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Hamilton's Discovery of Quaternions Author(s): B. L. van der Waerden Source: Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 227-234 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2689449 . Accessed: 27/02/2011 09:45 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=maa. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Hamilton's Discoveryof Quaternions Contemporarysources describe Hamilton'strail from repeated failures at multiplyingtriplets to the intuitiveleap into the fourthdimension. B. L.

VANDER WAERDEN

University of Zurich

Introduction Theordinary a + b\ - 1) areadded complexnumbers (a + ib) (or,as theywereformerly written, and multiplied readsas follows: rules.The ruleformultiplication accordingto definite Firstmultiplyaccordingto the rules of high school algebra: (a + ib)(c + id) = ac + adi + bci + bdi2

and then replace i' by (-1): (a + ib)(c + id) = (ac - bd) + (ad + bc)i.

canalso be defined as couples(a, b). The product oftwocouples(a, b) and(c,d) is Complexnumbers defined as thecouple(ac - bd,ad + bc). Thecouple(1,0) iscalled1,thecouple(0, 1) iscalledi. Then we also havetheresult P =(0,1)(0,1)=(-1,0)=

-1. of the means this definition i =/--I unit" i is simplythe losesall of itsmystery: By "imaginary couple(0, 1). a + bi + cj + dk whichWilliamRowan Hamiltondiscoveredon the 16thof The quaternions thatis to to fixedrules,inanalogyto thecomplexnumbers; October,1843,are multiplied according say: i2 =

j2

= k2

ij = k,

jk =i,

ki = j,

ji = - k,

kj= - i,

ik = -j.

as quadruples(a, b,c,d). Quaternions forma division Theycan also be defined algebra;thatis,they and multiplied, cannotonlybe added,subtracted, butalso divided(excluding divisionbyzero).All rulesofcalculation ofhighschoolschoolalgebrahold;onlythecommutative lawAB = BA doesnot holdsinceij is notthesameas ji. der Originallypublished in German as "Hamilton's Entdeckungder Quaternionen" in Veroffentlichungen JoachimJungius-Gesellschaft Hamburg by Vandenhoeck& Ruprecht,Gottingen,1974. Translatedinto English by F. V. Pohle, Adelphi University,and publishedhere by permissionof the originalpublisherVandenhoeck& Ruprecht. VOL. 49, NO. 5, NOVEMBER 1976

227

rules?Whatwashisproblemandhowdidhe find HowdidHamiltonarriveat thesemultiplication indocuments andpaperswhichappear We areaccurately informed aboutthesematters thesolution? Papers[3]: in thethirdvolumeof Hamilton'scollectedMathematical an entryin Hamilton'sNote Book dated 16 October1843[3, pp. First,through 103-105]; Second,througha letterto JohnGravesof the 17thof October1843 [3, pp. 106-110];

of theRoyalIrishAcademy(2 (1844) a paperin theProceedings Third,through on the 13thof November1843[3, pp. 111-116]; 424-434)presented thedetailedPrefaceto Hamilton's"Lectureson Quaternions", Fourth, through pp. 142-144]; datedJune1853[3, pp. 117-155,in particular whichHamilton beforehis wroteshortly a lettertohissonArchibald Fifth, through beforethe2ndof September1865[3, pp. xv-xvi]. death,thatis shortly Thisis a all ofthesedocuments. through We can followexactlyeachofHamilton'sstepsofthought as he posed inwhichwecanobservewhatflashed acrossthemindofa mathematician rareoccurrence a lightning strokeso the problem,as he approachedthe solutionstepby stepand thenthrough modified theproblemthatit becamesolvable. ofcomplexnumbers A briefhistory in the middleages in the Expressionsof the formA + -B had alreadybeen encountered surdi:absurd solutionof quadraticequations.Theywerecalled "impossiblesolutions"or numeri A + -B in too werecalled"impossible."Cardanusednumbers numbers. The negativenumbers inwhichall threerootsarereal. thesolutionofequationsofthethirddegreeinthecasusirreducibilis such as A + -B without Bombellishowedthatit was possibleto calculatewithexpressions without value. and apparently buthe did notlikethem:he calledthem"sophistical" contradiction, The expression "imaginary number"stemsfromDescartes. He proposed freelywithcomplexnumbers. Euler had no scruplesaboutoperatingaltogether such as cosa = (1/2)(e" + eC). The geometric of complexnumbersas formulas representation vectorsor as pointsin a planestemsfromArgand(1813),Warren(1828)and Gauss (1832). as directed segments intheplane.He took Thefirst thecomplexnumbers named,Argand,defined thebasisvectors1 and i as mutually unitvectors. Additionis theusualvectoraddition, perpendicular orofforces).The length ofa lawofvelocities (theparallelogram withwhichNewtonmadeus familiar vectorwas denotedat thattimeby theterm"modulus",theangleof thevectorwiththepositive ofcomplexnumbers, to x-axisas the"argument" according Multiplication ofthecomplexnumber. are added. Argand,then takes place so that the moduliare multipliedand the arguments and ofArgand,Warrenand Gauss also represented geometrically complexnumbers Independently theiradditionand multiplication geometrically. interpreted triplets?" "Papa, can youmultiply of complexnumbers.In his published Hamiltonknewand used the geometric representation of complexnumbersas the couple (a, b) which papers,however,he emphasizedthe definition Relatedto that,Hamiltonposedthisproblem definite rulesforadditionandmultiplication. followed in analogyto couples(a, b). to himself:To findhownumber-triplets (a, b,c) are to be multiplied as he himself rulefortriplets, Fora longtimeHamiltonhadhopedto discoverthemultiplication andmoreserious.He putitthiswayina stated.ButinOctober1843thishopebecamemuchstronger letterto hisson Archibald[3, p. xv]: 228

MATHEMATICS MAGAZINE

SIR WILLIAM ROWAN HAMILTON, a child prodigy whose maturity was all that his childhood promised, was born in Dublin, Ireland, in 1805. He was literate in seven languages and knowledgeable in halfa dozen more. In 1827 while still an undergraduate Hamilton was appointed Andrews Professor of Astronomyand Superintendent of the Observatory, and soon afterwards Astronomer Royal, a position he held for the rest of his extraordinarilyproductive life. His work in dynamics is probably most well-known today. "The Hamiltonian principle has become the cornerstone of modern physics", said Erwin Schrbdinger, "the thing with which a physicistexpects everyphysical phenomenon to be in conformity."Hamilton's other major discovery is the system of quaternions. The flash of insight which produced this discovery occurred in 1843 and is described in the accompanying article. A century later the Irish government commemorated this achievement with the stamp pictured at the right.

*

&

"... the desire to discoverthe law of multiplicationof tripletsregainedwith me a certain strengthand earnestness,..."

In analogyto the complexnumbers(a + ib) Hamiltonwrotehis triplets as (a + bi + cj). He hisunitvectors1,i,j as mutually represented "directedsegments" of unitlengthin perpendicular usedthewordvector, whichI alsoshalluse inthefollowing. space.LaterHamiltonhimself Hamilton suchas (a + bi + cj)(x + yi+ zj) againas vectors inthesamespace. thensoughtto represent products He required, thatitbe possibleto multiply outtermbyterm;andsecond,thatthelengthofthe first, product ofthevectors be equaltotheproduct Thislatterrulewascalledthe"lawofthe ofthelengths. moduli"by Hamilton. Today we knowthatthe two requirements of Hamiltoncan be fulfilled onlyin spaces of in three dimensions Hamilton'sattempt 1, 2, 4 and 8. Thiswas provedby Hurwitz[5]. Therefore hadto fail.His profound dimensions ideawastocontinue to4 dimensions sinceallofhisattempts in3 dimensions failedto reachthegoal. In thepreviously mentioned letterto hisson,Hamiltonwroteabouthisfirstattempt: "Every morningin the early part of the above-citedmonth[October 1843], on my coming down to breakfast,yourbrotherWilliamEdwin and yourselfused to ask me, 'Well, Papa, can you multiplytriplets?'WheretoI was always obliged to reply,witha sad shake of the head, 'No, I can only add and subtractthem.'"

Fromtheotherdocuments we learnmoreprecisely aboutHamilton's first To fulfill the"law attempts. of the moduli"at leastforthe complexnumbers(a + ib), Hamiltonset ii = - 1, as forordinary andsimilarly so thatthelawwouldalsoholdforthenumbers complexnumbers, (a + cj), jj = - 1. But whatwas ij and whatwas ji? At firstHamiltonassumedij = ji and calculatedas follows: (a + ib + jc)(x + iy + jz)

=

(ax

-

by- cz) + i(ay + bx) + j(az + cx) + ij(bz + cy).

Now,he asked,whatis one to do withij? Willit havetheforma + pi + yj? Thesquareofij hadto be 1,sincei2 = _ 1 andj2 = - 1. Therefore, Firstattempt. wroteHamilton, inthisattempt one wouldhaveto chooseij = 1 or ij = -1. Butinneither ofthesetwocaseswillthe law of themodulibe fulfilled, as calculation shows. Secondattempt. Hamiltonconsidered thesimplest case (a + ib +

jC)2 =

as - b2_ c2+ 2iab + 2jac + 2ijbc.

Thenhe calculatedthesumofthesquaresofthecoefficients of 1, i,andj on therighthandsideand found (a 2- b2 _ C2)2 +(2ab)2

VOL. 49, NO. 5, NOVEMBER 1976

+(2ac)2

= (a2 + b2+ c2)2.

229

he said,theproduct ifwe setij = 0. Andfurther: Therefore, ruleis fulfilled ifwe passa planethrough thepoints0, 1,anda + ib + jc, thentheconstruction oftheproductaccording to ArgandandWarren willholdin thisplane:thevector(a + bi + cj)2 liesinthesameplaneandtheanglewhichthisvector makeswiththevector1 istwiceas largeas theanglebetweenthevectors (a + bi + cj) and1. Hamilton verified thisby computing thetangents of thetwoangles. Thirdattempt. Hamiltonreports thattheassumption ij = 0, whichhe madeinthesecondattempt, didnotappeartobe quiteright subsequently tohim.He writes inthelettertoGraves[3,p. 107]: "Behold me thereforetemptedfora momentto fancythat ij = 0. But thisseemed odd and uncomfortable, and I perceivedthatthesame suppressionofthetermwhichwas de tropmight be attainedby assumingwhatseemed to me less harsh,namelythatji = - ij. I made therefore ij = k, ji = - k, reservingto myselfto inquirewhetherk was 0 or not."

Hamiltonwas entirely rightin givingup the assumption ij = 0 and takinginsteadij = - ji. For example,ifij = 0 thenthemodulusoftheproductij wouldbe zero,whichwouldcontradict thelawof themoduli. Fourthattempt. Somewhat moregenerally, Hamiltonmultiplied (a + ib + jc) and(x + ib + jc). In thiscasethetwosegments whicharetobe multiplied alsolieinoneplane,thatis,intheplanespanned - c2+ i(a + x)b + by the points 0, 1, and ib + jc. The result of the multiplicationwas ax j(a + x)c + k(bc - bc). Hamiltonconcludedfromthiscalculation[3, p. 107]that: .. . the coefficient of k stillvanishes;and ax - b2- c2, (a + x)b, (a + x)c are easilyfoundto be thecorrectcoordinatesof theproduct-point in thesense thattherotationfromtheunitline to the radius vectorof a, b,c being added in its own plane to the rotationfromthe same unit-lineto the radiusvectorof the otherfactor-point x,b,c conductsto the radiusvectorof the latelymentionedproduct-point;and thatthislatterradiusvectoris in lengththe product of the two former.Confirmationof ij = - ji; but no informationyet of the value of k."

The leap intothefourth dimension Afterthisencouraging resultHamiltonventured to attackthegeneralcase. ("Tryboldlythenthe

general productof two triplets,..." [3, p. 107].) He calculated (a + ib +jc)(x + iy+ jz)= (ax

-

by - cz)+ i(ay + bx)+j(az

+ cx)+ k(bz

-

cy).

In an exploratory In otherwords, he set k = 0 andasked:Is thelawofthemodulisatisfied? attempt does theidentity (a 2+ b2+ c2)(x2+

y2+

Z2) =

(ax

- by -

CZ)2+

(ay + bx)2+ (az + cx)2

hold? "No, the firstmemberexceeds the second by (bz - cy)2. But thisis just the square of the of k, in the developmentof the product(a + ib + ic)(x + iy + jz), ifwe grantthat coefficient ij = k, ji = - k, as before."

In theletterto Graves Andnowcomestheinsight a newdirection. whichgavetheentireproblem [3, p. 108],Hamiltonemphasizedtheinsight: "And here there dawned on me the notion that we must admit, in some sense, a fourth dimensionof space for the purpose of calculatingwithtriplets;"

dimension the Thisfourth andhe hastenedtotransfer appearedas a "paradox"to Hamiltonhimself paradoxto algebra[3, p. 108]: "...;or transferring theparadox to algebra,[we] mustadmita thirddistinctimaginarysymbol k, not to be confoundedwitheitheri or j, but equal to the productof the firstas multiplier, and the second as multiplicand;and therefore[I] was led to introducequaternionssuch as a + ib+ jc + kd, or (a, b,c, d)."

wasnotthefirst to thinkabouta multi-dimensional In a footnote totheletter Hamilton geometry. 230

MATHEMATICS MAGAZINE

to Graveshe wrote: "The writerhas thismomentbeen informed(in a letterfroma friend)thatin the Cambridge MathematicalJournalforMay last [18431a paper on AnalyticalGeometryof n dimensions has been publishedby Mr. Cayley, but regretshe does not yet know how far Mr. Cayley's views and his own may resembleor differfromeach other."

"Thismoment" caninthisconnection onlymeanthesamedayinwhichhewrotethelettertoGraves. ofthepaperbyCayley.Hamilton In theNoteBookofthe16thofOctober1843thereis no mention ofCayley. spaceindependently at theconceptofa 4-dimensional therefore appearstohavearrived independent basisvector,he continued AfterHamiltonhad introduced ij = -ji = k as a fourth thecalculation[3, p. 108]"I saw thatwe had probablyik = -j, because ik = iij,and i2= -1; and thatin like manner we mightexpect to findkj = ijj = -

continued. He scarcely Hamilton Fromtheuse oftheword"probably"itcan be seenhowcautiously = certain if the associative he was not yet (ii)j because associative law i(ij) himself to apply the trusted ki: law to determine the associative could have used Hamilton held for quaternions. Likewise law

ki= - (i)i = - j(ii) = (- j)(- i) = j. Insteadhe applieda conclusionbyanalogy.He wrote[3, p. 108] "... ; fromwhichI thoughtit likelythatki = j,jk = i, because itseemed likelythatifji = - ij,

we should have also kj = - jk, ik = -ki."

Hamiltonagainproceededcautiously: Finallyk2 had to be determined. we cannotinfer "And since the orderof multiplication of these imaginariesis not indifferent, that k2, or ijij, is = + 1, because i2Xj2= (- 1)(- 1) = + 1. It is more likelythat k2 = ijij =

- iijj = - 1." ifwe wishto fulfill the"law ofthe k2= _ 1,assertsHamilton, is also necessary Thislastassumption

moduli."He carriedthisout and concluded[3, p. 108]: "My assumptionswere now completed,namely, i2 =

j2 = k2= _1

ij = - ji = k jk = - kj =i ki = - ik =

And nowHamiltontestedifthelaw of themoduliwas actuallysatisfied. "But I consideredit essentialto trywhetherthese equations were consistentwiththe law of moduli ...., withoutwhichconsistencebeing verified,I should have regardedthe whole speculationas a failure."

to therulesjust formulated according He therefore twoarbitrary quaternions multiplied (a, b,c, d)(a', b', c', d') = (a", b",cd

calculated(a", b",c",d") and formedthesumof thesquares (a ')2 + (b")2 + (c,')2+

(dtt)2

and foundto hisgreatjoy thatthissumof squaresactuallywas equal to theproduct (a2+

b2+ c2+

d2)(at2+

bt2+ ct2+

dQ2).

whichbefell In Hamilton's circumstances letterto hissonwe learnevenmoreabouttheexternal citedwords,"No, I can onlyadd and himat thisflashof insight. afterthepreviously Immediately VOL. 49, NO. 5, NOVEMBER 1976

231

them."Hamiltoncontinued[3, p. xx-xvi]: subtract "But on the 16thday of the same month[October 1843]-which happened to be a Monday and a Council day of the Royal IrishAcademy-I was walkingin to attendand preside,and your motherwas walkingwithme, along the Royal Canal, to whichshe had perhaps been of thoughtwas driven;and althoughshe talked withme now and then,yetan under-current goingon in mymind,whichgave at last a result,whereofit is not too muchto say thatI feltat once the importance.An electriccircuitseemed to close; and a sparkflashedforth,theherald (as I foresawimmediately)of many long years to come of definitelydirectedthoughtand work,by myselfifspared,and at all eventson thepartofothers,ifI shouldever be allowed to live long enough distinctlyto communicate the discovery. I pulled out on the spot a pocket-book,whichstill exists,and made an entrythere and then. Nor could I resistthe impulse-unphilosophicalas it may have been-to cut witha knifeon a stone of Brougham Bridge, as we passed it, the fundamentalformulawiththe symbolsi,j, k; i2 =

j2 = k2= ijk= - 1,

whichcontainsthe solutionof the Problem,but of course as an inscription,has long since moulderedaway."

The entryin thepocketbook is reproduced on thetitlepage of [3]: it containstheformulas *

i2=

2

j2

ij=k,

jk=i,

ki=j

ji=-k,

kj=-i,

ik=-j.

I assumeas likelythatbeforehiswalkHamiltonhadalreadywritten on a pieceofpapertheresult of thesomewhattiresome whichshowedthatthesumof squares calculation (ax - bystilllacked(bz -

cy)2

cz)2 +

(ay + bx)2+ (az

+ cx)2

comparedwiththeproduct (a2+

+c2 )(x2+ y2+

).

Whatthenhappenedimmediately thatremarkable walkalongtheRoyalCanal,he beforeandduring describedagainon thesameday in hisNote Book, as follows: "I believe that I now remember the order of my thought.The equation ij = 0 was recommendedby the circumstancesthat (ax _

y2_

Z2)2+

(a

+ X)2(y2+

Z2) = (a2+

y2+

z2)(X2+

y2+

Z2)

I thereforetried whetherit mightnot turnout to be true that (a 2+ b2+

C2)(X2

+ y2 + Z2)=

(ax - by-

Cz)2

+

(ay + bx)2 +(az

+ cx)2,

but foundthatthisequation required,in order to make it true,the additionof (bz - cy)2 to the second member.This forcedon me the non-neglectof ij, and suggestedthatit mightbe equal to k, a new imaginary."

the italicizedwordsforced and suggested Hamiltonemphasizedthathe was By underscoring withtwoentirely different wasa compelling concerned facts.The first whichcame logicalconclusion, outofthecalculation: itwasnotpossibleto setij equaltozero,sincethenthelawofthe immediately moduliwouldnothold.Thesecondfactwasan insight whichcameoverhimina flashatthecanal("an circuit seemedto close,anda sparkflashed thatis,thatij couldbe takento be a new electric forth"), unit. imaginary Aftertheinsight else wasverysimple.The calculations ik= iij= - j was oncethere,everything and kj= ijj= ' i could be made easily enoughby Hamiltonin his head. The assumptions ki= - ik= j andjk = - kj= i wereimmediate. And k2couldbe easilycalculatedtoo: k2= j= -iii] = -1. 232

MATHEMATICS MAGAZINE

Andso during hiswalkHamilton alsodiscovered therulesofcalculation whichheenteredintothe pocketbook.The pocketbook also containstheformulas forthecoefficients of theproduct (a + bi + cj + dk)(a + 3i + yj + 8k),

thatis, aa - bl3- cy - d8 al3 + ba + c8 - dy ay - b8 + ca + do3 a8 + by - co3+ da

as wellas thesketchfortheverification ofthefactthatinthesumofthesquaresofthesecoefficients inthe allmixedterms(suchas ada8) cancelandonly(a2+ b2+ c2+ d2)(a2? 32? y2? 82) remains. Note Book of thesame dayeverything was againcompletely restated. Octonions thediscovery ofquaternions ThelettertoGravesinwhichHamilton announced waswritten on the 17thofOctober1843,onedayafterthediscovery. The seeds,whichHamilton sowed,felluponfertile soil,sincein December1843therecipient JohnT. Gravesalreadyfounda linearalgebrawith8 unit as elements 1,i,j, k,l,m,n,o, thealgebraof octavesor octonions. Gravesdefinedtheirmultiplication follows[3, p. 648]: i2 = i2

=k2 =12 = m 2 =

0

n2

2=_1

i = jk = Im= on = - kj = - ml = j = ki = In = mo = - ik = - nl =

-

no om

k = ij =lo =nm =-ji = -ol = -mn I = mi= nj = ok =-im= m = il = oj = kn =

-

- jn= - ko

li = -jo =

n = jl = io = mk =

-

Ij =

o = ni = jm = kl =

-

in =

-

oi =

-

-

mj =

nk km -

lk.

In thissystemthe"law of themoduli"also holds: (1)

(a2

2*

2 +

+ c2

2) = (C2 +

Hamiltonansweredon the8thofJuly1844[3, p. 650].He notedto Gravesthattheassociativelaw A BC = AB *C clearlyheldforquaternions butnotforoctaves. Octaveswererediscovered by Cayleyin 1845;becauseof thistheyare also knownas Cayley with16 unitelementsbutit was unsuccessful. It couldnot numbers. Gravesalso madean attempt oftheform(1) areonlypossibleforsumsof 1,2,4 and8 succeedsincewe knowtodaythatidentities aboutthehistory of theseidentities. squares.I shouldliketo close witha briefcomment Productformulas forthesumsofsquares It is likelythatthe"law of themoduli"forcomplexnumbers was alreadyknownto Euler: (a 2+ b2)(c2+ d2)= (ac - bd)2+ (ad + bc)2. A similarformula forthesumof 4 squares L

,

N2

NO. NOEME VOL.~49 ~ 5,~~~~(

(C2M M)

2+ 1976233

+

=

+

'

+

2)

fromEulertoGoldbachon May4th,1748[4]. isstatedina letter wasdiscovered byEuler;theformula waspreviously Theformula (1) for8 squares,whichGravesandCayleyprovedbymeansofoctonions, thought thathe couldgeneralizethetheorem to 2" foundbyDegen (1818)[6]. Degen erroneously squares. (a, b,c) and (x,y,z) be so The problem,whichstartedwithHamilton,reads:can twotriplets multiplied thatthelaw of themoduliholds?In otherwords:is it possibleso to define(u,v,w) as of (a, b,c) and (x,y,z) thattheidentity bilinearfunctions (2)

(a2+ b2+ c2)(x2 +

y2+ z2) = (u2 + v2+

w2)

results? des was Legendre.In hisgreatworkTheorie The first to showtheimpossibility forthisidentity as rationally 3 and21 can easilybe represented he remarked on page 198thatthenumbers nombres sumsof threesquares: 3 =1 + 1 + 1, 21 = 16+4+ 1,

of theform(8n + 7). It buttheproduct3 x 21 = 63 cannotbe so represented, since63 is an integer to theextentthatit is assumedthat followsfromthisthatan identity of theform(2) is impossible, IfHamilton hadknownof in(a, b,c) and(x,y,z) withrational coefficients. (u,v,w) arebilinearforms triplets. up the search to multiply have quickly given thisremarkby Legendrehe wouldprobably he did notreadLegendre:he was self-taught. Fortunately The questionforwhichvaluesof n a formula of thekind (a2+

*

2n)(21

+

+

2) = (C2l + ...+

C2n

he proved multiplication is possible,wasfinally in 1898.Withthehelpofmatrix decidedbyHurwitz historical accountsthereadermay Forfurther (in [5]) thatn = 1,2,4 and8 are theonlypossibilities. referto [I] or [2]. References [1] [2]

[31 [4] [5]

[61

C. W. Curtis,The fourand eightsquare problemand divisionalgebras,Studiesin Mathematics,vol. 2 (Ed. by A. A. Albert), p.100, MathematicalAssociation of America, 1963. L. E. Dickson, On quaternionsand theirgeneralizationsand the historyof the eightsquare theorem,Ann. of Math., 20 (1919) 155. The MathematicalPapers of Sir William Rowan Hamilton,vol. III, Algebra, Edited fbrthe Royal Irish Academy, by H. Halberstamand R. E. Ingram,CambridgeUniversityPress, 1967. P. H. Fuss (Editor), CorrespondenceMathematiqueet Physique I, St. Petersburg,1843. Hurwitz,Ueber die Compositionder quadratischenFormen von beliebig vielen Variabeln, Nachr. der k6niglichenGesellschaftder Wiss. Gottingen(189* 309-316, MathematischeWerke,Bd. II, Basel, 1932, pp. 565-571. C. P. Degen, AdumbratioDemonstrationisTheorematisArithmeticaemaxime generalis,Memoires de l'Academie de St. Petersbourg,VIII, (1822) 207.

The sun never sets on mathematics ofNewtonhavedonemoreforEnglandandfortherace,than The discoveries hasbeendonebywholedynasties ofBritish monarchs; andwedoubtnotthatinthe birthof 1843,theQuaternions of Hamilton, greatmathematical thereis as much to mankind realpromiseof benefit as in anyeventof Victoria'sreign. -

234

THOMAS HILL

MATHEMATICS MAGAZINE