160103007606367236222545016669447561299418309732535824392223168
544295936160989184607368449223648543670448900300763000733537289393832837
545297025161878625608369664224755712671450241302111711734538756395446904
546298116162771336609370881225866529672451584303464448735540225397065375
547299209163667323610372100226981000673452929304821217736541696398688256
548300304164566592611373321228099131674454276306182024737543169400315553
549301401165469149612374544229220928675455625307546875738544644401947272
550302500166375000613375769230346397676456976308915776739546121403583419
551303601167284151614376996231475544677458329310288733740547600405224000
552304704168196608615378225232608375678459684311665752741549081406869021
553305809169112377616379456233744896679461041313046839742550564408518488
554306916170031464617380689234885113680462400314432000743552049410172407
555308025170953875618381924236029032681463761315821241744553536411830784
556309136171879616619383161237176659682465124317214568745555025413493625
557310249172808693620384400238328000683466489318611987746556516415160936
558311364173741112621385641239483061684467856320013504747558009416832723
559312481174676879622386884240641848685469225321419125748559504418508992
560313600175616000623388129241804367686470596322828856749561001420189749
561314721176558481624389376242970624687471969324242703750562500421875000
562315844177504328625390625244140625688473344325660672751564001423564751
563316969178453547626391876245314376689474721327082769752565504425259008
564318096179406144627393129246491883690476100328509000753567009426957777
565319225180362125628394384247673152691477481329939371754568516428661064
566320356181321496629395641248858189692478864331373888755570025430368875
567321489182284263630396900250047000693480249332812557756571536432081216
|
Table of Square and Cubic Roots.
RootSquareCubeRootSquareCubeRootSquareCubeRootSquareCube
757573049433798093820672400551368000883779689688465387946894916846590536
758574564435519512821674041553387661884781456690807104947896809849378123
759576081437245479822675684555412248885783225693154125948898704851971392
760577600438976000823677329557441767886784996695506456949900601854670349
761579121440711081824678976559476224887786769697864103950902500857375000
762580644442450728825680625561515625888788544700227072951904401860085351
763582169444194947826682276563559976889790321702595369952906304862801408
764583696445943744827683920565609283890792100704969000953908209865523177
765585225447697125828685584567663552891793881707347971954910116868250664
766586756449455096829687241569722789892795664709732288955912025870983875
767588289451217663830688900571787000893797449712121957956913936873722816
768589824452984832831690561573856191894799236714516984957915849876467493
769591361454756609832692224575930368895801025716917375958917764879217912
770592900456533000833693889578009537896802816719323136959919681881974079
771594441458314011834695556580093704897804609721734273960921600884736000
772595984460099648835697225582182875898806404724150792961923521887503681
773597529461889917836698896584277056899808201726572699962925444890277128
774599076463684824837700569586376253900810000729000000963927369893056347
775600625465484375838702244588480472901811801731432701964929296895841344
776602176467288576839703921590589719902813604733870808965931225898632125
777603729469097433840705600592704000903815409736314327966933156901428696
778605284470910952841707281594823321904817216738763264967935089904231063
779606841472729139842708964596947688905819025741217625968937024907039232
780608400474552000843710649599077107906820836743677416969938961909853209
781609961476379541844712336601211584907822649746142643970940900912673000
782611524478211768845714025603351125908824464748613312971942841915498611
7836130894800486878467157166054957369098262817510894299729447849183300<*>8
784614656481890304847717409607645423910828100753571000973946729921167317
785616225483736625848719104609800192911829921756058031974948676924010424
786617796485587656849720801611960049912831744758550528975950625926859375
787619369487443403850722500614125000913833569761048497976952576929714176
788620944489303872851724201616295051914835396763551944977954529932574833
789622521491169069852725904618470208915837225766060875978956484935441352
790624100493039000853727609620650477916839056768575296979958441938313739
791625681494913671854729316622835864917840889771095213980960400941192001
792627264496793088855731025625026375918842724773620632981962361944076141
793628849498677257856732736627222016919844561776151559982964324946966168
794630436500566184857734449629422793920846400778688000983966289949862087
795632025502459875858736164631628712921848241781229961984968256952763904
796633616504358336859737881633839779922850084783777448985970225955671625
797635209506261573860739600636056000923851929786330467986972196958585256
798636804508169592861741321638277381924853776788889024987974169961504803
799638401510082399862743044640503928925855625791453125988976144964430272
800640000512000000863744769642735647926857476794022776989978121967<*>61669
801641601513922401864746496644972544927859329796597983990980100970299000
802643204515849608865748225647214625928861184799178752991982081973242271
803644809517781627866749956649461896929863041801765089992984064976191488
804646416519718464867751689651714363930864900804357000993986049979146657
805648025521660125868753424653972032931866761806954491994988036982107784
806649636523606616869755161656234909932868624809557568995990025985074875
807651249525557943870756900658503000933870489812166237996992016988047936
808652864527514112871758641660776311934872356814780504997994009991026973
809654481529475129872760384663054848935874225817400375998996004994011992
810656100531441000873762129665338617936876096820025856999993001997002999
811657721533411731874763876667627624937877969822656953100010000001000000000
812659344535387328875765625669921875938879844825293672100110020011003003001
813660969537366797876767376672221376939881721827936019100210040041006012008
814662596539353144877769129674526133940883600830584000100310060091009027027
815664225541343375878770884676836152941885481833237621100410080161012048064
816665856543338496879772641679151439942887364835896888100510100251015075125
817667489545338513880774400681472000943889249838561807100610120361018108216
818669124547343432881776161683797841944891136841232384100710140491021147343
819670761549353259882777924686128968945893025843908625100810160641024192512
|

The following is another Table of the Square Roots of the first 1000 Numbers to 10 places of decimal figures beside the integers, which needs no farther explanation, as Numbers stand always in the first column, and their Square Roots in the next.

Table of Square Roots to ten Decimal Places.
No.Square Root.No.Square Root.No.Square Root.No.Square Root.
11.0000000000648.000000000012711.269427669619013.7840487521
21.4142135624658.062257748312811.313708499019113.8202749611
31.7320508076668.124038404612911.357816691619213.8564064606
42.0000000000678.185352771913011.401754251019313.8924439894
52.2360679775688.246211251213111.445523142319413.9283882772
62.4494897428698.306623862913211.489125293119513.9642400438
72.6457513111708.366600265313311.532562594719614.0000000000
82.8284271247718.426149773213411.575836902819714.0356688441
93.0000000000728.485281374213511.618950038619814.0712472795
103.1622776602738.544003745313611.661903789719914.1067359797
113.3166247904748.602325267013711.704699911120014.1421356237
123.4641016151758.660254037813811.747344380820114.1774468788
133.6055512755768.717797887113911.789826122620214.2126704036
143.7416573868778.774964387414011.832159566220314.2478068488
153.8729833462788.831760866314111.874342087020414.2828568571
164.0000000000798.888194417314211.916375287820514.3178210633
174.1231056256808.944271910014311.958260743120614.3527000944
184.2426406871819.000000000014412.000000000020714.3874945699
194.3588989435829.055385138114512.041594578820814.4222051019
204.4721359550839.110433579114612.083045973620914.4568322948
214.5825756950849.165151389914712.124355653021014.4913767462
224.6904157598859.219544457314812.165525060621114.5258390463
234.7958315233869.273618495514912.206555615321214.5602197786
244.8989794856879.327379053115012.247448713921314.5945195193
255.0000000000889.380831519615112.288205727421414.6287388383
265.0990195136899.433981132115212.328828005921514.6628782986
275.1961524227909.486832980515312.369316876921614.6969384567
285.2915026221919.539392014215412.409673646021714.7309198627
295.3851648071929.591663046615512.449899598021814.7648230602
305.4772255751939.643650761015612.489995996821914.7986485869
315.5677643628949.695359714815712.529964086122014.8323969742
325.6568542495959.746794344815812.569805090022114.8660687473
335.7445626465969.797958971115912.609520212922214.8996644258
345.8309518948979.848857801816012.649110640722314.9331845231
355.9160797831989.899494936616112.688577540422414.9666295471
366.0000000000999.949874371116212.727922061422515.0000000000
376.082762530310010.000000000016312.767145334822615.0332963784
386.164414003010110.049875621116412.806248474922715.0665191733
396.244997998410210.099504938416512.845232578722815.0996688705
406.324555320310310.148891565116612.884098726722915.1327459504
416.403124237410410.198039027216712.922847983323015.1657508881
426.480740698410510.246950766016812.961481396823115.1986841536
436.557438524310610.295630141016913.000000000023215.2315462117
446.633249580710710.344080432817013.038404810423315.2643375225
456.708203932510810.392304845417113.076696830623415.2970585408
466.782329983110910.440306508917213.114877048623515.3297097168
476.855654600411010.488088481717313.152946438023615.3622914957
486.928203230311110.535653752917413.190905958323715.3948043183
497.000000000011210.583005244317513.228756555323815.4272486209
507.071067811911310.630145812717613.266499161423915.4596248337
517.141428428511410.677078252017713.304134695724015.4919333848
527.211102550911510.723805294817813.341664064124115.5241746963
537.280109889311610.770329614317913.379088160324215.5563491861
547.348469228311710.816653826418013.416407865024315.5884572681
557.416198487111810.862780491218113.458624047124415.6204993518
567.4833147<*>3511910.908712114618213.490737563224515.6524758425
577.549834435312010.954451150118313.527749258524615.6843871414
587.615773105912111.000000000018413.564659966324715.7162336455
597.681145747912211.045361017218513.601470508724815.7480157480
607.745966692412311.090536506418613.638181697024915.7797338381
617.810249675912411.135528725718713.674794331225015.8113883008
627.874007874012511.180339887518813.711309200825115.8429795178
637.937253933212611.224972160318913.747727084925215.8745078664
|
Table of Square Roots.
No.Square Root.No.Square Root.No.Square Root.No.Square Root.
25315.905973720631617.776388834637919.467922333944221.0237960416
25415.937377450531717.804493814838019.493588689644321.0475651798
25515.968719422731817.832554500138119.519221295944421.0713075057
25616.000000000031917.860571099538219.544820285744521.0950231097
25716.031219541932017.888543820038319.570385790844621.1187120819
25816.062378404232117.916472867238419.595917942344721.1423745119
25916.093476939432217.944358444938519.621416870344821.1660104885
26016.124515496632317.972200755638619.646882704444921.1896201004
26116.155494421432418.000000000038719.672315572945021.2132034356
26216.186414056232518.027756377338819.697715603645121.2367605816
26316.217274740232618.055470085338919.723082923145221.2602916255
26416.248076809232718.083141320039019.748417658145321.2837966538
26516.278820596132818.110770276339119.773719933345421.3072757527
26616.309506430332918.138357147239219.798989873245521.3307290077
26716.340134638433018.165902124639319.824227601645621.3541565041
26816.370705543733118.193405398739419.849433241345721.3775583264
26916.401219466933218.220867158339519.874606914445821.4009345590
27016.431676725233318.248287590939619.899748742145921.4242852856
27116.462077633233418.275666882539719.924858845246021.4476105895
27216.492422502533518.303005217739819.949937343346121.4709105536
27316.522711641933618.330302779839919.974984355446221.4941852579
27416.552945356933718.357559750740020.000000000046321.5174347914
27516.583123951833818.384776310940120.024984394546421.5406592285
27616.613247725833918.411952639540220.049937655846521.5638586528
27716.643316977134018.439088914640320.074859899946621.5870331449
27816.673332000534118.466185312640420.099751242246721.6101827850
27916.703293088534218.493242008940520.124611797546821.6333076528
28016.733200530734318.520259177540620.149441679646921.6564078277
28116.763054614234418.547236991040720.174241001847021.6794833887
28216.792855623734518.574175621040820.199009876747121.7025344142
28316.822603841334618.601075237740920.223748416247221.7255609824
28416.852299546434718.627936010241020.248456731347321.7485631709
28516.881943016134818.654758106241120.273134932747421.7715410571
28616.911534525334918.681541692341220.297783130247521.7944947177
28716.941074346135018.708286933941320.322401432947621.8174242293
28816.970562748535118.734993995241420.346989949447721.8403296678
28917.000000000035218.761663039341520.371548787547821.8632111091
29017.029386365935318.788294228141620.396078054447921.8860686282
29117.058722109235418.814887722241720.420577856748021.9089023002
29217.088007490635518.841443681441820.445048300348121.9317121995
29317.117242768635618.867962264141920.469489490548221.9544984024
29417.146428199535718.894443627742020.493901531948321.9772609758
29517.175564037335818.920887928442120.518284528748422.0000000000
29617.204650534135918.947295321542220.542638584248522.0227155455
29717.233687939636018.973665961042320.566963801248622.0454076850
29817.262676501636119.000000000042420.591260282048722.0680764907
29917.291616465836219.026297590442520.615528128148822.0907220344
30017.320508075736319.052558883342620.639767440648922.1133443875
30117.349351572936419.078784028342720.663978319849022.1359436212
30217.378147196936519.104973174542820.688160865649122.1585198062
30317.406895185536619.131126469742920.712315177249222.1810730128
30417.435595774236719.157244060743020.736441353349322.2036033112
30517.464249196636819.183326093343120.760539492049422.2261107709
30617.492855684536919.209372712343220.784609690849522.2485954613
30717.521415467937019.235384061743320.808652046749622.2710574513
30817.549928774837119.261360284343420.832666656049722.2934968096
30917.578395831237219.287301522043520.856653614649822.3159136044
31017.606816861737319.313207915843620.880613017849922.3383079039
31117.635192088537419.339053751443720.904544960450022.3606797750
31217.663521732737519.364916731043820.928449536550122.3830292856
31317.691806013037619.390719429743920.952326839850222.4053565024
31417.720045146737719.416487838944020.9761769<*>3450322.4276614920
31517.748239349337819.442222095244121.000000000050422.4499441206
|
Table of Square Roots.
No.Square Root.No.Square Root.No.Square Root.No.Square Root.
50522.472205054256823.832750575663125.119713374269426.3438797446
50622.494443758456923.853720883863225.139610180069526.3628526529
50722.516660498457023.874672772663325.159491250869626.3818119165
50822.538855339257123.895606290763425.179356620169726.4007575649
50922.561028345457223.916521486263525.199206336769826.4196896272
51022.583179581357323.937418407263625.219040425869926.4386081328
51122.605309110957423.958297101463725.238858928270026.4575131106
51222.627416998057523.979157616663825.258661880670126.4764045897
51322.649503305857624.000000000063925.278449319570226.4952825990
51422.671568097557724.020824298964025.298221281370326.5141471671
51522.693611435857824.041630560364125.317977802370426.5329983228
51622.715633383257924.062418831064225.337718918670526.5518360947
51722.737634001858024.083168396264325.357444666270626.4706605112
51822.759613353558124.103941586464425.377155080970726.5894716006
51922.781571499858224.124676163664525.396850198470826.6082693913
52022.803508502058324.145392935364625.416530054370926.6270539114
52122.825424421058424.166091947264725.436194684071026.6458251889
52222.847319317658524.186773244964825.455844122771126.6645832519
52322.869193252158624.207436873664925.475478405771226.6833281283
52422.891046284558724.228082879265025.495097568071326.7020598456
52522.912878474858824.248711306065125.514701644371426.7207784318
52622.934689882458924.269322199065225.534290669671526.7394839142
52722.956480566559024.289915603065325.553864678471626.7581763205
52822.978250586259124.310491562365425.573423705171726.7768556780
52923.000000000059224.331050121265525.592967784171826.7955220139
53023.021728866459324.351591323865625.612496949771926.8141753556
53123.043437243659424.372115213965725.632011236072026.8328157300
53223.065125189359524.392621835365825.651510676872126.8514431642
53323.086792761259624.413111231565925.670995306072226.8700576851
53423.108440016659724.433583445766025.690465157372326.8886593195
53523.130067012459824.454038521366125.709920264472426.9072480941
53623.151673805659924.474476501066225.729360660572526.9258240357
53723.173260452560024.494897427866325.748786379272626.9443871706
53823.194827009560124.515301344366425.768197453572726.9629375254
53923.216373532560224.535688292866525.787593916572826.9814751265
54023.237900077260324.556058315666625.806975801172927.0000000000
54123.259406699260424.576411454966725.826343140373027.0185121722
54223.280893453660524.596747752566825.845695966673127.0370116692
54323.302360395560624.617067250266925.865034312873227.0554985169
54423.323807579460724.637369989567025.884358211173327.0739727414
54523.345235059960824.657656011967125.903667694073427.0924343683
54623.366642891160924.677925358567225.922962793673527.1108834235
54723.388031127161024.698178070567325.942243542173627.1293199325
54823.409399821461124.718414188667425.961509971573727.1477439210
54923.430749027761224.738633753767525.980762113573827.1661554144
55023.452078799161324.758836806367626.000000000073927.1845544381
55123.473389188661424.779023386767726.019223662574027.2029410175
55223.494680248961524.799193535367826.038433132674127.2213151776
55323.515952032661624.819347292067926.057628441674227.2396769438
55423.537204591961724.839484696768026.076809620874327.2580263409
55523.558437978861824.859605789368126.095976701474427.2763633940
55623.579652245161924.879710609268226.115129714474527.2946881279
55723.600847442462024.899799196068326.134268690774627.3130005675
55823.622023622062124.919871588868426.153393661274727.3313007374
55923.643180835162224.939927826768526.172504656674827.3495886624
56023.664319132462324.959967948768626.191601707474927.3678643668
56123.685438564762424.979991993668726.210684844275027.3861278753
56223.706539182362525.000000000068826.229754097275127.4043792121
56323.727621035462625.019992006468926.248809496875227.4226184016
56423.748684174162725.039968051169026.267851073175327.4408454680
56523.769728648062825.059928172369126.286878856275427.4590604355
56623.790754506762925.079872408069226.305892875975527.4772633281
56723.811761799663025.099800796069326.324893162275627.4954541697
|
Table of Square Roots.
No.Square Root.No.Square Root.No.Square Root.No.Square Root.
75727.513632984481828.600699292287929.647932474394030.6594194335
75827.531799795981928.618176042588029.664793948494130.6757233004
75927.549954627982028.635642126688129.681644159394230.6920185064
76027.568097504282128.653097563888229.698484809894330.7083050656
76127.586228448382228.670542373788329.715315916294430.7245829915
76227.604347483782328.687976575688429.732137494694530.7408522979
76327.622454633982428.705400188888529.748949561394630.7571129985
76427.640549922282528.722813232788629.765752132394730.7733651069
76527.658633371982628.740215726488729.782545223794830.7896086367
76627.676705006282728.757607689188829.799328851594930.8058436015
76727.694764848382828.774989139988929.816103031895030.8220700148
76827.712812921182928.792360097889029.832867780495130.8382878902
76927.730849247783028.809720581889129.849623113295230.8544972417
77027.748873851083128.827070610889229.866369046195330.8706980809
77127.766886753883228.844410203789329.883105595095430.8868904230
77227.784887978983328.861739379389429.899832775595530.9030742807
77327.802877548983428.879058156489529.916550603395630.9192496675
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ABCDEFGHKLMNOPQRSTU and VWXYZABCEGLMN

Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

This text has been generated using commercial OCR software, and there are still many problems; it is slowly getting better over time. Please don't reuse the content (e.g. do not post to wikipedia) without asking liam at holoweb dot net first (mention the colour of your socks in the mail), because I am still working on fixing errors. Thanks!

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ROLLING
RONDEL
ROOD
ROOF
ROOKE (Lawrence)
* ROOT
ROTA
ROTATION
ROTONDO
ROUND
ROWNING (John)