In the Eighteenth Century

DEPARTMENT OF THE INTERIOR. BURCAUOF EDUCATION

BULLETIN,1924, No. 18

INTRODUCTION OF ALGEBRA INTO AMERIÇANSCHOOLS IN THE EIGHTEENTHCENTURY

.11

LAO GENEVRA SIMONS

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Page Preface `0410.... (*hoMer I. Foreign influencesleading tothe introductiim of algebra into American education__ _ 1 Englishinfluence 1 English algebra 1 II. Algebra atHarvard in 1730 .....J .. - -.. .,AW ,.. :...sm 0M. . 3 Arithmeticnotebooks_ .10 3

woo am ma Algebranotebooks om.. . =, ow. molo 3 Isaac Greenwood 4

am ow. am. Samuel Langdon a.m. era mom ,.. 'James Diman 5 Introduction of theHarvard manuscripts Symbols__ 7 Topics 8 Subject matterOfanallvanced nature 12 eometricproblems 13 Sources ofmaterial 15 Algebra in theHarvard course 17 III. Thenotebook of aPrincetonstudent' 1141 Philip VickersFithian, 1770-1772___ 18 Saundersonproblems 19 Hillproblems 19 WilliamChurchill Houston,profvsor of lees_ 20 IV. Amathematicalnotebook fromthe University of Pennsyl- varila 21 " Mathematics!Compendia 21 Robert Patterson,professor ofmatheratics_____ 22 V. Manuscriptmaterial frommiscelhmeoussources______. Manuscriptsby Robert Brooke 05 " PracticalMathematics" 26 NathanielBowditch 27 .End of thenotebook custom 28 VI.Commencementtheses 30 A commencementcustom 30 A Account ofthe custom ofprintingLlitincommence- menttheses 30 in 1718 32 O Yalemathematical theses Algebra theses In1718 7 33 Samuel Johnsonand algebra atYale in 34 Mathematical thesesitt Harvard 35 Algebra theses at'Harvard In 1721 37 Fluxions atHarvard sandYale.. 38 r- t

t. IV' CONTENTS

Page. l'hapterVII. Mathematkal theses of HarvardCollege______._. 41 General 41 Algebra theses_ 41 VIII. College records and writings of protessorsand presidents_ 45 Harvard requirements 45 Hugh Jones, professor of mathematicsat the College of WilliAm and Mary______41; Thomas Clair, president of YaleCollege______4S Records of University ofPennsylvania______49 William Smith,provost of the University of Pennsyl-

vania A.A. AAA 49 Laws and ordersat Kings College (Columbia Uni- .

versity)______:11) Ix. Evidences of theuseof foreign textbooks______51 Scarcity of printedbooks__ .______51 The Young Mathematician's Guide,JohnWar41___ 51 The Elements of Algebra,Nlthaniel Hammond__ 54 A Treatise of Algebra, ThomasSimpson__ 55 X. The first books containing algebrapublished in thenew world 57 A Mexican algebra 57 A Dutch algebraArithmetica of Cyfrer-Konst______58 Pieter Venema 59 XI. Eighteenth century bookson algebra by Americanau- , k thors t;-1.

Algebra in the work of Nicholas _ The American Youth.Consider and JohnSterry_ 4;5 XII. Algebra andadvertisements 6i Algebra in the publicpress. ______67

Privatetutors and schoolmasters__ - IMM 67 P.. .1 Solutions of algebra problems___ 4=1 Sale of algebra books I.

ammo *WO . OM. 74 A chronological list ofAmerican algebratextbooks to 18'20 75 Bibliography 77 INDEX MD MD mOM 79 0

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4 A V.

- PREFACE .. 4

-Certain periods in the making of history hav;e lyen deficient in contemporary chronicles.This is notably true in the history of American education during the eighteenthcentury.Such history' presentslessonsiothe educator ofalater generation. As we-follow the growth of the American peoplefrjmAbe status of settlers ina newcnuntry tothat ofadistinctive nation with itsownlife to provide for by training and education,we areled toanunder- standing of the American character and civilizatiorf ofour ownday. This understanding isnecessaryfor all those whoareengaged in the attempt,toprepareboys and girlstotake their places in the present social structure. The history of educatibn is madeupin part of accounts of various subjects which h:ive developed intocoursesof study.Mathematics ofsomekind has Always beenincl;dedin such courses..In the American Colonies arithmeticwas animportant subject forprac- ticalreasons.Itwasneeded--for trade andcommerce.With sailing .vessels plying betweeil Europe and America and the only.meansof communication with the " homeland," navigation and all the kinds of sailing that hadto be, put to dailyusecamel()beacontinuation of thecourse'in arithmetic. Astronomical observationswere an importad feature in layingouta courselasea,andsoastronomyis found in connection with arithmetic.Some elenwntary trigonometry, logarithms, andgeoinetric constructions playedA necessary part in the calculations incidentto both navigation and astronomy. With this list the practicalusesof mathematics in that daysare exhausted. It is the ptrpo§e of this studyto show thatalgebra, another branch I.of mathematics,entered, intq the American education of the eigh; teenthcentury, and to show further that We'must seeksomeother )weasonfor itspreseirthanapractical need for it. The research connected with this work has been made from originalsourcesfound inmanylibraries, 1)9th public and private, in v

-41t k. VI PREFACE

L. the East.It would beaWeasure, if itwerepossible,to acknowledge in detail the cordialhelpfilnessfhat has been extendedinevery one of these libraries.In particular the writcr is indebtedto Miss Isadore G.Miidge:reference librarian of Columbia -Upiversity Library, for herefforts in inany directions. To Prof. David EugeneSmith., of Columbia University, who suggested the problem, andwhose interest and appreciation have been unfailing, the writeracknowledges inspiration in this-studyas wellasin Iwr whole professional.life. LA() GENEvn SIMONS. Hunter Colleyc of the Cityof New York.

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s I INTRODUCTION OF ALGEBRA INTO AMERICAN SCHOOLS IN THE EIGHTEENTH CENTURY /

4. Chapter I FDREIGN INFLUENCES LEADING TO THEINTRODUCTION OF ALGEBRA INTO AMERICANEDUCATION . English infl wnee.Educat ion in the American Colonies forinore thanacentury afterits beginningswas anattempt atareproduction of the education of the countries from whichourforefatherscame. The influence offoreignuniversitiesappearsiaeveryaspect of the school life before the American Revolution. The first colleges were modeled ent irelyonthe English universities, sofarasthe limitedresourcesof tile founderallowed.Courses of study, textbooks, and organizationwereforalong time almostex- clusively English.-Presidents went to England to raisemoneyto carry onthe wprkonthe home field.Colle-ge professorswereeither importea from11:nglaildoil Scotland, and inmanyinstances returned t here,as wastieeasewithalong line ofmenat the College of Wil- liam and Ma ìy,orthey traveled to England to obtain the education

necessaryto.1411 their positiqns. . . Professors éducatedatOxford\pr Cambridgeor anyother university must have been interested inqheapplicalion of their foreign experiencesontheirreturn arid would naturally have transplanted the traditions of the day,sofarasitwaspossible,to the American institut ionstowhich theycame.At thesametime theywereengaged in correspondence -with foreignJeaders, andsoaliftrough the period under considerationreflection);of the intellectual life abroad will be apparent. Ewa&agebra.---From the middle of the' seventeenth century worksonalgebrawerebeing published, and prominent teacherswere presenting the subject.In 1707SirIsaac Newton'sworiconalgebra and the theory of equations, the Arithmetica Unireiwallg, appeared; and itwasfollowed byanEnglish translation in 1720.With the unparalleled reputation of Newton and the genius of his discoveries this workmust havetakenastrongboldonthe algebraically mindedli 2 ALGEBRÁ thTHE EIGHTEENTH CENTURY al teacher.It seems.inevitable that the firsiquarter= of the eighteenth century should have:Alen somebeginnings of algebraaswellas fluxions in American schoolsand colleges. The earliest formal pieceof work in American alObra growing out Ofthis English influence is found inaset*of notebooksprepared under the direceion ofanative ,kmerican who went. toEnglandto continue studies begun in his undergraduate days.

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a 1730 RN. ALGEBRAAT HARVARD IN

Arithmetic votebooks.The custom ofkeeping notelmoks for school subjectswaswell-nigh universal ¡hiringthe eighteenthcen- ury.One 'of the,se subjectswasarithmetie.and arithmetic note- Looksarecommonly found inlibraries and collections ofAmericana. They'aresometimessoexquisitely prepared byhand that theyap- pearto bethe work of_acraftsman in printing. One 'of the earliest of these books is thatof Robed Bale. and it hearsthe colophon "'HieEnd of this Treatise ofArithmetick, begun bymeRobert Hale, Feh. %23,17-18/9tzEnded Nov.5,11-19."It containstreatises onarithmetic-,logic,metaphysics, physics, geography,antiplow- try.'Only Occasionally isasectiononalgebra numberedaniangthe contents of suchahook. Alychlyt votebooN.With the comparative rarityOf algebraic material, the discovery' ofacomplete nainuscriptnotebookov algebraasearlyas1739 greatly enrichedpurknowledge of that subject in America. The manuscriptreferred towasfoun(kin the museum.housed in the old jailin rk Village. Maine. Itisa manuscript.onalgebra by Samuel Lang( 1.atonetime president of Harvard College: and theoriginal of itwasprepared,aslater evidence.will show,by isaac Greemwood, for .several years professor ofmatLematics ,findnatural philosophy ofthesameinsti- . talon. The value cif this manuscriptwasconsiderably enhanced by the further discovery, the Ntanuscripts Americana atHarvardUni- versity, of another manuscriptOnalgebra. whichsoclosely resembles the LaVonone astoleavenodoubt that the .twonotebooks w.ere

taken from. thesameoriginal. The HarvardmanuscriptASwritten

'American Antiquarian Society. 'Robert Hale wasgraduated 'from Harvard in11211and this probably representsthe nnithematleal work that he took duringhis freslunal or Sophomore year. 3 David Dugene Sn\ith."A tahapse atWarlyCcAnialAlgebpi."Nehool and Suefetp,. Jan. 5, 1918.This article fully describesOil manUscript, and description herein contained willnecesatirily-covor thefame ground, although itbethe result of a study

made. bv. the author. . 3 4 ALGEBRA IN THE EIGHTEENTHCENTURY by James Diman, whowasforatime the librarian of Harvard Coh lege, and contains the date 1730. The existence of two suchmanuscripts isa.matfer ofgreatim- portaricein the history of American mathematics. One manuscript might have been the work giventoa'private pupilby the professor at Harvard, but two similar manuscriptswritten at different times during thesameprofessorship afford unmistakable prooftl)at such workwasbaing taught during the period of this'particularman. The Langdon manuscript consists of 75 numberedpages,.with2 unnumberedpagesforming afrontcaverand18unnum- heredpages atthe end.The, 'fitter containnonotes, except oneleaf.Thepages are14.7cmiby 18.7cm.,and the bookcon- sists.of .48 leavels, three-fourths ofoneleaf having been cut out. by the author.On the frontcover appearsthe inscription: "Samuel Langdon's Book, July- 25, 1739," andonthereverseof this lealare the words: " Algebra by Isaac Greenwood, M. A.Began July 25, 1739."A colophon *reads: " Finished writing Algebra August, 17, 1739.Algebrae Finis." The. Diman manuscript consists of 4 unnumberedpagesfollowed by 125 munberedpages,16cm.by 19.3cm.In theupperright-hand cornerof the first.'pageis the: inscription:".Tames Diman's Book s. 1730/31." In large writingonthissame page appearsthe title: " Al- gebraorUniversal Mathematics reviewed 1738 with Notes and Addi- tions." The thirdpagehas the follo.wingnote: " Books perused in yereview ofmyVgebra made in 1738. 1. Harris Lexicon Technicum. 2. Chambers Cyclopaedia.. 3. Wolfius Elementa M4ffieseos Uni- vers." The work endson page1.25 simply with "Finis.'" Theap- parentdifference in the lengths of thetwo books is due toadiffer- encein the size and closeness of the Ntriting andnotioadifference in theampunt ofsubject matter, the Dimanmanusq4t 'having only sixpagesnot foil* in Langdon. . I8aac Greenwood.Interest always attachestothe.person'alities of menconnected withanywork, andso weshall first givesome accou of the threemenwhose name§appear Onthesetwo manuscripts. ban Greenwotid iras born in Boston in 1702, andwasgraduatedarHar- vard in 1721.Threeyearslater he received the A. M. degree. -In 1727aprofessorship.of mathematicswascreatedat Harvard through the benefaction of ThomasHollis, of London. As is still and probably eveewill bethecustom, the college authoritiessougheamofigtheir owngraduatesabrilliant studenttofill the chair. 'Theywereunani- mousin their choice of Greenwood.A. visittsoEngland about this. time enabled- himtoqualify himselfmoreperfectly for the expected appointment. He receivedit, andanentry in the " Minutes of the

704

. k tp ALGEBRA AT HARVARDIN 1130 College Officer"under date of1728, February13, reads, " Mr. Isaac (ireenwood installedprofesoir.'14 In 1729 Greenwood publishedanonymouslyaworkonarithmetic.° Thiswasthe second arithmetictoappearin the AmericanColonies and the first byanative American. The'collegénotes above referted tocontain' anotherentry, " N. B. dismissedJuly 13, 1138, diedat Oiarleston, S° CarolinaliOcto.22, 1745."Hisdismissafcame asthe result of his having ben" guilty ofmany actsofgrossintemperance, tothe dishonor ofGod and thegreat hurt and reproach ofthe society."Theieseemsto have beennoquestion aboutGreenwood's abilities.Both thecaiise of education which-hewasservingsowell and hisown careerin lifewerethe losers by hisweakness 9f will and unfortunateappetites. 'Samuel Langdon.SamuelLangdon6wasborn in Bostonin 1723, entered Harvardin 1736, grachiatedin 1740, andreceived thedegree of A. M. inthree 3Pears.He tudied divinityforatimeat the college itself,and in 1745waslicensedto preach. The Universityof Aber- deenconferreduponhim thehonorary degree ofS.';.D.In 1774 Langdonbecame presidentof HarvardColkge, and hisname appearsfor the firsttimeonthc Harvardcopmencementprogramfor 1776.Itmust have beenasatisfactionto have dedicatedto him aspresidentthe first,sheet ofcommencement thesestoappearafter the Declarati6nof Independence,for hewasheart and soul insym- pathy withthe principlesof theAmericanRevolution.These patri- oticsympathies ledin theyendtohis'forcedresignationaspresident in 1780.Hespent the remainderof his lifeasministerat Hampton Falls, N.H., wherebe die.d andwasburied in1797. James Diman.hiiiies Diman,7wasbornm1707 in EastHampton,l'i Long Isbuid.In 1730 1w,ffraduatcifrom Harvard,and in 1733re- ceived thedegree of A.; APF.Hewasappointedlibrarian in1735 and served until the springof 1737.In Februaryof thatyearhewas . calledto the pastorate of the Second,orEast, Churchin Salem, . his ministry 'Ittlre continuedforover50years. . Wesee,therefore, that Dimantookpartorall ()thiscollegecourseduringGreenwood's term of serviceand later w)is associatedwith himinanofficial . capacity. The earlierdate, 1730-31onthe Dimanmanuscriptoccurred dur- ingGreenwood'sfirstyearsin hisprofessorship,and hence - there is .4 HarvardUniversity Library. g Seepage 68 for conclusiveevidenceon the authorship of this of this mirk arithmetic.Thecopy in the New YorkPublic Librarycontains,. inseveral places," Willis.his Book1733."Willis graduated fromHarvard in1735 andno doubt used the bookin his collegecourse. I SeeF. B. Sanborn.Dr. Langdon(1723-1797),1904, fora full description of Langdon's lifeand connectionwith Harvard College. 1Account taken fromA, C. Potterand C. K. .Bolton.LibrariansofHarvard College; 2647-4877,calEbridge,1897. . e

le .6 ALGEBRA IN THE EIGHTEENTH CENTURY every reasonto believe that Greenwoodwasusing this algebra mate-. rialatleast.as earlyas1730.The other date, August, 1738,onThis samemanuscriptwasjust after Greenwood's withdrawal fromthe college and during Diman's pastorateatSalem.A. comparison of the two shows that the Diman manuscript contains justasmallsec- tion thiq isnotfound in the Langdonoile,andsoDiman added little to the original in his review.Langdon's bookwasmade abouta

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yearlater, in July, 1739.Both manuscriptsseemto becareful . copies of work ?one earlieratHarva.rd.' Introduction of the Harvard manascripts.---We turnnowto a description and comparison of the.Diman and Langdon manuscripts.8 Both algebrasopenwithan" Introduction," which beginsasfollows: 1. This Science is calledAlkebra fromtwo or three wordsenr Arabian Lankunge, w"maybe interpreted either the Art of Restitution, & Comparison; or),,' Art of Resolution & Equation.It 14 also known by various other Names. °The quotatioba givenare from the earlier manuscript,that by Diman; citationsfrom the Langdononewould be practically the,same.

.0* ALGEBRAAT HARVARDIN 1730 7 The first yt wroteupon this Subject, inEurope,termed it y'Rule of Restitu- tion & Opposition; Since,it hasbeen calléd bysome, the ApalytickArt; by others, SpeciousComputation;Regula Reiet 'Census;ye4great Art; Modern .Geometry; UniversalMathematicksRe.(Di/nan,p.1; Langdon, i p. 1.) This introductionshowsaninterestonthe pariof theauthor of the original inthe historyof the subject.For thishistory hedrew directly fromJohn Wallis,9orfromsomework foundedonWallis.

Symbols givenin the Langdon(173U) manuscriptfrom HarvardCollege The paragraghcited showsastrong resemblanceto the openingpara- graph inthe articleonalgebra inChambers'sCyclopaedia." Symbols.---Apageof "AlgebraicalCharacters"follows thein- Effr troduction.Attention ist (tiledtothe interestingfeatures ofthis tableof symbols;to the -barwhichis the onlysign ofaggregation

°John Wank'.A Treiatise of Algebra. both,Historical and Pructica London, 1685. 10E. Chambers,Cyclopaedia.Second edition,London, 1738. 8 ALGEBAA IN THEEIGITTLENTH CENTURY

used; to the symbol forcontinued proportion;to thoseTor inequality, unequal paraljel lines met atonesetof extremities byavertical trans- versal ;" to the capital S turned onits side to indicatethe difference Igetweenaand b;12 a2x5 isexplainedto rnéan"ainvold to ye 24x to ye5thpower& joined inoneproduct," although the more common .formaais found, and frequently evensuch formsas aaaaa,showing the difficulty in adopting therepresentation by the exponent.Later in the manuscripts, it iscurious to find thepowersofx y upto the sixth inthis'latter form,sothat the sixthpowerreadsxxxxxx Cxxxxxy-15xxxxyy-20xxxyyy+.15xxyyyy---Cxyyyyy+ yyyyy(sic) and then to find the seventh,eighth, and ninth powersin the present day form.(Diman.p.27; Langdon,p.19.) Topic8.The. topics reproducedastheyappearin the twomanu- scriptsaresetClowntoshow the similaritybetween them.Theyare paired except whendietopicsareidentical,ablankspaceindicating that the topic is omitted.The firstmembei. of _eachpair is taken from the Diman manuscript,the second fromthat by Langdon. Notation; Algebraical Caracters,Algebraical Characters ;Addition of Inte- gers;Subtraction, Substraction.;" Multiplication of Algebraic Integer,Multi- plication of AlgebraickIntegers; Division ;Algebraical Fractions,Algebraick Fractions; Addition. &Subtraction of Fractions,Addition & Substractionof Fractions; Multiplication ofFractions: Division ofFractions; Involutionof wholeQuantit.ies;Involution of FractionalQuantities, FractionalInvolution ; Evolution of wholeQuantities;Fractional Evolution; BinomialQuantities; Involution [of binomialquantities]; Promiscuous Examples[the examples areefound but not the heading] ;[Heading notgiven, but two fractions are included in the set of examples],Involution of BinomialFractional Quantities; Multinomial Quantities;Involution[of multinomialcompound quantities] ; [No heading but the statement :" Fractional CompoundQuantities are qlso Involved in37°same maner1, FractionalCompound Quantities;Evolution, Evolution of MultinomialQuantities ;Surd Quantities;Notation[(if surds]; Reduction of Surds;Multiplication of Surds;Division of Surds;Addition and Subtraction ofSurds, Addition andSubstraction of Surds;Compound Surds;Multiplication ofBinomial Surds, ;Division in Compound Surds, ;Equation;iteductiönof Equations;Reduction by Addition ; Reduction by Subtraction,Reduction by Substraction;Reduction by'Multi- , plication; Reduction by.Division-; Reductionby Involution;Reduction by of Evolution;Reduction hy Analogies toEquatlon & e Contra ;The Method Resolving Algebraical Questions;General RulesConcerning ye Reductionof Equations; SimpleEquations; The Solution ofAdfected Quadratick-Equations, ;Mr"Oughtreds method'ofsolving adfectedQuadraticks, Mr.Ought- reds method ofsolvingaxlfectedQuadratics; The SolutionofAdfected Equations by takingaWayYSecond Term,The Solution of adfectedEqua-

" These symbols areused by WilliamOughtred in his Clarislle(hentatice, p. 166, London, 1648. "This symbolWW1used flrgt by Oughtred,loc. cit., Eu. 2, 1652. "Diman 'uses the spelling"subtraction," while Langdon uses"substraction."The latter is usedtliroughoutGreenwood's Arithmetic.For a discussion ofthe spellingof these words, set)David Eugene Smith,History ufMathematics, 2 vole.,Boston, 1923-24, hereafter referred to asSmith, History, II, 95. ALGEBRA AT HARVARD IN 1730 9

tions by taking away the second term; The Solution of AdfectedQuadratick Equations by ye method of Compleating y' Square, The Solutionof Adfected OuadratkEquatiotisby ye Method of compleatingye square;Questions, Ques- tionsproducing adfected Quadratick Equations;The Resolution of Cubic Equations, The Resolution of Cubick Equations; Cubic Equationsby Substitu- tion. Cubkk Equations by Substitution; Cubic Equations by Tryalls& Depres- sion, Cubick Equations byTryals & depression;'The Solution of Irregular Cubic's, The Solution of irregularCubicks The Method of Converging Series, The method of.converging Seriesand Approximafion: ,I of Simple

Roots; . , II of Adfeeted Equations; Mr Itaphson's Theorems for Simple Powers [notsodesignated, hut all theRephso'nformulasaregiven]: Mr. ltaphson's Theorems for adfeeted Equations; Dr Halley'sTheorems for Solving Equations of allsorts, Dr. Halley's method for solvitt equatims of all sorts; Concerning the Methodof resolvingGeometiVIOProblems algebraically. Concerning ye methodof solving Geometrical Questiong Algebraically. Treatment of topie8.Manyinterestingpassagesshow the spirit and subjectmatter of these two notebooks, andat thesametime multiply the evidence boiringonfheircommon, sources , Some im- portantoneswill betotichedupon. The clearness of explanationthroughoutmaybe illustrated by the treatment of signs in addition and subtraction, whichbeginsas follows: Thereason of y* Operation in Algebraical Addition and Subtractionmaybe easily understood by considering yeaffirmative and negative Quantities like oppositesasy° Case is in Ballancingaccompts.(Diman,p.7; Langdon.,p.5).

Involution ofBinomial.Quantities....ConsequentlyifyeNumeral figures of Coefficientscould he found y° whole might he performedwithout multiplication and this is doneby y following Problem. To find ye Coefficients in Binomial Powers.Rule.Multiply ye Coefficient into ye Index of ye Power and Divide that Productby ye ,Number of terms, counting horn y°left hand, o and y° Quotient will he y Coefficient-orNumeral Figure of e next successive Quantity.(Diman,p.27; Langdon,p.18). Iri.ationnlQuantities.are noted thus.,4ï2 weh is 2 wm ye Sign of Irration- ality before It....There is also anotherwayof marking surd Quantities where Rootsareexpressed without ye Radical sign by theirIndex, this is foundeduponyemanner of expreskting Powerr, thusas J.' signifies ye Square, -Cube& -111quadratick of :so./j, aj, .11 will accordinglysignify y° Square,Cube and Riquadratick Root ofx....and w° atanytime there is y Sign of Irrationalityprefixt to mixt 'Quantities withy' sign of Insepera- tion (sic)overym thus:In 37+V itis callednuniversal Root.(Diman, p.33f; Lang(1on,p.23f.)

-.. Thereare sevenmethods of " Reduction of Equations.""The ,

seventh," Reduction of Analogiesto Equations & e Contra," is , ,

I illustratedby Ex. 1: 4, 111

Reduceye Analogyx :4:2x, 4 X 4=16,xX ex=2xx227=16=8per,16:6 ' 2 Euclid,za=8Dhnan,p.59; Langtion,p.31). Under" The, Method of Resolving Algebraical Questions "we find 10 ALGEBRA IN THE EIGHTEENTH CENTURY

This part of Algebra is wholly arbitrary &everyoneis left to himselfto pursuehisownparticular Genius andwayof thinking, which isso far from beingaDefect yt.it isoneof y Chief Excellencies of this Science, whichmay from hence not unjustly be calledasublimewayof Reasoning.(Munn,p.62; Lang(Ion,p.33). Eight rules precede theset. of questions under " Simpk Equations," and theseareaccompanied by illustrative examples, employing letters throughout.The method employed in the questionscanbest be understood by examiningoneof them.Thisonébears thesame number in thetwo manuscripts. } .34 -,2c. eeZeze,,z (: i, Siezez/Z7/..r. ., --:((-2---.4et--fet.f ItA 4.e, e..I11.ryÌi..;Itterefjpi. tI e / 111's. -t7 c ) r,4eZe1.)% AerctiJag;/le'try r eépetti 7,4A ).-4: eb. Aiqj ) # / , It(..C119114041g16Lilitle4 (it14(*44,4;61.tile 1 ai ) / 4 .? a. 414 oi/fi,1 tl het.t'/f/(A. II44jerwkit (4)0a 74tit L kttOta.1IL

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1 . 2. 14:-..).%% (PaiLbiLI0r /t.gaVtLIft21-1 1;t.A e44I:" 4 ji1t4LI4é't"b b,,k_ sxpit..44.4.A.i/&#i):1-)e-47,044 el'atfb-riett;if (49.7.ig14 AIM- 1./3 yeIAidt 4eio %.,2 fe:d4 ti-4 yiLlfs4 ,714.,t.A 405-1---ft. i

Z. e;:a..4a:r. eL4pt.-Ole tol(tpet-,tv fa ofr.dr-c? f )4:, .. .. %...- / . ii ./ 3. q 4211/40ex-cpt.dc:4.4, fe;piee ea reextretyV ,,,.44:,-4, .1 ...... ,....,, \./ / i ' 4, dt-figottatfilaier 444.) le-44.01,0dee firiCtrft¡se.' . . . (a4,-0.,,,;:-Aiiri ...... - ...t'- - I 111 l MN -Sit Introduction to cubic equations in the Diman (1730) manuScript from Harvard College

Quest. 25. Iam aBrazon Lyon,_inktwo Eyes, & y° Sole of my Right foot,are so manyseveral Pipes, which fillaCistern, y° Right Eye in 2 41,6 Days, 34 Left in 3, & ye Sole ofmyfoot in 4, hutmyMouthcanfill it in 6 hours, tellmein wt. time all these to-gether,mymouth,myEyes andmyfoot will fill )7° Cistern.(Diman,p.77; Langdon,p.41.) xxxxTheunknownxis taken for the number of hours sought,and 1 1-9 '94 for thepartof the cistern filled by the respective pipes. a oca From these fractional valuesanequation is formed andageneral ALGEBRA AT HARVARD IN 1730 11

solution in terms-Ofa,b,c,d effected. . In the value ofd sofc;und, numerical substitutionsaremade.This is the usual procedurein thesolutionof the questions Dimon gives 26 such problemstò be solved, while Langdon gives 30.Twenty-two problemsarealike in the two works.'The problemsaremixed in theirnature, Diman favoring somewhat themoremechanical kind and Langdon themore practicalpractical for the day in which theytrose,if not for theirs norforours.They includeage.merchants trading for linen and pepper,lumbers multiplied, divided. and operateduponinavariety ofways,the vintner. themanwho foundpoor personsat his door ready to receive alms, (:loc,ks, the shepherd in time ofwar,cisterns, noblemen traveling for pleasure, the gentleman Nvho hiredaservant, ageneral setting hisarmyinsquare array.t personsdiscoursing about theirmoney,and partnership. "Adfected Quadratick Equations areconsidered under three forms, and "Each of¡theseFormsmaybe Resolv'd 3 several ways." The first of the methods shown is:

Mr.Oughtreds" Method of Solving Quadraticks. . . Rule. Ili1t Iply y* ab- solute Number by four & add t hereunto ye Square of y" Differential Quantity, y Square Roat of y" minus y" Differential Quantity being divided by two, is ye Quantity Sought. Diman, p. 78; Langdon.p.431. The second methml is " The Solution, of Adfected Equations by takingawayyeSecond Term."(Dimon,p.81; Langdon,p.46.) In this method yV., (1 is substitt¡ted in the equation ,r2-Ed.r=m, whencey2.....yd+1/2 d2,,r 2 and y2771+14 d2or On _4_ 1/4d2. The labor involved in removingonetermis admissible only from the standpoint of the interest inherent inadifferentmannercifso- lut ion. The third of the, three methods given for the treatment of the quadratic equation is " The Solution of Adfected Quadratick Equa- tions byyemethod ofco'mpleatinj4-yeSquare."This method is thè familiaroneknown by tHesametitjeatthe present time. A set of 25 problems follows these three methods, andtlw two manuscripts agreein the problems tind in the order of themwithasingleexcep- tion.It is of especial interest to note that,in connection Nv.ith the solution of question 7 of this set, the LAngdonmanuscript (p. 49) givesanimaginary liumbtr in the result.This is the only approach toanimaginary in either of the manuscripts,and indicares scholar- shiponthe part of the autluirof the.notesaswellasabilityonthe part of hisNO.

" Oughtred'sClaris Ilathonatiterconsistsfor the aloe pnrt of thesolution of quadratic equations.On his treatment of these equations, seeF. Capri,William Otwhtred,p.29, Chicago,1916."N%444 95337°-24-2 .% 12 ALGEBRA IN THEEIGHTEENTH CENTURY

Subjectmatter ofanadvanced nature.--Upto this point thesubject matter hasbeen largely thatof the latter-daysecondary schools, although the -spiritof thetexts is inmanyrespectsmore mature. Thenext topic is usually includedto-day inacollegecoursein algebra and showsthat work ofahigh orderwasbeing doneat

Introductioirto cubic equations in the Langdon (1730)manuscript fromHarvard College

Harvard in 1730.It begins with" The Resolutionof Cubic Equa- tions. The first methodemployed in the solutionof théseequations is that of Substitution.(Diman,p.94 ;-Langdon,p.54.)An outline of the problem solved inmodern form readsasfollows: Given x)-1-120er-300=3714544.Let =b-f-c. Then (b-f-c)s-WO (b+c)*---300 (b+c)=874544 ALOEBRA AT HARVARD IN 1130 13

Or b1-1-3b2C-4-3bel+c'E 12(11)2+2 WK. +120c3--300b300e= 371 4.544 40 3714544b=100 Substituting in all terms :1)ntaining b alone, b3+120V-8001)= 2170000 3714544-2170000= 154 PP/ 1st Dividend Substituting for b Iii all terms which incwithsomeother11%tterornnmber 3b1 +31)+2401) + 120=5-'1120 Then 1544544 ±5-11120=20=c Substitutingvaliíesof b andein all the terms containing r, 31,c+Sbe-Fc3+240bc+120e:100r= 1250on0 1544544-1250000= 294544 2nd Dividend Then make b equal the former b+ror120 Substituting for b in the terms containing e, . 81)1+3b+240b+ 120-300=7 2180 294544+72180= ,a new o 'Substituting for b &eagain in the terms containing o, 81,c+Sbe+e-i-240bc+12001-30Ww 29 4544 294544-2945 Vi=0 Hence x=124. 116 The second method of solvin4r cubic equations is by " Tryalls and Depression," and this is'virtuallyanapplication of the remainder theorm.With the third method, " The Method of Converging Series," themoredifficult handling of these equationsappears.As defined in the book, .* The Method of Converging Series is an Approximation, or orderly approach nearer&nearere Truth y°more onewo?ksonad Intinitum, by wch %means any Equation wt soever either Quadratick, Cubic, Biquadratick, Sursolid &c may be answeed toanydegree of exactness.(Dirnan,p.102;1Langdon,p.60). Aftersomefurther general remarksonthis method, there follows "Mr Raphson's 25 Theorems for Simple Powers." The theorem for "ye Biquadratick " is applied to extract"yeBiquadrate of 90."In addition, there are" Mr Raphson's Theorems for adfected Equations. (Diman,p.108; Langdon,p.(2.)The biquadratic equationaaaa Parr= is solved." Tp last section of this part of the work is devotedto "Dr Halley's la Theorems for SOlving Equations of all sorts."(Diman,p.110; Langdon,p.64.)TheyaretheoPems for obtaining approximate roots of numericalhigher equations. Geometricprob8.The book closes withasection "Concern- ing the Method of resolving Geometrical ProblemsAlgebraically," asection which showshow completely Oometry 'wasunder thesway of algebra during theeightéenthcentury. Inthe openingpara- graph, the author writes: But in Geometrical Problems IRthol sufficient to note only such Particulars as; are necesivry tolead y Geometritian to some.knownThedrem, whereby 1===ftalIle a Joseph Raphson, whopublished in 1690 a workentitled "AnalyitiR nequationum swirersalia," in which he modified Newton'smethod of finding the approximate rootsofit numerical equation.. For a discussion, seeF. rajori, Oughtred, p. 140. Is10dmund 1itkiley, the greatastronomer.liewasalso deeplyInterestedin geometry andalgebra. 14 ALGEBRA IN THEtIG*TEENTIICENTURY

y" solutionmayhe made.And to facilitate this I must advise ye Student always In Geometrical affairs,to consider ye unknownQuantityasreally known. Then Comparing y°several Quantities in ye Problem, notehow they arerelated either directly orby Consequence ofanyof Euclid &c Demon- rations.(Diman,p.114; Langdon, p. 65). wenty-four problems ofageometric nature fóllow: Dimanuses is Latin throughout this set, while Langdoncontinues tousethe English language.The problemsarethesameandarenumbeNd alike in the two manusoripts.No statement of the problem in words precedes the solution, except inoneinstance.The diagram in each caseis clearly marked. a.nd the datagiven by referenceiothe dia-

AP k -i;;.' 24- (ceira.474 , itttt4: :: a ;.

! "4.7t4Amotemwridi4-4.4 .136.=x+1 At , Z::yi..??__=iiC. 1 a41 culft.2 tut 4-ma /..elcre; ift -74CI'.21.y=tzg

v 1) =ItX-ritA +ige 1111. 1-a caL0-f- 1.44ex '. j.=7 aioot .1E ,*\ 2 4.416.rig.44144.1e+tsofr. /AAetoil 4 oit 1.N1/41 v.. .1 45)0601 1CLAL7ert lAA.A.lc JUL - 4#1 #41.0 a

°looms.aLos 4444LAl4At../)14ZretALIL A. Xj/f t'11)00 411111010 1.44" ci-tra 4tAAP

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I. diA1117.4"1°° itift.,.4;Wit 14)

. . . 9/L,&FpYir.V.Mass=e.w..^ ...?: '..... 1 ...:,. : ,' .e.: .44.4 A %,0$$A,I! . '. 4. 4, r -1 ,..- . ".% 6 t II " '_. . . *. .:-. ; :i. .=. _.3411.1s.::1, t ** (La." nt. f » ii, , le i'. 1 ."4: !$ i- . i ' f . . ..% 09! : .. 'i :-.-34.. '4- ok..7'Att-; .'ve. to: l . ' a/ ..Of . . 4 '` , ..6 ..i. '...0'I iiikte.''AP afrt.- 44 ... ( Geometric Problems solved by nwans of algebra In theDimun (1730) manuscript film Harvard College alb gram: The problem shown would read, " Given thetriángle ABC, with the altitude CDuponthe base AB. Letacircle with center 40 and radius DA cut CDjillE, andtacircle with center D and radius DC cut DB in F." 7Given CE and FB equal toaand brespec- tively and AC:BC:: a :d.Required the value of CD in.terms of a,b, and d.AC-is represented byyand CD byx,and the problem is solved byapp1i6ht*s of the Pythagorean theorem and thegiven data.

11Langdon uses a triangle in which CD equals DB, and henceFR is Incorrect.

2 - ALGEBRA AT HARVARDIN 1730 15

Tlierearescattered throughout the exercisesknnny-facts from Euclid, both definitely referredtoand implied. Wecannot.doubt that the pupils who studied this algebra, writing out thesemanu- scriptsastextbooks, had already takenagoodcoursein geometry. iSources of material.Thesourcesfrom which Greenwood drew his materialcannot, ofcourse,be known withany assurance.Cera. tain -striking resemblancesto textsof the day whichappear$o be

j-/- :4 0 Iv :-r...... - ,,, a - I r--- Ir.: _' Am -Nie -7».4.-±; ti, ,r4-10Ve-, : .=-10 6 ' .,4;- -ST ... L _'' - 117 4t. 1\41. % , _il ;...... r..1íj-,---, ''-;*bl4Eri.o...irtu;s.,; Y. -... SY"-',e4 .. P e...e. ... . ' 111- - , ',- - .. - : 1.11:1L -.. .,, n.A.A. , An- .74. ,40 ptd"e . ,I 0. . e .,41,4144, 4, er*;...,-, . ,-- ., .> 46. . r -'t-L.,,....4,,,..4- .. .x._ '61. $ 'e --,4-t,..,_ riss,,,. en .. 1 t . , I e-r- A . I 41'..' 1P A A' . (' 37 ... 4 no .. 1,, .,d_ At- , ..t., 27F-.1 P,07 li7. A40:,-. '-....::71 nii Il ..;," .,,e .._--. r-,TeA Ei.,f:t10.74., ...._ JD ,... 4.2 .,:,,t,.. ',...... no- 1 -.-re. ill ! _ 401.16. 46.--,,e NI....- ' , ...... _ _.,. . vv,....,,....A.elneits#?iN. ;-i *II% . _. _. , .. r-v, el -t f 1_341 n'X'n' , A i-.- i A47A 1.;.406...... '..,,,4.,>:. ,..,L.6-,, .. -i-r.-,.. 0,-, - As r-; .,..gin,,Li ',I. .- . w ,- . . :.- ..- ,,..t., lag.,st'Vqt... _ 7so : ,.. V, t . i i% 1.. . . :A.. ; In., 4. . t i41- ...... ,. . ... v t-. k- , 11( 3,.,,,if %F.,:'''t_t '. :.';: . 4. : l'i ,;;', 1 f.;,» .,- ,...;., ( _i..1.1.4.4._ ..... Pi .. ,.. . 1 ,, s 174.'1 .6, iv, 6. . , . .,LL',.0.4 .. °'. . .- P. -- ,L'hiro%,,,...t.::.../ !. r-,A,1 ; ,:i I -.-'...04 5.11., Wu., na, . .- . . . ff.. "4% -II ,I n i . 0 ' :': , ...*;* * P 0 %Is Tr:I': N , * b A . ' -, t '.....r. nill.7 -.... o. t1.7..,.,.: 11111. I, a 6' .e: -.91 i r . 1, . . is ....-"at L "16 A

A . 17 046 ...1"'diV-1-.... .,.i. ...3.7,..-.., 4;4., i. kg v.'. 41, . -.... " ...k. rt.' ..' I., ,È.,;;...... 4.4, .7 0..t AP a..; '.4' '5- _17,...I.. 1 . I I t ,ja, M.- ..±: l'.., :1, rl ' - A:t ri -:' ,, , . .-:;7, r; - P 1- liA. , ., ,-.,-,. :.i , -L, a `F. e'tr ` t, .,f !"L.L..e.,,'.. ti 1 'nil I or, ir,' ''kS0 : r A....4.... ,:- -41L-.k, . i a e 0.0 , . % 1 .A04' ±. 04 '4.WP4 .-. ri -1.14 'IV .',Ii4, 1".41.--- l'4*-' ii; rikZ't4 .. .l'on ez eq.." . . . A ` 9 .14i- -14; L'.s "i A ,no 85,44.4io.3":,,,p4,1,3 -. a 1, 4 ame10:0',11.4 Inn. .. ..1 E'4' L,1,6, :ri ._. t4, .J1-- e n- 1 -' :.' rV`tt.,_11 Illati r . ' , e- -.,-_ ..R. 44' _..:9 .- 4. 1 sf rr.-' ,.,... ' s ;-:- it , . - .. 114.... en0 ,-,' ,,we ; a i*. .. .1,Pttlr,6,4,, !k' I i Iri--1-Z:at.t,..p.---0'.t 4 sl. 4.. :II a Pe...... - .,_____1 -* ess...,,i ., - . T. 0. . . . . : 4,...4F3''" iner ,,,,,,,,., e IIIF'. ...- -.Afa:rj .41 4 .... i' ._,....0..., i :.. ,"?' 1. -t," rVr r''' I- 'ote ryen-C .. 'm. .. -'.- 4 -.:i_,0"Z -4i , --,Nr.. i rhk''"' r., . ..-.1 I C.ik. i.., i, ilr -7.111.' y e I. _ .. ... D,;(-Pri"" , -.Y. -.7 .- 4Pore":1 w erirl_.1-,, ,,r)vg -1....r e ,0: t-'' jef'- ./ iii-:-/Let- i/..- - .'`,.'%-5 Tir ... ' -1 . -- . , ';I C . NO' .....a 4/41. w . . . fll.i Ilt_e I --k. ri ,anbiV2L. 7 _JAr9 ' _.. -,a,41 :v : An , ,:-:À . , .- -...,. .( .. R77e f -I 1# e e ...: " fln .si. -1 ":"'-'?L-t1411-2.-,17. ' .,1 4'. el, t, ; tior,-' a , sidlltNil I.7'f;_..,t, _.,' o o :4 ,-;...iA-k -1...... ,.--..----..; ....r. ..., In _!#...... ,, ..riA.?, ' --: .s..-'....S 4 4!'e-,. , . * 0 . 01"-" ...r 17f , ,..1-;, l,, .4.7,1., - 4 l'-',- i .t., ,..- '-:, ;15- .-.-, ..,.R I01-.--K-L,-,-tr, - 1116: '-'1`' il ."Z1...,r --`,. :,,c,.. k,.11..1)A.- .4-,....4t, --omo .it" . , , , J ..r-A 1-varr7 ,.-,- -. i.: Ada't y,lt,21, n _,Arie if--' .-- X. kr,i4 4. - r 1 II.. T.. ' v A, NN ,--I-....nj_ i .11_.1...it t '7IeH r r .1.-.7, "*. o ..11 ."-n7 7Z0V4 , ..;,, --13.4 I _ 4 le,O In i C 74 y_ 1 t --,' 4] -.. V -".). e : I ./ trt 4 A -" ,,i, 61 AO r:.7. .... --, t? 1. It11' a. t. - _ It.9. 4 7--, /- I . 1 ,!. fa' ^- ....0 P1 ei _Fr' rPi. rri---h-"--V..,1&,...-r_,. rrefn.:t% 4,-P., ' 11-tp I e * -4 1 , ,4:41 1 Tr.-11. ,Tal. % .& e . r ,:.1....,Firt,": 1 g..;_r. -.7.;1,A rk i * 1 0. . ..,.....wri21 , ,"Fp.-4:0. ..:.<;.)..'41,4ii`;'2'.

gcdv.7.-1 d . 11 - '¡Ter.°1111.....a -"--".-1, .._A-1 I *4100 - - ...... - .....- : ...... - . S' Li...r. ... , NOE j v.v. r°1 P.; .1, i! : :vpirLd.r to "----71--77li, -A _A 2a - , - ie I 4.I etali;,,,....41uK, / A,...7: le, .-.u. ;', te....:-,;-;, 1° i/1641$ . ...-... -. : iV 'tvd . ...,- AO toll.; pi,1$, P f0 . 1 ..1. , A E:1 r14, i ol ann- l 41 1 1'; 6r :41 .....sw In.--,- ,,A, .. III,P 44'.4 V?et - ...4 - 19.is.a- fl _Ageo.r.1...1- 44 .....tn'-: r a m1101_1 .tittorP., .r.a,..-* krt' t 'h.-.e,5 %3 , ,I1 . ,., I.-..Tr 7-7- .e.. .,44 4e. A::.'' :. ..,, S.4g ''' , A r. Ti AL -''L 4, a tJ'_A _ . -!. 4; r. re : 1$ 'WA r .. .., , , ....- ..1 A 0107 ..._ Al 71.. 4 n4 . o' 4 . dr- -'qeet* ..,, . I r-- saia.4 - fps r :07:1141;' 5Z1prot .:111 flettirk-fA, it ,ram - 1-..:iy-i r ,,,, , -Tni i , : E3 it_Li43. /1- e40-! KW- ti."' rte" -11q fEt-1 "'- ' ' . Geometric probleme solved bymeansdf algebra iu thp Langdon manuscript from Harvard College

morethau merè coincidencesmaybe indicated and indeedsomeof these- haiTe been referredto in the descriptiop of the manuscripts.. Hintsaregiven by the writers. As already noted, »iman states

' that heused in his review Harris, Wolfius, and Chambers's Cyclo- Nedia.Langdonatthe end of the manuscript,onthe inside of

, the lastcoverleaf, 'states that " The Diameter ofaCircle being supiposed,the Circumference is expressed by the following figures-- videChamb. Diet.," and then gives the ratio of the circumference t. I .

- r-s- -n2r_1 _ -a, 16 ALGEBRA INTHEEIGHTEENTHCENTURY et. ofacircle tp thvdiameter correctto 32decimal placesbtft without usinganysymbol forthe ratio. The bodiesof themanuscrilits furnishclues tofurthersoKees of material,for inthemappearthenamesof Raphson,Oughtred, and Dr.Halley.In allprobabilityGreenwood had accessto the original worksof thesealgebraistsandmayhavebrought copies of themback fromEngland.Inaddition tothesenames, welin4 " SuchDemonstrationsas maybe foundin approved.AutImrs such asEuclid,Apollolius,Archimedes."(Diman,p.62;Langdon, pMI)et A reference toChambers hasalready beennuNe, showing that thiscyclopediamayhave beendrawnonfor thehistory ofalgebrx. The meaningof universal rootis givenin practicallythesame wordsasthose usedinCb.ambers, viz "V : :2,which lktst isfcalledanuniversalroot." Under tlwtopic"Reduction ofSurds" and holnog.emal, are em- tworilther unusualtermshderoyeiwal s-, ployed, whichseemtohave beentaken frpmthissourcT. It iseasytobelieve thatGreenwooddrewsomeinspiration, if not actualsubject matter,for thesectionon"GeometricalProblems". from Raphson'stranslation "of Sir IsaacNewton'sArithmetical;pi- ver808.Acomparison ofCertainintroductory mattertothegeo- *metrical exercisesfound inthe t*oworks follows: In thefirstollace, therefore,the Calculus mybe assisted.bythe Additionand `Subtractioncif Lines, sothat fromthe Valuesof the Parts you mayfindthe Values of thewhole, or fromthe Valueof the wholpand one ofthe Parts, you mayobtaintilevalue of theotherPart." I. TheCalculus maybe assistedby y"Addition andSubtraction ofLineg. of y' soyt from y" valuesof ye Parts37" value ofthe whole, orfrom ye value whole and oneof 'yeParts, ye Valueof y" otherPart maylieobtained.(Diman, 61 p.114 t; Langdon, p. f.) a The other twopartsshow thesamesimilarities. One of thesuccessful andwell-knownteachers ofmathematicsof theseventeenth century wasan' Englishman bythenameof John -was s. Kersey(1616-1671).Hewasthe authorofanalgebrawhich entitled: TheElements of thatMathemat.iral Art commonlycalled Algebra.Thefirst editionappeardin 1673,and itwasfollowedby othereditions. Onitppublication,this workbecameastandard authority. An indicationthatKersey'salgebrawasused byGreenwoodin preparing hisalgebra notes wasgiven byareference in oneof the tivomiinuscripts.In statingthe rulefor thecoefficients intheex- Rapbson . froin the lAtinby the lateMr. 18Unirer8a1Arithmetick .. Translated and revisedand correctedby Mr.Cunn,London, 1720. ID Ibid., p.89 f.

ir;1- ; ALGEBRA ATHARVARD IN1730 17 pansion ofabinomial, Langdon (p.19)says :" This is doneby ye following rule,give9 by Sr. IsaacNewton,see p.139."-kersey's algebra iswithout doubt, theauthority referred to.Page 139of that workconsists of "A Table ofPowersproduced fromthe Binomial Boot a+e."Further, the orderof the chaptersand thesublli:adinks under themarein practicallyeverydetail thosegiven 1?y Kersey,and themanuscrijAsagreewith tlfis algebrainevery oneof the symbols used.'b All of the28.probleins under "Simple Equaons"in Kersey arefound either in DinkanorLangdon. Greenwoodwaseclectic in histeaching, gleaninghere and there the mateljalbest suited for theasttuction of his students.Evidence of his wideacquaintance Withwriters of booksonmathematics and of hiswisdon'i in the choiceof subject inat terareshowri by thiscomparison of his work withsomeof the authors of theday. Algebra in theHarvardeonI's-1%-T-*The Iwomanuscripts under discussion beara'close resemblance toeach other, Iota theydiffer enough toindicate that theindividualities of thestudents themselves played apartin the.fin-alproductions.Theyareboth unquestionablybased on asetof notes preparedby ProfessorGreenwood touseina cours( alHarvard. Had helived out, his-.professionalfe, it is altogether -probable thatweshould have had fromhAllpenthe firstprinted work on algeibrawritten byanative nAtican.

el P

4*. Chapter III

THE NOTEBOOK OF A PRINCETONSTUDENT1,

PhiliNFirkersFit The journal and lettersof Philip Vickers Fithian2 haveiwendrawnuponextensively for picturesof life in Virginia during thetime that hespent thereastutor in the hom6 of Colonel Carterat Nomini Hall. It isnotgenenillyknown, however,thatamongthepapers3relatingto Fithian there isto be found acollection of problemswhich givesaninsight into the work algebra thatwasbeing taught inthe College of NewJersey (now.; PrincetonUniversiV) while hewas astudent there.. Philip Fithianwas astudent, at the College ofNew 'Jersey from .171.0-1772,studied theology 1772-73, andtaught in Virginia forthe year1773-74.In July, MG, he enlistedas achhplain inthe Ameri- i canRevolution and served underWashington during thebattles of Long Island andHarlem Heights.Ile diednearFort Washing; ton, October 8, 1776, in his twenty-eighthyear. Fithian keptacareful- diaryup tothe time of hisentrance to college 4nd again'afterhe graduated, especiallyduring hisstay in Virginia.It is unfortunate that-there isrfotrace.ofajourhal kept duringtheyearsthat hefllent in college.Some incidental refer- (flCe,Sthrow lightonthe studies that hewaspursuing. That bewas familiar with several branchesof mathematics is shown'by. the following extra'ct from hisMiscellany: glt A neciamatipnonthe Dliffieulty of Composition;the second In tlie.1,unior Winterprononeed at Nasssti Hall. January10th Anne 1771 ...And when all Is done, I think Itfulla's hard to become a compleat Master ofthe first of those divisions, that isto be able to makea prOlierand concise Introduction to a Sn:jectasto becomeacompleat Proficient 6'inAlgebra and Fluctions: . asold Euclid tensus Inhis 16:prop; Lib. 3 ls sufficiently eVident.

9. I The earliest reference to algebra at Princeton is foundina set of letters writtenrill 1750, 1751. 1752,1 753, by Joseph Shippers.a student at the college, .to his father and otherfriendiCInaletter of the Sth 'of June (1750) hesays: " Laotian learn Horace in alittle.Nhile,"...butmy time isfilleduptin studying Virgil,Gr&kTestament, and Rhetoric,sothatIhaveno time hardly to 11nIover any French,orAlgebra, or any Engliphbook formy Improvement." John Mitcleitn.History oftheCo1lrfic of Nevi Jergra...1.p.141, Philadelphia, 1S77. 4.4. 2PhilipVickersIrv,.Journal and Letters176'7-1774: Edited by John Rogne . Williams. Princeton1900. 3 P. V. Fithian, letters and miscellaneouspapers. 1786-1776, MS. Princeton 1.7niversity lfbr,ary. - A I 16 IhO EARLYREFERENCESATPRINCETON *4 19 It isapparent fromthisspeech thatalgebra, Euclidand fluxions were a part of thecourseinmatlAnaticsat the Colletiieof New ,Iersey duringtheperiodfrom 1770to 1772. Fithian'spapersdonot shqganvsubjectmatter of thesebranches oafmathematicsexcepta setCitproblems inalgebra,but suchaset impliesanexcellentfoundation along othermathematicallines. I _ Thereare99 problemsaltogether.The first48areheáded"Prob- lems of Saunderson,"andproblems49 through99areiridicatedas cominfromHill. Saunderson problems.TlwSaundersonproblemsaretaken from a work entitled:SelectParts ofProfessorSaunderson'sElioments of Algebra.4 Thiswork isanabstract ofalong treatiseonalgebra by NiclolasSaunderson,1e *blindmathematician.The problems in theFithiannotebook'arefoundon pages 121-149 ofthe Saunder-: sonwork, underthe heading," Thesolutionófsome problemspro- ducingsimpleequations."Theyresemble,the usualproblems of the day, problemsrelatingto numbers,age, afish,distribution of 11 money, debtors, hiringa servant whowas to forfeitmoneythe days that he didnot work,couriers,shepherds intime ofwar,and sofiwth. prob.lem11.Thesecond grodpofproblemsis drawnfroma work bydrohnHill whichranthroughmanyeditions,although itisacuriouscollectionofmathematicaltopicsunder thetitle: "Aritlanetkk, both inthethebry.andpractice . . . with thead- ditionof several 'algebraicalbaustions.The likenot, extant."5 *f De Morgan"saysconcerningthe 1745edition of thisarithmetic: Thisis the seventhedition ofa work of muchcelebrity.Itseems to haw owed Itsfame partlyto a recommendationby HumphreyDitton,prefixedto the firstedition(about 1712),praising itin thestrongest terms. this time Perhapsat the onlythingswhich wouldcatch theeye are the_tablesof logarithms at the end,and thepowers of 2up to the 144th, very'usefulfor laying up grains of corn on thesquares ofa chessboard, ruiningpeople byhorse-shoe bargains,and otherapprovedproblems.' Hill gives,on pages 365-382, ninety-nine"ProblemsorQues- tions 'inAlgebra)"with neitherexplanationsnorsolutions.Fithian's set bqns a'ttheforty-ninthandcontinuesto the end,problem by problemagreeing withthelike-numberedonein Hill.Most of these problemsleadto quadraticequations.Othergeneralfeatures of themincInderadicals, $roOortion,fractions,andequations of the quadraticform. *Henceit isobviousthatwehave inFithian's noteMokagoodset of exercisesunder thetwo topics"Simple and QuadraticEquations."

4 Firstedition. Lolidon. 1756.The secondedition, withwhich thiscomparison appearedin1V1. .1. is made. 6l'ompar Nonmadeon Mats of the, eleventh 10% edition.London.1772. i *AugustusDeMorgan, . Arithincticat,Books,p.70, London,1847. L .. , . ..- 1 . , . 41 .. . .1.; % . 1 . _ , _._. ,....z,_..: ...... _____ , 20 ALGEBRA IN THEEIGHTEENTHCENTUrtY ChurchillBoueton,Fithian studied mathematics under William ChurchillHouston.'Houston received his earlyeduca- tion in North Carolina atAlexander's Academy. Hewasappointed ateacher in the PrincetonGrammar Schooljustas soon ashe had matriculated and retainedhis positionduring most ofhis college course. In 1768 Houston w;i'sappointed senior tutor,and thatsame year he graduated withthe' highest honors.In 1771 hereceived the A. M. degree.Now, .in 1768, JohnWitherspoonbecamepresident orthe College of New Jersey.Under hisleadership the curriculum wasenlarged by the introductionofnew coursesin Hebrew,French, and mathematics.Itwasatthis timethat theprofessorship of mathematics and naturalphilosophy,"asmostinunediately requisite,"wasestablished, andAfr. ITouston wasthe firstmanto occupyth.e jhair. %His work wentonfrom 1771until theoutbreak of theAmerican Revolutionwhen, owing tothe abandonmentin the autumn of MGof a4work at thecollege, itwasdiscontinued. liewas amember ofthe ContinentalCongress, clerkof thesupreme courtof NewJersey,and delegate totheConstitutionalConven- tion of 1787and toAnnapolis.By the winterof 1778 Houston wasback againinhis old position,which he helduntil 1783. . The scholarshipandcharacter ofthis mathematicsprofessor musthave madeadeepimpressionMIhis students,with whom he cameinto dailyand close contact.And thisis themanunder whom Fithian did theproblemswhich havebeenpreserved with hisjournal and letters.

11 illiautChurchill HouNlon,r,16-1,:s, ITAccounttakenfrom.IllonfiNAllen4;lenn, Norristown,l'a.,1903.

qt.

frr 171

* --_.÷ - - _ - s'

Chapter IV A MATHEMATICAL NOTEBOOK FRÓMTHE UNIVERSITY OF PENNSYLVANIA "Math rompendia."Mathematicsheldanimportant phial in the curriculum of theUniversity of Pennsylvaniaduring the eighteenthcentury,asis shown elsewhere, andthe existence ofa manuscript notebook whichbelongedtoastudentat,the collegeem- phasizes the strengthof this position.This notebook consistsoftwo II volumes and bearson*itsfirstpagethe words:"MathematicaCom- pendia: orA Short System ofMathematical Literattire, asit is taught by RobertPattersoq,..A. 'NI,Professor of Akthematics inThe Uni- versity of Philadelphia,Augyt 26, 1788.Sane. Miner."1 The first volumeof this work contains170pages,33 emmby 19.2cm. It begins with"Algebra " under the date "Aug-t.26, 1788."This is follow-NIby" Practical Geometry,"" Trigonometry,begun Sep- tember,1788,"five-place " Logarithms,"" Oblique SphericTrigo- nometry," "Afensurat ion of plane figures"including "Areti of ellip- sis," " Mensurationof solids."and" Stereographic Projection of the sphere."The second volumecontains 17)pagesof thesamedimen- sionsasthe first.The subjectsureated Tie"ConicSections," " Sur- veying."" Navigation," and"SphericGeometry."Fluionsare usedto find the length ofa curve,but there isnoformal presentation of the subject.

Out of 170pagesof the .first volume ofthis compendium, 71treat-e of algebra.The topics in the wordsof thetextare asfollows: Definitions,PraxisontheSigns.Addition,Subtraction, Muffiplication, Division,Jnvolution, Evolutipn; Formation& ResolutionId Equations, Method ofresolvihgQuestions that containtwo Equations and twounknown(puma- ties, Quadratic Equations,Method of resolvingquestions that containthree Equations and threeunknownquantities,itliL Resolution ofAdfected Equation"; by theuniversal methodof Converging Series,Themannerof resolving Equationswhere the unknownquantity is tose4ralpowersInbothEqua- tions, The methodof resolving questions whichcontain four3quations & fourunknown quantities, andPromiscuous Question; Throughoutthetext, rulesare.givtln, and thesearefollowed by examples.Under equations, "Simpleguadratic& Adfected "are definedin detail.Fractional sod -Mica]equationsarefound in

University ofPennsylltilalibrary.Another notebook in this library containsno nameordate but gives evidence ofbeing atleastas early as thIssote. 4k, 21 - 22 ALOEBRA IN THE EIGHTEENTHCENTURY 1 the questions which followthese definitions, butnoexplanation.is offered of the special treatmentwhiN Yuri( be accorded them. Eighteen 'questions exemplify simple equalions.The treatment of thle problems is made unnecessarilyhard by theinsistenceonthe useof letters in the first statementof the equationand its later workingout, withthe substitution ofthe numbersgiven in the problematthe end. Thereare manypure&literal equationsamong thesetsof problems.No attempt is made toplace the problems ofageometric nattire inaspecial section. The most advanced workis that under thetopic " The Resolu- tion of Adfected °Equationsby the universalmethod of Converging Series." The solutionof the ninety-fourthquestion illustrates this method and showsalso the kind ofwork in algebra ltwasbeing undertaken in theUniversity ofPennsylvania in 1788.Professor Patterson, who compiledthe materialfor this notebook, musthave believed in problemsmorethan in.mechanical work.This section onalgebra, covering71pages,contains 110problems.The solutions ofmanyof themare verylong, unnecessarilysoinsome instaiwes. But it is signihcantthasomuchspaceshould have been devoted tothe applicationof -thelgebra toproblems rather than to tedi- ouswork inoperations. RobertPilfter8on.This account.would be incompletewithoutsome record of themanwho worked outsuchagoodcoursein mathematics for his students.Robert Patterson 2 wasaScotchman, although he wasborn in Irelandin 1743.Robert early showedpromise of his mathematicalabildy.When lith hadcompleted his first formal schooling, hegaveevidence of -hisadventurous .and ambitiousspirit by setting sailfor the NewWorld. Hecametothe Colonies with the firm intentionof becofilingaschoolmaster andsoonsucceeded in im- pressing hisqualificationsontheproperauthorities in Buckingham, nearPhiladelphia. Helater removed toPhiladelphia and offered instruction tonavigators inmethods of calculatinglongitude from lunar observations,and related matters. During.theAmericah Revolution, Patterson devotedhimself to the causeof liberty,and actedforsometimeasmilitary instructor.'He entered the-University ofPennsylvania in 1..79, receivingan appointment first asprofessor and laterasvice prgvost, andremained there for 35years.The college honoredhim with the degree of IAL.D. in 1816. Afterholding various officesin the AmericanPhilosophical Society, Pattersonbecame its president in 1819,which office hefilled until hisdeath in 1824. Professor Pattersonwas agreatteacher, but heseemstohave written little'forpublicatton.In 1818 there appearedaworken-

1Account taken fromTransactions of the AmericanPhiloaophical Society,Vol.II, New Series,Philadelphia, 1825.Obituary tiotlets of RobertPatterson,11. I). AT THEUNIVERSITYOF PENNSYLVANIA 23

titled: " ATreatiseof PracticalArithmetic,intendedfor theuse of Schools; intwo parts." The" ExplanatoryNotes " of PartI have someinterest inconnect ioh with algebra. Pattersonedited American

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or r? , ..1E4A ;S.; . 57iA .1:` Or - r4 , .4, AP - Solution ofa cubic equation11.orn the notebookofa University of Pennsylvania student in 1788. Doctor Pellintroduced shown here the " methodof registeringthe steps" editionsof several Englishworksonarithmetic,natural andexperi- mentalphilosophyandastronomy, andinsome eases added *. an ap- 24 ALGEBRA IN THEEIGHTEENTHCENTURY pendix to the,edited work.3Besides theMathematica Compendia, he preparedaworkonnavigation for theuseof his students Alich is also extantin notebookform.The title-pageof this book carries t* superscription: "Note-book of SamuelHayes.Philadelphia, ID 1789.Navigation by Robert Patterson,A.M."' If Robert Pattersonhad lived inaday when paperand printing wererelatively inexpensive.he would havepublislwd both the "Ma- Navigation."The character of 4110 thernaticaCompendia"and the " these works calledfor theirpublication.In his long yearsof service asprofessor in theUniversity ofPennsylvania, bemight haveac- complished greaterthings by theusoofitextbooks than by themore tediousdictation of lectures.

I ProfessorCajori states thatPat terson wrote asmall astronomyentitled' TheNorton- ianSystem," whkh waspublished inISOM. 1'.Cajori.The Teaching and lifstury ot Mathematicx inthe United statrx, p.66,Vashington, I S90. 4Manuscript Department,Historical Socicty ofPeuusylvania.

410 t

a ChaperV MANUSCRIPTMATERIALFROMMISCELLANEOUS SOÜRCES

3Thrutcrripts byRobert Brooke.Anotherset of notebookswhich wereprobablyprepared,at least inpart, under thedirection ofPro- fessor Pattersonis that.byRobertBrooke.Ile isgivenas a non- graduate ofthe UniversityofPennsylvania,presumablyof theclass of 1793.1 The first2 f thesebooksbears thesuperscription," RobertBrooke, HisBookMarch10, 17s3.Arithmetick."It. contains180pages. .20.cm.by 33cm.,coveringthe subjectof arithmeticinacomplete fashion.A. secondbook,dated 179.2,ison"PracticalGeometry." WhileBrookewasgivento dating his workrather freely,he failed to set downanyindicationof thetimeat which he begantoprepare the materialonalgebra.Thispart of thenotebookconsists of41 pages, 19cm.by 33cm.,and hasacompleteset of algebratopics. Theybeararesemblanceto the MathematicaCompendiaby Rob- ert Patterson,and theworkwasprobablydoneunder him,although this isabriefercourse.The topicsare as follows: Definitions,Praxisonthe Signs,Addition,Subtraction,Multiplication,Divi- sion,Involution,Binomials,Niultinomhtls,Evolution.Fractions,Simple Equa- tions, Equations with TwoUnknowns,QuadraticEquations,AdfectedEquations. Thesurvival ofearlier symbolsand formsfrilly be noted inthis work.The signsforinequalityarethe Oughtredsymbcds foundin theI lorvardmanuscripts,but intheopposite Order,and hencethey resemblethepresent-d%y symbols.% Stillanother undated-bookcontainsno name,but isunmistakably inthe handwritingof RobertBrooke.Itconsists of36pagesand dealswith the"ApplicationofAlgebratotheInventionof Theorems."Theseapplicationsaremadeto Arithmetic* Progression.CrOsmetrieProgression,Annuities,CompoundInterest. Extractionof SquareHoot, Mensurationof Supertiehbs,Mensurationof Solids, andSphericTrigonometry. tUnivenityof PenteRplrattia. Biographical Cataloourof theMatriculatesof theCol- lege174.9-1893,XIII, Philadelphia,1894. 'Thisnotebook andall the othersby RobertBrookeherein in the describedare inonepackage manuscript department,HistoricalSociety ofPennsylvania,hereafter LiMat.k3uc. Pa. referredto 25 26 ALGEBRA INTHEEIGHTEENTHCENTURY Othernotebooks bythesame man aredevotO to:"Applications of PlainTrigonometry toAstronomy," "SphericGeometry,""Appli- cations ofAstronomy,"."Dialing," and"Fluxions." There is sufficientresemblancebetweenthenotebookonalgebra by Robertgrooke andoneby AndrewPorter, jr.. towarranttheindusion of the Porterbook at thispoint.The latter was"AndrewPorter jun.r.'sBOok1792." 3Thirty-ninepages,19cm.by 31cm.,show.a fair knowledgeof elementaryalgebra andconstitutean abbreviated edition of thealgebra workby Brooke.The orderof topics agrees with thissamebookasfaras" QuadraticEquations."Only the beading for thattopicappears,and t1work stopsthere. Siymbols and formsand, insome cases,exercisesarethesame. " PraeticillMathematies."Briefmentionwillnowbe madeof such othernotebookscontAingalgebraicmaterialashave been found duringacarefulsearch in manylibraries oftlw East.The these books extantdoes not comparewith thenunkrcon- number, of tainingarithmetic, and yet abit. ofalgebraicmaterialcrops upin the mostunlooked forplaces.A setof fivevolumesbeautifully printed byhand, by oneThomasSullivan, in1796, isentitled " PracticalMathematics." 4Virtuallyeverytopic ofinterest in appliedmathematics atthat timeis includedin thesevolumes,as wellassmile purelytheoreticalwork. Acollection ofalgebraic problems insuchaseriescould not provetobe other than an un- expected treasure.Algebra isused herefor itsapplications under thehehding:" TheApplicationof Algebra tothe Solutionof Problems."Thee 24problemsaredrawnfrom geometry,butare kitofareallyuseful nature. King,' Inagoodpiece ofworkontrigonometrybyoneJoseph whichcontains thedate1740,although hanother handthan that of King,familiaritywith theuseof lettersin algebraicApressions is shown.Theproportion .e:y::y:z appears,followed by "Then alz=y2and61-z=y."Tenprinciples oftrigonometryaregiven, and the tenth onereads: "By Algebra orAnalyticalInvestigation." Somewhat moreextended isthe algebrain " WilliamWinthrop's Book, beganJune 6,1769," and,carriedoverinto theyear1770. Thisbook has 20problems aqdaOw statements,suchas"jThe Sum andDifferenceofanytwoNumbersmultiply'dto-getherproduces the Differenceof theirsquares."Later in thebook thereareseveral pages Onalgebrairi Latin. Some ofthesenotebooksgiveus-siae lightsontheattitudes ofthe collegeboyswhowerecompelled topreparethem. One 7suchbook a Riot. Soc.Pa. 4 ManuscriptDepartmenj,Library ofCongreRs. I Hiat. Soc.Pa. "HarvardUniversityLibrary. I AmericanAntiquarianSociety. . .itte orm==.as...r--_ . - !IPPIPPP-P MATERIAL FROM MISCELLANEOUS SOURCES 27 whichstarts: " Bought Jan. 3, 1786, Doctor Webber Philomath..8 Samuel Haven's Book, Cambridge Univprsity," ends withatelling -Commentby Havenonthe well-known quotation from Pope,as follows: Alittle learning ksadangerous thing Drink deep,or taste not thepyerian Spring For shallow draughts intoxicate the brain But drinking largely sobersus again. N. It Ifindrather he intoxiated than drink deeply of marhematicallearning. The AUTHOR. The author hadnotdrunk deeply of algebra, because-only17 out of 167pagesreferto that subject ;and only definitions, axioms, and the foul fundamental operationsarecovered. A.later student.At Harvard 461794 lefteven abriefer record 9 of his work in algebra.This studentwasJoseph AlcKean, whowas afterwards professor of rhetoric andoratory at, thissamecollege. Xetthil1i;e1 Bowiltell.Anotebook " byN.athanielBowditch 11 should be mentionedonaccount, of its intrinsic interest rather than because of its bearingonthe mathematical education of his day. When Bowditchwasabout14yearsofagehe heardavague account ofamethod of workingout problems bylettersinstead -of figures: he succeeded in borrowing t1ì0book, andwas-sò mu6in- terested and excited by this, his first glanceatalgebra, that he could, not get the least sleep durintr the whole of thenext night." Bow- ditch transcribedaflimmense number of mathematicalpapersfrom the printed TrIftwletioos of theRoyalSocietyofLodonandmany other scientific worksein orderthat he might have this materialin his personal possession, buthis bookonalgebra givesnoevidence of beingsotranscribed.

. This notebook has the title:" XAT1Ie1xBOWDITCH his Book Aug.t 23d1788.Ile beganto learn algebraonthe 1st of August, 1787."Thecontentsa're theusualonesfrom " Additionof Al- gebra" through "Solution of highadfected Equations.% Thepe- culiar character of this notebookof 114pagesis that through 80 papsof it the catechism method is used.And thisseemsto indi- cate that Bowditch worked outatextbook of hisown, . using the sub- ,. jectmatter ofother warks.Indeed,'lle refersto "Mr. Ward" in

ftSnmuel Webber.Tutor of mathematics at Harvard from17R6-1789. Later professor of mathematics and president of the university.I " Bus tonPublic Library. '" BostonPublicLibrary. "NathanielBowdltch(1773-1838).Best knownasthe translator ofLaplace's. 'Monique relleteand by inother work thatWaR largely his COWILThe NewAmerican PracticalNavigator., 1802. t . -7'4-1Memoir of the translator of the Mécanique Celeste by hisSon, Nathaniel Ingersoll Boittlftch,in Vol. IV, Boston, 1839.. r, 953370-24-3

a 28 ALGEBRA INTrfie,EIGHTEENTHCENTURY severalplaces, to "Cocker " and to" JamesdeBilly's al gebra." Elsewhere in hismathematicalmaterial herefers also to"Dilworth," andDilworth didemploy thismethod. anearlier methodthatoneillustra- ;This is suchalate survivalof . tion of it iscited.- Dialogue 1. BetweenPhilomathes &Tyrannaculusconcerningaddition,Substruction,Multi- plication & Divisionof algebra. > (sic) P.as youunderstandVulgar fractions& knowthe Signs& characters you saythe veryfirst thingthat I shew youwill beaddition. T. howis additionin algebraperformed. 1. the same ascommonadditionprovided thesigns beboth affirmative orboth negative as youwill soonfind by thefour following cases. T. Iunderstand allbut thefifth forI cannot atpresentconceive that+42 added to 42 canbe equal tonothing.1 should'think rather thatsubstract- ing themthey wouldbe equal tonothing. P. that is yourmistake fortheirdifferencets 84 . . because the Negative signmakesvoid theaffirmative. T. I ask pardonbut I di) notrightlyapprehend it. P. Ithink you are alittleDull now.Do you notremember that Itold you times less than .that thisSign( )signifiesawant orDeficiency so many nothing asthe figuresafter it express. T. YesI do. P. Observethen supposeyoustoodIndebtedioa person£42 and lmd noeffeets than of any tiliorttq paythe debtthere it isplain youwould be 42times worse your own.Now ifa nothingthat is have42 timesless than arealproperty of friend shouldgive-you £42 to payof the debtand -you do sostill it is plain consequently youwouldhfive Nothingin hand tobegin theworld again with then+42 addedto42=* orNothing. T. I am verythankfulPhilomathes for soplainademonstration. devoted toproblems ofall d&generous'portion ofthenotebook is. possiblevarieties.Theonequoted isin rhyme,andsotheibook illus- tratesthe oldrhymingarithmeticsaswellasthe dialog&method. .14 Problem 114th.four VirtuousDamselswould be joinedfor life ComeBatchelors Come nowand chuse awife Phoebe is young.Stella hascharmbut hold Astrea hasVirtueolaiAurilea Gold Proportitn Geometril theirfortunesclaim Phoebe has leastAurilea, aRich Dame Their fortunes sumdoes in themargin Stand Take BeautyVirtue youth orhouse and land Divide eachfortune byjust twenty-four 9. And youltheir agesEasilyExplork q End of thenotebookcuetom.This customof keepingmanuscript mathematicalbooks does not seemtohave died ontuntil' wellinto& thenineteenth century." ConicSections.For theuseof students in'UnionCollege byWilliamAllen, A. M.Prof. Math. ét.NatAPhil. In, U. CollegeSchenectady stateof NewYork.Transcribedbyand for JohnS.Makon, Jan. 1st,1805,"contains 60 pages onadvanced PP' MATERIAL EZIM MISC.ELLANEOUSSOURCEg 29 algebra.""Ebenezer Gay's Property.Algeb--irir stu- (lent noteboV which extends throughcubic equations." A notebook whichrecor sonly the date when itwasfinished readsonits title page:"End of Algebra 4 February1813 Elisha Fuller Merrimack AD 1813."15Fuller's work is excellintand is carried through .the -.solution of cubic and higherdegree equations. By theyear 1814 seireral reprints of English algebras had.appeared, and in thatyear ana1gebra:15yanAmerican authorwas published:"Textbookswerein the hands of students,and the uni- versalcustom of student textbook making had ,practicallydiedout. Notebookswerekept,as astudent might keepabook to-day, for work supplementaryin. itscb. nature, or to meetthe whims ofa p- ticular professor.

7 '3Manuscriptcollectionf Mr. George A. Plimpton. "Ibid. "American Antiquarian Society. "Jeremiah Day, TheElementsof Algebra, New Haven, 1i14. -"14r-440.-

'101.1-

4 .

J. 40 - .; W:11., AI: 0

b. 4*

Chapter VI COMMENCEMENT THESES Nr-

Acommencement cutvtam.---Commencementtheses (!")nstitute otri.g- inalsourcematerial of quite adothernaturefrom that of student notebboks. A.thesis hascometomeangenerallyacompletN1essay ordissertation presented bya,candidate foradegire.) Another meaning, inlesis colimlonuse,attaches to it, however. This meanin( maksisathesistiobeastatementofaproposition which is to estafflished byargument, implying that objectionsmaybe raised and answered.It is with this latter significance that thecommence- menttheses' of Harvard. Yale., Rhode IslandCoAlege.and the College of New Jersey during the eighteenthcesntuFy willbe considered. Severalaccountsof these early commencements havecomedownto us.The earliestonerelatestothe.customalJlarvard in theseven- teethcentury,and,is found intworeferences from the_ .1(agnidia, by

Cotton Mather. It isasfollows: . .

When the Commencement arrived,....they that*were toproceed Bacjielors, held their t publickly inCambridge. ;whither. the Magistrates andMinisters, and otherGentlementhencatiiib,to putRespectupontheirExercises;And these Exercises were besidesanOration:usually made by the President, Orations both Salutatory amd Valedictory,made bysome orother of the Com- mencers,.. . But themain Exercfte*wereDisputationsuponQue8tion8, vherein theRespondentsfirst made their Theses.

11. At theCommencement,it hasbeen the Annualcustomfor the Batchelors to publishaSheet of Theses,proviriliDcfmulendae,uponallor mostof the Liberalt4t rt8;amongwhich they do, withiparticular character, distinguish tho:le thatare tobe the Subjects of thePublickDisputationsthenbeforethou; and"thoseThows theydedicateashandsomelyastheycan,to the Persons of Quality, but especially the Ciovernour of the Province,whose Patronnge the rolledge Wouldbe recommended unto!

t The hest collection of commencement theses of the variousuniversitiesare to liefound in theirlibiltrIesasfollows: Harvard, 1642-1810; Yale, 1718-1797;Brown, 1769-1811; Princeton, 1752. Onlyone commencement broadside could be located at Princeton.The library ))ossesses, however,amanuscript volume, compiled by John Rogers Williams, and containing all thenewspapernotices of commencements from1748 to 1865.The account for 1748 records that the candidafes entereduponpublic disputations in Latin(.A.theses which had been distributed in printed form.Similar Recountsfor many yearsshlwthat thesame custom prevailed nt this collewb As at the other colleges of theday. - 2 Magnalia (17(12) Wok IV, 128, 131.Quoted In,11enry W. Edes " Harvard Theses of 1663," Tranvietions of the Colonial Society cif Massachusetts.Vol. VI, 1897, 1898, Boston,1902,p.323. . ng Os ; lwp

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- 32 ALGEBRA IN THE EIGHTEENTH CENTURY

These theses more_thananyextant material show thecontent and character of undergraduatk studies duringthe earlyyearsin which 0'

s theywereprinted.It is admissible in this wòrk, however,to enlarge onlyuponthecoursein mathematics implied in them,at thesame °time laying uponthe earlyappearanceof .algebraic truths and ,thèir subsequent inclusion.inmost of thecomr;lencementpro- grams..

_ Yate,nbathernatkol theses in`1718.T1 .e.two broadsides which have been cliosenfor discussionaretheearlikonesfrom Yale and1-far- yard which-contain algebra.theses.The broadside for Yale of1718 is also the earliest-onefrom Yale extant.4The English translations of the Latin thesesonmathematicsareintroducedatthis p9int.

4 NRIEMATICAL THESES, YALE,1718 .1

1. Mathematics isa.set of rules for computing quantity. tita 2. Discrete quantity is the *Wed of arithmetic,continuous, however; -of geometry. 3. Unity isa partof numtier. 4. Ciphers to the left ofa-vhole number havenovalue, but they decrease the

4 value ofadecimal fraction. 5. Multiplication byadecimal fraction se§ the value ofany number; °division InCreasesit. 1 6. Algebra is the docttine in which by comparing know9quantities with unknown, difficultquestions of' arithmetic andgeometryareeasily resolv41.* o. 7. The.fliiiiamèntali parts of algebra are numeration and equation. 8. Algebraic number gives neither greatestnorleast. 9. Subtractinganegative fromanaffirmativeincreases quantity. 10. The product of two negative quantities producesanaffirmative. - U. When quantities in both the dividend and divisorEpvthësame,thequo- qient is unity. 12. An algebrarc fraction is multiplied by takingoutafactor of the denomi- . r. ptnator. 13. What is involved by Involution is resolved by evolution. 14. InaprOportion,the prpductofthe extremes is equal to the product of the mpans. 15. Ah equation is solved by transferting -all knownquantities tooneside of t . the equatim. .41it 10. Primary logarithmsarefprmed by the repeated extraction ofthe square root. a 17. Secondarylogaritansareproduced by the addition and gubtractIon of primary logarithms. 4- 18. All reOilinear triangles contain two right angles.

.. . 114' 19. Given' thé basesandaltitude, the angle .at thebas.ecan notbeioundby theuseofaline otesines. 20. Sines of anglpsareproportional to the opposite sides. 4 21. Trigonometric problemscan :be..aolved meet accurately by the use of logarithms.

+ Franklin B. Dexter.Piographical. Sketches of the Graduates of YaleCollegewith Annals of the College History, October, 1101May, 17451p.179, New York- MO.

. :4Q.tute4.1."-' ' = / COMMENCEMENT tHEsEs 33 22. A circlecanbe'measured in thesame manneras a right-angled triangle. 413.Thearea of a sectorcanbe foundwithout knowingthearea of the whole circle. 24. Surface is widthand lengthwithout depth. . e 25. The surface ofa sphere is four tin*stheareaof its largestcircle. 26. The declinationofa star is Its distancefromthe:tequator,the latitudeits distance fromthe ecliptic. 27. The rightascension ofa star is its meriliandistAnee from file beginning of. the Ram[vernal equinox] numberedby degreeson thegrquator, the longitudeon the ecliptic. 28. The amount oftime frommidday to middayis not alwaysthesame. 211. The angle aethebase ofa horizontal sundial mustagree with the eleva- tion'of the pole. . 30. The solar polemust be elevatedin agreementwiththecomplementof the latitude of thelocality. Letus nowexamine insome&tailthese mathematical.theses. They includetwo general [1 2],four a.rithmetic[3, 4,5, lll, eight Algebraic [6, 7, 8,9, 10, 12,, 13,15], seveiigeometric[14, 18, 19,22, 23, 24, 25],twologarithmic[16, 17],two trigonometric[20, 21], andfive nstronomical[26, 27, 28,29 30]statements.Here, then,wehave intheyear1718 algebraand trigonometryin thecollegecourseat Yale. along witharithmeticplane andsolidgeometry, andastron- omy,the last threeof whichhad been loolied.uponfor centuriesas es'sentialparts ofaliberal education. The theses'showawidedivergence indegree ofdifficulty.The fourth thesis,with itsstatement concerningthe placingof ciphers to the left ofsignificant figuresin wholenumbers anddecimals, is firobablyknowntoeveryfirst-yeargrammar-schoolchildtoLday, while thetwenty-fifth,which givestheareaof thesurface ofa .sphere, isknown onlyat the end ofahigh-schoolcourse or during thefirstyearin college. Thestatements in geomefryshow thatdemonstrationswerein- cluded in thatbranch ofmathematics,andnot memlygeometric ainstructions withnoaccompanyingproofssuchasformapart of manyof the studentnotebooks of dieighteenthcentury. The defense of thethesis that" allrectilineartrianglescontaintwo right angles"involvesageometricdemonstration,asdomost of the other geometricstatements in thisset.Logarithmsmighthave beentaught mechanically, butthe law ofsihesmust haye beendeveloped incon- nection witha courseintrigonometry,evenif itwas alimitedone. Declination,rightascension,latitude,longitude,andgnomon areall terms that implyfamiliaritywithmathematicalandnot merely I. descriptiveastronomy. AlgebraOwes in 17111.Thealgebrathesesonthis 1718program possesspeculiar interest.,andemphasemustbelaiaontheirpresen* among commencementthesesasbarlyasthatyear.This is,asfar asresearchsofat. conductedshows, thefirst directevidenceof the 34 ALGEBRA IN THE EIGHTEENTH CENTURY teaching of algebra in the American Colonies.By 1718aclassof boys had studied this subjectone ortwoorthre(*)years,and these studentswere nowreadytoargueforithe truth of -certainalgebraic relations. On this particularprogramthe number of algebraictruths is greaterthan that .ofanyother mathematical subject..Algebraap- pearsin the sixth statement in all itspowerwhen it takes theform ofamagic wand wavedoverabstrhie questionsof arithmeticand geometry.A modern dictum, that the equation is the.central topic of algebra,occursin the seventh thesis.In number Oght is given oneof the distinguishing characteristics.ofthe subje'ct, the extension of the number system to include positive and negative quantities without limit. The other statements leadtothe conclusion that only the initial steps had been taken at this early date.Butanybegin- ningatallwas ahopeful sign thgt the tutor of mathematicsatYale had thecouragetobreakawayfrom the accepted curriculum andto introduce thesubject which had developedsorapidly in the preceding century. 0Samuel Johmon: and AI gehra.At.this time Samuel Johnson (1696-1772), whowaslater calledtobe the first,president.of King's College(laierColumbia College and at.presentColumbia Univer- sity),wasthe sole tutor at Yale College:* He had been sole tutor for twoyearssince his appointment. in the fall of 1716, and hisname appears amongthe graduates of 1714as b Sannwl Johnson, Mr. Tutor,"onthe Yaleprogramof 1718. Certain notebooks and printed books 6 contribptetoourknowledgeoithe lines of work that hewas followingas'tutor. Onenotebook in Joh4nson's handwriting carries the words: " F Libris Same' Johnsoni Anno Dom. 1717."It contains the following scheme ofmathematicaNubjects,which he doubtlessgaveto his pupils: 1111 Arithmatie Is thp Art of Numbering both of Integers & Fractions.Append- ageshereuntoare1. Decimal Fractions.2. Logarithms.3. The Extraction of Roots.4. Algebra. Geometry is the Art of measuring heretobelong the Treatises 1. ofTrigonom- etry Plain &Spherical2. Geodesia of Surfaces.3. Stereometry ofSolids.

In another place in thissamenotebook, thereoccursthis list: " Mathematicks, Arithmatic, Geometry,Algebra, Trigonometry, of Numbering & Measuring." Sveralverydetailed outlines of the parts of mathematicsan writtenonthe first blankpagesofa copyof Sturm's 4tathe8i.B Jure- nili8 7 which hasonits lastnage,"Libris Samuelis Johnsoni. C91. f mellow I F. B. Dexter, Biographical. Sketehea, p: 123. Samuel Johnt4on Collection, Columbia Uniyeralty library. 7. Mathesis Juvenilia:Made English fromthe Latin of Jo. Christopher Sturmlus. By 44 (Wry Vaux, M. D., London, 1700, COMMENCEMENTTHESEg 85 legii YalensisNov. Port.Anno Dom. 11/18."rfhis workmust have yieldedmaterial forJohnson'sclassroomuse.It is quitecertain that .hispupils hadnotextbooks, butit isto Iw hoped thatno oneof them wrote concerninghim whathewrote while hewas astudentat Yale. This frank confession isfound inanotebook dated1714: . Ho when Iwas at Colledg Iwas taught nothingbut to bea 4bonceited Cox- combiik-ethose thattaughtme.Indeedwe had no Books &our4.inorance madeus thinkwe needednone So that.we dug everythingaltpost 6ut ifour onBrainsas a certain Gent ofthose times,used tosay was his way. In September.1718, DanielBrown.aclassmate andclose friend of Johnson,WaS chosen juniortutor at Yale, andtogether they carried onthe workofinstructionuntil thefollowingyear.On the first blankPageofacolVTof Euclid'sElements8 is found thefollowing interestingindication oftheownership ofthe book:

DanielBrown's BookDeer 25 1717'el. . Saìnuel Johnson''sBook 1718bought of Mr.Brown Most ofthepropositionsin BookII ofthe"Elements havemar- ginalnote'. in Brown'shandwriting,givingacompletealgebraic% treatment of them. Atthe timethat Brown ownedthis book,hewas rector of the HopkinsGrammarSchool, of 'NewHaven. Nothingcanbe saiddefinitelyabout thebearing of thisalgebraic materialonthecourseof studyin eitherthegrammar schoolor Yale,.BothJohnson andBrownwereexcellentscholars andstu- dents.Theywerealso successfulteachers, andenthusiasticstudy led, in Johnson'scase, toareaction ienhisclassroomwhich tookat. leastoneform,that of thefirstpresentationof algebraat Yale College. Mathematicalthesesat Harrard.Theearliest oftheextant theses of II:iitardCollege whichincludeanymatkematicswereprinted in 1653a.ndcover "Arithmeticae"and "Gilometricae."On theset of theses for1708,we.findoneswhich read:"Arithrneticaet. Geometria tantumsunt artespureAlathematicae"(Arithmeticandgeometry are pure,mathematicalarts), and"Astronomiaest Scientia Mathe- miticamixta"(Astronomy isacombinationof scienceand mathematics).Conicsectionsenter in 1711 withthestatement: " Nullus ExcentriciertisGradusCirculurilproducit,Indefinitogrado Parabola producitur" (Nodegree of ecceritrIc.i.typroducesa,circle, buta parabolais producedbyanindefinitedegree oftccentricity).In 1719twostatements showthat. fluxionshadOftena°foothold;one isto the effect. that" Fluxioest AugmentAktionisvelDiminutionis IEue/ide's Elements,...By IsaacBarrow, D.D.London, 1705.Yohnson's library contains also" The Elementsof Euclid... Writtenin French by...de Chalet'. Nowmade English...Orford, 1704."It hason the, frontpage " Thomas Prince 1707 Booksomitted viz. 7, 8, 9, 10, 13& Supra.Books containedviz:1..2, 3, 4, IF), 0,11, 12." Thep followsa list of omittedpropositions inthesame handWriting.Prince from Harvardin 1707. graduated Was tills thetextbookin geometryat Harvard Inthe "sameyowl

. e 36 ALGEBRA IN THE EIGHTEENTHCENTURt

ro..7 'N..' ., e,, V_.'.1 Theft:- M A'rHEM All C.Z. i 1 " hI'l MATtCA,(bantitascs Matcr:s,Spud.ct Motionis'rival:jot, Pend Matheruaticz funt duz, Puract mists. 1tt tidbit. :1E: )-`MaThematica Para, Quntjtatem,ablateria,Spats,« Maim* w, '4r...;u.;...ablicattain confident. ..''''" ..... sitiAtiobis Modis conficieranda quoadMousse,quoad.N4tislisil -no-&Mims ,Afirlymetiric, alterum Gemetrkss,ctActual=Akekra objeaumeft. 1 .r.-11 elicitNova.1,Qtantitaturn cujUiTisGesstris, iualia cjuideildQun ...... , :--% .-' .c. matc,conientos, cxponit. NI Arltimet lce InNumnis propettionalibuti, furnm4 . numccorum extremortuu -, : ;.,ts. legman imams' (Worm inecliorum.ieg,-literab Eurcmisdifiantium. 7Nat:InnUtet('ktproportIonaltbus,RcaangutumEat:moronssqualls !.: -,' quadrat() Tcratinorum medlorum. . . g'Lolefibmi Clot NutrirroltlITI artificiatiumferia, In PropottiontArithmetic*, ,..:.-,,....,. ...-ficutlnumcri rtfpondentes in GeonactricapioRecdientes. pAdditio tt Subtratlio in Arithmetic& Multi ',.,. n...... ,..,,. . t e Iti e r 11 o 11,10 trimmtliaTotuts,relaliquarn Pattern date indicat,etiNtesminat. IIParahclogramtna fupec *qualibus Bafibus,ct in ilsdcasiPaullclisvcocaksca irlice Cc cqualiafunt. IlLsaAlubtst eft Ars ratiodnandi, jaissfuttalkiisnotiseut arum Habit, Quantitatibusnotis definiat. .;3gfrilboutkaaQtautltatibus datis,ad quettas,4gebrsmitt=avials, ad dams progreditur. )4 MultipticatioQuantltatumconerctaruns, tRAumadoaliqua Qsuntltuinvolvis . talcm Rationtrn Multiplicando,qualm Multiplicator,unstari. is In Dairsoncpu)tictucandemRatIoncmDividcodo, eDsviCosunitatiknot*. 16 Moments sheujutQuantitarisonctate ¡quails fuat, securia omniumswam- tiumcum Indicibus tuarum Poteitatum,ccCocifscicuti ombuscinuomolts §u.si,!Asthmatics illpfna tittanninspatt,sd Manisa),tel Spadues, motioqcm, Applicatio. nl se Proicab (Oilsin piano,eft omniumCireuloeugsfemme five:alloqus CireullplanumDelinestioCkocnetslcs. .; :9dIrmariaetiScirotia CommasCceleflium,quoadtosuat hissnkudiaan, . Diesniias, Edipict,Porticoes,ct Mot o&lilies Lynam cfficiunturpci AtmofpbcramCorpus's. ablituantis. Longitudo Locasum, &gullsAftronornicis,faciledetcuulnatipotett. 23panes Naps(vaunt runt titsmeta diftinete.

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Theses from the commencement program of .Harvard College In 1721. Six of the thesesrelateto algebra V. COMMENCEMENTTHESES 37 quantitatumfluentiumVelocitas"(A. fluxionis thevelocityofan ificreasingor(liminishingflowingquantity),andthe otherthat " FluxioexquantitateUtienteInvenitur "(Afluxion isfound froma flowingquantity).The thesesof thissame year includeone onthe fourth subject ofthequadrivium,denoting thatmusicwasstill being taughtas abranchof mathematics.Thepropositionto be argued is: "Diaset Trias harmonicastint fundamenta contrapuncti.musici" (Harmonic diadsand triadsarethe foundationof musicalcounter- point.) The firstset ofextant Harvard thesesto contain algebra isthat of 1721.The thesesin thisbroadsideareclassified andarranged in thesameorderas/those ofYale in1718.Most ofthe mathe- n.iaticalthesesaregeneral intheirnature, with fewdetailed facts from thevarious subjects.Thesecoveraboutthe sanierange as thoseonthe Yalepaperalreadydiscussed.Theyarepresented in two divisions, "AlatthematicaPura," and" MathematicaMiNta." Under theformer therearetwo generalstatements and threesub- divisions,viz, "Arithmetic:I,""Geometria,"and"Algebra";under the latterarefound"Astronomia"and "Musica." Algehra. Moto?at Harvardin 1721.T1eEnglishtranslationsof thestatements relatingto algebraare asfollows: 44, Anyquantity hasto be consideredfrom twoviewpoints,viz. (hitheras a numberor as a magnitude;theoneis theoliject ofarithmetic,the other of geometry,and bothof algebra. 12. Algebrais theart of reasoning withunknown quaiititiesin orderto define their relationsto knownquantities. 13. Arithmeticproceeds fromgiven to requiredquantities;algebra,however, from quantitiessought to thosegiven. 9 14.Multiplication ofconcrete [" concretarum"means here " thatwhichIs grown together"1quantities is whensome quantity involvesthe multi- plicand insucha relationasthe multiplier[involves] unity. 15. Indivision thedIvIdendInvoLves thequotient in thesame relationas the divisor [involves]unity. 10. Themoments ofany generated quantityare equal to themoments of all thegenerators with theexponents of theirpowers and their coefficients in continuedmultiplication. These thesesgivenoibdication oftheaiwunt of subject.matter that wasbeingundertaken inalgebra.The &blusionof number16 showsthat algebraand fluxionswerecloselyassociated.Agreater degreeof maturityin the studentor.in hisfamiliaritywith thesub- jectwasneededto handle these truthsthan wouldbe neededtoprove, forinstance, thatthe productoftwo negative quantitiesisanaffirma-. fivequantity.Apowerofgeneralizationonthe basis ofalgebraic factswasrequiredto defend suchthesesas arehere included. Attentionhas alreadybeen calledto.onegraduate ofthis class of 17211"IsaacusGreenwood,"ashisname appearsontheprogram. 38 ALGEBRA IN THE EIGHTEENTH CENTURY

t Whatever his later study, Greenwood musthave gottenastarton algebra in his collegecoursethat awakenedalove for thesubject: This interest be passedontohis students when hebecame thewo- 'fissor of mathematicsatHarvard College; The recurience of certain thesesonthecommencement programof thilsamecollege and of the different colleges hasanelement of inter- esi.A favoriteonewhichoccurs,with slight variations in wording, atale in' 1720. Harvard ip 1731, Brown in 1773, andHarvard again in 1780, is the following: " Duo numeri bigumlratisummatim sumpti numerumquadratum constituerenonpossunt."(Two biquadratic numbers taken at randomcannot(onstitutea square,Yale, 17'20, 4).9 .Thee ctle'et'A iofpresenting thesesatcommencementtime continued in force Ill the principal colleges until well into thenineteenth century.A careful examination of the thesesof Harvard, Yale, Rhode Island, College, andth.e College of New eTei*y shows that algebraztppears amongthematfrequent intervds until the custom passedaway,Its omission is apparently due toagreateremphasis Illong another line in thatyear. At. Yale, from 1739 for seveKalyears,conic sections and higher plane curves-were stressed with such thesesas:" Cycloidisareatri- plex est ejuscirculi generantis" (theareaofacycloid is three times that of the generating circle. Yale, 1739, (s),and " Parallelogramma omnia, circa (hltae ellipseos vel hyperbolae diametros ylaeviscon- jugacas descripta,suntinterseaequalia."(All )arallelograms de- scribedonconjugate dianleters of ellipsesorhyperbolasareequal,

-Vale, 1740, 17.) . Begitining atanearlier date, 1735, Ibirvard pandlels thisXmale record. From thatyearthro,ugh 1750,aprominent place is given to statementsconcerning conic sections, cycloids, semicycloids, cissoids, spinds, and other higher Planecurves.One instance of this is the thesis: " Conchoides, Cissoides et Curva Logarithmica habentunum Asymptoton."(Conchoids, cissoids, and logarithmiccurveshave oneasymptote,Harvard, 17.1G, 24.) 14711.vions.Then 1751 showsaradical change anAa veryinterest- ing.one.Fluxionsoccupy aleading placeamongthe mathematical Commencement thesesto theexclusion of practically all the mathematical topics that had hitherto appeared, and this subject did not recede front its prominent position atHarvard during the eighteepth century.The fact. that all problems in fluxions requireafamiliarity with the handling Of algebraic expressions and opergtionsshould notbe lost sight of atanytine.

ViNINEMlb 11 This Is one of Fermat's theorems.See Leonard Eugene Dickson, Eistory ofthe Theory of Numbers, Vol. 14 p. 616, Washington, '1920. COMMENCEMENT THESES 89 At Yale, fluxionsappeared in 1758, andduring 25yearsthereafter onlyafewsets of these lack problems in this subject.Somestu-

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...4------1,V --- --._ f 40 ALGEBRA IN TILL!: EIGHTEENTH CENTURY is superiortoalit lunkic,sofar is the duct vine of fluxions superiorto algebra, Yale,1n_ib2. 1(3),athesis which require,daknowledge of all three branches ofmat1wmatics. This discussionmaywell conclude withapronouncement which the changes.inpsychological fashions have inno wayklisturbed.It isasfollows: " nathesis studiumdisciplinam mentis optimumprae- bet."(Trie'st tidy of mathematicsprovides the best mental discipline, Harvard, 1606, 30.)

ref

4 Chapter VU

MATHEMATICALTHESES OF HARVARDCOLLEGE ,

In quiteadifferenlcategory from 'thecommencement theses dis- cussed inanearlier chapter aréthetrot6msPresentedto the department of mathematicsat Harvard Collegeby themembers ofthe junior andsenior classes.Thereare406sets of tibese problemsex- tant,' the earliestonedated 1782and thelatest1839.2Manyof them show thesignatures ofmenwell known inlateryearsin various walks oflife.Some provideinformation ofconditionsat Harvard whichnolonger exist.All givetoanunusual degreeapicture of the work inmathematics doneby thestudents whosenamesthey bear. Eachset ofprobler»hsis workedon oneside ofasheet of heavy linenpaperNi-ith body enoughto make it resemblealight-weight. cardboard.The sheetsareof slightlyvarying sizes,18 by 14 inches, 19.5 by 15 inches,23.5 by 1R inches, beingthemeasurements ofnfew chosenat. random. The work is doneby hand inink, andat times the workmanshipis exquisite.Afanof thetrigonometry problems areaccompanied by beautifulillustrations incolor. Thereare79sets of Problems from 1782to 1800.The ihajority of these setearedevotedto calculations of eclipsesorto plans of buildings.A. grn(lllate of1796 took for hismathematicalthesis: "A North EastView of the Houseof Samuel Webber, A. A.S.,3 and of fhe Court House in Cambridgeby'anactual Survey."Of the 79 tshowever, 13aredesignated " AlgebraicProblems"or" Al- braical Solutions ofProblems"or"pplibation of Algebratothe GeometricalSolution of Problems," while3aresimply " Alathemati- cal Problems."Algebra problems continuedtooccupy aprominent placeamongthose inpuremathmatics formany years. Algebra the8es.The heading "AlgebraicProblems"appearsfor the first timein 1786, and in thisset of exercisesarefound four problems,asimpleoneingeometry, two in surveying, andonein trigonometry.In the follow¡ngyearfour students presented mathe- -matical theses,three of whichareentitled "Mathematical Problems Harvard TTniversity Library.

2 A completklist of the headingsfoundonthese sets *of problems isgiven in HenryV1/4 Badger, " Matqlnatical Thesesof the Junior and Senior Classes, 17S2-1S39,"in Bio- graphical Contributions;No. 32, Library of HarvardUniversity, Cambridge, 1888. °Samuel Webberwasthe piofessor of mathematicsat the time. 41 * -4.----- 42 .. ./4- O of four,exercisesisprobablyrepresentativehis and theirSolutions,"thefourthisagain"Algebraic mathematical thesis,gralluatedinthat type ofwork. This last matics atHarvard. expected from eighteenth century. Here The of Pennsylvan* in connectionwiththenotebooksfromHarvard and tities, The 1 ' JohnWard. . -, --_,,, .;. , 4;1 A..S; ;44 ..t. . 1! . 74 th..), treatment ofthealgebraproblemisthat which, next problem,"Tofind qh, ._--...-- a usage 1 , 11. I. . 711%I. . . ____.--____- *. ns set of1787willbediscussed . .11 4.C :t .. \ 4%; 4 t N 4. \I '1" 't st. N. b :-S% 4 k Nits*. The YouuMaihrmatiqion's Guide,London, 1709 . VIP Olio ALGitBRA INTHEEIGHTEENTHCENTURY 42: Nk% ....._ I -4".1..' 11%.4 *a , occurring inWard's Yeung Mathematician'sGuide.4 IN; V et% 40' ...,_-, '4 ..4.1) t-1 k 1 % a 111.1 § t% Samuel Willard,whosesignatureis . N/ 4 4'14. .! , . 11,44 4 t *4 4 %.4 ,b1 good secondaryschoolpupilofthe ; sit .1% ZiN "N-. tr *%. . 16; 4. N-as . - 1/4t . -x; .13 v. "1 t 4 It isin ION zu47,..." Nst. N - . Ni : *. I I accorded problems P.-N 4:% 5. .t ttj. le ; 4 v . : °I' .1. a. ..4""%5 ""-=',..., 1 - * "r4 .....,, --ZN and .. '...., many z. r* qf :w. 111.1111iiii.... I. 44).' .14 tfie l . _.. , Nat \,44. 1,C. `4.! .. A . !.1 t tft. 1..'s , '%, .114 .4 I hi "NI 41. 4 I $ r \.t: It e are Cube rootof 4 , if4 N' t %1. ,..,zsi tr. V e...) ?) 41 4 tr 4 . respects theworkthatwould o *.:NO 1114 NI .45 * N ; i - -. gl 4:- 11.1. ;11;' ' 111 NN1 ; .4 ",:). % ... 1\. xlf, %. .- Ft. , J used fortheunknown 4 t , f 4 jr ot %. :, 1 as an -:, * k year, éN I I , - ... 4 I 14 ''fl°4 NZ , / ; ' ...),/* - ''' . ; e so O .. 1 %, ..11b` t III 1 5 e. , 1. .. ; #1,.. any . and illustration ofthis universally inthe \ ..., i ,, ., 1 , 1 3. N7', .. ; an course ,,,,,,.... t'.;, i . .,, % I , i :t # So I es.: N. %. '1. 1 4' ..?. V 4 i proposed 11,1 1 so ali%"17 the University ... I. as affixed tothis h 11- V V. Vi -7, V .0.14,:01% r ... 14 ft... . 4N I. - 47 " --...: a Y. present 4 V..- 4 . . 4.: _this titer editions. pointed out 1 11 %11 1 Problems." 4 AI 4's :?..,: q 4 , in mathe- a 14 .%';... , .1,A.t *77' ; N. t", ``2 -4- ki 1 group I quan- gum- . 'P. day. ONES %goo r -. OM, awl 410 fm a. 1441, NO ;El IMP o :60 *AP a be

mathematics department 1 -a MATHEMATICAL THESES OF HARVARD COLLEGE 43

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:-..4- 4.---1. w -..,,i,---Lt- 1.:,..-;-f:- , $.47 ,. ..--5-,-_24,.... .4is!4.4* 4 ,.':;,-,;-;'.: 2 - . 4..._,-4....- "Algebraical solvtions of problems" from the thesis presentedby Luther Richaidson of liarvard Collegein. to the mathematics department. 1799 9533i°24-7---4 44 ALGEBRA IN THEEIGHTEENTHCENTÚRY

tity," might have been copied fi:omthatsomework.5The third problem, emplqs six-place logarithms,but showsno useof interpo- lation.It combinesaknowledge ofgeonwtric factswithcertain trigonometric relations.The fourthproblem isthe real applied work and showstheuseof the law ofsines.In the originalthe diagram is cohwed withexcellenttaste. Evidence drawnfrom the conimelcementthesesgoes toshow that this work by SI;inuel Witlaradoesnotcoverall of themathematical instructionat Harvard in 1781, forthe theses ofthatyearinclude a'number ofstatements concerning fluxions.It11111/ represent the required work, while themoredifficultcourses were open only lit exceptional students- Othersets of algebraic probiemsshow such differenCesaswould arise from the needof the professorof mathematicstovaryhis subjectmatter and from the inclinationsof the individualpupils. Francis CaboteLowell,onOctober30, 1792, presentedtwo prot ns -wiiichshowalove for longmechanical operations.In.one the.1 of the unknownis given inanumber containing38 digits.In the other, the equation45,r4 1800,r3+,,q1564,r2-324000=324000is broughttoasuccessful solution byamethod ofapproximation, but th6re i§agreat deal of ,tedious work beforethat end is attained. The pemistence ofthe Oughtred symbolsfor ipey!ality isshown by th9ir usé ina paperof 1793. Work ofaninvolvednature andPVCII ofan.4.dvanced characteris thatonthepaperof LutherRichardson in 1799. Such direct, evidenceasthe foregoing mathematicaltheses is worth moreinall accountof the mathematics,in American colNgesduring the eighteenthcentury than all thestatements about that subject in

catalogueorcollege president'sreport.It. warrants the cpnchision. that algel3ra had obtainedanestablished place in ithe cuhiculum, which place it hasfilled from that timeiuntilthepresentda§,.,L6. tig 5Ibid,p.238 (1719). .

4 41, 11. #

.30

- "F. 4

1 a

i 4 Chapter VIII COLLEGERECORDSAND WRITINGS OF PROFESSORS AND OP PRESIDENTS o

lgebra inmanuscriptsandpriated:worles.Referencestoalgebra in the Manuscript. and printed worksof cpllege presidents andpro- fessors,taitwellasin regulatitmsand laws governingcoursesof study, constitute another link inthe chainofevidehce showing the recognition grantedtothe subject duringth(' eighteenthcentitry. Harvard requirements.Theearliest la*s for Harvard College weeprepared in1642, by President-Dunster and-includedtli'estudy of mathematics but withnomention of specific .branches.'These At-4'. laws governed tito Harvard curriculumwithno.material changes duringthe seventeenthcentury.In the earlypart of theeighteenth centuryaprofessoiiship in mathematicswasestablished by Thomas Hollis. The first definite requiremnt ofalgebraatHarvard is in- chidedin the 1)rin6plesOnwhiclt. the chairwasfounded.These principlesareset down th,ps: Rules and Orders relatingto a Professor of the Mathematics, of Natural& Expe'rimentalPhilosophy in Harvard College in Cambridge in New England, appointed by Mr. Thomas Hollis of London Merchant. 1. That thePr4.fessorbeaMasfer of Arts and well acquainted with the severalparts of the111athentatles& Experimental Philosophy. 2. that his Province he to lIztruct the Students titft. System of Natural Philos4phy &a(Pollnow of ExperimentaLin which to be emnprehended, Pneu- matkks, llydrostatickm. Mechankelis,Stait's;ks.Opticks, &e in the Elements of Geotuetry.to-gether with theadoetrine of Proportions the Principles of Algrebia (sle)Conic Sections, plain & (s16) Trimmmetrywiththe general principles of .Mensuration, Plain & Solidsin the Principles of Astronomy & (:eowetry, 'viz, the Doetrlike of the Sphervs theusepf the Globes, the motions of theHerivenlyBodles according to thediete'rent Hypothesesof Ptolomy (sic) Tycho Brahe & Copernicus with the general PrinciOesof Dialling tbe DivIsion of the World into itg various, Kingdoms with theuseof the Maps, 84.c.3 Tipseconoitruleis the onl: efiv which bears directlyondie..sub- .. , . . ject in hand. It shows that the principles bf algebiliwereaccepfed as a necessary partoftiiinstrifaranin mathematics as.earlyasthe dateonthispaper,1726.IsaacGreenwood, the firstmanwhowas

t Louis Fttnklln Snow.The College Curriculum in.the Dated State*, p. 2 [n;a'' place], 1907. -

liarvari, College Papers, Vol.I,.1050-176A,fn. 18,1726,.. Hartardet-Un)verelty I Library.Thereare13 rules. "lb ., 45 46 ALGEBRA INTHEEIGHTEENTHCENTURY

OW calledupon to liveup tothis fineset of rules,may..have hadsome- thingto do with'formulatingthem.Hewas one of fiveAllen to respondto,a call from Mr.Hollisto furnish pjahsfor 'hisprojected chair ofmathematics.3Thetwo Harvard notebooksOnalgebra alreadyset forthattest thesuccessof Greenwoodin fulfillingthe requirementto teach algebra. These rulesguided theprofessoe ofmathematicsat Harvard for many years.Not until1781arethereany more- specificdirePtions adopted for hiscontrol. Acommitteeappointed .torevise thecourse of instructionvotedanAugust16( 17874 withrespect to the sopho- mores 41 thtit at eleven o'clockonFerdilythey attctid theProfessor ofNiatheitatics,to be instructedin Algebra. anll'iobe carriedforwardto other branchesof the Mathematics ifthe timeallow.

And againónQctober16, 1788,5 thatfuture HollisprofessorsOf mathematicswere tocarrythe classesforward bypriAoate lectures, .. in Algebraas faras through affected quadraticequations andinfinite series. Algebra,then, receivesspecific mentionin all thelaws andregula- tions-formulatedat Harvard in theeighteenthcentury. lliighJonesat William and.11ary.TheCollege ofWilliam and Mary,"the secondoldestcollege inthe UnitedStzites,wasfounded In 1693at Williamsburg. Va.,and thecharter providedat theoutset foeapresident andsix profelisas.The publishedrecol* until recently1ntmed theRev. illighJonesasthe 114professor ofmathematics. nearliernamein theftculty of the c.ollegeis thatpf Ali% Le Fevre, shown bythe followingextracts from the.letters of GovernorSpotswood, ofVirginia:7 i7IRGINo,Ju1y28, 1711. To Mr.BLATHWAYT: SIR:I havenot had thehonor ofany fromyou sincemy last, but having seen a Letter thatyou writt to rollo.Diggs inbehalf of Mr.Le kevre, Ivery gladly embracedthe Opportunityof doing hon'rto your Recommendationby getting theGovernor ofthe Collepto receive himas a Matimuntick Pro- fessuL... VIRGINIA,3hty 8111,1712. To theB'POF LONDON : ...Igave your Lord'pan account of Mr. LeFevre's admissioninto the Collegeupon your Lord'p's recommendation.andamno\vto acquaintyou that aftera Tryal of three-quarters ofnyew:he appearedso negligent in all the s 3 Josiah Quincy, The 114fitoryof Narrated Unirersily,Vol. I.p. 399, Boston, 1S60. 4 L. F. Snow, College Curriculum,p.84, Quoted fromCol. Book S.p. 243. gIbid.,p.270 ff. °The researches of Dr. LyonGardinerTyier,for 31years the president of William and Mary, have been moÇ helpfulin the study ofthat college.Theyare published in The William and Mary Quarterly. ?The Official Lettersof Alezander Rpotwood,Lieutenant Ooegruorol ihr (;ohisigi of l'Irginia,1710-1722. Now First-Printed from theManuscript the Collectiopsof the Virginia Historical Society withán introduction and notes by7it. A. Brock . . . I, pp. 108, 150, Richmond, 1882. COLLEGE RECORDS 47 pot of duty and guilty ofsomeOther+very great irregularities, that the Gov- ernorsof the College couldnolonger bear with him,andwere obliged to removehim from his Office, .*. However unworthy the gentlemanwas.Mr. Le Fievrewasthe first professor of mathematics inanAmerican college. Another letter8 fromGovernOTSpotswood settles definitelythe fact that the Rev. Hughones wasestablishedat'William and Mary in 1717,some10yearsbefore Hollisluld createdaprofessorship of matheniaticsatHarvard. Thepartdthe letter referringto Jon7 is asfollows: .jUNE13, 1717. To the BISHOPOF°LONDON: .. .and I (10414 not y'r Lord'p is already informedthat'Air. Jones is ad- mitted into the College accordirigto y'r Lo'p's Recommendation;.. 'Toneswas an"Englishman of university trainingand keptvery closely in touch with his nalive land:'Hisconnection with Williem andlitryasprofessor Of mathematicswas abriejone,for be left the Colonies for England in 1722.On hisreturn to America he devoted himself exclusivelyto theministry.In 17.24 Jones published in Lonionamostinteresting history in whose making he had Participated, entitled The Pre.1/4ent State of Virginia.. This isthe work which holds interest for the student of the historyof mathe- -nmticsaswellasthe student of colonial histoty. Referringto the

Virginians, Jonessaysin this work: . 1 e Theyare moreinclinable to read Men by Business and Conversation,ktllan to dive into Books, andaril jor the mostpart only desirous of Itbarning what is. absolutelynecessary,in tho shortest and best Method. Having this knowledge of their Capacities nd Inclinations from sufficient ExperienceIhave compose4onPurposesomeshort Treatises adapted- with*my best Judgment toarours of Education for the Gentlemen of the Plantations; conesistinginashort English 'Grammar:anAccidence toChristianity;an Accidence to the Mathcmaticks, especially to Arithmetick in all- its Parts and Applications, Algebra, Geometry, Surveying of Land, andNavigation. Thesearethe most useful Branches of Learning for them, andsuchasthey willingly and readil? master, if taught inaplain and short Method, truly air plichble to their Genius; which Ihave endeavoured to do, for theuseoelhem, and a1t91hers of their Temper and Parts." It will be noted that algebra istobe.foundamongthese treatises which Jonesstates that he composed.Since he holds "from sufficient. Experience" that the people of Virginiaarecapable of mastering this and other branches ofmathematics. " if taught ina

plain and- shott Metho,d." hemusthave reached thatopinion by II actual experience in teachingthe subjects epumeratedtothem. And

-a 8Spotsood's Letters, loc. eit..II,p.253. "Many letters of hisareIIIiì irr British Museum catalogues of manuscripts. 10Hugh Janes, The Prceent lateof1701tinia,p. 44,London, 1724. aye 48 ALGEBRA TN THEETOIITEENTIECENTURY

soit. isfairlyobvious that algebrawasincluded in the curriculum of the Çollege of William and Mary before theyear1722. The treat iseongtanaar " jextant, anda copyof it is in the British Museum.The "Accidenceto theMathematicks" does not-appear to have. been preserved in likemanner,but the British Museum does k. possess amanuscriptonmatkeniatics by Hugh Jones." entitled: "Reasohs andusesOfthe Georgian Calendar and of Octave Com- putationorNatural Arithmetic."Thecrateof h is evidently 1752, as" thisyear1752"occursin-the.text.The firstpart of this work is concerned withacalendar co1fl1)os0 of 13seasonsof lunations'of 28 days.Oneday for the Nativity andfta bissextileeveryfourthyear 1- for public!:prayer.The seconsi parts starts withadissertationOnthe disadvantages ofnumbering'bv 10 andProposessubstituting8as aradix. Mgivesacomplete 'numeration table. and lays down rules for the reduction of decadesto octavesand 'othermatters.It also pointsoutmethods for dividing weights andmeasures onthis basis, somakingauniversal standard."' Thoma8 Clap at Yale.Weturn'nktto the thirdcollege estab- lished in the Colonies.:. The keen interest of the president ofacollege inanysubject when that collegewaslittlemorethan,a c611egiate schoolwascertain to_baveastronginfluenceonthe place.which that subject,'6ccupiedin the curriculum. Thomas Clap brouight suchan interest in mathemat ics to Yale when he becamerectororpresident in 1-739.Fouryearslater, he publisheda.catalogue of the college library" which he had.prepared himself.In the preface to this catalogue, herecommendsaplan of studies for the Collegecourse, as follows: In the Firstyear tostudy principally the Tongues. Arithmetic. and Algebra: theSecond, Logic. Ithtoric, and Geometry; the Third. Mathematics and Natural Philosophy;and the Fourth, Ethics and Divinity. Mathematics is fowid in threeyearsof thisvitan,with algebra in the firstyear. ThatC14carriedoutItstillmoreambitious plan of mathematical instructiop than hee-had advocated is shosi;n in the history of Yale which 116 wrQte in 1766. Referring to the undergraduate students, hesays: Theyaredivide() into four classes: according to the respectiveyearsin which theyareadmittédAt their admission theyareable well to svonstrue andparse Tully's.Or3tic48.Virgil and the Greek Testament;and understand the Rules ofcombmonArithmetick.In the firstyearthey learn Hehrew, andprincipalli- 111' - u Jones (Hugh); A. M., Minister of James Town,Virginia.An A evidence to time gng- A.bah Tongue, etc.,London, 1724. Given in British Museum catalogue. 12 British Museum Additional Manuscript21,F93. Vs Account, of which the above is arésumé, furnished by B. F. Stevens & Brown,bón-- ** don, through the courtesy ofColumbiaUniverskylibrary. " [Thomas ClaRa] A Catalogue of the Library of Yale-College in New-Harm, N. London, 1748. . -

COLLEGE RECORDS 49 , pursuethe Study of the Languages,and makea.-Beginning in Logiek, and some Parts of theMathematicks.In the second year, they study theLanguages; butprincipally recite Logick,Rhetorick, Oratory, Geographyano:lnatural Philosophy.Andsomeof them make good Proficiencyin Trigonometry and Algebra.In the third year, they still pursuethe Study of NaturalPhilosophy, and mostBranches of the Mathematicks: manyof them well understandSur- veying, Navigation and the Calcuhutionof the Eclipses: and some of them are Oinsidernble Proficients in ConicSections: anti Flux!ons.In the fourth year Et hicks andDivinity." they vrincipally studyand recite NIetuphysidis, 41. The printed statement.ofaprojected planorofacompletedone mayalways 1)( taken withsomereservations, but the conunencement thesesduring Clap's presidency of Yalegive added confidence inthe acceptanceof thisoneatits face value. In 1753, duringClap's administration, the "Linonian Society "was founded. Atonetime itwasthe customtohaveacurious question brought in at each meeting. .Fromthe records " of this societyis tak.enonespecimen which relates toalgebra: Dec. 5th A D 1770. 2.How do you solve Questions.when the unknown quantity finsseveral Powers inoneEquat ion, and only the,firstPower In the other Equation. Thi recording of the questionand itsanswer meansthat algebra engaged the interest ofeighteenth century Yale boys. niversity of Pennsyl eania.---Thefirst.pageof ttrecords of the institutionnowknown tis the'Universityof l'ennsylvanw. shows that algebra is includedamongthe subjects intended to be, taught.The seventhpagecontains this entry, showing thefulfillment of the intention: Mr.'iiheophilus drewhaving offered himself as aMaster in the Academy to teach Writing. Aritleinmetie4k,lereZhes..ccounts. Algebra, Astronomy,Navi- gation, and all other Branchesof the Mathematicks,itis ordered that he be received . . . his Service to commvnee on theseventh day of January next

July 27,1750.1T -Or In 1753 ProvostWilliam Smith publishedawork entitled "A General I4en of the Collegeof Mirania, withanaccountof the Col- lege and Academy of Philadelphia " [Universityof Pennsylvania]. His belief in mathematics ingeneral is indicated in the preface to this work, where he. sets forth the plan of studie-covering five classes.He states further that he isnowendeavoring torealiie these plans in the seminaryoverwhich he has the honor to p.reside, andso leknow that hewasactuallywCwkingout hisown.fruitful i(leas.The portions relating .tomathematics-are found in the first twoclasses and read:

16 Thomas Clap,The Annal. or Ilbitorn ofYale.rotlegein New Harm.(New Haven,

1766.) Appendix, p. 81: . 16 Yale University libcary. Academy, Charitable Schools of Philadelphia, 17 Minutes of the Trustees of the Concoct', A Vol. I, 170-1768. 60 ALGEBRA IN THEEIGHTEENTH CENTURY

The FirstClass of .theCollege . . . In the afternoonthey learnarith- metic,u18rand (lerimal,merchant'saccounts.some parts t,f algebra,and some of the first hooks ofEuclid. The SecondClass. Thenextyear.jsspent in this class;the master ofwhich is styledProfessor of*Mathematics.Ile carxies theyouth forward inalgebra, teaches theremainder of thefirstsixhooksof Euclid.to-gether withthe eleventh andtwelfth, and alsothe elementsof geometry,astronimiy, chronology,

navigation. andother most usefulbranches 4wf the mathematics." The planhere suggestedwasformulatedat the request 'of the trustees.Itwasadopted inI76 and continuedinusewhile Smith was provost and,asfarasrecords show, untilthepail.-part of the next century.It influencedthe %later curriculaof all the American 'colleges.The prominencegivento al(rebra inthe mathematical scheme isindicated by theinclusion of thatsubjefq intwo yearsof the collegecourse. ColumbiaT nirergity.SanmelJohnson hroughtfrom hisex- perienceat Yale sufficient interestin mathematicstocausehimto giveto that snbjectaPlace int he plans which he,toalargeextent, formulated forKing's College[Columbia. University].The first professorshipestablishedwasthat of mathematics.'"The laws and Orders whichwereadopted'in June, 1755,placed b lathematics and the Mathematicaland ExperimentalPhilosophy in allthe 'several brsInches of it"in the secondand thirdyearsof thecourse.In 1785 the plan ofillincationincluded "Algebraasfarasquadraticequa- tions" for the'freshman classand the "higherbranches of Alge- brafor the sophomoreclass.In 1789 the professorof mathematics and naturalphilosophywasauthorizedto give.asthe algebracourse, "AlgebraasfarasCubic Equations"to the fresimukn class and "the higherparts of Algebra " with "the applicationof Algebra to Geometry" to the junior class.Thiswas anadvance in the subjectmatter in algebra which continueqin force until Owend of the centtry.2" Evidencegoestoshow then thatfrom theveryearlyyearsof the eighteenthcentury until its clo4b, algebra is mentionedbypro- fessor and presidentalike and is included inplans Proposedand plails adopted forthe curricula of the collegesof this period.

14ThWorksof lrillionr Rmith, D. D.. LateProvost of Ilu College and4%v:1114.11)yfir Phiiadelphia.Vol. I. Part II,p.183. Philadelphia, 1S03.

la .4 History of ColumbiaUn4rcr8itil (New York, ."Ai.I p. 22 . . . "the 4;overnorg.on the8ttof November (1757)appotnted Mr. Daniel. Troadwell,'ayoung gentleman of a very excellent character eilucafednt ilarvard College, and recommendedbyProfessor 7 Winthrop nii emlnently fitted forthat station.'" . 21° L. F. Snow,CollegeCurrichlutn,p.93 ff. I.

*MEN Chapter IX EVIDENCES OF THE USE OF FOREIGN TEXTBOOKS

Sare;ty of printrdbonZ.R.In the twentiethcentury.when inexpensive booksareprocurablv in literature aml sciencepalike, it is dif- ficulttorealize the almost total lack of printed- classroom subject matterduringpraclicallv the wholeextentof the eighteenthcentury. Authorship of texts had not yet becomea.trade. and the cost of printing andpapermade. the publication of works for pupils in schools well-nighanimpossibility.Bookswerein thi; hands ofpro- fesl.ors and tutors, and their contents.asalreadywointed Out,were . . passedonin the form of lectures taken downalmost verbatim by the members ofaclass. Butsomebookswere.imported fromiibroad . I"in.suflicientnumberstopermit of putting- them into the hands of

I , R the stmhbnts themselves. Such foreign textbooksas wereused in American schools of the colonial period and forsome yearsthereafter by the studentswere books by English authors, Published in England.In the first.guar- lerof the nineteenth century French and German authors began to exert. aninfluence, and reprintsi ortranslaiionsof works from Eng- land.France,- and Germanywerepublished in America. forusein Anierican colleges. -The Ymot/ildthematfr;an`s Gulde.We shall speak first of the English texts.Am'ong themost popularbutnotthemostworthy was "The Young Mathematician'sGuide. BeingaPlainaridEasie Introduction to the Mathematicksby John Ward. Thisseemsto have hada useoutof all proport ion to its merits. and direct evidence of the position held by Ward's book is extensive. An intimate record of the.use of this work is found inthecom- monpla-eo book of Eleazer May,' of the class of 1752, Yale.A long list of thisstudent's readings is givenonthe. seventhpageof his book.This list is headed: An Memorandum ofthe Books red inmyfreshmanshipvtz:A. D. 1749.

InmySoiltimoreShip viz. A. D. 1750

recited. Wards Mathematiek. A '",Elenzer May'sNotebofits."Interesting manuscripts of the middle and last quarter of the eighteenth century now in the university.library.Vole Alumni Weckly, January 19, 1923,p.501 f.Some additional material, as the result of a personal study is noted.

. 51 52 ALGEBRAIN THEEIGHTEENTHCENTURY In anotherplace youngNfay copiedaquotationfromWardthat hadstruck hisfancy.Itwas: Of timeit isnot an Easy thingto givea true Divinitinnof timeforaccord. ingto thephilosophickPoet Time ofitself isnothing butfrom thought Recieves itsrise bylabouringfancywrought fromthingsconsideredwhilstwee thinkon some as presentsome as pastor yet tOome no thoughtcan thinkon time that stllcontest but thinkson things in motionor at rest: Ward's Mat liema:' Agraduate of Harvard.class of1746, Dr.E: A.Holyoke,saysin aletter to Prof.BenjaminPeircethatWard'sMathematicsand Euclid'sGeometpwere used_during hiscollegecourse.Withoutre- ferringtoany otherauthority.Peircestates that: In theearlypart of this presidency(that .ofPresidentEdwardHolyoke, whichb(gan in1737) and probablyformany years before.the 'textbookswere the following . . . Ward's Mathematics . . . Euclid'sGeometry.' As lateas*1794aHarvardstudentnotebook'contains the Ward in nameof connectionwithbooksto be çonsultedonmathematicalsub- jects. Cataloguesof librariesofgentlemenof thisperiod showthat Ward's bookwas oneof thoseonmathematics tòbegenerallyac- quired.' e- , SamuelJohnsonprepared,"A CatalogueofmyLibrary with thevallue ofeach BookAng.15, 1726."6Thiscatalogue included "Mr, Wand'sYoungMathematician,"which ho mityhave been using while,he was a tutor at Yale.M.1743the'rewaspublisheda book witlithe title:" AnIntroductionto the Studyof Philosophy, Exhibitinga General Viewof allthe Artsand Sciences.Bya.Gen- tlemanEducatedat Yale-College."ThisgentlemanwasSamuel ohnson,and hegive's theadvice (p.28):"OnMathematics,read Ward'sYoungMathematician'sGuide, . . ." A 1764"Catalogue. ofBooksbelpngingto the PublicEfiglish School ofFriendsat Philadelphia"hasone copyof Ward'sMathe- mtatician'sGuide.In 1778a" Catalogueof Booksgrantedthe Col- lege [Harvard]from thesequesteredLibraries"contains Ward's Mathematics.In the librariesofRnivei-sitiesand inlibrariescon- nected withorganizations ofvarioussoilsare tobe foundto-day from one.tofour cdpies ofthispopular oldwork. IThecurriculum of thecollegesat certain tim(us'wasstated interms of thetexts used rather thaninterms of.subjects.In 1756,incon- , NEINwow I John Ward,YoungMathematician's Ouide,p. 37. Benjamin reirmA Historyof HarvardUntrerRity,p. 237,(=bridge, 1833. 4 Joseph McKean,BostonPublic Library. 5" Librariesof Colonial Virginia InOrilliant.an41 Mary Quarterly.Vol. III,pi133; Vol. IX,p.167, Vol.X,p. 232. s sNotebook inSamuelJohnsonCollection,Columbia UniversityLibrary. 7Harvard CollegePapers, Vol.II, 1704-1785.Harvard UniversityLibrary. TISE OP FOREIGN TEXTBOOXS 58 nection with the plan of studiesprepared for thetrustees of the Col- lege of Phjladelphia, Provost Smith recommendedcertain booksto be read " for improving youth in Variousbranches." He suggested "Maclaurin's -41gebra"(London; 1748) and " Ward's Mathematics" for the mathematics in the firstyear.' At Yale in 1778, PresidentStilesonassuming the executive office gave asthe mathematicalpart of theprogramof studies then in force: Freshman ClassWar41'sArithmetic Sophomore Classllammond's Algebra. Ward's 111W hemat ics

Junior (lassWard's Trigonometry9 Since the YoungMathematieian'NGuidewasconsidered worthy of aplace. in American schools forsolongatime, itmaybe W-ortha brief presentation."' Thisworkon" lAlathematicks "iS.in fiveparts, "I Ahmet ick, II Algebra,III The Elements of Geometry.IV Con- ick Sections, V The Aritlunet ick ofInfinites," andto these sections is addedan" Apj)endix- of PracticalGauging." The bookwasput out withaFecommendation that might wellhave started itupon asuccessfulcareer.This tocommendation reads: rpon Careful Peilisalof this Book,wethink itagood In- rolluct i4,n to the.ibmtrartcdParts ofMathematiekx, andas (IP suchwe recommend it to the Studious and Industrious -Header. .1. Raphson. A.M.& It. S. S. Ditton,"Magter of the New - Mathematical School in Christ's Hospital. Samuel Cunn, who revised andcorrected Raphson's translation of .theArithmetkvU1iircr8ali8by Sir Isaac Newton, expressed histkz, appreciation ina poem,"To the Ingenious Mr. JohnWard Upon His Most Useful Piece, theYoung Mathematician's Guide," which is printedatthe back of tit() book. PartII is entitled: Algebra,orArithmetickin Species; whereinthe Method of Raising and Resolving Equations:is rendered easie; and Illustrated withaVariety of Ex- amples, and Numerical Questions.Also the whole Business of 1ntibresLand Annuities, &c. perforad by the Pen, andasmall Table, with severalnew Improvements. K. The algebracovers pages143 to 277,or134pages,andsoranksas tospaceoccupied abouton afooting with the arithmeticof the book. The tableof contents gives the topics treatedasfollows: The Method of Notingdown Quantities, and Tracingof the Steps used In bringing themto an .Equation; The Six Principal Itule*A of Algebraic* Aritb-

'Horace W. Smith, Life andCorrespon4ence of the Rev. WilliamSmith,p.124 f., Phil- adelphia, 1879. . h,P Snow,CohegeCurriculum,p.79.Quoted from Stile's Diary,Nov. 0, 1779. °Accounttaken from the " Third Edition, Corrected,"1719. 11 Humphrey Dittonwashigbly warded by Sir IsaacNewton, and exercisedanin- fluenceon the Newtonian Philosophy.Bee Smith, History, I,p.456, Boston, 1923. , 54 ALGEBRA IN TILE EIGHTEENTH CENTURY metick In whole Quantitie: Of Algebraic':rravtions, or Broken Quantities; Of Surds, or IrrationalQuantities; Concerning the Nature of ;Equations, and Nowto prepare them for aS dution. &c.; Of ProportionalQuaentities,both Arithmetical and (:eometrical c4Intinued; Also ofAIusical Proportion; Of Pro- Port kmal Quantities Disjunct. bothSimple. Duplicate; and Triplicate; And bow to turn .EquationsintoAnAlogies, :Of Substitution; And ResolvingQuail- ratick .Equations; Of Analysis, or the Method ofResolving Problems, Exempli- fiedl, Forty Numerical Questions:The Solution of all Kind of Adfeeted .Equatimis in Numbers: of Simple Interest. andAnnuities in all their variou Cases: Of CompoundInterest, 311111 Annuities.1)4)th for yeaN and Lives: iiil of Purchasing Free-hold Estates. Stivp There is nothing ofanoriginalnature towhich attention need he called in the subject matter of this algebra.Rules and solutions of 1(1 Problems followoneanotherinapresentation that strikes the reader of to-dayastedious.No exercisesareadded. inanytopic for the sake of the student.The applicatiim of algebra to thesolution of geometric problems is found in the section onaeometry. l'extbw.)/..8entitled.1/f/e/H;/.The.first textbook designated alge- bro thatwasused inanyAnierican collegeseemstohave beeli Thv LWniéittx of Alfiebra, by NathanielHammond» In the list of studies given by President Stiles, of Yale, in ITN,'it will 1Kb n6ted that Hammond's algebraOccursin the soph4moreyear.1 4 HArVa rd in 177s received friini the sequesteredlibraries, in addition to Ward's llathematics 'already referred to,Hammond's algebra. These hookswere put atthe disposal of die professorof ,Inathe- matics, butnore(*.ord remainstoshow that, IIamniondIva!-;plac.edin the pupi.V hands: In the laws of Rhode Island College I Brown University] for1783, Hammond's algebra.was4re1idredin the thirdyear,.and in the hkws for 1793 thesame.requirementappearsin the secondyear.Memo- randa collected1.)yadescendant from thepapersof Solomon Drowne, of tlub class of 1773, give the gply knowledge of the curriculumbefore 1783.15 ThesePapersshow hat Drowne began Hill's arithmetic in October,.1771, and Hammond'salgebrietin December. of thesame year,with Euclid's- Elements and trigonometry in February,1772. From this informal ipn itappearsthat the first printed laws of Brown setdown requirements in algebra which had already been inopera- tion forover10years.

12 Nathaniel Hammond: The Elements(#fAlgebra. inaNew andEasy ifctluul,with their Use and Application in the Solution of a twat Varietyof Arithmetical and geometrical Questions; Bygeneral and universal Mars.To Which hi pretixedanIntroduction, con- tainingaSuccinct History of this Science.By Mr. Nathaniel Hammond, of th Ibuik. London, 1742.' si Seep. 53. 14 In "A partial list. of textbooks used in YaAtkCollege in the eighteenth century "(Nale University Library) occurs " Nathaniel Hammond. Elements of Algebra (used bySopho- mores,1774-17g1)." is Walter C. Bronson, History of BrownUniversity,1761-1914, p. 102,Arovidence, 1914,

ite se, USE OF FOREIGN TEXTBOOKS 55 ...

We 'openareal algebra whenweturn toanexamination of Ham- mond,eenthough thescopeof it is limitedasmeasured by present- claystandards.In the preface the author states the prerequisitefor the student of his book andhis confidence in ifs pedagogicalprin- ciples.A. good sketch of the historyof algebra follo.wthis preface. After the first-75pages,whirharegivenoyertothe presentation of the four fundamentalOperations, involution, evolution.andsttiva quantities, there is nota page(except fora veryoccasional digres-. sion to introduceaneeded process) in the whole extent of 328pages which doesnotcontainaproblem.,or somesteps inthe workingout ofaproblem.In thiswayHammond takesupsimple equations. equationswitht.\<'oorthree unknowns. quadratic. and adfected equations, and gives explanations forthOir solution in the greatest detail.Some 109 problemsaresolved. marking theprogressfrom the solution of the simplest equation tothe solution of cubic and hiquadratic equations by the mcbthod ofconvergiilgseries.There is the almost irreducible minimumof rules and the maximum of inns- trath'ematerial whichmusthave madeanappealtopupil and

.teacher alike.It is curious thata mallwho could writeaworkas goodasthisOneshould have beensoprejudicedastomakeno mention of exponents untilpage304. therebyperpetuating in themany (.110015 in which itwasusol the long form Of writingPowersofa base. . If this whole book was.covered inthe colleges wiliA required it, the pupils .hailasgooda coursein the solution of equationsasthqy Would ,get- to-clay inasecondary school, plussomeof thecoursein , , . college algebra. . Another good algebra thatas,ithout doubt. isn the hands of

, teachersas soon asit appetired is entitled .1:TrHttise ofAlgebra." Its authorwasThomas Simpson.1 amanof undoubted genius. The first mention of thiswopkas atextbookoccursinapamphlet pub- lkhed in1794 and containing the' charter andthe jaws of the Nlegeof New Jersey." In addition to the usual list. a topics,this workcoversthe following:Resolution of equations of several dimensions.Sir Isaac Newton's- method of divisors, Cardan'scubic and higher equation's,

Descartes's .fltiquadratic. equations. convergingsel:ies. indeterminate problems. investigation ofsumsofpowers,figurate numbers, interest and annuities, plane trigonometry. And application of algebra to the sdiitionof geometrical problems. .

.4vd. 16 Thomas Simpaon, ATreatise of Algebra: wherrin the Prineiples nn.it flenimmtrated, and. applied inninny umrfuland intereMing inquiriem, and in. the reRoliition of a great . rartety of. problomx of difTerOnt kinds . . . London, 1747). 17 Smith, Ilintorv, I, 457. bt I's John Maclean, Hietory ofoieCollege of New Jerxey, p. 307.

AO% 56 ALUERTIA IN THE EIGHTEENTH CENTURY

In the editions of this work after the firstonethe treatmentis apeculiar orie.Rules with illustrationsaregiven, and inafootnote arrangement " demonstrations."These demonstrationsarecomplete explanations of the rules. An American reprint of Simpsones algebra from the eighth London editioncamefromaPhiladelphiapressin 1809 andasecondonein 1821.The change firm theuseof imported books to theuseof books printedat home wasonlyaquestion of time.It involved the establishment of printingpressesand the demandonthe pvt ofa sufficient number of influentialpersonsthat theseprqsesbe putto workontextbooks.

a 1

e 0.

, t- e

Ate Chapter

THE FIRSTBOOKS CONTAININGALGEBRAYUBLISHED IN THE NEW WORLD dr .1 Ale.rieanalgebra.Inverywidely separated sections appeared O the first works printedpnthe American Continentand containing algebra.As earlyas1556 therewaspublished in Mexicoabook entitledTheSumarioCompenVioso.1The most interestin*feature of this workconsist.sof sixpagesdevoted to algebra. Under the title "Arte Mayor," theauthor givesanumberofex- amples generally involvingquadratic equations, ofwhich the.. fol- lowingaretypes: 1Filida squarefrom which if 15% is subtractedtileresult Is its own root. Rule: Let the number be cost'(it).Thesquareof half a cosa ls:oplal to 1/I Ofa zenso(2'2).Adding 15 and 34 to Vi makes16, of which the root is 4, and thi plus 1/2 is the root of therequired number. Proof: S(Iuare the square root, x)f 16, plus half a Cosa,Whiell is four and a half, giving 20 and %, which is the square numberrequired.From2014 subtract 15 and % and youhave 4 and 1/2, which is the root ofthe numberItself. 2AmantakespassageInaship and asks the master what hehas to pay. The mastersaysthat it will not be any morethan for the others.Tbe page sengt.:ronagain asking how much itwould be, the master replies: "Itwf1l be the number ofpesoswhich, multiplied by itself andadded to the number,will ; give 1260."Required to know how much the masterasked. Rule: Let the cost bea cosaofpesos.Then half ofa cosasqyared makes14 ofa zenso,and this\added to 1260 makes 1260 And a quarter,he root ofwhich less17:ofa cosais the numberrequired.Reduce 1260 andiXto fourths;this is equal to 5041 divided. by 4; the root of which is 71hoyA;subtract from it half ofa cosaand there remains' 70 halves,*which Is equal to 35pesos,and this iswhatwasasked for the passage. IEroof: Multiply 35 by itself and you have 1225 ;lidding to it 35, you have 1260, the requirednumber.. 4 -idt Far removed in time and languageaswellasplacewasthe next bookto be printedonthe WesternHeinisphere,which bears in part of itscontentsonthe subject of algebra.Thiswas aDitch textbook. of 1730, published in New York City, and itwaspreceded in the American Colonies by only two published worksonmathematics, both of whichwereflipmetics.

IForacomplete account and facsimile, see David EugeneSmith, The Somatic)Own- pendioeo of Brother Juan ,D4ez, Boston, 1921. ; 57 11 -rce. 4 58 ALCEBRA IN THE EIGHTEENXIT CENTURY'

A Dutch- alyebra.liI natun.d that arithmetic should have been the fh.st. mathematical subjecttoappearin print in the American. Colonies.Itis,Onthecontrary, surprising that, algebra should Occupy OverOne-third of the ';pace in the third bookonarithmetic published in thiscountry when nearly GOyears Nvereto elapse before theappea ramp.fanother bookcontaining ny aIrebra. )ine4,500 titles` of publications inPennsylvania beft4e 1785 'showsmanyal- manacs,but few worksoilmathematics, andnonecontaining alphra. coniTleté bibliography3 of all .knierican books up_to-179.2. reveals, amongthoseonmatirematics, only three WhitiCincludealgebra in theircontents.Oiw of the three is thiiibutchtextbook, and while theextent of its influencewasprobablyverylimited, it has interest. asthe earliest and, foralong period, the only workon'illgebraprinted here.

The title of the book is: 4.1o7lInut;ca: of .Cyffer-Kon8t,. 4Th Mole Eert heart witwerp rande A7ge1)ra.4 ThenameswhiiThAppear onthe title-pagearethenamesof threemenwho,as weshallsee, werokindred spirits in theirindependence of authority. The-nameof John Peter Zengeris inseparable fromthe history of the freedom of thepress.Hewasthe second piinter ofNew York, and hisnewspaper wasthe instrument bymeansof whia :wtive pr(Ostwasmade against thetyranny of the royalgovernors,Nvhich eventuated in the AmericanRevolution.The trial of Zenger is significant. in allhistory,.and theOutcome of itwasthat liberty of thepresswhichgavt people in thiscountry the right to freely.criti- cize the conduct of publicofficials.Zenger'spress wasestablished A in1726, and his'newspaperIleac its beginnink in1733. The Venema, book, appearedbetween thesetwo d4tes. Jacob appearsinaminorway asbreakingawayfrom the. authority of the church.Ile i§_a...effir.gdil_to--ag--*Nrotbessing ii[mself in oppositiontoastand taken by the ecclesitksticalbody in several instances. One of theseWaSconnected with the licensingof private- school teachers."Atanyrate, he succeeded in having printedthis textbook in Dutch.bya manwho,aswillappearlater,wasunder the ban of the localchurch leaders.

4 - 2 flimsier,' R. Iiildeburn. dtCentury of Printing.The Igsue.4-ot thel'egsin Penagy1- e- vawia,1685-478¡,Philadelphia, iss5. aCharlesEvans. morican Bibliography,1639-1N.:0, Chicago, 1903-1914. 4 Onlytwo.cvpler4have been located.Theyareboth in thelibrary ofthè New York Historical Society.Theeopy in the New York StatetIlbrary, listed in Pavans,loc.cit., was destroyed in the fire of March 29, 1911. 6Ecclesiastical Recordsof the State ofNeu,York. Vol. IV,p.2133,Albany, 1901-1916, 4 e H. W. DunsNie,History of the 'Schoolof the CollegiateReformed Dutch. Church from '1613-18,p.3g,New York, 1883.

!. --.--mTmwml.+=-'.-=n.=.m"."..^- FIRSTAIAEBRASOF THENÈWATORLD 59

PieeerVenema *as repeating history by makingtrouble14the 'church: Init letter' from the Rev. GaulterusDuBoistoME;Rev. Classis,of Amsterdam, May 14,- 17411.ive, find this-complaint:

OP . . a 11111/4r14141.111111. La 0le% .1 4 u v.. 0 4i, 0 1-,. .- 617077 --W.."'i 177 .."17.4 r. i Pl. A i L ,. _7 -12(41 ,i, --4 ' '' . r. , i' #s9'4:1400' e._.. "t.../ ,Ttps , in . :C., I - ''' Ars, A .! 9 s si, . §104 lik.;:t '1..* -s, . e--AX .-,-...,,, 7 , , Ix...... 44,4 ...... 9 t . e lt E 'Ll% i . , o t .: .f 411 6 lb ,._to y . , , 1',- 4 ' . rat ',1 016 q t''v CI 0 210 L.1,3 ..1t 'r , 0 . . . % '4.41 4 i ,'T.444- V 4 ib' .., lkk;:stoL,..4 .;44 ,, 44i di% IM A AI ..,- , ri 0 t. al .rd -1.' 4 ill- ?.. li, * . I 6I,'1 ...,...- t.'' et . 0.4 at 7;") .r , 4 . 7-t7'114g..a2 fla:,? .z II I. 411k i MtIIE5l9f k r'i -,,fti C .14; - .. :-'. . . - 4-. -.. ih_...,(._ '1:: AItl,' I r11..:"+....01. r,tt 41...1. . .4..4 . e r. s er . . a ... ri, 13; 110,-., Ile, ,rliL 1 . t., .* c ,1"11" t., 1-.., -:I. -I t S 64'; , - ...'.... ''.46:11' ..)'.ff...f.'''''*;"...,11.4",7,:3.'-47:;:fi_.1-il 11{1. ,.°..i:I... iiii"!1At,:iilielsi7:'::'2...--3,p-: .IL4';,.i c.*,1-4,-,..e.,., 3i g s. to., ., 4. -'"' - . ' f .7 .1¡: :S, ./e,.,.i. ''IlitiA - ' .' . .., ,.. -... - ' - - ",,, crIs.' op ... .1,' 9 ? p , e iS,."` % N...... 44, ' fAe- ;k .1.,ge "11.114jdre';" ' c,;#, eV/1 - I t;T, A J ih I ..., , i 7ft -:. .., ,8;...... *:.. . ' . .ell:. ft. gp -.,t-: -.4..* ::riva4 _ AO .:..' . k ; - If ,.. ,K .. . 9. ti ,1...7:` .44.,kr 1*. Ifa, af=-_,,,I 1.4.." t; I 41...te' I..* . Ctr?' OS' ; .. ., s!:...4. II .:.44"S)- e,'t 1 .,.,'. , f, v.,-r 10 t41 1,.a> ...,..t -4 .18 ,...3. oft( 1 ...4.1,4 .1 'A 714. ..Ar, . , ( Iv?i. ..-, .4. , .: ''.;. v.17 _': 7 74:t.: :.,- ...,...... -..',.a ." ,. 'f,:. . .. rAfAt 1%,:4. .. , 1 e6i-filr-il .--...- . ;tig 1 * .. "..1 i. t .s.r., 1 f E1,4 e. *Al pi, t."/"`1%!* I' r.) ,I : r.^ a

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O . "Arithmeticorthe art of ciphering, according to the coins,meas- ures and weights of New York, together with a short Treatise onalgebra drawnup bypeter Venema,master in mathematics and the art of wilting.New York, printed for Jacob Goelet, -near the Old Slip, by J. Peter Zenger, 1730."The titlepageof the first book containing algebra thatw/asprinted in the Anieri- canColonies. .1

Is Inasmuchasthe Rev. Consistory of NewYork severalyears agoexhorted theirministers to beontheir guard, andoppose. the Artful misleadingt ofone

411, et. M i! 7 EcclesiasticalRecord8, loc. Cit.,p.2756. a j 95337*---245 411IL 4. ' . -.I. SIP t :

1 'V I. 60 ALGEBRA IN THEEIGHTEENTH CENTURY Pieter Vetima, a craftyfree-thinker of Groeningen,who had previously Wen aReader and School-masterjust outside that city,I, therefore. determkvedls set myself againsthim.enderGod's blessing my efforts accomplished much good,althouglisomestill adhere to him.Among these is one Jacob Owlet, who witb his conventicles,endeavorsto doalLf)ossihle harm to imr church. IVe thereforeseethat Venemacamefromacity in Holland@ which offered universityprivileges, am' in which he badbeenaschool- teacher. More definite evidencethat Venemawas aschoolmaster is given in the dedication to awork of his published inHo 111nd in 1714.8In o this dedication,which is signed PieterVenema,appearsthestate- ment: Is de Gunst.vant7Ed. INIog. geweest dat ik eenigeJaren herwaarts iiiU. Ed. r1 Nlog. niet min vermaardeStad Groningen, mijn bedieninge nlsSchool-Wester hebbe waargenomen.(Throughyourfavor 1 had the clion('e (ifserving;is a schoolmaster at Groningen for several years.) oi Fartheronhe states that he had tlwhonor of enjoyingfor several yearsthe teachings of"Heer J. Berneulli." 9 The high regard inwhich he'witsheld by the mathematicians of hisow.ncountryand time is shown byaninscription to him in the work" already referred p. Thisinscription contains such phrases as: I.

...the talent which God has granted you. .. weneed no teacher, the hook is aguide in itself. We thank you, veiwma. wethankyou.brave toucher, give us moreofyourknowledge!You havewon somuch distinction atGroningen tiptitis impi)ssible thatyoti.sbould lie _forgotten.'

. Venemamust¡lavebeen known during theeightee`nthcenturyanti the early partof the nineteentirventury.'because heis cited repeat- edly in collectionsof problems solved and published by Dutchmathe- maticians and societies of thatperiod." Venema's ivasons forwriting the book under consideration appear in the " Konst LievendeLeser." Hesays: Because I realized that there washerenociphering book In the Low Dutch coricerning tradeormerchandise, nno for the sake Aftile teaching of inquiring

.youth andof all lok-ers of theteaCbhingof grithnletic.Ihave undertaken to makeaclear and succinct cipheringhookuponthat excellent sciencewhich flourishes in this city and country.To thisareadded the elemenis of nigebni. a Pieter Venema.." Een kort enklbreOndenrygiagr in. de Beginselenrande Algebru ofte egtet-konst."Te Groningen. 1714. Other editions Amsterdam 1730, 1756,1768, 1783,1.1.91111803. This 'VMS Jean (I), who wasprofessor at Grilningen, 169511.1705. to P Tenema, 1714, loc.cit.No -461bIlograp4lyoontulted revealed the existenceof this edition,' but a copy wasfound in the co1leeti1m of Mr. George A.Plimpton, to whomthe writer is indebted for theprivilege of examining the hook.Mr. Plimpton also has a copy of the 1756' edition.

e ti illakunatige verhmliffing.etc.. Vol.1.pp.29, (13, 70, 79, 81OntitindIngen, etc..P. .218 ff.Vol. II.p. 8(Amsterdam 1793, 1795), and other references. . s. J.i4111.4 SVilir,CYIV AO HILL .11:311 TIMOAt 19

0.10.01111 IWO lo11 10111s.101MII 11! 4,1111111pi; 111!.) ISWIt1101) 1,13411.1 0111 9C11.)111M .6)1 ail St1 .11:411J s1).1(1,1k JO '141410.:111: ().: Wu, 1,1(10;110 .! t)111 .01 nj 010 oati.qp stioutsoduad Jo ...)!0111111!.11: pinuti_s .0111 olio .1.1!.01) IQ t):41 .11)10.11j 1:111 mil soaping pornitotli 11! s!ql T)t)(1 ail ut;.) alutit ostl J4) Stu opltuls tutla:Ity 14) Isuompis patiseinii ut J gig .11:). L 61-1.1 11! .111 0.1111M 6.1!.) LiIIbII!II)O.I)

a al Itoug s1.s!suO) Jo 6s,1711:(I Jo ti.)1.qm (.:24 adu pajoAap 0} IpLIE - IOW .)t puu ¶t,. 01 vacioritt: ,)ILL 1101.4,)!):-; p())().)1) yl o9oultopt: kq.it:4s tinm uoi.01)1)1: bsaptut pRai -

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1)11,1a 1? ).1 -;11:11h4 j() 1.10.(1() 1101.1 iti u,-).t;1 Inuit alp uoiloos tio pavuo.I Satu, OM: flfJ j111S11 SellighS JOJ 'Itotoppl: 'ilot431:1Ilqns Pun . .1111:11ba puu L tion¡At Itiapiaq loi salouop) tut 4:4dtua .(amds () 1:-; tc1 si iì .10j *tioir..)11(1tIplut plq 1 U Hot)ituricha st oputu aol stopoi goy() riu!moiloj amp° -suri0..litog)tm tu, U1S .10j Splutilh) .1 p1111.1(1 tty.m. .111uttiuti .711101 ip.ti:(1 1J1 .sauil aaup000ml clana .)1(10) Ovsi ap:ls 'atu Ituoutd `alai 11.10.11 Ulu tiu uotpu4stiut. Jo %.11 Pm: a.0.1(1 ail) ssatiloau().) Jo alp iinsai .kri IU.A.,101111111 1111ft441114 .suol sj.) j() sopitiluxo 111: sapll siltioluv ¡Hp satnill'aj JO alp looq oat: is)t.tios ijo!ti. pitiom 10(1 mott ,)(1 Tunoj

ail() JO I) tN31 1.10.1 1 1 .10J Mil (ioistAil) jo) R '1101).-MIJ %HU..

JI) 4- Josimp * alp puoptAiI) pm: oto omi quo 1) ,40raimitd d poluntdas St1 tut paluaatixo X ampolly aitlytal st ati4 tioloadai jo v aallat .10j alp puoios OM 'A p11 1 .).111:-thj ol oi() lo 'nu asug (0 -- aztu:i 411zhi.t ,i tiv Ito! :01 .111 r 'tritium(). tut 11.1410.1111: vimotis loin %)Iii vom jo 04,2,1 WI au pappamti laatuicaJi JO mil 3.)IpIN) ',VOA% $,Ittritlao.0 Inca 'savItu putt tictomoati jo a0 Aanitti aa) -imam . ,. u . pp cui stalk! JO Wit %loam.; , i r i . *I cvc ii: d %. e V i A ; . - ALGEBRA IN .THE EIGHTEENTH CENTURY 62 .

is the representation of higherpowers;thus a3 is writtena3.13 ,The lowestcommonmultiple of several expressionsis found bya

.0 1 a d...... !.t.. ...4 .. .-01....07-"...v.v. qv 2 « II, :1 . I on '' Ikilli'711 I. ;... LI , - a ir .... L'AV«au .- a ",;-.g g 1 set--, 1!-' i ...,-c ,. c,-.... 41 ow '1,- . .. o } -1! , , r'-'," .C.-do cal a- -r ,,,. .. ,_, . ts rer, --, q..n,.., y.

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ori

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4' el ') PIrN4°- 1 1 .-- ,1 ,L- 2 7 <3._ :r ' or i 1, 1111r, AP III . e ,..- L, g /1- '-4... II ;.-. pa- , Ni,-,..e.,... , " i. .- `.. , 1 -. '. '.*I I. ¡ trlai; *10_0, Ot` 4-1.t 15,1 '1 ' qau ..:LI L ' -[_ i L ,.... i - gj a ., a I As, 1 A , a- 4, .IAA 44 %e t.,._:. - s , --...4.,14:...... ,,,,I,-1' 41IP I. ..-. .. __,....-.., ._,_..._._,.. _t6c...,:Xt... 0---.--4 -_

Division of fractionsfrom Venema.The word for fractionis "broken"; for numerator, "numberer"; and fordenominator, 'snamer." a method whichis associated to-day with finding the lowestcommon multiple of smallnulubers.

, ...... ______L. 4 .4.- -...... 4-...... M.....4...... r.,_. . "This form of' thè eitponentmay have been due to the convenience of the printer, since Venemalutes'the present-day formIn.-hlsearlier algebra.[P. Venema, 1714, . cit.,p.74], except-for the secondpower. However, suchaform of the exponent may have been known to him.Itis found In Pierre HOrlgone,Curitu4 Mathernaticus, nora brevi et elara, etc.Vol. II, sectiononalgebra,p. 4, and consistently throughout the

tire work, Paris, 1644. , 10,

ir _.1.-_- Nome. . --- I FIRST ALGEBRAS OF.THE NEWWORLD 63 The bookends with 24problems,and thefamiliarageproblem isamongthem.'Ittakes the formofacurioussonwho asked his *father hisage: The hitheranswered,your age with the secondpart, the third part,and the fpurth part ofitself inerear.(1by 241/2yvarI is equalto mine]. Iam as much over 40years as youare. under 40.How oldwas the son? Ans. 18 years. 410 o Two unknownquantitiesareused in thesolution. Otherproblems lead toindeterminateequations.One reads: Threewomen l( night apples, the firsti IN),the second110, andthe third 1:20.They silk!, eacha different number,. the firstday,at thesame price, and the remainder the second day,also nta Worn) price.In counting their r, money.they foundthat they had equalamounts. Howmany apples were sold. on each day? The unknowns,r, y, 2,respectively,areassumed for thenumber of apples soldthe first day,pandtr, respectively, forthe priceon 10w thetwodays. Bytheconditionsof the problem tr+ and qr%-t 1. 2,0.tr 2=111+V0areobtained.With theusual ingenuityin such problems, r,andtr aretakenso asto giveoneset of values forx, y, and2. As simple,asall this workseems,there isgood stuffin it.The question arisesasto the schools in whichitwasused,for theremust have beenadefinitereasonfor printingit withthe arithmetic.It isto be noted in thepreface by theauthor thathe lookedupon algebraas necessary to the clearivupof doubt fulpoints iiiarith- metic.But he,as apracticalschoolmaster,nnist have knownthat the bookwasneethd forinstruction.Did Venalikhimself havea private school,and did hesucceed,soearly inthe historyof printing, inthiscountry, in puttinginto printthe materialthat he lieeded1 Itseems tobeasafe conjectureto imt himat the head of suchaschool, andthe algebraofat leastonesecondary schoolof thatperiodwas not unworthy.

14111411~s"` p.

I. so

41, dalamip Ix Itila I1LN:4311 AklifI1N:43 SNOW NO Vat:13.9'1V NVOITANV szlinufw

Imidfiar its/ sy/ ZW.ir sivt))./x 100([ 110111.1M 11111! tilwistatily .71(111111?)110.) It (101).)t) 110 1?,1411).1111! SUM JIEl puoJty. jùoq 41111.1a.o.) 1.)tq(pi-.4 o) pothivin(1 tti Imp si .011 1)11) 10)111.1 11 1)).11;4)(1(11i Ul I II: '1.10(1.1..III(pl) "4SI:IV R UAW) S11100; .110 alp 110114(1 I 1.11:.1 JO bt10111:.)1110 (HU) j(J oll!ok.; imp) 11.)rqm stt.w1. slooq 1).14A1 4101111(111(1 10"1 \\ '1! .it Or) /111/) I did/dim), ///,,/sfiNf ./0 s.)./o/tHi/f./ 11111: .11. .1011)111; -4:10.)1N N- "Iv u o1I:111)11.1;1 luttA.lvii 0:11,110.) tu *(,)91,1

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otu, auji:A .11111I sbolid 09att1tIo..1 ,:41)toti Ithy.;).1(i )1.1o.kt st gnip 0) (IT ItoKtipiti (II o Jo u tioipas tio .101:j0.1(1 1s.19 Pt) P. 4)11d ) .44111 4"1 91.1)4) 1111"111 ") 4111(h)61(1 111)4111" *(1 :iffurSus

. 1 red.\ put) d)ildiut,,) ,plis p) IjJU JiIJUi pili(Ml1fw.) J(1) .11/1 Jrin jo NUJ:W.) JO 1 P6Mt1.1 strIo.)!N H y to.Z.t v ilmotts pa;litquo 11011W) pOSIAda pop.11.1(,) Jazolo41:4 .stuipy aopl000mi jo Joissupvi 'Stlit)Inov aunt.) Ino lalgo,uom U I 2,(; Hp" it 'fi(eitipa bpas!.taJ i,h).1j4).) .1k) I PU (III 1)3A4m4I fq)ItrUIDUN . "IV tiLLU P411,4111110 ni sosi c uulsolt )tinua.)y [Mitt.) 1110.1.) '1° JI1,) ati3,1 II 4.184111 JIJ j.Wild.in "NNI011. !,ti I ...fail I it; 91 81 tit) EARLY ALGEBRASBY AMERICANAUTHORS 65 The short introduction toalgebra, which is subjoined.was abstracted principally from Itonnycastle, and that ofConic Sections. fromEmerson's works. The.,sect iononalgebrais designated`"An Introductiohto Algebra. Desigtiedfor theuseof academies,"andcoversonly 39out of 512 pagesin the wholework. Thismaterial would benegligiblewete not itspresencesignificant ofsomedemand whichled its z)uthi);itoin- chide it:The usualstartwith definitions is made.'The si.operations follow, Withall examplesunder them completelyworkedout. "Sir Isaac Newton'sRule for raisingabinomialorresidual quantity to anypowerwhatever" isstated./ Infinite series,arithmetical and geometricalproportion, simpleandquaoalratic equations, allreceive brieftreat ment.Onlyls problemsaregiven,12 under simple and 6 underquadraticequations.Thesection concludes witha" Re- capitulation oofthe principlesof Arithmetic& Algebra " under9 so-calledaxiom's. Pike'sarithmeticWasthe. first work writtenbyanAmericanto haeanyextendedusein the UnitedStates.itmustbe regarded alsoasthe first' printedworkonalgebra,written byan:inerican, ,thatwasplaced in thehands ofstudents in collegesand academies. Anotherquarter ofa century was toelapse after thefirstappearance of Pike's book bkforeabookonalebra 3 aloneand bearingthat titlewas tobe conipiledbyanAmerican professor , and publishedfor theuseofludentAin hisclasses and elsewhere. 1 T he Am( rie(In,) out h.Anotherwork contai.ningalgella and published intheeihteenthcentury deservedmorepopularity tlían extant evidence sho.sitto have attained.Its authors followedthe- cus(om, quifecomm nin these earlyyears,Of usingageneral title. The bookappeared A TheAnil ripan Y 0011.4The authors,Consider and JohnSterry,were .3° apparently outsideof universitycircles and engaged entirely inwork withprivate pupils.Theymust have felt justified in goingto theeNpenseof publication,but it tookcourage ontheirpartto putaboalike thisonthe marketin 1790. Volume 1is dividedinto books,muchasgeometry volumesare divided.Book II of thisvolume extendsfrompage241 topage387, the end of thevolume.All of thesubjectmatter inanelementary algebra of thepresent day is covered,with theomission ofinvolved exercises in factoringand fractions.Themoreadvancedtopicsare quoted:

Infinite Series, BinomialTheorem. Pioportionor Analogy Algebraicallycon- sidered, Arithuwtical,Geinnetrical.Harmonica! Proportion,Genesisor Forma- tion of Equations inGeneral,Concerning theTransformationof Equations

3Jeremiah Day,Thor Elernenta ofAlgebra,toe. Cit. 4The AmericanYouth: beinga twits isn4i completecourseof introductorymathematic*: dcmignedfor theuse of private Rbudents. By Considerand JohnMerry,v.1 .Provi- deuce.Printedby B. Wheeler,for the authors,1790. 66 ALGEBRA IN THEEIGHTEENTHCENTURY

and Exterminatingtheir 'IntermediateTerms, Resolutionof Equationsby Divisors,Finding the Rootsof NumericalEquations inGeneral, by theMethod of Approximations,Concerning unlimitedProblems undDiophantine Problems. It'isanambitiouscoursein algebraset forth in thistext ata time whenstudents insomecollegeswerestill dependentontaking mathematicalnotes from lectures and settingthem down innote- books.Perhaps itsinfluenceas morewidespread thanhistorical testimonyshows.Atany rate,copies of thebookareto be found rather generallyin the librariesof New England. The findingsof thesetwo chapters leadtothe conclusionthat only three bookscontaining algebraappeared. in Printin the.American Colonies andtheyoungAmerican Republicduring theeighteenth century.In eachoneof these books,it is treated inasection along with sectionsonother mathematictilsubjects.

. a

14.4.

IRV

.010' Chap ter,XII ALGEBRA ANDADVERTISEMENTS

Algebra inthe publicpre8s.Perha1)sthemost unlikelysourceof information bearingonthe teachi-ngof algebrain the AmericanGbh oniesduring theeighteenthcentury wouldseem tobe the filesof earlynewspapers. And yetanumber ofadvertisementsrelatingto different phasesof the subjectareto be foundamongthose dealing ith the datesof the sailingof vesselsand of thearrival of thepost. from Philadelphia,Boston,orNew. York,withrunaway servantsor slaves for sale,Nvith theimportation ofgood Cheshirecheese,orwith lotteries forwharf,church andcollege. /Wrotehdornand8choolnupiteex.One formof advertisement which indicateseducationalactivity istlott in 'whichaprivatetutor orthemaster ofaschool otterssubjectstobe taught.The earliest advertisement of thisnature. and Matingto mathematics. locatedis the following: noRtonNews-Letter, Mch.21, 1709. Oppositeto the MitreTavern inFifth- treet next to Scarlet'sWharff, Boston,are Taught, Writing.Arithmetiek. lii ail itsparts; And alsoGeometry, Trigonometry,Plain andSphaericalSurvey- ing. Dialling,Gauging, Navigation,Astronomy; TheProjection ofthe Sphaeee, and theuse of Mathematical Instruments:-By OwenHarris.Who Teaches ataseasie Rates,andas speedy asmay be. IsaacGreenwood,before bebecame the firstprofessor ofmathematicsat Harvard College, usedthenewspaper asa meansof obtain-..% ing pupils.asshown intheseextracts: R000n-NewsLetter.Jan. 12, 1727.An ExperimentalCourse of Mechanical Philosophy,wherein thePrinciples of thatNoble Science,with the. discoveries =111111 .01....mmolme IThis studyhas been made fromrepresentativenewspapers of Boston, New York, Phila- delphia.and Virginia.Thepapers have been examined systematicallyillbomtheir begin- uingmto the dates indicated,guch dates being inseveral instancesthe time at whichpub- lication of thepaper ceased.The files examinedconsist of theoriginalsortacidmilesIn the NewYork Public Library,New YorkHistorical Society,New YortSocietIiistorical societyof Pennsylvania,Library Companyof Pennsylvania,and litirginiatate Library. Thefollowing 11,1 ofnewspapersindicatesthe extent ofthe inventigatian: Boston Newg-Letter,Apr. 24, 1704-toee.29, 1757. New-YorkGazette, Feb. 28,1726--Oct. 15,1744. New-York Weekly Post-Boy, Jan. 10,1747-Dec. 18. 1752. n01' New-York WeeklyJournal, Oct 5,1733-Mar. 18..1751. New-York *EveningPost, Dec. 17,1744-Dec. 30, 1751. (Philadelphia)American WeeklyMercury, Dee. 22,1719-Jan. 1, 1748. (Philadelphia)Pennsylvania Gazette,Oct. 1, 1728-Dee.31, 1754. VlPgInIa Gaiette,Jan. 1. 1767-Dec.81, WO. 07 89 VaarzitYlV NI HILL HINtaraddword .111ILLN1O jo alp udtuo.)ui ILI 1 al( .1!S alMS )1MON j L11441011 4/3 )S11°11111) P4)/1" Sti a.togn Q.111,14 poapunH snowl,) putt injos,1 slu,At1!.4)(11::.1 po!umitu(o.m 11:1110wpailxa

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O vaElatriv ONY SINHIVHSLIAIHAGY / 69

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Another series of advertisementswas runby AlexanderBuller, s. the firstoneof which follows: Pennsylpania Gazrtle,Nov.5, 1741.Writing, Arithmetic,Nterchfint's_ Ac- counts, Navigatifon.:11ge-bra,and other parts.ofthe mathematicsaretaught by AlexanderBuller, at the Public Schoolln StrawberryAlley. .. (Repeated Nov.12,19, 26.) 4 Buller had receivedpermission inOctober, 1738.to leach mathe- !woks,amongother subjects,in thePublick School"I AVilliani Penn Charter School 1.fiIle had evidentlyLeenapupil of Thomas Simpson.Ak letter dated" Philadelphia, Oct. 27, 1741,"reads: VrivadSimps4)11 . .1111. years and half ago I gotaninsight Intosome difficult4jia1't4()fypMathematics Tromthee .. thyoldfriendAleix- 1:uller.', This teacherin Philadelphia carriedinto his professionthe in- Tiration recei %-cd frokn hisstudy under " thatstrange mathemat igal genius." Thomas Simpson.

Otiotr.4advertisements ofthe teaching of algebrato be citedare: O

(Pempolraniyakettr.Aug.. 13,1747...are taughtthese Mathematick Sciencps, V Arithmetick, algebra.geometry, plain and the sperical(sic) trigonometry,conickSPt'tiOnS,arithmetickof intinites... 1,)yJohn Clare. ( Repented Aug. 20'and:27.)

/bi(fNOV. 4:2, 175.2 . . .?.irtbstiii taught, theseMathematical Sciences,viz., .\i'it Iiiiìtt ick in allilsparts,. Algebra, Geometry... by .JohnClare.(Re- peated N4IV.9 and M.)

Ibi41.I lee.8,1748.. .arithmetic*, vulgar anddecimal'... algehra,all rarefuIly titught . .IyThomas Craven. (Ad.ranthrfpgh Apr. 13, 1149.) irginia Gazette. May2,1771. A Clergyman offlit% Church ofÌ,ng1nn(1,n soberyoungMan.. proposes to teach...Algebra, GeometrMSurveying, 0 Mechanics ..the Reverend W. S....Potowmack, Virginia.. . In additiontothese advertisements in whichalgebra is definitely namedsome scoredifferent instances of the pbxaseotheroarts of the MagAnatieks" have been-found. With thelong list ofmathematical \,.;libjects usually ,preceding thisphrase algebrawas un- doubtedly,covered by it,747. , Teadier.s\ofmathematics wanted.The advertig4ementslocated in which the shvices ofateacher of mathematicsarecalled forareonly three in number. They lire foundin the PenmylvaqoGazette, Oc- tober 9,1740, and in the Viryinia Gazette,October15, 1767, and

6 Minutes of the Publick School, 1712-1770, Vol.1,p. 26. 6.From Simpsonlana in the possession of Prof. DavidEugene Smith. 7 A number of acIrertisements ofalgebra have been located innewspapers between 1783 .4 and 1800.On account, however, of the desultorynature of the searchmade.duringthee yeam they will not be cited.

Il 7;71- 7-7 34 VIREVVIV NI HILL IlltiTeLLIMIR Al11.N11

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40. 4 . ; ALGEBRA ANDADVERTISEMENT'S 73

.matick8(Penivyk'aidaGazette, Apr.12,1729);Ward's ....Y.dliny 41b1themuti,eian's Guide(PenMyllVa.,r,;(1Zette,May25, 1738); Woitius'sAlgebra(Pcntolylviliiiaz(tte,Aug. 4, 174S). Wesee,then, thatthe publicpressof theeighteenthvtintury. beArs witnosstoasctivity,inthe, teachingof algebra,andin the sale of:ilgebratextbooks, activitywhichmust be interpreted sinthe a light of demand for thisbranch oftmathematics. I.3

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f- eeN Chapter XIII SUMMARY

The direct evidence in defense of the thesis that.algebritwas an impokantpartof the American education of the eighteenthcen-

turymaybe summarizedasfollow's: b

Manusceipt.notebook.. )tialgebra: Min.:m.(1College,73R-31. 1739. 17711 0.° College of _New.1e.sey(Princeton Universitj.1770. University of Pennsylvania, 1788. Miscellaneous. Commencement theses containing algebraic truths: Yale College. 1718-1797. Harvard College, 1721-18111 College ot New Jersey, 1752. RhodeIslaint(»liege [Brown Iniv(brsity I. 17M-1811. a111ematie8 theses Harvard College, 178r)-1839. Statements from(171 records and writingsof college presidents and pro- . fessors: . Hugh JOnes, professor of truathrmatles at the CollegeifWM11:flu anti Mary, 1724. Rules governing Hollis professorship at Harvard Colh;ge. 1726. 17S7, 178S. Thomas Clap, president of Yale College, 1743. 1766. 177S. A *EzraStiles, president of YaleColiegre, . Minutes of the Trustees ofthe College, 'Academy, Charitable Schook

--e [University Pennsylvanitt 1,17494177)0. William Smith,rftvost of theshInecollege,1753, 1756. ifothessiaffiálrequirements in terms oftextbooks. <7* . .1ohn Ward'snicYoung .itathematitan'ss Guide. 174n). Nathahiel Hammond's.TheElements ofeAlgebra,1742. ," Thonas Simpson'sATrw;tise of..11ffebra...1145. Neetiorm ritlgebra in.tvlbooks7r1imAmerican printingpresses: . Pieter Venema'a .1ri1hmeticaof Cyffer-/Con4 . . Ab Mede Penkort onttrcrprail.de Algebra,1730. Nicolas Pike's, iComplete Nystemof.;frillimetielett h An In trodueliowto di 1 sellgrbra,z17g8. OD Consider andJohtSterry'sThe American Youth, 1790. Advertptrmentxin the public preo fran tcacherm inextabli.shed ooland privalelutorn: Boston.FirstdattJuly 6, 1727. t.

A .New York City.FirKt date. September 7. 1730. or . Philadelphia.First date, August 15, 1734. e own.Rake.--t-Nowhere $ Algebra for its ;. arethere,ktind indiscations thatapracticalne'alfor algebra actlated the teiiching of it during " iubject inthetciirriculupof.6 this miod. The inclusion of t4is,.. .

7. 4." ...Id A

, SUMMARY 75 tfoegeof theeighteenthcentury,orthe teaching . . of itas aspecial iubjedby-some enthusiasticteacher,must be accountedforonthe ,groundthat itwasdope forthe sakeof thesubject itselforfor the theoreticalaspects .of. fluxions.Thefascination ofthis kindof analysis attrae1e/1teAcher andpupil alike,and thesimple joyof the intellectual life thafitAtordedwasrewardenough forits study,:t reasonthat liesattileveryheart ofprogressalonganyline of mentalactivity.

hronoloyica 1LiNt.4)/AMericantlget» Textbook* to1 S20,' theYear in whichAlgeto-aJtax ist Mequiredfor Admixxi.ontoan American College ; 1730 Venema.Vietc:r. 169 A ri t htnetjcaiif I KunstVolgens iIMunteil liatenen Gewigten. te Nithu-Yirk. gebruykely1;AlsMedeEen kortontwerpvan de Algebr:i . opgesteltd4)or PieterVenema, Mr. in deMathesisen Schryt-Konst.Neu- Y4Irk. Gedrucki door JacHilG(Htlet. Iyjlp by.J. PeterZenger. MI*WCXXX. 17SM l'ikv. cohs. A NOW and (14 MOO4' SteinaArithmeticCM11POSOilfort ht% useof theCitizens (ti.the UnitcdStates: PoNiciOas Pike.A. M.Newbury-Port.

MDC('LXXXVIII. . Skondedition.Enlailie;1.hevised,and corrected.By EbenezerAdams, A.M..Prece1;t4)1*411 LeicesterAcademy.Worcester, Mass.,179q. TiiirdeditionsRevised. corrected,and improved.By NathanielLord. A. NI.. Boston,1808.

1790 Sterry,Consider andJohn. S O TheAmerican Yowl): beinga aftv and completecoursof introductory 'mathematics:designed fortheuso:of privatestudents. 11.yConsider and John Sterry,V. 1 . . Provhlenee.Printed by B.Wheeler, for ; the

authors, 1790. e 1798 Gough. John a &Practical Arithmetick.Ity John ( wo.carefullyreviseil byThomAs Telfair. l'hfk)math. Withan Appendix of Algebra. By tIit.V. Atkin- won, of Billfast, Dublin:Printed.Wilmington: ite.printed an4sold by Peter Tirynberg.111.INIC.X(XIII.Second edition.1S00. 18016We1rer,Samuel Mathematicscompiled fromthe BestAuthors andintendedto be the Text-book ofthe Courseof Private LAturesonthe.seSciences in the o the .Universityat4 la nil)rIdge. pulerthe directionOf 7SamuelWebber, *.A. M.A. A. 8:Hollis Professorof Mathematicsand NaturalPhliomphy. In 2 Vols.Boston.Printed byl'lionias& Andrews.1801. Setowl edithm,Cambridge. W.Hilliard. 1808. 1844Bonnyeasitle, John . An Introductionti) Algebra;withnotes and(;bservntionsdeignedfor 4heugeof schoolsand placeAof publiceducation.First Aniericanedi- tion.Philadelphia:Published byJosephCrukshank, 1806. . 'SecondAmericanedition.Philadelphia:Kimher and Canrad;1811. [Titleas above]...tp whleh is addedan appmplixon ,the application

ofalgebra . 6 to geometry.First'NewYork,tteinthetenth London édition. 0:Now,y4;)rk: . t.Dwickinck, D. D.Smithj&G. Long, 1818.

i'Conipetebibliugitpkyto 1,850 In wept:maim:1. 4 e

, ' 053870.6-24-----743 a . s-r 0. 111 w-w7: 76 ALGEBRA IN THEEIGHTEENTH CENTURY 806 Chevigne, L. I. M. .Mathematical Manual for the useof St Mary's College Of Baltimore* containing fopr parts Itattonal ArithmeticII ElemeRts of AlgebraIII Practical Arithmeticfi'-'PracticalAlgebra [L.I.M. Chevignel Baltimore.Printed for St. Mtv's(4ollege.By John WeRt Butler.. 1S06. 1806 Vyse, Charles The Tutor's Guide.By Charles Vyse.i'lliladelphia:Joseph rruk- shank. 18041. 1807 Chevigne. L. 1. M. AlathematicalMantalfor the use of Colleges;ími Academie.Volume First.(Rest of the title sameas1s04; NlithinlPrinted by Aoki.) West Butler.1S07. 1809 Simpson, Thomas A Treatise of Algebra: whereinlint Principles are demonstratedand applied in many useful and interestinginquiries, and in the resolution of n great variety of problems Ofdifferent kinds.To whichis added, the geometricalconsfruction of ìgreat numberof linear and plane°problems, with the method ofresolving the same numerically.Ity Thomas Simpson. F It.S. First American. from the eighthLondon edition. Philadelphia: Printed for MathewCarey hy 1'. & (;. rainier.1St/9. 1812 Hutton. Charles A Course oflathemativs in two volumes for the useof academies. as wellasprivate tuition.By Charle.- Hutton. 1.1..I).Is'.R. S,Late Pro- fessor OfMdthematic4 in the Royal Military Academy.From the Fifth and Sixth London Editions.Iteyised and corrected byRobert Admin. A. 31.Fellow of the AmericanPhilosophical Society andprofessor of Mathematics in Queen's College, New ersey.New York, Samuel Camp-

bell,... 1812. Second edition, New York: Publishedby Samuel camphell,...1816. ,Third Edition. New York: Published bySamuel Campbell, . . . 1818. . 1814 Day, Jeremiah An Introduction toAlgebra. being the first partofaCourse of Mathe- matics adaptOd tothe method of instructionAnthe American colleges. By Jeremiah Day. New'Raven: Howe & Spalding.1814. Second edition.New Hayen: Howe &Spalding, 1820. 1818 Euler, Leonard . o, An Introduction tothe Elements of /11(},rebradesigned for the use of those who areacqua4nted onlywittrthe Pirst Principles of Arithmetic. Selectedby John Farrar)from the Algebw ofEuler.Cambridge, N. -Eng.: Hilliard& Metcalf.1818. 1818 Lacroix,Sfilvestrel Firancolsi Elements ofAlgebra, by S. F.'Lacroix.Translated from theFrench for the useof the Students ofthe University atCambrklge, Nvw England. Cambridge. N. E.:Printed By Hilliard &Metcalf,181e 1810 Day, Jeremiah g An Inttodbetion toAlgebra, being the WitrstPart ofaCourse of Math& in tbe higher chool 104 adaptedtothe method ofinstriletion 4., males and achdemiesInth4United States. ByJeremiah Day, IL D., President Ft:. of "feltcollege, New liven:Publishidby HoveSpalang.1819.4

. 'a 4

, fo'fle . r4 , SUMMARY 041. I 77

1820 Lacroix (&B6zout) . An elementrytreatiseon Plane andSphericalTrigonometryandon the application of Algebrato Geometryfrom themathematicsof Lacroixand Mow.Translated fromthe Frenchfor theuse of the studentsat the University at Cambridge,New England.Cambridge,N. E.: Printedby Hilliard andMetcalf, 1820.

BIBLIOGRAPHY .16

This studyhas beenmade largelyfromoriginalsourcesand printed worksin the librariesof thefollowinginstitutions: AnfericanAssociationof.Artsaria NewburyportPublicLibrary. Sciences;. New YorkHistoricalociety. American AntiquarianSociety. New YorkPublic Librhy. Boston Athenaeum. New YorkSociety. Boston Public Library. PeabodyInstitute. c. -Bowdoin College. Plimpton,George A.,PrivateLibrary Brown University. PrincetonUniversity. BureauofEducation.Washington. Rhode IslandHistoricalSociety. D. C. - Sal411Athenaeum. ColumbiaUniversity. Smith,David Eugene.PrivateLibrary, Essex Institute. Universityof Pennsylvania. John Carter131-ownLibrary. University ofVirginia.. HarvardUniversity. Vi?giniaStateLibrary. Historical Societyof Pennsylvania. WatkinsonLibrary. \- LibraryCompany ofPennsylvapia. Williamand MaryCollege. Libraryof Congress. Yale University. MaineHistorical Society. York Village,Maine, Museum. MassachusettsHistorical Soclety. Furthersourcesof materialarelisted below: o AmericanPhilosophicalSociety.Transactions.Vol., II.New Series,Phila- delphia, 1825. ------..., 1. Badger, a H. C.,MathematicalTheses ofJunior andSeniorCla8ses,,1782-1839, Cambridge,Mass., 188. . s Bowditch, N.,MtraniqueCeleNte, Bythe Marquisde La Pce, Translated by NathanielBowditch, LLD., Vol. IVAMemoir ofthtranslator, by his son, Nathaniel IngersollBowditch,Boston, 1839. Bronion, W. C.1 The HistoryQf BrownUniversity,1764-1914, Providence,1914. P.ajori,F., The Teaching*and Historyof Mathematicsin the.United State*, Washington, 1890. (lajori, F.,William Oughtred,Chicago,191Q. Chambers,PI, Cyclopdia.Secondedition, London,1738. Clap, Thomas, ThAnnalsor History ofYaleCollegein New-Haven,New Haven, 1766. 4 (lam Thomas,ACatalogueof the Libraryof Yale-College I in Newflaven,N. London, 1743. ColumbiaUniversity, AHisoryof ColumbiaUniversity,1781-1904, New York, .4 1904. _ I Ourrier,J.J., Iffittoisyof affrfiport;~Manct,784-1905.-New1ti7PortPOO, . 1). MorganiAugustus,ArithetkaiBorges, London,1847. 4 . t , . Dexter,P. B., 1 Biographicalketches ofthe arlOatIs-2Pile College itgl" , tugs of the _ , . .00 College. Hisdry, . , October,. 11 `. - . New Tork;18854, C;*S 1 ¡:t 1':. . . IsV°7% . -1:... . : . . , "1. . . ". . . "' 5.ft. :'4141. . s- ...... :*. ' ! - 7., .- ;\ . % . . . , f.¡.: "- ..41V le.P *4 ;14 *>_.1-j-4*- .... ' jes'g ..41;-^''Y. V ' .1 it 4.i"¡: s XL:VC '',.7% i " .' 1 .5 ....:.:%4""r-' "tg,4.45d:* ... i 2:::' s Nh :..: '...4 ifIrA' :MI.:A:ate:I:* a l 78 ALGEBRA IN THEEIGHTEEKTII CENTURY

Dickson, 11. E.,History of the Theory of Numbers,Vol. II, Washington.1920. Dunshee, H. W..Hixtory ofthe Schoolof the CollegiateReformed Dutch Church from16A-18.13.New York. 1883. EeelestiaxtialRecordsof the State of Neu. York.Allmny, 1901-1916. Edes. H. H.. " HarvardTheses of 1683."Transactionsof The Colonial Sorietw of Alaxxachnscttx, Vol.V, 1897. 1898,Boston, 1902. Ilvans, Charles. AmericanBibliography, 1639-1s20. Chicago,1903-1914 Olenn, T. A.. William ChurchillIlouNton, 17'16-17R8, Norristown,Pa.. 1903. Ilkigone, Pierre. rurvux Hatlicnoilicux, noralweri et clam. Vol. II, Paris, 1644. Hildehurn. C. IL.1 Century ofPrinting.The lgsueN of the Pressin Pennxyl- ranig, 1685- 17.'34.Philadelphia, 1885. Jones, Hugh. ThePre8ent State ofVirginia, 144nidiiii. 1724. Lane, W.C.."Early HarvardBroadsides." Proceedings(ifAmericanInti- quarian 'Society. Worcester,1914. NTadean, John. History of the Collegeof New erxcy, 1.1com itx origin in r46 to theCommencement of 18.5.Phillidelphia. 1877. Oughtred, William,rlarik Mathematic(G,London, 1(131. Peirce, Benjamin..1History ofllarrarPrnirersily.cambridge, 1833. Potter, A.C..and Bolton,C. K.. Librarians of IbirrardCollege, 1667 -lsr. I. Cambridge. 1897. Pratt, D. .1.. AvnalN ofPublic Education in theState of Veil. York,From 1626 lo17411, AThany,..1872. Quincy, 4001h.TheHistoryof Harvard University.itrston,180). 64biip:ph;rnirermal Arithmetick: or. aTreatise ofArithmetical Corn- Raphson, . _

position_frfi Resolution.London, 1720. Sanborn¡IVA,Doctor 1,angdon(1723-17971, 1914. Smith, D. E.. "A Glimpse atEarly ColonhdAlgebra inSchool andSociety, January 5, -1918. History ofVathematics, 2 vols.. Boston.1923-24. The SumarioConpendioRo of BrotherJuan Die:. Boston.1921. Smith, H. W.'LifeandCorrespondence' of the Rer.William'miIh, D.D., Philadelphhi. 1879. Smith, The Workm ofWilliamNmah. n. D., Late Provost ofthe Collegeand Academy ofPhiladelphia. 1803. Snow, L. F.,The CollegeCurriculum in theUnfted State'? [NoWave], 1907. Spotgwood, TheOfficial Lciters ofAlexanderSpotxtroud. LieutenantGovernor of the Colony ofVirginia, 1710-1722,Richmond, 1882, t . University of Pennsylvania,BiographicalCatalogue ofthe Hatrfrulatex ofthe College,170-1893, Philadelphia. 1894. . Wallis, John,A Treatiseof Algebra bothhistorical and Practical,London,1685. William and MaryQuarterly, Vols.I-XXVII. Williams, J.R., PhilipVickers Fithian,Journal and Letters,1767-1774.Baited by JohnRogers Williams,Princeton, TheUniversityLibrary, 1900.

p. 0 *.

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IN DEX

Addition, 8, 9, 21, 25, 61 Euclid, 15, 16, 18, 19, 50, 54 Advertisement of arithmetic by ¡glut('Evolution, 8, '45, V Greenwood, GS Exponents, 8, 55, 02 dvertisements, algebra problems, 72: First Algebra hook in American Colo- sale of algebra books, 72.: tutors. nies, 58 67-71 Firstprofessorofmathematicsin Aggregation. S Aintrican4,11olonies, 47 Algebra in Tliç.111Seriegn Youth,tri Fithian, Philip Vickers, 18 Algebra in Pike's Arithmetic,tri Fluxions, 18, 19, 21, 26, 35, 37, 38, 40, American Youth. The. s1. 49, 68 Analogies. 9, 54 Fractions, 8, 19, 21, 25, A2, 54,51. 62 .Appiications ofIgebni,625.2ebe 41. 50, Geometry, 15,21, 25, 32--35,37, 41, 4 48,50,52, 68 Arithmetic, 3.71,.)5.:).-) -35,:17.40. .43,Greenwood. Isaac, 3-6,15-17.437,, 45, 49, 68 67, 68 Arithmetic(' of rilfter-Konxt,58, 59, 61Greenwomrs arithmetic, 5.f;,4 Arithmetic(' Ifnir(1rvolis, t).1(1, 23,q5,Halley, Edmund, 9, 13, 16 53 Hammond, Nathaniel, 53, 54, 72 Bibliography fit.Vmerivantextbooks toHarvard University, 3-5.17. 30. 35, 1820, .75 36, 38, 41, 45, 46, 52, 68 Binomial theorem, 9, 17,65 Ileterogeneal, 16 Bowditch, Nathaniel, 27 Hill, John. 19, 54 Brooke. Robert, 25 History of algebra, 6,1,55 Brown University, 30, 38, 54 Hollis professorship, 4, 45 l'olculus, 16, 68.Ser (ilsoFluxions llomogeneal, 16 Catechism method, 2fik Houston,WilfiamChurchIll, 20 ChambersCrlopedia.4. 7, 15, 14; Imaginary number, 11

Circle, 15, 16, 33 Inequality, 8. 25 s

Clip,Thomas, 48 4 Involuiks 8, 0, 13, 25, 32 . , s. College of New Jersey. scePrincetonJohnson, Samuel, 34, 50, 52 University Jones, Hugh. 46, 47;manusitriptof. 48 (olumbia University, 34, 54) Kersey, John, 16, 72 De Morgan, Augustus, 19 b. King's College. ecColumbia Univer- Difference symbol, 8 sity Diman, James, 4-6 Langdon, Samuel, 3, 4, 5. Ditton, Humphrey, 19, 53 Le Fevre, 46, 47 . Division, 8, 21, 25, 37, 61 Unonian Society at Yale, 49 Dutch algebra, 58 Magnalin. 30 English influences, 1 Manuscript, Diman, e if; Laudon, Equations, affected, 8,11, 21, 25, 27, 6 ff 46, 54, 55; biquadratie, 13, 55; cu-Mathematica Compendia, 21,, 25 bic, 9, 10, 12, 50, 55, 65; quOratietMay, Eleazer, commonplace book, 51

8, 11, 19, 21, 25, 54, 55; radical, 21;Mentaldiicipline,40 ,

simple, 8, 10, 19, 21, 25, 82,01;simul-Mexican algebra, 57 al taneous, ex, 25, 81 Multiplipition, 8. ZAK 37, 81 . 19

9 -*. " . x lo - a , - : -*, . - .; v'T . A' 1Z 1r% *. *- e# '' -* ";17",1"-;.:. 6 AT .4% . 46 6.. , 6. 6-6;.. -s`14.% ... .;- .^. 6, ' p - . ' ' . -'1. *- . an. . .1 , . # ; f.:-#t; -.",2::1416-,% . t ; 6 V.' 6. ,.. - . 4 -,- .4'''+: izr - *A.'" :; '1:!` . . - ?.- V. . . ' t ' "! 7 ...r . .? V..,, s- ..4 *--st11.7 it- ata-t t. -- I ` qLt.%. itte k 1/4..4sk f.n7 . . 80 INDEX Negative numbers, p, 28, 32 'Smith, William, 49, 53 Newton, Sir Isaac, 1, 16, 17, 53, 55, 72Sterry, Consider and Joitm, 65 Notebooks, algebra, 3, 18, 21, 25-28;Substitution, 9, 12, 2254 arithmetic, 3 Subtraction, 8, 9, 21, 25. 61 Oughtred, William, 8. 11, 16, 25 Sumario Compendioso, 57 Patterson, Robert, 21-23 Summary, 74 Pike, Nicolas, 64 Surds, 8, 9, 16, 19, 54 Positive numbers, 9, 28, 32 'Symbols,6, 7 Praxis, 21, 25 Teaching ofalgebra,evidences. 4, 17. Princeton' University,18, 38, 55 18, '21, 25, 34 Problems, algebraic, .1)-11, 14,17, 19, Textbooks, 29, 51, 54, 55, 57. 64, 75 22, 28, 41, 44, 55, 57, 63, 72;geo- Theses, algebraic, 33, 37; commence. metric, 9, 13, 14, 16, 22, 25, 26, 41, ment,30;mathematical,41-43: 50, 55 translations of, 32, 37 Progressions, 25 Topics in Harvard manuscripts, 8 Proportion, 8, 19, 32, 54, 65 " Tryalls and Depression," 9, 13 Raphson, Joseph, 9, 13, 16 University of Pennsylvania, 21, 25, 49 Reduction, 8; 9 Veuema, Pieter, 59, 72 .Remainder theorem, 13 Wallis, John, 7 Rhode IslandCollege.SeeBrown Ward's Young Mathematician's Guide, University 42, 51, 5:3.73 Rhyme, problems in, 28 Willard,Sainiuel.42 Saunderson, Nicholas, 19, 72 William and Mary, Collegq of, 46 Series, 9, 21, 46, 55, 65 Yale University, 30-32, 34,-38,39, 48, Simpson, Thomas, 35, 72 49, 53, 54

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IIVASlititoTON tlevitaNKSter PRINT! so winner:tom . . 4. ,- . `.. t4,,,,:,.-2f-:...4 ' '7%!;- -.A ...... 4 4rsr ' ir I . 0,t1:...-. .,. ' r . -...:-...t --.;....,....44:-.... . ,. t.-.-...... ,:*,; 1 ---...- r--,.... . 'AI; -'4-.- , . ... 1,4*Ar... 741,*. Ibti-g.4` \111. 'T C.v. let r ",4 a - --e 'r - 1' '..»..... 1144, :....4,44,....,.., Lz.:.....,..:74ii....;:;11) 27r; .,);"'. 4%1 ..-Zrz. r : ¡- ,t,,,,,,,...t-,;,,, ,.-,..,.. ...- .,. , . t r 4.a-_.r .4% 4. qk ": .2